# Patent

## Title
Signal processing method and data processing method and apparatus

## Abstract

The present invention discloses a signal processing method and a data processing method and apparatus. A time-domain to frequency-domain signal processing method includes: pre-processing time-domain data; pre-rotating the pre-processed data by using a rotation factor a·WNn+0.5; performing a discrete Fourier transform (DFT) of N/4 points on the pre-rotated data; and post-rotating the data transformed by the DFT by using a rotation factor b·WNk+0.5 to obtain frequency-domain data. A frequency-domain to time-domain signal processing method includes: twiddling frequency-domain data; pre-rotating the twiddled data by using a rotation factor c·WNk+0.5; performing a DFT of N/4 points on the pre-rotated data; and post-rotating the data transformed by the DFT by using a rotation factor d·WNn+0.5; and post-processing the post-rotated data to obtain time-domain data. The present invention increases the efficiency of signal processing.

## Background
CROSS-REFERENCE TO RELATED APPLICATIONS
This application is a continuation of U.S. patent application Ser. No. 12/980,001, filed on Dec. 28, 2010, which is a continuation of International Application No. PCT/CN2010/072029, filed on Apr. 22, 2010. The International Application claims priority to Chinese Patent Application No. 200910150720.5, filed on Jun. 24, 2009. The afore-mentioned patent applications are hereby incorporated by reference in their entireties.


FIELD OF THE INVENTION
The present invention relates to digital signal processing technologies, and in particular, to a signal processing method and a data processing method and apparatus.
BACKGROUND OF THE INVENTION
Orthogonal transforms as Fourier transform, discrete cosine transform (DCT), and discrete sine transform (DST) are widely used in digital signal processing, especially in spectrum analysis, image coding and speech coding.
A DCT is a spatial transform and has a strong energy compaction property so that a DCT-based coding system provides good compression performance.
A modified discrete cosine transform (MDCT) is a DCT-based improvement. The MDCT is designed to avoid blocking effect between consecutive blocks. It plays a vital role in wideband and super wideband digital audio coding.
A direct application of the forward MDCT and inverse MDCT according to the transform formula causes high complexity of computation. Especially, an MDCT on a large number of points cannot be performed in real time due to the high complexity of computation. Because of the wide application of the MDCT in real-time communications and especially in audio coding, a fast MDCT method becomes an urgent need.
The widely used FFT-based fast MDCT of N points includes the fast MDCT based on an N/2-point FFT and the fast MDCT based on an N/4-point FFT.
In the research and practice of the conventional art, the following weaknesses are found:
In the application of a fast MDCT algorithm based on an N/4-point FFT, pre- and post-rotation processing requires the storing of at least cosine values of N/4 points and sine values of N/4 points, altogether N/2 values. When pre- and post-rotation steps are asymmetrical, N values need to be stored. In the case of a large N, the amount of additional storage is large, and therefore a lot of storage resources are consumed, and the transform efficiency is affected.
When a fast Fourier transform algorithm based on coprime factor is adopted, a lot of addressing operations are required and the algorithm does not take into account sequential access of the data sequence. The data obtained through a transform needs to be rearranged to obtain the final output data. Thus, the amount of computation is larger.
SUMMARY OF THE INVENTION
Embodiments of the present invention provide a signal processing method and a data processing method and apparatus to help improve the efficiency of signal processing.
A time-domain to frequency-domain signal processing method includes:

    
    
        pre-processing time-domain data;
        pre-rotating the pre-processed data by using a rotation factor a·WNn+0.5;
        performing a discrete Fourier transform (DFT) of N/4 points on the pre-rotated data; and
        post-rotating the data transformed by the DFT by using a rotation factor b·WNk+0.5 to obtain frequency-domain data; where:
        a and b are constants, N is the length of the time-domain data,
    
    






  
    
      W
      N
    
    =
    
      e
      
        
          -
          j
        
        ⁢
        
          
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
      
    
  
  ,
  
    
      W
      N
      
        n
        +
        0.5
      
    
    =
    
      e
      
        
          -
          j
        
        ⁢
        
          
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            n
            +
            0.5
          
          )
        
      
    
  
  ,
  
    
      W
      N
      
        k
        +
        0.5
      
    
    =
    
      e
      
        
          -
          j
        
        ⁢
        
          
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          
        
        ⁢
        
          (
          
            k
            +
            0.5
          
          )
        
      
    
  
  ,




    
    
         and n and k are respectively all non-negative integers not larger than N/4.
    
    


A frequency-domain to time-domain signal processing method includes:

    
    
        twiddling frequency-domain data;
        pre-rotating the twiddled data by using a rotation factor c·WNk+0.5;
        performing a DFT of N/4 points on the pre-rotated data;
        post-rotating the data transformed by the DFT by using a rotation factor d·WNn+0.5; and
        post-processing the post-rotated data to obtain time-domain data; where:
        c and d are constants, N is the double length of the frequency-domain data,
    
    






  
    
      W
      N
    
    =
    
      e
      
        
          -
          j
        
        ⁢
        
          
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
      
    
  
  ,
  
    
      W
      N
      
        n
        +
        0.5
      
    
    =
    
      e
      
        
          -
          j
        
        ⁢
        
          
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            n
            +
            0.5
          
          )
        
      
    
  
  ,
  
    
      W
      N
      
        k
        +
        0.5
      
    
    =
    
      e
      
        
          -
          j
        
        ⁢
        
          
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            k
            +
            0.5
          
          )
        
      
    
  
  ,




    
    
         and n and k are respectively all non-negative integers not larger than N/4.
    
    


A data processing method includes:

    
    
        performing P-point DFT on data for Q times based on the address table; and.
        performing Q-point DFT on the data obtained after P-point DFT for Q times for P times based on the address table.
    
    


A time-domain to frequency-domain signal processing apparatus includes:

    
    
        a pre-processing unit, configured to pre-process time-domain data;
        a first pre-rotating unit, configured to pre-rotate the data obtained by the pre-processing unit by using a rotation factor a·WNn+0.5;
        a first Fourier transforming unit, configured to perform a DFT of N/4 points on the data processed by the first pre-rotating unit; and
        a first post-rotating unit, configured to post-rotate the data transformed by the first Fourier transforming unit by using a rotation factor b·WNk+0.5 to obtain frequency-domain data, where
    
    






  
    
      W
      N
      
        n
        +
        0.5
      
    
    =
    
      e
      
        
          -
          j
        
        ⁢
        
          
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            n
            +
            0.5
          
          )
        
      
    
  
  ,
  
    
      W
      N
      
        k
        +
        0.5
      
    
    =
    
      e
      
        
          -
          j
        
        ⁢
        
          
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            k
            +
            0.5
          
          )
        
      
    
  
  ,




    
    
         and n and k are respectively all non-negative integers not larger than N/4.
    
    


A frequency-domain to time-domain signal processing apparatus includes:

    
    
        a twiddling unit, configured to twiddle frequency-domain data;
        a second pre-rotating unit, configured to pre-rotate the data obtained by the twiddling unit by using a rotation factor c·WNk+0.5;
        a second Fourier transforming unit, configured to perform a DFT of N/4 points on the data processed by the second pre-rotating unit;
        a third post-rotating unit, configured to post-rotate the data transformed by the second Fourier transforming unit by using a rotation factor d·WNn+0.5; and
        a post-processing unit, configured to post-process the data processed by the third post-rotating unit to obtain time-domain data, where
    
    






  
    
      W
      N
      
        n
        +
        0.5
      
    
    =
    
      e
      
        
          -
          j
        
        ⁢
        
          
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            n
            +
            0.5
          
          )
        
      
    
  
  ,
  
    
      W
      N
      
        k
        +
        0.5
      
    
    =
    
      e
      
        
          -
          j
        
        ⁢
        
          
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            k
            +
            0.5
          
          )
        
      
    
  
  ,




    
    
         and n and k are respectively all non-negative integers not larger than N/4.
    
    


A data processing apparatus includes:

    
    
        an address table unit, configured to create or store an address table;
        a fifth transforming unit, configured to perform P-point DFT on data for Q times based on the address table created or stored by the address table unit; and
        a sixth transforming unit, configured to perform Q-point DFT on the data transformed by the fifth transforming unit for P times based on the address table created or stored by the address table unit.
    
    


In the above technical solution, the rotation factors adopted by the signal processing methods provided in the embodiments of the present invention are products of a constant and WNn+0.5. Because WNn+0.5 is symmetrical, it is only necessary to store a data table of N/4 points to complete the pre-rotation and post-rotation. Compared with the prior art, the signal processing methods provided in the embodiments of the present invention reduce the amount of storage, reduce the consumption of storage resources and increase the transform efficiency. Further, the signal processing methods in the embodiments of the present invention do not require a normalization factor when the post-rotation is performed so that related processing steps are spared and therefore the transform efficiency is further improved.



BRIEF DESCRIPTION OF THE DRAWINGS
To better explain the technical solution of the embodiments of the present invention or the prior art, the drawings required in the description of the embodiments of the present invention or the prior art will be described in brief. It is apparent that the drawings described below are only some embodiments of the present invention. Those skilled in the art can obtain other drawings based on the drawings without creative work.
FIG. 1 is a flowchart of a time-domain to frequency-domain signal processing method in a first embodiment of the present invention;
FIG. 2 is a flowchart of a data processing method that adopts an address table for FFT operations in an embodiment of the present invention;
FIG. 3 is a flowchart of a frequency-domain to time-domain signal processing method in a second embodiment of the present invention;
FIG. 4 is a flowchart of a time-domain to frequency-domain signal processing method in a third embodiment of the present invention;
FIG. 5 is a flowchart of a frequency-domain to time-domain signal processing method in a fourth embodiment of the present invention;
FIG. 6 is a flowchart of a time-domain to frequency-domain signal processing method in a fifth embodiment of the present invention;
FIG. 7 is a flowchart of a frequency-domain to time-domain signal processing method in a sixth embodiment of the present invention;
FIG. 8 shows a structure of a time-domain to frequency-domain signal processing apparatus in an embodiment of the present invention;
FIG. 9 shows a structure of a frequency-domain to time-domain signal processing apparatus in an embodiment of the present invention; and
FIG. 10 shows a structure of a data processing apparatus in an embodiment of the present invention.



DETAILED DESCRIPTION OF THE INVENTION
The technical solution of the present invention is hereinafter described in detail with reference to the accompanying drawings. It is evident that the embodiments are exemplary only and the present invention is not limited to such embodiments. Other embodiments that those skilled in the art obtain based on the embodiments of the present invention also fall within the scope of protection of the present invention.
The embodiments of the present invention provide a signal processing method and a data processing method and apparatus to help improve the efficiency of signal processing.
In the prior art, to achieve a fast MDCT based on the FFT of N/4 points, the data of N points is first windowed, twiddled and pre-rotated so that an MDCT of N points is changed to an FFT of N/4 points. Data obtained through the FFT operation is post-rotated to obtain a final MDCT coefficient.
The definition of the MDCT is:



  
    
      X
      k
    
    =
    
      A
      ⁢
      
        
          ∑
          
            n
            =
            0
          
          
            N
            -
            1
          
        
        ⁢
        
          
            y
            n
          
          ⁢
          
            cos
            ⁡
            
              [
              
                
                  
                    2
                    ⁢
                    π
                  
                  N
                
                ⁢
                
                  (
                  
                    n
                    +
                    
                      1
                      2
                    
                    +
                    
                      N
                      4
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    k
                    +
                    
                      1
                      2
                    
                  
                  )
                
              
              ]
            
          
        
      
    
  




  
    
      k
      =
      0
    
    ,
    1
    ,
    ...
    ⁢
    
      
    
    ,
    
      
        N
        2
      
      -
      1
    
  



    
    
        where, A is a normalization factor and a constant.
    
    


According to the definition of the MDCT, the following can be obtained:



  
    
      X
      
        2
        ⁢
        k
      
    
    =
    
      
        A
        ·
        Re
      
      ⁢
      
        {
        
          
            W
            N
            
              k
              +
              0.125
            
          
          ⁢
          
            
              ∑
              
                n
                =
                0
              
              
                
                  N
                  /
                  4
                
                -
                1
              
            
            ⁢
            
              
                u
                n
              
              ⁢
              
                W
                N
                
                  n
                  +
                  0.125
                
              
              ⁢
              
                W
                
                  N
                  /
                  4
                
                nk
              
            
          
        
        }
      
    
  




  
    
      k
      =
      0
    
    ,
    ...
    ⁢
    
      
    
    ,
    
      
        N
        /
        4
      
      -
      1
    
  




  
    
      X
      
        
          N
          /
          2
        
        -
        1
        -
        
          2
          ⁢
          k
        
      
    
    =
    
      
        
          -
          A
        
        ·
        Im
      
      ⁢
      
        {
        
          
            W
            N
            
              k
              +
              0.125
            
          
          ⁢
          
            
              ∑
              
                n
                =
                0
              
              
                
                  N
                  /
                  4
                
                -
                1
              
            
            ⁢
            
              
                u
                n
              
              ⁢
              
                W
                N
                
                  n
                  +
                  0.125
                
              
              ⁢
              
                W
                
                  N
                  /
                  4
                
                nk
              
            
          
        
        }
      
    
  




  
    
      k
      =
      0
    
    ,
    ...
    ⁢
    
      
    
    ,
    
      
        N
        /
        4
      
      -
      1
    
  




  
    
      where
      ⁢
      
        
      
      ⁢
      
        W
        N
        x
      
    
    =
    
      
        e
        
          
            -
            j
          
          ⁢
          
            
          
          ⁢
          
            
              2
              ⁢
              π
              ⁢
              
                
              
              ⁢
              x
            
            N
          
        
      
      .
    
  



The two equations can be combined to obtain the following equation:



  
    
      
        X
        
          2
          ⁢
          k
        
      
      -
      
        j
        ⁢
        
          
        
        ⁢
        
          X
          
            
              N
              /
              2
            
            -
            1
            -
            
              2
              ⁢
              k
            
          
        
      
    
    =
    
      
        A
        ·
        
          
            W
            N
            
              k
              +
              0.125
            
          
          
            post
            ⁢
            
              -
            
            ⁢
            rotation
          
        
      
      ⁢
      
        
          ∑
          
            n
            =
            0
          
          
            
              N
              /
              4
            
            -
            1
          
        
        ⁢
        
          
            {
            
              
                u
                n
              
              ⁢
              
                
                  W
                  N
                  
                    n
                    +
                    0.125
                  
                
                
                  pre
                  ⁢
                  
                    -
                  
                  ⁢
                  rotation
                
              
            
            }
          
          ⁢
          
            
              W
              
                N
                /
                4
              
              nk
            
            
              
                N
                /
                4
              
              ⁢
              
                
              
              ⁢
              points
              ⁢
              
                
              
              ⁢
              DFT
            
          
        
      
    
  




  
    
      k
      =
      0
    
    ,
    ...
    ⁢
    
      
    
    ,
    
      
        N
        /
        4
      
      -
      1
    
  



Then, the forward MDCT based on the N/4-point FFT includes the following steps:
1. Windowing:

yn=hnxn, n=0, . . . ,N−1

hn is a window function that satisfies the perfect reconstruction condition but unnecessarily a symmetrical window. That is, analysis and synthesis may use different windows so long as the analysis window and synthesis window can together achieve a perfect reconstruction. xn is the input data. Normally,




  x
  
    
      
        0
        ~
        N
      
      /
      2
    
    -
    1
  




is data of the cached previous frame and





  x
  
    
      N
      /
      
        2
        ~
        N
      
    
    -
    1
  




is data of the current frame.

2. Twiddling:



  
    
      u
      n
    
    =
    
      (
      
        
          z
          
            2
            ⁢
            n
          
        
        +
        
          j
          ⁢
          
            
          
          ⁢
          
            z
            
              
                N
                /
                2
              
              -
              1
              -
              
                2
                ⁢
                n
              
            
          
        
      
      )
    
  




  
    
      n
      =
      0
    
    ,
    ...
    ⁢
    
      
    
    ,
    
      
        N
        /
        4
      
      -
      1
    
  




  
    where
    ⁢
    
      :
    
  




  
    {
    
      
        
          
            
              
                
                  
                    
                      z
                      
                        n
                        +
                        
                          N
                          /
                          4
                        
                      
                    
                    =
                    
                      
                        y
                        n
                      
                      -
                      
                        y
                        
                          
                            N
                            /
                            2
                          
                          -
                          1
                          -
                          n
                        
                      
                    
                  
                  ⁢
                  
                    
                  
                
              
            
            
              
                
                  
                    z
                    
                      
                        N
                        /
                        4
                      
                      -
                      1
                      -
                      n
                    
                  
                  =
                  
                    
                      -
                      
                        y
                        
                          N
                          -
                          1
                          -
                          n
                        
                      
                    
                    -
                    
                      y
                      
                        
                          N
                          /
                          2
                        
                        +
                        n
                      
                    
                  
                
              
            
          
          ⁢
          
            

          
          ⁢
          n
        
        =
        0
      
      ,
      ...
      ⁢
      
        
      
      ,
      
        
          N
          /
          4
        
        -
        
          1
          ⁢
          
            

          
          ⁢
          or
        
      
      ,
      
        

      
      ⁢
      
        {
        
          
            
              
                
                  z
                  n
                
                =
                
                  
                    -
                    
                      y
                      
                        n
                        +
                        
                          3
                          ⁢
                          
                            N
                            /
                            4
                          
                        
                      
                    
                  
                  -
                  
                    y
                    
                      
                        3
                        ⁢
                        
                          N
                          /
                          4
                        
                      
                      -
                      1
                      -
                      n
                    
                  
                
              
            
            
              
                
                  n
                  =
                  0
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  
                    N
                    /
                    4
                  
                  -
                  1
                
              
            
          
          
            
              
                
                  z
                  n
                
                =
                
                  
                    y
                    
                      n
                      -
                      
                        N
                        /
                        4
                      
                    
                  
                  -
                  
                    y
                    
                      
                        3
                        ⁢
                        
                          N
                          /
                          4
                        
                      
                      -
                      1
                      -
                      n
                    
                  
                
              
            
            
              
                
                  n
                  =
                  
                    N
                    /
                    4
                  
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  
                    N
                    /
                    2
                  
                  -
                  1
                
              
            
          
        
      
    
  



3. Pre-rotate un by using the rotation factor WNn+0.125.
4. Perform an N/4-point FFT on the rotated data.
5. Post-rotate the transformed data by using the rotation factor WNk+0.125 and the normalization factor A.
6. The real part of the complex number obtained through the post-rotation is X2k which denotes the odd frequencies of the MDCT spectrum and the imaginary part is




  -
  
    X
    
      
        N
        /
        2
      
      -
      1
      -
      
        2
        ⁢
        k
      
    
  




which denotes the even frequencies of the MDCT spectrum.

