Transient Analysis of the Set-Membership LMS Algorithm

Abstract

Adaptive filtering algorithms, which adopt the set-membership strategy, are able to attain good steady-state performance with low computational burden. In general, such advantages are obtained by defining a bounded-error. This specification translates into a time-variant step size, chosen in each iteration according to a nonlinear function of the instantaneous error. Unfortunately, this type of nonlinear behaviour hampers the stochastic modelling of these algorithms. This work devises a novel transient analysis of the Set-Membership Least Mean Squares algorithm. Additionally, a new interpretation is advanced about the implicit optimization problem solved by the algorithm. This explanation is important since it can contribute to the design of new adaptive algorithms. The proposed theoretical analysis provides predictions for: (i) steady-state performance; (ii) transient performance; and (iii) evolution of the update probability. It is noteworthy that the latter influences the computational complexity of the algorithm. Furthermore, we perform a novel comprehensive transient analysis of a set-membership algorithm. In addition, both time-variant transfer functions and deficient-length configurations are addressed. The resulting theoretical estimates are confirmed by simulations.