# Patent

## Title
Signal transformation apparatus applied hybrid architecture, signal transformation method, and recording medium

## Abstract

A signal transformation apparatus, a signal transformation method, and a recording medium are provided. The signal transformation apparatus selects any one of DCT-II, DST-II, DFT, and HWT, and transforms and outputs the input signal according to the selected transform. Therefore, the hybrid architecture of low complexity can be attained.

## Background
CROSS-REFERENCE TO RELATED APPLICATION(S) AND CLAIM OF PRIORITY
The present application claims the benefit under 35 U.S.C. §119(a) to a Korean patent application filed in the Korean Intellectual Property Office on Oct. 19, 2011, and assigned Serial No. 10-2011-0107029, the entire disclosure of which is hereby incorporated by reference.
TECHNICAL FIELD OF THE INVENTION
The present invention relates generally to a signal transformation apparatus, a signal transformation method, and a recording medium. More particularly, the present invention relates to a signal transformation apparatus with a hybrid architecture applied, a signal transformation method, and a recording medium.
BACKGROUND OF THE INVENTION
Digital video technology goes through a hybrid fusion revolution over the last decade. Digital videos are applied to many fields such as DVD player, computer, and mobile handset. Nowadays, High Efficiency Video Coding (HEVC) is under development in combination with ISO/IEC Moving Picture Experts Group (MPEG) and ITU-T Video Coding Experts Group (VCEG). The MPEG and the VCEG create Joint Collaborative Team on Video Coding (JCT-VC) for HEVC standardization.
Recently, it is possible to easily catch news on the web using a smart TV or an iPhone. Video compression is essential to all of these applications. Discrete Cosine Transform (DCT)-II is a popular compression structure and accepted as the best suboptimal transformation exhibiting the performance quite close to the optimal Karhunen-Loeve transform for MPEG 4 and H.264 standard. Discrete orthogonal transform is used for applications of signal classification and representation. Discrete signal processing of Discrete Fourier Transform (DFT) is a popular transformation for Orthogonal Frequency Division Multiplexing 4th Generation (OFDM-4G) and communication. The OFDM is a key technology for next-generation mobile communication (3GPP-LTE, mobile WiMAX, IMT-Advanced) as well as wireless LAN (IEEE 802.11a, IEEE 802.11n), wireless PAN (multiband OFDM), and DFT-based broadcasting (DAB, DVB, DMB). Further, discrete wavelet transform based on Haar Wavelet Transform (HWT) is very useful in JPEG 2000 standard and signal analysis. What is needed is a method for applying the four transforms more easily.
SUMMARY OF THE INVENTION
To address the above-discussed deficiencies of the prior art, it is a primary aspect of the present invention to provide a signal transformation apparatus for selecting one of DCT-II, DST-II, DFT, and HWT, transforming and outputting an input signal according to the selected transform, a signal transformation method, and a recording medium.
According to one aspect of the present invention, a signal transformation apparatus applying a hybrid architecture for Discrete Cosine Transform (DCT)-II, Discrete Sine Transform (DST)-II, Discrete Fourier Transform (DFT), and Haar Wavelet Transform (HWT) includes a switching part for selecting any one of the DCT-II, the DST-II, the DFT, and the HWT; and a transformation part for transforming and outputting an input signal according to the transform selected by the switching part.
The switching part may include a submatrix selector for selecting submatrix computation for any one of the DCT-II, the DST-II, the DFT, and the HWT; and a permutation matrix selector for selecting permutation matrix computation for any one of the DCT-II, the DST-II, the DFT, and the HWT.
When the switching part selects the DCT-II, the submatrix selector may select [L]N as a submatrix and the permutation matrix selector selects [D]N as a permutation matrix.
The transformation part may apply the DCT-II to the input signal by computing Equation (27) using the submatrix [L]N and the permutation matrix [D]N:




  
    
      
        
          
            [
            C
            ]
          
          N
        
        =
        
          
            
              
                
                  
                    
                      
                        
                          2
                          N
                        
                      
                      ⁡
                      
                        [
                        Pr
                        ]
                      
                    
                    N
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            L
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      I
                      2
                    
                    ⊗
                    
                      Pr
                      
                        N
                        /
                        2
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    I
                    2
                  
                  ⊗
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              4
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            L
                            
                              N
                              /
                              4
                            
                          
                        
                      
                    
                    ]
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        I
                        4
                      
                      ⊗
                    
                  
                
                
                  
                    
                      Pr
                      
                        N
                        /
                        4
                      
                    
                  
                
              
              ]
            
          
          ⁢
          
             
            
              
                [
                
                  
                    I
                    4
                  
                  ⊗
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              8
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            L
                            
                              N
                              /
                              8
                            
                          
                        
                      
                    
                    ]
                  
                
                ]
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    
                      
                        
                          ...
                          ⁢
                          
                            
                          
                          [
                          
                            
                              I
                              
                                N
                                /
                                4
                              
                            
                            ⊗
                            
                              Pr
                              4
                            
                          
                          ]
                        
                        ⁡
                        
                          [
                          
                            
                              I
                              
                                N
                                /
                                4
                              
                            
                            ⊗
                            
                              [
                              
                                
                                  
                                    
                                      I
                                      2
                                    
                                  
                                  
                                    0
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    
                                      L
                                      2
                                    
                                  
                                
                              
                              ]
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                          ⊗
                          
                            C
                            2
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          I
                          
                            N
                            /
                            4
                          
                        
                        ⊗
                        
                          [
                          
                            
                              
                                
                                  I
                                  2
                                
                              
                              
                                0
                              
                            
                            
                              
                                0
                              
                              
                                
                                  D
                                  2
                                
                              
                            
                          
                          ]
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        I
                        
                          N
                          /
                          4
                        
                      
                      ⊗
                      
                        [
                        
                          
                            
                              
                                I
                                2
                              
                            
                            
                              
                                I
                                2
                              
                            
                          
                          
                            
                              
                                I
                                2
                              
                            
                            
                              
                                -
                                
                                  I
                                  2
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      I
                      
                        N
                        /
                        4
                      
                    
                    ⊗
                    
                      Pc
                      4
                    
                  
                  ]
                
              
              ⁢
              
                
              
              ⁢
              
                ...
                ⁢
                
                  
                
                [
                
                  
                    I
                    4
                  
                  ⊗
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              8
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            D
                            
                              N
                              /
                              8
                            
                          
                        
                      
                    
                    ]
                  
                
                ]
              
              ⁢
              
                 
                
                  
                    
                      
                        
                          
                            
                              
                                
                                  [
                                  
                                    
                                      I
                                      4
                                    
                                    ⊗
                                    
                                      [
                                      
                                        
                                          
                                            
                                              I
                                              
                                                N
                                                /
                                                8
                                              
                                            
                                          
                                          
                                            
                                              I
                                              
                                                N
                                                /
                                                8
                                              
                                            
                                          
                                        
                                        
                                          
                                            
                                              I
                                              
                                                N
                                                /
                                                8
                                              
                                            
                                          
                                          
                                            
                                              -
                                              
                                                I
                                                
                                                  N
                                                  /
                                                  8
                                                
                                              
                                            
                                          
                                        
                                      
                                      ]
                                    
                                  
                                  ]
                                
                                ⁡
                                
                                  [
                                  
                                    
                                      I
                                      4
                                    
                                    ⊗
                                    
                                      Pc
                                      
                                        N
                                        /
                                        4
                                      
                                    
                                  
                                  ]
                                
                              
                              ⁡
                              
                                [
                                
                                  
                                    I
                                    2
                                  
                                  ⊗
                                  
                                    [
                                    
                                      
                                        
                                          
                                            I
                                            
                                              N
                                              /
                                              4
                                            
                                          
                                        
                                        
                                          0
                                        
                                      
                                      
                                        
                                          0
                                        
                                        
                                          
                                            D
                                            
                                              N
                                              /
                                              4
                                            
                                          
                                        
                                      
                                    
                                    ]
                                  
                                
                                ]
                              
                            
                            ⁡
                            
                              [
                              
                                
                                  I
                                  4
                                
                                ⊗
                                
                                  [
                                  
                                    
                                      
                                        
                                          I
                                          
                                            N
                                            /
                                            4
                                          
                                        
                                      
                                      
                                        
                                          I
                                          
                                            N
                                            /
                                            4
                                          
                                        
                                      
                                    
                                    
                                      
                                        
                                          I
                                          
                                            N
                                            /
                                            4
                                          
                                        
                                      
                                      
                                        
                                          -
                                          
                                            I
                                            
                                              N
                                              /
                                              4
                                            
                                          
                                        
                                      
                                    
                                  
                                  ]
                                
                              
                              ]
                            
                          
                          ⁡
                          
                            [
                            
                              
                                I
                                2
                              
                              ⊗
                              
                                Pc
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            ]
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                              
                                0
                              
                            
                            
                              
                                0
                              
                              
                                
                                  D
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              
                                -
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      Pc
                      ]
                    
                  
                  N
                
              
            
          
        
      
    
    
      
        (
        27
        )
      
    
  




When the switching part selects the DST-II, the submatrix selector may select [U]N as a submatrix and the permutation matrix selector selects [D]N as a permutation matrix.
The transformation part may apply the DST-II to the input signal by computing Equation (45) by using the submatrix [U]N and the permutation matrix [D]N and multiplying every output by a matrix [M1] [M2]:



  
    
      
    
    ⁢
    
      (
      45
      )
    
  




  
    
      
        
          
            
              [
              S
              ]
            
            N
          
          =
          
          ⁢
          
            
              
                
                  2
                  N
                
              
              ⁡
              
                [
                S
                ]
              
            
            N
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              
                
                  
                    
                      2
                      N
                    
                  
                  ⁡
                  
                    [
                    Pr
                    ]
                  
                
                N
              
              ⁡
              
                [
                
                  
                    
                      
                        U
                        
                          N
                          /
                          2
                        
                      
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        I
                        
                          N
                          /
                          2
                        
                      
                    
                  
                
                ]
              
            
            [
            
              
                I
                2
              
              ⊗
              
                
                  
                    [
                    
                      
                    
                    ⁢
                    
                      
                        ...
                        ⁢
                        
                          
                            
                          
                          ⁢
                          
                            
                          
                        
                        [
                        
                          
                            I
                            2
                          
                          ⊗
                          
                            
                              
                                [
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                [
                                                Pr
                                                ]
                                              
                                              4
                                            
                                            ⁡
                                            
                                              [
                                              
                                                
                                                  
                                                    
                                                      U
                                                      2
                                                    
                                                  
                                                  
                                                    0
                                                  
                                                
                                                
                                                  
                                                    0
                                                  
                                                  
                                                    
                                                      I
                                                      2
                                                    
                                                  
                                                
                                              
                                              ]
                                            
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                I
                                                2
                                              
                                              ⊗
                                              
                                                S
                                                2
                                              
                                            
                                            ]
                                          
                                        
                                        ⁡
                                        
                                          [
                                          
                                            
                                              
                                                
                                                  D
                                                  2
                                                
                                              
                                              
                                                0
                                              
                                            
                                            
                                              
                                                0
                                              
                                              
                                                
                                                  I
                                                  2
                                                
                                              
                                            
                                          
                                          ]
                                        
                                      
                                      ⁡
                                      
                                        [
                                        
                                          
                                            
                                              
                                                I
                                                2
                                              
                                            
                                            
                                              
                                                I
                                                2
                                              
                                            
                                          
                                          
                                            
                                              
                                                I
                                                2
                                              
                                            
                                            
                                              
                                                -
                                                
                                                  I
                                                  2
                                                
                                              
                                            
                                          
                                        
                                        ]
                                      
                                    
                                    ⁡
                                    
                                      [
                                      Pc
                                      ]
                                    
                                  
                                  4
                                
                                ]
                              
                              }
                            
                            
                              S
                              4
                            
                          
                        
                        ⁢
                        
                          
                        
                        ]
                      
                      ⁢
                      
                        
                      
                      ⁢
                      ...
                    
                    ⁢
                    
                      
                    
                    ]
                  
                  }
                
                
                  S
                  
                    N
                    /
                    2
                  
                
              
            
            ]
          
        
      
    
    
      
        
          
          ⁢
          
            
              
                
                  
                    [
                    
                      
                        
                          
                            D
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          
                            -
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  Pc
                  ]
                
              
              N
            
            .
          
        
      
    
  



When the switching part selects the DFT, the submatrix selector may select [Pr]N as a submatrix and the permutation matrix selector selects [W]n as a permutation matrix.
The transformation part may apply the DFT to the input signal by computing Equation (56) by using the submatrix [Pr]N and the permutation matrix [W]n, multiplying the input signal by a matrix [M3] during first h steps of 2h steps, and multiplying a matrix [M4] during last h steps:




  
    
      
        
          
            
              
                
                  [
                  F
                  ]
                
                N
              
              =
              
              ⁢
              
                
                  
                    
                      (
                      
                        
                          [
                          Pr
                          ]
                        
                        N
                      
                      )
                    
                    
                      -
                      1
                    
                  
                  ⁡
                  
                    [
                    
                      F
                      ~
                    
                    ]
                  
                
                N
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          (
                          
                            
                              [
                              Pr
                              ]
                            
                            N
                          
                          )
                        
                        
                          -
                          1
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                Pr
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              
                                F
                                ~
                              
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              
                                F
                                ~
                              
                              
                                N
                                /
                                2
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            W
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                    
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          -
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      [
                      Pr
                      ]
                    
                    N
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            Pr
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁢
                
                   
                  
                    
                      
                        [
                        
                          
                            I
                            2
                          
                          ⊗
                          
                            Pr
                            
                              N
                              /
                              2
                            
                          
                        
                        ]
                      
                      ⁡
                      
                        [
                        
                          
                            I
                            2
                          
                          ⊗
                          
                            [
                            
                              
                                
                                  
                                    I
                                    
                                      N
                                      /
                                      4
                                    
                                  
                                
                                
                                  0
                                
                              
                              
                                
                                  0
                                
                                
                                  
                                    Pr
                                    
                                      N
                                      /
                                      4
                                    
                                  
                                
                              
                            
                            ]
                          
                        
                        ]
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      ...
                      ⁢
                      
                        
                      
                      [
                      
                        
                          I
                          
                            N
                            /
                            4
                          
                        
                        ⊗
                        
                          Pr
                          4
                        
                      
                      ]
                    
                  
                  ⁢
                  
                    
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                 
                
                  
                    [
                    
                      
                        I
                        
                          N
                          /
                          4
                        
                      
                      ⊗
                      
                        [
                        
                          
                            
                              
                                I
                                2
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                Pr
                                2
                              
                            
                          
                        
                        ]
                      
                    
                    ]
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                     
                    
                      
                        
                          [
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                            ⊗
                            
                              F
                              2
                            
                          
                          ]
                        
                        ⁡
                        
                          [
                          
                            
                              I
                              
                                N
                                /
                                4
                              
                            
                            ⊗
                            
                              [
                              
                                
                                  
                                    
                                      I
                                      2
                                    
                                  
                                  
                                    0
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    
                                      W
                                      2
                                    
                                  
                                
                              
                              ]
                            
                          
                          ]
                        
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                         
                        
                          
                        
                      
                    
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  [
                  
                    
                      I
                      
                        N
                        /
                        4
                      
                    
                    ⊗
                    
                      [
                      
                        
                          
                            
                              I
                              2
                            
                          
                          
                            
                              I
                              2
                            
                          
                        
                        
                          
                            
                              I
                              2
                            
                          
                          
                            
                              -
                              
                                I
                                2
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ]
                
                ⁢
                
                  
                
                ⁢
                
                  ...
                  ⁢
                  
                    
                  
                  [
                  
                    
                      I
                      2
                    
                    ⊗
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                4
                              
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              W
                              
                                N
                                /
                                4
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ]
                
              
              ⁢
              
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    [
                    
                      
                        I
                        2
                      
                      ⊗
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  4
                                
                              
                            
                            
                              
                                I
                                
                                  N
                                  /
                                  4
                                
                              
                            
                          
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  4
                                
                              
                            
                            
                              
                                -
                                
                                  I
                                  
                                    N
                                    /
                                    4
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            W
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                    
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          -
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      [
                      Pr
                      ]
                    
                    N
                    
                      -
                      1
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            Pr
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
                [
                
                  
                    I
                    2
                  
                  ⊗
                  
                    
                      
                        [
                        
                          
                        
                        ⁢
                        
                          ...
                          ⁢
                          
                            
                              
                            
                            ⁢
                            
                              
                            
                          
                          ⁢
                          
                            
                              
                                
                                  
                                    
                                      
                                        
                                          [
                                          Pr
                                          ]
                                        
                                        4
                                        
                                          -
                                          1
                                        
                                      
                                      ⁡
                                      
                                        [
                                        
                                          
                                            
                                              
                                                I
                                                2
                                              
                                            
                                            
                                              0
                                            
                                          
                                          
                                            
                                              0
                                            
                                            
                                              
                                                Pr
                                                2
                                              
                                            
                                          
                                        
                                        ]
                                      
                                    
                                    ⁡
                                    
                                      [
                                      
                                        
                                          I
                                          2
                                        
                                        ⊗
                                        
                                          
                                            F
                                            ~
                                          
                                          2
                                        
                                      
                                      ]
                                    
                                  
                                  ⁡
                                  
                                    [
                                    
                                      
                                        
                                          
                                            I
                                            2
                                          
                                        
                                        
                                          0
                                        
                                      
                                      
                                        
                                          0
                                        
                                        
                                          
                                            W
                                            2
                                          
                                        
                                      
                                    
                                    ]
                                  
                                
                                ⁡
                                
                                  [
                                  
                                    
                                      
                                        
                                          I
                                          2
                                        
                                      
                                      
                                        
                                          I
                                          2
                                        
                                      
                                    
                                    
                                      
                                        
                                          I
                                          2
                                        
                                      
                                      
                                        
                                          -
                                          
                                            I
                                            2
                                          
                                        
                                      
                                    
                                  
                                  ]
                                
                              
                              }
                            
                            
                              
                                F
                                ~
                              
                              4
                            
                          
                          ⁢
                          
                            
                          
                          ⁢
                          ...
                        
                        ⁢
                        
                          
                        
                        ]
                      
                      }
                    
                    
                      
                        F
                        ~
                      
                      
                        N
                        /
                        2
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            W
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          
                            -
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                        
                      
                    
                    ]
                  
                
                .
              
            
          
        
      
    
    
      
        (
        56
        )
      
    
  




When the switching part selects the HWT, the submatrix selector may select [Pa]N as a submatrix and the permutation matrix selector selects [Pb]N as a permutation matrix.
The transformation part may apply the HWT to the input signal by computing Equation (64) by using the submatrix [Pa]N and the permutation matrix [Pb]N and multiplying the input signal by a matrix [M5] during last h steps of 2h steps:




  
    
      
        
          
            
              
                
                  [
                  H
                  ]
                
                N
              
              =
              
              ⁢
              
                
                  
                    [
                    Pi
                    ]
                  
                  N
                  
                    -
                    1
                  
                
                ⁢
                
                  
                    
                      
                        
                          [
                          H
                          ⁢
                          
                            
                          
                          ]
                        
                        •
                      
                      N
                    
                    ⁡
                    
                      [
                      Pj
                      ]
                    
                  
                  N
                  
                    -
                    1
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          1
                          r
                        
                        ⁡
                        
                          [
                          Pa
                          ]
                        
                      
                      N
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              Pa
                              
                                N
                                /
                                2
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              3
                              ⁢
                              
                                N
                                /
                                4
                              
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            Pa
                            
                              N
                              /
                              4
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁢
                
                  
                
                ⁢
                ...
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  -
                                  2
                                
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                Pa
                                2
                              
                            
                          
                        
                        ]
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  -
                                  2
                                
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                
                                  H
                                  2
                                
                                •
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                -
                                4
                              
                            
                          
                          
                            
                              
                            
                          
                          
                            0
                          
                        
                        
                          
                            
                              
                            
                          
                          
                            
                              r
                              ⁢
                              
                                
                              
                              ⁢
                              
                                I
                                2
                              
                            
                          
                          
                            
                              
                            
                          
                        
                        
                          
                            0
                          
                          
                            
                              
                            
                          
                          
                            
                              Pb
                              2
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              -
                              4
                            
                          
                        
                        
                          
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          
                            
                          
                        
                        
                          
                            I
                            2
                          
                        
                        
                          
                            I
                            2
                          
                        
                      
                      
                        
                          0
                        
                        
                          
                            I
                            2
                          
                        
                        
                          
                            -
                            
                              I
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁢
                
                  
                
                ⁢
                ...
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          
                            
                          
                        
                        
                          
                            r
                            ⁢
                            
                              
                            
                            ⁢
                            
                              I
                              
                                N
                                /
                                4
                              
                            
                          
                        
                        
                          
                            
                          
                        
                      
                      
                        
                          0
                        
                        
                          
                            
                          
                        
                        
                          
                            Pb
                            
                              N
                              /
                              4
                            
                          
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          
                            
                          
                        
                        
                          
                            I
                            
                              N
                              /
                              4
                            
                          
                        
                        
                          
                            I
                            
                              N
                              /
                              4
                            
                          
                        
                      
                      
                        
                          0
                        
                        
                          
                            I
                            
                              N
                              /
                              4
                            
                          
                        
                        
                          
                            -
                            
                              I
                              
                                N
                                /
                                4
                              
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          r
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          Pb
                          
                            N
                            /
                            2
                          
                        
                      
                    
                  
                  ]
                
              
              ⁢
              
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                        
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            
                              -
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                        
                      
                      ]
                    
                    ⁢
                    
                      
                    
                    [
                    Pb
                    ]
                  
                  N
                
                .
              
              ⁢
              
                
              
            
          
        
      
    
    
      
        (
        64
        )
      
    
  




A signal transformation method may be applied to the signal transformation apparatus.
A computer-readable recording medium may contain a computer program for executing functions of the signal transformation apparatus.



BRIEF DESCRIPTION OF THE DRAWINGS
The above and other aspects, features, and advantages of certain exemplary embodiments of the present invention will be more apparent from the following description taken in conjunction with the accompanying drawings, in which:
FIG. 1A-1C is a butterfly data flow graph of Equation (27) according to an exemplary embodiment of the present invention;
FIG. 2A-2C is a butterfly data flow graph of Equation (45) according to an exemplary embodiment of the present invention;
FIG. 3A-3C is a butterfly data flow graph of Equation (56) according to an exemplary embodiment of the present invention;
FIG. 4 is a butterfly data flow graph of Equation (64) according to an exemplary embodiment of the present invention;
FIG. 5 is a block diagram of a signal transformation apparatus adopting a hybrid architecture according to an exemplary embodiment of the present invention;
FIG. 6A to 6C is a detailed block diagram of the signal transformation apparatus according to an exemplary embodiment of the present invention;
FIG. 6D is a graph of variance comparison of the KLT/DCT/DST/DFT/HWT
FIG. 7A is a diagram of a one-way video coding scenario according to an exemplary embodiment of the present invention; and
FIG. 7B is a diagram of a two-way video coding scenario according to an exemplary embodiment of the present invention.



Throughout the drawings, like reference numerals will be understood to refer to like parts, components and structures.
DETAILED DESCRIPTION OF THE INVENTION
The following description with reference to the accompanying drawings is provided to assist in a comprehensive understanding of exemplary embodiments of the invention as defined by the claims and their equivalents.
I. Introduction
Exemplary embodiments of the present invention provide a fast DCT-II/DST-II/DFT/HWT hybrid transform architecture for new digital videos and fusion mobile handsets based on Jacket-like sparse matrix decomposition. The fast hybrid architecture includes source coding standard such as MPEG4 and JPEG 2000, and digital filtering discrete Fourier transform. The fast hybrid architecture includes two operations. One operation is Block-wise Inverse Jacket Matrix (BIJM) for DCT-II/DST-II and the other operation is Element-wise Inverse Jacket Matrix (EIJM) for DFT/HWT. These operations have similar recursive computational fashion, which implies that they can be decomposed to Kronecker products of an identity Hadamard matrix and a successively lower-order sparse matrix. Based on this, it is possible to develop a single chip applying the signal transformation apparatus of the fast hybrid algorithm architecture for intelligent mobile handsets.
For doing so, the present invention provides sparse unified matrix factorization for the unified chip based on the Jacket matrix.
Analysis and decomposition of the sparse matrix is a useful tool for fast computations and character generation. DCT-II, DST-II, DFT and HWT matrices can be decomposed to one orthogonal character matrix and a special sparse matrix. The inverse of the sparse matrix is a block-wise inverse or an element-wise inverse. Mathematically, when [A]N=[aij]N is a matrix and




  
    
      [
      J
      ]
    
    N
    
      -
      1
    
  
  =
  
    
      
        1
        N
      
      ⁡
      
        [
        
          a
          ij
          
            -
            1
          
        
        ]
      
    
    N
    T
  




is its inverse, the matrix A corresponds to the Jacket matrix. The special sparse matrix belongs to the Jacket matrix.

Hereafter, the hybrid architecture for combining the unified architecture of the sparse matrix decomposition and the DCT-II, the DST-II, the DFT, and the HWT is explained.
II. Block-Wise Inverse Sparse Matrix Decomposition for DCT-II Transform
First, the block-wise inverse sparse matrix decomposition for the DCT-II transform is described. A DCT-II transform matrix is given as follows.




  
    
      
        
          
            
              [
              
                C
                
                  m
                  ,
                  n
                
              
              ]
            
            N
          
          =
          
            
              
                c
                m
              
              ⁢
              
                
                  2
                  N
                
              
              ⁢
              cos
              ⁢
              
                
                  
                    m
                    ⁡
                    
                      (
                      
                        
                          2
                          ⁢
                          n
                        
                        +
                        1
                      
                      )
                    
                  
                  ⁢
                  π
                
                
                  2
                  ⁢
                  N
                
              
            
            =
            
              
                
                  
                    2
                    N
                  
                
                ⁡
                
                  [
                  
                    C
                    
                      m
                      ,
                      n
                    
                  
                  ]
                
              
              N
            
          
        
        ,
        
          

        
        ⁢
        where
        ,
        
          
            c
            m
          
          =
          
            {
            
              
                
                  
                    
                      
                        1
                        ,
                      
                    
                    
                      
                        m
                        ≠
                        0
                      
                    
                  
                  
                    
                      
                        1
                        /
                        
                          
                            2
                            ,
                          
                        
                      
                    
                    
                      
                        
                          m
                          =
                          0
                        
                        ,
                      
                    
                  
                
                ⁢
                
                  

                
                ⁢
                m
              
              ,
              
                n
                =
                0
              
              ,
              1
              ,
              ...
              ⁢
              
                
              
              ,
              
                N
                -
                1
              
            
          
        
      
    
    
      
        (
        1
        )
      
    
  




A 2×2 matrix which is a basic matrix is given by




  
    
      
        
          
            
              
                
                  
                    [
                    C
                    ]
                  
                  2
                
                =
                
                ⁢
                
                  [
                  
                    
                      
                        
                          1
                          
                            2
                          
                        
                      
                      
                        
                          1
                          
                            2
                          
                        
                      
                    
                    
                      
                        
                          C
                          4
                          1
                        
                      
                      
                        
                          C
                          4
                          3
                        
                      
                    
                  
                  ]
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  [
                  
                    
                      
                        
                          1
                          
                            2
                          
                        
                      
                      
                        
                          1
                          
                            2
                          
                        
                      
                    
                    
                      
                        
                          1
                          
                            2
                          
                        
                      
                      
                        
                          -
                          
                            1
                            
                              2
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    1
                    
                      2
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          1
                        
                        
                          1
                        
                      
                      
                        
                          1
                        
                        
                          
                            -
                            1
                          
                        
                      
                    
                    ]
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    [
                    C
                    ]
                  
                  2
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          where
          ⁢
          
            
          
          ⁢
          
            1
            
              2
            
          
        
      
    
    
      
        (
        2
        )
      
    
  




can be seen as a special element-wise inverse matrix of order 1, and its inverse is √{square root over (2)}, and Cli=cos(iπ/l) is the cosine unit for DCT computations.

