Transfer matrix

In <a href="/facts/Applied_mathematics/qyqklXXm">applied mathematics</a>, the transfer matrix is a formulation in terms of a <a href="/facts/Block-Toeplitz_matrix/WAjSdlaW">block-Toeplitz matrix</a> of the two-scale equation, which characterizes <a href="/facts/Refinable_function/K8G1hMEk">refinable functions</a>. Refinable functions play an important role in <a href="/facts/Wavelet/3XcX4uIP">wavelet</a> theory and <a href="/facts/Finite_element/eUcfP1vn">finite element</a> theory.
For the mask 
 
 
 
 h
 
 
 {\displaystyle h}
 
, which is a vector with component indexes from 
 
 
 
 a
 
 
 {\displaystyle a}
 
 to 
 
 
 
 b
 
 
 {\displaystyle b}
 
,
the transfer matrix of 
 
 
 
 h
 
 
 {\displaystyle h}
 
, we call it 
 
 
 
 
 T
 
 h
 
 
 
 
 {\displaystyle T_{h}}
 
 here, is defined as

(
        
          T
          
            h
          
        
        
          )
          
            j
            ,
            k
          
        
        =
        
          h
          
            2
            ⋅
            j
            −
            k
          
        
        .
      
    
    {\displaystyle (T_{h})_{j,k}=h_{2\cdot j-k}.}

More verbosely

T
          
            h
          
        
        =
        
          
            (
            
              
                
                  
                    h
                    
                      a
                    
                  
                
                
                
                
                
                
              
              
                
                  
                    h
                    
                      a
                      +
                      2
                    
                  
                
                
                  
                    h
                    
                      a
                      +
                      1
                    
                  
                
                
                  
                    h
                    
                      a
                    
                  
                
                
                
                
              
              
                
                  
                    h
                    
                      a
                      +
                      4
                    
                  
                
                
                  
                    h
                    
                      a
                      +
                      3
                    
                  
                
                
                  
                    h
                    
                      a
                      +
                      2
                    
                  
                
                
                  
                    h
                    
                      a
                      +
                      1
                    
                  
                
                
                  
                    h
                    
                      a
                    
                  
                
                
              
              
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
                
                  ⋱
                
              
              
                
                
                  
                    h
                    
                      b
                    
                  
                
                
                  
                    h
                    
                      b
                      −
                      1
                    
                  
                
                
                  
                    h
                    
                      b
                      −
                      2
                    
                  
                
                
                  
                    h
                    
                      b
                      −
                      3
                    
                  
                
                
                  
                    h
                    
                      b
                      −
                      4
                    
                  
                
              
              
                
                
                
                
                  
                    h
                    
                      b
                    
                  
                
                
                  
                    h
                    
                      b
                      −
                      1
                    
                  
                
                
                  
                    h
                    
                      b
                      −
                      2
                    
                  
                
              
              
                
                
                
                
                
                
                  
                    h
                    
                      b
                    
                  
                
              
            
            )
          
        
        .
      
    
    {\displaystyle T_{h}={\begin{pmatrix}h_{a}&&&&&\\h_{a+2}&h_{a+1}&h_{a}&&&\\h_{a+4}&h_{a+3}&h_{a+2}&h_{a+1}&h_{a}&\\\ddots &\ddots &\ddots &\ddots &\ddots &\ddots \\&h_{b}&h_{b-1}&h_{b-2}&h_{b-3}&h_{b-4}\\&&&h_{b}&h_{b-1}&h_{b-2}\\&&&&&h_{b}\end{pmatrix}}.}

The effect of 
 
 
 
 
 T
 
 h
 
 
 
 
 {\displaystyle T_{h}}
 
 can be expressed in terms of the <a href="/facts/Downsampling/gN5Wwrhv">downsampling</a> operator "
 
 
 
 ↓
 
 
 {\displaystyle \downarrow }
 
":

T
          
            h
          
        
        ⋅
        x
        =
        (
        h
        ∗
        x
        )
        ↓
        2.
      
    
    {\displaystyle T_{h}\cdot x=(h*x)\downarrow 2.}

Transfer matrix open-in-new

Transfer matrix