11.8 Continuous time filter design  (Page 3/3)

The form for the magnitude squared of the frequency-response function for the Chebyshev filter is

where $C_N\left(\omega \right)$ is an Nth-order Chebyshev polynomial and $ϵ$ is a parameter that controls the ripple size. This polynomial in $\omega$ has very special characteristics that result in the optimality of the response function ( [link] ).

Bessel filters

Insert bessel filter information

Elliptic filters

There is yet another method that has been developed that uses a Chebyshev error criterion in both the passband and the stopband.This is the fourth possible combination of Chebyshev and Taylor's series approximations in the passband and stopband. The resultingfilter is called an elliptic-function filter, because elliptic functions are normally used to calculate the pole and zerolocations. It is also sometimes called a Cauer filter or a rational Chebyshev filter, and it has equal ripple approximation error inboth pass and stopbands [link] , [link] , [link] , [link] .

The error criteria of the elliptic-function filter are particularly well suited to the way specifications for filtersare often given. For that reason, use of the elliptic-function filter design usually gives the lowest order filter of the fourclassical filter design methods for a given set of specifications. Unfortunately, the design of this filter is themost complicated of the four. However, because of the efficiency of this class of filters, it is worthwhile gaining someunderstanding of the mathematics behind the design procedure.

This section sketches an outline of the theory of elliptic- function filter design. The details and properties of the ellipticfunctions themselves should simply be accepted, and attention put on understanding the overall picture. A more complete development isavailable in [link] , [link] .

Because both the passband and stopband approximations are over the entire bands, a transition band between the two must bedefined. Using a normalized passband edge, the bands are defined by

This is illustrated in Figure .

The characteristics of the elliptic function filter are best described in terms of the four parameters thatspecify the frequency response:

1. The maximum variation or ripple in the passband ${\delta }_1$ ,
2. The width of the transition band $({\omega }_s-1)$ ,
3. The maximum response or ripple in the stopband ${\delta }_2$ , and
4. The order of the filter $N$ .

The result of the design is that for any three of the parameters given, the fourth is minimum. This is a very flexible andpowerful description of a filter frequency response.

The form of the frequency-response function is a generalization of that for the Chebyshev filter

where

with $F\left(s\right)$ being the prototype analog filter transfer function similar to that for the Chebyshev filter. $G\left(\omega \right)$ is a rational function that approximates zero in the passband and infinity inthe stopband. The definition of this function is a generalization of the definition of the Chebyshev polynomial.

Filter design demonstration

Conclusion

As can be seen, there is a large amount of information available in filter design, more than an introductory module can cover. Even for designing Discrete-time IIR filters, it is important to remember that there is a far larger body of literature for design methods for the analog signal processing world than there is for the digital. Therefore, it is often easier and more practical to implement an IIR filter using standard analog methods, and then discretize it using methods such as the Bilateral Transform.