The rotation factor WNn+0.125 may be expanded to:




  
    W
    N
    
      n
      +
      0.125
    
  
  =
  
    
      e
      
        
          -
          j
        
        ⁢
        
          
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            n
            +
            0.125
          
          )
        
      
    
    =
    
      
        cos
        ⁡
        
          (
          
            
              2
              ⁢
              
                π
                ⁡
                
                  (
                  
                    n
                    +
                    0.125
                  
                  )
                
              
            
            N
          
          )
        
      
      -
      
        j
        ⁢
        
          
        
        ⁢
        
          sin
          ⁡
          
            (
            
              
                2
                ⁢
                
                  π
                  ⁡
                  
                    (
                    
                      n
                      +
                      0.125
                    
                    )
                  
                
              
              N
            
            )
          
        
      
    
  




Rotation factors WNn+0.125 and WNk+0.125 are required for pre-rotation and post-rotation processing and therefore at least cosine values of N/4 points and sine values of N/4 points, altogether N/2 values, need to be stored. When pre- and post-rotation steps are asymmetrical, N values need to be stored. In the case of a large N, the amount of additional storage is large and therefore a lost of storage resources are consumed and the transform efficiency is affected.
A time-domain to frequency-domain signal processing method provided in an embodiment of the present invention includes:
1. Pre-process the time-domain data.
2. Pre-rotate the pre-processed data by using the rotation factor a·WNn+0.5.
3. Perform a DFT of N/4 points on the pre-rotated data.
4. Post-rotate the data after the DFT by using the rotation factor b·WNk+0.5 to obtain the frequency-domain data,

    
    
        where, a and b are constants, N is the length of the time-domain data,
    
    






  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      /
      4
    
    -
    1
  
  ,
  
    
  
  ⁢
  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      /
      4
    
    -
    1
  
  ,
  
    
      and
      ⁢
      
        
      
      ⁢
      
        W
        N
      
    
    =
    
      
        e
        
          
            -
            j
          
          ⁢
          
            
          
          ⁢
          
            
              2
              ⁢
              π
            
            N
          
        
      
      .
    
  




All processing on the data before the pre-rotation may be described as pre-processing. In this embodiment, for example, pre-processing may be windowing and twiddling or only twiddling.
A frequency-domain to time-domain signal processing method provided in an embodiment of the present invention includes:
1. Twiddle the frequency-domain data.
2. Pre-rotate the twiddled data by using the rotation factor c·WNk+0.5.
3. Perform a DFT of N/4 points on the pre-rotated data.
4. Post-rotate the data after the DFT by using the rotation factor d·WNn+0.5.
5. Post-process the post-rotated data to obtain the time-domain data.

    
    
        where, c and d are constants, N is twice the length of the frequency-domain data,
    
    






  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      /
      4
    
    -
    1
  
  ,
  
    
  
  ⁢
  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      /
      4
    
    -
    1
  
  ,
  
    
      and
      ⁢
      
        
      
      ⁢
      
        W
        N
      
    
    =
    
      
        e
        
          
            -
            j
          
          ⁢
          
            
          
          ⁢
          
            
              2
              ⁢
              π
            
            N
          
        
      
      .
    
  




All processing on the data after the post-rotation may be described as post-processing. The content of post-processing depends on the content of pre-processing in the corresponding time-domain to frequency-domain signal processing method. In this embodiment, for example, if the pre-processing is windowing and twiddling, the post-processing is also windowing and twiddling; if the pre-processing is just twiddling, the post-processing is also just twiddling.
The time-domain to frequency-domain signal processing may also be called a forward transform and the frequency-domain to time-domain signal processing may also be called an inverse transform. When a×b×c×d=4/N in the forward transform and the inverse transform, using the output data of the forward transform as the input data of the inverse transform, a perfect reconstruction can be achieved, because the result of the inverse transform can recover the data before the forward transform. In practice, the perfect reconstruction is unnecessarily required.
WNn+0.5 in the rotation factor adopted by the signal processing method provided in an embodiment of the present invention may be expanded to:



  
    
      W
      N
      
        n
        +
        0.5
      
    
    =
    
      
        e
        
          
            -
            j
          
          ⁢
          
            
          
          ⁢
          
            
              2
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              n
              +
              0.5
            
            )
          
        
      
      =
      
        
          cos
          ⁡
          
            (
            
              
                2
                ⁢
                
                  π
                  ⁡
                  
                    (
                    
                      n
                      +
                      0.5
                    
                    )
                  
                
              
              N
            
            )
          
        
        -
        
          j
          ⁢
          
            
          
          ⁢
          
            sin
            ⁡
            
              (
              
                
                  2
                  ⁢
                  
                    π
                    ⁡
                    
                      (
                      
                        n
                        +
                        0.5
                      
                      )
                    
                  
                
                N
              
              )
            
          
        
      
    
  




  
    where
    ,
    
      

    
    ⁢
    
      
        
          cos
          ⁡
          
            (
            
              
                
                  2
                  ⁢
                  π
                
                N
              
              ⁢
              
                (
                
                  n
                  +
                  0.5
                
                )
              
            
            )
          
        
        =
        
          sin
          ⁡
          
            (
            
              
                
                  2
                  ⁢
                  π
                
                N
              
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    N
                    /
                    4
                  
                  -
                  1
                  -
                  n
                  +
                  0.5
                
                )
              
            
            )
          
        
      
      ;
    
  




  
    
      
        sin
        ⁡
        
          (
          
            
              
                2
                ⁢
                π
              
              
                N
                ⁢
                
                  
                
              
            
            ⁢
            
              (
              
                n
                +
                0.5
              
              )
            
          
          )
        
      
      =
      
        cos
        ⁡
        
          (
          
            
              
                2
                ⁢
                π
              
              N
            
            ⁢
            
              (
              
                
                  N
                  /
                  4
                
                -
                1
                -
                n
                +
                0.5
              
              )
            
          
          )
        
      
    
    ;
  



From




  
    
      cos
      ⁡
      
        (
        
          
            
              2
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              n
              +
              0.5
            
            )
          
        
        )
      
    
    =
    
      sin
      ⁡
      
        (
        
          
            
              2
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              
                N
                /
                4
              
              -
              1
              -
              n
              +
              0.5
            
            )
          
        
        )
      
    
  
  ,




one can know that




  
    
      
        n
        +
        
          (
          
            
              N
              /
              4
            
            -
            1
            -
            n
          
          )
        
      
      =
      
        
          N
          /
          4
        
        -
        1
      
    
    ;
    as
  




  
    
      n
      =
      0
    
    ,
    ...
    ⁢
    
      
    
    ,
    
      
        
          N
          /
          4
        
        -
        
          1
          ⁢
          
            
          
          ⁢
          and
          ⁢
          
            
          
          ⁢
          k
        
      
      =
      0
    
    ,
    ...
    ⁢
    
      
    
    ,
    
      
        N
        /
        4
      
      -
      1
    
    ,
  



the first value of





  cos
  ⁡
  
    (
    
      
        
          2
          ⁢
          π
        
        N
      
      ⁢
      
        (
        
          n
          +
          0.5
        
        )
      
    
    )
  




(when n=0) equals the last value of





  sin
  ⁡
  
    (
    
      
        
          2
          ⁢
          π
        
        N
      
      ⁢
      
        (
        
          n
          +
          0.5
        
        )
      
    
    )
  




(when





  
    
      n
      =
      
        
          N
          /
          4
        
        -
        1
      
    
    )
  
  .




The second value of





  cos
  ⁡
  
    (
    
      
        
          2
          ⁢
          π
        
        N
      
      ⁢
      
        (
        
          n
          +
          0.5
        
        )
      
    
    )
  




(when n=1) equals the next to last value of





  sin
  ⁡
  
    (
    
      
        
          2
          ⁢
          π
        
        N
      
      ⁢
      
        (
        
          n
          +
          0.5
        
        )
      
    
    )
  




(when





  
    n
    =
    
      
        N
        4
      
      -
      2
    
  
  )




and so on.

Therefore, WNn+0.5 is symmetrical. It is only necessary to store any




  N
  4




points data table of





  
    cos
    ⁡
    
      (
      
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            n
            +
            0.5
          
          )
        
      
      )
    
  
  ⁢
  
    
  
  ⁢
  or
  ⁢
  
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            n
            +
            0.5
          
          )
        
      
      )
    
  




to complete the pre-rotation and post-rotation.

In the traditional MDCT, to obtain the MDCT spectrum coefficient, the rotation factors WNn+0.125 and WNk+0.125 need to be adopted and WNn+0.125 and WNk+0.125 are not symmetrical. Therefore, at least cosine values of N/4 points and sine values of N/4 points need to be stored during pre-rotation and post-rotation. The rotation factor adopted by the signal processing method provided in the embodiment of the present invention is the product of a constant and WNn+0.5. Because WNn+0.5 is symmetrical, it is only necessary to store a data table of N/4 points to complete the pre-rotation and post-rotation. Compared with the prior art, the signal processing method in the embodiment of the present invention significantly reduces the amount of storage and the consumption of storage resources, and increases the transform efficiency. Further, the signal processing method in the embodiment of the present invention does not require a normalization factor so that related processing steps are spared and therefore the transform efficiency is further improved.
Exemplary embodiments of the signal processing method of the present invention will be described. FIG. 1 is a flowchart of the time-domain to frequency-domain signal processing method in a first embodiment of the present invention. The method includes the following steps:
101. Pre-process the time-domain data.
In this step, pre-processing includes windowing and twiddling. In practice, other pre-processing means may be adopted.
Assume the time-domain data xn is the input data. Normally,




  x
  
    0
    ∼
    
      
        N
        2
      
      -
      1
    
  




is the cached previous frame data and





  x
  
    
      N
      2
    
    ∼
    
      N
      -
      1
    
  




is the current frame data. N is the length of the time-domain data, and may be 1280 in this embodiment. The data is first windowed:

yn=hnxn n=0, . . . , N−1 ,

hn is a window function that satisfies the perfect reconstruction condition and may be referred to as an analysis window function. The window function used in the inverse transform may be called a synthesis window function. The analysis window function may be different from the synthesis window function so long as the analysis window function and the synthesis window function can complete the perfect reconstruction together. The windowed data is twiddled and the twiddled data un satisfies:



  
    
      u
      n
    
    =
    
      (
      
        
          z
          
            2
            ⁢
            n
          
        
        +
        
          j
          ⁢
          
            
          
          ⁢
          
            z
            
              
                N
                2
              
              -
              1
              -
              
                2
                ⁢
                n
              
            
          
        
      
      )
    
  




  
    
      n
      =
      0
    
    ,
    ...
    ⁢
    
      
    
    ,
    
      
        N
        4
      
      -
      1
    
  




  
    where
    ⁢
    
      :
    
  




  
    {
    
      
        
          
            
              
                
                  
                    
                      
                        z
                        
                          n
                          +
                          
                            N
                            4
                          
                        
                      
                      =
                      
                        
                          y
                          n
                        
                        -
                        
                          y
                          
                            
                              N
                              2
                            
                            -
                            1
                            -
                            n
                          
                        
                      
                    
                  
                
                
                  
                    
                      
                        z
                        
                          
                            N
                            4
                          
                          -
                          1
                          -
                          n
                        
                      
                      =
                      
                        
                          -
                          
                            y
                            
                              N
                              -
                              1
                              -
                              n
                            
                          
                        
                        -
                        
                          y
                          
                            
                              N
                              2
                            
                            +
                            n
                          
                        
                      
                    
                  
                
              
            
            
              
                
                  n
                  =
                  0
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  
                    N
                    4
                  
                  -
                  1
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        or
      
      ,
      
        

      
      ⁢
      
        {
        
          
            
              
                
                  z
                  n
                
                =
                
                  
                    -
                    
                      y
                      
                        n
                        +
                        
                          
                            3
                            ⁢
                            N
                          
                          4
                        
                      
                    
                  
                  -
                  
                    y
                    
                      
                        
                          3
                          ⁢
                          N
                        
                        4
                      
                      -
                      1
                      -
                      n
                    
                  
                
              
            
            
              
                
                  n
                  =
                  0
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  
                    N
                    4
                  
                  -
                  1
                
              
            
          
          
            
              
                
                  z
                  n
                
                =
                
                  
                    y
                    
                      n
                      -
                      
                        N
                        4
                      
                    
                  
                  -
                  
                    y
                    
                      
                        
                          3
                          ⁢
                          N
                        
                        4
                      
                      -
                      1
                      -
                      n
                    
                  
                
              
            
            
              
                
                  n
                  =
                  
                    N
                    4
                  
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  
                    N
                    2
                  
                  -
                  1
                
              
            
          
        
      
    
  



102. Pre-rotate the pre-processed data by using the rotation factor a·WNn+0.5.
The twiddled data un is pre-rotated by using the rotation factor a·WNn+0.5,




  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      4
    
    -
    1
  
  ,
  
    

  
  ⁢
  where
  ,
  
    

  
  ⁢
  
    
      W
      N
    
    =
    
      
        e
        
          
            -
            j
          
          ⁢
          
            
              2
              ⁢
              π
            
            N
          
        
      
      =
      
        
          cos
          ⁡
          
            (
            
              
                2
                ⁢
                π
              
              N
            
            )
          
        
        -
        
          j
          ⁢
          
            
          
          ⁢
          
            sin
            ⁡
            
              (
              
                
                  2
                  ⁢
                  π
                
                N
              
              )
            
          
        
      
    
  




and a is a constant.

The data obtained after the rotation is a·WNn+0.5·un.
In this embodiment, the constant may be




  a
  =
  
    
      
        2
      
      
        N
        4
      
    
    .
  




Because WNn+0.5 in the rotation factor may be expressed in the following form:





  
    
      W
      N
      
        n
        +
        0.5
      
    
    =
    
      
        e
        
          
            -
            j
          
          ⁢
          
            
              2
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              n
              +
              0.5
            
            )
          
        
      
      =
      
        
          cos
          ⁡
          
            (
            
              
                2
                ⁢
                
                  π
                  ⁡
                  
                    (
                    
                      n
                      +
                      0.5
                    
                    )
                  
                
              
              N
            
            )
          
        
        -
        
          j
          ⁢
          
            
          
          ⁢
          
            sin
            ⁡
            
              (
              
                
                  2
                  ⁢
                  
                    π
                    ⁡
                    
                      (
                      
                        n
                        +
                        0.5
                      
                      )
                    
                  
                
                N
              
              )
            
          
        
      
    
  
  ,




which satisfies



  
    
      cos
      ⁡
      
        (
        
          
            
              2
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              n
              +
              0.5
            
            )
          
        
        )
      
    
    =
    
      
        sin
        ⁡
        
          (
          
            
              
                2
                ⁢
                π
              
              N
            
            ⁢
            
              (
              
                
                  N
                  4
                
                -
                1
                -
                n
                +
                0.5
              
              )
            
          
          )
        
      
      ⁢
      
        
          
        
        ⁢
        
          
        
      
      ⁢
      and
    
  




  
    
      
        sin
        ⁡
        
          (
          
            
              
                2
                ⁢
                π
              
              N
            
            ⁢
            
              (
              
                n
                +
                0.5
              
              )
            
          
          )
        
      
      =
      
        cos
        ⁡
        
          (
          
            
              
                2
                ⁢
                π
              
              N
            
            ⁢
            
              (
              
                
                  N
                  4
                
                -
                1
                -
                n
                +
                0.5
              
              )
            
          
          )
        
      
    
    ,
  



only a sine or cosine data table of N/4 points is required in the implementation of the method. In this embodiment, a data table of only N/4 points is stored. Values in the data table are expressed by





  
    
      1
      
        320
        4
      
    
    ⁢
    
      cos
      ⁡
      
        (
        
          
            
              2
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              n
              +
              0.5
            
            )
          
        
        )
      
    
  
  ,
  
    

  
  ⁢
  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  319.




103. Perform a DFT of N/4 points on the pre-rotated data.
In this step, a common DFT of N/4 points may be used or other N/4-point FFTs may be used.
Further, in step 103, an FFT based on an address table may be used when M=N/4=P×Q, where P and Q are positive integers and P and Q are coprime and the length of the address table is M.
FIG. 2 is a flowchart of the data processing method that adopts an address table for the FFT in an embodiment of the present invention. The method includes the following steps:
1031. Create an address table I. The address table I satisfies:

I=<K1×n1+K2×n2>M n1=0, . . . , P−1

n2=0, . . . , Q−1

K1 and K2 are coprime positive integers that satisfy <K1K2>M=0, where K1K2=Z and <Z>M means Z modulo M, which is the remainder of division of Z by M. This step may be performed earlier and has no sequential relation with step 101 or 102. That is, address table I may be calculated and stored in advance. Addresses stored in address table I need to traverse 0, 1, . . . , M−1 to form a one-to-one mapping with the input data so as to determine the sequential relations of the data. When address table I satisfies the one-to-one mapping, the following three conditions need to be met:
(1) K1 is a positive integral multiple of Q and/or K2 is a positive integral multiple of P.
(2) K1 and P are coprime.
(3) K2 and Q are coprime.
Assuming P=64, Q=5, K1=65, K2=256, and M=N/4=320, address table I may be expressed by:

I=<65n1+256n2>320 

    
    
        where, n1=0, . . . , 63 and n2=0, . . . , 4.
    
