Performance analysis of the DCT-LMS adaptive filtering algorithm

Abstract

This paper presents the convergence analysis result of the discrete cosine transform-least-mean-square (DCT-LMS) adaptive filtering algorithm which is based on a well-known interpretation of the variable stepsize algorithm. The time-varying stepsize of the DCT-LMS algorithm is implemented by the modified power estimator to redistribute the spread power after the DCT. The performance analysis is considerably simplified by the modification of a power estimator. First of all, the proposed DCT-LMS algorithm has a fast convergence rate when compared to the LMS, the normalised LMS (NLMS), the variable stepsize LMS (VSLMS) algorithm for a highly correlated input signal, whilst constraining the level of the misadjustment required by a specification. The main contribution of this paper is the statistical performance analysis in terms of the mean and mean-squared error of the weight error vector. In addition, the decorrelation property of the DCT-LMS is derived from the lower and upper bounds of the eigenvalue spread ratio, λ_max/λ_min. It is also shown that the shape of sidelobes affecting the decorrelation of the input signal is governed by the location of two zeros. Theoretical analysis results are validated by the Monte Carlo simulation. The proposed algorithm is also applied in the system identification and the inverse modelling for a channel equalisation in order to verify its applicability.

Zusammenfassung

In dieser Arbeit wird eine Konvergenzanalyse des “discrete cosine transform-least mean square” (DCT-LMS) adaptiven Filteralgorithmus präsentiert, welche auf einer bekannten Interpretation des Variablen Schrittgröße-Algorithmus beruht. Die zeitvariante Schrittgröße des DCT-LMS-Algorithmus wird durch den modifizierten Leistungsschätzer implementiert, um die gestreute Leistung nach der DCT umzuverteilen. Die Analyse der Leistungsfähigkeit wird durch die Modifikation eines Leistungsschätzers erheblich vereinfacht. Der vorgeschlagene DCT-LMS-Algorithmus besitzt verglichen mit dem LMS-Algorithmus, dem normierten LMS-Algorithmus (NLMS-Algorithmus) und dem LMS-Algorithmus mit variabler Schrittgröße (VSLMS-Algorithmus) eine schnelle Konvergenzrate für ein stark korreliertes Eingangssignal, wobei die durch eine Spezifikation geforderte Größe der Fehleinstellung eingeschränkt wird. Der Hauptbeitrag dieser Arbeit liegt in der statistischen Analyse der Leistungsfähigkeit hinsichtlich des Mittelwerts und des mittleren quadratischen Fehlers des Gewichts-Fehlervektors. Zusätzlich wird die Dekorrelationseigenschaft des DCT-LMS aus der unteren und oberen Schranke der Konditionszahl λ_max/λ_min abgeleitet. Es wird weiters gezeigt, daß die Form von Nebenmaxima, die die Dekorrelation des Eingangssignals beeinflussen, durch die Lage zweier Nullstellen bestimmt wird. Die Ergebnisse der theoretischen Analyse werden durch Monte Carlo-Simulation überprüft. Der vorgeschlagene Algorithmus wird zur Verifizierung seiner Anwendbarkeit auch auf die Systemidentifikation und inverse Modellierung im Rahmen einer Kanalentzerrung angewandt.

Résumé

Cet article présente le résultat de l'analyse de convergence de l'algorithme de filtrage adaptatif par transformation en cosinus discret et moindre carrés moyens (DCT-LMS), qui repose sur l'interprétation bien connue de l'algorithme à taille de pas variable. La taille de pas variant dans le temps de l'algorithme DCT-LMS est mise en œuvre par l'estimateur de puissance modifié pour redistribuer la puissance étalée après la DCT. L'analyse de performance est considérablement simplifiée par la modification de l'estimateur de puissance. Tout d'abord, l'algorithme DCT-LMS proposé a un taux de convergence rapide en comparaison avec les algorithmes LMS, LMS normalisé, et LMS à taille de pas variable pour un signal d'entrée hautement corrélé, tout en contraignant le niveau de mésajustement demandé par les spécifications. La principale contribution de cet article est l'analyse de performances statistique en termes de l'erreur moyenne et de l'erreur quadratique moyenne du vecteur d'erreur des poids. De plus, la propriété de décorrélation du DCT-LMS est dérivée des bornes inférieures et supérieures du rapport d’étalement des valeurs propres λ_max/λ_min. Nous montrons aussi que la forme des lobes latéraux affectant la décorrélation du signal d'entrée est gouvernée par la position de deux zéros. Des résultats d'analyse théorique sont validés par une simulation Monte Carlo. L'algorithme proposé est aussi appliqué en identification de systèmes et en modélisation inverse pour l’égalisation de canal afin de vérifier son applicabilité.

Introduction

Adaptive filtering algorithms based on the stochastic gradient method are widely used in many applications such as system identification, noise cancellation, active noise control and communication channel equalisation. The least mean square (LMS) which belongs to the stochastic gradient-type algorithm has been the focus of much study due to its simplicity and robustness. However, it is well known that the convergence rate is seriously affected by the correlation of an input signal. To circumvent this inherent limitation, many algorithms have been implemented.

As one of popular approaches, the transform domain least-mean-square (TDLMS) adaptive filtering algorithms [3], [11], [18], [19], [21], [8] have been developed to improve a slow convergence rate caused by an ill-conditioned input signal.

