Orthogonal Filterbanks

Abstract

In this chapter, we explore the use of positive polynomials in the design of FIR filterbanks (FB) . The study is confined to a single class, that of orthogonal FBs. Two-channel FBs are discussed first, as the simplest instance of the problem; naturally related with it are the design of compaction filters or of signal-adapted wavelets. We go then to DFT-modulated FBs, with an arbitrary number of channels; similarly to the two-channel case, the free parameters of the whole FB are the coefficients of a single prototype filter. A typical requirement on FBs is that of perfect reconstruction (PR): the output signal is a delayed version of the input one. The connection between orthogonal FBs and positive polynomials is eased by the fact that PR amounts to simple (Nyquist) conditions on the squared magnitude of the prototype filter. Optimization problems that are nonconvex in the coefficients of the prototype filter become convex once expressed using its squared magnitude, which is a nonegative polynomial described by an appropriate Gram matrix parameterization. After solving the equivalent SDP problem, the prototype filter is recovered by spectral factorization.