    


Address table I may be stored in a table in the form of arrays, such as one-dimensional arrays of I[n2×n1] or two-dimensional arrays of I[n1][n2].
Assuming that address table I is the address table of M=320 points, K1=65, K2=256, Q=5, P=64, and Q×P=5×64 points; if address table I is stored as one-dimensional arrays of I[n2×n1]:

    
    
        Id×[Q×P]=
        {0,65,130,195,260,5,70,135,200,265,10,75,140,205,270,15,80,145,210,275,20,85,150, 215,280,25,90,155,220,285,30,95,160,225,290,35,100,165,230,295,40,105,170,235,300,45, 110,175,240,305,50,115,180,245,310,55,120,185,250,315,60,125,190,255,
        256,1,66,131,196,261,6,71,136,201,266,11,76,141,206,271,16,81,146,211,276,21,86,1 51,216,281,26,91,156,221,286,31,96,161,226,291,36,101,166,231,296,41,106,171,236,301, 46,111,176,241,306,51,116,181,246,311,56,121,186,251,316,61,126,191,
        192,257,2,67,132,197,262,7,72,137,202,267,12,77,142,207,272,17,82,147,212,277,22, 87,152,217,282,27,92,157,222,287,32,97,162,227,292,37,102,167,232,297,42,107,172,237, 302,47,112,177,242,307,52,117,182,247,312,57,122,187,252,317,62,127,
        128,193,258,3,68,133,198,263,8,73,138,203,268,13,78,143,208,273,18,83,148,213,27 8,23,88,153,218,283,28,93,158,223,288,33,98,163,228,293,38,103,168,233,298,43,108,173, 238,303,48,113,178,243,308,53,118,183,248,313,58,123,188,253,318,63,
        64,129,194,259,4,69,134,199,264,9,74,139,204,269,14,79,144,209,274,19,84,149,214, 279,24,89,154,219,284,29,94,159,224,289,34,99,164,229,294,39,104,169,234,299,44,109,1 74,239,304,49,114,179,244,309,54,119,184,249,314,59,124,189,254,319};
    
    


If address table I is stored as two-dimensional arrays of I[n1n2]:

    
    
        Idx[Q][P]=
        {{0,65,130,195,260,5,70,135,200,265,10,75,140,205,270,15,80,145,210,275,20,85,150, 215,280,25,90,155,220,285,30,95,160,225,290,35,100,165,230,295,40,105,170,235,300,45, 110,175,240,305,50,115,180,245,310,55,120,185,250,315,60,125,190,255},
    
    


{256,1,66,131,196,261,6,71,136,201,266,11,76,141,206,271,16,81,146,211,276,21,86, 151,216,281,26,91,156,221,286,31,96,161,226,291,36,101,166,231,296,41,106,171,236,301, 46,111,176,241,306,51,116,181,246,311,56,121,186,251,316,61,126,191},

    
    
        {192,257,2,67,132,197,262,7,72,137,202,267,12,77,142,207,272,17,82,147,212,277,2 2,87,152,217,282,27,92,157,222,287,32,97,162,227,292,37,102,167,232,297,42,107,172,23 7,302,47,112,177,242,307,52,117,182,247,312,57,122,187,252,317,62,127},
        {128,193,258,3,68,133,198,263,8,73,138,203,268,13,78,143,208,273,18,83,148,213,2 78,23,88,153,218,283,28,93,158,223,288,33,98,163,228,293,38,103,168,233,298,43,108,17 3,238,303,48,113,178,243,308,53,118,183,248,313,58,123,188,253,318,63},
        {64,129,194,259,4,69,134,199,264,9,74,139,204,269,14,79,144,209,274,19,84,149,21 4,279,24,89,154,219,284,29,94,159,224,289,34,99,164,229,294,39,104,169,234,299,44,109, 174,239,304,49,114,179,244,309,54,119,184,249,314,59,124,189,254,319}};
    
    


1032. Perform P-Point DFT for Q Times Based on Address Table I.
The input data of the ith (i=0, . . . , Q−1) P-point DFT is the data associated with the P address indexes corresponding to n2=i, and n1=0, . . . , P−1 in address table I. The result of the ith P-point DFT need to be applied a circular shift with a step of x to obtain the final output.
Here, x is the inverse of a cyclic modulus of




  <
  
    
      K
      1
      2
    
    Q
  
  ⁢
  
    >
    P
  




over P and satisfies





  
    <
    
      x
      ·
    
    <
    
      
        K
        1
        2
      
      Q
    
    ⁢
    
      >
      P
    
    ⁢
    
      >
      P
    
  
  =
  1.




If address table I is stored in the form of one-dimensional arrays of I[n2×n1], the P-point DFT for Q times based on address table I may be expressed by:

DFT—P(z(I+iP))x i=0, . . . , Q−1;

The input data of each P-point DFT is the data corresponding to the P continuous indexes staring from I+iP. The result of each P-point DFT is applied a circular shift with a step of x.
In this embodiment, 64-point DFT are performed for five times. The input data of the ith (i=0, . . . , 4) 64-point DFT is that associated with the 64 continuous address indexes starting from the address table array I[64i] (data after rotation). The result of the ith 64-point DFT is applied a circular shift with a step of five. For example, the result of circular shift with a step of two on the vector Z=[z0 z1 z2 z3 z4] is Z=[z0 z2 z4 z1 z3]. In addition, to further increase the efficiency of computation, the FFT may be used in stead of the DFT.
1033. Perform Q-Point DFT for P Times Based on Address Table I.
The input data of the ith (i=0, . . . , P−1) Q-point DFT is the data associated with the Q address indexes corresponding to n1=i, n2=0, . . . , Q−1 in address table I. The result of the ith Q-point DFT need to be applied a circular shift with a step of y to obtain the final output.
Here, y is the inverse of a cyclic modulus of




  <
  
    
      K
      2
      2
    
    P
  
  ⁢
  
    >
    Q
  




over Q and satisfies





  
    <
    
      y
      ·
    
    <
    
      
        K
        2
        2
      
      P
    
    ⁢
    
      >
      Q
    
    ⁢
    
      >
      Q
    
  
  =
  1.




If address table I is stored in the form of one-dimensional arrays of I[n2×n1], the Q-point DFT for P times based on address table I may be expressed by:

DFT—Q(z(I+i))y i=0, . . . , P−1;

The input data of each Q-point DFT is that corresponding to the Q indexes starting from I+i in table I each of which are separated by a step of P. The result of each Q-point DFT is applied a circular shift with a step of y.

In this embodiment, 5-point DFT are performed for 65 times. The input data of the ith (i=0, . . . 63) 5-point DFT is that associated with the 5 address indexes starting from the address table array I[i] (data after rotation) each of which are separated by a step of 64. The ith 5-point DFT is applied a circular shift with a step of four.
In the prior art, most DFTs based on coprime factor do not take into account the sequential access of the data sequence. The data obtained through a transform needs to be rearranged to obtain the final output data. Thus, the amount of computation is larger. In this embodiment, an address table is adopted in a DFT, where addressing is based on the address table, the input data is read according to the associated addresses in the address table and the output data is stored according to the associated addresses in the address table. The sequence of the data obtained after a transform is ready without any special rearrangement. Thus, the complexity of computation is reduced and efficiency of computation is higher.
104. Post-rotate the data after the DFT by using the rotation factor b·WNk+0.5 to obtain the frequency-domain data.
The transformed data is post-rotated by using the rotation factor b·WNk+0.5,




  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      4
    
    -
    1
  
  ,
  
    

  
  ⁢
  
    
      where
      ⁢
      
        
      
      ⁢
      
        W
        N
      
    
    =
    
      
        e
        
          
            -
            j
          
          ⁢
          
            
              2
              ⁢
              π
            
            N
          
        
      
      =
      
        
          cos
          ⁡
          
            (
            
              
                2
                ⁢
                π
              
              N
            
            )
          
        
        -
        
          j
          ⁢
          
            
          
          ⁢
          
            sin
            ⁡
            
              (
              
                
                  2
                  ⁢
                  π
                
                N
              
              )
            
          
        
      
    
  




and b is a constant. In this embodiment, the constant may be





  b
  =
  
    
      
        2
      
      
        N
        4
      
    
    .
  




Because WNk+0.5 in the rotation factor may be expressed in the following form:





  
    
      W
      N
      
        k
        +
        0.5
      
    
    =
    
      
        e
        
          
            -
            j
          
          ⁢
          
            
              2
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              k
              +
              0.5
            
            )
          
        
      
      =
      
        
          cos
          ⁡
          
            (
            
              
                2
                ⁢
                
                  π
                  ⁡
                  
                    (
                    
                      +
                      0.5
                    
                    )
                  
                
              
              N
            
            )
          
        
        -
        
          j
          ⁢
          
            
          
          ⁢
          
            sin
            ⁡
            
              (
              
                
                  2
                  ⁢
                  
                    π
                    ⁡
                    
                      (
                      
                        k
                        +
                        0.5
                      
                      )
                    
                  
                
                N
              
              )
            
          
        
      
    
  
  ,
  
    

  
  ⁢
  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      4
    
    -
    1
  




    
    
        and b=a, the
    
    






  N
  4




    
    
         -point data table used in step 102 may be reused in the implementation of the step.
    
    


The frequency-domain data, also the final spectrum, is




  
    X
    k
  
  ,
  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      2
    
    -
    1
  




and may be expressed by:




  
    
      
        
          
            X
            
              2
              ⁢
              k
            
          
          =
          
            Re
            ⁢
            
              {
              
                
                  b
                  ·
                  
                    W
                    N
                    
                      k
                      +
                      0.5
                    
                  
                
                ⁢
                
                  
                    ∑
                    
                      n
                      =
                      0
                    
                    
                      
                        N
                        4
                      
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      
                        u
                        n
                      
                      ·
                      a
                      ·
                      
                        W
                        N
                        
                          n
                          +
                          0.5
                        
                      
                    
                    ⁢
                    
                      W
                      
                        N
                        4
                      
                      nk
                    
                  
                
              
              }
            
          
        
      
      
        
          
            k
            =
            0
          
          ,
          ...
          ⁢
          
            
          
          ,
          
            
              N
              4
            
            -
            1
          
        
      
    
  




  
    
      
        
          
            X
            
              
                N
                2
              
              -
              1
              -
              
                2
                ⁢
                k
              
            
          
          =
          
            
              -
              Im
            
            ⁢
            
              {
              
                
                  b
                  ·
                  
                    W
                    N
                    
                      k
                      +
                      0.5
                    
                  
                
                ⁢
                
                  
                    ∑
                    
                      n
                      =
                      0
                    
                    
                      
                        N
                        4
                      
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      
                        u
                        n
                      
                      ·
                      a
                      ·
                      
                        W
                        N
                        
                          n
                          +
                          0.5
                        
                      
                    
                    ⁢
                    
                      W
                      
                        N
                        4
                      
                      nk
                    
                  
                
              
              }
            
          
        
      
      
        
          
            k
            =
            0
          
          ,
          ...
          ⁢
          
            
          
          ,
          
            
              N
              4
            
            -
            1
          
        
      
    
  



According to the time-domain to frequency-domain signal processing method in the first embodiment of the present invention, a frequency-domain to time-domain signal processing method is provided in the second embodiment of the present invention. If a forward transform is performed by using the time-domain to frequency-domain signal processing method provided in the first embodiment of the present invention, a corresponding inverse transform may be performed by using the frequency-domain to time-domain signal processing method provided in the second embodiment of the present invention.
FIG. 3 is a flowchart of the frequency-domain to time-domain signal processing method in the second embodiment of the present invention. The method includes the following steps:
301. Twiddle the frequency-domain data.
Specifically, the frequency-domain data




  
    X
    k
  
  ,
  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      2
    
    -
    1
  




is twiddled. In this embodiment, N=1280.

An intermediate variable obtained after the twiddling is




  
    (
    
      
        X
        
          2
          ⁢
          k
        
      
      +
      
        j
        ⁢
        
          
        
        ⁢
        
          X
          
            
              N
              2
            
            -
            1
            -
            
              2
              ⁢
              k
            
          
        
      
    
    )
  
  ,
  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      4
    
    -
    1.
  




302. Pre-rotate the twiddled data by using the rotation factor c·WNk+0.5.
The twiddled data is pre-rotated by using the rotation factor




  
    c
    ·
    
      W
      N
      
        k
        +
        0.5
      
    
  
  ,
  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      4
    
    -
    1.
  




    
    
        where,
    
    






  
    W
    N
  
  =
  
    
      e
      
        
          -
          j
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
      
    
    =
    
      
        cos
        ⁡
        
          (
          
            
              2
              ⁢
              π
            
            N
          
          )
        
      
      -
      
        j
        ⁢
        
          
        
        ⁢
        sin
        ⁢
        
          (
          
            
              2
              ⁢
              π
            
            N
          
          )
        
      
    
  




    
    
         and c is a constant.
    
    


In this embodiment, the constant may be




  c
  =
  
    
      2
    
    
      N
      4
    
  




and therefore the





  N
  4




-point data table used in steps 102 and 104 may also be reused in this step.

303. Perform a DFT of N/4 points on the pre-rotated data.
In this step, a common DFT of N/4 points may be used or other N/4-point FFTs may be used.
In this step, a 320-point FFT shown in FIG. 2 may be performed.
304. Post-rotate the data after the DFT by using the rotation factor d·WNn+0.5.
The transformed data is post-rotated by using the rotation factor d·WNn+0.5,




  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      4
    
    =
    1
  
  ,
  
    

  
  ⁢
  where
  ,
  
    
  
  ⁢
  
    
      W
      N
    
    =
    
      
        e
        
          
            -
            j
          
          ⁢
          
            
              2
              ⁢
              π
            
            N
          
        
      
      =
      
        
          cos
          ⁡
          
            (
            
              
                2
                ⁢
                π
              
              N
            
            )
          
        
        -
        
          j
          ⁢
          
            
          
          ⁢
          
            sin
            ⁡
            
              (
              
                
                  2
                  ⁢
                  π
                
                N
              
              )
            
          
        
      
    
  




and d is a constant.

In this embodiment, the constant may be




  d
  =
  
    
      2
    
    
      N
      4
    
  




and therefore the





  N
  4




-point data table used in steps 102 and 104 may also be reused in this step.

The data obtained after the above processing is yn:




  
    y
    n
  
  =
  
    {
    
      
        
          
            
              
                (
                
                  -
                  j
                
                )
              
              ·
              d
              ·
              
                W
                N
                
                  n
                  +
                  0.5
                
              
            
            ⁢
            
              
                ∑
                
                  k
                  =
                  0
                
                
                  
                    N
                    4
                  
                  -
                  1
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          
                            (
                            
                              
                                X
                                
                                  2
                                  ⁢
                                  k
                                
                              
                              +
                              
                                j
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  X
                                  
                                    
                                      N
                                      2
                                    
                                    -
                                    1
                                    -
                                    
                                      2
                                      ⁢
                                      k
                                    
                                  
                                
                              
                            
                            )
                          
                          ·
                        
                      
                    
                    
                      
                        
                          c
                          ·
                          
                            W
                            N
                            
                              k
                              +
                              0.5
                            
                          
                        
                      
                    
                  
                  ]
                
                ⁢
                
                  W
                  
                    N
                    4
                  
                  nk
                
              
            
          
        
        
          
            
              n
              =
              0
            
            ,
            ...
            ⁢
            
              
            
            ,
            
              
                N
                8
              
              -
              1
            
          
        
      
      
        
          
            
              d
              ·
              
                W
                N
                
                  n
                  +
                  0.5
                
              
            
            ⁢
            
              
                ∑
                
                  k
                  =
                  0
                
                
                  
                    N
                    4
                  
                  -
                  1
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          
                            (
                            
                              
                                X
                                
                                  2
                                  ⁢
                                  k
                                
                              
                              +
                              
                                j
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  X
                                  
                                    
                                      N
                                      2
                                    
                                    -
                                    1
                                    -
                                    
                                      2
                                      ⁢
                                      k
                                    
                                  
                                
                              
                            
                            )
                          
                          ·
                        
                      
                    
                    
                      
                        
                          c
                          ·
                          
                            W
                            N
                            
                              k
                              +
                              0.5
                            
                          
                        
                      
                    
                  
                  ]
                
                ⁢
                
                  W
                  
                    N
                    4
                  
                  nk
                
              
            
          
        
        
          
            
              n
              =
              
                N
                8
              
            
            ,
            ...
            ⁢
            
              
            
            ,
            
              
                N
                4
              
              -
              1
            
          
        
      
    
  




305. Post-process the post-rotated data to obtain the time-domain data.
In this step, post-processing includes windowing and twiddling. In practice, other post-processing means may be adopted.
After windowing and twiddling, the time-domain data




  
    
      x
      ^
    
    n
  
  ,
  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      2
    
    -
    1
  




is obtained.





