0.8 Filter banks and transmultiplexers  (Page 23/23)

Filter banks and wavelets—summary

Filter banks are structures that allow a signal to be decomposed into subsignals—typically at a lower sampling rate.If the original signal can be reconstituted from the subsignals, the filter bank is said to be aperfect reconstruction (PR) filter bank. For PR, the analysis and synthesis filters have to satisfy a set of bilinear constraints.These constraints can be viewed from three perspectives, viz., the direct, matrix, and polyphase formulations. In PR filter bankdesign one chooses filters that maximize a “goodness" criterion and satisfy the PR constraints.

Unitary filter banks are an important class of PR filter banks—they give orthogonal decompositions of signals. For unitary filter banks,the PR constraints are quadratic in the analysis filters since the synthesis filters are index-reverses of the analysis filters.All FIR unitary filter banks can be explicitly parameterized. This leads to easy design (unconstrained optimization) and efficientimplementation. Sometimes one can impose structural constraints compatible with the goodness criterion. For example, modulated filter banks require that theanalysis and synthesis filters are modulates of single analysis and synthesis prototype filter respectively. Unitary modulated filterbanks exist and can be explicitly parameterized. This allows one to design and implement filter banks with hundreds of channels easilyand efficiently. Other structural constraints on the filters (e.g., linear phase filters) can be imposed and lead toparameterizations of associated unitary filter banks. Cascades of filter banks (used in a tree structure) can be used torecursively decompose signals.

Every unitary FIR filter bank with an additional linear constraint on the lowpass filter is associatedwith a wavelet tight frame. The lowpass filter is associated with the scaling function, and the remaining filters are each associated withwavelets. The coefficients of the wavelet expansion of a signal can be computed using a tree-structure where the filter bank isapplied recursively along the lowpass filter channel. The parameterization of unitary filter banks, with a minor modification,gives a parameterization of all compactly supported wavelet tight frames. In general, wavelets associated with a unitary filter bank areirregular (i.e., not smooth). By imposing further linear constraints (regularity constraints) on the lowpass filter,one obtains smooth wavelet bases. Structured filter banks give rise to associated structured waveletbases; modulated filter banks are associated with modulated wavelet bases and linear phase filter banks are associated with linear-phasewavelet bases. Filter banks cascade—where all the channels are recursively decomposed, they are associated with wavelet packet bases.

From a time-frequency analysis point of view filter banks trees can be used to give arbitraryresolutions of the frequency. In order to obtain arbitrary temporal resolutions one has to use local bases or switch between filter banktrees at points in time. Techniques for time-varying filter banks can be used to generate segmented wavelet bases (i.e., a different waveletbases for disjoint segments of the time axis). Finally, just as unitary filter banks are associated with wavelet tight frames,general PR filter banks, with a few additional constraints, are associated with wavelet frames (or biorthogonal bases).