Transform-domain adaptive algorithms have been proposed to reduce the eigenvalue spread of the matrix governing their convergence, thus improving the convergence rate. However, a classical problem arises from the conflicting requirements between algorithm improvement requiring rather long transforms and the need to keep the input/output delay as small as possible, thus imposing short transforms. This dilemma has been alleviated by the so-called "short-block transform domain algorithms" but is still apparent. This paper proposes an adaptive algorithm compatible with the use of rectangular orthogonal transforms (e.g., critically subsampled, lossless, perfect reconstruction filter banks), thus allowing better tradeoffs between algorithm improvement, arithmetic complexity, and input/output delay. The method proposed makes a direct connection between the minimization of a specific weighted least squares criterion and the convergence rate of the corresponding stochastic gradient algorithm. This method leads to improvements in the convergence rate compared with both LMS and classical frequency domain algorithms.

/pdf/adaptive-filtering-in-subbands-using-a-weighted-criterion-3613kkynar.pdf

Adaptive filtering in subbands using a weighted criterion

It is shown how linear time-varying systems can be modeled in several different ways by discrete-time wavelets or, more generally, by some set of functions. Interpretation of physical meanings, possible efficiency, and other characteristics of the modeling are considered. System identification minimizing the mean square output error is studied. Optimal coefficients and the corresponding minimum mean square error are found, and they are, in general, time varying. Least-mean-square adaptive filtering algorithms are derived for on-line filtering and system Identification. Theoretically and by simulations, the advantages of using wavelet-based filtering are shown: separation of adaptation effects from unknown time-varying system behavior and fast convergence. Adaptive coefficients estimated by a recursive-least-square algorithm can tend toward constants, even in the case of time-varying systems. Time-invariant system identification and adaptive filtering is given as a special case of the general time-varying setting.

Wavelet-based linear system modeling and adaptive filtering

A multi-resolution filtered-x LMS algorithm based on discrete wavelet transform for active noise control

The wavelet transform-domain least-mean square (WTDLMS) algorithm is known to have, in general, a faster convergence rate than the time-domain LMS algorithm, and can find many applications in signal processing and communications areas. However, the computational complexity of the wavelet filter bank is relatively high. In this work, we show how to exploit the redundancy which exists in the computation of the wavelet coefficients between successive iterations so as to significantly reduce the computational load of the algorithm.

The wavelet transform-domain LMS algorithm: a more practical approach

Wavelet transform domain LMS algorithm

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Wavelet transform domain LMS algorithm