Analytical solutions for design of IIR equiripple filters

A purely analytical technique for the design of recursive digital filters with equiripple magnitude behavior is presented. The design is accomplished in a transformed variable denoted by w and defined as w=(z+1/z)/2. The method can be attributed to the class of rational Chebyshev approximations. It uses design steps corresponding to those for elliptical filters in the analog domain. Using Jacobian elliptic functions, a conformal mapping of the w-plane is introduced which describes the solutions of equiripple symmetric specifications. These are not halfband specifications, since the attenuation and the bandripple in the passband are chosen independently of those in the stopband. Examples are given to demonstrate the efficiency of the approach. It is suggested that this sort of algorithm can be used advantageously in adaptive filtering because of the simplicity of the analytical solutions. The algorithms are well behaved, since the infinite series expansion of elliptic functions and the consecutive manipulation with abridged forms of these series have been avoided.