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.