Elsevier

Signal Processing

Volume 89, Issue 5, May 2009, Pages 894-900
Signal Processing

A generalized data windowing scheme for adaptive conjugate gradient algorithms

https://doi.org/10.1016/j.sigpro.2008.11.007Get rights and content

Abstract

The performance of the modified adaptive conjugate gradient (CG) algorithms based on the iterative CG method for adaptive filtering is highly related to the ways of estimating the correlation matrix and the cross-correlation vector. The existing approaches of implementing the CG algorithms using the data windows of exponential form or sliding form result in either loss of convergence or increase in misadjustment. This paper presents and analyzes a new approach to the implementation of the CG algorithms for adaptive filtering by using a generalized data windowing scheme. For the new modified CG algorithms, we show that the convergence speed is accelerated, the misadjustment and tracking capability comparable to those of the recursive least squares (RLS) algorithm are achieved. Computer simulations demonstrated in the framework of linear system modeling problem show the improvements of the new modifications.

Introduction

Adaptive conjugate gradient (CG) algorithms based on the CG method have been successfully introduced to solve the adaptive filtering problems. The advantages are that they have convergence properties superior to those of least mean square (LMS) algorithms and computational cost less than that of the classic recursive least squares (RLS) algorithm. Moreover, the instability problems existing in the RLS algorithm are not likely to occur in the CG algorithm [1], [2]. In the reported modified adaptive CG algorithms [1], [2], [3], [4], [5], [6], two data windowing schemes, i.e., the finite sliding data windowing scheme and the exponentially decaying data windowing scheme, of estimating the correlation matrix and the cross-correlation vector have been applied. However, the analysis and simulations presented in [1], [2] show that the modified CG algorithms, which are implemented using the finite sliding data windowing scheme and run several iterations per data update, have a convergence behavior and misadjustment which depend on the length of data window. A small window size produces slow convergence, whereas a large window size introduces high computational cost and high misadjustment. On the other hand, the modified CG algorithms, which are implemented using the exponentially decaying data windowing scheme and run one iteration per coefficient and data update, have the convergence dependent of the input eigenvalue spread. When the input with large eigenvalue spread is applied, the convergence is considerably slow.

To improve the convergence performance as well as misadjustment of the modified CG algorithms, this paper presents and analyzes a new approach of implementing the CG algorithm using a generalized data windowing scheme which combines the features of the finite sliding data windowing scheme and the exponentially decaying data windowing scheme. In Section 2, we first introduce the generalized data windowing scheme, and analyze the computational complexity and its advantages for CG algorithms. The numerical problems and the convergence behavior of the modified CG algorithms are then studied. Section 3 presents computer simulations to compare the convergence speed, misadjustment and tracking capabilities of the new modified CG algorithms with those of the existing approaches as well as the RLS algorithm. Finally, Section 4 concludes this paper.

Section snippets

Generalized data windowing scheme and accelerated CG algorithms

Consider a minimization problem for the following quadratic performance function:J(w)=12wTRw-bTw,where R is N×N square matrix and positive definite, b and w are vectors of dimension N. Solving for the vector wo that minimizes the quadratic performance function (1) is equivalent to solving the linear equation Rwo=b. It has been shown that the CG algorithm can be used to solve this linear equation iteratively and efficiently in [7], [8]. The algorithm uses the following weight update equation:w(k)

Application of adaptive system modeling

The performance of the proposed algorithms is evaluated by carrying out the computer simulation in the framework of adaptive system modeling problem. We use a Hamming window to generate the unknown system as a finite impulse response (FIR) lowpass plant with cutoff frequency of 0.5. The adaptive filter and the unknown system are assumed to have the same number of taps. In the simulations, both the low order filter (N=20) and the high order filter (N=200) with stationary and nonstationary input

Conclusion

In this paper, we have presented and analyzed a new approach to the implementation of the CG algorithm for adaptive filtering based on a generalized data windowing scheme which combines the features of the finite sliding window scheme and the exponentially decaying data window scheme. The modified adaptive CG algorithms show improved filter performance of accelerated convergence rate and low misadjustment. Besides the application of adaptive system modeling, we also tested the proposed scheme

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