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SUMMARYIn this paper, a novel adaptive filter for sparse systems is proposed. The proposed algorithm incorporates a log‐sum penalty into the cost function of the standard leaky least mean square (LMS) algorithm, which results in a shrinkage in the update equation. This shrinkage, in turn, enhances the performance of the adaptive filter, especially, when the majority of unknown system coefficients are zero. Convergence analysis of the proposed algorithm is presented, and a stability criterion for the algorithm is derived. This algorithm is given a name of zero‐attracting leaky‐LMS (ZA‐LLMS) algorithm. The performance of the proposed ZA‐LLMS algorithm is compared to those of the standard leaky‐LMS and ZA‐LMS algorithms in sparse system identification settings, and it shows superior performance compared to the aforementioned algorithms. Copyright © 2013 John Wiley & Sons, Ltd.

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Abstract-The leaky least-mean-square (LLMS) algorithm was first proposed to mitigate the drifting problem of the leastmean-square (LMS) algorithm. Though the LLMS algorithm solves this problem, its performance is similar to that of the LMS algorithm. In this paper, we propose an improved version of the LLMS algorithm that brings better performance to the LLMS algorithm and similarly solves the problem of drifting in the LMS algorithm. This better performance is achieved at a negligible increase in the computational complexity. The performance of the proposed algorithm is compared to that of the conventional LLMS algorithm in a system identification and a noise cancellation settings in additive white and correlated, Gaussian and impulsive, noise environments. IndexTerms-Leaky least-mean-square, system identification, noise cancellation. I. INTRODUCTIONThe least-mean-square (LMS) algorithm [1] is one of the most famous adaptive filtering algorithms because of its simplicity and ease of analysis. This has made most researchers to improve the LMS algorithm and also to find solutions to some of its drawbacks. Some of these improved algorithms include: the normalized least-mean-square (NLMS) [2], variable step-size least-mean-square (VSSLMS) [3], etc. These improved algorithms generally improve the performance of the LMS algorithm in terms of convergence rate and mean-square-error (mse) value.One of the main drawbacks of the LMS algorithm is the drifting problem as analyzed in [4]. This is a situation where the LMS algorithm generates unbounded parameter estimates for a bounded input sequence. This may drive the LMS weight update to diverge as a result of inadequate input sequence [4]. The drifting problem has been shown in [5]-[7] in details.The leaky least-mean-square (LLMS) algorithm is one of the improved LMS-based algorithms that use a leakage factor to control the weight update of the LMS algorithm [5], [6]. This leakage factor solves the problem of drifting in the LMS algorithm by bounding the parameter estimate. It also improves the tracking capability of the algorithm, convergence and stability of the LMS algorithm.One of the main drawbacks of the LLMS algorithm is its low convergence rate compared to the other improved LMSbased algorithms. In this paper, we propose a new algorithm that improves the convergence rate of the LLMS algorithm. This is achieved by employing the sum of exponentials of the error as the cost function; this cost function is a generalized of the stochastic gradient algorithm as proposed by Boukis et al. [8]. A leakage factor is added to the sum of exponential cost function which makes the proposed algorithm a combination of the generalized of the mixednorm stochastic gradient algorithm with a leaky factor. This paper is organized as follows. In Section II, a review of the LLMS is introduced. In Section III, the proposed algorithm is introduced. In Section IV, experimental results are presented and discussed. Finally, the conclusions are drawn. II. LEAKY LEAST MEAN SQUARE ALGORITHMIn sy...