Short communicationConvergence behavior of the LMS algorithm in subband adaptive filtering
Introduction
In applications such as adaptive echo canceler and adaptive noise control, adaptive filtering with an LMS-type algorithm suffers from slow convergence and high computational complexity [4], since the adaptive filter requires hundreds or thousands of taps and the input signal is often highly correlated. Subband adaptive filtering has been proposed to avoid these drawbacks [1], [2], [3].
The convergence speed of the subband adaptive digital filters (ADF) with LMS-type algorithms was investigated, respectively, in [6], [9] by means of the condition numbers of correlation matrices of the subband tap-input vectors. It is said that, since the condition numbers of correlation matrices of the subband tap-input vectors are smaller than those for the corresponding fullband ADF, the subband ADF would converge faster and hence the subband ADF is attractive in convergence speed. However, it has also been shown by some computer simulations that the cancellation error of the subband ADF does not always decay faster than that of the fullband ADF.
To get more insights into the convergence speed of the subband ADF, we derive a theoretical but approximate expression that describes the convergence behavior of the cancellation error of the critically sampled subband ADF. It is shown that the critically sampled subband ADF do not always have advantages over the fullband ADF in convergence speed. Computer simulations are presented to see the validity of the expression.
Section snippets
A review of subband adaptive digital filters
First of all, let us review subband ADF with the LMS algorithm. Throughout this paper, signals and coefficients of filters are assumed to be real. We only consider the critically sampled subband ADF with M-channel paraunitary filter banks [8]. Let us denote the delay associated to the filter bank as D and express the z-transform of a sequence denoted by a small letter by the corresponding capital letter.
Fig. 1 shows a block diagram of the subband ADF where the decimation rate is two for
Convergence behavior of critically sampled subband ADF
Let us consider the fullband ADF. We assume that the unknown system has a finite impulse response of length L and denote the correlation function and the spectral density of the input signal as r(n) and as S(ejω), respectively.
The tap-weight vector is adjusted bywhere is the tap-input vector at time n, μ is a convergence parameter, ε(n) is the adaptation (cancellation) error given by
Let us assume that the convergence parameter is small enough
Numerical examples
Simulation results and numerical examples are presented to confirm our analysis. We considered the normalized LMS algorithm with . The frequency response of the unknown system F(z) is plotted in Fig. 2. The length of the impulse response of the unknown system was 512. We put the length of the fullband adaptive filter to be 512. As input signals, first-order AR processes with coefficients 0.9 and −0.5 driven by white processes were used. The output signals of the unknown system
Conclusions
An approximate expression that describes the convergence behavior of the cancellation error of the critically sampled subband ADF has been derived. From the expression, the convergence speed of the subband ADF has been investigated. It has been shown theoretically and empirically that the critically sampled subband ADF does not always have advantages over the fullband ADF in convergence speed. Computer simulations have also been presented to see the validity of the expression.
Acknowledgements
The authors wish to thank the anonymous reviewers for their valuable comments and suggestions.
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