Wavelets and filter banks: theory and design

Wavelets, filter banks and multiresolution signal analysis, have been used independently in the fields of applied mathematics, computer vision and signal processing. It is interesting to note that they performed similar functions in different fields. It is recently, that they converged to form a single theory. In the paper, it is shown that the fundamental idea was the same in all the fields i.e. breaking down of the signal or functions into component signals at different resolutions. The Wavelet transform are first compared to classical short-time Fourier Transform (STFT) and it is shown that the wavelet transform approach solves the shortcomings of STFT. Then, the relation between wavelets, filter banks and multiresolution analysis are explored. The author explains how regular perfect reconstruction digital filter banks (PRFB) can be used to compute continuous wavelet transform and perform discrete wavelet transform. The set of conditions necessary to achieve PRFB are then derived in the paper. The author then discusses different design techniques for PRFB. Orthogonal filter banks are briefly discussed and design techniques for the same are mentioned in the paper. Next, a comparison between perfect reconstruction conditions and Bezout identity is made. Using the Bezout identity, the author formulates a set of constraints, for a set of filters to achieve perfect reconstruction. Then, with an analogy to Diophantine equations, he shows that for a given low pass filter, how one can find all the complementary filters satisfying perfect reconstruction conditions. Also, an alternative approach of using the theory of continued fractions is discussed. The author specifically addresses the regularity condition and proposes new techniques to produce regular designs. He finally uses the derived conditions and constraints to design highly regular filter banks and presents the results in the paper. In particular, biorthogonal compactly supported wavelets bases with symmetries using regular FIR are derived.