Further, the 4-by-4 DCT-II matrix has the form as




  
    
      
        
          
            
              
                
                  [
                  C
                  ]
                
                4
              
              =
              
              ⁢
              
                
                  
                    2
                    4
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          1
                          
                            2
                          
                        
                      
                      
                        
                          1
                          
                            2
                          
                        
                      
                      
                        
                          1
                          
                            2
                          
                        
                      
                      
                        
                          1
                          
                            2
                          
                        
                      
                    
                    
                      
                        
                          C
                          8
                          1
                        
                      
                      
                        
                          C
                          8
                          3
                        
                      
                      
                        
                          C
                          8
                          5
                        
                      
                      
                        
                          C
                          8
                          7
                        
                      
                    
                    
                      
                        
                          C
                          8
                          2
                        
                      
                      
                        
                          C
                          8
                          6
                        
                      
                      
                        
                          C
                          8
                          6
                        
                      
                      
                        
                          C
                          8
                          2
                        
                      
                    
                    
                      
                        
                          C
                          8
                          3
                        
                      
                      
                        
                          C
                          8
                          7
                        
                      
                      
                        
                          C
                          8
                          1
                        
                      
                      
                        
                          C
                          8
                          5
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  1
                  
                    2
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          1
                          
                            2
                          
                        
                      
                      
                        
                          1
                          
                            2
                          
                        
                      
                      
                        
                          1
                          
                            2
                          
                        
                      
                      
                        
                          1
                          
                            2
                          
                        
                      
                    
                    
                      
                        
                          C
                          8
                          1
                        
                      
                      
                        
                          C
                          8
                          3
                        
                      
                      
                        
                          -
                          
                            C
                            8
                            3
                          
                        
                      
                      
                        
                          -
                          
                            C
                            8
                            1
                          
                        
                      
                    
                    
                      
                        
                          C
                          8
                          2
                        
                      
                      
                        
                          -
                          
                            C
                            8
                            2
                          
                        
                      
                      
                        
                          -
                          
                            C
                            8
                            2
                          
                        
                      
                      
                        
                          C
                          8
                          2
                        
                      
                    
                    
                      
                        
                          C
                          8
                          3
                        
                      
                      
                        
                          -
                          
                            C
                            8
                            1
                          
                        
                      
                      
                        
                          C
                          8
                          1
                        
                      
                      
                        
                          -
                          
                            C
                            8
                            3
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    1
                    
                      2
                    
                  
                  ⁡
                  
                    [
                    C
                    ]
                  
                
                4
              
            
          
        
      
    
    
      
        (
        3
        )
      
    
  




We can write




  
    
      
        
          
            
              
                
                  
                    
                      
                        [
                        Pr
                        ]
                      
                      4
                    
                    ⁡
                    
                      [
                      C
                      ]
                    
                  
                  4
                
                =
                
                ⁢
                
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                        
                          0
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          0
                        
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          0
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          0
                        
                        
                          0
                        
                        
                          1
                        
                      
                    
                    ]
                  
                  ⁢
                  
                    
                      1
                      
                        2
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              1
                              
                                2
                              
                            
                          
                          
                            
                              1
                              
                                2
                              
                            
                          
                          
                            
                              1
                              
                                2
                              
                            
                          
                          
                            
                              1
                              
                                2
                              
                            
                          
                        
                        
                          
                            
                              C
                              8
                              1
                            
                          
                          
                            
                              C
                              8
                              3
                            
                          
                          
                            
                              -
                              
                                C
                                8
                                3
                              
                            
                          
                          
                            
                              -
                              
                                C
                                8
                                1
                              
                            
                          
                        
                        
                          
                            
                              C
                              8
                              2
                            
                          
                          
                            
                              -
                              
                                C
                                8
                                2
                              
                            
                          
                          
                            
                              -
                              
                                C
                                8
                                2
                              
                            
                          
                          
                            
                              C
                              8
                              2
                            
                          
                        
                        
                          
                            
                              C
                              8
                              3
                            
                          
                          
                            
                              -
                              
                                C
                                8
                                1
                              
                            
                          
                          
                            
                              C
                              8
                              1
                            
                          
                          
                            
                              -
                              
                                C
                                8
                                3
                              
                            
                          
                        
                      
                      ]
                    
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    1
                    
                      2
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            1
                            
                              2
                            
                          
                        
                        
                          
                            1
                            
                              2
                            
                          
                        
                        
                          
                            1
                            
                              2
                            
                          
                        
                        
                          
                            1
                            
                              2
                            
                          
                        
                      
                      
                        
                          
                            C
                            8
                            2
                          
                        
                        
                          
                            -
                            
                              C
                              8
                              2
                            
                          
                        
                        
                          
                            -
                            
                              C
                              8
                              2
                            
                          
                        
                        
                          
                            C
                            8
                            2
                          
                        
                      
                      
                        
                          
                            C
                            8
                            1
                          
                        
                        
                          
                            C
                            8
                            3
                          
                        
                        
                          
                            -
                            
                              C
                              8
                              3
                            
                          
                        
                        
                          
                            -
                            
                              C
                              8
                              1
                            
                          
                        
                      
                      
                        
                          
                            C
                            8
                            3
                          
                        
                        
                          
                            -
                            
                              C
                              8
                              1
                            
                          
                        
                        
                          
                            C
                            8
                            1
                          
                        
                        
                          
                            -
                            
                              C
                              8
                              3
                            
                          
                        
                      
                    
                    ]
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              where
              ⁢
              
                
              
              [
              Pr
              ]
            
            4
          
          =
          
            [
            
              
                
                  1
                
                
                  0
                
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  1
                
                
                  0
                
              
              
                
                  0
                
                
                  1
                
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  0
                
                
                  1
                
              
            
            ]
          
        
      
    
    
      
        (
        4
        )
      
    
  




is permutation matrix. [Pr]N Permutation matrix is a special case, which has the form





  
    
      
        [
        Pr
        ]
      
      2
    
    =
    
      
        
          
            [
            I
            ]
          
          2
        
        ⁢
        
          
        
        ⁢
        
          
            and
            ⁢
            
              

            
            [
            Pr
            ]
          
          N
        
      
      =
      
        [
        
          
            
              1
            
            
              0
            
            
              0
            
            
              ...
            
            
              0
            
            
              0
            
            
              ...
            
            
              0
            
          
          
            
              0
            
            
              0
            
            
              0
            
            
              ...
            
            
              1
            
            
              0
            
            
              ...
            
            
              0
            
          
          
            
              0
            
            
              1
            
            
              0
            
            
              
                
              
            
            
              0
            
            
              0
            
            
              ...
            
            
              0
            
          
          
            
              0
            
            
              0
            
            
              0
            
            
              
                
              
            
            
              0
            
            
              1
            
            
              
                
              
            
            
              0
            
          
          
            
              0
            
            
              0
            
            
              1
            
            
              
                
              
            
            
              0
            
            
              0
            
            
              
                
              
            
            
              0
            
          
          
            
              0
            
            
              0
            
            
              0
            
            
              
                
              
            
            
              0
            
            
              0
            
            
              ⋱
            
            
              0
            
          
          
            
              0
            
            
              0
            
            
              0
            
            
              ⋱
            
            
              0
            
            
              0
            
            
              
                
              
            
            
              0
            
          
          
            
              0
            
            
              0
            
            
              0
            
            
              0
            
            
              0
            
            
              0
            
            
              
                
              
            
            
              1
            
          
        
        ]
      
    
  
  ,
  
    N
    ≥
    4
  
  ,
  
    

  
  ⁢
  
    
      
        where
        ⁢
        
          
        
        [
        Pr
        ]
      
      N
    
    =
    
      
        [
        
          pr
          
            i
            ,
            j
          
        
        ]
      
      N
    
  
  ,
  
    

  
  ⁢
  
    with
    ⁢
    
      
    
    ⁢
    
      {
      
        
          
            
              
                
                  
                    
                      pr
                      
                        i
                        ,
                        j
                      
                    
                    =
                    1
                  
                  ,
                
              
              
                
                  
                    
                      if
                      ⁢
                      
                        
                      
                      ⁢
                      i
                    
                    =
                    
                      2
                      ⁢
                      j
                    
                  
                  ,
                  
                    0
                    ≤
                    j
                    ≤
                    
                      
                        N
                        2
                      
                      -
                      1
                    
                  
                  ,
                
              
            
            
              
                
                  
                    
                      pr
                      
                        i
                        ,
                        j
                      
                    
                    =
                    1
                  
                  ,
                
              
              
                
                  
                    
                      if
                      ⁢
                      
                        
                      
                      ⁢
                      i
                    
                    =
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            j
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      mod
                      ⁢
                      
                        
                      
                      ⁢
                      N
                    
                  
                  ,
                  
                    
                      N
                      2
                    
                    ≤
                    j
                    ≤
                    
                      N
                      -
                      1
                    
                  
                  ,
                
              
            
            
              
                
                  
                    
                      pr
                      
                        i
                        ,
                        j
                      
                    
                    =
                    0
                  
                  ,
                
              
              
                
                  others
                  .
                
              
            
          
          ⁢
          
            

          
          ⁢
          where
          ⁢
          
            
          
          ⁢
          i
        
        ,
        
          j
          ∈
          
            
              {
              
                0
                ,
                1
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  N
                  -
                  1
                
              
              }
            
            .
          
        
      
    
  




Let us define a column permutation matrix




  
    
      
        [
        Pc
        ]
      
      4
    
    =
    
      [
      
        
          
            1
          
          
            0
          
          
            0
          
          
            0
          
        
        
          
            0
          
          
            1
          
          
            0
          
          
            0
          
        
        
          
            0
          
          
            0
          
          
            0
          
          
            1
          
        
        
          
            0
          
          
            0
          
          
            1
          
          
            0
          
        
      
      ]
    
  
  ,




and [Pc]N is a reversible permutation matrix, which is defined by





  
    
      
        
          
            
              [
              Pc
              ]
            
            2
          
          =
          
            
              [
              I
              ]
            
            2
          
        
        ,
        
          
            
              and
              ⁢
              
                

              
              [
              Pc
              ]
            
            N
          
          =
          
            [
            
              
                
                  
                    I
                    
                      N
                      /
                      4
                    
                  
                
                
                  0
                
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    I
                    
                      N
                      /
                      4
                    
                  
                
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  0
                
                
                  
                    I
                    
                      N
                      /
                      4
                    
                  
                
              
              
                
                  0
                
                
                  0
                
                
                  
                    I
                    
                      N
                      /
                      4
                    
                  
                
                
                  0
                
              
            
            ]
          
        
        ,
        
          N
          ≥
          4.
        
      
    
    
      
        (
        5
        )
      
    
  




Thus the matrix decomposition is the form of the matrix product of diagonal block-wise inverse Jacket sparse and Hadamard matrix,




  
    
      
        
          
            
              
                
                  [
                  
                    C
                    ~
                  
                  ]
                
                4
              
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          [
                          Pr
                          ]
                        
                        4
                      
                      ⁡
                      
                        [
                        C
                        ]
                      
                    
                    4
                  
                  ⁡
                  
                    [
                    Pc
                    ]
                  
                
                4
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    1
                    
                      2
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            1
                            
                              2
                            
                          
                        
                        
                          
                            1
                            
                              2
                            
                          
                        
                        
                          
                            1
                            
                              2
                            
                          
                        
                        
                          
                            1
                            
                              2
                            
                          
                        
                      
                      
                        
                          
                            C
                            8
                            2
                          
                        
                        
                          
                            -
                            
                              C
                              8
                              2
                            
                          
                        
                        
                          
                            -
                            
                              C
                              8
                              2
                            
                          
                        
                        
                          
                            C
                            8
                            2
                          
                        
                      
                      
                        
                          
                            C
                            8
                            1
                          
                        
                        
                          
                            C
                            8
                            3
                          
                        
                        
                          
                            -
                            
                              C
                              8
                              3
                            
                          
                        
                        
                          
                            -
                            
                              C
                              8
                              1
                            
                          
                        
                      
                      
                        
                          
                            C
                            8
                            3
                          
                        
                        
                          
                            -
                            
                              C
                              8
                              1
                            
                          
                        
                        
                          
                            C
                            8
                            1
                          
                        
                        
                          
                            -
                            
                              C
                              8
                              3
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  1
                  
                    2
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          1
                          
                            2
                          
                        
                      
                      
                        
                          1
                          
                            2
                          
                        
                      
                      
                        
                          1
                          
                            2
                          
                        
                      
                      
                        
                          1
                          
                            2
                          
                        
                      
                    
                    
                      
                        
                          C
                          8
                          2
                        
                      
                      
                        
                          -
                          
                            C
                            8
                            2
                          
                        
                      
                      
                        
                          C
                          8
                          2
                        
                      
                      
                        
                          -
                          
                            C
                            8
                            2
                          
                        
                      
                    
                    
                      
                        
                          C
                          8
                          1
                        
                      
                      
                        
                          C
                          8
                          3
                        
                      
                      
                        
                          C
                          8
                          1
                        
                      
                      
                        
                          -
                          
                            C
                            8
                            3
                          
                        
                      
                    
                    
                      
                        
                          C
                          8
                          3
                        
                      
                      
                        
                          -
                          
                            C
                            8
                            1
                          
                        
                      
                      
                        
                          -
                          
                            C
                            8
                            3
                          
                        
                      
                      
                        
                          C
                          8
                          1
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  1
                  
                    2
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          C
                          2
                        
                      
                      
                        
                          C
                          2
                        
                      
                    
                    
                      
                        
                          B
                          2
                        
                      
                      
                        
                          -
                          
                            B
                            2
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              
                =
                
                ⁢
                
                  
                    
                      1
                      
                        2
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              C
                              2
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              B
                              2
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            2
                          
                        
                        
                          
                            I
                            2
                          
                        
                      
                      
                        
                          
                            I
                            2
                          
                        
                        
                          
                            -
                            
                              I
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
              
              ,
            
          
        
      
    
    
      
        (
        6
        )
      
    
  




Where





  
    
      
        [
        C
        ]
      
      2
    
    =
    
      [
      
        
          
            
              1
              
                2
              
            
          
          
            
              1
              
                2
              
            
          
        
        
          
            
              1
              
                2
              
            
          
          
            
              -
              
                1
                
                  2
                
              
            
          
        
      
      ]
    
  
  ,




the 2-by-2 DCT-II matrix and





  
    
      [
      B
      ]
    
    2
  
  =
  
    
      [
      
        
          
            
              C
              8
              1
            
          
          
            
              C
              8
              3
            
          
        
        
          
            
              C
              8
              3
            
          
          
            
              -
              
                C
                8
                1
              
            
          
        
      
      ]
    
    .
  




Then we can write that it is clear that





   
  
    [
    
      
        
          
            C
            2
          
        
        
          0
        
      
      
        
          0
        
        
          
            B
            2
          
        
      
    
    ]
  




is a diagonal block-wise inverse Jacket matrix, which has





  
    
      
        
          
            [
            
              
                
                  
                    C
                    2
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    B
                    2
                  
                
              
            
            ]
          
          
            -
            1
          
        
        =
        
          
            [
            
              
                
                  
                    
                      (
                      
                        C
                        2
                      
                      )
                    
                    
                      -
                      1
                    
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    
                      (
                      
                        B
                        2
                      
                      )
                    
                    
                      -
                      1
                    
                  
                
              
            
            ]
          
          .
        
      
    
    
      
        (
        7
        )
      
    
  




In a general, the permuted DCT-II matrix └{tilde over (C)}┘N can be constructed recursively by using




  
    
      
        
          
            
              
                
                  [
                  
                    C
                    ~
                  
                  ]
                
                N
              
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          [
                          Pr
                          ]
                        
                        N
                      
                      ⁡
                      
                        [
                        C
                        ]
                      
                    
                    N
                  
                  ⁡
                  
                    [
                    Pc
                    ]
                  
                
                N
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    [
                    Pr
                    ]
                  
                  N
                
                ⁢
                
                  
                    
                      
                        
                          
                            2
                            N
                          
                        
                        ⁡
                        
                          [
                          C
                          ]
                        
                      
                      N
                    
                    ⁡
                    
                      [
                      Pc
                      ]
                    
                  
                  N
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          
                            
                              2
                              N
                            
                          
                          ⁡
                          
                            [
                            Pr
                            ]
                          
                        
                        N
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                C
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                B
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                        
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            
                              -
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    Pc
                    ]
                  
                
                N
              
            
          
        
      
    
    
      
        (
        8
        )
      
    
  




where [C]N/2 denotes the





  
    N
    2
  
  ×
  
    N
    2
  




DCT-II matrix, and [B]N/2 can be calculated by using





  
    
      
        
          
            
              [
              B
              ]
            
            
              N
              /
              2
            
          
          =
          
            
              [
              
                
                  (
                  
                    C
                    
                      2
                      ⁢
                      N
                    
                    
                      f
                      ⁡
                      
                        (
                        
                          m
                          ,
                          n
                        
                        )
                      
                    
                  
                  )
                
                
                  m
                  ,
                  n
                
              
              ]
            
            
              N
              /
              2
            
          
        
        ,
      
    
    
      
        (
        9
        )
      
    
  
  
    
      
        where
        ⁢
        
          
        
        ⁢
        
          {
          
            
              
                
                  
                    
                      
                        
                          f
                          ⁡
                          
                            (
                            
                              m
                              ,
                              1
                            
                            )
                          
                        
                        =
                        
                          
                            2
                            ⁢
                            m
                          
                          -
                          1
                        
                      
                      ,
                    
                  
                
                
                  
                    
                      
                        
                          f
                          ⁡
                          
                            (
                            
                              m
                              ,
                              
                                n
                                +
                                1
                              
                            
                            )
                          
                        
                        =
                        
                          
                            f
                            ⁡
                            
                              (
                              
                                m
                                ,
                                n
                              
                              )
                            
                          
                          +
                          
                            
                              f
                              ⁡
                              
                                (
                                
                                  m
                                  ,
                                  1
                                
                                )
                              
                            
                            ×
                            2
                          
                        
                      
                      ,
                    
                  
                
              
              ⁢
              
                

              
              ⁢
              m
            
            ,
            
              n
              ∈
              
                
                  {
                  
                    1
                    ,
                    2
                    ,
                    ...
                    ⁢
                    
                      
                    
                    ,
                    
                      N
                      /
                      2
                    
                  
                  }
                
                .
              
            
          
        
      
    
    
      
        (
        10
        )
      
    
  




For example, in the 4-by-4 permuted DCT-II matrix └{tilde over (C)}┘4, B2 could be calculated by using ƒ(1,1)=1, ƒ(2,1)=3, ƒ(1,2)=ƒ(1,1)+ƒ(1,1)×2=3, and ƒ(2,2)=ƒ(2,1)+ƒ(2,2)=ƒ(2,1)+ƒ(2,1)×2=9,




  
    
      
        
          
            
              
                
                  [
                  B
                  ]
                
                2
              
              =
              
              ⁢
              
                
                  [
                  
                    
                      (
                      
                        C
                        8
                        
                          f
                          ⁡
                          
                            (
                            
                              m
                              ,
                              n
                            
                            )
                          
                        
                      
                      )
                    
                    
                      m
                      ,
                      n
                    
                  
                  ]
                
                4
              
            
          
        
        
          
            
              =
              
              ⁢
              
                [
                
                  
                    
                      
                        
                          (
                          
                            C
                            8
                            
                              f
                              ⁡
                              
                                (
                                
                                  1
                                  ,
                                  1
                                
                                )
                              
                            
                          
                          )
                        
                        
                          1
                          ,
                          2
                        
                      
                    
                    
                      
                        
                          (
                          
                            C
                            8
                            
                              f
                              ⁡
                              
                                (
                                
                                  1
                                  ,
                                  2
                                
                                )
                              
                            
                          
                          )
                        
                        
                          1
                          ,
                          2
                        
                      
                    
                  
                  
                    
                      
                        
                          (
                          
                            C
                            8
                            
                              f
                              ⁡
                              
                                (
                                
                                  2
                                  ,
                                  1
                                
                                )
                              
                            
                          
                          )
                        
                        
                          2
                          ,
                          1
                        
                      
                    
                    
                      
                        
                          (
                          
                            C
                            8
                            
                              f
                              ⁡
                              
                                (
                                
                                  2
                                  ,
                                  2
                                
                                )
                              
                            
                          
                          )
                        
                        
                          2
                          ,
                          2
                        
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              =
              
              ⁢
              
                [
                
                  
                    
                      
                        C
                        8
                        1
                      
                    
                    
                      
                        C
                        8
                        3
                      
                    
                  
                  
                    
                      
                        C
                        8
                        3
                      
                    
                    
                      
                        C
                        8
                        9
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          C
                          8
                          1
                        
                      
                      
                        
                          C
                          8
                          3
                        
                      
                    
                    
                      
                        
                          C
                          8
                          3
                        
                      
                      
                        
                          -
                          
                            C
                            8
                            1
                          
                        
                      
                    
                  
                  ]
                
                .
              
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




From Eq. (8), the inverse of └{tilde over (C)}┘N can be computed from the block-wise inverse




  
    
      
        
          
            
              
                
                  [
                  C
                  ]
                
                N
                
                  -
                  1
                
              
              =
              
              ⁢
              
                
                  (
                  
                    
                      
                        
                          
                            [
                            Pr
                            ]
                          
                          N
                        
                        ⁡
                        
                          [
                          C
                          ]
                        
                      
                      N
                    
                    ⁡
                    
                      [
                      Pc
                      ]
                    
                  
                  )
                
                
                  -
                  1
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  (
                  
                    
                      
                        
                          
                            
                              
                                
                                  2
                                  N
                                
                              
                              ⁡
                              
                                [
                                Pr
                                ]
                              
                            
                            N
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  
                                    C
                                    
                                      N
                                      /
                                      2
                                    
                                  
                                
                                
                                  0
                                
                              
                              
                                
                                  0
                                
                                
                                  
                                    B
                                    
                                      N
                                      /
                                      2
                                    
                                  
                                
                              
                            
                            ]
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                              
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                            
                            
                              
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                              
                                
                                  -
                                  
                                    I
                                    
                                      N
                                      /
                                      2
                                    
                                  
                                
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        Pc
                        ]
                      
                    
                    N
                  
                  )
                
                
                  -
                  1
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          
                            
                              2
                              N
                            
                            ⁡
                            
                              [
                              Pc
                              ]
                            
                          
                          N
                          
                            -
                            1
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                              
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                            
                            
                              
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                              
                                
                                  -
                                  
                                    I
                                    
                                      N
                                      /
                                      2
                                    
                                  
                                
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                C
                                
                                  N
                                  /
                                  2
                                
                                
                                  -
                                  1
                                
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                B
                                
                                  N
                                  /
                                  2
                                
                                
                                  -
                                  1
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      Pr
                      ]
                    
                  
                  N
                  
                    -
                    1
                  
                
                .
              
            
          
        
      
    
    
      
        (
        12
        )
      
    
  




Moreover, the matrix




  
    
      [
      B
      ]
    
    2
  
  =
  
    [
    
      
        
          
            C
            8
            1
          
        
        
          
            C
            8
            3
          
        
      
      
        
          
            C
            8
            3
          
        
        
          
            -
            
              C
              8
              1
            
          
        
      
    
    ]
  




can be decomposed by using the 2-by-2 DCT-II matrix as





  
    
      
        
          
            
              
                
                  
                    [
                    B
                    ]
                  
                  2
                
                =
                
                ⁢
                
                  [
                  
                    
                      
                        
                          C
                          8
                          1
                        
                      
                      
                        
                          C
                          8
                          3
                        
                      
                    
                    
                      
                        
                          C
                          8
                          3
                        
                      
                      
                        
                          -
                          
                            C
                            8
                            1
                          
                        
                      
                    
                  
                  ]
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    
                      
                        
                          
                            [
                            L
                            ]
                          
                          2
                        
                        ⁡
                        
                          [
                          C
                          ]
                        
                      
                      2
                    
                    ⁡
                    
                      [
                      D
                      ]
                    
                  
                  2
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    
                      [
                      
                        
                          
                            
                              2
                            
                          
                          
                            0
                          
                        
                        
                          
                            
                              -
                              
                                2
                              
                            
                          
                          
                            2
                          
                        
                      
                      ]
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              1
                              
                                2
                              
                            
                          
                          
                            
                              1
                              
                                2
                              
                            
                          
                        
                        
                          
                            
                              C
                              8
                              2
                            
                          
                          
                            
                              -
                              
                                C
                                8
                                2
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            C
                            8
                            1
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            C
                            8
                            3
                          
                        
                      
                    
                    ]
                  
                
              
            
          
        
        ,
        
          

        
        ⁢
        
          
        
        ⁢
        where
      
    
    
      
        (
        13
        )
      
    
  
  
    
      
        
          
            [
            L
            ]
          
          2
        
        =
        
          [
          
            
              
                
                  2
                
              
              
                0
              
            
            
              
                
                  -
                  
                    2
                  
                
              
              
                2
              
            
          
          ]
        
      
    
    
      
        
      
    
  




is a lower triangular matrix,





  
    
      [
      D
      ]
    
    2
  
  =
  
    [
    
      
        
          
            C
            8
            1
          
        
        
          0
        
      
      
        
          0
        
        
          
            C
            8
            3
          
        
      
    
    ]
  




is a diagonal matrix, and we use the cosine related function

cos(2k+1)φm=2 cos(2kφm)cos φm−cos(2k−1)φm. where, φm is m-th angle  (14)

We note that




  
    
      
        
          
            
              [
              B
              ]
            
            N
          
          =
          
            
              
                
                  
                    
                      [
                      L
                      ]
                    
                    N
                  
                  ⁡
                  
                    [
                    C
                    ]
                  
                
                N
              
              ⁡
              
                [
                D
                ]
              
            
            N
          
        
        ⁢
        
          

        
        ⁢
        
          where
          ,
          
            

          
          ⁢
          
            
              
                [
                L
                ]
              
              N
            
            =
            
              [
              
                
                  
                    
                      2
                    
                  
                  
                    0
                  
                  
                    0
                  
                  
                    ...
                  
                
                
                  
                    
                      -
                      
                        2
                      
                    
                  
                  
                    2
                  
                  
                    0
                  
                  
                    ...
                  
                
                
                  
                    
                      2
                    
                  
                  
                    
                      -
                      2
                    
                  
                  
                    2
                  
                  
                    ...
                  
                
                
                  
                    ⋮
                  
                  
                    ⋮
                  
                  
                    
                      
                    
                  
                  
                    ⋱
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
              
                [
                D
                ]
              
              N
            
            =
            
              [
              
                
                  
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        Φ
                        0
                      
                    
                  
                  
                    0
                  
                  
                    ...
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        Φ
                        1
                      
                    
                  
                  
                    
                      
                    
                  
                  
                    ⋮
                  
                
                
                  
                    ⋮
                  
                  
                    
                      
                    
                  
                  
                    ⋱
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    ...
                  
                  
                    0
                  
                  
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        Φ
                        
                          N
                          -
                          1
                        
                      
                    
                  
                
              
              ]
            
          
          ,
        
      
    
    
      
        (
        15
        )
      
    
  




and Φi=2i+1, ε{0,1,2, . . . ,N−1}.  (16)

Proof: In case of the N×N DCT-II matrix, [C]N can be represented by using the form as




  
    
      
        
          
            [
            C
            ]
          
          N
        
        =
      
    
    
      
        (
        17
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            [
            
              
                
                  
                    1
                    
                      2
                    
                  
                
                
                  
                    1
                    
                      2
                    
                  
                
                
                  
                    1
                    
                      2
                    
                  
                
                
                  ...
                
                
                  
                    1
                    
                      2
                    
                  
                
                
                  
                    1
                    
                      2
                    
                  
                
              
              
                
                  
                    C
                    
                      4
                      ⁢
                      N
                    
                    
                      2
                      ⁢
                      
                        k
                        0
                      
                      ⁢
                      
                        Φ
                        0
                      
                    
                  
                
                
                  
                    C
                    
                      4
                      ⁢
                      N
                    
                    
                      2
                      ⁢
                      
                        k
                        0
                      
                      ⁢
                      
                        Φ
                        1
                      
                    
                  
                
                
                  
                    C
                    
                      4
                      ⁢
                      N
                    
                    
                      2
                      ⁢
                      
                        k
                        0
                      
                      ⁢
                      
                        Φ
                        2
                      
                    
                  
                
                
                  ...
                
                
                  
                    C
                    
                      4
                      ⁢
                      N
                    
                    
                      2
                      ⁢
                      
                        k
                        0
                      
                      ⁢
                      
                        Φ
                        
                          N
                          -
                          2
                        
                      
                    
                  
                
                
                  
                    C
                    
                      4
                      ⁢
                      N
                    
                    
                      2
                      ⁢
                      
                        k
                        0
                      
                      ⁢
                      
                        Φ
                        
                          N
                          -
                          1
                        
                      
                    
                  
                
              
              
                
                  
                    C
                    
                      4
                      ⁢
                      N
                    
                    
                      2
                      ⁢
                      
                        k
                        1
                      
                      ⁢
                      
                        Φ
                        0
                      
                    
                  
                
                
                  
                    C
                    
                      4
                      ⁢
                      N
                    
                    
                      2
                      ⁢
                      
                        k
                        1
                      
                      ⁢
                      
                        Φ
                        1
                      
                    
                  
                
                
                  
                    C
                    
                      4
                      ⁢
                      N
                    
                    
                      2
                      ⁢
                      
                        k
                        1
                      
                      ⁢
                      
                        Φ
                        2
                      
                    
                  
                
                
                  ...
                
                
                  
                    C
                    
                      4
                      ⁢
                      N
                    
                    
                      2
                      ⁢
                      
                        k
                        1
                      
                      ⁢
                      
                        Φ
                        
                          N
                          -
                          2
                        
                      
                    
                  
                
                
                  
                    C
                    
                      4
                      ⁢
                      N
                    
                    
                      2
                      ⁢
                      
                        k
                        1
                      
                      ⁢
                      
                        Φ
                        
                          N
                          -
                          1
                        
                      
                    
                  
                
              
              
                
                  
                    C
                    
                      4
                      ⁢
                      N
                    
                    
                      2
                      ⁢
                      
                        k
                        2
                      
                      ⁢
                      
                        Φ
                        0
                      
                    
                  
                
                
                  
                    C
                    
                      4
                      ⁢
                      N
                    
                    
                      2
                      ⁢
                      
                        k
                        2
                      
                      ⁢
                      
                        Φ
                        1
                      
                    
                  
                
                
                  
                    C
                    
                      4
                      ⁢
                      N
                    
                    
                      2
                      ⁢
                      
                        k
                        2
                      
                      ⁢
                      
                        Φ
                        2
                      
                    
                  
                
                
                  ...
                
