Principal component filter banks: existence issues, and application to modulated filter banks

Principal component lter banks (PCFB's) sequentially compress most of the input signal energy into the rst few subbands, and are mathematically de ned using the notion of majorization. In a series of recent works, we have exploited connections between majorization and convexity theory to provide a uni ed explanation of PCFB optimality for numerous signal processing problems, involving compression, noise suppression and progressive transmission. However PCFB's are known to exist for all input spectra only for three special classes of orthonormal lter banks (FB's): Any class of two c hannelFB's, the transform coder class and the unconstrained class. This paper uses the developed theory to describe techniques to examine existence of PCFB's. We prove that the classes of DFT and cosine{modulated FB's do not have PCFB's for large families of input spectra. This result is new and quite di erent from most known facts on nonexistence of PCFB's, which usually involve very speci c examples and proofs with numerical optimizations.