Stochastic Modeling of the Transform-Domain $\varepsilon {\rm LMS}$ Algorithm

This paper presents a statistical analysis of the transform-domain least-mean-square (TDLMS) algorithm, resulting in a more accurate model than those discussed in the current open literature. The motivation to analyze such an algorithm comes from the fact that the TDLMS presents a higher convergence speed for correlated input signals, as compared with other adaptive algorithms possessing a similar computational complexity. Such a fact makes it a highly competitive alternative to some applications. Approximate analytical models for the first and second moments of the filter weight vector are obtained. The TDLMS algorithm has an orthonormal transformation stage, accomplishing a decomposition of the input signal into distinct frequency bands, in which the interband samples are practically uncorrelated. On the other hand, the intraband samples are correlated; the larger the number of bands, the higher their correlation. The model is then derived taking into account such a correlation, requiring that a high-order hyperelliptic integral be computed. In addition to the proposed model, an approximate procedure for computing high-order hyperelliptic integrals is presented. A regularization parameter is also considered in the model expressions, permitting to assess its impact on the adaptive algorithm behavior. An upper bound for the step-size control parameter is also obtained. Through simulation results, the accuracy of the proposed model is assessed.