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 lter bank of a xed nite degree that maximizes the coding gain does not always contain an optimum compaction lter. In other words, in general, there does not exist a principal component lter bank (PCFB) of a given nonzero degree. This is sharply in contrast to the cases of transfom 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 lter corresponding to the largest subband variance to be a compaction lter does not result in a signi cant loss of performance for practical input signals. Since there exist very e cient methods to design FIR compaction lters and since the best completion of the lter bank given the rst lter is trivially done by a KLT, we see that this is an exteremely e cient method despite the fact that it is suboptimum.