On the design of recursive low-pass digital filters

The magnitude-squared characteristic of a low-pass filter is approximated, over the finite interval [-1, 1 ], by the ratio φ(x)/[φ(x) + P(x)] of two polynomials. For elliptic filter design, a special case, the polynomials φ(x) and P(x) (of the same order) are chosen such that the ratios P(x)/φ(x) and φ(x)/P(x) approximate, in the Chebyshev sense, the zero function over the passband [x
p
, 1] and stopband [-1, x
s
], respectively. The passband and stopband form two disjoint intervals. The polynominals are determined by repeated applications of Darlington's technique for obtaining a rational function generalization of Chebyshev polynominals.