                
                  
                    C
                    
                      4
                      ⁢
                      N
                    
                    
                      2
                      ⁢
                      
                        k
                        2
                      
                      ⁢
                      
                        Φ
                        
                          N
                          -
                          2
                        
                      
                    
                  
                
                
                  
                    C
                    
                      4
                      ⁢
                      N
                    
                    
                      2
                      ⁢
                      
                        k
                        2
                      
                      ⁢
                      
                        Φ
                        
                          N
                          -
                          1
                        
                      
                    
                  
                
              
              
                
                  ⋮
                
                
                  
                    
                  
                
                
                  
                    
                  
                
                
                  ⋮
                
                
                  
                    
                  
                
                
                  ⋮
                
              
              
                
                  
                    C
                    
                      4
                      ⁢
                      N
                    
                    
                      2
                      ⁢
                      
                        k
                        
                          N
                          -
                          2
                        
                      
                      ⁢
                      
                        Φ
                        0
                      
                    
                  
                
                
                  
                    C
                    
                      4
                      ⁢
                      N
                    
                    
                      2
                      ⁢
                      
                        k
                        
                          N
                          -
                          2
                        
                      
                      ⁢
                      
                        Φ
                        1
                      
                    
                  
                
                
                  
                    C
                    
                      4
                      ⁢
                      N
                    
                    
                      2
                      ⁢
                      
                        k
                        
                          N
                          -
                          2
                        
                      
                      ⁢
                      
                        Φ
                        2
                      
                    
                  
                
                
                  ...
                
                
                  
                    C
                    
                      4
                      ⁢
                      N
                    
                    
                      2
                      ⁢
                      
                        k
                        
                          N
                          -
                          2
                        
                      
                      ⁢
                      
                        Φ
                        
                          N
                          -
                          2
                        
                      
                    
                  
                
                
                  
                    C
                    
                      4
                      ⁢
                      N
                    
                    
                      2
                      ⁢
                      
                        k
                        
                          N
                          -
                          2
                        
                      
                      ⁢
                      
                        Φ
                        
                          N
                          -
                          1
                        
                      
                    
                  
                
              
            
            ]
          
          ,
        
      
    
    
      
        
      
    
  




where ki=i+1, iε{0, 1, 2, . . . }.

According to (10), a N×N matrix [B]N from [C]2N can be represented




  
    
      
        
          
            [
            B
            ]
          
          N
        
        =
        
           
          
            
              [
              
                
                  
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        Φ
                        0
                      
                    
                  
                  
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        Φ
                        1
                      
                    
                  
                  
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        Φ
                        2
                      
                    
                  
                  
                    ...
                  
                  
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        Φ
                        
                          N
                          -
                          1
                        
                      
                    
                  
                
                
                  
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                          Φ
                          0
                        
                      
                    
                  
                  
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                          Φ
                          1
                        
                      
                    
                  
                  
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                          Φ
                          2
                        
                      
                    
                  
                  
                    ...
                  
                  
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                          Φ
                          
                            N
                            -
                            1
                          
                        
                      
                    
                  
                
                
                  
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                k
                                1
                              
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                          Φ
                          0
                        
                      
                    
                  
                  
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                k
                                1
                              
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                          Φ
                          1
                        
                      
                    
                  
                  
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                k
                                1
                              
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                          Φ
                          2
                        
                      
                    
                  
                  
                    ...
                  
                  
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                k
                                1
                              
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                          Φ
                          
                            N
                            -
                            1
                          
                        
                      
                    
                  
                
                
                  
                    ⋮
                  
                  
                    
                      
                    
                  
                  
                    ⋮
                  
                  
                    
                      
                    
                  
                  
                    ⋮
                  
                
                
                  
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                k
                                
                                  N
                                  -
                                  2
                                
                              
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                          Φ
                          0
                        
                      
                    
                  
                  
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                k
                                
                                  N
                                  -
                                  2
                                
                              
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                          Φ
                          1
                        
                      
                    
                  
                  
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                k
                                
                                  N
                                  -
                                  2
                                
                              
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                          Φ
                          2
                        
                      
                    
                  
                  
                    ...
                  
                  
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                k
                                
                                  N
                                  -
                                  2
                                
                              
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                          Φ
                          
                            N
                            -
                            1
                          
                        
                      
                    
                  
                
              
              ]
            
            ,
          
        
      
    
    
      
        (
        18
        )
      
    
  




Based on Eq. (14), we have the formula

C4N(2ki+1)Φm=2C4N2kiΦmC4N101 m−C4N(2ki−1)Φm=−C4N(2ki−1)Φm+2C4N2kiΦmC4N101 m,  (19)

where, m ε{0, 1, 2, . . . }.

Thus we can calculate




  
    
      
        
          
            
              
                
                  
                    
                      
                        
                          [
                          L
                          ]
                        
                        N
                      
                      ⁡
                      
                        [
                        C
                        ]
                      
                    
                    N
                  
                  ⁡
                  
                    [
                    D
                    ]
                  
                
                N
              
              =
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          2
                        
                      
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                      
                        ...
                      
                      
                        0
                      
                    
                    
                      
                        
                          -
                          
                            2
                          
                        
                      
                      
                        2
                      
                      
                        0
                      
                      
                        0
                      
                      
                        
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        
                          2
                        
                      
                      
                        
                          -
                          2
                        
                      
                      
                        2
                      
                      
                        0
                      
                      
                        
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        
                          -
                          
                            2
                          
                        
                      
                      
                        2
                      
                      
                        
                          -
                          2
                        
                      
                      
                        2
                      
                      
                        ...
                      
                      
                        ⋮
                      
                    
                    
                      
                        
                          2
                        
                      
                      
                        
                          -
                          2
                        
                      
                      
                        2
                      
                      
                        
                          -
                          2
                        
                      
                      
                        2
                      
                      
                        
                          
                        
                      
                    
                    
                      
                        ⋮
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        ⋱
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        
                          1
                          
                            2
                          
                        
                      
                      
                        
                          1
                          
                            2
                          
                        
                      
                      
                        
                          1
                          
                            2
                          
                        
                      
                      
                        ...
                      
                      
                        
                          1
                          
                            2
                          
                        
                      
                      
                        
                          1
                          
                            2
                          
                        
                      
                    
                    
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              0
                            
                            ⁢
                            
                              Φ
                              0
                            
                          
                        
                      
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              0
                            
                            ⁢
                            
                              Φ
                              1
                            
                          
                        
                      
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              0
                            
                            ⁢
                            
                              Φ
                              2
                            
                          
                        
                      
                      
                        ...
                      
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              0
                            
                            ⁢
                            
                              Φ
                              
                                N
                                -
                                2
                              
                            
                          
                        
                      
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              0
                            
                            ⁢
                            
                              Φ
                              
                                N
                                -
                                1
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              1
                            
                            ⁢
                            
                              Φ
                              0
                            
                          
                        
                      
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              1
                            
                            ⁢
                            
                              Φ
                              1
                            
                          
                        
                      
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              1
                            
                            ⁢
                            
                              Φ
                              2
                            
                          
                        
                      
                      
                        ...
                      
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              1
                            
                            ⁢
                            
                              Φ
                              
                                N
                                -
                                2
                              
                            
                          
                        
                      
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              1
                            
                            ⁢
                            
                              Φ
                              
                                N
                                -
                                1
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              2
                            
                            ⁢
                            
                              Φ
                              0
                            
                          
                        
                      
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              2
                            
                            ⁢
                            
                              Φ
                              1
                            
                          
                        
                      
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              2
                            
                            ⁢
                            
                              Φ
                              2
                            
                          
                        
                      
                      
                        ...
                      
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              2
                            
                            ⁢
                            
                              Φ
                              
                                N
                                -
                                2
                              
                            
                          
                        
                      
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              2
                            
                            ⁢
                            
                              Φ
                              
                                N
                                -
                                1
                              
                            
                          
                        
                      
                    
                    
                      
                        ⋮
                      
                      
                        
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        ⋮
                      
                      
                        
                          
                        
                      
                      
                        ⋮
                      
                    
                    
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              
                                N
                                -
                                2
                              
                            
                            ⁢
                            
                              Φ
                              0
                            
                          
                        
                      
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              
                                N
                                -
                                2
                              
                            
                            ⁢
                            
                              Φ
                              1
                            
                          
                        
                      
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              
                                N
                                -
                                2
                              
                            
                            ⁢
                            
                              Φ
                              2
                            
                          
                        
                      
                      
                        ...
                      
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              
                                N
                                -
                                2
                              
                            
                            ⁢
                            
                              Φ
                              
                                N
                                -
                                2
                              
                            
                          
                        
                      
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              
                                N
                                -
                                2
                              
                            
                            ⁢
                            
                              Φ
                              
                                N
                                -
                                1
                              
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              
              ⁢
              
                [
                
                  
                    
                      
                        C
                        
                          4
                          ⁢
                          N
                        
                        
                          Φ
                          0
                        
                      
                    
                    
                      0
                    
                    
                      ...
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        C
                        
                          4
                          ⁢
                          N
                        
                        
                          Φ
                          1
                        
                      
                    
                    
                      
                        
                      
                    
                    
                      ⋮
                    
                  
                  
                    
                      ⋮
                    
                    
                      
                        
                      
                    
                    
                      ⋱
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      ...
                    
                    
                      0
                    
                    
                      
                        C
                        
                          4
                          ⁢
                          N
                        
                        
                          Φ
                          
                            N
                            -
                            1
                          
                        
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              =
              
              ⁢
              
                [
                
                  
                    
                      1
                    
                    
                      1
                    
                    
                      ...
                    
                    
                      1
                    
                  
                  
                    
                      
                        
                          -
                          1
                        
                        +
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                              ⁢
                              
                                Φ
                                0
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          -
                          1
                        
                        +
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                              ⁢
                              
                                Φ
                                1
                              
                            
                          
                        
                      
                    
                    
                      ...
                    
                    
                      
                        
                          -
                          1
                        
                        +
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                              ⁢
                              
                                Φ
                                
                                  N
                                  -
                                  1
                                
                              
                            
                          
                        
                      
                    
                  
                  
                    
                      
                        1
                        -
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                              ⁢
                              
                                Φ
                                0
                              
                            
                          
                        
                        +
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                1
                              
                              ⁢
                              
                                Φ
                                0
                              
                            
                          
                        
                      
                    
                    
                      
                        1
                        -
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                              ⁢
                              
                                Φ
                                1
                              
                            
                          
                        
                        +
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                1
                              
                              ⁢
                              
                                Φ
                                1
                              
                            
                          
                        
                      
                    
                    
                      ...
                    
                    
                      
                        1
                        -
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                              ⁢
                              
                                Φ
                                
                                  N
                                  -
                                  1
                                
                              
                            
                          
                        
                        +
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                1
                              
                              ⁢
                              
                                Φ
                                
                                  N
                                  -
                                  2
                                
                              
                            
                          
                        
                      
                    
                  
                  
                    
                      ⋮
                    
                    
                      
                        
                      
                    
                    
                      
                        
                      
                    
                    
                      ⋮
                    
                  
                
                ]
              
            
          
        
        
          
            
              
              ⁢
              
                [
                
                  
                    
                      
                        C
                        
                          4
                          ⁢
                          N
                        
                        
                          Φ
                          0
                        
                      
                    
                    
                      0
                    
                    
                      ...
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        C
                        
                          4
                          ⁢
                          N
                        
                        
                          Φ
                          1
                        
                      
                    
                    
                      
                        
                      
                    
                    
                      ⋮
                    
                  
                  
                    
                      ⋮
                    
                    
                      
                        
                      
                    
                    
                      ⋱
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      ...
                    
                    
                      0
                    
                    
                      
                        C
                        
                          4
                          ⁢
                          N
                        
                        
                          Φ
                          
                            N
                            -
                            1
                          
                        
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              =
              
              ⁢
              
                [
                
                  
                    
                      
                        C
                        
                          4
                          ⁢
                          N
                        
                        
                          Φ
                          0
                        
                      
                    
                    
                      
                        C
                        
                          4
                          ⁢
                          N
                        
                        
                          Φ
                          1
                        
                      
                    
                    
                      ...
                    
                    
                      
                        C
                        
                          4
                          ⁢
                          N
                        
                        
                          Φ
                          
                            N
                            -
                            1
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          -
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              Φ
                              0
                            
                          
                        
                        +
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                              ⁢
                              
                                Φ
                                0
                              
                            
                          
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              Φ
                              0
                            
                          
                        
                      
                    
                    
                      
                        
                          -
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              Φ
                              1
                            
                          
                        
                        +
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                              ⁢
                              
                                Φ
                                1
                              
                            
                          
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              Φ
                              1
                            
                          
                        
                      
                    
                    
                      ...
                    
                    
                      
                        
                          -
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              Φ
                              
                                N
                                -
                                1
                              
                            
                          
                        
                        +
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                              ⁢
                              
                                Φ
                                
                                  N
                                  -
                                  1
                                
                              
                            
                          
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              Φ
                              
                                N
                                -
                                1
                              
                            
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            Φ
                            0
                          
                        
                        -
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                              ⁢
                              
                                Φ
                                0
                              
                            
                          
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              Φ
                              0
                            
                          
                        
                        +
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                1
                              
                              ⁢
                              
                                Φ
                                0
                              
                            
                          
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              Φ
                              0
                            
                          
                        
                      
                    
                    
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            Φ
                            1
                          
                        
                        -
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                              ⁢
                              
                                Φ
                                1
                              
                            
                          
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              Φ
                              1
                            
                          
                        
                        +
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                1
                              
                              ⁢
                              
                                Φ
                                1
                              
                            
                          
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              Φ
                              1
                            
                          
                        
                      
                    
                    
                      ...
                    
                    
                      
                        
                      
                    
                  
                  
                    
                      ⋮
                    
                    
                      
                        
                      
                    
                    
                      
                        
                      
                    
                    
                      ⋮
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        (
        20
        )
      
    
  




Since k0=1, we get

−C4NΦm+2C4N2k0ΦmC4NΦm=−C4N(2k0−1)Φm+2C4N2k0ΦmC4NΦm=C4N(2k0+1)Φm, and  (21)

C4NΦm−2C4N2k0ΦmC4NΦm=−(−C4N(2k0−1)Φm+2C4N2k0ΦmC4NΦm)=−C4N(2k0+1)Φm  (22)

In case of ki=i+1, we have (2ki−1+1)Φm=(2(ki−1)+1)Φm=(2ki−1)Φm, then we get

C4NΦm−2C4N2ki−1ΦmC4NΦm+2C4N2kiΦmC4NΦm=−C4N(2ki−1+1)Φm+2C4N2kiΦmC4NΦm=−C4N(2ki−1)Φm+2C4N2kiΦmC4NΦm=C4N(2ki+1)Φm  (23)

Taking the (21)-(23) to (20), we can rewrite




  
    
      
        
          
            
              
                
                  
                    
                      
                        
                          [
                          L
                          ]
                        
                        N
                      
                      ⁡
                      
                        [
                        C
                        ]
                      
                    
                    N
                  
                  ⁡
                  
                    [
                    D
                    ]
                  
                
                N
              
              =
              
              ⁢
              
                [
                
                  
                    
                      
                        C
                        
                          4
                          ⁢
                          N
                        
                        
                          Φ
                          0
                        
                      
                    
                    
                      
                        C
                        
                          4
                          ⁢
                          N
                        
                        
                          Φ
                          1
                        
                      
                    
                    
                      ...
                    
                    
                      
                        C
                        
                          4
                          ⁢
                          N
                        
                        
                          Φ
                          
                            N
                            -
                            1
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          -
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              Φ
                              0
                            
                          
                        
                        +
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                              ⁢
                              
                                Φ
                                0
                              
                            
                          
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              Φ
                              0
                            
                          
                        
                      
                    
                    
                      
                        
                          -
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              Φ
                              1
                            
                          
                        
                        +
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                              ⁢
                              
                                Φ
                                1
                              
                            
                          
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              Φ
                              1
                            
                          
                        
                      
                    
                    
                      ...
                    
                    
                      
                        
                          -
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              Φ
                              
                                N
                                -
                                1
                              
                            
                          
                        
                        +
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                              ⁢
                              
                                Φ
                                
                                  N
                                  -
                                  1
                                
                              
                            
                          
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              Φ
                              
                                N
                                -
                                1
                              
                            
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            Φ
                            0
                          
                        
                        -
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                              ⁢
                              
                                Φ
                                0
                              
                            
                          
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              Φ
                              0
                            
                          
                        
                        +
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                1
                              
                              ⁢
                              
                                Φ
                                0
                              
                            
                          
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              Φ
                              0
                            
                          
                        
                      
                    
                    
                      
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            Φ
                            1
                          
                        
                        -
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                              ⁢
                              
                                Φ
                                1
                              
                            
                          
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              Φ
                              1
                            
                          
                        
                        +
                        
                          2
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                1
                              
                              ⁢
                              
                                Φ
                                1
                              
                            
                          
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              Φ
                              1
                            
                          
                        
                      
                    
                    
                      ...
                    
                    
                      
                        
                      
                    
                  
                  
                    
                      ⋮
                    
                    
                      
                        
                      
                    
                    
                      
                        
                      
                    
                    
                      ⋮
                    
                  
                
                ]
              
            
          
        
        
          
            
              =
              
              ⁢
              
                [
                
                  
                    
                      
                        C
                        
                          4
                          ⁢
                          N
                        
                        
                          Φ
                          0
                        
                      
                    
                    
                      
                        C
                        
                          4
                          ⁢
                          N
                        
                        
                          Φ
                          1
                        
                      
                    
                    
                      ...
                    
                    
                      
                        C
                        
                          4
                          ⁢
                          N
                        
                        
                          Φ
                          
                            N
                            -
                            1
                          
                        
                      
                    
                  
                  
                    
                      
                        C
                        
                          4
                          ⁢
                          N
                        
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  k
                                  0
                                
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          
                            Φ
                            0
                          
                        
                      
                    
                    
                      
                        C
                        
                          4
                          ⁢
                          N
                        
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  k
                                  0
                                
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          
                            Φ
                            1
                          
                        
                      
                    
                    
                      ...
                    
                    
                      
                        C
                        
                          4
                          ⁢
                          N
                        
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  k
                                  0
                                
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          
                            Φ
                            
                              N
                              -
                              1
                            
                          
                        
                      
                    
                  
                  
                    
                      
                        C
                        
                          4
                          ⁢
                          N
                        
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  k
                                  1
                                
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          
                            Φ
                            0
                          
                        
                      
                    
                    
                      
                        C
                        
                          4
                          ⁢
                          N
                        
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  k
                                  1
                                
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          
                            Φ
                            1
                          
                        
                      
                    
                    
                      ...
                    
                    
                      
                        C
                        
                          4
                          ⁢
                          N
                        
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  k
                                  1
                                
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          
                            Φ
                            
                              N
                              -
                              1
                            
                          
                        
                      
                    
                  
                  
                    
                      ⋮
                    
                    
                      
                        
                      
                    
                    
                      
                        
                      
                    
                    
                      ⋮
                    
                  
                
                ]
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    [
                    B
                    ]
                  
                  N
                
                .
              
            
          
        
      
    
    
      
        (
        24
        )
      
    
  




The proof of (15) is completed.
By using the results from (8) and (15), we have a new form for the DCT-II matrix




  
    
      
        
          
            
              
                [
                
                  C
                  ~
                
                ]
              
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          
                            
                              2
                              N
                            
                          
                          ⁡
                          
                            [
                            Pr
                            ]
                          
                        
                        N
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                C
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                
                                  L
                                  
                                    N
                                    /
                                    2
                                  
                                
                                ⁢
                                
                                  C
                                  
                                    N
                                    /
                                    2
                                  
                                
                                ⁢
                                
                                  D
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                        
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            
                              -
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    Pc
                    ]
                  
                
                N
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          
                            2
                            N
                          
                        
                        ⁡
                        
                          [
                          Pr
                          ]
                        
                      
                      N
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              L
                              
                                N
                                /
                                2
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        I
                        2
                      
                      ⊗
                      
                        C
                        
                          N
                          /
                          2
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          D
                          
                            N
                            /
                            2
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          
                            -
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    Pc
                    ]
                  
                
                N
              
            
          
        
      
    
    
      
        (
        25
        )
      
    
  




Then, based on the recursive 4×4 kernel c4 form, we have




  
    
      
        
          
            
              
                
                  
                    [
                    C
                    ]
                  
                  N
                
                =
                
                ⁢
                
                  
                    
                      
                        
                          
                            
                              
                                [
                                Pr
                                ]
                              
                              N
                            
                            ⁡
                            
                              [
                              
                                
                                  
                                    
                                      I
                                      
                                        N
                                        /
                                        2
                                      
                                    
                                  
                                  
                                    0
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    
                                      L
                                      
                                        N
                                        /
                                        2
                                      
                                    
                                  
                                
                              
                              ]
                            
                          
                          ⁡
                          
                            [
                            
                              
                                I
                                2
                              
                              ⊗
                              
                                C
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            ]
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                              
                                0
                              
                            
                            
                              
                                0
                              
                              
                                
                                  D
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              
                                -
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      Pc
                      ]
                    
                  
                  N
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    
                      
                        [
                        Pr
                        ]
                      
                      N
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              L
                              
                                N
                                /
                                2
                              
                            
                          
                        
                      
                      ]
                    
                  
                  [
                  
                    
                      I
                      2
                    
                    ⊗
                    
                      
                        [
                        
                          
                            ...
                            [
                            
                              
                                I
                                2
                              
                              ⊗
                              
                                
                                  [
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  [
                                                  Pr
                                                  ]
                                                
                                                4
                                              
                                              ⁡
                                              
                                                [
                                                
                                                  
                                                    
                                                      
                                                        I
                                                        2
                                                      
                                                    
                                                    
                                                      0
                                                    
                                                  
                                                  
                                                    
                                                      0
                                                    
                                                    
                                                      
                                                        L
                                                        2
                                                      
                                                    
                                                  
                                                
                                                ]
                                              
                                            
                                            ⁡
                                            
                                              [
                                              
                                                
                                                  I
                                                  2
                                                
                                                ⊗
                                                
                                                  C
                                                  2
                                                
                                              
                                              ]
                                            
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                
                                                  
                                                    I
                                                    2
                                                  
                                                
                                                
                                                  0
                                                
                                              
                                              
                                                
                                                  0
                                                
                                                
                                                  
                                                    D
                                                    2
                                                  
                                                
                                              
                                            
                                            ]
                                          
                                        
                                        ⁡
                                        
                                          [
                                          
                                            
                                              
                                                
                                                  I
                                                  2
                                                
                                              
                                              
                                                
                                                  I
                                                  2
                                                
                                              
                                            
                                            
                                              
                                                
                                                  I
                                                  2
                                                
                                              
                                              
                                                
                                                  -
                                                  
                                                    I
                                                    2
                                                  
                                                
                                              
                                            
                                          
                                          ]
                                        
                                      
                                      ⁡
                                      
                                        [
                                        Pc
                                        ]
                                      
                                    
                                    4
                                  
                                  ]
                                
                                
                                  }
                                  
                                    C
                                    4
                                  
                                
                              
                            
                            ]
                          
                          ⁢
                          ...
                        
                        ]
                      
                      
                        }
                        
                          C
                          
                            N
                            /
                            2
                          
                        
                      
                    
                  
                  ]
                
              
            
          
          
            
              
                
                ⁢
                
                  
                    
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                D
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                        
                        ]
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              
                                -
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      Pc
                      ]
                    
                  
                  N
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          Finally
          ,
        
      
    
    
      
        (
        26
        )
      
    
  
  
    
      
        
          
            [
            C
            ]
          
          N
        
        =
        
          
            
              
                
                  
                    
                      
                        
                          2
                          N
                        
                      
                      ⁡
                      
                        [
                        Pr
                        ]
                      
                    
                    N
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            L
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          
                            I
                            2
                          
                          ⊗
                        
                      
                    
                    
                      
                        
                          Pr
                          
                            N
                            /
                            2
                          
                        
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          I
                          2
                        
                        ⊗
                      
                    
                  
                  
                    
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  4
                                
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                L
                                
                                  N
                                  /
                                  4
                                
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        I
                        4
                      
                      ⊗
                    
                  
                
                
                  
                    
                      Pr
                      
                        N
                        /
                        4
                      
                    
                  
                
              
              ]
            
          
          ⁢
          
             
            
              
                [
                
                  
                    
                      
                        
                          I
                          4
                        
                        ⊗
                      
                    
                  
                  
                    
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  8
                                
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                L
                                
                                  N
                                  /
                                  8
                                
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                ]
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    
                      
                        
                          ...
                          ⁢
                          
                            
                          
                          [
                          
                            
                              
                                
                                  
                                    I
                                    
                                      N
                                      /
                                      4
                                    
                                  
                                  ⊗
                                
                              
                            
                            
                              
                                
                                  Pr
                                  4
                                
                              
                            
                          
                          ]
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  
                                    I
                                    
                                      N
                                      /
                                      4
                                    
                                  
                                  ⊗
                                
                              
                            
                            
                              
                                
                                  [
                                  
                                    
                                      
                                        
                                          I
                                          2
                                        
                                      
                                      
                                        0
                                      
                                    
                                    
                                      
                                        0
                                      
                                      
                                        
                                          L
                                          2
                                        
                                      
                                    
                                  
                                  ]
                                
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                                ⊗
                              
                            
                          
                          
                            
                              
                                C
                                2
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  4
                                
                              
                              ⊗
                            
                          
                        
                        
                          
                            
                              [
                              
                                
                                  
                                    
                                      I
                                      2
                                    
                                  
                                  
                                    0
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    
                                      D
                                      2
                                    
                                  
                                
                              
                              ]
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                4
                              
                            
                            ⊗
                          
                        
                      
                      
                        
                          
                            [
                            
                              
                                
                                  
                                    I
                                    2
                                  
                                
                                
                                  
                                    I
                                    2
                                  
                                
                              
                              
                                
                                  
                                    I
                                    2
                                  
                                
                                
                                  
                                    -
                                    
                                      I
                                      2
                                    
                                  
                                
                              
                            
                            ]
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              4
                            
                          
                          ⊗
                        
                      
                    
                    
                      
                        
                          Pc
                          4
                        
                      
                    
                  
                  ]
                
              
              ⁢
              
                
              
              ⁢
              
                ...
                ⁢
                
                  
                
                [
                
                  
                    
                      
                        
                          I
                          4
                        
                        ⊗
                      
                    
                  
                  
                    
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  8
                                
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                D
                                
                                  N
                                  /
                                  8
                                
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                ]
              
              ⁢
              
                 
                
                  
                    
                      
                        
                          
                            
                              
                                
                                  [
                                  
                                    
                                      
                                        
                                          
                                            I
                                            4
                                          
                                          ⊗
                                        
                                      
                                    
                                    
                                      
                                        
                                          [
                                          
                                            
                                              
                                                
                                                  I
                                                  
                                                    N
                                                    /
                                                    8
                                                  
                                                
                                              
                                              
                                                
                                                  I
                                                  
                                                    N
                                                    /
                                                    8
                                                  
                                                
                                              
                                            
                                            
                                              
                                                
                                                  I
                                                  
                                                    N
                                                    /
                                                    8
                                                  
                                                
                                              
                                              
                                                
                                                  -
                                                  
                                                    I
                                                    
                                                      N
                                                      /
                                                      8
                                                    
                                                  
                                                
                                              
                                            
                                          
                                          ]
                                        
                                      
                                    
                                  
                                  ]
                                
                                ⁡
                                
                                  [
                                  
                                    
                                      
                                        
                                          
                                            I
                                            4
                                          
                                          ⊗
                                        
                                      
                                    
                                    
                                      
                                        
                                          Pc
                                          
                                            N
                                            /
                                            4
                                          
                                        
                                      
                                    
                                  
                                  ]
                                
                              
                              ⁡
                              
                                [
                                
                                  
                                    
                                      
                                        
                                          I
                                          2
                                        
                                        ⊗
                                      
                                    
                                  
                                  
                                    
                                      
                                        [
                                        
                                          
                                            
                                              
                                                I
                                                
                                                  N
                                                  /
                                                  4
                                                
                                              
                                            
                                            
                                              0
                                            
                                          
                                          
                                            
                                              0
                                            
                                            
                                              
                                                D
                                                
                                                  N
                                                  /
                                                  4
                                                
                                              
                                            
                                          
                                        
                                        ]
                                      
                                    
                                  
                                
                                ]
                              
                            
                            ⁡
                            
                              [
                              
                                
                                  
                                    
                                      
                                        I
                                        4
                                      
                                      ⊗
                                    
                                  
                                
                                
                                  
                                    
                                      [
                                      
                                        
                                          
                                            
                                              I
                                              
                                                N
                                                /
                                                4
                                              
                                            
                                          
                                          
                                            
                                              I
                                              
                                                N
                                                /
                                                4
                                              
                                            
                                          
                                        
                                        
                                          
                                            
                                              I
                                              
                                                N
                                                /
                                                4
                                              
                                            
                                          
                                          
                                            
                                              -
                                              
                                                I
                                                
                                                  N
                                                  /
                                                  4
                                                
                                              
                                            
                                          
                                        
                                      
                                      ]
                                    
                                  
                                
                              
                              ]
                            
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  
                                    
                                      I
                                      2
                                    
                                    ⊗
                                  
                                
                              
                              
                                
                                  
                                    Pc
                                    
                                      N
                                      /
                                      2
                                    
                                  
                                
                              
                            
                            ]
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                              
                                0
                              
                            
                            
                              
                                0
                              
                              
                                
                                  D
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              
                                -
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      Pc
                      ]
                    
                  
                  N
                
              
            
          
        
      
    
    
      
        (
        27
        )
      
    
  




FIG. 1A-1C is a butterfly data flow graph of Equation (27) according to an exemplary embodiment of the present invention. As shown in FIG. 1A-1C, the data flow of Equation (27) is represented.
III. Block-Wise Inverse Sparse Matrix Decomposition for DST-II
Similar fashion as the DCT-II, the DST-II matrix is given by




  
    
      
        
          
            
              
                
                  
                    [
                    
                      S
                      
                        m
                        ,
                        n
                      
                    
                    ]
                  
                  N
                
                =
                
                ⁢
                
                  
                    s
                    m
                  
                  ⁢
                  
                    
                      2
                      N
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                      
                        π
                        ⁡
                        
                          (
                          
                            m
                            +
                            1
                          
                          )
                        
                      
                      ⁢
                      
                        (
                        
                          
                            2
                            ⁢
                            n
                          
                          +
                          1
                        
                        )
                      
                    
                    
                      2
                      ⁢
                      N
                    
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    
                      
                        2
                        N
                      
                    
                    ⁡
                    
                      [
                      
                        S
                        
                          m
                          ,
                          n
                        
                      
                      ]
                    
                  
                  N
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            s
            m
          
          =
          
            {
            
              
                
                  
                    
                      1
                      ,
                    
                  
                  
                    
                      m
                      ≠
                      
                        N
                        -
                        1
                      
                    
                  
                
                
                  
                    
                      
                        1
                        /
                        
                          2
                        
                      
                      ,
                    
                  
                  
                    
                      m
                      =
                      
                        N
                        -
                        1
                      
                    
                  
                
              
              ,
              
                

              
              ⁢
              m
              ,
              
                n
                =
                0
              
              ,
              1
              ,
              ...
              ⁢
              
                
              
              ,
              
                N
                -
                1.
              
