The author investigates the convergence characteristics of the last mean square (LMS) and the recursive least squares (RLS) adaptive algorithms when the correlation matrix of the input signal does not have a full rank. It is shown that the initial convergence rate of the LMS algorithm is inversely proportional to the rank of correlation matrix, or equivalently, the number of nonzero eigenvalues. The same conclusion holds for the RLS algorithms if the minimum norm solution (MNS) is used in each iteration. A simple time-recursive method to obtain approximate MNSs in each iteration is presented and proven. The effect of additive noise is discussed.
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