Are nonuniform principal component filter banks optimal?

The notion of a principal component filter bank (PCFB) for a given class of uniform filter banks (FB's) has been well studied. Recent work by the authors has shown that PCFB's are optimal orthonormal FB's whenever the minimization objective is a concave function of the vector of subband variances of the FB. This result gives a unified explanation of PCFB optimality for progressive transmission, compression, noise suppression, and as shown more recently, for use in DMT (discrete multi-tone modulation) systems. This paper generalizes such results to nonuniform FB's. We propose two distinct definitions of nonuniform PCFB's. Each definition results in PCFB optimality for certain types of concave objectives whose form is somewhat more restricted than in the case of uniform FB's. We study existence of the defined PCFB's, and observe that it can be very delicate: Small perturbations of the input spectra can sometimes destroy the existence of nonuniform PCFB's.