Filter Banks and DWT

Abstract

The study of digital signal processing normally concentrates on the design, realization, and application of single-input, single-output digital filters. There are applications, as in the case of spectrum analyzer, where it is desired to separate a signal into a set of sub-band signals occupying, usually nonoverlapping, portions of the original frequency band. In other applications, it may be desired to combine many such sub-band signals into a single composite signal occupying the whole Nyquist range. To this end, digital filter banks play an important role. Implementation of a filter bank on a processor with finite precision arithmetic necessitates quantization of filter coefficients [95]. This results in loss of perfect reconstruction (PR) property. The theory of filter banks were developed much before modern discrete wavelet transform (DWT) analysis became popular [127, 134]. The study of literature reveals a close relationship between the DWT and digital filter banks. It turns out that a tree of digital filter banks, without computing mother wavelets, can simply achieve the wavelet transform. Hence, the filter banks have been playing a central role in the area of wavelet analysis. It is therefore of interest to study the filter bank theory before addressing the implementation issues of finite precision wavelet transforms. In this chapter, fundamental concept of filter bank theory leading to new implementation issues described in latter chapters is introduced. The material presented in this chapter will be useful in discussing error modeling and parallel computing techniques discussed in the book. In present chapter, the filter bank concept related to DWT is revisited in Sect. 2.1. Section 2.2 presents two-channel PR filter bank. Section 2.3 presents derivation of parallel filter DWT from pyramid DWT structure. Section 2.4 presents frequency response of generated parallel filters followed by conclusion in Sect. 2.5.