  {
  
    
      
        
          
            
              
                
                  
                    x
                    ^
                  
                  
                    2
                    ⁢
                    n
                  
                
                =
                
                  
                    
                      m
                      
                        2
                        ⁢
                        n
                      
                    
                    ⁢
                    
                      g
                      
                        
                          N
                          2
                        
                        +
                        
                          2
                          ⁢
                          n
                        
                      
                    
                  
                  +
                  
                    
                      Re
                      (
                      
                        u
                        n
                      
                      }
                    
                    ⁢
                    
                      g
                      
                        2
                        ⁢
                        n
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                    x
                    ^
                  
                  
                    
                      N
                      2
                    
                    -
                    1
                    -
                    
                      2
                      ⁢
                      n
                    
                  
                
                =
                
                  
                    
                      m
                      
                        
                          N
                          2
                        
                        -
                        1
                        -
                        
                          2
                          ⁢
                          n
                        
                      
                    
                    ⁢
                    
                      g
                      
                        N
                        -
                        1
                        -
                        
                          2
                          ⁢
                          n
                        
                      
                    
                  
                  -
                  
                    
                      Re
                      (
                      
                        u
                        n
                      
                      }
                    
                    ⁢
                    
                      g
                      
                        
                          N
                          2
                        
                        -
                        1
                        -
                        
                          2
                          ⁢
                          n
                        
                      
                    
                  
                
              
            
          
        
      
      
        
          
            n
            =
            0
          
          ,
          ...
          ⁢
          
            
          
          ,
          
            
              N
              4
            
            -
            1
          
        
      
    
  




    
    
        where,
    
    






  {
  
    
      
        
          
            
              
                
                  
                    
                      u
                      n
                    
                    =
                    
                      y
                      
                        n
                        +
                        
                          N
                          8
                        
                      
                    
                  
                
              
              
                
                  
                    
                      u
                      
                        n
                        +
                        
                          N
                          8
                        
                      
                    
                    =
                    
                      y
                      n
                    
                  
                
              
            
          
          
            
              
                n
                =
                0
              
              ,
              ...
              ⁢
              
                
              
              ,
              
                N
                8
              
            
          
        
      
      -
      1
    
    ,
  




    
    
         and gn is a synthesis window, which satisfies the perfect reconstruction condition together with the analysis window hn, that is,
    
    






  
    
      
        
          h
          n
        
        ⁢
        
          g
          n
        
      
      +
      
        
          h
          
            n
            +
            
              N
              2
            
          
        
        ⁢
        
          g
          
            n
            +
            
              N
              2
            
          
        
      
    
    =
    1
  
  ,
  
    n
    =
    0
  
  ,
  ...
  ,
  
    N
    -
    
      N
      2
    
    -
    1.
  




    
    
         mn is the buffered data of the previous frame and the update of mn satisfies:
    
    






   
  
    {
    
      
        
          
            
              m
              
                
                  N
                  4
                
                -
                1
                -
                
                  2
                  ⁢
                  n
                
              
            
            =
            
              
                m
                
                  
                    N
                    4
                  
                  +
                  
                    2
                    ⁢
                    n
                  
                
              
              =
              
                Im
                ⁢
                
                  {
                  
                    u
                    n
                  
                  }
                
              
            
          
        
        
          
            
              n
              =
              0
            
            ,
            ...
            ⁢
            
              
            
            ,
            
              
                N
                8
              
              -
              1
            
          
        
      
      
        
          
            
              m
              
                
                  2
                  ⁢
                  n
                
                -
                
                  N
                  4
                
              
            
            =
            
              
                m
                
                  
                    
                      3
                      ⁢
                      N
                    
                    4
                  
                  -
                  1
                  -
                  
                    2
                    ⁢
                    n
                  
                
              
              =
              
                Im
                ⁢
                
                  {
                  
                    u
                    n
                  
                  }
                
              
            
          
        
        
          
            
              n
              =
              
                N
                8
              
            
            ,
            ...
            ⁢
            
              
            
            ,
            
              
                N
                4
              
              -
              1
            
          
        
      
    
  




When the time-domain to frequency-domain signal processing method provided in the first embodiment of the present invention and the frequency-domain to time-domain signal processing method provided in the second embodiment of the present invention are used, if the constants a, b, c, and d in the forward transform and the inverse transform satisfy a×b×c×d=4/N, the result of the inverse transform is ready for a perfect reconstruction. In practice, the perfect reconstruction is unnecessarily required.
For example, one can choose




  a
  =
  
    b
    =
    
      c
      =
      
        d
        =
        
          
            
              2
            
            
              N
              4
            
          
          .
        
      
    
  




Then, the forward transform and the inverse transform require only storage of any





  N
  4




-point data table in





  
    
      
        2
      
      
        N
        4
      
    
    ⁢
    
      cos
      ⁡
      
        (
        
          
            
              2
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              n
              +
              0.5
            
            )
          
        
        )
      
    
    ⁢
    
      
    
    ⁢
    and
    ⁢
    
      
    
    ⁢
    
      
        2
      
      
        N
        4
      
    
    ⁢
    
      sin
      ⁡
      
        (
        
          
            
              2
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              n
              +
              0.5
            
            )
          
        
        )
      
    
  
  ⁢
  
    
  




to complete the pre-rotation and post-rotation. If the constants a, b, c, and d are not equal, an appropriate data table needs to be stored according to the practical conditions to complete the pre-rotation and post-rotation.

Another time-domain to frequency-domain signal processing method provided in an embodiment of the present invention includes:
1. Pre-process the time-domain data.
2. Pre-rotate the pre-processed data by using the rotation factor a·WNn+0.5.
3. Perform a DFT of N/4 points on the pre-rotated data.
4. Post-rotate the data after the DFT by using the rotation factor b·WNk+0.5 and normalization factor A to obtain the frequency-domain data,
where, a and b are constants, N is the length of the time-domain data,




  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      4
    
    -
    1
  
  ,
  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      4
    
    -
    1
  
  ,
  
    
      and
      ⁢
      
        
      
      ⁢
      
        W
        N
      
    
    =
    
      
        e
        
          *
          
            -
            j
          
          ⁢
          
            
              2
              ⁢
              π
            
            N
          
        
      
      .
    
  




All processing on the data before the pre-rotation may be described as pre-processing. In this embodiment, for example, pre-processing may be windowing and twiddling or only twiddling.
Another frequency-domain to time-domain signal processing method provided in an embodiment of the present invention includes:
1. Twiddle the frequency-domain data.
2. Pre-rotate the twiddled data by using the rotation factor c·WNk+0.5.
3. Perform a DFT of N/4 points on the pre-rotated data.
4. Post-rotate the data after the DFT by using the rotation factor d·WNn+0.5 and the normalization factor B.
5. Post-process the post-rotated data to obtain the time-domain data,
where, c and d are constants, N is twice the length of the frequency-domain data,




  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      4
    
    -
    1
  
  ,
  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      4
    
    -
    1
  
  ,
  
    
      and
      ⁢
      
        
      
      ⁢
      
        W
        N
      
    
    =
    
      
        e
        
          
            -
            j
          
          ⁢
          
            
              2
              ⁢
              π
            
            N
          
        
      
      .
    
  




All processing on the data after the post-rotation may be described as post-processing. The content of post-processing depends on the content of pre-processing in the corresponding time-domain to frequency-domain signal processing method. In this embodiment, for example, if the pre-processing is windowing and twiddling, the post-processing is also windowing and twiddling; if the pre-processing is twiddling, the post-processing is also twiddling.
The time-domain to frequency-domain signal processing is also called a forward transform and the frequency-domain to time-domain signal processing is also called an inverse transform. When a×b×c×d=4/N in the forward transform and the inverse transform, using the output data of the forward transform as the input data of the inverse transform. A perfect reconstruction can be achieved, because the result of the inverse transform can recover the data before the forward transform. For example, one may choose



  
    
      
        a
        =
        
          b
          =
          
            c
            =
            
              d
              =
              1
            
          
        
      
      ,
      
        
          A
          ·
          B
        
        =
        
          4
          N
        
      
      ,
      
        

      
      ⁢
      or
    
    ⁢
    
      
    
  




  
    
      a
      =
      
        b
        =
        
          c
          =
          
            d
            =
            
              
                2
              
              
                N
                4
              
            
          
        
      
    
    ,
    
      A
      =
      
        B
        =
        1.
      
    
  



In practice, the perfect reconstruction is unnecessarily required.

WNn+0.5 in the rotation factor adopted by the signal processing method provided in this embodiment may be expanded to:



  
    
      W
      N
      
        n
        +
        0.5
      
    
    =
    
      
        e
        
          
            -
            j
          
          ⁢
          
            
              2
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              n
              +
              0.5
            
            )
          
        
      
      =
      
        
          cos
          ⁡
          
            (
            
              
                2
                ⁢
                
                  π
                  ⁡
                  
                    (
                    
                      n
                      +
                      0.5
                    
                    )
                  
                
              
              N
            
            )
          
        
        -
        
          j
          ⁢
          
            
          
          ⁢
          
            sin
            ⁡
            
              (
              
                
                  2
                  ⁢
                  
                    π
                    ⁡
                    
                      (
                      
                        n
                        +
                        0.5
                      
                      )
                    
                  
                
                N
              
              )
            
          
        
      
    
  




  
    where
    ,
    
      
        
          cos
          ⁡
          
            (
            
              
                
                  2
                  ⁢
                  π
                
                N
              
              ⁢
              
                (
                
                  n
                  +
                  0.5
                
                )
              
            
            )
          
        
        =
        
          sin
          ⁡
          
            (
            
              
                
                  2
                  ⁢
                  π
                
                N
              
              ⁢
              
                (
                
                  
                    N
                    4
                  
                  -
                  1
                  -
                  n
                  +
                  0.5
                
                )
              
            
            )
          
        
      
      ⁢
      
        
      
      ;
    
  




  
    
      
        sin
        ⁡
        
          (
          
            
              
                2
                ⁢
                π
              
              N
            
            ⁢
            
              (
              
                n
                +
                0.5
              
              )
            
          
          )
        
      
      ⁢
      
        
      
      =
      
        cos
        ⁡
        
          (
          
            
              
                2
                ⁢
                π
              
              N
            
            ⁢
            
              (
              
                
                  N
                  4
                
                -
                1
                -
                n
                +
                0.5
              
              )
            
          
          )
        
      
    
    ;
  



Therefore, WNn+0.5 is symmetrical. It is only necessary to store any




  N
  4




points data table of





  
    cos
    ⁡
    
      (
      
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            n
            +
            0.5
          
          )
        
      
      )
    
  
  ⁢
  
    
  
  ⁢
  or
  ⁢
  
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            n
            +
            0.5
          
          )
        
      
      )
    
  




to complete the pre-rotation and post-rotation.

In the traditional MDCT, to obtain the MDCT spectrum coefficients, the rotation factor WNk+0.125 needs to be adopted and WNk+0.125 is not symmetrical. Therefore, at least cosine values of N/4 points and sine values of N/4 points need to be stored during pre-rotation and post-rotation. The rotation factors adopted by the signal processing method provided in the embodiment of the present invention are products of a constant and WNk+0.125. Because WNk+0.125 is symmetrical, it is only necessary to store a data table of N/4 points to complete the pre-rotation and post-rotation. Compared with the prior art, the signal processing method in this embodiment significantly reduces the amount and complexity of storage, reduces the consumption of storage resources, and increases the transform efficiency.
Exemplary embodiments of the signal processing method of the present invention will be described.
Assume that the sampling frequency of a wideband audio codec is 16 kHz and that the processed frame size is 20 ms. One frame contains 320 sampling points. The time-domain to frequency-domain signal processing method provided in an embodiment of the present invention is used to perform forward transforms at the encoder. The input data of a forward transform is




  
    x
    n
  
  ;
  
    x
    
      0
      ∼
      
        
          N
          2
        
        -
        1
      
    
  




is the cached 320-point data of the previous frame and





  x
  
    
      N
      2
    
    ∼
    
      N
      -
      1
    
  




is the 320-point data of the current frame; the transform section length N is 640 points. FIG. 4 is a flowchart of the signal processing method provided in the third embodiment of the present invention. The method includes:

401. Pre-process the time-domain input data.
In this step, pre-processing includes windowing and twiddling.
If xn is 640-point time-domain input data and hn is a 640-point window function that satisfies the perfect reconstruction condition, the windowed data yn satisfies:

yn=hnxn n=0, . . . , N−1,

    
    
        where, hn is a window function that satisfies the perfect reconstruction condition, which may be described as an analysis window function.
    
    


The windowed data is twiddled and the obtained data un satisfies:



  
    
      u
      n
    
    =
    
      (
      
        
          z
          
            2
            ⁢
            n
          
        
        +
        
          j
          ⁢
          
            
          
          ⁢
          
            z
            
              
                N
                2
              
              -
              1
              -
              
                2
                ⁢
                n
              
            
          
        
      
      )
    
  




  
    
      n
      =
      0
    
    ,
    ...
    ⁢
    
      
    
    ,
    
      
        N
        4
      
      -
      1
    
  




  
    where
    ⁢
    
      :
    
  




  
    {
    
      
        
          
            
              
                
                  
                    
                      
                        z
                        
                          n
                          +
                          
                            N
                            4
                          
                        
                      
                      =
                      
                        
                          y
                          n
                        
                        -
                        
                          y
                          
                            
                              N
                              2
                            
                            -
                            1
                            -
                            n
                          
                        
                      
                    
                  
                
                
                  
                    
                      
                        z
                        
                          
                            N
                            4
                          
                          -
                          1
                          -
                          n
                        
                      
                      =
                      
                        
                          -
                          
                            y
                            
                              N
                              -
                              1
                              -
                              n
                            
                          
                        
                        -
                        
                          y
                          
                            
                              N
                              2
                            
                            +
                            n
                          
                        
                      
                    
                  
                
              
            
            
              
                
                  n
                  =
                  0
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  
                    N
                    4
                  
                  -
                  1
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        or
      
      ,
      
        

      
      ⁢
      
        {
        
          
            
              
                
                  z
                  n
                
                =
                
                  
                    -
                    
                      y
                      
                        n
                        +
                        
                          
                            3
                            ⁢
                            N
                          
                          4
                        
                      
                    
                  
                  -
                  
                    y
                    
                      
                        
                          3
                          ⁢
                          N
                        
                        4
                      
                      -
                      1
                      -
                      n
                    
                  
                
              
            
            
              
                
                  n
                  =
                  0
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  
                    N
                    4
                  
                  -
                  1
                
              
            
          
          
            
              
                
                  z
                  n
                
                =
                
                  
                    y
                    
                      n
                      -
                      
                        N
                        4
                      
                    
                  
                  -
                  
                    y
                    
                      
                        
                          3
                          ⁢
                          N
                        
                        4
                      
                      -
                      1
                      -
                      n
                    
                  
                
              
            
            
              
                
                  n
                  =
                  
                    N
                    4
                  
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  
                    N
                    2
                  
                  -
                  1
                
              
            
          
        
      
    
  



402. Pre-rotate the pre-processed data by using the rotation factor a·WNn+0.5.
The twiddled data un is pre-rotated by using the rotation factor a·WNn+0.5,




  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      4
    
    -
    1
  
  ,
  
    

  
  ⁢
  where
  ,
  
    

  
  ⁢
  
    
      W
      N
    
    =
    
      
        e
        
          
            -
            j
          
          ⁢
          
            
              2
              ⁢
              π
            
            N
          
        
      
      =
      
        
          cos
          ⁡
          
            (
            
              
                2
                ⁢
                π
              
              N
            
            )
          
        
        -
        
          j
          ⁢
          
            
          
          ⁢
          
            sin
            ⁡
            
              (
              
                
                  2
                  ⁢
                  π
                
                N
              
              )
            
          
        
      
    
  




and a is a constant.

In this embodiment, a=1 and the rotation factor may be expressed by:




  
    W
    N
    
      n
      +
      0.5
    
  
  =
  
    
      e
      
        
          -
          j
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            n
            +
            0.5
          
          )
        
      
    
    =
    
      
        cos
        ⁡
        
          (
          
            
              2
              ⁢
              
                π
                ⁡
                
                  (
                  
                    n
                    +
                    0.5
                  
                  )
                
              
            
            N
          
          )
        
      
      -
      
        j
        ⁢
        
          
        
        ⁢
        
          sin
          ⁡
          
            (
            
              
                2
                ⁢
                
                  π
                  ⁡
                  
                    (
                    
                      n
                      +
                      0.5
                    
                    )
                  
                
              
              N
            
            )
          
        
      
    
  




The equation satisfies



  
    
      cos
      ⁡
      
        (
        
          
            
              2
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              n
              +
              0.5
            
            )
          
        
        )
      
    
    =
    
      
        sin
        ⁡
        
          (
          
            
              
                2
                ⁢
                π
              
              N
            
            ⁢
            
              (
              
                
                  N
                  4
                
                -
                1
                -
                n
                +
                0.5
              
              )
            
          
          )
        
      
      ⁢
      
        
          
        
        ⁢
        
          
        
      
      ⁢
      and
    
  




  
    
      sin
      ⁡
      
        (
        
          
            
              2
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              n
              +
              0.5
            
            )
          
        
        )
      
    
    =
    
      
        cos
        ⁡
        
          (
          
            
              
                2
                ⁢
                π
              
              N
            
            ⁢
            
              (
              
                
                  N
                  4
                
                -
                1
                -
                n
                +
                0.5
              
              )
            
          
          )
        
      
      .
    
  



It is only necessary to store any




  N
  4




points data table of





  
    cos
    ⁡
    
      (
      
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            n
            +
            0.5
          
          )
        
      
      )
    
  
  ⁢
  
    
  
  ⁢
  or
  ⁢
  
    
  
  ⁢
  
    sin
    ⁡
    
      (
      
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            n
            +
            0.5
          
          )
        
      
      )
    
  




in the specific implementation. In this embodiment, a sine table of N/4 points is stored. Values in the sine table are expressed by





  
    sin
    ⁡
    
      (
      
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            n
            +
            0.5
          
          )
        
      
      )
    
  
  ,




n=0, . . . , 159.

