Least Mean Squares Algorithms For Sparse System Identification

The least mean squares?LMS?algorithm has been widely used in system identification.In practical application,many unknown systems are sparse in nature.Dispersed sparse systems,single-clustering sparse systems and multi-clustering sparse systems are three typical sparse systems.In the identification of sparse systems,the traditional LMS algorithm suffers from a slow convergence rate,and is unable to quickly track changes of the external environment,since it does not make full use of the prior knowledge of sparseness.The objective of this paper is to accelerate the convergence rate of LMS type algorithms in the identification of sparse systems.The research content of this paper is divided into the following three aspects,according to different sparse characteristics of the unknown impulse responses:In the identification of general sparse systems,a normalized LMS?NLMS?algorithm with a tap selection matrix is proposed,where the tap selection matrix is utilized to adaptively locate the inactive coefficients in real time.The selected inactive coefficients are then set to zero directly during the iteration to improve the convergence speed and reduce the steady-state error.On the one hand,the proposed algorithm is suitable for the identification of both clustering-sparse systems and dispersed sparse systems.On the other hand,the proposed scheme is also applicable to the problem of unknown system identification with low sparseness or even non-sparseness,in which case the proposed algorithm will be reduced to the traditional NLMS algorithm through the adaptive adjustment of the tap selection matrix.In addition,the steady-state performance of the proposed algorithm is analyzed,and the closed-form expression of the steady-state mean square deviation?MSD?is derived.Finally,simulation results validate the efficiency of the proposed algorithm and the correctness of theoretical analysis.In the identification of clustering sparse systems,a mixed7)_2,-like-norm penalized LMS algorithm is proposed.Simulation results show that the proposed algorithm has a faster convergence speed than existing algorithms in the application of clustering sparse system identification.Then,the generalized7)_,-like-norm penalized LMS?7?_,-like-LMS)algorithm is proposed.The influence of the values ofandon the7)_,-like-LMS algorithm is analyzed.It is proved that existing norm penalized LMS algorithms are special cases when the parametersandtake certain values in the proposed7)_,-like-LMS algorithm.Finally,for single-clustering sparse system identification,a single-clustering sparse LMS algorithm is proposed,which uses the difference between the7)_2,1-norm and7)₂-norm of the tap-weight vector as a single-clustering sparse constraint.An adaptive non-uniform grouping method is developed to further improve performance of the proposed scheme.Simulation results show that the proposed algorithm performs well in single-clustering sparse system identification.In the identification of dispersed sparse systems,a dispersed sparse LMS algorithm is proposed by using the difference between the7)₁-norm and7)_?,1-norm of the tap-weight vector as a constraint.Based on that dispersed sparse constraint,the proposed algorithm imposes an attraction to zero on zero coefficients of the adaptive filter during the iteration to accelerate their convergence.The stability and steady-state performance of the proposed scheme are analyzed.The mean and mean-square convergence conditions of the proposed algorithm and the closed-form expression of the steady-state MSD are derived.Finally,simulation results validate the efficiency of the proposed algorithm and the correctness of theoretical analysis.