            
          
        
      
    
    
      
        (
        28
        )
      
    
  




The basic form of the DCT-II 2×2 matrix is given by




  
    
      
        
          
            [
            S
            ]
          
          2
        
        =
        
          
            
              1
              
                2
              
            
            ⁡
            
              [
              
                
                  
                    1
                  
                  
                    1
                  
                
                
                  
                    1
                  
                  
                    
                      -
                      1
                    
                  
                
              
              ]
            
          
          =
          
            
              
                [
                S
                ]
              
              2
            
            .
          
        
      
    
    
      
        (
        29
        )
      
    
  




The general N×N DST-II matrix is given by




  
    
      
        
          
            
              
                
                  [
                  S
                  ]
                
                N
              
              =
              
              ⁢
              
                
                  
                    2
                    N
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          S
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              0
                            
                            ⁢
                            
                              φ
                              0
                            
                          
                        
                      
                      
                        
                          S
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              0
                            
                            ⁢
                            
                              φ
                              1
                            
                          
                        
                      
                      
                        ...
                      
                      
                        
                          S
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              0
                            
                            ⁢
                            
                              φ
                              
                                N
                                -
                                1
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          S
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              1
                            
                            ⁢
                            
                              φ
                              0
                            
                          
                        
                      
                      
                        
                          S
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              1
                            
                            ⁢
                            
                              φ
                              1
                            
                          
                        
                      
                      
                        ...
                      
                      
                        
                          S
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              1
                            
                            ⁢
                            
                              φ
                              
                                N
                                -
                                1
                              
                            
                          
                        
                      
                    
                    
                      
                        ⋮
                      
                      
                        ⋮
                      
                      
                        ⋱
                      
                      
                        ⋮
                      
                    
                    
                      
                        
                          S
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              
                                N
                                -
                                2
                              
                            
                            ⁢
                            
                              φ
                              0
                            
                          
                        
                      
                      
                        
                          S
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              
                                N
                                -
                                2
                              
                            
                            ⁢
                            
                              φ
                              1
                            
                          
                        
                      
                      
                        ...
                      
                      
                        
                          S
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              
                                N
                                -
                                2
                              
                            
                            ⁢
                            
                              φ
                              
                                N
                                -
                                1
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          1
                          /
                          
                            2
                          
                        
                      
                      
                        
                          
                            -
                            1
                          
                          /
                          
                            2
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        
                          
                            -
                            1
                          
                          ⁢
                          
                            2
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      2
                      N
                    
                  
                  ⁡
                  
                    [
                    S
                    ]
                  
                
                N
              
            
          
        
      
    
    
      
        (
        30
        )
      
    
  




where Slk=sin(kπ/l), ki=i+1, Φj=2j+1, i=0, 1, . . . , N−2, j=0, 1, . . . , N−1

As such, the DST-II matrix is decomposed based on the Jacket matrix by multiplying the same permutation matrices. Herein, the row permutation matrix is defined as




  
    
      [
      Ps
      ]
    
    N
  
  =
  
    
      
        [
        
          Ps
          
            i
            ,
            j
          
        
        ]
      
      N
    
    =
    
      {
      
        
          
            
              
                1
                ,
                if
              
            
            
              
                
                  j
                  =
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          i
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    mod
                    ⁢
                    
                      
                    
                    ⁢
                    N
                  
                
                ,
                
                  0
                  ≤
                  i
                  ≤
                  
                    
                      N
                      /
                      2
                    
                    -
                    1
                  
                
              
            
          
          
            
              
                1
                ,
                if
              
            
            
              
                
                  j
                  =
                  
                    
                      (
                      
                        2
                        ⁢
                        i
                      
                      )
                    
                    ⁢
                    mod
                    ⁢
                    
                      
                    
                    ⁢
                    N
                  
                
                ,
                
                  
                    N
                    /
                    2
                  
                  ≤
                  i
                  ≤
                  
                    N
                    -
                    1
                  
                
              
            
          
          
            
              
                0
                ,
              
            
            
              others
            
          
        
        ,
        
          

        
        ⁢
        i
        ,
        
          j
          =
          0
        
        ,
        
          1
          ⁢
          
            
          
          ⁢
          ...
        
        ⁢
        
          
        
        ,
        
          N
          -
          1.
        
      
    
  




Using Equation (30), we can obtain the 4×4 sparse matrix similar fashion the DCT-II of Equation (6) as follows:




  
    
      
        
          
            
              
                
                  
                    
                      [
                      S
                      ]
                    
                    4
                  
                  •
                
                =
                
                ⁢
                
                  
                    
                      
                        
                          
                            [
                            Pr
                            ]
                          
                          4
                        
                        ⁡
                        
                          [
                          S
                          ]
                        
                      
                      4
                    
                    ⁡
                    
                      [
                      Pc
                      ]
                    
                  
                  4
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    1
                    
                      2
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            A
                            2
                          
                        
                        
                          
                            A
                            2
                          
                        
                      
                      
                        
                          
                            S
                            2
                          
                        
                        
                          
                            -
                            
                              S
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    
                      1
                      
                        2
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              A
                              2
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              S
                              2
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            2
                          
                        
                        
                          
                            I
                            2
                          
                        
                      
                      
                        
                          
                            I
                            2
                          
                        
                        
                          
                            -
                            
                              I
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              where
              ⁢
              
                

              
              [
              A
              ]
            
            2
          
          =
          
            [
            
              
                
                  
                    S
                    8
                    1
                  
                
                
                  
                    S
                    8
                    1
                  
                
              
              
                
                  
                    S
                    8
                    3
                  
                
                
                  
                    -
                    
                      S
                      8
                      1
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        (
        31
        )
      
    
  




Likewise, an 8×8 permuted matrix can be given by




  
    
      
        
          
            
              
                
                  
                    
                      [
                      S
                      ]
                    
                    8
                  
                  •
                
                =
                
                ⁢
                
                  
                    
                      
                        
                          
                            [
                            Pr
                            ]
                          
                          8
                        
                        ⁡
                        
                          [
                          S
                          ]
                        
                      
                      8
                    
                    ⁡
                    
                      [
                      Pc
                      ]
                    
                  
                  8
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    1
                    2
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            A
                            4
                          
                        
                        
                          
                            -
                            
                              A
                              4
                            
                          
                        
                      
                      
                        
                          
                            S
                            4
                          
                        
                        
                          
                            S
                            4
                          
                        
                      
                    
                    ]
                  
                
              
            
          
          
            
              
                
                  =
                  
                  ⁢
                  
                    
                      
                        1
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                A
                                4
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                S
                                4
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              I
                              4
                            
                          
                          
                            
                              I
                              4
                            
                          
                        
                        
                          
                            
                              I
                              4
                            
                          
                          
                            
                              -
                              
                                I
                                4
                              
                            
                          
                        
                      
                      ]
                    
                  
                
                ,
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              where
              ⁢
              
                

              
              [
              A
              ]
            
            4
          
          =
          
            
              [
              
                
                  
                    
                      S
                      16
                      1
                    
                  
                  
                    
                      S
                      16
                      3
                    
                  
                  
                    
                      S
                      16
                      5
                    
                  
                  
                    
                      S
                      16
                      7
                    
                  
                
                
                  
                    
                      S
                      16
                      3
                    
                  
                  
                    
                      S
                      16
                      7
                    
                  
                  
                    
                      S
                      16
                      1
                    
                  
                  
                    
                      -
                      
                        S
                        16
                        5
                      
                    
                  
                
                
                  
                    
                      S
                      16
                      5
                    
                  
                  
                    
                      S
                      16
                      1
                    
                  
                  
                    
                      -
                      
                        S
                        16
                        7
                      
                    
                  
                  
                    
                      S
                      16
                      3
                    
                  
                
                
                  
                    
                      S
                      16
                      7
                    
                  
                  
                    
                      -
                      
                        S
                        16
                        5
                      
                    
                  
                  
                    
                      S
                      16
                      3
                    
                  
                  
                    
                      -
                      
                        S
                        16
                        1
                      
                    
                  
                
              
              ]
            
            .
          
        
      
    
    
      
        (
        32
        )
      
    
  




In general, the N×N DST-II matrix can be given by




  
    
      
        
          
            
              
                
                  
                    [
                    S
                    ]
                  
                  N
                
                =
                
                ⁢
                
                  
                    
                      2
                      N
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            S
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                              ⁢
                              
                                φ
                                0
                              
                            
                          
                        
                        
                          
                            S
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                              ⁢
                              
                                φ
                                1
                              
                            
                          
                        
                        
                          ...
                        
                        
                          
                            S
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                              ⁢
                              
                                φ
                                
                                  N
                                  -
                                  1
                                
                              
                            
                          
                        
                      
                      
                        
                          
                            S
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                1
                              
                              ⁢
                              
                                φ
                                0
                              
                            
                          
                        
                        
                          
                            S
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                1
                              
                              ⁢
                              
                                φ
                                1
                              
                            
                          
                        
                        
                          ...
                        
                        
                          
                            S
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                1
                              
                              ⁢
                              
                                φ
                                
                                  N
                                  -
                                  1
                                
                              
                            
                          
                        
                      
                      
                        
                          ⋮
                        
                        
                          ⋮
                        
                        
                          ...
                        
                        
                          ⋮
                        
                      
                      
                        
                          
                            S
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                
                                  N
                                  -
                                  2
                                
                              
                              ⁢
                              
                                φ
                                0
                              
                            
                          
                        
                        
                          
                            S
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                
                                  N
                                  -
                                  2
                                
                              
                              ⁢
                              
                                φ
                                1
                              
                            
                          
                        
                        
                          ...
                        
                        
                          
                            S
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                
                                  N
                                  -
                                  2
                                
                              
                              ⁢
                              
                                φ
                                
                                  N
                                  -
                                  1
                                
                              
                            
                          
                        
                      
                      
                        
                          
                            1
                            /
                            
                              2
                            
                          
                        
                        
                          
                            
                              -
                              1
                            
                            /
                            
                              2
                            
                          
                        
                        
                          ...
                        
                        
                          
                            
                              -
                              1
                            
                            /
                            
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    
                      
                        2
                        N
                      
                    
                    ⁡
                    
                      [
                      S
                      ]
                    
                  
                  N
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        where
        ⁢
        
          

        
        ⁢
        
          
            
              
                
                  
                    [
                    
                      S
                      
                        m
                        ,
                        n
                      
                    
                    ]
                  
                  N
                
                =
                
                ⁢
                
                  [
                  
                    
                      s
                      
                        i
                        ,
                        j
                      
                    
                    ⁢
                    
                      S
                      
                        2
                        ⁢
                        n
                      
                      
                        
                          (
                          
                            m
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                          (
                          
                            
                              2
                              ⁢
                              n
                            
                            +
                            1
                          
                          )
                        
                      
                    
                  
                  ]
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  [
                  
                    
                      s
                      
                        i
                        ,
                        j
                      
                    
                    ⁢
                    
                      S
                      
                        4
                        ⁢
                        N
                      
                      
                        
                          (
                          
                            m
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                          (
                          
                            
                              2
                              ⁢
                              n
                            
                            +
                            1
                          
                          )
                        
                      
                    
                  
                  ]
                
              
            
          
          
            
              
                
                  =
                  
                  ⁢
                  
                    
                      [
                      
                        
                          s
                          
                            i
                            ,
                            j
                          
                        
                        ⁢
                        
                          S
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              i
                            
                            ⁢
                            
                              φ
                              j
                            
                          
                        
                      
                      ]
                    
                    N
                  
                
                ,
              
            
          
        
        ⁢
        
          

        
        ⁢
        where
        ⁢
        
          

        
        ⁢
        
          {
          
            
              
                
                  
                    
                      k
                      i
                    
                    =
                    
                      i
                      +
                      1
                    
                  
                
              
              
                
                  
                    
                      φ
                      j
                    
                    =
                    
                      
                        2
                        ⁢
                        j
                      
                      +
                      1
                    
                  
                
              
            
            ,
            
              
                s
                
                  i
                  ,
                  j
                
              
              =
              
                {
                
                  
                    
                      
                        
                          1
                          ,
                        
                      
                      
                        
                          i
                          ≠
                          
                            N
                            -
                            1
                          
                        
                      
                    
                    
                      
                        
                          
                            1
                            /
                            
                              2
                            
                          
                          ,
                        
                      
                      
                        
                          i
                          =
                          
                            N
                            -
                            1
                          
                        
                      
                    
                  
                  ,
                  
                    

                  
                  ⁢
                  m
                  ,
                  n
                  ,
                  i
                  ,
                  
                    j
                    =
                    0
                  
                  ,
                  
                    1
                    ⁢
                    
                      
                    
                    ⁢
                    ...
                  
                  ⁢
                  
                    
                  
                  ,
                  
                    N
                    -
                    1.
                  
                
              
            
          
        
      
    
    
      
        (
        33
        )
      
    
  




The N×N permuted DST-II matrix [S]N can be recursively formed by using




  
    
      
        
          
            
              
                
                  
                    [
                    S
                    ]
                  
                  N
                
                •
              
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          [
                          Pr
                          ]
                        
                        N
                      
                      ⁡
                      
                        [
                        S
                        ]
                      
                    
                    N
                  
                  ⁡
                  
                    [
                    Pc
                    ]
                  
                
                N
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      2
                      N
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            A
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            S
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                    
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          -
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
      
    
    
      
        (
        34
        )
      
    
  




A submatrix [A]N is given by




  
    
      
        
          
            
              
                
                  
                    [
                    
                      A
                      
                        m
                        ,
                        n
                      
                    
                    ]
                  
                  N
                
                =
                
                ⁢
                
                  [
                  
                    S
                    
                      4
                      ⁢
                      N
                    
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            m
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      
                        (
                        
                          
                            2
                            ⁢
                            n
                          
                          +
                          1
                        
                        )
                      
                    
                  
                  ]
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  [
                  
                    S
                    
                      4
                      ⁢
                      N
                    
                    
                      
                        [
                        
                          
                            2
                            ⁢
                            
                              (
                              
                                m
                                +
                                1
                              
                              )
                            
                          
                          -
                          1
                        
                        ]
                      
                      ⁢
                      
                        (
                        
                          
                            2
                            ⁢
                            n
                          
                          +
                          1
                        
                        )
                      
                    
                  
                  ]
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    [
                    
                      S
                      
                        4
                        ⁢
                        N
                      
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                k
                                1
                              
                            
                            -
                            1
                          
                          )
                        
                        ⁢
                        
                          φ
                          j
                        
                      
                    
                    ]
                  
                  N
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          {
          
            
              
                
                  
                    
                      k
                      i
                    
                    =
                    
                      i
                      +
                      1
                    
                  
                
              
              
                
                  
                    
                      φ
                      j
                    
                    =
                    
                      
                        2
                        ⁢
                        j
                      
                      +
                      1
                    
                  
                
              
            
            ,
            
              

            
            ⁢
            m
            ,
            n
            ,
            i
            ,
            
              j
              =
              0
            
            ,
            
              1
              ⁢
              
                
              
              ⁢
              ...
            
            ⁢
            
              
            
            ,
            
              N
              -
              1.
            
          
        
      
    
    
      
        (
        35
        )
      
    
  




through multiplying the row and column permutation matrices, we can obtain the decomposed matrix as the diagonal block-wise inverse Jacket sparse matrix as below:




  
    
      
        
          
            
              [
              
                
                  
                    
                      A
                      4
                    
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    
                      S
                      4
                    
                  
                
              
              ]
            
            
              -
              1
            
          
          =
          
            [
            
              
                
                  
                    A
                    4
                    
                      -
                      1
                    
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    S
                    4
                    
                      -
                      1
                    
                  
                
              
            
            ]
          
        
        ,
        
          

        
        ⁢
        then
        ,
        
          

        
        ⁢
        
          
            
              [
              
                
                  
                    
                      A
                      
                        N
                        /
                        2
                      
                    
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    
                      S
                      
                        N
                        /
                        2
                      
                    
                  
                
              
              ]
            
            
              -
              1
            
          
          =
          
            [
            
              
                
                  
                    A
                    
                      N
                      /
                      2
                    
                    
                      -
                      1
                    
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    S
                    
                      N
                      /
                      2
                    
                    
                      -
                      1
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        (
        36
        )
      
    
  




The inverse of Equation (36) is simply calculated as below.




  
    
      
        
          
            
              
                
                  
                    [
                    S
                    ]
                  
                  N
                  
                    -
                    1
                  
                
                •
              
              =
              
              ⁢
              
                
                  (
                  
                    
                      
                        
                          2
                          N
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                A
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              o
                            
                          
                          
                            
                              o
                            
                            
                              
                                S
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                        
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            
                              -
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                        
                      
                      ]
                    
                  
                  )
                
                
                  -
                  1
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    N
                    2
                  
                
                ×
                
                  
                    
                      
                        1
                        2
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              
                                -
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              A
                              
                                N
                                /
                                2
                              
                              
                                -
                                1
                              
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              S
                              
                                N
                                /
                                2
                              
                              
                                -
                                1
                              
                            
                          
                        
                      
                      ]
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        37
        )
      
    
  




Furthermore, it was provided follow that the submatrix [A]N can be represented by:




  
    
      
        
          
        
        ⁢
        
          
            
              
                [
                A
                ]
              
              N
            
            =
            
              
                
                  
                    
                      
                        [
                        U
                        ]
                      
                      N
                    
                    ⁡
                    
                      [
                      S
                      ]
                    
                  
                  N
                
                ⁡
                
                  [
                  D
                  ]
                
              
              N
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              
                
                  where
                  ⁢
                  
                    

                  
                  ⁢
                  
                    
                  
                  [
                  U
                  ]
                
                N
              
              =
              
                
                  [
                  
                    
                      
                        2
                      
                      
                        
                          -
                          2
                        
                      
                      
                        2
                      
                      
                        
                          -
                          
                            2
                          
                        
                      
                    
                    
                      
                        0
                      
                      
                        2
                      
                      
                        
                          -
                          2
                        
                      
                      
                        
                          2
                        
                      
                    
                    
                      
                        ⋮
                      
                      
                        ⋱
                      
                      
                        2
                      
                      
                        ⋮
                      
                    
                    
                      
                        0
                      
                      
                        ...
                      
                      
                        0
                      
                      
                        
                          2
                        
                      
                    
                  
                  ]
                
                N
              
            
            ,
            
              

            
            ⁢
            
              
                
                  [
                  D
                  ]
                
                N
              
              =
              
                
                  
                    
                      diag
                      ⁡
                      
                        [
                        
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              Φ
                              0
                            
                          
                          ,
                          ...
                          ⁢
                          
                            
                          
                          ,
                          
                            C
                            4
                            
                              Φ
                              
                                N
                                -
                                1
                              
                            
                          
                        
                        ]
                      
                    
                    N
                  
                  ⁢
                  
                    Φ
                    i
                  
                
                =
                
                  
                    2
                    ⁢
                    i
                  
                  +
                  1
                
              
            
            ,
            
              i
              =
              0
            
            ,
            1
            ,
            ...
            ⁢
            
              
            
            ,
            
              N
              -
              1.
            
          
        
      
    
    
      
        (
        38
        )
      
    
  




and [U]N is an upper triangular matrix.

Similar fashion as the DCT lower triangular matrix of Eq. (16), also the matrix of the DST is upper triangular matrix and we can obtain the relationship as below:




  
    
      
        
          
            
              [
              L
              ]
            
            N
          
          =
          
            
              
                
                  [
                  
                    I
                    _
                  
                  ]
                
                N
              
              ⁡
              
                [
                U
                ]
              
            
            N
          
        
        ,
        
          

        
        ⁢
        
          
            
              where
              ⁢
              
                

              
              [
              
                I
                _
              
              ]
            
            2
          
          =
          
            [
            
              
                
                  0
                
                
                  1
                
              
              
                
                  1
                
                
                  0
                
              
            
            ]
          
        
      
    
    
      
        (
        39
        )
      
    
  




Equation (38) is proved as follows.
Using relationship of sum formula and difference formula of sine function, the following equation is expressed.




  
    
      
        
          
            
              
                
                  C
                  
                    4
                    ⁢
                    N
                  
                  
                    φ
                    j
                  
                
                =
                
                ⁢
                
                  S
                  
                    4
                    ⁢
                    N
                  
                  
                    
                      φ
                      ⁢
                      
                        
                      
                    
                    
                      N
                      -
                      j
                      -
                      1
                    
                  
                
              
            
          
          
            
              
                
                  =
                  
                  ⁢
                  
                    S
                    
                      4
                      ⁢
                      N
                    
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                              k
                              
                                N
                                -
                                1
                              
                            
                          
                          -
                          1
                        
                        )
                      
                      ⁢
                      
                        φ
                        j
                      
                    
                  
                
                ,
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                2
                ⁢
                
                  S
                  
                    4
                    ⁢
                    N
                  
                  
                    2
                    ⁢
                    
                      k
                      i
                    
                    ⁢
                    
                      φ
                      j
                    
                  
                
                ⁢
                
                  C
                  
                    4
                    ⁢
                    N
                  
                  
                    φ
                    j
                  
                
              
              -
              
                S
                
                  4
                  ⁢
                  N
                
                
                  
                    (
                    
                      
                        2
                        ⁢
                        
                          k
                          
                            i
                            +
                            1
                          
                        
                      
                      -
                      1
                    
                    )
                  
                  ⁢
                  
                    φ
                    j
                  
                
              
            
            =
            
              S
              
                4
                ⁢
                N
              
              
                
                  (
                  
                    
                      2
                      ⁢
                      
                        k
                        i
                      
                    
                    -
                    1
                  
                  )
                
                ⁢
                
                  φ
                  j
                
              
            
          
          ,
          
            

          
          ⁢
          
            
              
                2
                ⁢
                
                  S
                  
                    4
                    ⁢
                    N
                  
                  
                    2
                    ⁢
                    
                      k
                      i
                    
                    ⁢
                    
                      φ
                      j
                    
                  
                
                ⁢
                
                  C
                  
                    4
                    ⁢
                    N
                  
                  
                    φ
                    j
                  
                
              
              -
              
                S
                
                  4
                  ⁢
                  N
                
                
                  
                    (
                    
                      
                        2
                        ⁢
                        
                          k
                          i
                        
                      
                      -
                      1
                    
                    )
                  
                  ⁢
                  
                    φ
                    j
                  
                
              
            
            =
            
              S
              
                4
                ⁢
                N
              
              
                
                  (
                  
                    
                      2
                      ⁢
                      
                        k
                        
                          i
                          +
                          1
                        
                      
                    
                    -
                    1
                  
                  )
                
                ⁢
                
                  φ
                  j
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          {
          
            
              
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            S
                            
                              4
                              ⁢
                              N
                            
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    
                                      k
                                      0
                                    
                                  
                                  -
                                  1
                                
                                )
                              
                              ⁢
                              
                                φ
                                j
                              
                            
                          
                        
                        -
                        
                          2
                          ⁢
                          
                            S
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                1
                              
                              ⁢
                              
                                φ
                                j
                              
                            
                          
                        
                        +
                        ...
                        +
                        
                          2
                          ⁢
                          
                            S
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                
                                  N
                                  -
                                  2
                                
                              
                              ⁢
                              
                                φ
                                j
                              
                            
                          
                        
                        -
                        1
                      
                      )
                    
                    ⁢
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        φ
                        j
                      
                    
                  
                  =
                  
                    
                      
                        2
                        ⁢
                        
                          S
                          
                            4
                            ⁢
                            N
                          
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  
                                    k
                                    0
                                  
                                
                                -
                                1
                              
                              )
                            
                            ⁢
                            
                              φ
                              j
                            
                          
                        
                        ⁢
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            φ
                            j
                          
                        
                      
                      -
                      
                        S
                        
                          4
                          ⁢
                          N
                        
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  k
                                  1
                                
                              
                              -
                              1
                            
                            )
                          
                          ⁢
                          
                            φ
                            j
                          
                        
                      
                    
                    =
                    
                      S
                      
                        4
                        ⁢
                        N
                      
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                k
                                0
                              
                            
                            -
                            1
                          
                          )
                        
                        ⁢
                        
                          φ
                          j
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            S
                            
                              4
                              ⁢
                              N
                            
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    
                                      k
                                      1
                                    
                                  
                                  -
                                  1
                                
                                )
                              
                              ⁢
                              
                                φ
                                j
                              
                            
                          
                        
                        -
                        
                          2
                          ⁢
                          
                            S
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                1
                              
                              ⁢
                              
                                φ
                                j
                              
                            
                          
                        
                        +
                        ...
                        -
                        
                          2
                          ⁢
                          
                            S
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                
                                  N
                                  -
                                  2
                                
                              
                              ⁢
                              
                                φ
                                j
                              
                            
                          
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        φ
                        j
                      
                    
                  
                  =
                  
                    
                      
                        2
                        ⁢
                        
                          S
                          
                            4
                            ⁢
                            N
                          
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  
                                    k
                                    1
                                  
                                
                                -
                                1
                              
                              )
                            
                            ⁢
                            
                              φ
                              j
                            
                          
                        
                        ⁢
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            φ
                            j
                          
                        
                      
                      -
                      
                        S
                        
                          4
                          ⁢
                          N
                        
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  k
                                  2
                                
                              
                              -
                              1
                            
                            )
                          
                          ⁢
                          
                            φ
                            j
                          
                        
                      
                    
                    =
                    
                      S
                      
                        4
                        ⁢
                        N
                      
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                k
                                1
                              
                            
                            -
                            1
                          
                          )
                        
                        ⁢
                        
                          φ
                          j
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    
                  
                  ⁢
                  
                    ⋰
                    ⁢
                    
                      
                    
                    ⁢
                    ⋰
                    ⁢
                    
                      
                    
                    ⁢
                    ⋰
                  
                
              
            
            
              
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            S
                            
                              4
                              ⁢
                              N
                            
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    
                                      k
                                      
                                        N
                                        -
                                        3
                                      
                                    
                                  
                                  -
                                  1
                                
                                )
                              
                              ⁢
                              
                                φ
                                j
                              
                            
                          
                        
                        -
                        
                          2
                          ⁢
                          
                            S
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                
                                  N
                                  -
                                  2
                                
                              
                              ⁢
                              
                                φ
                                j
                              
                            
                          
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        φ
                        j
                      
                    
                  
                  =
                  
                    
                      
                        2
                        ⁢
                        
                          S
                          
                            4
                            ⁢
                            N
                          
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  
                                    k
                                    
                                      N
                                      -
                                      3
                                    
                                  
                                
                                -
                                1
                              
                              )
                            
                            ⁢
                            
                              φ
                              j
                            
                          
                        
                        ⁢
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            φ
                            j
                          
                        
                      
                      -
                      
                        S
                        
                          4
                          ⁢
                          N
                        
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  k
                                  
                                    N
                                    -
                                    2
                                  
                                
                              
                              -
                              1
                            
                            )
                          
                          ⁢
                          
                            φ
                            j
                          
                        
                      
                    
                    =
                    
                      S
                      
                        4
                        ⁢
                        N
                      
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                k
                                
                                  N
                                  -
                                  3
                                
                              
                            
                            -
                            1
                          
                          )
                        
                        ⁢
                        
                          φ
                          j
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            S
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                
                                  n
                                  -
                                  2
                                
                              
                              ⁢
                              
                                φ
                                j
                              
                            
                          
                        
                        -
                        1
                      
                      )
                    
                    ⁢
                    
                      C
                      
                        4
                        ⁢
                        N
                      
                      
                        φ
                        j
                      
                    
                  
                  =
                  
                    
                      
                        2
                        ⁢
                        
                          S
                          
                            4
                            ⁢
                            N
                          
                          
                            2
                            ⁢
                            
                              k
                              
                                N
                                -
                                2
                              
                            
                            ⁢
                            
                              φ
                              j
                            
                          
                        
                        ⁢
                        
                          C
                          
                            4
                            ⁢
                            N
                          
                          
                            φ
                            j
                          
                        
                      
                      -
                      
                        S
                        
                          4
                          ⁢
                          N
                        
                        
                          φ
                          
                            N
                            -
                            j
                            -
                            1
                          
                        
                      
                    
                    =
                    
                      
                        
                          2
                          ⁢
                          
                            S
                            
                              4
                              ⁢
                              N
                            
                            
                              2
                              ⁢
                              
                                k
                                
                                  N
                                  -
                                  2
                                
                              
                              ⁢
                              
                                φ
                                j
                              
                            
                          
                          ⁢
                          
                            C
                            
                              4
                              ⁢
                              N
                            
                            
                              φ
                              j
                            
                          
                        
                        -
                        
                          S
                          
                            4
                            ⁢
                            N
                          
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  
                                    k
                                    
                                      N
                                      -
                                      1
                                    
                                  
                                
                                -
                                1
                              
                              )
                            
                            ⁢
                            
                              φ
                              j
                            
                          
                        
                      
                      =
                      
                        S
                        
                          4
                          ⁢
                          N
                        
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  k
                                  
                                    N
                                    -
                                    2
                                  
                                
                              
                              -
                              1
                            
                            )
                          
                          ⁢
                          
                            φ
                            j
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        40
        )
      
    
  




where ki=i+1, Φj=2j+1, i,j=0, 1, . . . , N−1.

