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Simple and efficient computational algorithms for nonparametric wavelet-based identification of nonlinearities in Hammerstein systems driven by random signals are proposed. They exploit binary grid interpolations of compactly supported wavelet functions. The main contribution consists in showing how to use the wavelet values from the binary grid together with the fast wavelet algorithms to obtain the practical counterparts of the wavelet-based estimates for irregularly and randomly spaced data, without any loss of the asymptotic accuracy. The convergence and the rates of convergence are examined for the new algorithms and, in particular, conditions for the optimal convergence speed are presented. Efficiency of the algorithms for a finite number of data is also illustrated by means of the computer simulations.