Results on principal component filter banks: Colored noise suppression andexistence issues

We have recently made explicit the precise connection between the optimizat ion of orthonormal filter banks (FBs) and the principal component property: The principal component filter bank (PCFB) is optimal whenever the minimiz ation objective is a concave function of the subband variances of the FB. T his explains PCFB optimality for compression, progressive transmission, and various hitherto unnoticed white-noise suppression applications such as su bband Wiener filtering. The present work examines the nature of the FB opti mization problems for such schemes when PCFBs do not exist. Using the geome try of the optimization search spaces, we explain exactly why these problem s are usually analytically intractable. We show the relation between compac tion tilter design (i.e., variance maximization) and optimum FBs. A sequent ial maximization of subband variances produces a PCFB if one exists, but is otherwise suboptimal for several concave objectives. We then study PCFB op timality for colored noise suppression. Unlike the case when the noise is w hite, here the minimization objective is a function of both the signal and the noise subband variances. We show that for the transform coder class, if a common signal and noise PCFB (KLT) exists, it is optimal for a large cla ss of concave objectives, Common PCFBs for general FB classes have a consid erably more restricted optimality, as we show using the class of unconstrai ned orthonormal FBs. For this class, we also show how to find an optimum FB when the signal and noise spectra are both piecewise constant with all dis continuities at rational multiples of pi.