By taking Equation (40) and into the right hand side of Equation (38), the following equation is obtained.




  
    
      
        
          
            
              
                
                  
                    [
                    U
                    ]
                  
                  N
                
                ⁡
                
                  [
                  S
                  ]
                
              
              N
            
            ⁡
            
              [
              D
              ]
            
          
          N
        
        =
        
          [
          
            
              
                
                  S
                  
                    4
                    ⁢
                    N
                  
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            k
                            0
                          
                        
                        -
                        1
                      
                      )
                    
                    ⁢
                    
                      φ
                      0
                    
                  
                
              
              
                
                  S
                  
                    4
                    ⁢
                    N
                  
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            k
                            0
                          
                        
                        -
                        1
                      
                      )
                    
                    ⁢
                    
                      φ
                      1
                    
                  
                
              
              
                ...
              
              
                
                  S
                  
                    4
                    ⁢
                    N
                  
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            k
                            0
                          
                        
                        -
                        1
                      
                      )
                    
                    ⁢
                    
                      φ
                      
                        N
                        -
                        1
                      
                    
                  
                
              
            
            
              
                
                  S
                  
                    4
                    ⁢
                    N
                  
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            k
                            1
                          
                        
                        -
                        1
                      
                      )
                    
                    ⁢
                    
                      φ
                      0
                    
                  
                
              
              
                
                  S
                  
                    4
                    ⁢
                    N
                  
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            k
                            1
                          
                        
                        -
                        1
                      
                      )
                    
                    ⁢
                    
                      φ
                      1
                    
                  
                
              
              
                ...
              
              
                
                  S
                  
                    4
                    ⁢
                    N
                  
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            k
                            1
                          
                        
                        -
                        1
                      
                      )
                    
                    ⁢
                    
                      φ
                      
                        N
                        -
                        1
                      
                    
                  
                
              
            
            
              
                ⋮
              
              
                ⋮
              
              
                ⋱
              
              
                ⋮
              
            
            
              
                
                  S
                  
                    4
                    ⁢
                    N
                  
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            k
                            
                              N
                              -
                              1
                            
                          
                        
                        -
                        1
                      
                      )
                    
                    ⁢
                    
                      φ
                      0
                    
                  
                
              
              
                
                  S
                  
                    4
                    ⁢
                    N
                  
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            k
                            
                              N
                              -
                              1
                            
                          
                        
                        -
                        1
                      
                      )
                    
                    ⁢
                    
                      φ
                      1
                    
                  
                
              
              
                ...
              
              
                
                  S
                  
                    4
                    ⁢
                    N
                  
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            k
                            
                              N
                              -
                              1
                            
                          
                        
                        -
                        1
                      
                      )
                    
                    ⁢
                    
                      φ
                      
                        N
                        -
                        1
                      
                    
                  
                
              
            
          
          ]
        
      
    
    
      
        (
        41
        )
      
    
  




According to Equation (35), the left hand side of Equation (38) of [A]N from [S]2N can be expressed by




  
    
      
        
          
            [
            A
            ]
          
          N
        
        =
        
          [
          
            
              
                
                  S
                  
                    4
                    ⁢
                    N
                  
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            k
                            0
                          
                        
                        -
                        1
                      
                      )
                    
                    ⁢
                    
                      φ
                      0
                    
                  
                
              
              
                
                  S
                  
                    4
                    ⁢
                    N
                  
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            k
                            0
                          
                        
                        -
                        1
                      
                      )
                    
                    ⁢
                    
                      φ
                      1
                    
                  
                
              
              
                ...
              
              
                
                  S
                  
                    4
                    ⁢
                    N
                  
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            k
                            0
                          
                        
                        -
                        1
                      
                      )
                    
                    ⁢
                    
                      φ
                      
                        N
                        -
                        1
                      
                    
                  
                
              
            
            
              
                
                  S
                  
                    4
                    ⁢
                    N
                  
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            k
                            1
                          
                        
                        -
                        1
                      
                      )
                    
                    ⁢
                    
                      φ
                      0
                    
                  
                
              
              
                
                  S
                  
                    4
                    ⁢
                    N
                  
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            k
                            1
                          
                        
                        -
                        1
                      
                      )
                    
                    ⁢
                    
                      φ
                      1
                    
                  
                
              
              
                ...
              
              
                
                  S
                  
                    4
                    ⁢
                    N
                  
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            k
                            1
                          
                        
                        -
                        1
                      
                      )
                    
                    ⁢
                    
                      φ
                      
                        N
                        -
                        1
                      
                    
                  
                
              
            
            
              
                ⋮
              
              
                ⋮
              
              
                ⋱
              
              
                ⋮
              
            
            
              
                
                  S
                  
                    4
                    ⁢
                    N
                  
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            k
                            
                              N
                              -
                              1
                            
                          
                        
                        -
                        1
                      
                      )
                    
                    ⁢
                    
                      φ
                      0
                    
                  
                
              
              
                
                  S
                  
                    4
                    ⁢
                    N
                  
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            k
                            
                              N
                              -
                              1
                            
                          
                        
                        -
                        1
                      
                      )
                    
                    ⁢
                    
                      φ
                      1
                    
                  
                
              
              
                ...
              
              
                
                  S
                  
                    4
                    ⁢
                    N
                  
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                            k
                            
                              N
                              -
                              1
                            
                          
                        
                        -
                        1
                      
                      )
                    
                    ⁢
                    
                      φ
                      
                        N
                        -
                        1
                      
                    
                  
                
              
            
          
          ]
        
      
    
    
      
        (
        42
        )
      
    
  




It is proved that Equation (41) and Equation (42) are the same and the expression of Equation (38) is correct.
Then, the 4×4 of DST-II matrices are as shown




  
    
      
        
          
            
              
                
                  [
                  S
                  ]
                
                4
              
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          
                            1
                            
                              2
                            
                          
                          ⁡
                          
                            [
                            Pr
                            ]
                          
                        
                        4
                        
                          -
                          1
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                A
                                2
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                S
                                2
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              I
                              2
                            
                          
                          
                            
                              I
                              2
                            
                          
                        
                        
                          
                            
                              I
                              2
                            
                          
                          
                            
                              -
                              
                                I
                                2
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    Pc
                    ]
                  
                
                4
                
                  -
                  1
                
              
            
          
        
        
          
            
              
                =
                
                ⁢
                
                  
                    
                      
                        
                          
                            
                              1
                              
                                2
                              
                            
                            ⁡
                            
                              [
                              Pr
                              ]
                            
                          
                          4
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  A
                                  2
                                
                              
                              
                                0
                              
                            
                            
                              
                                0
                              
                              
                                
                                  S
                                  2
                                
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                I
                                2
                              
                            
                            
                              
                                I
                                2
                              
                            
                          
                          
                            
                              
                                I
                                2
                              
                            
                            
                              
                                -
                                
                                  I
                                  2
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      Pc
                      ]
                    
                  
                  4
                
              
              ,
            
          
        
      
    
    
      
        (
        43
        )
      
    
  




We note that




  
    
      
        
          
            
              
                
                  [
                  S
                  ]
                
                N
              
              =
              
              ⁢
              
                
                  
                    
                      N
                      2
                    
                  
                  ⁡
                  
                    [
                    S
                    ]
                  
                
                N
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          [
                          Pr
                          ]
                        
                        N
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                A
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                S
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                        
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            
                              -
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    Pc
                    ]
                  
                
                N
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          [
                          Pr
                          ]
                        
                        N
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                
                                  U
                                  
                                    N
                                    /
                                    2
                                  
                                
                                ⁢
                                
                                  S
                                  
                                    N
                                    /
                                    2
                                  
                                
                                ⁢
                                
                                  D
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                S
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                        
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            
                              -
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    Pc
                    ]
                  
                
                N
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          
                            
                              [
                              Pr
                              ]
                            
                            N
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  
                                    U
                                    
                                      N
                                      /
                                      2
                                    
                                  
                                
                                
                                  0
                                
                              
                              
                                
                                  0
                                
                                
                                  
                                    I
                                    
                                      N
                                      /
                                      2
                                    
                                  
                                
                              
                            
                            ]
                          
                        
                        ⁡
                        
                          [
                          
                            
                              I
                              2
                            
                            ⊗
                            
                              S
                              
                                N
                                /
                                2
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                D
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                        
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            
                              -
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    Pc
                    ]
                  
                
                N
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      [
                      Pr
                      ]
                    
                    N
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            U
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
                [
                
                  
                    I
                    2
                  
                  ⊗
                  
                    
                      
                        [
                        
                          
                        
                        ⁢
                        
                          ...
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              I
                              2
                            
                            ⊗
                            
                              
                                
                                  [
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  [
                                                  Pr
                                                  ]
                                                
                                                4
                                              
                                              ⁡
                                              
                                                [
                                                
                                                  
                                                    
                                                      
                                                        U
                                                        2
                                                      
                                                    
                                                    
                                                      0
                                                    
                                                  
                                                  
                                                    
                                                      0
                                                    
                                                    
                                                      
                                                        I
                                                        2
                                                      
                                                    
                                                  
                                                
                                                ]
                                              
                                            
                                            ⁡
                                            
                                              [
                                              
                                                
                                                  I
                                                  2
                                                
                                                ⊗
                                                
                                                  S
                                                  2
                                                
                                              
                                              ]
                                            
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                
                                                  
                                                    D
                                                    2
                                                  
                                                
                                                
                                                  0
                                                
                                              
                                              
                                                
                                                  0
                                                
                                                
                                                  
                                                    I
                                                    2
                                                  
                                                
                                              
                                            
                                            ]
                                          
                                        
                                        ⁡
                                        
                                          [
                                          
                                            
                                              
                                                
                                                  I
                                                  2
                                                
                                              
                                              
                                                
                                                  I
                                                  2
                                                
                                              
                                            
                                            
                                              
                                                
                                                  I
                                                  2
                                                
                                              
                                              
                                                
                                                  -
                                                  
                                                    I
                                                    2
                                                  
                                                
                                              
                                            
                                          
                                          ]
                                        
                                      
                                      ⁡
                                      
                                        [
                                        Pc
                                        ]
                                      
                                    
                                    4
                                  
                                  ]
                                
                                }
                              
                              
                                S
                                4
                              
                            
                          
                          ⁢
                          
                            
                          
                          ⁢
                          ...
                        
                        ]
                      
                      }
                    
                    
                      S
                      
                        N
                        /
                        2
                      
                    
                  
                
                ⁢
                
                  
                
                ]
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      [
                      
                        
                          
                            
                              D
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                        
                      
                      ]
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                        
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            
                              -
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    Pc
                    ]
                  
                
                N
              
            
          
        
      
    
    
      
        (
        44
        )
      
    
  




As a result, the general recursive form for DST-II matrix can be represented by




  
    
      [
      S
      ]
    
    N
  
  =
  
    
      
        
          
            
              
                
                  
                    2
                    N
                  
                
                ⁡
                
                  [
                  Pr
                  ]
                
              
              N
            
            ⁡
            
              [
              
                
                  
                    
                      U
                      
                        N
                        /
                        2
                      
                    
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    
                      I
                      
                        N
                        /
                        2
                      
                    
                  
                
              
              ]
            
          
          ⁡
          
            [
            
              
                I
                2
              
              ⊗
              
                Pr
                
                  N
                  /
                  2
                
              
            
            ]
          
        
        ⁡
        
          [
          
            
              I
              2
            
            ⊗
            
              [
              
                
                  
                    
                      U
                      
                        N
                        /
                        4
                      
                    
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    
                      I
                      
                        N
                        /
                        4
                      
                    
                  
                
              
              ]
            
          
          ]
        
      
      ⁡
      
        [
        
          
            I
            4
          
          ⊗
          
            Pr
            
              N
              /
              4
            
          
        
        ]
      
    
    ⁢
    
       
      
        
          [
          
            
              I
              4
            
            ⊗
            
              [
              
                
                  
                    
                      U
                      
                        N
                        /
                        8
                      
                    
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    
                      I
                      
                        N
                        /
                        8
                      
                    
                  
                
              
              ]
            
          
          ]
        
        ⁢
        
          
        
        ⁢
        
          
            
              ...
              ⁢
              
                
              
              [
              
                
                  I
                  
                    N
                    /
                    4
                  
                
                ⊗
                
                  Pr
                  4
                
              
              ]
            
            ⁡
            
              [
              
                
                  I
                  
                    N
                    /
                    4
                  
                
                ⊗
                
                  [
                  
                    
                      
                        
                          U
                          2
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          I
                          2
                        
                      
                    
                  
                  ]
                
              
              ]
            
          
          ⁡
          
            [
            
              
                I
                
                  N
                  /
                  2
                
              
              ⊗
              
                S
                2
              
            
            ]
          
        
        ⁢
        
           
          
            
              
                
                  [
                  
                    
                      I
                      
                        N
                        /
                        4
                      
                    
                    ⊗
                    
                      [
                      
                        
                          
                            
                              D
                              2
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              I
                              2
                            
                          
                        
                      
                      ]
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      I
                      
                        N
                        /
                        4
                      
                    
                    ⊗
                    
                      [
                      
                        
                          
                            
                              I
                              2
                            
                          
                          
                            
                              I
                              2
                            
                          
                        
                        
                          
                            
                              I
                              2
                            
                          
                          
                            
                              -
                              
                                I
                                2
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    I
                    
                      N
                      /
                      4
                    
                  
                  ⊗
                  
                    Pc
                    4
                  
                
                ]
              
            
            ⁢
            
              
            
            ⁢
            
              
                
                  ...
                  ⁢
                  
                    
                  
                  [
                  
                    
                      I
                      4
                    
                    ⊗
                    
                      [
                      
                        
                          
                            
                              D
                              
                                N
                                /
                                8
                              
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              I
                              
                                N
                                /
                                8
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      I
                      4
                    
                    ⊗
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                8
                              
                            
                          
                          
                            
                              I
                              
                                N
                                /
                                8
                              
                            
                          
                        
                        
                          
                            
                              I
                              
                                N
                                /
                                8
                              
                            
                          
                          
                            
                              -
                              
                                I
                                
                                  N
                                  /
                                  8
                                
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    I
                    4
                  
                  ⊗
                  
                    Pc
                    
                      N
                      /
                      4
                    
                  
                
                ]
              
            
            ⁢
            
               
              
                
                  
                    
                      [
                      
                        
                          I
                          2
                        
                        ⊗
                        
                          [
                          
                            
                              
                                
                                  D
                                  
                                    N
                                    /
                                    4
                                  
                                
                              
                              
                                0
                              
                            
                            
                              
                                0
                              
                              
                                
                                  I
                                  
                                    N
                                    /
                                    4
                                  
                                
                              
                            
                          
                          ]
                        
                      
                      ]
                    
                    ⁡
                    
                      [
                      
                        
                          I
                          4
                        
                        ⊗
                        
                          [
                          
                            
                              
                                
                                  I
                                  
                                    N
                                    /
                                    4
                                  
                                
                              
                              
                                
                                  I
                                  
                                    N
                                    /
                                    4
                                  
                                
                              
                            
                            
                              
                                
                                  I
                                  
                                    N
                                    /
                                    4
                                  
                                
                              
                              
                                
                                  -
                                  
                                    I
                                    
                                      N
                                      /
                                      4
                                    
                                  
                                
                              
                            
                          
                          ]
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        I
                        2
                      
                      ⊗
                      
                        Pc
                        
                          N
                          /
                          2
                        
                      
                    
                    ]
                  
                
                ⁢
                
                   
                  
                    
                      
                        
                          
                            [
                            
                              
                                
                                  
                                    D
                                    
                                      N
                                      /
                                      2
                                    
                                  
                                
                                
                                  0
                                
                              
                              
                                
                                  0
                                
                                
                                  
                                    I
                                    
                                      N
                                      /
                                      2
                                    
                                  
                                
                              
                            
                            ]
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  
                                    I
                                    
                                      N
                                      /
                                      2
                                    
                                  
                                
                                
                                  
                                    I
                                    
                                      N
                                      /
                                      2
                                    
                                  
                                
                              
                              
                                
                                  
                                    I
                                    
                                      N
                                      /
                                      2
                                    
                                  
                                
                                
                                  
                                    -
                                    
                                      I
                                      
                                        N
                                        /
                                        2
                                      
                                    
                                  
                                
                              
                            
                            ]
                          
                        
                        ⁡
                        
                          [
                          Pc
                          ]
                        
                      
                      N
                    
                    ,
                  
                
              
            
          
        
      
    
  




also, 4×4 kernel s4 form, we have



  
    
      
    
    ⁢
    
      (
      45
      )
    
  




  
    
      
        
          
            
              [
              S
              ]
            
            N
          
          =
          
          ⁢
          
            
              
                
                  2
                  N
                
              
              ⁡
              
                [
                S
                ]
              
            
            N
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              
                
                  
                    
                      2
                      N
                    
                  
                  ⁡
                  
                    [
                    Pr
                    ]
                  
                
                N
              
              ⁡
              
                [
                
                  
                    
                      
                        U
                        
                          N
                          /
                          2
                        
                      
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        I
                        
                          N
                          /
                          2
                        
                      
                    
                  
                
                ]
              
            
            [
            
              
                I
                2
              
              ⊗
              
                
                  
                    [
                    
                      
                    
                    ⁢
                    
                      
                        ...
                        ⁢
                        
                          
                            
                          
                          ⁢
                          
                            
                          
                        
                        [
                        
                          
                            I
                            2
                          
                          ⊗
                          
                            
                              
                                [
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                [
                                                Pr
                                                ]
                                              
                                              4
                                            
                                            ⁡
                                            
                                              [
                                              
                                                
                                                  
                                                    
                                                      U
                                                      2
                                                    
                                                  
                                                  
                                                    0
                                                  
                                                
                                                
                                                  
                                                    0
                                                  
                                                  
                                                    
                                                      I
                                                      2
                                                    
                                                  
                                                
                                              
                                              ]
                                            
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                I
                                                2
                                              
                                              ⊗
                                              
                                                S
                                                2
                                              
                                            
                                            ]
                                          
                                        
                                        ⁡
                                        
                                          [
                                          
                                            
                                              
                                                
                                                  D
                                                  2
                                                
                                              
                                              
                                                0
                                              
                                            
                                            
                                              
                                                0
                                              
                                              
                                                
                                                  I
                                                  2
                                                
                                              
                                            
                                          
                                          ]
                                        
                                      
                                      ⁡
                                      
                                        [
                                        
                                          
                                            
                                              
                                                I
                                                2
                                              
                                            
                                            
                                              
                                                I
                                                2
                                              
                                            
                                          
                                          
                                            
                                              
                                                I
                                                2
                                              
                                            
                                            
                                              
                                                -
                                                
                                                  I
                                                  2
                                                
                                              
                                            
                                          
                                        
                                        ]
                                      
                                    
                                    ⁡
                                    
                                      [
                                      Pc
                                      ]
                                    
                                  
                                  4
                                
                                ]
                              
                              }
                            
                            
                              S
                              4
                            
                          
                        
                        ⁢
                        
                          
                        
                        ]
                      
                      ⁢
                      
                        
                          
                        
                        ⁢
                        
                          
                        
                      
                      ⁢
                      ...
                    
                    ⁢
                    
                      
                    
                    ]
                  
                  }
                
                
                  S
                  
                    N
                    /
                    2
                  
                
              
            
            ⁢
            
              
                
              
              ⁢
              
                
              
            
            ]
          
        
      
    
    
      
        
          =
          
          ⁢
          
            
              
                
                  
                    [
                    
                      
                        
                          
                            D
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          
                            -
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  Pc
                  ]
                
              
              N
            
            .
          
        
      
    
  



FIG. 2A-2C is a butterfly data flow graph of Equation (45) according to an exemplary embodiment of the present invention. As shown in FIG. 2A-2C, the data flow of Equation (45) is expressed.
IV. Element-Wise Inverse Sparse Matrix Decomposition for DFT Transform
The DFT is a Fourier representation of a given sequence x(m),




  
    
      
        
          
            X
            ⁡
            
              (
              n
              )
            
          
          =
          
            
              ∑
              
                m
                =
                0
              
              
                N
                -
                1
              
            
            ⁢
            
              
                x
                ⁡
                
                  (
                  m
                  )
                
              
              ⁢
              
                W
                nm
              
            
          
        
        ,
        
          

        
        ⁢
        
          0
          ≤
          n
          ≤
          
            N
            -
            1
          
        
        ,
        
          

        
        ⁢
        
          
            where
            ⁢
            
              
            
            ⁢
            W
          
          =
          
            
              e
              
                j
                ⁢
                
                  
                    2
                    ⁢
                    π
                  
                  N
                
              
            
            .
          
        
      
    
    
      
        (
        46
        )
      
    
  




The N-point DFT matrix can be denoted by [F]N=[Wnm]N, n,m={0, 1, 2, . . . , N−1}, and the N×N Sylvester Hadamard matrix as [H]N, respectively. The Sylvester Hadamard matrix is generated recursively by using the successive Kronecker products,

[H]N=[H]2[H]N/2  (47)

for N=4, 8, 16, . . . and





  
    
      [
      H
      ]
    
    2
  
  =
  
    
      [
      
        
          
            1
          
          
            1
          
        
        
          
            1
          
          
            
              -
              1
            
          
        
      
      ]
    
    .
  




For the remainder of this section, analysis will be concerned only with N=2k, k=1, 2, 3, . . . as the dimensionality of 2×2 Fourier matrices,





  
    
      
        
          
            [
            F
            ]
          
          2
        
        =
        
          
            
              1
              
                2
              
            
            ⁡
            
              [
              
                
                  
                    1
                  
                  
                    1
                  
                
                
                  
                    1
                  
                  
                    
                      e
                      jπ
                    
                  
                
              
              ]
            
          
          =
          
            
              
                1
                
                  2
                
              
              ⁡
              
                [
                
                  
                    
                      1
                    
                    
                      1
                    
                  
                  
                    
                      1
                    
                    
                      
                        -
                        1
                      
                    
                  
                
                ]
              
            
            .
          
        
      
    
    
      
        (
        48
        )
      
    
  




As an example for N=4, the DFT matrix [F]4 can be represented as




  
    
      
        
          
            
              
                
                  [
                  F
                  ]
                
                4
              
              =
              
              ⁢
              
                [
                
                  
                    
                      
                        W
                        0
                      
                    
                    
                      
                        W
                        0
                      
                    
                    
                      
                        W
                        0
                      
                    
                    
                      
                        W
                        0
                      
                    
                  
                  
                    
                      
                        W
                        0
                      
                    
                    
                      
                        W
                        1
                      
                    
                    
                      
                        W
                        2
                      
                    
                    
                      
                        W
                        3
                      
                    
                  
                  
                    
                      
                        W
                        0
                      
                    
                    
                      
                        W
                        2
                      
                    
                    
                      
                        W
                        4
                      
                    
                    
                      
                        W
                        6
                      
                    
                  
                  
                    
                      
                        W
                        0
                      
                    
                    
                      
                        W
                        3
                      
                    
                    
                      
                        W
                        6
                      
                    
                    
                      
                        W
                        9
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              =
              
              ⁢
              
                [
                
                  
                    
                      1
                    
                    
                      1
                    
                    
                      1
                    
                    
                      1
                    
                  
                  
                    
                      1
                    
                    
                      
                        e
                        
                          
                            -
                            j
                          
                          ⁢
                          
                            π
                            2
                          
                        
                      
                    
                    
                      
                        e
                        
                          
                            -
                            j
                          
                          ⁢
                          
                            π
                            2
                          
                          ×
                          2
                        
                      
                    
                    
                      
                        e
                        
                          
                            -
                            j
                          
                          ⁢
                          
                            π
                            2
                          
                          ×
                          3
                        
                      
                    
                  
                  
                    
                      1
                    
                    
                      
                        e
                        
                          
                            -
                            j
                          
                          ⁢
                          
                            π
                            2
                          
                          ×
                          2
                        
                      
                    
                    
                      
                        e
                        
                          
                            -
                            j
                          
                          ⁢
                          
                            π
                            2
                          
                          ×
                          4
                        
                      
                    
                    
                      
                        e
                        
                          
                            -
                            j
                          
                          ⁢
                          
                            π
                            2
                          
                          ×
                          6
                        
                      
                    
                  
                  
                    
                      1
                    
                    
                      
                        e
                        
                          
                            -
                            j
                          
                          ⁢
                          
                            π
                            2
                          
                          ×
                          3
                        
                      
                    
                    
                      
                        e
                        
                          
                            -
                            j
                          
                          ⁢
                          
                            π
                            2
                          
                          ×
                          6
                        
                      
                    
                    
                      
                        e
                        
                          
                            -
                            j
                          
                          ⁢
                          
                            π
                            2
                          
                          ×
                          9
                        
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        1
                      
                      
                        1
                      
                      
                        1
                      
                    
                    
                      
                        1
                      
                      
                        
                          -
                          j
                        
                      
                      
                        
                          -
                          1
                        
                      
                      
                        j
                      
                    
                    
                      
                        1
                      
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                      
                        
                          -
                          1
                        
                      
                    
                    
                      
                        1
                      
                      
                        j
                      
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          j
                        
                      
                    
                  
                  ]
                
                .
              
