On the stochastic modeling of the LMS algorithm operating with bilinear forms

This paper presents a stochastic model of the least-mean-square for bilinear forms (LMS-BF) algorithm in which the bilinear term is defined with respect to the temporal and spatial impulse responses of a multiple-input/single-output (MISO) spatiotemporal system. Specifically, taking into account uncorrelated and correlated Gaussian input data, an analytical stochastic model is derived describing the behavior of the temporal and spatial (and, consequently, spatiotemporal) adaptive filters for both transient and steady-state phases. Based on the proposed model, some interesting insights about convergence and steady-state characteristics of the algorithm are discussed, thereby providing useful design guidelines. Moreover, an analytical relationship is established between the LMS and the LMS-BF algorithms, which makes it possible to obtain appropriate performance comparisons confirming the improved convergence characteristics achieved by the latter. Through simulation results, the accuracy of the proposed model as well as some features of the algorithm are verified under different operating scenarios.