Elliptic filter

An elliptic filter (also known as a Cauer filter, named after <a href="/facts/Wilhelm_Cauer/TfLzRRAc">Wilhelm Cauer</a>, or as a Zolotarev filter, after <a href="/facts/Yegor_Zolotarev/pCXfRy4x">Yegor Zolotarev</a>) is a <a href="/facts/Filter_(signal_processing)/vydJCNle">signal processing filter</a> with equalized <a href="/facts/Ripple_(filters)/T9mzYBAh">ripple</a> (equiripple) behavior in both the <a href="/facts/Passband/VebMrazm">passband</a> and the <a href="/facts/Stopband/rfIhELus">stopband</a>. The amount of ripple in each band is independently adjustable, and no other filter of equal order can have a faster transition in <a href="/facts/Gain_(electronics)/CWbehg27">gain</a> between the <a href="/facts/Passband/VebMrazm">passband</a> and the <a href="/facts/Stopband/rfIhELus">stopband</a>, for the given values of ripple (whether the ripple is equalized or not). Alternatively, one may give up the ability to adjust independently the passband and stopband ripple, and instead design a filter which is maximally insensitive to component variations.
As the ripple in the stopband approaches zero, the filter becomes a type I <a href="/facts/Chebyshev_filter/bMllKQan">Chebyshev filter</a>. As the ripple in the passband approaches zero, the filter becomes a type II <a href="/facts/Chebyshev_filter/bMllKQan">Chebyshev filter</a> and finally, as both ripple values approach zero, the filter becomes a <a href="/facts/Butterworth_filter/3sQkLaqS">Butterworth filter</a>.
The gain of a <a href="/facts/Lowpass/QFsUjw7h">lowpass</a> elliptic filter as a function of angular frequency ω is given by:

G
          
            n
          
        
        (
        ω
        )
        =
        
          
            1
            
              
                1
                +
                
                  ϵ
                  
                    2
                  
                
                
                  R
                  
                    n
                  
                  
                    2
                  
                
                (
                ξ
                ,
                ω
                
                  /
                
                
                  ω
                  
                    0
                  
                
                )
              
            
          
        
      
    
    {\displaystyle G_{n}(\omega )={1 \over {\sqrt {1+\epsilon ^{2}R_{n}^{2}(\xi ,\omega /\omega _{0})}}}}

where Rn is the nth-order <a href="/facts/Elliptic_rational_functions/hazXgDo5">elliptic rational function</a> (sometimes known as a Chebyshev rational function) and

ω
          
            0
          
        
      
    
    {\displaystyle \omega _{0}}
  
  is the cutoff frequency

ϵ
      
    
    {\displaystyle \epsilon }
  
 is the ripple factor

ξ
 
 
 {\displaystyle \xi }
 
 is the selectivity factor
The value of the ripple factor specifies the passband ripple, while the combination of the ripple factor and the selectivity factor specify the stopband ripple.

Elliptic filter open-in-new

Elliptic filter