A Unified Analysis of Proposed Wavelet Transform Domain LMS-algorithm for ARMA Process

A unified analysis of Wavelet Transform (WT) domain Least Mean Square (LMS) adaptive filter is presented in this work for highly correlated Autoregressive-Moving-Average (ARMA) input process. It is well known that the Unitary transform (UT) domain LMS (UT-LMS) adaptive filter for Autoregressive (AR) process with power normalization improves the filter performance, where DCT provides best performance among them. In this work, we apply the UT-LMS algorithm for time-varying ARMA process, and the analytical result shows that the lower decorrelation property of UT degrades the LMS performance. As a result, Unitary transform is not applicable for LMS as a transform algorithm for ARMA process, and this outcome has not been explored in early published work. In this paper, we propose Discrete Wavelet domain LMS (DWT-LMS) for ARMA process to enhance the basic performances of LMS such as misadjustment, convergence, and tracking properties, and the theoretical and simulation result of this work show that DWT-LMS provides better performance than that of DCT-LMS for 1st and $2^{\mathrm{n}\mathrm{d}}$ order AR, Moving-average (MA), and ARMA process. This paper concludes with the MATLAB simulation for the proposed method with various inputs for demonstrating the validity of the derived mathematical algorithm.