Designing N-band biorthogonal FIR systems given several of the analysis and synthesis filters

Abstract

The problem of designing an N-band biorthogonal FIR analysis/synthesis system, given K of its analysis and synthesis filters, is addressed in this paper. A necessary and sufficient condition on the K filters, that guides their selection, is derived. The solution set is completely parameterized via ladder-type structures that guarantee the perfect reconstruction property under both coefficient quantization and roundoff errors. Both nonlinear- and linear-phase systems are considered. A design example is included to illustrate the theory.

Introduction

Designing an N-band analysis/synthesis perfect reconstruction (PR) FIR filter bank (FB) given $K⩾1$ of the analysis filters constitutes a powerful means of constructing scalar- and vector-valued wavelet bases with desired properties in their analysis functions, and has also been shown to facilitate the design and realization of PRFBs [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. A complete solution to the above so-called analysis $(N,K)$-problem has been reported in [20] both for the paraunitary and the general biorthogonal linear-phase (LP) and non-LP cases.¹ Related works include [22], [23], [24], [25]. However, such a design approach provides no direct control on the quality of the synthesis FB [15]. Of course, this is of no concern in the case of orthogonal analysis/synthesis systems where the synthesis filters are simply time-reversed versions of the analysis ones. For nonorthogonal systems, however, this becomes an important issue, especially in applications like wavelet design, where it is well-known that the regularity of the analysis and synthesis wavelets has to be separately enforced [26]. Furthermore, it has been pointed out that the regularity of the synthesis wavelets may be equally or even more important than that of the analysis ones [5], [27], [28], [29]. Moreover, the interest in considering biorthogonal multi-band wavelets was demonstrated in [26], among others.

Hence the question arises of whether one can build an N-band biorthogonal analysis/synthesis system with $1⩽K<N$ filters being selected a priori in both the analysis and synthesis filter banks. This problem was posed in [26] in the context of wavelet design, though no solution was provided. A complete solution to the above problem (to be referred to hereafter as the analysis/synthesis (A/S) ($N,K$)-problem) is presented in this paper. Based on existing results on the analysis $(N,K)$-problem, a procedure of determining a particular set of the filters that complete the given analysis and synthesis filters to a PR A/S system is developed. The set of all possible completions is also parameterized. A ladder-type realization scheme for the system is suggested by the solution process, which generalizes the well-known lifting scheme [30], to multiple channels. The design of the K filters in the two banks as well as the optimization of the frequency characteristics of the $N-K$ complementary filters are also studied.

The rest of this paper is organized as follows. In Section 2 the analysis $(N,K)$-problem is recalled along with results on its solution. These are employed to arrive at the solution of the corresponding A/S problem in Section 3, for both the general (non-LP) and LP cases. The problem of designing the K given analysis and synthesis filters is also addressed. The optimization of the complementary filters is discussed in Section 4, where a design example is presented in detail. Section 5 concludes the paper.