Elliptic Filter - Poles and Zeroes

Poles and Zeroes

The zeroes of the gain of an elliptic filter will coincide with the poles of the elliptic rational function, which are derived in the article on elliptic rational functions.

The poles of the gain of an elliptic filter may be derived in a manner very similar to the derivation of the poles of the gain of a type I Chebyshev filter. For simplicity, assume that the cutoff frequency is equal to unity. The poles of the gain of the elliptical filter will be the zeroes of the denominator of the gain. Using the complex frequency this means that:

Defining where cd is the Jacobi elliptic cosine function and using the definition of the elliptic rational functions yields:

where and . Solving for w

where the multiple values of the inverse cd function are made explicit using the integer index m.

The poles of the elliptic gain function are then:

As is the case for the Chebyshev polynomials, this may be expressed in explicitly complex form (Lutovac & et al. 2001, § 12.8)

where is a function of and and are the zeroes of the elliptic rational function. is expressible for all n in terms of Jacobi elliptic functions, or algebraically for some orders, especially orders 1,2, and 3. For orders 1 and 2 we have

where

The algebraic expression for is rather involved (See Lutovac & et al. (2001, § 12.8.1)).

The nesting property of the elliptic rational functions can be used to build up higher order expressions for :

where .