Design of FIR Paraunitary Filter Banks for Subband Coding Using a Polynomial Eigenvalue Decomposition

Abstract

The problem of paraunitary filter bank design for subband coding has received considerable attention in recent years, not least because of the energy preserving property of this class of filter banks. In this paper, we consider the design of signal-adapted, finite impulse response (FIR), paraunitary filter banks using polynomial matrix EVD (PEVD) techniques. Modifications are proposed to an iterative, time-domain PEVD method, known as the sequential best rotation (SBR2) algorithm, which enables its effective application to the problem of FIR orthonormal filter bank design for efficient subband coding. By choosing an optimisation scheme that maximises the coding gain at each stage of the algorithm, it is shown that the resulting filter bank behaves more and more like the infiniteorder principle component filter bank (PCFB). The proposed method is compared to state-of-the-art techniques, namely the iterative greedy algorithm (IGA), the approximate EVD (AEVD), standard SBR2 and a fast algorithm for FIR compaction filter design, called the window method (WM). We demonstrate that for the calculation of the subband coder, the WM approach offers a low-cost alternative at lower coding gains, while at moderate to high complexity, the proposed approach outperforms the benchmarkers. In terms of run-time complexity, AEVD performs well at low orders, while the proposed algorithm offers a better coding gain than the benchmarkers at moderate to high filter order for a number of simulation scenarios. Index Terms-Orthonormal subband coders, paraunitary matrix, principal component filter banks, polynomial matrix eigenvalue decomposition, sequential best rotation. I. INTRODUCTION P ARAUNITARY filter banks have been extensively studied for subband coding and applied to an increasing number of applications, including noise reduction [1], audio and image coding [2] and digital communications [3], [4]. For the case where the order of the filters is unconstrained, it is known that a principal component filter bank (PCFB) [5], [6] exists and is an orthonormal or paraunitary (PU) filter bank that is simultaneously optimal for a number of objectives [7], including mean-squared error and coding gain for subband coding in data compression applications [8]. This is also true when the filter orders are constrained to be not greater

Impact of source model matrix conditioning on PEVD algorithms

“…A number of PEVD algorithms have been introduced [4,[6][7][8][9][10], and offer various performance characteristics. The algorithms in [4,6,10] have been demonstrated on parahermitian matrices R(z) ∈ C M×M derived from random A(z) ∈ C M×K as R(z) = A(z)Ã(z).…”

“…For K < M , R(z) is guaranteed to be rank deficient, but when K ≥ M it is possible for R(z) to have full rank. In [7,8], subband coding was considered as an application, and the parahermitian matrices that need to be factorised by the algorithms were based on auto-regressive functions generating auto-correlation functions with infinite support but potentially permitting finite order paraunitary factors (for a justification, see the factorisation in Sec. IV.B.3 in [8]).…”

“…In [7,8], subband coding was considered as an application, and the parahermitian matrices that need to be factorised by the algorithms were based on auto-regressive functions generating auto-correlation functions with infinite support but potentially permitting finite order paraunitary factors (for a justification, see the factorisation in Sec. IV.B.3 in [8]). In [9], a source model convolutively mixes spectrally majorised sources by means of an arbitrary paraunitary matrix, such that the ground truth PEVD with finite order factors and equality in (1) is guaranteed.…”

“…In [9], a source model convolutively mixes spectrally majorised sources by means of an arbitrary paraunitary matrix, such that the ground truth PEVD with finite order factors and equality in (1) is guaranteed. Since these publications [4,[6][7][8][9][10] use differently conditioned problems, a direct comparison between algorithms proposed in individual papers is not always straightforward.…”

“…An ensemble of randomised parahermitian matrices with different dynamic ranges and types of majorisation are factorised by a number of PEVD algorithms belonging to the second order sequential best rotation (SBR2 family, [4,8] and the sequential matrix diagonalisation (SMD family, [9,10]). …”