403. Perform a DFT of N/4 inputs on the pre-rotated data.
In this step, a common DFT of M=N/4=160 points may be used or other FFTs of M=N/4=160 points may be used.
404. Post-rotate the data after the DFT by using the rotation factor b·WNk+0.5 and the normalization factor A to obtain the frequency-domain output data.
The transformed data is post-rotated by using the rotation factor b·WNk+0.5,




  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      4
    
    -
    1
  
  ,
  
    

  
  ⁢
  where
  ,
  
    

  
  ⁢
  
    
      W
      N
    
    =
    
      
        e
        
          
            -
            j
          
          ⁢
          
            
              2
              ⁢
              π
            
            N
          
        
      
      =
      
        
          cos
          ⁡
          
            (
            
              
                2
                ⁢
                π
              
              N
            
            )
          
        
        -
        
          j
          ⁢
          
            
          
          ⁢
          
            sin
            ⁡
            
              (
              
                
                  2
                  ⁢
                  π
                
                N
              
              )
            
          
        
      
    
  




and b is a constant. In this embodiment, b=1 and the rotation factor may be expressed by:





  
    
      W
      N
      
        k
        +
        0.5
      
    
    =
    
      
        e
        
          
            -
            j
          
          ⁢
          
            
              2
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              k
              +
              0.5
            
            )
          
        
      
      =
      
        
          cos
          ⁡
          
            (
            
              
                2
                ⁢
                
                  π
                  ⁡
                  
                    (
                    
                      k
                      +
                      0.5
                    
                    )
                  
                
              
              N
            
            )
          
        
        -
        
          j
          ⁢
          
            
          
          ⁢
          
            sin
            ⁡
            
              (
              
                
                  2
                  ⁢
                  
                    π
                    ⁡
                    
                      (
                      
                        k
                        +
                        0.5
                      
                      )
                    
                  
                
                N
              
              )
            
          
        
      
    
  
  ,
  
    

  
  ⁢
  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      4
    
    -
    1
  




As b=a, the




  N
  4




-point data table used in step 402 may be used in the implementation of this step.

The final data




  
    X
    k
  
  ,
  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      2
    
    -
    1
  




is obtained.




  
    
      
        
          
            X
            
              2
              ⁢
              k
            
          
          =
          
            
              A
              ·
              Re
            
            ⁢
            
              {
              
                
                  b
                  ·
                  
                    W
                    N
                    
                      k
                      +
                      0.5
                    
                  
                
                ⁢
                
                  
                    ∑
                    
                      n
                      =
                      0
                    
                    
                      
                        N
                        4
                      
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      
                        u
                        n
                      
                      ·
                      a
                      ·
                      
                        W
                        N
                        
                          n
                          +
                          0.5
                        
                      
                    
                    ⁢
                    
                      W
                      
                        N
                        4
                      
                      nk
                    
                  
                
              
              }
            
          
        
      
      
        
          
            k
            =
            0
          
          ,
          ...
          ⁢
          
            
          
          ,
          
            
              N
              4
            
            -
            1
          
        
      
    
  




  
    
      
        
          
            X
            
              
                N
                2
              
              -
              1
              -
              
                2
                ⁢
                k
              
            
          
          =
          
            
              A
              ·
              
                -
                Im
              
            
            ⁢
            
              {
              
                
                  b
                  ·
                  
                    W
                    N
                    
                      k
                      +
                      0.5
                    
                  
                
                ⁢
                
                  
                    ∑
                    
                      n
                      =
                      0
                    
                    
                      
                        N
                        4
                      
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      
                        u
                        n
                      
                      ·
                      a
                      ·
                      
                        W
                        N
                        
                          n
                          +
                          0.5
                        
                      
                    
                    ⁢
                    
                      W
                      
                        N
                        4
                      
                      nk
                    
                  
                
              
              }
            
          
        
      
      
        
          
            k
            =
            0
          
          ,
          ...
          ⁢
          
            
          
          ,
          
            
              N
              4
            
            -
            1
          
        
      
    
  



    
    
        where, A is a normalization factor and a constant.
    
    


In this embodiment, the constant A may be




  
    2
    
      N
      2
    
  
  .




According to the time-domain to frequency-domain signal processing method provided in the third embodiment of the present invention, a frequency-domain to time-domain signal processing method is provided in the fourth embodiment of the present invention. When the time-domain to frequency-domain signal processing method provided in the third embodiment of the present invention is used to perform a transform, the frequency-domain to time-domain signal processing method provided in the fourth embodiment of the present invention may be used to perform a corresponding inverse transform.
FIG. 5 is a flowchart of the frequency-domain to time-domain signal processing method in the fourth embodiment of the present invention. The method includes:
501. Twiddle the frequency-domain input data.
In this embodiment, the data




  
    X
    k
  
  ,
  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      /
      2
    
    -
    1
  




is twiddled to obtain the intermediate variable





  
    (
    
      
        X
        
          2
          ⁢
          
            
          
          ⁢
          k
        
      
      +
      
        j
        ⁢
        
          
        
        ⁢
        
          X
          
            
              N
              /
              2
            
            -
            1
            -
            
              2
              ⁢
              
                
              
              ⁢
              k
            
          
        
      
    
    )
  
  ,
  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      /
      4
    
    -
    1
  
  ,




where N=640.

502. Pre rotate the twiddled data by using the rotation factor c·WNk+0.5.
The twiddled data is pre-rotated by using the rotation factor c·WNk+0.5,




  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      /
      4
    
    -
    1
  
  ,
  where
  ,
  
    

  
  ⁢
  
    
      W
      N
    
    =
    
      
        e
        
          
            -
            j
          
          ⁢
          
            
              2
              ⁢
              π
            
            N
          
        
      
      =
      
        
          cos
          ⁡
          
            (
            
              
                2
                ⁢
                π
              
              N
            
            )
          
        
        -
        
          j
          ⁢
          
            
          
          ⁢
          
            sin
            ⁡
            
              (
              
                
                  2
                  ⁢
                  π
                
                N
              
              )
            
          
        
      
    
  




and c is a constant. In this embodiment, c=1 and the





  N
  /
  4




-point data table used in steps 402 and 404 in the time-domain to frequency-domain signal processing method provided in the third embodiment of the present invention may also be reused.

503. Perform a DFT of N/4 points on the pre-rotated data.
A DFT of M=N/4=160 points is performed on the rotated data. In this step, a common DFT of 160 points may be used or other 160-point FFTs may be used.
504. Post-rotate the data after the DFT by using the rotation factor d·WNn+0.5 and the normalization factor B.
The transformed data is post-rotated by using the rotation factor d·WNn+0.5,




  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      /
      4
    
    -
    1
  
  ,
  where
  ,
  
    

  
  ⁢
  
    
      W
      N
    
    =
    
      
        e
        
          
            -
            j
          
          ⁢
          
            
              2
              ⁢
              π
            
            N
          
        
      
      =
      
        
          cos
          ⁡
          
            (
            
              
                2
                ⁢
                π
              
              N
            
            )
          
        
        -
        
          j
          ⁢
          
            
          
          ⁢
          
            sin
            ⁡
            
              (
              
                
                  2
                  ⁢
                  π
                
                N
              
              )
            
          
        
      
    
  




and d is a constant. The data obtained after the above processing is yn:





  
    y
    n
  
  =
  
    {
    
      
        
          
            
              B
              ·
              
                (
                
                  -
                  j
                
                )
              
              ·
              d
              ·
              
                W
                N
                
                  n
                  +
                  0.5
                
              
            
            ⁢
            
              
                ∑
                
                  k
                  =
                  0
                
                
                  
                    N
                    /
                    4
                  
                  -
                  1
                
              
              ⁢
              
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          
                            (
                            
                              
                                X
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  k
                                
                              
                              +
                              
                                j
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  X
                                  
                                    
                                      N
                                      /
                                      2
                                    
                                    -
                                    1
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      k
                                    
                                  
                                
                              
                            
                            )
                          
                          ·
                        
                      
                    
                    
                      
                        
                          c
                          ·
                          
                            W
                            N
                            
                              k
                              +
                              0.5
                            
                          
                        
                      
                    
                  
                  ]
                
                ⁢
                
                  W
                  
                    N
                    /
                    4
                  
                  nk
                
              
            
          
        
        
          
            
              n
              =
              0
            
            ,
            ...
            ⁢
            
              
            
            ,
            
              
                N
                /
                8
              
              -
              1
            
          
        
      
      
        
          
            
              B
              ·
              d
              ·
              
                W
                N
                
                  n
                  +
                  0.5
                
              
            
            ⁢
            
              
                ∑
                
                  k
                  =
                  0
                
                
                  
                    N
                    /
                    4
                  
                  -
                  1
                
              
              ⁢
              
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          
                            (
                            
                              
                                X
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  k
                                
                              
                              +
                              
                                j
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  X
                                  
                                    
                                      N
                                      /
                                      2
                                    
                                    -
                                    1
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      k
                                    
                                  
                                
                              
                            
                            )
                          
                          ·
                        
                      
                    
                    
                      
                        
                          c
                          ·
                          
                            W
                            N
                            
                              k
                              +
                              0.5
                            
                          
                        
                      
                    
                  
                  ]
                
                ⁢
                
                  W
                  
                    N
                    /
                    4
                  
                  nk
                
              
            
          
        
        
          
            
              
                
                  
                    n
                    =
                    
                      N
                      /
                      8
                    
                  
                  ,
                  ...
                  ⁢
                  
                    
                  
                  ,
                
              
            
            
              
                
                  
                    N
                    /
                    4
                  
                  -
                  1
                
              
            
          
        
      
    
  




B is a normalization factor and a constant. In this embodiment, the constant B may be




  
    2
    
      N
      2
    
  
  .




In this embodiment, d=1 and the





  N
  /
  4




-point data table used in steps 402 and 404 in the time-domain to frequency-domain signal processing method provided in the third embodiment of the present invention may also be reused.

505. Post-process the post-rotated data to obtain the time-domain output data.
In this step, post-processing includes windowing and twiddling. In practice, other post-processing means may be adopted.
After the windowing and twiddling, the output data




  
    
      x
      ^
    
    n
  
  ,
  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      /
      2
    
    -
    1
  




is obtained, which satisfies:





  {
  
    
      
        
          
            
              
                
                  
                    x
                    ^
                  
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    n
                  
                
                =
                
                  
                    
                      m
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        n
                      
                    
                    ⁢
                    
                      g
                      
                        
                          N
                          /
                          2
                        
                        +
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          n
                        
                      
                    
                  
                  +
                  
                    Re
                    ⁢
                    
                      {
                      
                        u
                        n
                      
                      }
                    
                    ⁢
                    
                      g
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        n
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                    x
                    ^
                  
                  
                    
                      N
                      /
                      2
                    
                    -
                    1
                    -
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      n
                    
                  
                
                =
                
                  
                    
                      m
                      
                        
                          N
                          /
                          2
                        
                        -
                        1
                        -
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          n
                        
                      
                    
                    ⁢
                    
                      g
                      
                        N
                        -
                        1
                        -
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          n
                        
                      
                    
                  
                  -
                  
                    Re
                    ⁢
                    
                      {
                      
                        u
                        n
                      
                      }
                    
                    ⁢
                    
                      g
                      
                        
                          N
                          /
                          2
                        
                        -
                        1
                        -
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          n
                        
                      
                    
                  
                
              
            
          
        
        ⁢
        n
      
      =
      0
    
    ,
    ...
    ⁢
    
      
    
    ,
    
      
        N
        /
        4
      
      -
      1
    
  




    
    
        where,
    
    






  {
  
    
      
        
          
            
              
                
                  u
                  n
                
                =
                
                  y
                  
                    n
                    +
                    
                      N
                      /
                      8
                    
                  
                
              
            
          
          
            
              
                
                  u
                  
                    n
                    +
                    
                      N
                      /
                      8
                    
                  
                
                =
                
                  y
                  n
                
              
            
          
        
        ⁢
        n
      
      =
      0
    
    ,
    ...
    ⁢
    
      
    
    ,
    
      
        N
        /
        8
      
      -
      1
    
    ,
  




    
    
         and gn is a synthesis window, which satisfies the perfect reconstruction condition together with the analysis window hn, that is,
    
    






  
    
      
        
          h
          n
        
        ⁢
        
          g
          n
        
      
      +
      
        
          h
          
            n
            +
            
              N
              /
              2
            
          
        
        ⁢
        
          g
          
            n
            +
            
              N
              /
              2
            
          
        
      
    
    =
    1
  
  ,
  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    N
    -
    
      N
      /
      2
    
    -
    1.
  




    
    
         mn is the buffered data of the previous frame and the update of mn satisfies:
    
    






   
  
    {
    
      
        
          
            
              m
              
                
                  N
                  /
                  4
                
                -
                1
                -
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  n
                
              
            
            =
            
              
                m
                
                  
                    N
                    /
                    4
                  
                  +
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    n
                  
                
              
              =
              
                Im
                ⁢
                
                  {
                  
                    u
                    n
                  
                  }
                
              
            
          
        
        
          
            
              n
              =
              0
            
            ,
            ...
            ⁢
            
              
            
            ,
            
              
                N
                /
                8
              
              -
              1
            
          
        
      
      
        
          
            
              m
              
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  n
                
                -
                
                  N
                  /
                  4
                
              
            
            =
            
              
                m
                
                  
                    3
                    ⁢
                    
                      N
                      /
                      4
                    
                  
                  -
                  1
                  -
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    n
                  
                
              
              =
              
                Im
                ⁢
                
                  {
                  
                    u
                    n
                  
                  }
                
              
            
          
        
        
          
            
              n
              =
              
                N
                /
                8
              
            
            ,
            ...
            ⁢
            
              
            
            ,
            
              
                N
                /
                4
              
              -
              1
            
          
        
      
    
  




To satisfy the perfect reconstruction condition, a forward, inverse, or forward-inverse transform may be multiplied by a specific normalization factor. In this embodiment, the forward and inverse transforms are multiplied by a same normalization factor. Optionally, only the forward transform or the inverse transform is multiplied by the specific normalization factor or the forward transform and the inverse transform are multiplied by different normalization factors. A perfect reconstruction can be completed so long as a×b×c×d×A×B=4/N.
Further, when the normalization factor is used, an address table may also be adopted to perform FFTs. Assume that in a super wideband audio codec, the sampling frequency of the super wideband audio encoder is 32 kHz and that the processed frame size is 20 ms. That is, one frame contains 640 sampling points. A forward transform is performed on the data. If xn is the input data of the forward transform,




  x
  
    0
    ∼
    
      
        N
        /
        2
      
      -
      1
    
  




is the cached 640-point data of the previous frame and





  X
  
    
      N
      2
    
    ∼
    
      N
      -
      1
    
  




is the 640-point data of the current frame.

FIG. 6 is a flowchart of a time-domain to frequency-domain signal processing method in the fifth embodiment of the present invention. The method includes the following steps:
601. Pre-process the time-domain data.
The pre-processing includes windowing and twiddling.
If xn is 1280-point time-domain input data and hn is a 1280-point window function that satisfies the perfect reconstruction condition, the windowed data yn satisfies:

yn=hnxn n=0, . . . , N−1,

The windowed data is twiddled and the obtained data un satisfies:



  
    
      u
      n
    
    =
    
      (
      
        
          z
          
            2
            ⁢
            
              
            
            ⁢
            n
          
        
        +
        
          j
          ⁢
          
            
          
          ⁢
          
            z
            
              
                N
                2
              
              -
              1
              -
              
                2
                ⁢
                n
              
            
          
        
      
      )
    
  




  
    
      n
      =
      0
    
    ,
    ...
    ⁢
    
      
    
    ,
    
      
        N
        4
      
      -
      
        1
        ⁢
        
          

        
        ⁢
        where
        ⁢
        
          :
        
      
    
  




  
    {
    
      
        
          
            
              
                
                  
                    z
                    
                      n
                      +
                      
                        N
                        4
                      
                    
                  
                  =
                  
                    
                      y
                      n
                    
                    -
                    
                      y
                      
                        
                          N
                          2
                        
                        -
                        1
                        -
                        n
                      
                    
                  
                
              
            
            
              
                
                  
                    z
                    
                      
                        N
                        4
                      
                      -
                      1
                      -
                      n
                    
                  
                  =
                  
                    
                      -
                      
                        y
                        
                          N
                          -
                          1
                          -
                          n
                        
                      
                    
                    -
                    
                      y
                      
                        
                          N
                          2
                        
                        +
                        n
                      
                    
                  
                
              
            
          
          ⁢
          n
        
        =
        0
      
      ,
      ...
      ⁢
      
        
      
      ,
      
        
          N
          4
        
        -
        
          1
          ⁢
          
            

          
          ⁢
          or
        
      
      ,
      
        

      
      ⁢
      
        {
        
          
            
              
                
                  z
                  n
                
                =
                
                  
                    -
                    
                      y
                      
                        n
                        +
                        
                          
                            3
                            ⁢
                            N
                          
                          4
                        
                      
                    
                  
                  -
                  
                    y
                    
                      
                        
                          3
                          ⁢
                          N
                        
                        4
                      
                      -
                      1
                      -
                      n
                    
                  
                
              
            
            
              
                
                  n
                  =
                  0
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  
                    N
                    4
                  
                  -
                  1
                
              
            
          
          
            
              
                
                  z
                  n
                
                =
                
                  
                    y
                    
                      n
                      -
                      
                        N
                        4
                      
                    
                  
                  -
                  
                    y
                    
                      
                        
                          3
                          ⁢
                          N
                        
                        4
                      
                      -
                      1
                      -
                      n
                    
                  
                
              
            
            
              
                
                  n
                  =
                  
                    N
                    4
                  
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  
                    N
                    2
                  
                  -
                  1
                
              
            
          
        
      
    
  



