Classical adaptive algorithms (LMS, RLS, CMA, decision directed) seen as recursive structures

Any finite impulse response (FIR) adaptive algorithm has an inherent recursive structure, since the error at time n is fed back into the algorithm to provide the next tap vector. The authors explicitly provide this recursive structure and show that a least mean square (LMS) adaptive algorithm can be expressed in terms of a recursive filter, the coefficients of this recurrence being the correlation coefficients of the input signal. Corresponding structures are proposed for the recursive least squares (RLS) algorithm, the constant modulus algorithm (CMA), and the decision-directed (DD) algorithm. These schemes provide block algorithms that are equivalent to the corresponding sample-by-sample algorithm, while requiring fewer computations. The corresponding structures are easily used for studying equilibrium states of the adaptive algorithms.<>