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In order to solve the system identification problems, the normalized least mean absolute deviation (NLMAD) algorithm was developed as an effective and robust method. In this paper, aiming at the system identification problems with sparsity characteristic, and taking the advantage of the NLMAD algorithm to suppress impulsive output measurement noise interference, we introduce the L<sub>1</sub>-norm as a sparse penalty constraint into the NLMAD algorithm to design a robust sparse adaptive filtering algorithm. Furthermore, considering the biased estimation caused by the input noise, we employ an unbiasedness criterion to derive an effective bias-compensated vector which can compensate the bias efficiently for the proposed sparse NLMAD algorithm. The desirable performance of the new method is measured with simulations of two stages. The proposed bias-compensated sparse NLMAD algorithm achieves better performance compared to other existing methods in both stages. Simulation results demonstrate the excellent performance of the proposed algorithm in solving sparse system identification problems. The promising results in this paper suggest that the bias-compensated sparse NLMAD algorithm may become a useful tool for system identification with noisy input and impulsive output noise.

Sparse Normalized Least Mean Absolute Deviation Algorithm Based on Unbiasedness Criterion for System Identification With Noisy Input