離散正交轉換
數學上，任一的
  
    
      
        M
        ×
        N
      
    
    {\displaystyle M\times N}
  離散線性轉換皆可表示成矩陣（Matrix） 的型式：
  
    
      
        
          Y
          =
          A
          X
        
      
    
    {\displaystyle {\boldsymbol {Y=AX}}}
  

  
    
      
        
          
            [
            
              
                
                  y
                  
                    [0]
                  
                
              
              
                
                  y
                  
                    [1]
                  
                
              
              
                
                  y
                  
                    [2]
                  
                
              
              
                
                  ⋮
                
              
              
                
                  y
                  
                    [
                    
                      M
                      −
                      1
                    
                    ]
                  
                
              
            
            ]
          
        
        =
        
          
            [
            
              
                
                  
                    
                      a
                      
                        0
                      
                    
                    
                      [0]
                    
                  
                
                
                  
                    
                      a
                      
                        0
                      
                    
                    
                      [1]
                    
                  
                
                
                  
                    
                      a
                      
                        0
                      
                    
                    
                      [2]
                    
                  
                
                
                  ⋯
                
                
                  
                    
                      a
                      
                        0
                      
                    
                    
                      [
                      
                        N
                        −
                        1
                      
                      ]
                    
                  
                
              
              
                
                  
                    
                      a
                      
                        1
                      
                    
                    
                      [0]
                    
                  
                
                
                  
                    
                      a
                      
                        1
                      
                    
                    
                      [1]
                    
                  
                
                
                  
                    
                      a
                      
                        1
                      
                    
                    
                      [2]
                    
                  
                
                
                  ⋯
                
                
                  
                    
                      a
                      
                        1
                      
                    
                    
                      [
                      
                        N
                        −
                        1
                      
                      ]
                    
                  
                
              
              
                
                  
                    
                      a
                      
                        2
                      
                    
                    
                      [0]
                    
                  
                
                
                  
                    
                      a
                      
                        2
                      
                    
                    
                      [1]
                    
                  
                
                
                  
                    
                      a
                      
                        2
                      
                    
                    
                      [2]
                    
                  
                
                
                  ⋯
                
                
                  
                    
                      a
                      
                        2
                      
                    
                    
                      [
                      
                        N
                        −
                        1
                      
                      ]
                    
                  
                
              
              
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  
                    
                      a
                      
                        M
                        −
                        1
                      
                    
                    
                      [0]
                    
                  
                
                
                  
                    
                      a
                      
                        M
                        −
                        1
                      
                    
                    
                      [1]
                    
                  
                
                
                  
                    
                      a
                      
                        M
                        −
                        1
                      
                    
                    
                      [2]
                    
                  
                
                
                  ⋯
                
                
                  
                    
                      a
                      
                        M
                        −
                        1
                      
                    
                    
                      [
                      
                        N
                        −
                        1
                      
                      ]
                    
                  
                
              
            
            ]
          
        
        
          
            [
            
              
                
                  x
                  
                    [0]
                  
                
              
              
                
                  x
                  
                    [1]
                  
                
              
              
                
                  x
                  
                    [2]
                  
                
              
              
                
                  ⋮
                
              
              
                
                  x
                  
                    [
                    
                      N
                      −
                      1
                    
                    ]
                  
                
              
            
            ]
          
        
        .
      
    
    {\displaystyle {\begin {bmatrix} y\left [{0}\right]\\y\left [{1}\right]\\y\left [{2}\right]\\\vdots \\y\left [{M-1}\right]\end {bmatrix}}={\begin {bmatrix}{a_{0}\left [0\right]}&{a_{0}\left [1\right]}&{a_{0}\left [2\right]}&\cdots &{a_{0}\left [N-1\right]}\\{a_{1}\left [0\right]}&{a_{1}\left [1\right]}&{a_{1}\left [2\right]}&\cdots &{a_{1}\left [N-1\right]}\\{a_{2}\left [0\right]}&{a_{2}\left [1\right]}&{a_{2}\left [2\right]}&\cdots &{a_{2}\left [N-1\right]}\\\vdots &\vdots &\vdots &\ddots &\vdots \\{a_{M-1}\left [0\right]}&{a_{M-1}\left [1\right]}&{a_{M-1}\left [2\right]}&\cdots &{a_{M-1}\left [N-1\right]}\end {bmatrix}}{\begin {bmatrix} x\left [{0}\right]\\x\left [{1}\right]\\x\left [{2}\right]\\\vdots \\x\left [{N-1}\right]\end {bmatrix}}.}
  

