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This paper proposes the Set-Membership sign-NLMS (SM-sign-NLMS) adaptive filter, which combines the ability for data censoring (offered by Set-Membership schemes) with robustness against impulsive noise (provided by signed schemes). The algorithm can present a much lower steady-state probability of update than the standard SM-NLMS algorithm when impulsive noise is present in the system. It is derived from a local deterministic optimization problem modulated by a minimum disturbance cost function combined with a bounded error criterion. Several stochastic models are proposed in order to extract insights and a time-variant step size extension of the algorithm. The first of them, based on energy conservation arguments, leads to a fixed-point analytic equation whose solution predicts the asymptotic performance of the algorithm. Further, a transient analysis based on a statistical decoupling of the radial and (discrete) angular distributions of the input vector is derived. Based on such an analysis, an efficient time-variant step-size version of the algorithm is proposed. Additionally, such an analysis is also utilized to obtain a fixed-point formula whose solution describes the asymptotic performance when the unknown plant that the filter intends to match varies according to a first-order Markovian model. Lastly, a novel stochastic model is advanced for the description of the algorithm learning behavior under a deficient-length scenario for a white input signal, which provides some insights about the asymptotic performance of the algorithm. The findings are confirmed by extensive simulations.

Stochastic Modelling of the Set-Membership-sign-NLMS Algorithm