A comparative study of different wavelets used in transformed domain LMS filter

In order to improve the convergence performance of the LMS filter, transform domain adaptive filters have been widely used. The transformation chosen include Discrete Fourier Transform(DFT), Discrete Sine Transform(DST), Discrete Cosine Transform(DCT) etc. Some comparisons among these transforms, based on their effectiveness in improving convergence of the resulting LMS filter, have also been reported. While discrete wavelet transform too has been used in some of the cases, comparison of it with other transforms is rarely found. Moreover, as there exist a number of wavelets leaving us with the choice to select one of them, there is a necessity to investigate convergence performance of different wavelets. In this paper, three different wavelets (Haar, Daubechies-4 and Symlet-4) have been taken and their performance in improving the convergence of the LMS filter has been studied. It is found that Symlet-4 yields the best convergence for the cases taken. This observation is in agreement with the conclusion drawn from the point of eigenvalue spread of the correlation matrix of the transformed input. In addition, a general expression that relates minimal mean square in the transform domain to the LMS minimal mean square error (without transformation), has been established in this paper. An explicit expression relating the eigenvalues of the correlation matrix of the transformed input with the eigenvalues of the correlation matrix of the original input has also been arrived at.