Hammerstein uniform cubic spline adaptive filters: Learning and convergence properties

Robust Hammerstein Adaptive Filtering under Maximum Correntropy Criterion

“…The derived bound on step-sizes is only of theoretical importance as in general, Equation (27) cannot be verified in a practical situation. Similar theoretical results can be found in [7].…”

“…In addition, the coefficient vectors are initialized with the first coefficient equal to 1 and the others equal to zero [7].…”

“…In this work, for simplicity we consider only the stochastic gradient based algorithm (i.e., L = 1). In this case, we have The proposed algorithm is in form similar to the traditional Hammerstein adaptive filters under MSE criterion [7], but the step-sizes are different.…”

“…Using the Taylor series expansion of the error e(n + 1) around the instant n and keeping only the linear term, we have [4,7,40] …”

“…The Wiener model and the Hammerstein model are two typical block-oriented nonlinear models [3]. Specifically, the Wiener model consists of a cascade of a linear time invariant (LTI) filter followed by a static nonlinear function, indicated as a linear-nonlinear (LN) model [4][5][6], and the Hammerstein model consists of a cascade of a static nonlinear function follow by a LTI filter, known as a nonlinear-linear (NL) model [7][8][9][10][11][12][13][14][15][16][17][18][19]. Other nonlinear models include neural networks (NNs) [20], Volterra adaptive filters (VAFs) [21], kernel adaptive filters (KAF) [22][23][24][25], among others.…”