FIR (q) filter design without the linear phase constraint

The authors present a novel approach to the problem of designing a finite impulse response (FIR) filter of specified length q, which approximates in uniform frequency (L_∞) norm a given desired (possibly infinite impulse response) filter transfer function. They derive an algorithm-independent lower bound on the achievable approximation error and then present an approximation method which involves the solution of a fixed number of all-pass (Nehari) extension problems and so is called the Nehari shuffle. An upper bound on the approximation error for the algorithm is derived. As this bound is calculable a priori, the length of filter required to satisfy a given maximum error can be found before designing the filter. Examples indicate that the method closely approaches the derived global lower bound. The method is compared with the Parks-McClellan (Remez exchange) algorithm in some examples