On existence of FIR principal component filter banks

In this paper we have two interesting results. One is of theoretical interest and the other practical. The theoretical result is that the optimum FIR orthonormal filter bank of a fixed finite degree that maximizes the coding gain does not always contain an optimum compaction filter. In other words, in general, there does not exist a principal component filter bank (PCFB) of a given nonzero degree. This is sharply in contrast to the cases of transform coders and ideal subband coders where the existence of PCFB's are assured by their very construction. The practical result of the paper is that constraining the filter corresponding to the largest subband variance to be a compaction filter does not result in a significant loss of performance for practical input signals. Since there exist very efficient methods to design FIR compaction filters and since the best completion of the filter bank given the first filter is trivially done by a KLT, we see that this is an extremely efficient method despite the fact that it is suboptimum.