Polyphase matrix

In <a href="/facts/Signal_processing/Npaja6zb">signal processing</a>, a polyphase matrix is a matrix whose elements are <a href="/facts/Linear_filter/3W5oDlVq">filter masks</a>. It represents a <a href="/facts/Filter_bank/HXvTRgFN">filter bank</a> as it is used in <a href="/facts/Sub-band_coder/HWMqu3DR">sub-band coders</a> alias <a href="/facts/Discrete_wavelet_transform/19DVnk4n">discrete wavelet transforms</a>.
If 
 
 
 
 
 h
 ,
 
 g
 
 
 
 {\displaystyle \scriptstyle h,\,g}
 
 are two filters, then one level the traditional wavelet transform maps an input signal 
 
 
 
 
 
 a
 
 0
 
 
 
 
 
 {\displaystyle \scriptstyle a_{0}}
 
 to two output signals 
 
 
 
 
 
 a
 
 1
 
 
 ,
 
 
 d
 
 1
 
 
 
 
 
 {\displaystyle \scriptstyle a_{1},\,d_{1}}
 
, each of the half length:

a
                  
                    1
                  
                
              
              
                
                =
                (
                h
                ⋅
                
                  a
                  
                    0
                  
                
                )
                ↓
                2
              
            
            
              
                
                  d
                  
                    1
                  
                
              
              
                
                =
                (
                g
                ⋅
                
                  a
                  
                    0
                  
                
                )
                ↓
                2
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}a_{1}&=(h\cdot a_{0})\downarrow 2\\d_{1}&=(g\cdot a_{0})\downarrow 2\end{aligned}}}

Note, that the dot means <a href="/facts/Polynomial_multiplication/Lzak8VVx">polynomial multiplication</a>; i.e., <a href="/facts/Convolution/PVPrdz9J">convolution</a> and 
 
 
 
 
 ↓
 
 
 
 {\displaystyle \scriptstyle \downarrow }
 
 means <a href="/facts/Downsampling/gN5Wwrhv">downsampling</a>.
If the above formula is implemented directly, you will compute values that are subsequently flushed by the down-sampling. You can avoid their computation by splitting the filters and the signal into even and odd indexed values before the wavelet transformation:

h
                  
                    
                      e
                    
                  
                
              
              
                
                =
                h
                ↓
                2
              
              
                
                  a
                  
                    0
                    ,
                    
                      
                        e
                      
                    
                  
                
              
              
                
                =
                
                  a
                  
                    0
                  
                
                ↓
                2
              
            
            
              
                
                  h
                  
                    
                      o
                    
                  
                
              
              
                
                =
                (
                h
                ←
                1
                )
                ↓
                2
              
              
                
                  a
                  
                    0
                    ,
                    
                      
                        o
                      
                    
                  
                
              
              
                
                =
                (
                
                  a
                  
                    0
                  
                
                ←
                1
                )
                ↓
                2
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}h_{\mbox{e}}&=h\downarrow 2&a_{0,{\mbox{e}}}&=a_{0}\downarrow 2\\h_{\mbox{o}}&=(h\leftarrow 1)\downarrow 2&a_{0,{\mbox{o}}}&=(a_{0}\leftarrow 1)\downarrow 2\end{aligned}}}

The arrows 
 
 
 
 
 ←
 
 
 
 {\displaystyle \scriptstyle \leftarrow }
 
 and 
 
 
 
 
 →
 
 
 
 {\displaystyle \scriptstyle \rightarrow }
 
 denote left and right shifting, respectively. They shall have the same <a href="/facts/Operator_precedence/sWsoTzbH">precedence</a> like convolution, because they are in fact convolutions with a shifted discrete <a href="/facts/Kronecker_delta/gcGAy0bm">delta impulse</a>.

δ
        =
        (
        …
        ,
        0
        ,
        0
        ,
        
          
            1
            
              0
              −
              
                
                  th position
                
              
            
          
        
        ,
        0
        ,
        0
        ,
        …
        )
      
    
    {\displaystyle \delta =(\dots ,0,0,{\underset {0-{\mbox{th position}}}{1}},0,0,\dots )}

The wavelet transformation reformulated to the split filters is:

a
                  
                    1
                  
                
              
              
                
                =
                
                  h
                  
                    
                      e
                    
                  
                
                ⋅
                
                  a
                  
                    0
                    ,
                    
                      
                        e
                      
                    
                  
                
                +
                
                  h
                  
                    
                      o
                    
                  
                
                ⋅
                
                  a
                  
                    0
                    ,
                    
                      
                        o
                      
                    
                  
                
                →
                1
              
            
            
              
                
                  d
                  
                    1
                  
                
              
              
                
                =
                
                  g
                  
                    
                      e
                    
                  
                
                ⋅
                
                  a
                  
                    0
                    ,
                    
                      
                        e
                      
                    
                  
                
                +
                
                  g
                  
                    
                      o
                    
                  
                
                ⋅
                
                  a
                  
                    0
                    ,
                    
                      
                        o
                      
                    
                  
                
                →
                1
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}a_{1}&=h_{\mbox{e}}\cdot a_{0,{\mbox{e}}}+h_{\mbox{o}}\cdot a_{0,{\mbox{o}}}\rightarrow 1\\d_{1}&=g_{\mbox{e}}\cdot a_{0,{\mbox{e}}}+g_{\mbox{o}}\cdot a_{0,{\mbox{o}}}\rightarrow 1\end{aligned}}}

This can be written as <a href="/facts/Matrix_multiplication/lYDB6Ro2">matrix-vector-multiplication</a>

P
              
              
                
                =
                
                  
                    (
                    
                      
                        
                          
                            h
                            
                              
                                e
                              
                            
                          
                        
                        
                          
                            h
                            
                              
                                o
                              
                            
                          
                          →
                          1
                        
                      
                      
                        
                          
                            g
                            
                              
                                e
                              
                            
                          
                        
                        
                          
                            g
                            
                              
                                o
                              
                            
                          
                          →
                          1
                        
                      
                    
                    )
                  
                
              
            
            
              
                
                  
                    (
                    
                      
                        
                          
                            a
                            
                              1
                            
                          
                        
                      
                      
                        
                          
                            d
                            
                              1
                            
                          
                        
                      
                    
                    )
                  
                
              
              
                
                =
                P
                ⋅
                
                  
                    (
                    
                      
                        
                          
                            a
                            
                              0
                              ,
                              
                                
                                  e
                                
                              
                            
                          
                        
                      
                      
                        
                          
                            a
                            
                              0
                              ,
                              
                                
                                  o
                                
                              
                            
                          
                        
                      
                    
                    )
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}P&={\begin{pmatrix}h_{\mbox{e}}&h_{\mbox{o}}\rightarrow 1\\g_{\mbox{e}}&g_{\mbox{o}}\rightarrow 1\end{pmatrix}}\\{\begin{pmatrix}a_{1}\\d_{1}\end{pmatrix}}&=P\cdot {\begin{pmatrix}a_{0,{\mbox{e}}}\\a_{0,{\mbox{o}}}\end{pmatrix}}\end{aligned}}}

This matrix 
 
 
 
 
 P
 
 
 
 {\displaystyle \scriptstyle P}
 
 is the polyphase matrix.
Of course, a polyphase matrix can have any size, it need not to have square shape. That is, the principle scales well to any <a href="/facts/Filterbank/HXvTRgFN">filterbanks</a>, multiwavelets, wavelet transforms based on fractional <a href="/facts/Refinable_function/K8G1hMEk">refinements</a>.

Polyphase matrix open-in-new

Polyphase matrix