Recursive digital filters with maximally flat group delay

A well-known limitation of the recursive digital filter, when compared to the nonrecursive filter, is its incapability of having a strictly linear phase characteristic; thus it may only approximate a constant group delay. For the analog filters the choice of the maximally flat criterion leads to the use of the Bessel polynomials. Yet digital approximations of these continuous filter functions are inadequate to yield the true maximally flat delay approximation of the recursive filters. Our purpose is to provide for the problem at hand a solution obtained by a direct approximation procedure in the
z
plane. The denominator of the transfer function turns out to be a Gaussian hypergeometric function, more particularly connected with the Legendre functions. The stability of the filters is discussed and some numerical results in regard to the amplitude and phase responses as well as the pole loci are given.