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This paper proposes an adaptive step-size approach based on the stochastic gradient-based algorithm for Hammerstein spline adaptive filtering (HSAF).With the good convergence and low complexity computation, a normalized least mean square (NLMS) algorithm is developed with HSAF. A nonlinear Hammerstein adaptive filter consists of memoryless function modified during learning and the spline control point is automatically controlled by gradient-based method. An adaptive step-size approach based on NLMS algorithm with the HSAF is introduced how to derive using the direction of steepest descent method for tracking performance. Hence, the stability and steady-state performance are detailed. The simulation results of trajectories of both update step-size parameters are shown that the initial step-size are reduced, all convergence rates are correspondingly reduced to each own optimum over 100 independent trials of the experiment. Experimental plots of learning curves of mean-squared error of the proposed algorithm for varying the initial step-size parameters are obtained that the proposed algorithm conducts more robust performance and fast convergence compared with the HSAF based on least mean square algorithm, even the large variations of 100-fold initial step-size parameters are used.

Stability and Steady-State Performance of Hammerstein Spline Adaptive Filter Based on Stochastic Gradient Algorithm