On the design of recursive lowpass digital filters

The magnitude-squared characteristic of an ideal lowpass filter is approximated, over the finite interval [-1,1], by the ratio Φ(x)/Φ(x)+P(x) of two n-th-degree polynomials. The polynomials Φ(x) and P(x) are chosen such that the ratios P(x)/ Φ(x) and Φ(x)/P(x) approximate, in a Chebyshev sense, the zero function over the passband [xp,1] and the stopband [-1,xs], respectively. The passband and stopband form two disjoint intervals. The polynomials are iteratively determined by repeated applications of Darlington´s technique for obtaining rational function generalizations of Chebyshev polynomials. The efficacy of the iterative method is demonstrated by examples.