  
    
      
        
          
            C
            A
            S
            E
          
          
          
            1
          
        
      
    
    {\displaystyle \mathbf {{CASE}\,{1}} }
  
再進一步假設，若矩陣
  
    
      
        
          A
        
      
    
    {\displaystyle {\boldsymbol {A}}}
   by 正交基底 (Orthogonal basis) 列向量（Row vector） 所組成：

  
    
      
        
          
            [
            
              
                
                  
                    
                      
                        ϕ
                      
                      
                        0
                      
                      
                        ∗
                      
                    
                  
                
              
              
                
                  
                    
                      
                        ϕ
                      
                      
                        1
                      
                      
                        ∗
                      
                    
                  
                
              
              
                
                  
                    
                      
                        ϕ
                      
                      
                        2
                      
                      
                        ∗
                      
                    
                  
                
              
              
                
                  ⋮
                
              
              
                
                  
                    
                      
                        ϕ
                      
                      
                        M
                        −
                        1
                      
                      
                        ∗
                      
                    
                  
                
              
            
            ]
          
        
        =
        
          
            [
            
              
                
                  
                    
                      ϕ
                      
                        0
                      
                      
                        ∗
                      
                    
                    
                      [0]
                    
                  
                
                
                  
                    
                      ϕ
                      
                        0
                      
                      
                        ∗
                      
                    
                    
                      [1]
                    
                  
                
                
                  
                    
                      ϕ
                      
                        0
                      
                      
                        ∗
                      
                    
                    
                      [2]
                    
                  
                
                
                  ⋯
                
                
                  
                    
                      ϕ
                      
                        0
                      
                      
                        ∗
                      
                    
                    
                      [
                      
                        N
                        −
                        1
                      
                      ]
                    
                  
                
              
              
                
                  
                    
                      ϕ
                      
                        1
                      
                      
                        ∗
                      
                    
                    
                      [0]
                    
                  
                
                
                  
                    
                      ϕ
                      
                        1
                      
                      
                        ∗
                      
                    
                    
                      [1]
                    
                  
                
                
                  
                    
                      ϕ
                      
                        1
                      
                      
                        ∗
                      
                    
                    
                      [2]
                    
                  
                
                
                  ⋯
                
                
                  
                    
                      ϕ
                      
                        1
                      
                      
                        ∗
                      
                    
                    
                      [
                      
                        N
                        −
                        1
                      
                      ]
                    
                  
                
              
              
                
                  
                    
                      ϕ
                      
                        2
                      
                      
                        ∗
                      
                    
                    
                      [0]
                    
                  
                
                
                  
                    
                      ϕ
                      
                        2
                      
                      
                        ∗
                      
                    
                    
                      [1]
                    
                  
                
                
                  
                    
                      ϕ
                      
                        2
                      
                      
                        ∗
                      
                    
                    
                      [2]
                    
                  
                
                
                  ⋯
                
                
                  
                    
                      ϕ
                      
                        2
                      
                      
                        ∗
                      
                    
                    
                      [
                      
                        N
                        −
                        1
                      
                      ]
                    
                  
                
              
              
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋮
                
              
              
                
                  
                    
                      ϕ
                      
                        M
                        −
                        1
                      
                      
                        ∗
                      
                    
                    
                      [0]
                    
                  
                
                
                  
                    
                      ϕ
                      
                        M
                        −
                        1
                      
                      
                        ∗
                      
                    
                    
                      [1]
                    
                  
                
                
                  
                    
                      ϕ
                      
                        M
                        −
                        1
                      
                      
                        ∗
                      
                    
                    
                      [2]
                    
                  
                
                
                  ⋯
                
                
                  
                    
                      ϕ
                      
                        M
                        −
                        1
                      
                      
                        ∗
                      
                    
                    
                      [
                      
                        N
                        −
                        1
                      
                      ]
                    
                  
                
              
            
            ]
          
        
        =
        
          A
        
        .
      