            
          
        
      
    
    
      
        (
        49
        )
      
    
  




Similar to the DCT case, we can rewrite (49) by using permutations as




  
    
      
        
          
            
              
                
                  
                    [
                    
                      F
                      ~
                    
                    ]
                  
                  4
                
                =
                
                ⁢
                
                  
                    
                      [
                      Pr
                      ]
                    
                    4
                  
                  ⁡
                  
                    [
                    
                      F
                      4
                    
                    ]
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                        
                          0
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          0
                        
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          0
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          0
                        
                        
                          0
                        
                        
                          1
                        
                      
                    
                    ]
                  
                  ⁢
                  
                  [
                  
                    
                      
                        
                          W
                          0
                        
                      
                      
                        
                          W
                          0
                        
                      
                      
                        
                          W
                          0
                        
                      
                      
                        
                          W
                          0
                        
                      
                    
                    
                      
                        
                          W
                          0
                        
                      
                      
                        
                          W
                          1
                        
                      
                      
                        
                          W
                          2
                        
                      
                      
                        
                          W
                          3
                        
                      
                    
                    
                      
                        
                          W
                          0
                        
                      
                      
                        
                          W
                          2
                        
                      
                      
                        
                          W
                          4
                        
                      
                      
                        
                          W
                          6
                        
                      
                    
                    
                      
                        
                          W
                          0
                        
                      
                      
                        
                          W
                          3
                        
                      
                      
                        
                          W
                          6
                        
                      
                      
                        
                          W
                          9
                        
                      
                    
                  
                  ]
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                        
                          0
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          0
                        
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          0
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          0
                        
                        
                          0
                        
                        
                          1
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          1
                        
                        
                          1
                        
                        
                          1
                        
                        
                          1
                        
                      
                      
                        
                          1
                        
                        
                          
                            -
                            j
                          
                        
                        
                          
                            -
                            1
                          
                        
                        
                          j
                        
                      
                      
                        
                          1
                        
                        
                          
                            -
                            1
                          
                        
                        
                          1
                        
                        
                          
                            -
                            1
                          
                        
                      
                      
                        
                          1
                        
                        
                          j
                        
                        
                          
                            -
                            1
                          
                        
                        
                          
                            -
                            j
                          
                        
                      
                    
                    ]
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  [
                  
                    
                      
                        1
                      
                      
                        1
                      
                      
                        1
                      
                      
                        1
                      
                    
                    
                      
                        1
                      
                      
                        
                          -
                          1
                        
                      
                      
                        1
                      
                      
                        
                          -
                          1
                        
                      
                    
                    
                      
                        1
                      
                      
                        
                          -
                          j
                        
                      
                      
                        
                          -
                          1
                        
                      
                      
                        j
                      
                    
                    
                      
                        1
                      
                      
                        j
                      
                      
                        
                          -
                          1
                        
                      
                      
                        
                          -
                          j
                        
                      
                    
                  
                  ]
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  [
                  
                    
                      
                        
                          F
                          2
                        
                      
                      
                        
                          F
                          2
                        
                      
                    
                    
                      
                        
                          E
                          2
                        
                      
                      
                        
                          -
                          
                            E
                            2
                          
                        
                      
                    
                  
                  ]
                
              
            
          
          
            
              
                
                  =
                  
                  ⁢
                  
                    
                      (
                      
                        
                          [
                          
                            
                              
                                
                                  I
                                  2
                                
                              
                              
                                
                                  I
                                  2
                                
                              
                            
                            
                              
                                
                                  I
                                  2
                                
                              
                              
                                
                                  -
                                  
                                    I
                                    2
                                  
                                
                              
                            
                          
                          ]
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  F
                                  2
                                
                              
                              
                                0
                              
                            
                            
                              
                                0
                              
                              
                                
                                  E
                                  2
                                
                              
                            
                          
                          ]
                        
                      
                      )
                    
                    T
                  
                
                ,
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                where
                ⁢
                
                  
                
                [
                E
                ]
              
              2
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    
                      -
                      j
                    
                  
                
                
                  
                    1
                  
                  
                    j
                  
                
              
              ]
            
          
          ,
        
      
    
    
      
        (
        50
        )
      
    
  




its inverse matrix is from element-wise inverse, such that





  
    
      
        
          
            [
            E
            ]
          
          2
          
            -
            1
          
        
        =
        
          
            [
            
              
                
                  1
                
                
                  1
                
              
              
                
                  j
                
                
                  
                    -
                    j
                  
                
              
            
            ]
          
          =
          
            
              
                (
                
                  [
                  
                    
                      
                        
                          1
                          /
                          1
                        
                      
                      
                        
                          
                            -
                            1
                          
                          /
                          j
                        
                      
                    
                    
                      
                        
                          1
                          /
                          1
                        
                      
                      
                        
                          1
                          /
                          j
                        
                      
                    
                  
                  ]
                
                )
              
              T
            
            .
          
        
      
    
    
      
        (
        51
        )
      
    
  




We can write that sparse matrix decompose similar fashion of Eq. (6) & Eq. (31),




  
    
      
        
          
            
              
                
                  [
                  
                    F
                    ~
                  
                  ]
                
                N
              
              =
              
              ⁢
              
                
                  
                    
                      [
                      Pr
                      ]
                    
                    N
                  
                  ⁡
                  
                    [
                    F
                    ]
                  
                
                N
              
            
          
        
        
          
            
              =
              
              ⁢
              
                [
                
                  
                    
                      
                        
                          F
                          ~
                        
                        
                          N
                          /
                          2
                        
                      
                    
                    
                      
                        
                          F
                          ~
                        
                        
                          N
                          /
                          2
                        
                      
                    
                  
                  
                    
                      
                        E
                        
                          N
                          /
                          2
                        
                      
                    
                    
                      
                        -
                        
                          E
                          
                            N
                            /
                            2
                          
                        
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  (
                  
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                        
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            
                              -
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                        
                      
                      ]
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              
                                F
                                ~
                              
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              E
                              
                                N
                                /
                                2
                              
                            
                          
                        
                      
                      ]
                    
                  
                  )
                
                T
              
            
          
        
      
    
    
      
        (
        52
        )
      
    
  




where [F]2=[{tilde over (F)}]2. The submatrix EN could be written by

[E]N=[Pr]N[{tilde over (F)}]N[W]N.  (53)

Therefore,




  
    
      
        [
        W
        ]
      
      N
    
    =
    
      [
      
        
          
            
              W
              0
            
          
          
            0
          
          
            ...
          
          
            0
          
        
        
          
            0
          
          
            
              W
              1
            
          
          
            
              
            
          
          
            0
          
        
        
          
            ⋮
          
          
            
              
            
          
          
            ⋱
          
          
            ⋮
          
        
        
          
            0
          
          
            ...
          
          
            0
          
          
            
              W
              
                N
                -
                1
              
            
          
        
      
      ]
    
  
  ,




and W is the diagonal complex unit for 2N point DFT matrix.

For example,




  
    
      [
      E
      ]
    
    2
  
  =
  
    [
    
      
        
          1
        
        
          
            -
            j
          
        
      
      
        
          1
        
        
          j
        
      
    
    ]
  




can be calculated by using





  
    
      
        
          
            
              
                
                  [
                  E
                  ]
                
                2
              
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          [
                          Pr
                          ]
                        
                        2
                      
                      ⁡
                      
                        [
                        
                          F
                          ~
                        
                        ]
                      
                    
                    2
                  
                  ⁡
                  
                    [
                    W
                    ]
                  
                
                2
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          1
                        
                        
                          1
                        
                      
                      
                        
                          1
                        
                        
                          
                            -
                            1
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          W
                          0
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          W
                          1
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        1
                      
                    
                    
                      
                        1
                      
                      
                        
                          -
                          1
                        
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          -
                          j
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        
                          -
                          j
                        
                      
                    
                    
                      
                        1
                      
                      
                        j
                      
                    
                  
                  ]
                
                .
              
            
          
        
      
    
    
      
        (
        54
        )
      
    
  




Thus, similar to Eq. (25), we write the DFT matrix as




  
    
      
        
          
            
              
                
                  [
                  
                    F
                    ~
                  
                  ]
                
                N
              
              =
              
              ⁢
              
                
                  
                    
                      [
                      
                        P
                        ⁢
                        
                          
                        
                        ⁢
                        r
                      
                      ]
                    
                    N
                  
                  ⁡
                  
                    [
                    F
                    ]
                  
                
                N
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  (
                  
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                        
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            
                              -
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                        
                      
                      ]
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              
                                F
                                ~
                              
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              E
                              
                                N
                                /
                                2
                              
                            
                          
                        
                      
                      ]
                    
                  
                  )
                
                T
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          
                            F
                            ~
                          
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          E
                          
                            N
                            /
                            2
                          
                        
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                    
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          -
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          
                            F
                            ~
                          
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          P
                          ⁢
                          
                            
                          
                          ⁢
                          
                            r
                            
                              N
                              /
                              2
                            
                          
                          ⁢
                          
                            
                              F
                              ~
                            
                            
                              N
                              /
                              2
                            
                          
                          ⁢
                          
                            W
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                    
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          -
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            P
                            ⁢
                            
                              
                            
                            ⁢
                            
                              r
                              
                                N
                                /
                                2
                              
                            
                          
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            
                              F
                              ~
                            
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            
                              F
                              ~
                            
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          W
                          
                            N
                            /
                            2
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                    
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          -
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                  
                  ]
                
                .
              
            
          
        
      
    
    
      
        (
        55
        )
      
    
  




Finally, based on the recursive 4×4 kernel F4 form, we have




  
    
      
        
          
            
              
                
                  [
                  F
                  ]
                
                N
              
              =
              
              ⁢
              
                
                  
                    
                      (
                      
                      ⁢
                      
                        
                          [
                          
                            P
                            ⁢
                            
                              
                            
                            ⁢
                            r
                          
                          ]
                        
                        N
                      
                      )
                    
                    
                      -
                      1
                    
                  
                  ⁡
                  
                    [
                    
                      F
                      ~
                    
                    ]
                  
                
                N
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          (
                          
                          ⁢
                          
                            
                              [
                              
                                P
                                ⁢
                                
                                  
                                
                                ⁢
                                r
                              
                              ]
                            
                            N
                          
                          )
                        
                        
                          -
                          1
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                P
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  r
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              
                                F
                                ~
                              
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              
                                F
                                ~
                              
                              
                                N
                                /
                                2
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            W
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                    
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          -
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          [
                          
                            P
                            ⁢
                            
                              
                            
                            ⁢
                            r
                          
                          ]
                        
                        N
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                P
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  r
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            I
                            2
                          
                          ⊗
                          P
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          r
                          
                            N
                            /
                            2
                          
                        
                      
                      ]
                    
                  
                  ⁢
                  
                  [
                  
                    
                      I
                      2
                    
                    ⊗
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                4
                              
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              P
                              ⁢
                              
                                
                              
                              ⁢
                              
                                r
                                
                                  N
                                  /
                                  4
                                
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ]
                
                ⁢
                
                  
                
                ⁢
                
                  
                    ...
                    ⁢
                    
                      
                    
                    [
                    
                      
                        
                          I
                          
                            N
                            /
                            4
                          
                        
                        ⊗
                        P
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        r
                        4
                      
                    
                    ]
                  
                  [
                  
                    
                  
                  ⁢
                  
                    
                      I
                      
                        N
                        /
                        4
                      
                    
                    ⊗
                    
                      [
                      
                        
                          
                            
                              I
                              2
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              P
                              ⁢
                              
                                
                              
                              ⁢
                              
                                r
                                2
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ]
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      [
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                        ⊗
                        
                          F
                          2
                        
                      
                      ]
                    
                    ⁡
                    
                      [
                      
                        
                          I
                          
                            N
                            /
                            4
                          
                        
                        ⊗
                        
                          [
                          
                            
                              
                                
                                  I
                                  2
                                
                              
                              
                                0
                              
                            
                            
                              
                                0
                              
                              
                                
                                  W
                                  2
                                
                              
                            
                          
                          ]
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        I
                        
                          N
                          /
                          4
                        
                      
                      ⊗
                      
                        [
                        
                          
                            
                              
                                I
                                2
                              
                            
                            
                              
                                I
                                2
                              
                            
                          
                          
                            
                              
                                I
                                2
                              
                            
                            
                              
                                -
                                
                                  I
                                  2
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ]
                  
                
                ⁢
                
                  
                
                ⁢
                
                  ...
                  ⁢
                  
                    
                  
                  [
                  
                    
                      I
                      2
                    
                    ⊗
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                4
                              
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              W
                              
                                N
                                /
                                4
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ]
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    [
                    
                      
                        I
                        2
                      
                      ⊗
                      
                      ⁢
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  4
                                
                              
                            
                            
                              
                                I
                                
                                  N
                                  /
                                  4
                                
                              
                            
                          
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  4
                                
                              
                            
                            
                              
                                -
                                
                                  I
                                  
                                    N
                                    /
                                    4
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            W
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                    
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          -
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    [
                    
                      P
                      ⁢
                      
                        
                      
                      ⁢
                      r
                    
                    ]
                  
                  N
                  
                    -
                    1
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          P
                          ⁢
                          
                            
                          
                          ⁢
                          
                            r
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              
              ⁢
              
                [
                
                  
                    I
                    2
                  
                  ⊗
                  
                    
                      
                        [
                        
                          
                        
                        ⁢
                        
                          ...
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            [
                                            
                                              P
                                              ⁢
                                              
                                                
                                              
                                              ⁢
                                              r
                                            
                                            ]
                                          
                                          4
                                          
                                            -
                                            1
                                          
                                        
                                        ⁡
                                        
                                          [
                                          
                                            
                                              
                                                
                                                  I
                                                  2
                                                
                                              
                                              
                                                0
                                              
                                            
                                            
                                              
                                                0
                                              
                                              
                                                
                                                  P
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  
                                                    r
                                                    2
                                                  
                                                
                                              
                                            
                                          
                                          ]
                                        
                                      
                                      ⁡
                                      
                                        [
                                        
                                          
                                            I
                                            2
                                          
                                          ⊗
                                          
                                            
                                              F
                                              ~
                                            
                                            2
                                          
                                        
                                        ]
                                      
                                    
                                    ⁡
                                    
                                      [
                                      
                                        
                                          
                                            
                                              I
                                              2
                                            
                                          
                                          
                                            0
                                          
                                        
                                        
                                          
                                            0
                                          
                                          
                                            
                                              W
                                              2
                                            
                                          
                                        
                                      
                                      ]
                                    
                                  
                                  ⁡
                                  
                                    [
                                    
                                      
                                        
                                          
                                            I
                                            2
                                          
                                        
                                        
                                          
                                            I
                                            2
                                          
                                        
                                      
                                      
                                        
                                          
                                            I
                                            2
                                          
                                        
                                        
                                          
                                            -
                                            
                                              I
                                              2
                                            
                                          
                                        
                                      
                                    
                                    ]
                                  
                                
                                ⁢
                                
                                  
                                
                              
                              }
                            
                            
                              
                                F
                                ~
                              
                              4
                            
                          
                          ⁢
                          
                            
                          
                          ⁢
                          ...
                        
                        ⁢
                        
                          
                        
                        ]
                      
                      }
                    
                    
                      
                        F
                        ~
                      
                      
                        N
                        /
                        2
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            W
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          
                            -
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                        
                      
                    
                    ]
                  
                
                .
              
            
          
        
      
    
    
      
        (
        56
        )
      
    
  




It is clear that the form of Eq. (56) is the same as that of Eq. (27), where we only need change Kl to Prl and Dl to Wl, with lε{2, 4, 8, . . . , N/2}. Consequently, the butterfly data flow of the DFT matrix could be drawn as FIG. 3A-3C.
FIG. 3A-3C is a butterfly data flow graph of Equation (56) according to an exemplary embodiment of the present invention. The data flow of Equation (56) is expressed as shown in FIG. 3A-3C.
V. Element-Wise Inverse Sparse Matrix Decomposition for HWT Transform
The discrete wavelet transform based on the Haar matrix (HWT), from N components of the signal to N wavelet coefficients, is expressed by a N×N matrix S. Here come the two directions: Synthesis in discrete time: x=Sb, Analysis in discrete time: y=Hx, where H=S−1=ST. As we know




  
    
      
        
          
            [
            H
            ]
          
          2
        
        =
        
        ⁢
        
          [
          
            
              
                r
              
              
                r
              
            
            
              
                r
              
              
                
                  -
                  r
                
              
            
          
          ]
        
      
    
  
  
    
      
        =
        
        ⁢
        
          
            [
            S
            ]
          
          2
        
      
    
  




and the scaled coefficient: r=1√{square root over (2)}. Its inverse matrix is from element-wise inverse, such that





  
    
      
        
          
            
              
                
                  
                    [
                    H
                    ]
                  
                  2
                
                =
                
                ⁢
                
                  [
                  
                    
                      
                        r
                      
                      
                        r
                      
                    
                    
                      
                        r
                      
                      
                        
                          -
                          r
                        
                      
                    
                  
                  ]
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    1
                    
                      2
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          1
                        
                        
                          1
                        
                      
                      
                        
                          1
                        
                        
                          
                            -
                            1
                          
                        
                      
                    
                    ]
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              and
              ⁢
              
                

              
              [
              H
              ]
            
            2
            
              -
              1
            
          
          =
          
            
              
                
                  1
                  2
                
                ⁡
                
                  [
                  
                    
                      
                        
                          1
                          ⁢
                          
                            /
                          
                          ⁢
                          r
                        
                      
                      
                        
                          1
                          ⁢
                          
                            /
                          
                          ⁢
                          r
                        
                      
                    
                    
                      
                        
                          1
                          ⁢
                          
                            /
                          
                          ⁢
                          r
                        
                      
                      
                        
                          
                            -
                            1
                          
                          ⁢
                          
                            /
                          
                          ⁢
                          r
                        
                      
                    
                  
                  ]
                
              
              T
            
            .
          
        
      
    
    
      
        (
        57
        )
      
    
  




Similar fashion as Eqs. (6, 31, 50), then we have




  
    
      
        
          
            
              
                
                  
                    
                      [
                      H
                      ]
                    
                    •
                  
                  2
                
                =
                
                ⁢
                
                  
                    
                      
                        
                          
                            [
                            
                              P
                              ⁢
                              
                                
                              
                              ⁢
                              i
                            
                            ]
                          
                          2
                        
                        ⁡
                        
                          [
                          H
                          ]
                        
                      
                      2
                    
                    ⁡
                    
                      [
                      
                        P
                        ⁢
                        
                          
                        
                        ⁢
                        j
                      
                      ]
                    
                  
                  2
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    
                      [
                      
                        
                          
                            1
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            1
                          
                        
                      
                      ]
                    
                    ⁢
                    
                    [
                    
                      
                        
                          r
                        
                        
                          r
                        
                      
                      
                        
                          r
                        
                        
                          
                            -
                            r
                          
                        
                      
                    
                    ]
                  
                  ⁢
                  
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          -
                          1
                        
                      
                    
                  
                  ]
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    [
                    
                      
                        
                          r
                        
                        
                          
                            -
                            r
                          
                        
                      
                      
                        
                          r
                        
                        
                          r
                        
                      
                    
                    ]
                  
                  .
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        Where
      
    
    
      
        (
        58
        )
      
    
  
  
    
      
        
          
            
              [
              
                P
                ⁢
                
                  
                
                ⁢
                i
              
              ]
            
            N
          
          =
          
            [
            
              
                
                  0
                
                
                  
                    I
                    
                      N
                      /
                      2
                    
                  
                
              
              
                
                  
                    I
                    
                      N
                      /
                      2
                    
                  
                
                
                  0
                
              
            
            ]
          
        
        ,
        
          

        
        ⁢
        
          
            
              [
              
                P
                ⁢
                
                  
                
                ⁢
                j
              
              ]
            
            N
          
          =
          
            
              
                
                  [
                  
                    P
                    ⁢
                    
                      
                    
                    ⁢
                    r
                  
                  ]
                
                N
              
              ⁡
              
                [
                
                  
                    
                      
                        I
                        
                          N
                          /
                          2
                        
                      
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        -
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                    
                  
                
                ]
              
            
            .
          
        
      
    
    
      
        (
        59
        )
      
    
  




Moreover, set




  
    
      
        
          
            
              
                
                  
                    [
                    
                      P
                      ⁢
                      
                        
                      
                      ⁢
                      a
                    
                    ]
                  
                  N
                
                =
                
                ⁢
                
                  
                    [
                    
                      P
                      ⁢
                      
                        
                      
                      ⁢
                      i
                    
                    ]
                  
                  N
                  
                    -
                    1
                  
                
              
            
          
          
            
              
                
                  =
                  
                  ⁢
                  
                    
                      [
                      
                        P
                        ⁢
                        
                          
                        
                        ⁢
                        i
                      
                      ]
                    
                    N
                  
                
                ,
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                
                  
                    [
                    
                      P
                      ⁢
                      
                        
                      
                      ⁢
                      b
                    
                    ]
                  
                  N
                
                =
                
                ⁢
                
                  
                    r
                    ⁡
                    
                      [
                      
                        P
                        ⁢
                        
                          
                        
                        ⁢
                        j
                      
                      ]
                    
                  
                  N
                  
                    -
                    1
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    
                      
                        r
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                              
                                0
                              
                            
                            
                              
                                0
                              
                              
                                
                                  -
                                  
                                    I
                                    
                                      N
                                      /
                                      2
                                    
                                  
                                
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          P
                          ⁢
                          
                            
                          
                          ⁢
                          r
                        
                        ]
                      
                    
                    N
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        60
        )
      
    
  




We can write a permuted 4×4 HWT matrix




  
    
      
        
          
            
              
                
                  
                    [
                    H
                    ]
                  
                  •
                
                4
              
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          [
                          
                            P
                            ⁢
                            
                              
                            
                            ⁢
                            i
                          
                          ]
                        
                        4
                      
                      ⁡
                      
                        [
                        H
                        ]
                      
                    
                    4
                  
                  ⁡
                  
                    [
                    
                      P
                      ⁢
                      
                        
                      
                      ⁢
                      j
                    
                    ]
                  
                
                4
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    [
                    
                      
                        
                          0
                        
                        
                          0
                        
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          0
                        
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          1
                        
                        
                          0
                        
                        
                          0
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          0
                        
                        
                          0
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            r
                            2
                          
                        
                        
                          
                            r
                            2
                          
                        
                        
                          
                            r
                            2
                          
                        
                        
                          
                            r
                            2
                          
                        
                      
                      
                        
                          
                            r
                            2
                          
                        
                        
                          
                            r
                            2
                          
                        
                        
                          
                            -
                            
                              r
                              2
                            
                          
                        
                        
                          
                            -
                            
                              r
                              2
                            
                          
                        
                      
                      
                        
                          r
                        
                        
                          
                            -
                            r
                          
                        
                        
                          0
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          0
                        
                        
                          r
                        
                        
                          
                            -
                            r
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        
                          -
                          1
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                      
                        
                          -
                          1
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                r
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          2
                        
                      
                      
                        
                          I
                          2
                        
                      
                    
                    
                      
                        
                          H
                          2
                        
                      
                      
                        
                          -
                          
                            H
                            2
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  r
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            2
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            H
                            2
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          2
                        
                      
                      
                        
                          I
                          2
                        
                      
                    
                    
                      
                        
                          I
                          2
                        
                      
                      
                        
                          -
                          
                            I
                            2
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  r
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            2
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            P
                            ⁢
                            
                              
                            
                            ⁢
                            
                              i
                              2
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              
                                H
                                •
                              
                              2
                            
                            ⁢
                            P
                            ⁢
                            
                              
                            
                            ⁢
                            
                              j
                              2
                              
                                -
                                1
                              
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          2
                        
                      
                      
                        
                          I
                          2
                        
                      
                    
                    
                      
                        
                          I
                          2
                        
                      
                      
                        
                          -
                          
                            I
                            2
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      
                        [
                        
                          
                            
                              
                                I
                                2
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                P
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  a
                                  2
                                
                              
                            
                          
                        
                        ]
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                I
                                2
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                
                                  H
                                  •
                                
                                2
                              
                            
                          
                        
                        ]
                      
                    
                    ⁢
                    
                    [
                    
                      
                        
                          
                            r
                            ⁢
                            
                              
                            
                            ⁢
                            
                              I
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            P
                            ⁢
                            
                              
                            
                            ⁢
                            
                              b
                              2
                            
                          
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            2
                          
                        
                        
                          
                            I
                            2
                          
                        
                      
                      
                        
                          
                            I
                            2
                          
                        
                        
                          
                            -
                            
                              I
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
                .
              
            
          
        
      
    
    
      
        (
        61
        )
      
    
  




For the 8-point HWT, the permuted 8×8 HWT is




  
    
      
        
          
            
              
                
                  
                    [
                    H
                    ]
                  
                  •
                
                8
              
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          [
                          
                            P
                            ⁢
                            
                              
                            
                            ⁢
                            i
                          
                          ]
                        
                        8
                      
                      ⁡
                      
                        [
                        H
                        ]
                      
                    
                    8
                  
                  ⁡
                  
                    [
                    
                      P
                      ⁢
                      
                        
                      
                      ⁢
                      j
                    
                    ]
                  
                
                8
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  r
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            4
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            H
                            4
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          4
                        
                      
                      
                        
                          I
                          4
                        
                      
                    
                    
                      
                        
                          I
                          4
                        
                      
                      
                        
                          -
                          
                            I
                            4
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  r
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            4
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            P
                            ⁢
                            
                              
                            
                            ⁢
                            
                              i
                              4
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              
                                H
                                •
                              
                              4
                            
                            ⁢
                            P
                            ⁢
                            
                              
                            
                            ⁢
                            
                              j
                              4
                              
                                -
                                1
                              
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          4
                        
                      
                      
                        
                          I
                          4
                        
                      
                    
                    
                      
                        
                          I
                          4
                        
                      
                      
                        
                          -
                          
                            I
                            4
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      
                        [
                        
                          
                            
                              
                                I
                                4
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                P
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  a
                                  4
                                
                              
                            
                          
                        
                        ]
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                I
                                6
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                P
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  a
                                  2
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              I
                              6
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              
                                H
                                •
                              
                              2
                            
                          
                        
                      
                      ]
                    
                  
                  ⁢
                  
                  [
                  
                    
                      
                        
                          I
                          4
                        
                      
                      
                        
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        
                          
                        
                      
                      
                        
                          r
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            2
                          
                        
                      
                      
                        
                          
                        
                      
                    
                    
                      
                        0
                      
                      
                        
                          
                        
                      
                      
                        
                          P
                          ⁢
                          
                            
                          
                          ⁢
                          
                            b
                            2
                          
                        
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        
                          r
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            4
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          P
                          ⁢
                          
                            
                          
                          ⁢
                          
                            b
                            4
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          I
                          4
                        
                      
                      
                        
                          I
                          4
                        
                      
                    
                    
                      
                        
                          I
                          4
                        
                      
                      
                        
                          -
                          
                            I
                            4
                          
                        
                      
                    
                  
                  ]
                
                .
              
            
          
        
      
    
    
      
        (
        62
        )
      
    
  




We can write that




  
    
      
        
          
            
              
                
                  
                    [
                    H
                    ]
                  
                  •
                
                N
              
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          [
                          
                            P
                            ⁢
                            
                              
                            
                            ⁢
                            i
                          
                          ]
                        
                        N
                      
                      ⁡
                      
                        [
                        H
                        ]
                      
                    
                    N
                  
                  ⁡
                  
                    [
                    
                      P
                      ⁢
                      
                        
                      
                      ⁢
                      j
                    
                    ]
                  
                
                N
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  r
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            H
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                    
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          -
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  r
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            P
                            ⁢
                            
                              
                            
                            ⁢
                            
                              i
                              
                                N
                                /
                                2
                              
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              
                                H
                                •
                              
                              
                                N
                                /
                                2
                              
                            
                            ⁢
                            P
                            ⁢
                            
                              
                            
                            ⁢
                            
                              j
                              
                                N
                                /
                                2
                              
                              
                                -
                                1
                              
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                    
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          -
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                P
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  a
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                            
                          
                        
                        ]
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                
                                  H
                                  •
                                
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁢
                    
                    [
                    
                      
                        
                          
                            r
                            ⁢
                            
                              
                            
                            ⁢
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            P
                            ⁢
                            
                              
                            
                            ⁢
                            
                              b
                              
                                N
                                /
                                2
                              
                            
                          
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          
                            -
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                        
                      
                    
                    ]
                  
                
                .
              
            
          
        
      
    
    
      
        (
        63
        )
      
    
  




The Eq. (63) of general recursive form for the HWT matrix can be rewritten as




  
    
      
        
          
            
              
                
                  [
                  H
                  ]
                
                N
              
              =
              
              ⁢
              
                
                  
                    [
                    
                      P
                      ⁢
                      
                        
                      
                      ⁢
                      i
                    
                    ]
                  
                  N
                  
                    -
                    1
                  
                
                ⁢
                
                  
                    
                      
                        
                          [
                          H
                          ]
                        
                        •
                      
                      N
                    
                    ⁡
                    
                      [
                      
                        P
                        ⁢
                        
                          
                        
                        ⁢
                        j
                      
                      ]
                    
                  
                  N
                  
                    -
                    1
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          1
                          r
                        
                        ⁡
                        
                          [
                          
                            P
                            ⁢
                            
                              
                            
                            ⁢
                            a
                          
                          ]
                        
                      
                      N
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              P
                              ⁢
                              
                                
                              
                              ⁢
                              
                                a
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              3
                              ⁢
                              
                                N
                                /
                                4
                              
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            P
                            ⁢
                            
                              
                            
                            ⁢
                            
                              a
                              
                                N
                                /
                                4
                              
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁢
                
                  
                
                ⁢
                
                  ...
                  ⁢
                  
                    
                  
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            -
                            2
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          P
                          ⁢
                          
                            
                          
                          ⁢
                          
                            a
                            2
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            -
                            2
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          
                            H
                            •
                          
                          
                            N
                            /
                            2
                          
                        
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            -
                            4
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        
                          
                        
                      
                      
                        
                          r
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            2
                          
                        
                      
                      
                        
                          
                        
                      
                    
                    
                      
                        0
                      
                      
                        
                          
                        
                      
                      
                        
                          P
                          ⁢
                          
                            
                          
                          ⁢
                          
                            b
                            2
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            -
                            4
                          
                        
                      
                      
                        0
                      
                      
                        
                          
                        
                      
                    
                    
                      
                        0
                      
                      
                        
                          I
                          2
                        
                      
                      
                        
                          I
                          2
                        
                      
                    
                    
                      
                        
                          
                        
                      
                      
                        
                          I
                          2
                        
                      
                      
                        
                          -
                          
                            I
                            2
                          
                        
                      
                    
                  
                  ]
                
                ⁢
                
                  
                
                ⁢
                
                  ...
                  ⁢
                  
                    
                  
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        
                          
                        
                      
                      
                        
                          r
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              N
                              /
                              4
                            
                          
                        
                      
                      
                        
                          
                        
                      
                    
                    
                      
                        0
                      
                      
                        
                          
                        
                      
                      
                        
                          P
                          ⁢
                          
                            
                          
                          ⁢
                          
                            b
                            
                              N
                              /
                              4
                            
                          
                        
                      
                    
                  
                  ]
                
              
              ⁢
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      
                        
                          [
                          
                            
                              
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                              
                                0
                              
                              
                                
                                  
                                
                              
                            
                            
                              
                                0
                              
                              
                                
                                  I
                                  
                                    N
                                    /
                                    4
                                  
                                
                              
                              
                                
                                  I
                                  
                                    N
                                    /
                                    4
                                  
                                
                              
                            
                            
                              
                                
                                  
                                
                              
                              
                                
                                  I
                                  
                                    N
                                    /
                                    4
                                  
                                
                              
                              
                                
                                  -
                                  
                                    I
                                    
                                      N
                                      /
                                      4
                                    
                                  
                                
                              
                            
                          
                          ]
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  r
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    I
                                    
                                      N
                                      /
                                      2
                                    
                                  
                                
                              
                              
                                0
                              
                            
                            
                              
                                0
                              
                              
                                
                                  P
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    b
                                    
                                      N
                                      /
                                      2
                                    
                                  
                                
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              
                                -
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        P
                        ⁢
                        
                          
                        
                        ⁢
                        b
                      
                      ]
                    
                  
                  N
                
                .
              
