There are many different ways that one may compare the performance of adaptive filtering algorithms. Suppose that we are interested in adaptive linear prediction and our measure of performance is the number of arithmetic operations required for the adaptive filter to converge. Let the time constant r be used as the convergence time of the LMS algorithm. For the RLS algorithm, it is often stated that the rate of convergence is an order of magnitude faster than the LMS algorithm. Therefore, assume that the time constant for the RLS algorithm is one tenth that of the LMS algorithm.
(a) If the eigenvalues of the p x p autocorrelation matrix for x(n) are and if we use a step size µ, = 0.1 for the LMS algorithm, for what order filter, p, are the RLS and LMS adaptive filters equal in terms of their computational requirements to reach convergence?
(b) For high order filters, the computational requirements of the RLS filter become large, and the LMS algorithm becomes an attractive alternative. For what reasons might you prefer to use the RLS algorithm in spite of its increased computational cost?