Transform domain LMS algorithms for sparse system identification

This paper proposes a new adaptive algorithm to improve the least mean square (LMS) performance for the sparse system identification in the presence of the colored inputs. The l
₁
norm penalty on the filter coefficients is incorporated into the quadratic LMS cost function to improve the LMS performance in sparse systems. Different from the existing algorithms, the adaptive filter coefficients are updated in the transform domain (TD) to reduce the eigenvalue spread of the input signal correlation matrix. Correspondingly, the l
₁
norm constraint is applied to the TD filter coefficients. In this way, the TD zero-attracting LMS (TD-ZA-LMS) and TD reweighted-zero-attracting LMS (TD-RZA-LMS) algorithms result. Compared to ZA-LMS and RZA-LMS algorithms, the proposed TD-ZA-LMS and TD-RZA-LMS algorithms have been proven to have the same steady-state behavior, but achieve faster convergence rate with non-white system inputs. Effectiveness of the proposed algorithms is demonstrated through computer simulations.