Nonlinear Adaptive Filtering With Kernel Set-Membership Approach

This paper develops nonlinear kernel adaptive filtering algorithms based on the set-membership filtering (SMF) framework. The set-membership-based filtering approach is distinct from the conventional adaptive filtering approaches in that it aims for the filtering error being bounded in magnitude, as opposed to seeking to minimize the time average or ensemble average of the squared errors. The proposed kernel SMF algorithms feature selective updates of their parameter estimates by making discerning use of the input data, and selective increase of the dimension in the kernel expansion. These result in less computational cost and faster tracking without compromising the mean-squared error performance. We show, through convergence analysis, that the sequences of parameter estimates of our proposed algorithms are convergent, and the filtering error is asymptotically upper bounded in magnitude. Simulations are performed which show clearly the advantages of the proposed algorithms in terms of lower computational complexity, reduced dictionary size, and steady-state mean-squared errors comparable to existing algorithms.