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.