TABLE OF CONTENTS The elliptic or Cauer approximation function provides the best selectivity characteristic of any of the approximation methods discussed thus far. No other approximation method will be able to provide a lower-order filter for the specifications provided. The elliptic filter combines ripple in the passband and stopband in order to accomplish this feat. However, the elliptic approximation is also the most difficult to design. It involves the most sophisticated mathematical functions of any of the methods discussed in this text. Luckily, many good minds have laid the foundation for this work and their results will be presented here so that we can put the design procedure into a workable algorithm.
The elliptic approximation's magnitude frequency response function is shown in (2.55), where R n is the Chebyshev rational function of order n. R n is composed of both numerator and denominator portions, which allow an equiripple response in both the passband and stopband.
The Chebyshev rational function R n and much of elliptic approximation theory is based on the elliptic integral and the Jacobian elliptic functions. These functions can be evaluated via advanced mathematical packages available for most computers and are discussed in Appendix D. The incomplete elliptic integral of the first kind is shown in (2.57), where k is referred to as the modulus and ? is the amplitude of the integral. The modulus k must be less than or equal to 1 for the...