In 1983, Narayan [19] first introduced the TDLMS algorithm which uses the orthogonal transform matrices of the discrete Fourier transform (DFT) and the discrete cosine transform (DCT). The enhanced convergence rate when compared with the conventional LMS algorithm was verified empirically. However, focus was not placed on theoretical analysis. The performance was judged purely by computer simulation. In 1988, Florian [11] analysed the performance of the weighted normalised LMS algorithm via exponential weighted parameters. It was analysed only for the mean behaviour of weights. However, a general derivation was not obtained. In 1989, Marshall [18] investigated the convergence property through the computer simulation for several unitary transform matrices. In his work, transform domain processing was characterised by the effect of the transform on the shape of the error performance surface. In 1995, Beaufay [3] also studied analytically the behaviour of the eigenvalue spread for a first-order Markov process in the discrete Fourier transform least-mean-square (DFT-LMS) and the discrete cosine transform least-mean-square (DCT-LMS) algorithms. In most recent work (1997) [21], Parikh proposed the modified escalator structure to improve the performance of the LMS adaptive filter. The algorithm utilised the sparse structure of the correlation matrix. The sparse structure is extracted from the unitary transform matrix of the DCT to be applied in the escalator structure of the lattice model. This filter is not an efficient filtering structure in that it employs two transform layers: a unitary transform matrix and an escalator structure of the lattice model. The first transform layer by a unitary transform matrix does not affect the convergence speed because a correlated input signal is decorrelated by the escalator structure of the second layer.

As another alternative technique to overcome a slow convergence rate, the variable step-size LMS (VSLMS) algorithms [1], [7], [13], [17] have been developed to enhance the convergence rate and to reduce the misadjustment error in the state space. However, they might not be also effective for a highly correlated input signal. This is because the dynamic range of the variable stepsize is restricted by a directional convergence nature. In addition, the normalised LMS (NLMS) might be efficient and robust algorithm for the nonstationary input process, however, it also suffers from a slow convergence speed if driven by a highly correlated input signal. To resolve this problem, Ozeki [20] and Rupp [22] proposed the so-called affine projection algorithms to decorrelate an input signal.

In the first part of this paper, we analyse the decorrelation properties by the measure of the eigenvalue spread ratio, the complementary spectrum principle and the pole-zero location. It has been known that the DCT decorrelates effectively the input signal whose power spectrum lies in the low-frequency band. Boroujeny [9] explained intuitively the decorrelation feature of the DCT for the lowpass input process from the filtering viewpoint. However, this work did not show analytically how the DCT can decorrelate the input signals with the low-frequency input spectrum, similarly to the Karhunen–Loève transform (KLT).

In the second part of this paper, we analyse the convergence behaviour of the DCT-LMS adaptive filtering algorithm which is based on a well-known interpretation of the variable stepsize algorithm. A time-varying stepsize is implemented by the modified power estimator to redistribute the spread power after the transformation. This modification makes the performance analysis simple.

As we have investigated the previous work relevant to the transform domain adaptive filtering structure [3], [11], [18], [19], [21], so far, only a limited analysis of the TDLMS algorithm has been performed due to the difficulty of the analytical derivation for the normalisation term. The exponential weighted method is generally used for obtaining the convergence parameter μ_i(n) which is

$$\text{μ}{}_{\mathrm{i}}\text{(n)=}\text{μ}{}_{\mathrm{o}}\text{P}\text{̂}{}_{\mathrm{i}}\text{(n)}\text{=}\text{μ}{}_{\mathrm{o}}\text{(1−β)∑}{}_{\mathrm{k=0}}{}^{\mathrm{\infty }}\text{β}{}^{\mathrm{k}}\text{|x}{}_{\mathrm{i}}\text{(n−k)|}{}^2$$

at the ith bin of transform domain, where β∈[0,1], $\text{P}\text{̂}{}_{\mathrm{i}}$ is the power estimator, μ_i denotes elements of the diagonal matrix defined as $\text{diag}\text{[μ}{}_{\mathrm{i}}\text{(n),}\text{i=0,…,N−1]}$ and x_i(n) is the transformed input signal at the ith bin. The exponential weighted parameter also has the recursive form

$$\text{P}\text{̂}{}_{\mathrm{i}}\text{(n)=β}\text{P}\text{̂}{}_{\mathrm{i}}\text{(n−1)+(1−β)|x}{}_{\mathrm{i}}\text{(n)|}{}^2\text{.}$$

In this paper, we propose the modified power estimator based upon (1)

$$\text{u}{}_{\mathrm{i}}\text{(n)=γ(1−β)}\text{∑}\text{k=0}\text{∞}\text{β}{}^{\mathrm{k}}\text{1}\text{ε+(1/M)}\text{x}{}_{\mathrm{i}}{}^{\mathrm{<mtext>T</mtext>}}\text{(n−k)}\text{x}{}_{\mathrm{i}}\text{(n−k)}\text{,}$$

where $\text{β∈[0,1],}\text{γ∈[0,1],}\text{0<ε≪1,}\text{i=0,…,N−1}$, and M denotes the size of sample to estimate the power at the ith bin after transformation.

The main contribution of this paper is the statistical performance analysis of the DCT-LMS adaptive filtering algorithm based on the modified power estimator. In addition, the decorrelation properties of the DCT is described from the lower and upper bounds of the eigenvalue spread ratio. In particular, it is shown that the shape of sidelobes affecting the decorrelation of the input signal is governed by the location of two zeros. The theoretical analysis results are validated by the Monte Carlo simulation.

The rest of this paper is organised as follows. In Section 2, the decorrelation properties are investigated. In Section 3, the DCT-LMS adaptive filter via the modified power estimator is described. In Section 4, the convergence behaviour of the proposed algorithm is analysed. In Section 5, the computer simulation is undertaken in the system identification and the channel equalisation examples to verify the performance of the proposed DCT-LMS algorithm. The simulation results are compared to the standard LMS, the NLMS and the VSLMS algorithms [17]. Conclusions are then given in Section 6.