Analyses and applications of generalized transformed domain least-mean square (LMS) adaptive filters

Abstract

<p>In this thesis, a comprehensive analysis of the transformed domain adaptive filtering algorithms is presented. The idea behind this study is to expand the time domain filter impulse response as a set of weighted orthogonal functions. The least-mean square (LMS) adaptation algorithm is applied to the weights. In brief, we systematize and generalize the existing transformed domain algorithms and study their convergence properties. Based on the concept described above, we develop two types of transformed domain LMS (TDLMS) adaptive algorithm and the block transformed domain LMS (BTDLMS) adaptive algorithm. For the purpose of comparison, time domain expressions of various transformed domain adaptive algorithms are also obtained. We show that the TDLMS algorithm with the Karhunen-Loève (K-L) transform is equivalent to the time domain LMS adaptive algorithm. Whereas, the time domain expression of the TDLMS algorithm with the normalized K-L transform is equivalent to the generalized Newton-Raphson method. To study the effect of the normalized K-L TD:MS adaptive algorithm, we have derived an analytic expression of mean-square error (MSE) in terms of weight vector, weight covarience matrix, input autocorrelation matrix and the minimum MSE in the application of adaptive line-enhancer (ALE). An analytic expression of MSE using the time domain LMS algorithm in the ALE will also be obtained for comparison. It will be proved that the convergence rate, in terms of MSE, of the time domain LMS algorithm not only depends on the eigenvalue spread of the input autocorrelation matrix, but also depends highly on the power ratio between the sinusoidal signals in the ALE. However, the convergence rate of the normalized K-L TDLMS algorithm is independent of the eigenvalue spread and power ratio but depends only on the stepsize. The problem of employing different orthogonal transforms is also studied. The discrete cosine transform (DCT) is used to investigate the effect on convergence rate due to incomplete diagonalization of the autocorrelation matrix. To reduce the computational-effort, a recursive equation for evaluating the DCT output is obtained. Furthermore, in order to apply the transmultiplexer transform (TMT) to the transformed domain adaptive algorithms, the simplified form of TMT is also obtained. Finally, for limited supply of input data, an algorithm of re-circulating the input data in the BTDLMS adaptive algorithm with the discrete Fourier transform (DFT) is proposed. An analysis of the performance, in terms of MSE, of such an algorithm indicates that superior performance can be achieved compared to the algorithm without re-circulation.</p>