Variable Step-Size $\ell _{0}$-Norm Constraint NLMS Algorithms Based on Novel Mean Square Deviation Analyses

Abstract

This paper proposes variable step-size <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{0}$</tex-math></inline-formula> -norm constraint normalized least mean square ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{0}$</tex-math></inline-formula> -NLMS) algorithms for sparse channel identification. The mean square deviation of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{0}$</tex-math></inline-formula> -NLMS algorithm is newly analyzed to obtain the optimal step size by reflecting the correlated property of the inputs. The mean square deviation estimations are derived in both the absence and presence of information on measurement noises. The variable step-size schemes are derived by minimizing the forthcoming mean square deviations. A reset algorithm is introduced to consider the sudden change in the unknown system in practical situations. Simulations demonstrate that the proposed variable step-size algorithms have better performances than the existing algorithms in sparse system identification and acoustic echo cancellation scenarios.