    
    {\displaystyle {\begin {bmatrix}{{\boldsymbol {\phi}}_{0}^{*}}\\{{\boldsymbol {\phi}}_{1}^{*}}\\{{\boldsymbol {\phi}}_{2}^{*}}\\\vdots \\{{\boldsymbol {\phi}}_{M-1}^{*}}\end {bmatrix}}={\begin {bmatrix}{\phi _{0}^{*}\left [0\right]}&{\phi _{0}^{*}\left [1\right]}&{\phi _{0}^{*}\left [2\right]}&\cdots &{\phi _{0}^{*}\left [N-1\right]}\\{\phi _{1}^{*}\left [0\right]}&{\phi _{1}^{*}\left [1\right]}&{\phi _{1}^{*}\left [2\right]}&\cdots &{\phi _{1}^{*}\left [N-1\right]}\\{\phi _{2}^{*}\left [0\right]}&{\phi _{2}^{*}\left [1\right]}&{\phi _{2}^{*}\left [2\right]}&\cdots &{\phi _{2}^{*}\left [N-1\right]}\\\vdots &\vdots &\vdots &\vdots &\vdots \\{\phi _{M-1}^{*}\left [0\right]}&{\phi _{M-1}^{*}\left [1\right]}&{\phi _{M-1}^{*}\left [2\right]}&\cdots &{\phi _{M-1}^{*}\left [N-1\right]}\end {bmatrix}}={\boldsymbol {A}}.}
  
也可表示成級數和形式（Summation form）：

  
    
      
        y
        
          [m]
        
        =
        
          ⟨
          
            
              
                
                  ϕ
                
                
                  m
                
                
                  ∗
                
              
            
            ,
            x
            
              [n]
            
          
          ⟩
        
        =
        
          ∑
          
            n
            =
            0
          
          
            N
            −
            1
          
        
        
          
            ϕ
            
              m
            
            
              ∗
            
          
          
            [n]
          
        
        x
        
          [n]
        
        ,
        
        
          
            for 
          
        
        m
        =
        0
        ,
        
        1
        ,
        
        ⋯
        ,
        
        M
        −
        1.
      
    
    {\displaystyle y\left [m\right]=\left\langle {{\boldsymbol {\phi}}_{m}^{*}},x\left [n\right]\right\rangle =\sum _{n=0}^{N-1}{\phi _{m}^{*}\left [n\right]} x\left [n\right],\quad {\mbox {for}} m=0,\,1,\,\cdots ,\,M-1.}
  
其中
  
    
      
        
          ⟨
          
            ⋅
            
            ,
            ⋅
          
          ⟩
        
      
    
    {\displaystyle \left\langle \cdot \,,\cdot \right\rangle}
  代表內積運算（Inner product）。

  
    
      
        
          
            C
            A
            S
            E
          
          
          
            2
          
        
      
    
    {\displaystyle \mathbf {{CASE}\,{2}} }
  
同理也可假設，若矩陣
  
    
      
        
          A
        
      
    
    {\displaystyle {\boldsymbol {A}}}
   by 正交基底行向量（Column vector） 所組成：

  
    
      
        
          
            [
            
              
                
                  
                    
                      
                        β
                      
                      
                        0
                      
                      
                        ∗
                      
                    
                  
                
                
                  
                    
                      
                        β
                      
                      
                        1
                      
                      
                        ∗
                      
                    
                  
                
                
                  
                    
                      
                        β
                      
                      
                        2
                      
                      
                        ∗
                      
                    
                  
                
                
                  ⋯
                
                
                  
                    
                      
                        β
                      
                      
                        N
                        −
                        1
                      
                      
                        ∗
                      
                    
                  
                
              
            
            ]
          
        
        =
        
          
            [
            
              
                
                  
                    
                      β
                      
                        0
                      
                      
                        ∗
                      
                    
                    
                      [0]
                    
                  
                
                
                  
                    
                      β
                      
                        1
                      
                      
                        ∗
                      
                    
                    
                      [0]
                    
                  
                
                
                  
                    
                      β
                      
                        2
                      
                      
                        ∗
                      
                    
                    
                      [0]
                    
                  
                
                
                  ⋯
                
                
                  
                    
                      β
                      
                        N
                        −
                        1
                      
                      
                        ∗
                      
                    
                    
                      [0]
                    
                  
                
              
              
                
                  
                    
                      β
                      
                        0
                      
                      
                        ∗
                      
                    
                    
                      [1]
                    
                  
                
                
                  
                    
                      β
                      
                        1
                      
                      
                        ∗
                      
                    
                    
                      [1]
                    
                  
                
                
                  
                    
                      β
                      
                        2
                      
                      
                        ∗
                      
                    
                    
                      [1]
                    
                  
                
                
                  ⋯
                
                
                  
                    
                      β
                      
                        N
                        −
                        1
                      
                      
                        ∗
                      
                    
                    
