The LMS algorithm under arbitrary linearly filtered processes

In this paper the mean square convergence of the LMS algorithm is shown for a large class of linearly filtered random driving processes. In particular this paper contains the following contributions: i) The parameter error vector covariance matrix can be decomposed into two parts, a first part that exists in the modal space of the driving process of the LMS filter and a second part, existing in its orthogonal complement space, not contributing to the performance measures (misadjustment, mismatch) of the algorithm. ii) The LMS updates force the initial values of the parameter error vector covariance matrix to remain essentially in the modal space of the driving process and components of the orthogonal complement die out. iii) The impact of additive noise is shown to contribute only to the modal space of the driving process independent of the noise statistic and thus defines the steady-state of the filter. In particular it will be shown that the joint fourth order moment m_x^(2,2) of the decorrelated driving process is a more relevant parameter for the step-size bound and not as often believed the second order moment m_x⁽²⁾.