Perfect reconstruction time-varying IIR filter banks

We consider here the general problem of implementing perfect reconstruction IIR filter banks, allowing the use of both causal and noncausal filtering stages. Past work including that of Smith & Chung (1995) and Chen & Vaidyanathan (1993) has focused exclusively on filter banks having only noncausal filtering stages, probably because the computationally efficient modulated polyphase allpass filter bank falls into this category. We use a canonical direct form II framework here which is closely related to that of Chen and Vaidyanathan. In addition to considering extensions of this framework to include causal filtering stages, we point out some constraints analogous to those found by Smith and Chung for their direct form I implementation. Finally, we consider in detail the problem of selecting initial filter states to guarantee perfect cancellation of startup transients for finite length sequences in the time-varying case. We derive here a result for circular convolution with time-varying IIR filters analogous to that found by Smith & Eddins (1990) for time-invariant filters.