                      [1]
                    
                  
                
              
              
                
                  
                    
                      β
                      
                        0
                      
                      
                        ∗
                      
                    
                    
                      [2]
                    
                  
                
                
                  
                    
                      β
                      
                        1
                      
                      
                        ∗
                      
                    
                    
                      [2]
                    
                  
                
                
                  
                    
                      β
                      
                        2
                      
                      
                        ∗
                      
                    
                    
                      [2]
                    
                  
                
                
                  ⋯
                
                
                  
                    
                      β
                      
                        N
                        −
                        1
                      
                      
                        ∗
                      
                    
                    
                      [2]
                    
                  
                
              
              
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋮
                
              
              
                
                  
                    
                      β
                      
                        0
                      
                      
                        ∗
                      
                    
                    
                      [
                      
                        M
                        −
                        1
                      
                      ]
                    
                  
                
                
                  
                    
                      β
                      
                        1
                      
                      
                        ∗
                      
                    
                    
                      [
                      
                        M
                        −
                        1
                      
                      ]
                    
                  
                
                
                  
                    
                      β
                      
                        2
                      
                      
                        ∗
                      
                    
                    
                      [
                      
                        M
                        −
                        1
                      
                      ]
                    
                  
                
                
                  ⋯
                
                
                  
                    
                      β
                      
                        N
                        −
                        1
                      
                      
                        ∗
                      
                    
                    
                      [
                      
                        M
                        −
                        1
                      
                      ]
                    
                  
                
              
            
            ]
          
        
        =
        
          A
        
        .
      
    
    {\displaystyle {\begin {bmatrix}{{\boldsymbol {\beta}}_{0}^{*}}&{{\boldsymbol {\beta}}_{1}^{*}}&{{\boldsymbol {\beta}}_{2}^{*}}&\cdots &{{\boldsymbol {\beta}}_{N-1}^{*}}\end {bmatrix}}={\begin {bmatrix}{\beta _{0}^{*}\left [0\right]}&{\beta _{1}^{*}\left [0\right]}&{\beta _{2}^{*}\left [0\right]}&\cdots &{\beta _{N-1}^{*}\left [0\right]}\\{\beta _{0}^{*}\left [1\right]}&{\beta _{1}^{*}\left [1\right]}&{\beta _{2}^{*}\left [1\right]}&\cdots &{\beta _{N-1}^{*}\left [1\right]}\\{\beta _{0}^{*}\left [2\right]}&{\beta _{1}^{*}\left [2\right]}&{\beta _{2}^{*}\left [2\right]}&\cdots &{\beta _{N-1}^{*}\left [2\right]}\\\vdots &\vdots &\vdots &\vdots &\vdots \\{\beta _{0}^{*}\left [M-1\right]}&{\beta _{1}^{*}\left [M-1\right]}&{\beta _{2}^{*}\left [M-1\right]}&\cdots &{\beta _{N-1}^{*}\left [M-1\right]}\end {bmatrix}}={\boldsymbol {A}}.}
  
也可表示成級數和：

  
    
      
        y
        
          [m]
        
        =
        
          ∑
          
            n
            =
            0
          
          
            N
            −
            1
          
        
        x
        
          [n]
        
        
          
            β
            
              n
            
            
              ∗
            
          
          
            [m]
          
        
        ,
        
        
          
            for 
          
        
        m
        =
        0
        ,
        
        1
        ,
        
        ⋯
        ,
        
        M
        −
        1.
      
    
    {\displaystyle y\left [m\right]=\sum _{n=0}^{N-1} x\left [n\right]{\beta _{n}^{*}\left [m\right]},\quad {\mbox {for}} m=0,\,1,\,\cdots ,\,M-1.}
  
假若
  
    
      
        M
        =
        N
      
    
    {\displaystyle M=N}
  時，則
  
    
      
        d
        e
        t
        (A)
        ≠
        0
      
    
    {\displaystyle det (A)\neq 0}
  可得

  
    
      
        
          
            {
            
              
                
                  Y
                  =
                  A
                  X
                
                
                  (Forward transform)
                
              
              
                
                  X
                  =
                  
                    A
                    
                      −
                      1
                    
                  
                  X
                
                
                  (Inverse transform)
                
              
            
            
          
        
        .
      
    
    {\displaystyle {\begin {cases} Y=AX&({\mbox {Forward transform}})\\X=A^{-1} X&({\mbox {Inverse transform}})\end {cases}}.}