In the previous two examples we have used Butterworth filter. The Butterworth filter of order n is described by the magnitude square frequency response of

It has the following properties

1.

2.

4. As n gets larger, approaches an ideal low pass filter

5. is called maximally flat at origin, since all order derivative exist and they are zero at

The poles of a Butterworth filter lie on circle of radius in s-plane.
There are two types of Chebyshev filters, one containing ripples in the passband (type I) and the other containing a ripple in the stopband (type II). A Type I low pass normalizer Chebyshev filter has the magnitude squared frequency response.

where is n^th order Chebyshev polynomial. We have the relationship

Chebyshev filters have the following properties

1. The magnitude squared frequency response oscillates between 1 and within the passband, the so called equiripple and has a value of at , the normalized cut off frequency.
2. The magnitude response is monotonic outside the passband including transitionand stopband.
3. The poles of the Chebysher filter lie on an ellipse in s-plane.

An elliptic filter has ripples both in passband and in stopband. The square magnitude frequency response is given by

where is Chebyshev rational function of O determined from specified ripple characteristics.

An n^th order Chebyshev filter has sharper cutoff than a Butterworth filter, that is, has a narrower transition bandwidth. Elliptic filter provides the smallest transition width.