A design procedure for optimal energy compaction IIR filters

In this brief we consider the energy compaction problem for filters belonging to a restricted class of infinite-impulse response (IIR) filters, having the transfer function h(z)=(H/sub 0/(z/sup 2/)+z/sup -1/ H/sub 1/(z/sup 2/))//spl radic/2, where both polyphase components H/sub 0/(z) and H/sub 1/(z) are stable all pass filters. Assuming rational input spectra, we reduce the infinite series expansion of the compaction gain criterion to a finite sum with terms depending on the parameters of the filter H(z). Simple expressions for computing the gradient of the criterion are derived, and finally a gradient ascent maximization routine is proposed, which provides the IIR filter with optimum parameters. Simulation results illustrate the good compaction gain of the optimal IIR filters, which despite their constrained nature (allpass polyphase components), may well outperform finite-impulse response (FIR) filters with much more parameters. The main envisaged application of our work is building QMF perfect reconstruction filterbanks with maximum coding gain, but the above optimization problem and simple extensions of it can be solved in the same manner and also be used in other filter design applications.