            
          
        
      
    
    
      
        (
        64
        )
      
    
  




Corresponding to Eq. (64), we can draw the butterfly data flow graph in FIG. 4.
FIG. 4 is a butterfly data flow graph of Equation (64) according to an exemplary embodiment of the present invention. The data flow of Equation (64) is represented as shown in FIG. 4.
VI. Signal Transformation Apparatus Applying Hybrid Architecture
We derive the recursive formulas for the DCT-II/DST-II/DFT and HWT. The results show that the DCT-II/DST-II/DFT and HWT matrices can be unified by using the same sparse matrix decomposition algorithm based on Jacket matrix, and recursive architecture with in some characters changed.
Clearly, the butterfly data flow graphs corresponding to the Eqs. (27), (45), (56) and (64) have the similar recursive flows. What's more, all of the four graphs have 2h(h=log2 N) steps, and we can derive in FIGS. 2A ̃2C, FIGS. 3A ̃3C and FIG. 4 from in FIG. 1A ̃1C by the following steps.
To get the equation of Eq. (45) is the same as that of Eq. (27), we only need to change [L]n to [U]n, with the parameters nε{2, 4, 8, . . . , N/2}, and multiply diagonal block-wise basis identity matrix as [M1] [M2] at the every step output DCT-II, and then we can get the DST-II matrix as we need. If we change from [L]n to [Pr]n and [D]n to [W]n and multiply the matrix [M3] during the first h steps, when the last h steps coming, multiply the [M4] matrix, then we can get the Eq. (56). For the Eq. (64), we can change the switching from [L]n to [Pa]n and [D]n to [Pb]n, after input the data, multiply [M5] at the last h steps. These diagonal block matrices are given Eqs. (65-70) as shown in FIG. 6A,




  
    
      
        
          
            
              [
              M
              ]
            
            N
          
          =
          
            
              
                [
                I
                ]
              
              
                N
                /
                
                  2
                  
                    h
                    +
                    1
                  
                
              
            
            ⊗
            
              [
              
                
                  
                    
                      I
                      
                        2
                        h
                      
                    
                  
                  
                    
                      I
                      
                        2
                        h
                      
                    
                  
                
                
                  
                    
                      I
                      
                        2
                        h
                      
                    
                  
                  
                    
                      -
                      
                        I
                        
                          2
                          h
                        
                      
                    
                  
                
              
              ]
            
          
        
        ,
      
    
    
      
        (
        65
        )
      
    
  
  
    
      
        
          
            
              [
              
                M
                1
              
              ]
            
            N
          
          =
          
            
              
                [
                I
                ]
              
              
                N
                /
                
                  2
                  
                    h
                    +
                    1
                  
                
              
            
            ⊗
            
              (
              
                
                  
                    [
                    I
                    ]
                  
                  
                    2
                    h
                  
                
                ⊗
                
                  
                    [
                    
                      I
                      _
                    
                    ]
                  
                  2
                
              
              )
            
          
        
        ,
        
          

        
        ⁢
        
          
            
              where
              ⁢
              
                

              
              [
              
                I
                _
              
              ]
            
            2
          
          =
          
            [
            
              
                
                  0
                
                
                  1
                
              
              
                
                  1
                
                
                  0
                
              
            
            ]
          
        
      
    
    
      
        (
        66
        )
      
    
  
  
    
      
        
          
            
              [
              
                M
                2
              
              ]
            
            N
          
          =
          
            
              
                [
                I
                ]
              
              
                N
                /
                
                  2
                  
                    h
                    +
                    1
                  
                
              
            
            ⊗
            
              (
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          -
                          1
                        
                      
                    
                  
                  ]
                
                ⊗
                
                  
                    [
                    I
                    ]
                  
                  
                    2
                    h
                  
                
              
              )
            
          
        
        ,
      
    
    
      
        (
        67
        )
      
    
  
  
    
      
        
          
            
              [
              
                M
                3
              
              ]
            
            N
          
          =
          
            
              
                
                  [
                  I
                  ]
                
                
                  
                    2
                    
                      h
                      -
                      1
                    
                  
                  -
                  1
                
              
              ⊗
              
                (
                
                  
                    
                      [
                      I
                      ]
                    
                    
                      N
                      /
                      
                        2
                        h
                      
                    
                  
                  ⊕
                  
                    
                      [
                      
                        P
                        ⁢
                        
                          
                        
                        ⁢
                        a
                      
                      ]
                    
                    
                      N
                      /
                      
                        2
                        h
                      
                    
                    
                      -
                      1
                    
                  
                
                )
              
            
            ⊕
            
              
                [
                I
                ]
              
              
                N
                /
                
                  2
                  
                    h
                    -
                    1
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        68
        )
      
    
  
  
    
      
        
          
            
              [
              
                M
                4
              
              ]
            
            N
          
          =
          
            
              
                [
                I
                ]
              
              
                N
                -
                
                  2
                  
                    h
                    +
                    1
                  
                
              
            
            ⊕
            
              
                r
                ⁡
                
                  [
                  I
                  ]
                
              
              
                2
                h
              
            
            ⊕
            
              
                [
                I
                ]
              
              
                2
                h
              
            
          
        
        ,
      
    
    
      
        (
        69
        )
      
    
  
  
    
      
        
          
            
              [
              
                M
                5
              
              ]
            
            N
          
          =
          
            
              
                
                  [
                  I
                  ]
                
                
                  
                    N
                    /
                    
                      2
                      
                        h
                        +
                        1
                      
                    
                  
                  -
                  1
                
              
              ⊗
              
                [
                
                  
                    
                      
                        I
                        
                          2
                          h
                        
                      
                    
                    
                      
                        -
                        
                          I
                          
                            2
                            h
                          
                        
                      
                    
                  
                  
                    
                      
                        I
                        
                          2
                          h
                        
                      
                    
                    
                      
                        I
                        
                          2
                          h
                        
                      
                    
                  
                
                ]
              
            
            ⊕
            
              
                [
                I
                ]
              
              
                2
                
                  h
                  +
                  1
                
              
            
          
        
        ,
      
    
    
      
        (
        70
        )
      
    
  




where  is the Kronecker and ⊕ is the direct sum. For example, we calculate the dimension of Eq. (68)

(2h−1−1)×(N/2h+N/2h)+N/2h−1=(2h−1−1)×N/2h−1+N/2h−1=N−N/2h−1+N/2h−1=N. 

As an illustrated in FIG. 1A ̃1C, FIG. 2A ̃2C, FIG. 3A ̃3C and FIG. 4 we find that the DST-II computation can be from the computation of DCT-II by replacing the submatrix [L]N to [U]N. As to the DFT, it computation can be from the computation of the DCT-II matrix by replacing the submatrix [D]N to [W]N, and the permutation matrix [L]N to [Pr]N as to the HWT, we not only need to replace submatrix [D]N by [Pb]N, and the permutation matrix [L]N by [Pa]N, but also multiply some special matrices at the output of HWT. As a result, a simple generalized block diagram for the DCT-II/DST-II/DFT/HWT hybrid fast algorithms can be shown in FIG. 6A.
FIGS. 6B and 6C shows variance comparison of the KLT/DCT/DST/DFT/Haar. The overall scheme switches between the directions as horizontal, vertical or bounding in intra mode depending on the prediction. The compare of computational complexity of conventional independent the DCT-II, DST-II, DFT, Haar transform and hybrid DCT-II/DST-II/DFT/HW are shown in Table I & II. The computational complexity of the proposed hybrid algorithms are slower than the conventional ones, due to the hybrid architecture algorithm.
But the proposed hybrid algorithms are much more easily to extend because of the diagonal block-wise inverse operation. Moreover, it is really extensible to larger transform sizes based on 4×4 kernel such as □4, S4, {tilde over (F)}4 and {tilde over (H)}4. These diagonal block-wise are well suitable to without ISI OFDM, diagonal block zero forcing singular value decomposition (SVD) for multiuser MIMO and subspace diagonal channel interference alignment as shown in FIG. 6D.
As a result, the signal transformation apparatus applying the hybrid architecture of FIGS. 5, 6A, and 6B can be provided, which is explained now.
FIG. 5 is a block diagram of a signal transformation apparatus 500 adopting the hybrid architecture according to an exemplary embodiment of the present invention. FIGS. 6A and 6B are a detailed block diagram of the signal transformation apparatus 500 according to an exemplary embodiment of the present invention. The signal transformation apparatus 500 of FIG. 5 adopts the hybrid architecture for the DCT-II, the DST-II, the DFT, and the HWT.
In detail, the signal transformation apparatus 500 includes a switching part 510 and a transformation part 520. The switching part 510 selects one of the DCT-II, the DST-II, the DFT, and the HWT. The transformation part 520 transforms and outputs the input signal using the transform selected by the switching part 510.
As shown in FIG. 6A, the switching part 510 includes a submatrix selector 518 and a permutation matrix selector 519. The transformation part 520 includes a first multiply 521, a second multiply 522, a third multiply 523, a fourth multiply, and a fifth multiply 525.
In detail, the submatrix selector 518 selects the submatrix computation for any one of the DCT-II, the DST-II, the DFT, and the HWT. The permutation matrix selector 519 selects the permutation matrix computation for any one of the DCT-II, the DST-II, the DFT, and the HWT.
First, when the DCT-II transform is selected by the switching part 510, the submatrix selector 518 selects [L]N 511 as the submatrix and the permutation matrix selector 519 selects [D]N 515 as the permutation matrix.
The transformation part 520 computes Equation (18) using the submatrix [L]N and the permutation matrix [D]N and thus applies the DCT-II to the input signal. More specifically, the first multiply 521 of the transformation part 520 multiplies the input signal by the submatrix [L]N and the permutation matrix [D]N. Next, the second multiply 522 of the transformation part 520 multiplies the result by the matrix M. The transformation part 520 can transform the input signal using the DCT-II for the MPEG-4 HEVC.
Secondly, when the switching part 510 selects the DST-II transform, the submatrix selector 518 selects [U]N 512 as the submatrix and the permutation matrix selector 519 selects [D]N 515 as the permutation matrix.
The transformation part 520 computes Equation (34) by using the submatrix [U]N and the permutation matrix [D]N and multiplying the matrix [M1] [M2] in each output, and thus applies the DST-II to the input signal. More specifically, the first multiply 521 of the transformation part 520 multiplies the input signal by the submatrix [U]N and the permutation matrix [D]N. Next, the second multiply 522 of the transformation part 520 multiplies the product by the matrix M. The third multiply 523 multiplies the matrix [M1] [M2] in each output of the previous product. Next, the transformation part 520 can generate the HEVC signal by applying the DST-II to the input signal.
Thirdly, when the switching part 510 selects the DFT transform, the submatrix selector 518 selects [Pr]N 513 as the submatrix and the permutation matrix selector 519 selects [W]n 516 as the permutation matrix.
The transformation part 520 computes Equation (45) by using the submatrix [Pr]N and the permutation matrix [W]n, multiplying the input signal by the matrix [M3] during the first h steps of the 2h steps, and multiplying the matrix [M4] during the last h steps, and thus DFT-transforms the input signal. In more detail, the first multiply 521 of the transformation part 520 multiplies the input signal by the submatrix [Pr]N and the permutation matrix [W]n. Next, the second multiply 522 of the transformation part 520 multiplies the product by the matrix M. The fourth multiply 524 multiplies the input signal by the matrix [M3] during the first h steps of the 2h steps and multiplies the matrix [M4] during the last h steps. Thus, the transformation part 520 can transform the input signal to the 3GPP, LTE, and DVB signal through the DFT.
Fourthly, when the switching part 510 selects the HWT transform, the submatrix selector 518 selects [Pa]N 514 as the submatrix and the permutation matrix selector 519 selects [Pb]N 517 as the permutation matrix.
The transformation part 520 computes Equation (53) by using the submatrix [Pa]N and the permutation matrix [Pb]N and multiplying the input signal by the matrix [M5] during the last h steps of the 2h steps, and thus HWT-transforms the input signal. In more detail, the first multiply 521 of the transformation part 520 multiplies the input signal by the submatrix [Pa]N and the permutation matrix [Pb]N. Next, the second multiply 522 of the transformation part 520 multiplies the product by the matrix M. The fifth multiply 525 multiplies the input signal by the matrix [M5] during the last h steps of the 2h steps. Thus, the transformation part 520 can transform the input signal to the JPEG-2000 signal through the HWT.
As such, the signal transformation apparatus 500 can selectively perform any one of the DCT-II, the DST-II, the DFT and the HWT by applying the hybrid architecture. Also, the signal transformation apparatus 500 can attain the hybrid architecture of low complexity by jointly applying the recursive structure of the DCT-II, the DST-II, the DFT, and the HWT. The present hybrid architecture exhibits the lower computational complexity than the direct scheme as shown in the following table.














Conventional methods
Proposed












Addition
Multiplication
Addition
Multiplication














DCT-II
Chen et al.
N (3log2 N + N − 1)/4
(N2/32 + 2N)log2 N − N2/16 − N/2












 3/2N (log2 N − 1) + 2
N log2 N − 3/2N + 4












DST-II
Z. Wang
N (3log2 N + N − 1)/4
(N2/32 + 2N)log2 N − N2/16 − N/2












 3/2N (log2 N − 1) + 2
N log2 N − 3/2N + 4












DFT
Cooley & Tukey
N log2 N
N (log2 N − 1/2)












N log2 N
1/2N (log2N)












HWT
Andrews & Caspari
log2 N (2 + N/2) − N/2
(3 + 3N/4)log2 N + 1 + N












2(N − 1)
N










Now, the signal transformation apparatus 500 applied as a video encoder and a video encoder is explained by referring to FIGS. 7A and 7B. FIG. 7A is a diagram of a one-way video coding scenario according to an exemplary embodiment of the present invention.
As shown in FIG. 7A, the video source encodes the video signal and broadcasts the signal to the TV or transmits the signal to the PC or the mobile display device via the server or the network. The video source may compress and provide the video as a DVD. In so doing, the video source encodes the signal using one of the DCT-II, the DST-II, the DFT, and the HWT by use of the signal transformation apparatus 500. The TV, the PC, and the mobile display device decode the signal using one of the DCT-II, the DST-II, the DFT, and the HWT with the signal transformation apparatus 500.
As such, the signal transformation apparatus 500 can be applied to the one-way video coding scenario.
FIG. 7B is a diagram of a two-way video coding scenario according to an exemplary embodiment of the present invention.
As shown in FIG. 7B, for video chat using the PC, the PC encodes the video signal captured by the camera and transmits the encoded video signal to the other PC. The PC receives and decodes the video signal of the other PC, and then displays the video on the display device. In so doing, the PC performs the encoding and the decoding using one of the DCT-II, the DST-II, the DFT, and the HWT with the signal transformation apparatus 500.
As such, the signal transformation apparatus 500 can be applied to the two-way video coding scenario.
Meanwhile, the present invention can be applied to a signal transformation method of the signal transformation apparatus 500. Also, the present invention can be applied to a computer-readable recording medium containing a computer program for executing the functions of the signal transformation apparatus 500. The present invention according to various embodiments can be embodied as a computer-readable code recorded to the computer-readable recording medium. The computer-readable recording medium can employ any data storage device for reading and storing data through the computer. For example, the computer-readable recording medium can include ROM, RAM, CD-ROM, magnetic tape, floppy disc, optical disc, hard disc drive, and so on. A computer-readable code or program stored to the computer-readable recording medium may be transmitted over the network connected between computers.
As set forth above, with the signal transformation apparatus, the signal transformation method, and the recording medium for select any one of the DCT-II, the DST-II, the DFT, and the HWT and transforming and outputting the input signal according to the selected transform, the hybrid architecture of the low complexity can be attained.
While the invention has been shown and described with reference to certain exemplary embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims and their equivalents.

## Claims

1. A signal transformation apparatus applying a hybrid architecture for Discrete Cosine Transform (DCT)-II, Discrete Sine Transform (DST)-II, Discrete Fourier Transform (DFT), and Haar Wavelet Transform (HWT), comprising:
a switching part for selecting any one of the DCT-II, the DST-II, the DFT, and the HWT; and
a transformation part for transforming and outputting an input signal according to the transform selected by the switching part,
wherein the switching part comprises:
a submatrix selector for selecting submatrix computation for any one of the DCT-II, the DST-II, the DFT, and the HWT; and
a permutation matrix selector for selecting permutation matrix computation for any one of the DCT-II, the DST-II, the DFT, and the HWT,
wherein when the switching part selects the DCT-II, the submatrix selector selects [L]N as a submatrix and the permutation matrix selector selects [D]N as a permutation matrix,
wherein the transformation part applies the DCT-II to the input signal by computing Equation (27) using the submatrix [L]N and the permutation matrix [D]N;





  
    
      
        
          
            [
            C
            ]
          
          N
        
        =
        
          
            
              
                
                  
                    
                      
                        2
                        N
                      
                    
                    ⁡
                    
                      [
                      
                        P
                        ⁢
                        
                          
                        
                        ⁢
                        r
                      
                      ]
                    
                  
                  N
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          L
                          
                            N
                            /
                            2
                          
                        
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      I
                      2
                    
                    ⊗
                    P
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    r
                    
                      N
                      /
                      2
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  I
                  2
                
                ⊗
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            /
                            4
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          L
                          
                            N
                            /
                            4
                          
                        
                      
                    
                  
                  ]
                
              
              ]
            
          
          ⁢
          
            
          
          ⁢
          
             
            
              
                
                  [
                  
                    
                      
                        I
                        4
                      
                      ⊗
                      P
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      r
                      
                        N
                        /
                        4
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      I
                      4
                    
                    ⊗
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                8
                              
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              L
                              
                                N
                                /
                                8
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ]
                
              
              ⁢
              
                
              
              ⁢
              
                ...
                ⁢
                
                  
                
                [
                
                  
                    
                      I
                      
                        N
                        /
                        4
                      
                    
                    ⊗
                    P
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    r
                    4
                  
                
                ]
              
            
             
          
          ⁢
          
             
            
              
                 
                
                  
                    
                      [
                      
                        
                          I
                          
                            N
                            /
                            4
                          
                        
                        ⊗
                        
                          [
                          
                            
                              
                                
                                  I
                                  2
                                
                              
                              
                                0
                              
                            
                            
                              
                                0
                              
                              
                                
                                  L
                                  2
                                
                              
                            
                          
                          ]
                        
                      
                      ]
                    
                    ⁡
                    
                      [
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                        ⊗
                        
                          C
                          2
                        
                      
                      ]
                    
                  
                  [
                  
                    
                      I
                      
                        N
                        /
                        4
                      
                    
                    ⊗
                    
                      [
                      
                        
                          
                            
                              I
                              2
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              D
                              2
                            
                          
                        
                      
                      ]
                    
                  
                   
                
                ]
              
              ⁢
              
                 
                
                  
                    
                      [
                      
                        
                          I
                          
                            N
                            /
                            4
                          
                        
                        ⊗
                        
                          [
                          
                            
                              
                                
                                  I
                                  2
                                
                              
                              
                                
                                  I
                                  2
                                
                              
                            
                            
                              
                                
                                  I
                                  2
                                
                              
                              
                                
                                  -
                                  
                                    I
                                    2
                                  
                                
                              
                            
                          
                          ]
                        
                      
                      ]
                    
                    ⁡
                    
                      [
                      
                        
                          
                            I
                            
                              N
                              /
                              4
                            
                          
                          ⊗
                          P
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          c
                          4
                        
                      
                      ]
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ...
                    ⁢
                    
                      
                    
                    [
                    
                      
                        I
                        4
                      
                      ⊗
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  8
                                
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                D
                                
                                  N
                                  /
                                  8
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ]
                  
                
                 
              
              ⁢
              
                 
                
                  
                    
                      
                        [
                        
                          
                            I
                            4
                          
                          ⊗
                          
                            [
                            
                              
                                
                                  
                                    I
                                    
                                      N
                                      /
                                      8
                                    
                                  
                                
                                
                                  
                                    I
                                    
                                      N
                                      /
                                      8
                                    
                                  
                                
                              
                              
                                
                                  
                                    I
                                    
                                      N
                                      /
                                      8
                                    
                                  
                                
                                
                                  
                                    -
                                    
                                      I
                                      
                                        N
                                        /
                                        8
                                      
                                    
                                  
                                
                              
                            
                            ]
                          
                        
                        ]
                      
                      ⁡
                      
                        [
                        
                          
                            
                              I
                              4
                            
                            ⊗
                            P
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            c
                            
                              N
                              /
                              4
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          I
                          2
                        
                        ⊗
                        
                          [
                          
                            
                              
                                
                                  I
                                  
                                    N
                                    /
                                    4
                                  
                                
                              
                              
                                0
                              
                            
                            
                              
                                0
                              
                              
                                
                                  D
                                  
                                    N
                                    /
                                    4
                                  
                                
                              
                            
                          
                          ]
                        
                      
                      ]
                    
                  
                  ⁢
                  
                     
                    
                      
                        
                          
                            [
                            
                              
                                I
                                4
                              
                              ⊗
                              
                                [
                                
                                  
                                    
                                      
                                        I
                                        
                                          N
                                          /
                                          4
                                        
                                      
                                    
                                    
                                      
                                        I
                                        
                                          N
                                          /
                                          4
                                        
                                      
                                    
                                  
                                  
                                    
                                      
                                        I
                                        
                                          N
                                          /
                                          4
                                        
                                      
                                    
                                    
                                      
                                        -
                                        
                                          I
                                          
                                            N
                                            /
                                            4
                                          
                                        
                                      
                                    
                                  
                                
                                ]
                              
                            
                            ]
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  I
                                  2
                                
                                ⊗
                                P
                              
                              ⁢
                              
                                
                              
                              ⁢
                              
                                c
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            ]
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                              
                                0
                              
                            
                            
                              
                                0
                              
                              
                                
                                  D
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                            
                          
                          ]
                        
                      
                      ⁢
                      
                         
                        
                          
                            
                              
                                [
                                
                                  
                                    
                                      
                                        I
                                        
                                          N
                                          /
                                          2
                                        
                                      
                                    
                                    
                                      
                                        I
                                        
                                          N
                                          /
                                          2
                                        
                                      
                                    
                                  
                                  
                                    
                                      
                                        I
                                        
                                          N
                                          /
                                          2
                                        
                                      
                                    
                                    
                                      
                                        -
                                        
                                          I
                                          
                                            N
                                            /
                                            2
                                          
                                        
                                      
                                    
                                  
                                
                                ]
                              
                              ⁡
                              
                                [
                                
                                  P
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  c
                                
                                ]
                              
                            
                            N
                          
                          ,
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        27
        )
      
    
  




wherein the submatrix [L]N and the permutation matrix [D]N are defined as



  
    
      
        
          [
          L
          ]
        
        N
      
      =
      
        [
        
          
            
              
                2
              
            
            
              0
            
            
              0
            
            
              ⋯
            
          
          
            
              
                -
                
                  2
                
              
            
            
              2
            
            
              0
            
            
              ⋯
            
          
          
            
              
                2
              
            
            
              
                -
                2
              
            
            
              2
            
            
              ⋯
            
          
          
            
              ⋮
            
            
              ⋮
            
            
              
                
              
            
            
              ⋱
            
          
        
        ]
      
    
    ,
    
      
        
          [
          D
          ]
        
        N
      
      =
      
        [
        
          
            
              
                C
                
                  4
                  ⁢
                  
                    
                  
                  ⁢
                  N
                
                
                  Φ
                  0
                
              
            
            
              0
            
            
              ⋯
            
            
              0
            
          
          
            
              0
            
            
              
                C
                
                  4
                  ⁢
                  N
                
                Φ
              
            
            
              
                
              
            
            
              ⋮
            
          
          
            
              ⋮
            
            
              
                
              
            
            
              ⋱
            
            
              0
            
          
          
            
              0
            
            
              ⋯
            
            
              0
            
            
              
                C
                
                  4
                  ⁢
                  
                    
                  
                  ⁢
                  N
                
                
                  Φ
                  
                    N
                    -
                    1
                  
                
              
            
          
        
        ]
      
    
    ,
    
      
    
    ⁢
    and
  




  
    
      
        Φ
        i
      
      =
      
        
          2
          ⁢
          i
        
        +
        1
      
    
    ,
    
      i
      ∈
      
        {
        
          0
          ,
          1
          ,
          2
          ,
          ...
          ⁢
          
            
          
          ,
          
            N
            -
            1
          
        
        }
      
    
  




  
    
      
        
          and
          ⁢
          
            

          
          [
          Pr
          ]
        
        2
      
      =
      
        
          
            
              [
              I
              ]
            
            2
          
          ⁢
          
            
          
          ⁢
          
            
              and
              ⁢
              
                
              
              [
              Pr
              ]
            
            N
          
        
        =
        
          [
          
            
              
                1
              
              
                0
              
              
                0
              
              
                ...
              
              
                0
              
              
                0
              
              
                ⋯
              
              
                0
              
            
            
              
                0
              
              
                0
              
              
                0
              
              
                ⋯
              
              
                1
              
              
                0
              
              
                ⋯
              
              
                0
              
            
            
              
                0
              
              
                1
              
              
                0
              
              
                
                  
                
              
              
                0
              
              
                0
              
              
                ⋯
              
              
                0
              
            
            
              
                0
              
              
                0
              
              
                0
              
              
                
                  
                
              
              
                
                  
                    
                  
                  ⁢
                  0
                
              
              
                1
              
              
                
                  
                
              
              
                0
              
            
            
              
                0
              
              
                0
              
              
                1
              
              
                
                  
                
              
              
                0
              
              
                0
              
              
                
                  
                
              
              
                0
              
            
            
              
                0
              
              
                0
              
              
                0
              
              
                
                  
                
              
              
                0
              
              
                0
              
              
                ⋱
              
              
                0
              
            
            
              
                0
              
              
                0
              
              
                0
              
              
                ⋱
              
              
                0
              
              
                0
              
              
                
                  
                
              
              
                0
              
            
            
              
                0
              
              
                0
              
              
                0
              
              
                0
              
              
                0
              
              
                0
              
              
                
                  
                
              
              
                1
              
            
          
          ]
        
      
    
    ,
    
      
    
    ⁢
    
      N
      ≥
      4
    
    ,
    
      
    
    ⁢
    
      
        
          where
          ⁢
          
            

          
          [
          Pr
          ]
        
        N
      
      =
      
        
          [
          
            pr
            
              i
              ,
              j
            
          
          ]
        
        N
      
    
    ,
    
      

    
    ⁢
    
      with
      ⁢
      
        
      
      ⁢
      
        {
        
          
            
              
                
                  
                    
                      
                        pr
                        
                          i
                          ,
                          j
                        
                      
                      =
                      1
                    
                    ,
                  
                
                
                  
                    
                      
                        if
                        ⁢
                        
                          
                        
                        ⁢
                        i
                      
                      =
                      
                        2
                        ⁢
                        j
                      
                    
                    ,
                    
                      
                    
                    ⁢
                    
                      0
                      ≤
                      
                        
                      
                      ⁢
                      j
                      ≤
                      
                        
                          N
                          2
                        
                        -
                        1
                      
                    
                    ,
                  
                
              
              
                
                  
                    
                      
                        pr
                        
                          i
                          ,
                          j
                        
                      
                      =
                      1
                    
                    ,
                  
                
                
                  
                    
                      
                        if
                        ⁢
                        
                          
                        
                        ⁢
                        i
                      
                      =
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              j
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        mod
                        ⁢
                        
                          
                        
                        ⁢
                        N
                      
                    
                    ,
                    
                      
                        N
                        2
                      
                      ≤
                      j
                      ≤
                      
                        N
                        -
                        1
                      
                    
                    ,
                  
                
              
              
                
                  
                    
                      
                        pr
                        
                          i
                          ,
                          j
                        
                      
                      =
                      0
                    
                    ,
                  
                
                
                  
                    others
                    ,
                  
                
              
            
            ⁢
            where
            ⁢
            
              
            
            ⁢
            
              

            
            ⁢
            
              
            
            ⁢
            i
          
          ,
          
            
              j
              ∈
              
                
                  {
                  
                    0
                    ,
                    1
                    ,
                    ...
                    ⁢
                    
                      
                    
                    ,
                    
                      N
                      -
                      1
                    
                  
                  }
                
                ⁢
                
                  

                
                ⁢
                
                  
                    and
                    ⁢
                    
                      

                    
                    [
                    Pc
                    ]
                  
                  2
                
              
            
            =
            
              
                [
                I
                ]
              
              2
            
          
          ,
          
            
          
          ⁢
          
            
              
                and
                ⁢
                
                  
                
                [
                Pc
                ]
              
              N
            
            =
            
              [
              
                
                  
                    
                      I
                      
                        N
                        /
                        4
                      
                    
                  
                  
                    0
                  
                  
                    0
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    
                      I
                      
                        N
                        /
                        4
                      
                    
                  
                  
                    0
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    0
                  
                  
                    
                      I
                      
                        N
                        /
                        4
                      
                    
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    
                      I
                      
                        N
                        /
                        4
                      
                    
                  
                  
                    0
                  
                
              
              ]
            
          
          ,
          
            N
            ≥
            4.

