Matched Z-transform method

The matched Z-transform method, also called the pole–zero mapping or pole–zero matching method, and abbreviated MPZ or MZT, is a technique for converting a <a href="/facts/Continuous-time/RR8wpS9e">continuous-time</a> filter design to a <a href="/facts/Discrete-time/RR8wpS9e">discrete-time</a> filter (<a href="/facts/Digital_filter/2wcxCRdh">digital filter</a>) design.
The method works by mapping all poles and zeros of the <a href="/facts/Laplace_transform/0Ge4MIc9">s-plane</a> design to <a href="/facts/Z-transform/ERNvDJqt">z-plane</a> locations 
 
 
 
 z
 =
 
 e
 
 s
 T
 
 
 
 
 {\displaystyle z=e^{sT}}
 
, for a sample interval 
 
 
 
 T
 =
 1
 
 /
 
 
 f
 
 
 s
 
 
 
 
 
 {\displaystyle T=1/f_{\mathrm {s} }}
 
. So an analog filter with transfer function:

H
        (
        s
        )
        =
        
          k
          
            
              a
            
          
        
        
          
            
              
                ∏
                
                  i
                  =
                  1
                
                
                  M
                
              
              (
              s
              −
              
                ξ
                
                  i
                
              
              )
            
            
              
                ∏
                
                  i
                  =
                  1
                
                
                  N
                
              
              (
              s
              −
              
                p
                
                  i
                
              
              )
            
          
        
      
    
    {\displaystyle H(s)=k_{\mathrm {a} }{\frac {\prod _{i=1}^{M}(s-\xi _{i})}{\prod _{i=1}^{N}(s-p_{i})}}}

is transformed into the digital transfer function

H
        (
        z
        )
        =
        
          k
          
            
              d
            
          
        
        
          
            
              
                ∏
                
                  i
                  =
                  1
                
                
                  M
                
              
              (
              1
              −
              
                e
                
                  
                    ξ
                    
                      i
                    
                  
                  T
                
              
              
                z
                
                  −
                  1
                
              
              )
            
            
              
                ∏
                
                  i
                  =
                  1
                
                
                  N
                
              
              (
              1
              −
              
                e
                
                  
                    p
                    
                      i
                    
                  
                  T
                
              
              
                z
                
                  −
                  1
                
              
              )
            
          
        
      
    
    {\displaystyle H(z)=k_{\mathrm {d} }{\frac {\prod _{i=1}^{M}(1-e^{\xi _{i}T}z^{-1})}{\prod _{i=1}^{N}(1-e^{p_{i}T}z^{-1})}}}

The gain 
 
 
 
 
 k
 
 
 d
 
 
 
 
 
 {\displaystyle k_{\mathrm {d} }}
 
 must be adjusted to normalize the desired gain, typically set to match the analog filter's gain at DC by <a href="/facts/Final_value_theorem/uQ2hdoUY">setting 
 
 
 
 s
 =
 0
 
 
 {\displaystyle s=0}
 
 and 
 
 
 
 z
 =
 1
 
 
 {\displaystyle z=1}
 
</a> and solving for 
 
 
 
 
 k
 
 
 d
 
 
 
 
 
 {\displaystyle k_{\mathrm {d} }}
 
.
Since the mapping wraps the s-plane's 
 
 
 
 j
 ω
 
 
 {\displaystyle j\omega }
 
 axis around the z-plane's unit circle repeatedly, any zeros (or poles) greater than the Nyquist frequency will be mapped to an aliased location.
In the (common) case that the analog transfer function has more poles than zeros, the zeros at 
 
 
 
 s
 =
 ∞
 
 
 {\displaystyle s=\infty }
 
 may optionally be shifted down to the Nyquist frequency by putting them at 
 
 
 
 z
 =
 −
 1
 
 
 {\displaystyle z=-1}
 
, causing the transfer function to drop off as 
 
 
 
 z
 →
 −
 1
 
 
 {\displaystyle z\rightarrow -1}
 
 in much the same manner as with the <a href="/facts/Bilinear_transform/hstbQ7y8">bilinear transform</a> (BLT).
While this transform preserves <a href="/facts/BIBO_stability/RxD8h2f5">stability</a> and <a href="/facts/Minimum_phase/a9WcqXXc">minimum phase</a>, it preserves neither time- nor frequency-domain response and so is not widely used. More common methods include the BLT and <a href="/facts/Impulse_invariance/zwtuq5FY">impulse invariance</a> methods. MZT does provide less high frequency response error than the BLT, however, making it easier to correct by adding additional zeros, which is called the MZTi (for "improved").
A specific application of the matched Z-transform method in the digital control field is with the <a href="/facts/Ackermann%2527s_formula/lkNgjJQW">Ackermann's formula</a>, which changes the poles of the <a href="/facts/Controllability/p1TLxgLf">controllable</a> system; in general from an unstable (or nearby) location to a stable location.

Matched Z-transform method open-in-new

Matched Z-transform method