The window functions satisfy the perfect reconstruction condition when the analysis window function hn in this step and the synthesis window function gn used in the inverse transform satisfy the following equation:




  
    
      
        
          h
          n
        
        ⁢
        
          g
          n
        
      
      +
      
        
          h
          
            n
            +
            
              N
              2
            
          
        
        ⁢
        
          g
          
            n
            +
            
              N
              2
            
          
        
      
    
    =
    1
  
  ,
  
    n
    +
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    N
    -
    
      N
      2
    
    -
    1
  




602. Pre-rotate the pre-processed data by using the rotation factor a·WNn+0.5.
The twiddled data un is pre-rotated by using the rotation factor a·WNn+0.5. The data obtained after the rotation is un·a·WNn+0.5,




  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      4
    
    -
    1
  
  ,




where,





  
    w
    N
  
  =
  
    
      e
      
        
          -
          j
        
        ⁢
        
          
            2
            ⁢
            
              
            
            ⁢
            π
          
          N
        
      
    
    =
    
      
        cos
        ⁡
        
          (
          
            
              2
              ⁢
              
                
              
              ⁢
              π
            
            N
          
          )
        
      
      -
      
        j
        ⁢
        
          
        
        ⁢
        
          sin
          ⁡
          
            (
            
              
                2
                ⁢
                
                  
                
                ⁢
                π
              
              N
            
            )
          
        
      
    
  




and a is a constant. To further reduce the complexity of computation, the normalization factor may be directly combined with the rotation factor. In this embodiment, the constant a may be





  a
  =
  
    
      
        2
      
      
        N
        4
      
    
    ·
    
      W
      N
      
        n
        +
        0.5
      
    
  




in the rotation factor may be expressed in the following form:




  
    
      
        
          
            W
            N
            
              n
              +
              0.5
            
          
          =
          
            e
            
              
                -
                j
              
              ⁢
              
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  π
                
                N
              
              ⁢
              
                (
                
                  n
                  +
                  0.5
                
                )
              
            
          
        
      
    
    
      
        
          =
          
            
              cos
              ⁡
              
                (
                
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    
                      π
                      ⁡
                      
                        (
                        
                          n
                          +
                          0.5
                        
                        )
                      
                    
                  
                  N
                
                )
              
            
            -
            
              j
              ⁢
              
                
              
              ⁢
              
                sin
                ⁡
                
                  (
                  
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      
                        π
                        ⁡
                        
                          (
                          
                            n
                            +
                            0.5
                          
                          )
                        
                      
                    
                    N
                  
                  )
                
              
            
          
        
      
    
  




  
    and
    ⁢
    
      :
    
  




  
    
      cos
      ⁡
      
        (
        
          
            
              2
              ⁢
              
                
              
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              n
              +
              0.5
            
            )
          
        
        )
      
    
    =
    
      sin
      ⁡
      
        (
        
          
            
              2
              ⁢
              
                
              
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              
                N
                4
              
              -
              1
              -
              n
              +
              0.5
            
            )
          
        
        )
      
    
  




  
    
      sin
      ⁡
      
        (
        
          
            
              2
              ⁢
              
                
              
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              n
              +
              0.5
            
            )
          
        
        )
      
    
    =
    
      cos
      ⁡
      
        (
        
          
            
              2
              ⁢
              
                
              
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              
                N
                4
              
              -
              1
              -
              n
              +
              0.5
            
            )
          
        
        )
      
    
  



Therefore, it is only necessary to store a data table of




  N
  4




points. In this embodiment, a data table of





  N
  4




points is stored and the values in the table are expressed by





  
    
      1
      
        320
        4
      
    
    ⁢
    
      cos
      ⁡
      
        (
        
          
            
              2
              ⁢
              
                
              
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              n
              +
              0.5
            
            )
          
        
        )
      
    
  
  ,
  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  319.




603. Perform a DFT of N/4 points on the pre-rotated data using an FFT based on the address table.
In this embodiment, an FFT of




  M
  =
  
    
      N
      4
    
    =
    
      P
      ×
      Q
    
  




points is performed where P and Q are coprime positive integers. The FFT includes the following steps:

i. Let P=64, Q=5, and P and Q be coprime and satisfy




  
    M
    =
    
      
        N
        4
      
      =
      
        
          P
          ×
          Q
        
        =
        320
      
    
  
  ;




create an address table I, which satisfies:

I=<K1×n1+K2×n2>M n1=0, . . . , P−1

n2=0, . . . , Q−1

K1 and K2 are coprime positive integers that satisfy <K1K2>M=0, where K1K2=Z and <Z>M means Z modulo M, which is the remainder of division of Z by M.
Addresses stored in address table I need to traverse 0, 1, . . . , M−1 to form a one-to-one mapping with the input data so as to determine the sequential relations of the data. When address table I satisfies the one-to-one mapping, the following three conditions need to be met:
(1) K1 is a positive integral multiple of Q and/or K2 is a positive integral multiple of P.
(2) K1 and P are coprime.
(3) K2 and Q are coprime.
In this embodiment, K1=65, K2=256 and the address table I may be expressed by:

I=<65n1+256n2>320 

    
    
        where, n1=0, . . . , 63 and n2=0, . . . , 4.
    
    


The address table I may be calculated and stored in advance. This step has no sequential relation with steps 601 and 602. In practice, the address table I may be stored in a table. In this embodiment, the table is stored as one-dimensional arrays of n2×n1.
ii. Perform P-point DFT for Q times based on the address table I.

DFT—P(z(I+iP))x i=0, . . . , Q−1;

The input data of each P-point DFT is the data corresponding to the P continuous indexes starting from I+iP. Each P-point DFT is applied a circular shift with a step of x. Here, x is the inverse number of a cyclic modulus of




  <
  
    
      K
      1
      2
    
    Q
  
  ⁢
  
    >
    P
  




over P and satisfies





  
    <
    
      x
      ·
    
    <
    
      
        K
        1
        2
      
      Q
    
    ⁢
    
      >
      P
    
    ⁢
    
      >
      P
    
  
  =
  1.




In this embodiment, 64-point DFTs are performed for five times. The input data of the ith (i=0, . . . 4) 64-point DFT is that associated with the 64 continuous address indexes starting from the address table array I[64i] (data after rotation). The ith 64-point DFT is applied a circular shift with a step of five. For example, the result of circular shift with a step of two on the vector Z=[z0 z1 z2 z3 z4] is Z=[z0 z2 z4 z1 z3]. In addition, to further increase the efficiency of computation, the FFT may be used in stead of the DFT.

iii. Perform Q-point DFT for P times based on the address table I.

DFT—Q(z(I+i))y i=0, . . . , P−1;

The input data of each Q-point DFT is that corresponding to the Q indexes starting from point I+i in table I each of which are separated by a step of P. The result of each Q-point DFT is applied a circular shift with a step of y. Here, y is the inverse of a cyclic modulus of




  <
  
    
      K
      2
      2
    
    P
  
  ⁢
  
    >
    Q
  




over Q and satisfies





  
    <
    
      y
      ·
    
    <
    
      
        K
        2
        2
      
      P
    
    ⁢
    
      >
      Q
    
    ⁢
    
      >
      Q
    
  
  =
  1.




In this embodiment, 5-point DFTs are performed for 64 times. The input data of the ith (i=0, . . . 63) 5-point DFT is that associated with the 5 address indexes at spacing of 64 points starting from the address table array I[i] (data after rotation). The ith 5-point DFT is applied a circular shift with a step of four.

604. Post-rotate the data after the DFT by using the rotation factor b·WNk+0.5 and the normalization factor A to obtain the frequency-domain data.
The transformed data is post-rotated by using the rotation factor b·WNk+0.5,




  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      /
      4
    
    -
    1
  
  ,
  
    
      where
      ⁢
      
        
      
      ⁢
      
        W
        N
      
    
    =
    
      
        e
        
          
            -
            j
          
          ⁢
          
            
              2
              ⁢
              π
            
            N
          
        
      
      =
      
        
          cos
          (
          
            
              2
              ⁢
              π
            
            N
          
          )
        
        -
        
          j
          ⁢
          
            
          
          ⁢
          
            sin
            (
            
              
                2
                ⁢
                π
              
              N
            
            )
          
        
      
    
  




and b is a constant. In this embodiment, the constant b may be





  b
  =
  
    
      
        2
      
      
        N
        4
      
    
    .
  




When





  
    b
    =
    
      
        2
      
      
        N
        4
      
    
  
  ,




because WNk+0.5 in the rotation factor may be expressed in the following form:





  
    
      W
      N
      
        k
        +
        0.5
      
    
    =
    
      
        e
        
          
            -
            j
          
          ⁢
          
            
              2
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              k
              +
              0.5
            
            )
          
        
      
      =
      
        
          cos
          (
          
            
              2
              ⁢
              
                π
                ⁡
                
                  (
                  
                    k
                    +
                    0.5
                  
                  )
                
              
            
            N
          
          )
        
        -
        
          j
          ⁢
          
            
          
          ⁢
          
            sin
            (
            
              
                2
                ⁢
                
                  π
                  ⁡
                  
                    (
                    
                      k
                      +
                      0.5
                    
                    )
                  
                
              
              N
            
            )
          
        
      
    
  
  ,
  
    

  
  ⁢
  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      /
      4
    
    -
    1
  




    
    
        and the value of b equals a in step 602, the data table of
    
    






  N
  /
  4




    
    
         points stored in step 602 may also be reused in this step.
    
    


The output data, also the final spectrum, is




  
    X
    k
  
  ,
  
    
  
  ⁢
  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      /
      2
    
    -
    1
  




and may be expressed by:




  
    
      X
      
        2
        ⁢
        k
      
    
    =
    
      
        A
        ·
        Re
      
      ⁢
      
        {
        
          
            b
            ·
            
              W
              N
              
                k
                +
                0.5
              
            
          
          ⁢
          
            
              ∑
              
                n
                =
                0
              
              
                
                  N
                  /
                  4
                
                -
                1
              
            
            ⁢
            
              
                
                  u
                  n
                
                ·
                a
                ·
                
                  W
                  N
                  
                    n
                    +
                    0.5
                  
                
              
              ⁢
              
                W
                
                  N
                  /
                  4
                
                nk
              
            
          
        
        }
      
    
  




  
    
      k
      =
      0
    
    ,
    ...
    ⁢
    
      
    
    ,
    
      
        N
        /
        4
      
      -
      1
    
  




  
    
      X
      
        
          N
          /
          2
        
        -
        1
        -
        
          2
          ⁢
          k
        
      
    
    =
    
      
        A
        ·
        
          -
          Im
        
      
      ⁢
      
        {
        
          
            b
            ·
            
              W
              N
              
                k
                +
                0.5
              
            
          
          ⁢
          
            
              ∑
              
                n
                =
                0
              
              
                
                  N
                  /
                  4
                
                -
                1
              
            
            ⁢
            
              
                
                  u
                  n
                
                ·
                a
                ·
                
                  W
                  N
                  
                    n
                    +
                    0.5
                  
                
              
              ⁢
              
                W
                
                  N
                  /
                  4
                
                nk
              
            
          
        
        }
      
    
  




  
    
      k
      =
      0
    
    ,
    ...
    ⁢
    
      
    
    ,
    
      
        N
        /
        4
      
      -
      1
    
  



    
    
        where, A is a normalization factor and a constant.
    
    


According to the time-domain to frequency-domain signal processing method provided in the fifth embodiment of the present invention, a frequency-domain to time-domain signal processing method is provided in the sixth embodiment of the present invention. When the time-domain to frequency-domain signal processing method provided in the fifth embodiment of the present invention is used to perform a transform, the frequency-domain to time-domain signal processing method provided in the sixth embodiment of the present invention may be used to perform a corresponding inverse transform.
FIG. 7 is a flowchart of the frequency-domain to time-domain signal processing method in the sixth embodiment of the present invention, and




  
    X
    k
  
  ,
  
    
  
  ⁢
  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      /
      2
    
    -
    1
  




is the input data of the inverse transform. The method includes the following steps:

701. Twiddle the frequency-domain data.
Specifically, the input data




  
    X
    k
  
  ,
  
    
  
  ⁢
  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      /
      2
    
    -
    1
  




is twiddled.

An intermediate variable obtained after the twiddling is




  
    (
    
      
        X
        
          2
          ⁢
          k
        
      
      +
      
        j
        ⁢
        
          
        
        ⁢
        
          X
          
            
              N
              /
              2
            
            -
            1
            -
            
              2
              ⁢
              k
            
          
        
      
    
    )
  
  ,
  
    
  
  ⁢
  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      /
      4
    
    -
    1.
  




702. Pre-rotate the twiddled data by using the rotation factor c·WNk+0.5,




  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      /
      4
    
    -
    1.
  








  
    W
    N
  
  =
  
    
      e
      
        
          -
          j
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
      
    
    =
    
      
        cos
        ⁡
        
          (
          
            
              2
              ⁢
              π
            
            N
          
          )
        
      
      -
      
        j
        ⁢
        
          
        
        ⁢
        
          sin
          ⁡
          
            (
            
              
                2
                ⁢
                π
              
              N
            
            )
          
        
      
    
  




and c is a constant.

In this embodiment,




  c
  =
  
    
      2
    
    
      N
      4
    
  




and c equals a or b. Therefore, the





  N
  /
  4




-point data table used in steps 602 and 604 in the time-domain to frequency-domain signal processing method provided in the fifth embodiment of the present invention may also be reused in this step.

703. Perform a DFT of N/4 points on the pre-rotated data using an FFT based on the address table.
When




  M
  =
  
    
      N
      4
    
    =
    
      P
      ×
      Q
    
  




and P and Q are coprime, the data processing method in step 603 of the time-domain to frequency-domain signal processing method provided in the fifth embodiment of the present invention may be used to perform this step.

704. Post-rotate the data after the transform by using the rotation factor d·WNn+0.5 and the normalization factor B.
The transformed data is post-rotated by using the rotation factor d·WNn+0.5,




  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      4
    
    -
    1
  
  ,
  where
  ,
  
    
      w
      N
    
    =
    
      
        e
        
          
            -
            j
          
          ⁢
          
            
              2
              ⁢
              π
            
            N
          
        
      
      =
      
        
          cos
          ⁡
          
            (
            
              
                2
                ⁢
                π
              
              N
            
            )
          
        
        -
        
          j
          ⁢
          
            
          
          ⁢
          
            sin
            ⁡
            
              (
              
                
                  2
                  ⁢
                  π
                
                N
              
              )
            
          
        
      
    
  




and d is a constant. The data obtained after the above processing is yn:





  
    y
    n
  
  =
  
    {
    
      
        
          
            
              B
              ·
              
                (
                
                  -
                  j
                
                )
              
              ·
              d
              ·
              
                W
                N
                
                  n
                  +
                  0.5
                
              
            
            ⁢
            
              
                ∑
                
                  k
                  =
                  0
                
                
                  
                    N
                    4
                  
                  -
                  1
                
              
              ⁢
              
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          
                            (
                            
                              
                                
                                  
                                    
                                      X
                                      
                                        2
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        k
                                      
                                    
                                    +
                                  
                                
                              
                              
                                
                                  
                                    j
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      X
                                      
                                        
                                          N
                                          2
                                        
                                        -
                                        1
                                        -
                                        
                                          2
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          k
                                        
                                      
                                    
                                  
                                
                              
                            
                            )
                          
                          ·
                        
                      
                    
                    
                      
                        
                          c
                          ·
                          
                            W
                            N
                            
                              k
                              +
                              0.5
                            
                          
                        
                      
                    
                  
                  ]
                
                ⁢
                
                  W
                  
                    N
                    4
                  
                  nk
                
              
            
          
        
        
          
            
              n
              =
              0
            
            ,
            ...
            ⁢
            
              
            
            ,
            
              
                N
                8
              
              -
              1
            
          
        
      
      
        
          
            
              B
              ·
              d
              ·
              
                W
                N
                
                  n
                  +
                  0.5
                
              
            
            ⁢
            
              
                ∑
                
                  k
                  =
                  0
                
                
                  
                    N
                    4
                  
                  -
                  1
                
              
              ⁢
              
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          
                            (
                            
                              
                                X
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  k
                                
                              
                              +
                              
                                j
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  X
                                  
                                    
                                      N
                                      2
                                    
                                    -
                                    1
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      k
                                    
                                  
                                
                              
                            
                            )
                          
                          ·
                        
                      
                    
                    
                      
                        
                          c
                          ·
                          
                            W
                            N
                            
                              k
                              +
                              0.5
                            
                          
                        
                      
                    
                  
                  ]
                
                ⁢
                
                  W
                  
                    N
                    4
                  
                  nk
                
              
            
          
        
        
          
            
              n
              =
              
                N
                8
              
            
            ,
            ...
            ⁢
            
              
            
            ,
            
              
                N
                4
              
              -
              1
            
          
        
      
    
  




B is a normalization factor and a constant.
In this embodiment,




  d
  =
  
    
      2
    
    
      N
      4
    
  




and d equals a, b, or c. Therefore, the





  N
  4




-point data table used in steps 602 and 604 in the time-domain to frequency-domain signal processing method provided in the fifth embodiment of the present invention may also be reused in this step.

705. Windowing and twiddling the post-rotated data to obtain the time-domain data.
After the windowing and twiddling, the output data




  
    
      x
      ^
    
    n
  
  ,
  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      2
    
    -
    1
  




is obtained.