2. A signal transformation method applied to a signal transformation apparatus as claimed in claim 1.

3. A non-transitory computer-readable recording medium containing a computer program for executing functions of a signal transformation apparatus as claimed in claim 1.

4. A signal transformation apparatus applying a hybrid architecture for Discrete Cosine Transform (DCT)-II, Discrete Sine Transform (DST)-II, Discrete Fourier Transform (DFT), and Haar Wavelet Transform (HWT), comprising:
a switching part for selecting any one of the DCT-II, the DST-II, the DFT, and the HWT; and
a transformation part for transforming and outputting an input signal according to the transform selected by the switching part,
wherein the switching part comprises:
a submatrix selector for selecting submatrix computation for any one of the DCT-II, the DST-II, the DFT, and the HWT; and
a permutation matrix selector for selecting permutation matrix computation for any one of the DCT-II, the DST-II, the DFT, and the HWT,
wherein when the switching part selects the DST-II, the submatrix selector selects matrix [U]N as a submatrix and the permutation matrix selector selects [D]N as a permutation matrix,
wherein the transformation part applies the DST-II to the input signal by computing Equation (45) by using the submatrix [U]N and the permutation matrix [D]N and multiplying every output by a matrix:





  
    
      
        
          
            
              
                
                  [
                  S
                  ]
                
                N
              
              =
              
              ⁢
              
                
                  
                    
                      2
                      N
                    
                  
                  ⁡
                  
                    [
                    S
                    ]
                  
                
                N
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      
                        2
                        N
                      
                    
                    ⁡
                    
                      [
                      
                        P
                        ⁢
                        
                          
                        
                        ⁢
                        r
                      
                      ]
                    
                  
                  N
                
                ⁡
                
                  [
                  
                    
                      
                        
                          U
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              
              ⁢
              
                [
                
                  
                    I
                    2
                  
                  ⊗
                  
                    
                      
                        [
                        
                          
                        
                        ⁢
                        
                          
                            ...
                            ⁢
                            
                              
                            
                            [
                            
                              
                                I
                                2
                              
                              ⊗
                              
                                
                                  
                                    [
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    [
                                                    
                                                      P
                                                      ⁢
                                                      
                                                        
                                                      
                                                      ⁢
                                                      r
                                                    
                                                    ]
                                                  
                                                  4
                                                
                                                ⁡
                                                
                                                  [
                                                  
                                                    
                                                      
                                                        
                                                          U
                                                          2
                                                        
                                                      
                                                      
                                                        0
                                                      
                                                    
                                                    
                                                      
                                                        0
                                                      
                                                      
                                                        
                                                          I
                                                          2
                                                        
                                                      
                                                    
                                                  
                                                  ]
                                                
                                              
                                              ⁡
                                              
                                                [
                                                
                                                  
                                                    I
                                                    2
                                                  
                                                  ⊗
                                                  
                                                    S
                                                    2
                                                  
                                                
                                                ]
                                              
                                            
                                            ⁡
                                            
                                              [
                                              
                                                
                                                  
                                                    
                                                      D
                                                      2
                                                    
                                                  
                                                  
                                                    0
                                                  
                                                
                                                
                                                  
                                                    0
                                                  
                                                  
                                                    
                                                      I
                                                      2
                                                    
                                                  
                                                
                                              
                                              ]
                                            
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                
                                                  
                                                    I
                                                    2
                                                  
                                                
                                                
                                                  
                                                    I
                                                    2
                                                  
                                                
                                              
                                              
                                                
                                                  
                                                    I
                                                    2
                                                  
                                                
                                                
                                                  
                                                    -
                                                    
                                                      I
                                                      2
                                                    
                                                  
                                                
                                              
                                            
                                            ]
                                          
                                        
                                        ⁡
                                        
                                          [
                                          
                                            P
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            c
                                          
                                          ]
                                        
                                      
                                      4
                                    
                                    ]
                                  
                                  }
                                
                                
                                  S
                                  4
                                
                              
                            
                            ]
                          
                          ⁢
                          
                            
                          
                          ⁢
                          ...
                        
                        ⁢
                        
                          
                        
                        ]
                      
                      }
                    
                    
                      S
                      
                        N
                        /
                        2
                      
                    
                  
                
                ]
              
               
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      
                        [
                        
                          
                            
                              
                                D
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                        
                        ]
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              
                                -
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        P
                        ⁢
                        
                          
                        
                        ⁢
                        c
                      
                      ]
                    
                  
                  
                    N
                    *
                  
                
                ,
              
            
          
        
      
    
    
      
        (
        45
        )
      
    
  



wherein




  
    
      
        
          [
          U
          ]
        
        N
      
      =
      
        
          [
          
            
              
                2
              
              
                
                  -
                  2
                
              
              
                2
              
              
                
                  -
                  
                    2
                  
                
              
            
            
              
                0
              
              
                2
              
              
                
                  -
                  2
                
              
              
                
                  2
                
              
            
            
              
                ⋮
              
              
                ⋱
              
              
                2
              
              
                ⋮
              
            
            
              
                0
              
              
                ⋯
              
              
                0
              
              
                
                  2
                
              
            
          
          ]
        
        N
      
    
    ,
    
      
        
          [
          D
          ]
        
        N
      
      =
      
        
          diag
          [
          
            
              C
              
                4
                ⁢
                
                  
                
                ⁢
                N
              
              
                Φ
                0
              
            
            ,
            ...
            ⁢
            
              
            
            ,
            
              C
              
                4
                ⁢
                
                  
                
                ⁢
                N
              
              
                Φ
                
                  N
                  -
                  1
                
              
            
          
          ]
        
        N
      
    
  




  
    
      
        Φ
        i
      
      =
      
        
          2
          ⁢
          
            
          
          ⁢
          i
        
        +
        1
      
    
    ,
    
      i
      =
      0
    
    ,
    1
    ,
    ⋯
    ⁢
    
      
    
    ,
    
      
        N
        -
        
          1
          ⁢
          
            
          
          ⁢
          
            

          
          ⁢
          
            
              and
              ⁢
              
                

              
              [
              Pr
              ]
            
            2
          
        
      
      =
      
        
          
            
              [
              I
              ]
            
            2
          
          ⁢
          
            
          
          ⁢
          
            
              and
              ⁢
              
                
              
              [
              Pr
              ]
            
            N
          
        
        =
        
          [
          
            
              
                1
              
              
                0
              
              
                0
              
              
                ⋯
              
              
                0
              
              
                0
              
              
                ⋯
              
              
                0
              
            
            
              
                0
              
              
                0
              
              
                0
              
              
                ⋯
              
              
                1
              
              
                0
              
              
                ⋯
              
              
                0
              
            
            
              
                0
              
              
                1
              
              
                0
              
              
                
                  
                
              
              
                0
              
              
                0
              
              
                ⋯
              
              
                0
              
            
            
              
                0
              
              
                0
              
              
                0
              
              
                
                  
                
              
              
                0
              
              
                1
              
              
                
                  
                
              
              
                0
              
            
            
              
                0
              
              
                0
              
              
                1
              
              
                
                  
                
              
              
                0
              
              
                0
              
              
                
                  
                
              
              
                0
              
            
            
              
                0
              
              
                0
              
              
                0
              
              
                
                  
                
              
              
                0
              
              
                0
              
              
                ⋱
              
              
                0
              
            
            
              
                0
              
              
                0
              
              
                0
              
              
                ⋱
              
              
                0
              
              
                0
              
              
                
                  
                
              
              
                0
              
            
            
              
                0
              
              
                0
              
              
                0
              
              
                0
              
              
                0
              
              
                0
              
              
                
                  
                
              
              
                1
              
            
          
          ]
        
      
    
    ,
    
      
    
    ⁢
    
      N
      ≥
      4
    
    ,
    
      
    
    ⁢
    
      
        
          where
          ⁢
          
            
          
          ⁢
          
            

          
          [
          Pr
          ]
        
        N
      
      =
      
        
          [
          
            pr
            
              i
              ,
              j
            
          
          ]
        
        N
      
    
    ,
    
      

    
    ⁢
    
      with
      ⁢
      
        
      
      ⁢
      
        {
        
          
            
              
                
                  
                    
                      
                        pr
                        
                          i
                          ,
                          j
                        
                      
                      =
                      1
                    
                    ,
                  
                
                
                  
                    
                      
                        if
                        ⁢
                        
                          
                        
                        ⁢
                        i
                      
                      =
                      
                        2
                        ⁢
                        j
                      
                    
                    ,
                    
                      
                    
                    ⁢
                    
                      0
                      ≤
                      
                        
                      
                      ⁢
                      j
                      ≤
                      
                        
                          N
                          2
                        
                        -
                        1
                      
                    
                    ,
                  
                
              
              
                
                  
                    
                      
                        pr
                        
                          i
                          ,
                          j
                        
                      
                      =
                      1
                    
                    ,
                  
                
                
                  
                    
                      
                        if
                        ⁢
                        
                          
                        
                        ⁢
                        i
                      
                      =
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              j
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        mod
                        ⁢
                        
                          
                        
                        ⁢
                        N
                      
                    
                    ,
                    
                      
                        N
                        2
                      
                      ≤
                      j
                      ≤
                      
                        N
                        -
                        1
                      
                    
                    ,
                  
                
              
              
                
                  
                    
                      
                        pr
                        
                          i
                          ,
                          j
                        
                      
                      =
                      0
                    
                    ,
                  
                
                
                  
                    others
                    ,
                  
                
              
            
            ⁢
            where
            ⁢
            
              
                
              
              ⁢
              
                
              
            
            ⁢
            
              

            
            ⁢
            i
          
          ,
          
            
              j
              ∈
              
                
                  {
                  
                    0
                    ,
                    1
                    ,
                    ...
                    ⁢
                    
                      
                    
                    ,
                    
                      N
                      -
                      1
                    
                  
                  }
                
                ⁢
                
                  
                
                ⁢
                
                  

                
                ⁢
                
                  
                    and
                    ⁢
                    
                      

                    
                    [
                    Pc
                    ]
                  
                  2
                
              
            
            =
            
              
                [
                I
                ]
              
              2
            
          
          ,
          
            
          
          ⁢
          
            
              
                and
                ⁢
                
                  
                
                [
                Pc
                ]
              
              N
            
            =
            
              [
              
                
                  
                    
                      I
                      
                        N
                        /
                        4
                      
                    
                  
                  
                    0
                  
                  
                    0
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    
                      I
                      
                        N
                        /
                        4
                      
                    
                  
                  
                    0
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    0
                  
                  
                    
                      I
                      
                        N
                        /
                        4
                      
                    
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    
                      I
                      
                        N
                        /
                        4
                      
                    
                  
                  
                    0
                  
                
              
              ]
            
          
          ,
          
            N
            ≥
            4.

5. A signal transformation apparatus applying a hybrid architecture for Discrete Cosine Transform (DCT)-II, Discrete Sine Transform (DST)-II, Discrete Fourier Transform (DFT), and Haar Wavelet Transform (HWT), comprising:
a switching part for selecting any one of the DCT-II, the DST-II, the DFT, and the HWT; and
a transformation part for transforming and outputting an input signal according to the transform selected by the switching part,
wherein the switching part comprises:
a submatrix selector for selecting submatrix computation for any one of the DCT-II, the DST-II, the DFT, and the HWT; and
a permutation matrix selector for selecting permutation matrix computation for any one of the DCT-II, the DST-II, the DFT, and the HWT,
wherein when the switching part selects the DFT, the submatrix selector selects [Pr]N as a submatrix and the permutation matrix selector selects [W]n as a permutation matrix,
wherein the transformation part applies the DFT to the input signal by computing Equation (56) by using the submatrix [Pr]N and the permutation matrix [W]n and multiplying the input signal by a first matrix during first h steps of 2h steps and multiplying the input signal by a second matrix during last h steps:





  
    
      
        
          
            
              
                
                  [
                  F
                  ]
                
                N
              
              =
              
              ⁢
              
                
                  
                    
                      (
                      
                        
                          [
                          
                            P
                            ⁢
                            
                              
                            
                            ⁢
                            r
                          
                          ]
                        
                        N
                      
                      )
                    
                    
                      -
                      1
                    
                  
                  ⁡
                  
                    [
                    
                      F
                      ~
                    
                    ]
                  
                
                N
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          (
                          
                            
                              [
                              
                                P
                                ⁢
                                
                                  
                                
                                ⁢
                                r
                              
                              ]
                            
                            N
                          
                          )
                        
                        
                          -
                          1
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                P
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  r
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              
                                F
                                ~
                              
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              
                                F
                                ~
                              
                              
                                N
                                /
                                2
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            W
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                    
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          -
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      [
                      
                        P
                        ⁢
                        
                          
                        
                        ⁢
                        r
                      
                      ]
                    
                    N
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            P
                            ⁢
                            
                              
                            
                            ⁢
                            
                              r
                              
                                N
                                /
                                2
                              
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        I
                        2
                      
                      ⊗
                      P
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      r
                      
                        N
                        /
                        2
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  [
                  
                    
                      I
                      2
                    
                    ⊗
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                4
                              
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              P
                              ⁢
                              
                                
                              
                              ⁢
                              
                                r
                                
                                  N
                                  /
                                  4
                                
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ]
                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      
                        ...
                        ⁢
                        
                          
                        
                        [
                        
                          
                            
                              I
                              
                                N
                                /
                                4
                              
                            
                            ⊗
                            P
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            r
                            4
                          
                        
                        ]
                      
                      ⁡
                      
                        [
                        
                          
                            I
                            
                              N
                              /
                              4
                            
                          
                          ⊗
                          
                            [
                            
                              
                                
                                  
                                    I
                                    2
                                  
                                
                                
                                  0
                                
                              
                              
                                
                                  0
                                
                                
                                  
                                    P
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      r
                                      2
                                    
                                  
                                
                              
                            
                            ]
                          
                        
                        ]
                      
                    
                    ⁢
                    
                    [
                    
                      
                        I
                        
                          N
                          /
                          2
                        
                      
                      ⊗
                      
                        F
                        2
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        I
                        
                          N
                          /
                          4
                        
                      
                      ⊗
                      
                        [
                        
                          
                            
                              
                                I
                                2
                              
                            
                            
                              0
                            
                          
                          
                            
                              0
                            
                            
                              
                                W
                                2
                              
                            
                          
                        
                        ]
                      
                    
                    ]
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  [
                  
                    
                      I
                      
                        N
                        /
                        4
                      
                    
                    ⊗
                    
                      [
                      
                        
                          
                            
                              I
                              2
                            
                          
                          
                            
                              I
                              2
                            
                          
                        
                        
                          
                            
                              I
                              2
                            
                          
                          
                            
                              -
                              
                                I
                                2
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ]
                
                ⁢
                
                  
                
                ⁢
                
                  ...
                  ⁢
                  
                    
                  
                  [
                  
                    
                      I
                      2
                    
                    ⊗
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                4
                              
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              W
                              
                                N
                                /
                                4
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ]
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    [
                    
                      
                        I
                        2
                      
                      ⊗
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  4
                                
                              
                            
                            
                              
                                I
                                
                                  N
                                  /
                                  4
                                
                              
                            
                          
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  4
                                
                              
                            
                            
                              
                                -
                                
                                  I
                                  
                                    N
                                    /
                                    4
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            W
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                    
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          -
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    [
                    
                      P
                      ⁢
                      
                        
                      
                      ⁢
                      r
                    
                    ]
                  
                  N
                  
                    -
                    1
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          P
                          ⁢
                          
                            
                          
                          ⁢
                          
                            r
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              
              ⁢
              
                [
                
                  
                    I
                    2
                  
                  ⊗
                  
                    
                      
                        [
                        
                          
                        
                        ⁢
                        
                          ...
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              
                                
                              
                              ⁢
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            [
                                            
                                              P
                                              ⁢
                                              
                                                
                                              
                                              ⁢
                                              r
                                            
                                            ]
                                          
                                          4
                                          
                                            -
                                            1
                                          
                                        
                                        ⁡
                                        
                                          [
                                          
                                            
                                              
                                                
                                                  I
                                                  2
                                                
                                              
                                              
                                                0
                                              
                                            
                                            
                                              
                                                0
                                              
                                              
                                                
                                                  P
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  
                                                    r
                                                    2
                                                  
                                                
                                              
                                            
                                          
                                          ]
                                        
                                      
                                      ⁡
                                      
                                        [
                                        
                                          
                                            I
                                            2
                                          
                                          ⊗
                                          
                                            
                                              F
                                              ~
                                            
                                            2
                                          
                                        
                                        ]
                                      
                                    
                                    ⁡
                                    
                                      [
                                      
                                        
                                          
                                            
                                              I
                                              2
                                            
                                          
                                          
                                            0
                                          
                                        
                                        
                                          
                                            0
                                          
                                          
                                            
                                              W
                                              2
                                            
                                          
                                        
                                      
                                      ]
                                    
                                  
                                  ⁡
                                  
                                    [
                                    
                                      
                                        
                                          
                                            I
                                            2
                                          
                                        
                                        
                                          
                                            I
                                            2
                                          
                                        
                                      
                                      
                                        
                                          
                                            I
                                            2
                                          
                                        
                                        
                                          
                                            -
                                            
                                              I
                                              2
                                            
                                          
                                        
                                      
                                    
                                    ]
                                  
                                
                                }
                              
                            
                            
                              
                                F
                                ~
                              
                              4
                            
                          
                          ⁢
                          
                            
                          
                          ⁢
                          ...
                        
                        ⁢
                        
                          
                        
                        ]
                      
                      }
                    
                    
                      
                        F
                        ~
                      
                      
                        N
                        /
                        2
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            W
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                      
                      
                        
                          
                            I
                            
                              N
                              /
                              2
                            
                          
                        
                        
                          
                            -
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                        
                      
                    
                    ]
                  
                
                ,
              
            
          
        
      
    
    
      
        (
        56
        )
      
    
  



wherein





  
    
      
        [
        Pr
        ]
      
      2
    
    =
    
      
        
          
            [
            I
            ]
          
          2
        
        ⁢
        
          
        
        ⁢
        
          
            and
            ⁢
            
              
            
            [
            Pr
            ]
          
          N
        
      
      =
      
        [
        
          
            
              1
            
            
              0
            
            
              0
            
            
              ⋯
            
            
              0
            
            
              0
            
            
              ⋯
            
            
              0
            
          
          
            
              0
            
            
              0
            
            
              0
            
            
              ⋯
            
            
              1
            
            
              0
            
            
              ⋯
            
            
              0
            
          
          
            
              0
            
            
              1
            
            
              0
            
            
              
                
              
            
            
              0
            
            
              0
            
            
              
                
                  
                
                ⁢
                ⋯
              
            
            
              0
            
          
          
            
              0
            
            
              0
            
            
              0
            
            
              
                
              
            
            
              0
            
            
              1
            
            
              
                
              
            
            
              0
            
          
          
            
              0
            
            
              0
            
            
              1
            
            
              
                
              
            
            
              0
            
            
              0
            
            
              
                
              
            
            
              0
            
          
          
            
              0
            
            
              0
            
            
              0
            
            
              
                
              
            
            
              0
            
            
              0
            
            
              
                
                  
                
                ⁢
                ⋱
              
            
            
              0
            
          
          
            
              0
            
            
              0
            
            
              0
            
            
              ⋱
            
            
              0
            
            
              0
            
            
              
                
              
            
            
              0
            
          
          
            
              0
            
            
              0
            
            
              0
            
            
              0
            
            
              0
            
            
              0
            
            
              
                
              
            
            
              1
            
          
        
        ]
      
    
  
  ,
  
    
  
  ⁢
  
    N
    ≥
    4
  
  ,
  
    
  
  ⁢
  
    
      
        where
        ⁢
        
          

        
        [
        Pr
        ]
      
      N
    
    =
    
      
        [
        
          pr
          
            i
            ,
            j
          
        
        ]
      
      N
    
  
  ,
  
    

  
  ⁢
  
    with
    ⁢
    
      
    
    ⁢
    
      {
      
        
          
            
              
                
                  
                    
                      pr
                      
                        i
                        ,
                        j
                      
                    
                    =
                    1
                  
                  ,
                
              
              
                
                  
                    
                      if
                      ⁢
                      
                        
                      
                      ⁢
                      i
                    
                    =
                    
                      2
                      ⁢
                      j
                    
                  
                  ,
                  
                    
                  
                  ⁢
                  
                    0
                    ≤
                    
                      
                    
                    ⁢
                    j
                    ≤
                    
                      
                        N
                        2
                      
                      -
                      1
                    
                  
                  ,
                
              
            
            
              
                
                  
                    
                      pr
                      
                        i
                        ,
                        j
                      
                    
                    =
                    1
                  
                  ,
                
              
              
                
                  
                    
                      if
                      ⁢
                      
                        
                      
                      ⁢
                      i
                    
                    =
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            j
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      mod
                      ⁢
                      
                        
                      
                      ⁢
                      N
                    
                  
                  ,
                  
                    
                      N
                      2
                    
                    ≤
                    j
                    ≤
                    
                      N
                      -
                      1
                    
                  
                  ,
                
              
            
            
              
                
                  
                    
                      pr
                      
                        i
                        ,
                        j
                      
                    
                    =
                    0
                  
                  ,
                
              
              
                
                  others
                  ,
                
              
            
          
          ⁢
          where
          ⁢
          
            
              
            
            ⁢
            
              
            
          
          ⁢
          
            

          
          ⁢
          i
        
        ,
        
          j
          ∈
          
            {
            
              0
              ,
              1
              ,
              ...
              ⁢
              
                
              
              ,
              
                N
                -
                1
              
            
            }
          
        
        ,
        
          

        
        ⁢
        
          
            
              and
              ⁢
              
                

              
              [
              W
              ]
            
            N
          
          =
          
            [
            
              
                
                  
                    W
                    0
                  
                
                
                  0
                
                
                  ⋯
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    W
                    1
                  
                
                
                  
                    
                  
                
                
                  0
                
              
              
                
                  ⋮
                
                
                  
                    
                  
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  0
                
                
                  ⋯
                
                
                  0
                
                
                  
                    W
                    
                      N
                      -
                      1
                    
                  
                
              
            
            ]
          
        
      
    
  



W is a diagonal complex unit for 2N point DFT matrix.

6. A signal transformation apparatus applying a hybrid architecture for Discrete Cosine Transform (DCT)-II, Discrete Sine Transform (DST)-II, Discrete Fourier Transform (DFT), and Haar Wavelet Transform (HWT), comprising:
a switching part for selecting any one of the DCT-II, the DST-II, the DFT, and the HWT; and
a transformation part for transforming and outputting an input signal according to the transform selected by the switching part,
wherein the switching part comprises:
a submatrix selector for selecting submatrix computation for any one of the DCT-II, the DST-II, the DFT, and the HWT; and
a permutation matrix selector for selecting permutation matrix computation for any one of the DCT-II, the DST-II, the DFT, and the HWT,
wherein when the switching part selects the HWT, the submatrix selector selects [Pa]N as a submatrix and the permutation matrix selector selects [Pb]N as a permutation matrix,
wherein the transformation part applies the HWT to the input signal by computing Equation (64) by using the submatrix [Pa]N and the permutation matrix [Pb]N and multiplying the input signal by a matrix during last h steps of 2h steps:





  
    
      
        
          
            
              
                
                  [
                  H
                  ]
                
                N
              
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          [
                          
                            P
                            ⁢
                            
                              
                            
                            ⁢
                            i
                          
                          ]
                        
                        N
                        
                          -
                          1
                        
                      
                      [
                      
                        H
                        •
                      
                      ]
                    
                    N
                  
                  ⁡
                  
                    [
                    
                      P
                      ⁢
                      
                        
                      
                      ⁢
                      j
                    
                    ]
                  
                
                N
                
                  -
                  1
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      
                        
                          1
                          r
                        
                        ⁡
                        
                          [
                          
                            P
                            ⁢
                            
                              
                            
                            ⁢
                            a
                          
                          ]
                        
                      
                      N
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              I
                              
                                N
                                /
                                2
                              
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              P
                              ⁢
                              
                                
                              
                              ⁢
                              
                                a
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            I
                            
                              3
                              ⁢
                              
                                N
                                /
                                4
                              
                            
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            P
                            ⁢
                            
                              
                            
                            ⁢
                            
                              a
                              
                                N
                                /
                                4
                              
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁢
                
                  
                
                ⁢
                
                  ...
                  ⁢
                  
                    
                  
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            -
                            2
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          P
                          ⁢
                          
                            
                          
                          ⁢
                          
                            a
                            
                              N
                              /
                              2
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            -
                            2
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          
                            H
                            •
                          
                          2
                        
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            -
                            4
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        
                          
                        
                      
                      
                        
                          r
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            2
                          
                        
                      
                      
                        
                          
                        
                      
                    
                    
                      
                        0
                      
                      
                        
                          
                        
                      
                      
                        
                          P
                          ⁢
                          
                            
                          
                          ⁢
                          
                            b
                            2
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            -
                            4
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          I
                          2
                        
                      
                      
                        
                          I
                          2
                        
                      
                    
                    
                      
                        
                          
                        
                      
                      
                        
                          I
                          2
                        
                      
                      
                        
                          -
                          
                            I
                            2
                          
                        
                      
                    
                  
                  ]
                
                ⁢
                
                  
                
                ⁢
                
                  ...
                  ⁢
                  
                    
                  
                  [
                  
                    
                      
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                      
                        
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        
                          
                        
                      
                      
                        
                          r
                          ⁢
                          
                            
                          
                          ⁢
                          
                            I
                            
                              N
                              /
                              4
                            
                          
                        
                      
                      
                        
                          
                        
                      
                    
                    
                      
                        0
                      
                      
                        
                          
                        
                      
                      
                        
                          P
                          ⁢
                          
                            
                          
                          ⁢
                          
                            b
                            
                              N
                              /
                              4
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      
                        
                          [
                          
                            
                              
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                              
                                
                                  
                                
                              
                              
                                0
                              
                            
                            
                              
                                0
                              
                              
                                
                                  I
                                  
                                    N
                                    /
                                    4
                                  
                                
                              
                              
                                
                                  I
                                  
                                    N
                                    /
                                    4
                                  
                                
                              
                            
                            
                              
                                
                                  
                                
                              
                              
                                
                                  I
                                  
                                    N
                                    /
                                    4
                                  
                                
                              
                              
                                
                                  -
                                  
                                    I
                                    
                                      N
                                      /
                                      4
                                    
                                  
                                
                              
                            
                          
                          ]
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  r
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    I
                                    
                                      N
                                      /
                                      2
                                    
                                  
                                
                              
                              
                                0
                              
                            
                            
                              
                                0
                              
                              
                                
                                  P
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    b
                                    
                                      N
                                      /
                                      2
                                    
                                  
                                
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                          
                          
                            
                              
                                I
                                
                                  N
                                  /
                                  2
                                
                              
                            
                            
                              
                                -
                                
                                  I
                                  
                                    N
                                    /
                                    2
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        P
                        ⁢
                        
                          
                        
                        ⁢
                        b
                      
                      ]
                    
                  
                  
                    N
                    *
                  
                
                ,
              
            
          
        
      
    
    
      
        (
        64
        )
      
    
  








  
    
      
        wherein
        ⁢
        
          
        
        [
        Pa
        ]
      
      N
    
    =
    
      
        
          [
          Pi
          ]
        
        N
        
          -
          1
        
      
      =
      
        
          [
          Pi
          ]
        
        N
      
    
  
  ,
  
    
      
        [
        Pi
        ]
      
      N
    
    =
    
      
        
          [
          
            
              
                0
              
              
                
                  I
                  
                    N
                    /
                    2
                  
                
              
            
            
              
                
                  I
                  
                    N
                    /
                    2
                  
                
              
              
                0
              
            
          
          ]
        
        ⁢
        
          
        
        ⁢
        
          
            and
            ⁢
            
              
            
            ⁢
            
              

            
            [
            Pb
            ]
          
          N
        
      
      =
      
        
          
            r
            ⁡
            
              [
              Pj
              ]
            
          
          N
          
            -
            1
          
        
        =
        
          
        
        ⁢
        
          
            
              r
              ⁡
              
                [
                
                  
                    
                      
                        I
                        
                          N
                          /
                          2
                        
                      
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        -
                        
                          I
                          
                            N
                            /
                            2
                          
                        
                      
                    
                  
                
                ]
              
            
            ⁡
            
              [
              Pr
              ]
            
          
          N
        
      
    
  
  ,
  
    
      
        [
        Pj
        ]
      
      N
    
    =
    
      
        
          
            [
            Pr
            ]
          
          N
        
        ⁡
        
          [
          
            
              
                
                  I
                  
                    N
                    /
                    2
                  
                
              
              
                0
              
            
            
              
                0
              
              
                
                  -
                  
                    I
                    
                      N
                      /
                      2
                    
                  
                
              
            
          
          ]
        
      
      .