Approximating noncausal IIR digital filters having arbitrary poles, including new Hilbert transformer designs, via forward/backward block recursion

In this paper, we consider the design, use, and recursive implementation of noncausal infinite-impulse response (IIR) digital filters. Forward/backward realization of zero-phase IIR filters is well known for finite data lengths and is also applicable for arbitrary pole locations both inside and outside the unit circle. For systems processing indefinitely long inputs, this can be accomplished by separately filtering blocks of input that are much longer than the effective impulse response duration and combining the block outputs using either the overlap-add method or overlap-save method. Of course, some approximation is required because the corresponding impulse responses have theoretically infinite duration, but the associated error can be made arbitrarily small. In addition to traditional frequency selective filters and arbitrary system designs, we describe new IIR design methods for Hilbert transformers, differentiators, and interpolation networks.