Direct approach to design of PCAS filters with combined gain and phase specification

New approaches have been reported for the design of recursive digital filters with prescribed gain and delay using classical optimisation, simulated annealing and genetic algorithms. The filter structure used consisted of two allpass subfilters in parallel (PCAS). PCAS filters are used because of their low complexity and roundoff noise as well as their ability to realise nonminimum-phase transfer functions. An approach is described in which a set of linear equations are formed for each subfilter using a result due to Gregorian and Temes (1978). By imposing certain constraints on these equations, which relate to the filtering characteristics required, they may be solved yielding the coefficients of the two transfer functions. Several examples are given for the case of approximately linear phase. The L₂ norm is used as a metric to enable comparison with optimisation techniques. General comments are made on the relationship between the various parameters such as a_p, a_s, filter order, phase slope, etc. The design of PCAS filters with arbitrary phase is discussed