  {
  
    
      
        
          
            
              
                
                  
                    x
                    ^
                  
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    n
                  
                
                =
                
                  
                    
                      m
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        n
                      
                    
                    ⁢
                    
                      g
                      
                        
                          N
                          2
                        
                        +
                        
                          2
                          ⁢
                          n
                        
                      
                    
                  
                  +
                  
                    Re
                    ⁢
                    
                      {
                      
                        u
                        n
                      
                      }
                    
                    ⁢
                    
                      g
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        n
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                    x
                    ^
                  
                  
                    
                      N
                      2
                    
                    -
                    1
                    -
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      n
                    
                  
                
                =
                
                  
                    
                      m
                      
                        
                          N
                          2
                        
                        -
                        1
                        -
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          n
                        
                      
                    
                    ⁢
                    
                      g
                      
                        N
                        -
                        1
                        -
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          n
                        
                      
                    
                  
                  -
                  
                    Re
                    ⁢
                    
                      {
                      
                        u
                        n
                      
                      }
                    
                    ⁢
                    
                      g
                      
                        
                          N
                          2
                        
                        -
                        1
                        -
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          n
                        
                      
                    
                  
                
              
            
          
        
      
      
        
          
            n
            =
            0
          
          ,
          ...
          ⁢
          
            
          
          ,
          
            
              N
              4
            
            -
            1
          
        
      
    
  




    
    
        where,
    
    






  {
  
    
      
        
          
            
              
                
                  u
                  n
                
                =
                
                  y
                  
                    n
                    +
                    
                      N
                      8
                    
                  
                
              
            
          
          
            
              
                
                  u
                  
                    n
                    +
                    
                      N
                      8
                    
                  
                
                =
                
                  y
                  n
                
              
            
          
        
      
      
        
          
            n
            =
            0
          
          ,
          ...
          ⁢
          
            
          
          ,
          
            
              N
              8
            
            -
            1
          
          ,
        
      
    
  




    
    
         and gn is a synthesis window, which satisfies the perfect reconstruction condition together with the analysis window hn, that is,
    
    






  
    
      
        
          h
          n
        
        ⁢
        
          g
          n
        
      
      +
      
        
          h
          
            n
            +
            
              N
              /
              2
            
          
        
        ⁢
        
          g
          
            n
            +
            
              N
              /
              2
            
          
        
      
    
    =
    1
  
  ,
  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    N
    -
    
      N
      2
    
    -
    1.
  




    
    
         mn is the buffered data of the previous frame and the update of mn satisfies:
    
    






   
  
    {
    
      
        
          
            
              m
              
                
                  N
                  /
                  4
                
                -
                1
                -
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  n
                
              
            
            =
            
              
                m
                
                  
                    N
                    /
                    4
                  
                  +
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    n
                  
                
              
              =
              
                Im
                ⁢
                
                  {
                  
                    u
                    n
                  
                  }
                
              
            
          
        
        
          
            
              n
              =
              0
            
            ,
            ...
            ⁢
            
              
            
            ,
            
              
                N
                8
              
              -
              1
            
          
        
      
      
        
          
            
              m
              
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  n
                
                -
                
                  N
                  /
                  4
                
              
            
            =
            
              
                m
                
                  
                    3
                    ⁢
                    
                      
                    
                    ⁢
                    
                      N
                      /
                      4
                    
                  
                  -
                  1
                  -
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    n
                  
                
              
              =
              
                Im
                ⁢
                
                  {
                  
                    u
                    n
                  
                  }
                
              
            
          
        
        
          
            
              n
              =
              
                N
                8
              
            
            ,
            ...
            ⁢
            
              
            
            ,
            
              
                N
                4
              
              -
              1
            
          
        
      
    
  




In the prior art, most DFTs based on coprime factor do not take into account the sequential access of the data sequence. The data obtained through a transform needs to be rearranged to obtain the final output data. Thus, the amount of computation is increased. In the embodiment of the present invention, an address table is adopted in an FFT, where addressing is based on the address table, the input data is read according to the associated addresses in the address table and the output data is stored according to the associated addresses in the address table. Thus, the sequence of the data obtained after a transform is ready without any special rearrangement.
A data processing method provided in an embodiment of the present invention includes:
1. Create an address table.
2. Perform P-point DFT for Q times on the input data based on the address table.
3. Perform Q-point DFT for P times on the data obtained after the P-point DFT for Q times based on the address table.
P and Q are coprime positive integers; M is the length of the address table and M=P×Q; the addresses stored in the address table can traverse 0, 1, . . . , M−1 and form a one-to-one mapping with the input data.
In practice, the address table may be obtained in multiple approaches. In this embodiment, the address table is obtained by using the formula <K1×n1+K2×n2>M.
The address table I=<K1×n1+K2×n2>M, n1=0, . . . , P−1, n2=0, . . . , Q−1, K1 and K2 are coprime positive integers satisfying <K1K2>M=0. <Z>M means Z modulo M.
When the address table I satisfies the one-to-one mapping, the following three conditions need to be met:
(1) K1 is a positive integral multiple of Q and/or K2 is a positive integral multiple of P.
(2) K1 and P are coprime.
(3) K2 and Q are coprime.
The data processing method in this embodiment is applicable to FFTs where M=P×Q and P and Q are coprime positive integers. Taking a 320-point FFT where P=64, Q=5, and P and Q are coprime as an example, a flowchart of the data processing method provided in this embodiment is shown in FIG. 2. The method includes the following steps:
1031. Create an address table I.
The address table I satisfies:

I=<K1×n1+K2×n2>M n1=0, . . . , P−1

n2=0, . . . , Q−1

K1 and K2 are coprime positive integers that satisfy <K1K2>M=0<Z>M means Z modulo M.
In this embodiment, K1=5, K1=64, and the address table I may be expressed by:

I=<5n1+64n2>320 

    
    
        where, n1=0, . . . , 63 and n2=0, . . . , 4.
    
    


The address table I may be calculated and stored in advance. In practice, the address table I may be stored in a table. In this embodiment, the table is stored as two-dimensional arrays of I[n1][n2].
1032. Perform P-point DFT for Q times on the input data according to the address table I.
The input data of the ith (i=0, . . . , Q−1) P-point DFT is the data associated with the P address indexes corresponding to n2=i, n1=0, . . . , P−1 in the address table I. The result of the ith P-point DFT is applied a circular shift with a step of x to obtain the final output.
Here, x is the inverse of a cyclic modulus of




  <
  
    
      K
      1
      2
    
    Q
  
  ⁢
  
    >
    P
  




over P and satisfies





  
    <
    
      x
      ·
    
    <
    
      
        K
        1
        2
      
      Q
    
    ⁢
    
      >
      P
    
    ⁢
    
      >
      P
    
  
  =
  1.




For example, if the address table I is stored in the form of two-dimensional arrays of the input data of I[n1][n2], the ith P-point DFT is that associated with the P address indexes stored in the address table I[n1], n1=0, . . . , P−1. The result of each P-point DFT is applied a circular shift with a step of x. In this embodiment, because K1=5, P=64, and Q=5, x=13.
1033. Perform Q-point DFT for P times on the transformed data according to the address table I.
The input data of the ith (i=0, . . . , P−1) Q-point DFT is the data associated with the Q address indexes corresponding to n1=i, n2=0, . . . , Q−1 in the address table I. The result of the ith Q-point DFT is applied a circular shift with a step of y to obtain the final output.
Here, y is the inverse of a cyclic modulus of




  <
  
    
      K
      2
      2
    
    P
  
  ⁢
  
    >
    Q
  




over Q and satisfies





  
    <
    
      y
      ·
    
    <
    
      
        K
        2
        2
      
      P
    
    ⁢
    
      >
      Q
    
    ⁢
    
      >
      Q
    
  
  =
  1.




For example, if the address table I is stored in the form of two-dimensional arrays of I[n1n2], the input data of the ith Q-point DFT is that associated with the Q address indexes stored in the address table I[n1n2], n2=0, . . . , Q−1. The result of each Q-point DFT is applied a circular shift with a step of y. In this embodiment, because K2=64, P=64, and Q=5, y=4.
In the data processing method provided in the embodiment of the present invention, because the address table is adopted for addressing, when a DFT is performed, the input data is read according to the associated addresses in the address table and the output data is stored according to the associated addresses in the address table. The transformed data is in a ready sequence and no special rearrangement is required by an inverse transform. Thus, the addressing cost in the computation is reduced and the efficiency of computation is increased.
Those skilled in the art understand that part or all steps in the method in the preceding embodiments may be implemented by related hardware instructed by a program. The program may be stored in a computer readable storage medium. The execution of the program includes the following steps:
A time-domain to frequency-domain signal processing method includes:

    
    
        pre-processing the time-domain data;
        pre-rotating the pre-processed data by using the rotation factor a·WNn+0.5;
        performing a DFT of N/4 points on the pre-rotated data; and
        post-rotating the data after the DFT by using the rotation factor b·WNk+0.5 to obtain the frequency-domain data;
        where, a and b are constants, N is twice the length of the frequency-domain data, and
    
    






  
    w
    N
  
  =
  
    
      e
      
        
          -
          j
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
      
    
    .
  




A frequency-domain to time-domain signal processing method includes:

    
    
        twiddling the frequency-domain data;
        pre-rotating the twiddled data by using the rotation factor c·WNk+0.5;
        performing a DFT of N/4 points on the pre-rotated data;
        post-rotating the data after the DFT by using the rotation factor d·WNn+0.5; and
        post-processing the post-rotated data to obtain the time-domain data;
        where, c and d are constants, N is twice the length of the frequency-domain data, and
    
    






  
    W
    N
  
  =
  
    
      e
      
        
          -
          j
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
      
    
    .
  




A data processing method includes:

    
    
        performing P-point DFT for Q times on the input data according to an address table; and
        performing Q-point DFT for P times on the data obtained after the P-point DFT for Q times according to the address table.
    
    


The storage medium mentioned above may be a read-only memory (ROM), a magnetic disk or a compact disk.
FIG. 8 shows a structure of a time-domain to frequency-domain signal processing apparatus in an embodiment of the present invention. The apparatus includes:

    
    
        a pre-processing unit 801, configured to pre-process the time-domain data;
        a first pre-rotating unit 802, configured to pre-rotate the data obtained by the pre-processing unit 801 by using the rotation factor a·WNn+0.5;
        a first Fourier transforming unit 803, configured to perform a DFT of N/4 points on the data processed by the first pre-rotating unit 802; and
        a first post-rotating unit 804, configured to post-rotate the data after the DFT performed by the first Fourier transforming unit 803 to obtain the frequency-domain data.
        a and b are constants, N is the length of the time-domain data,
    
    






  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      /
      4
    
    -
    1
  
  ,
  
    
  
  ⁢
  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      /
      4
    
    -
    1
  
  ,
  
    

  
  ⁢
  
    
      and
      ⁢
      
        
      
      ⁢
      
        W
        N
      
    
    =
    
      
        e
        
          
            -
            j
          
          ⁢
          
            
              2
              ⁢
              π
            
            N
          
        
      
      .
    
  




The first post-rotating unit 804 includes:

    
    
        a second post-rotating unit, configured to post-rotate the data after the DFT is performed by using the rotation factor b·WNk+0.5 and the normalization factor A to obtain the frequency-domain data.
    
    


The first Fourier transforming unit 803 includes:

    
    
        a first transforming unit, configured to perform P-point DFT for Q times on the data processed by the first pre-rotating unit according to an address table; and
        a second transforming unit, configured to perform Q-point DFT for P times on the data transformed by the first transforming unit according to the address table.
    
    


P and Q are coprime positive integers and




  M
  =
  
    
      N
      /
      4
    
    =
    
      P
      ×
      
        Q
        .
      
    
  




M is the length of the address table. The address table is <K1×n1+K2×n2>M, where n1=0, . . . , P−1 and n2=0, . . . , Q−1. K1 and K2 are coprime positive integers and satisfy <K1K2>M=0. <Z>M means Z modulo M.

According to the time-domain to frequency-domain signal processing apparatus provided in the embodiment of the present invention, a frequency-domain to time-domain signal processing apparatus is provided in an embodiment of the present invention. FIG. 9 shows a structure of the frequency-domain to time-domain signal processing apparatus. The apparatus includes:

    
    
        a twiddling unit 901, configured to twiddle the frequency-domain data;
        a second pre-rotating unit 902, configured to pre-rotate the data obtained by the twiddling unit 901 by using the rotation factor c·WNk+0.5;
        a second Fourier transforming unit 903, configured to perform a DFT of N/4 points on the data processed by the second pre-rotating unit 902;
        a third post-rotating unit 904, configured to post-rotate the data after the DFT performed by the second Fourier transforming unit 903; and
        a post-processing unit 905, configured to post-process the data processed by the third post-rotating unit 904 to obtain the time-domain data.
        c and d are constants, N is twice the length of the frequency-domain data,
    
    






  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      /
      4
    
    -
    1
  
  ,
  
    
  
  ⁢
  
    k
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      /
      4
    
    -
    1
  
  ,
  
    

  
  ⁢
  
    
      and
      ⁢
      
        
      
      ⁢
      
        W
        N
      
    
    =
    
      
        e
        
          
            -
            j
          
          ⁢
          
            
              2
              ⁢
              π
            
            N
          
        
      
      .
    
  




The third post-rotating unit 904 includes:

    
    
        a fourth post-rotating unit, configured to post-rotate the data after the DFT is performed by using the rotation factor d·WNn+0.5 and the normalization factor B.
    
    


The second Fourier transforming unit 903 includes:

    
    
        a third transforming unit, configured to perform P-point DFT for Q times on the data processed by the second pre-rotating unit according to an address table; and
        a fourth transforming unit, configured to perform Q-point DFT for P times on the data transformed by the third transforming unit according to the address table.
    
    


P and Q are coprime positive integers and




  M
  =
  
    
      N
      /
      4
    
    =
    
      P
      ×
      
        Q
        .
      
    
  




M is the length of the address table. The address table is <K1×n1+K2×n2>M, where n1=0, . . . , P−1 and n2=0, . . . , Q−1, K1 and K2 are coprime positive integers and satisfy <K1K2>M=0. <Z>M means Z modulo M.

For specific applications of the time-domain to frequency-domain signal processing apparatus and the frequency-domain to time-domain signal processing apparatus provided in the embodiments of the present invention, refer to descriptions of the signal processing methods provided in the embodiments of the present invention.
In the time-domain to frequency-domain signal processing apparatus and the frequency-domain to time-domain signal processing method provided in the embodiments of the present invention, because WNn+0.5 in the rotation factor may be expanded to




  
    W
    N
    
      n
      +
      0.5
    
  
  =
  
    
      e
      
        
          -
          j
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            n
            +
            0.5
          
          )
        
      
    
    =
    
      
        cos
        ⁡
        
          (
          
            
              2
              ⁢
              
                π
                ⁡
                
                  (
                  
                    n
                    +
                    0.5
                  
                  )
                
              
            
            N
          
          )
        
      
      -
      
        j
        ⁢
        
          
        
        ⁢
        
          sin
          ⁡
          
            (
            
              
                2
                ⁢
                
                  π
                  ⁡
                  
                    (
                    
                      n
                      +
                      0.5
                    
                    )
                  
                
              
              N
            
            )
          
        
      
    
  




and satisfies the




  
    
      cos
      ⁡
      
        (
        
          
            
              2
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              n
              +
              0.5
            
            )
          
        
        )
      
    
    =
    
      sin
      ⁡
      
        (
        
          
            
              2
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              
                N
                /
                4
              
              -
              1
              -
              n
              +
              0.5
            
            )
          
        
        )
      
    
  




  and




  
    
      sin
      ⁡
      
        (
        
          
            
              2
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              n
              +
              0.5
            
            )
          
        
        )
      
    
    =
    
      cos
      ⁡
      
        (
        
          
            
              2
              ⁢
              π
            
            N
          
          ⁢
          
            (
            
              
                N
                /
                4
              
              -
              1
              -
              n
              +
              0.5
            
            )
          
        
        )
      
    
  



condition, and WNk+0.5 in the rotation factor satisfies the same symmetrical condition, the time-domain to frequency-domain signal processing apparatus and the frequency-domain to time-domain signal processing apparatus provided in the embodiments of the present invention only need to store a data table of N/4 points. Compared with the prior art, where at least a sine table of N/4 points and a cosine table of N/4 points, altogether N/2 points, need to be stored, the embodiments of the present invention significantly reduce the complexity and amount of storage, reduce the consumption of storage resources, and increase the efficiency of signal processing.

In the data processing apparatus provided in an embodiment of the present invention, an address table is adopted for addressing so as to guarantee a ready sequence of the output data and reduce the addressing cost. FIG. 10 shows a structure of the data processing apparatus provided in this embodiment. The apparatus includes an address table unit 1001, a fifth transforming unit 1002, and a sixth transforming unit 1003.
The address table unit 1001 is configured to create or store an address table.
The addresses stored in the address table can traverse 0, 1, . . . , M−1 to form a one-to-one mapping with the input data.
In practice, the address table may be obtained in multiple approaches. In this embodiment, the address table is obtained by using the formula <K1×n1+K2×n2>M.
The address table I=<K1×n1+K2×n2>M, where n1=0, . . . , P−1, n2=0, . . . , Q−1, P and Q are coprime positive integers, M=P×Q, K1 and K2 are coprime positive integers and satisfy <K1K2>M=0. <Z>M means Z modulo M.
When the address table I satisfies the one-to-one mapping, the following three conditions need to be met:
(1) K1 is a positive integral multiple of Q and/or K2 is a positive integral multiple of P.
(2) K1 and P are coprime.
(3) K2 and Q are coprime.
The fifth transforming unit 1002 is configured to perform P-point DFT for Q times on the input data according to the address table created or stored by the address table unit 1001.
The input data of the ith (i=0, . . . , Q−1) P-point DFT is the data associated with the P address indexes corresponding to n2=i, n1=0, . . . , P−1 in the address table I. The result of the ith P-point DFT is applied a circular shift with a step of x to obtain the final output.
Here, x is the inverse of a cyclic modulus of




  <
  
    
      K
      1
      2
    
    Q
  
  ⁢
  
    >
    P
  




over P and satisfies





  
    <
    
      x
      ·
    
    <
    
      
        K
        1
        2
      
      Q
    
    ⁢
    
      >
      P
    
    ⁢
    
      >
      P
    
  
  =
  1.




For example, if the address table I is stored in the form of two-dimensional arrays of I[n1][n2], the input data of the ith P-point DFT is that associated with the P address indexes stored in the address table I[n1][n2], n1=0, . . . , P−1. The result of each P-point DFT is applied a circular shift with a step of x.
The sixth transforming unit 1003 is configured to perform Q-point DFT for P times on the data transformed by the fifth transforming unit 1002 according to the address table created or stored by the address table unit 1001.
The input data of the ith (i=0, . . . , P−1) Q-point DFT is the data associated with the Q address indexes corresponding to n1=i, n2=0, . . . , Q−1 in the address table I. The result of the ith Q-point DFT is applied a circular shift with a step of y to obtain the final output.
Here, y is the inverse of a cyclic modulus of




  <
  
    
      K
      2
      2
    
    P
  
  ⁢
  
    >
    Q
  




over Q and satisfies





  
    <
    
      y
      ·
    
    <
    
      
        K
        2
        2
      
      P
    
    ⁢
    
      >
      Q
    
    ⁢
    
      >
      Q
    
  
  =
  1.




For example, if the address table I is stored in the form of two-dimensional arrays of I[n1][n2], the input data of the ith Q-point DFT is that associated with the Q address indexes stored in the address table I[n1][n2], n2=0, . . . , Q−1. The result of each Q-point DFT is applied a circular shift with a step of y.
In the data processing apparatus provided in the embodiment of the present invention, because the address table is adopted for addressing, when an FFT is performed, the input data is read according to the associated addresses in the address table and the output data is stored according to the associated addresses in the address table. The transformed data is in a ready sequence and no special rearrangement is required by an inverse transform. Thus, the addressing cost in the computation is reduced and the efficiency of computation is increased.
The following will prove that the transform based on the address table I ensures a ready sequence of the output data.
The definition of an FFT is:




  
    X
    ⁡
    
      (
      k
      )
    
  
  =
  
    
      ∑
      
        n
        =
        0
      
      
        M
        -
        1
      
    
    ⁢
    
      
    
    ⁢
    
      
        x
        ⁡
        
          (
          n
          )
        
      
      ·
      
        
          W
          M
          nk
        
        .
      
    
  




M=P×Q and P and Q are coprime positive integers; if:

n=<K1×n1+K2×n2>M n1=0, . . . , P−1

n2=0, . . . , Q−1,

k=<K1×k1+K2×k2>M k1=0, . . . , P−1

k2=0, . . . , Q−1,

X(k) may be written as and X′(k1,k2) and X′(k1,k2)=X(<K1×k1+K2×k2>M); x(n) may be written as x′(n1,n2) and x′(n1,n2)=X(<K1×n1+K2×n2>M).
Therefore the FFT formula may be expanded to:




  
    
      X
      ′
    
    ⁡
    
      (
      
        
          k
          1
        
        ,
        
          k
          2
        
      
      )
    
  
  =
  
    
      ∑
      
        
          n
          2
        
        =
        0
      
      
        Q
        -
        1
      
    
    ⁢
    
      
    
    ⁢
    
      
        ∑
        
          
            n
            1
          
          =
          0
        
        
          P
          -
          1
        
      
      ⁢
      
        
          
            x
            ′
          
          ⁡
          
            (
            
              
                n
                1
              
              ,
              
                n
                2
              
            
            )
          
        
        ·
        
          W
          M
          
            
              K
              1
            
            ·
            
              K
              1
            
            ·
            
              n
              1
            
            ·
            
              k
              1
            
          
        
        ·
        
          W
          M
          
            
              K
              2
            
            ·
            
              K
              2
            
            ·
            
              n
              2
            
            ·
            
              k
              2
            
          
        
        ·
        
          W
          M
          
            
              K
              1
            
            ·
            
              K
              2
            
            ·
            
              n
              1
            
            ·
            
              k
              2
            
          
        
        ·
        
          W
          M
          
            
              K
              2
            
            ·
            
              K
              1
            
            ·
            
              n
              2
            
            ·
            
              k
              1
            
          
        
      
    
  




When <K1K2>M=0, WMK1·K2·n1·k2=WMK2·K1·n2·k1=1; therefore:




  
    
      
        
          
            X
            ′
          
          ⁡
          
            (
            
              
                k
                1
              
              ,
              
                k
                2
              
            
            )
          
        
        =
        
        ⁢
        
          
            ∑
            
              
                n
                2
              
              =
              0
            
            
              Q
              -
              1
            
          
          ⁢
          
            
          
          ⁢
          
            
              [
              
                
                  ∑
                  
                    
                      n
                      1
                    
                    =
                    0
                  
                  
                    P
                    -
                    1
                  
                
                ⁢
                
                  
                    
                      x
                      ′
                    
                    ⁡
                    
                      (
                      
                        
                          n
                          1
                        
                        ,
                        
                          n
                          2
                        
                      
                      )
                    
                  
                  ·
                  
                    W
                    M
                    
                      
                        K
                        1
                        2
                      
                      ·
                      
                        n
                        1
                      
                      ·
                      
                        k
                        1
                      
                    
                  
                
              
              ]
            
            ⁢
            
              W
              M
              
                
                  K
                  2
                  2
                
                ·
                
                  n
                  2
                
                ·
                
                  k
                  2
                
              
            
          
        
      
    
  
  
    
      
        =
        
        ⁢
        
          
            ∑
            
              
                n
                2
              
              =
              0
            
            
              Q
              -
              1
            
          
          ⁢
          
            
          
          ⁢
          
            
              [
              
                
                  ∑
                  
                    
                      n
                      1
                    
                    =
                    0
                  
                  
                    P
                    -
                    1
                  
                
                ⁢
                
                  
                    
                      x
                      ′
                    
                    ⁡
                    
                      (
                      
                        
                          n
                          1
                        
                        ,
                        
                          n
                          2
                        
                      
                      )
                    
                  
                  ·
                  
                    W
                    P
                    
                      
                        
                          K
                          1
                          2
                        
                        Q
                      
                      ·
                      
                        n
                        1
                      
                      ·
                      
                        k
                        1
                      
                    
                  
                
              
              ]
            
            ⁢
            
              W
              Q
              
                
                  
                    K
                    2
                    2
                  
                  P
                
                ·
                
                  n
                  2
                
                ·
                
                  k
                  2
                
              
            
          
        
      
    
  




Because




  
    
      W
      P
      
        
          
            K
            1
            2
          
          Q
        
        ·
        
          n
          1
        
        ·
        
          k
          1
        
      
    
    =
    
      W
      P
      
        <
        
          
            K
            1
            2
          
          Q
        
        ⁢
        
          >
          P
        
        ⁢
        
          ·
          
            n
            1
          
          ·
          
            k
            1
          
        
      
    
  
  ,
  
    
  
  ⁢
  
    
      W
      Q
      
        
          
            K
            2
            2
          
          P
        
        ·
        
          n
          2
        
        ·
        
          k
          2
        
      
    
    =
    
      
        W
        Q
        
          <
          
            
              K
              2
              2
            
            P
          
          ⁢
          
            >
            Q
          
          ⁢
          
            ·
            
              n
              2
            
            ·
            
              k
              2
            
          
        
      
      .
    
  




The input data of the ith (i=0, . . . , Q−1) P-point DFT is the data associated with the P address indexes corresponding to n2=i, n1=0, . . . , P−1 in the address table I. The result of the ith P-point DFT is applied a circular shift with a step of x to obtain the final output.
Here, x is the inverse of a cyclic modulus of




  <
  
    
      K
      1
      2
    
    Q
  
  ⁢
  
    >
    P
  




over P and satisfies





  
    <
    
      x
      ·
    
    <
    
      
        K
        1
        2
      
      Q
    
    ⁢
    
      >
      P
    
    ⁢
    
      >
      P
    
  
  =
  1.








  
    
      ∑
      
        
          n
          1
        
        =
        0
      
      
        P
        -
        1
      
    
    ⁢
    
      
        
          x
          ′
        
        ⁡
        
          (
          
            
              n
              1
            
            ,
            
              n
              2
            
          
          )
        
      
      ·
      
        W
        P
        
          
            
              
                K
                1
                2
              
              Q
            
            ·
            
              n
              1
            
          
          ⁢
          
            k
            1
          
        
      
    
  
  =
  
    
      ∑
      
        
          n
          1
        
        =
        0
      
      
        P
        -
        1
      
    
    ⁢
    
      
        
          x
          ′
        
        ⁡
        
          (
          
            
              n
              1
            
            ,
            
              n
              2
            
          
          )
        
      
      ·
      
        
          W
          P
          
            <
            
              
                K
                1
                2
              
              Q
            
            ⁢
            
              >
              P
            
            ⁢
            
              ·
              
                n
                1
              
              ·
              
                k
                1
              
            
          
        
        .
      
    
  




When n1=0,





  
    
      W
      P
      
        <
        
          
            K
            1
            2
          
          Q
        
        ⁢
        
          >
          P
        
        ⁢
        
          ·
          
            n
            1
          
          ·
          
            k
            1
          
        
      
    
    =
    1
  
  ;




when n1=1,





  
    
      W
      P
      
        <
        
          
            K
            1
            2
          
          Q
        
        ⁢
        
          >
          P
        
        ⁢
        
          ·
          
            n
            1
          
          ·
          
            k
            1
          
        
      
    
    =
    
      W
      P
      
        <
        
          
            K
            1
            2
          
          Q
        
        ⁢
        
          >
          P
        
        ⁢
        
          ·
          
            k
            1
          
        
      
    
  
  ;




by analogy, when n1=x,





  
    W
    P
    
      <
      
        
          K
          1
          2
        
        Q
      
      ⁢
      
        >
        P
      
      ⁢
      
        ·
        
          n
          1
        
        ·
        
          k
          1
        
      
    
  
  =
  
    
      W
      P
      
        <
        
          
            K
            1
            2
          
          Q
        
        ⁢
        
          >
          P
        
        ⁢
        
          ·
          x
          ·
          
            k
            1
          
        
      
    
    .
  




In the FFT formula





  
    ∑
    
      
        n
        1
      
      =
      0
    
    
      P
      -
      1
    
  
  ⁢
  
    
      
        x
        ′
      
      ⁡
      
        (
        
          n
          1
        
        )
      
    
    ·
    
      W
      P
      
        
          n
          1
        
        ·
        
          k
          1
        
      
    
  




of sequential outputs, the rotation factor when n1=1 is WPn1·k1=WPk1. In the ith P-point DFT,





  
    W
    P
    
      <
      
        
          K
          1
          2
        
        Q
      
      ⁢
      
        >
        P
      
      ⁢
      
        ·
        
          n
          1
        
        ·
        
          k
          1
        
      
    
  
  =
  
    
      W
      P
      
        <
        
          
            K
            1
            2
          
          Q
        
        ⁢
        
          >
          P
        
        ⁢
        
          ·
          x
          ·
          
            k
            1
          
        
      
    
    =
    
      W
      P
      
        k
        1
      
    
  




when n1=x only when x satisfies





  
    <
    
      x
      ·
    
    <
    
      
        K
        1
        2
      
      Q
    
    ⁢
    
      >
      P
    
    ⁢
    
      >
      P
    
  
  =
  1.




To enable sequential outputs of the ith P-point DFT, circular shift with a step of x need to be performed on the output corresponding to n1=0, . . . , P−1.

Likewise, the input data of the ith (i=0, . . . , P−1) Q-point DFT is the data associated with the Q address indexes corresponding to n1=i, n2=0, . . . , Q−1 in the address table I. The result of the ith Q-point DFT is applied a circular shift with a step of y to obtain the final output.
Here, y is the inverse of a cyclic modulus of




  <
  
    
      K
      2
      2
    
    P
  
  ⁢
  
    >
    Q
  




over Q and satisfies





  
    <
    
      y
      ·
    
    <
    
      
        K
        2
        2
      
      P
    
    ⁢
    
      >
      Q
    
    ⁢
    
      >
      Q
    
  
  =
  1.




Although the present invention has been described through exemplary embodiments, the invention is not limited to such embodiments. It is apparent that those skilled in the art can make various modifications and variations to the invention without departing from the spirit and scope of the present invention. The invention is intended to cover the modifications and variations provided that they fall within the scope of protection defined by the claims or their equivalents.

## Claims

1. A time-domain to frequency-domain signal processing method, comprising:
pre-processing time-domain data by a pre-processing unit programmed on a super wideband audio encoder;
pre-rotating the pre-processed data by a pre-rotating unit, programmed on a super wideband audio encoder using a rotation factor a·WNn+0.5;
performing a discrete Fourier transform (DFT) of N/4 points on the pre-rotated data; and
post-rotating the data transformed by the DFT by using a rotation factor b·WNk+0.5 to obtain frequency-domain data; where:
a and b are constants, N is the length of the time-domain data,





  
    
      W
      N
    
    =
    
      e
      
        
          -
          j
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
      
    
  
  ,








  
    
      W
      N
      
        n
        +
        0.5
      
    
    =
    
      e
      
        
          -
          j
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            n
            +
            0.5
          
          )
        
      
    
  
  ,
  
    
  
  ⁢
  
    
      W
      N
      
        k
        +
        0.5
      
    
    =
    
      e
      
        
          -
          j
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            k
            +
            0.5
          
          )
        
      
    
  
  ,
  
    

  
  ⁢
  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      
        N
        4
      
      -
      
        1
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        k
      
    
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      4
    
    -
    1.

2. The method of claim 1, wherein the pre-processing comprises windowing and twiddling, or twiddling.

3. The method of claim 1, wherein the step of performing the DFT of N/4 points on the pre-rotated data comprises:
performing P-point DFT for Q times on the pre-rotated data based on an address table; and
performing Q-point DFT for P times on the data obtained after the P-point DFT for Q times based on the address table; where:
P and Q are coprime positive integers and M=N/4=P×Q; M is the length of the address table; the address table is <K1×n1+K2×n2>M, where n1=0, . . . , P−1 and n2=0, . . . , Q−1, and K1 and K2 are coprime positive integers and satisfy <K1K2>M=0.

4. The method of claim 3, wherein the pre-processing comprises windowing and twiddling, or twiddling.

5. The method of claim 3, wherein the address table is stored in a table in the form of arrays.

6. The method of claim 5, wherein the pre-processing comprises windowing and twiddling, or twiddling.

7. A time-domain to frequency-domain signal processing apparatus in super wideband digital audio encoding, comprising:
a pre-processing unit, configured on a digital signal processor to pre-process time-domain data;
a first pre-rotating unit, configured on a digital signal processor to pre-rotate the data obtained by the pre-processing unit by using a rotation factor a·WNn+0.5;
a first Fourier transforming unit, configured on a digital signal processor to perform a discrete Fourier transform (DFT) of N/4 points on the data processed by the first pre-rotating unit; and
a first post-rotating unit, configured on a digital signal processor to post-rotate the data transformed by the first Fourier transforming unit by using a rotation factor b·WNk+0.5 to obtain frequency-domain data; where:
a and b are constants, N is the length of the time-domain data,





  
    
      W
      N
    
    =
    
      e
      
        
          -
          j
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
      
    
  
  ,
  
    
  
  ⁢
  
    
      W
      N
      
        n
        +
        0.5
      
    
    =
    
      e
      
        
          -
          j
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            n
            +
            0.5
          
          )
        
      
    
  
  ,
  
    
  
  ⁢
  
    
      W
      N
      
        k
        +
        0.5
      
    
    =
    
      e
      
        
          -
          j
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            k
            +
            0.5
          
          )
        
      
    
  
  ,
  
    

  
  ⁢
  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      
        N
        4
      
      -
      
        1
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        k
      
    
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      4
    
    -
    1.

8. The apparatus of claim 7, wherein the first Fourier transforming unit comprises:
a first transforming unit, configured to perform P-point DFT for Q times on the data processed by the first pre-rotating unit based on an address table; and
a second transforming unit, configured to perform Q-point DFT for P times on the data transformed by the first transforming unit based on the address table.

9. The apparatus of claim 8, wherein P and Q are coprime positive integers and M=N/4=P×Q; M is the length of the address table; the address table is <K1×n1+K2×n2>M, where n1=0, . . . , P−1 and n2=0, . . . , Q−1, and K1 and K2 are coprime positive integers and satisfy <K1K2>M=0.

10. A non-transitory computer readable storage medium including computer program for processing time-domain to frequency-domain signal, when executed is operable to: pre-processing time-domain data; pre-rotating the pre-processed data by using a rotation factor a·WNn+0.5; performing a discrete Fourier transform (DFT) of N/4 points on the pre-rotated data; and post-rotating the data transformed by the DFT by using a rotation factor b·WNk+0.5; to obtain frequency-domain data; where: a and b are constants, N is the length of the time-domain data,




  
    
      W
      N
    
    =
    
      e
      
        
          -
          j
        
        ⁢
        
          
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
      
    
  
  ,
  
    
      W
      N
      
        n
        +
        0.5
      
    
    =
    
      e
      
        
          -
          j
        
        ⁢
        
          
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            n
            +
            0.5
          
          )
        
      
    
  
  ,
  
    
      W
      N
      
        k
        +
        0.5
      
    
    =
    
      e
      
        
          -
          j
        
        ⁢
        
          
        
        ⁢
        
          
            2
            ⁢
            π
          
          N
        
        ⁢
        
          (
          
            k
            +
            0.5
          
          )
        
      
    
  
  ,
  
    

  
  ⁢
  
    n
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      
        N
        4
      
      -
      
        1
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        k
      
    
    =
    0
  
  ,
  ...
  ⁢
  
    
  
  ,
  
    
      N
      4
    
    -
    1.