New normalized LMS adaptive filter with a variable regularization factor

doi:10.21629/JSEE.2019.02.05

Journal of Systems Engineering and Electronics ›› 2019, Vol. 30 ›› Issue (2): 259-269.doi: 10.21629/JSEE.2019.02.05

• Electronics Technology • Previous Articles Next Articles

New normalized LMS adaptive filter with a variable regularization factor

Zhoufan LI(), Dan LI*(), Xinlong XU(), Jianqiu ZHANG()

School of Information Science and Technology, Fudan University, Shanghai 200433, China

Received:2018-06-07 Online:2019-04-01 Published:2019-04-28
Contact: Dan LI E-mail:15210720031@fudan.edu.cn;lidan@fudan.edu.cn;11210720032@fudan.edu.cn;jqzhang@ieee.org
About author:LI Zhoufan was born in 1992. He received his B.Sc. degree in electrical engineering from Fudan University, Shanghai, China, in 2015. He is currently working towards his M.S. degree in Department of Electronic Engineering, Fudan University, Shanghai, China. His research interests include digital signal processing and its application in wireless communication. E-mail:15210720031@fudan.edu.cn|LI Dan was born in 1982. He received his B.Sc., M.S., and Ph.D. degrees in electrical engineering from Fudan University, Shanghai, China, in 2003, 2006, and 2013, respectively. Since 2006, he has been an lecturer at Department of Electronic Engineering, Fudan University, Shanghai, China. His research interests include digital signal processing and its application to nondestructive testing. E-mail:lidan@fudan.edu.cn|XU Xinlong was born in 1989. He received his B.Sc. and M.S. degrees in electrical engineering from Fudan University, Shanghai, China, in 2011 and 2014, respectively. His research interests include signal processing and adaptive filtering. E-mail:11210720032@fudan.edu.cn|ZHANG Jianqiu was born in 1962. He received his B.Sc. degree from East of China Institute of Engineering, Nanjing, and his M.S. and Ph.D. degrees from Harbin Institute of Technology (HIT), Harbin, China, in 1992 and 1996, respectively. He is currently a professor with the Department of Electronic Engineering, Fudan University, Shanghai, China. From 1999 to 2002, he was a senior research fellow at the School of Engineering, University of Greenwich, Chatham Maritime, U.K. In 1998, he was a visiting research scientist at the Institute of Intelligent Power Electronics, Helsinki University of Technology, Espoo, Finland. He was an associate professor from 1995 to 1997 and a lecturer from 1989 to 1994 in the Department of Electrical Engineering, HIT. During 1982 to 1987, he was an assistant electronic engineer at the 544th Factory, Hunan, China. His main research interests are signal processing and its application. Email:jqzhang@ieee.org
Supported by:
the National Natural Science Foundation of China(61571131);the National Natural Science Foundation of China(11604055);This work was supported by the National Natural Science Foundation of China (61571131; 11604055)

Abstract

Abstract:

A new normalized least mean square (NLMS) adaptive filter is first derived from a cost function, which incorporates the conventional one of the NLMS with a minimum-disturbance (MD) constraint. A variable regularization factor (RF) is then employed to control the contribution made by the MD constraint in the cost function. Analysis results show that the RF can be taken as a combination of the step size and regularization parameter in the conventional NLMS. This implies that these parameters can be jointly controlled by simply tuning the RF as the proposed algorithm does. It also demonstrates that the RF can accelerate the convergence rate of the proposed algorithm and its optimal value can be obtained by minimizing the squared noise-free posteriori error. A method for automatically determining the value of the RF is also presented, which is free of any prior knowledge of the noise. While simulation results verify the analytical ones, it is also illustrated that the performance of the proposed algorithm is superior to the state-of-art ones in both the steady-state misalignment and the convergence rate. A novel algorithm is proposed to solve some problems. Simulation results show the effectiveness of the proposed algorithm.

Key words: adaptive filtering, normalized least mean square (NLMS), minimum-disturbance (MD) constraint, variable regularization, variable step-size NLMS

Zhoufan LI, Dan LI, Xinlong XU, Jianqiu ZHANG. New normalized LMS adaptive filter with a variable regularization factor[J]. Journal of Systems Engineering and Electronics, 2019, 30(2): 259-269.

Figures/Tables 3

Table 1

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References 33

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Algorithm	Variable step	Parameter
SM-NLMS [13]	$\mu (n)=\left\{\!\!\begin{array}{*{20}l} {1-\dfrac{\gamma }{\|e(n)\|}}, ~~~ {\|e(n)\|>\gamma } \\ 0, ~~~ {\rm else}\\ \end{array}\!\!\right.$	$\gamma $ = 0.04 ${\ldots}$
NPVSS-NLMS [15]	$\mu(n)=\left\{\!\!\begin{array}{*{20}l} {1-\dfrac{\sigma _v }{\widehat {\sigma }_e (n)+\varepsilon }}, ~~~ {\widehat {\sigma }_e (n)\ge \widehat {\sigma }_e (n)} \\ 0, ~~~ {\rm else} \\ \end{array}\!\!\right.$ $\widehat {\sigma }_e (n)=\lambda \widehat {\sigma }_e (n-1)+(1-\lambda)e^2(n)$	$\lambda = 1-1/(LK)$, $\delta = 20\sigma _{x}^{2}$, $\varepsilon = 10^{-3}$, $K = 2$ for a white input and $K = 6$ for an AR(1) process. $\sigma _{v}^{2}$ and $\sigma _{x}^{2}$ are known as the priori.
NVSS-NLMS [16]	$\mu (n)=\left\{\!\!\begin{array}{*{20}l} \left[\alpha \mu (n-1)+(1-\alpha)\dfrac{\widehat {\sigma }_e^2 (n)}{\beta \widehat {\sigma }_v^2 (n)} \right] \bigg\|_{\mu _{\min } }^{\mu _{\max } }, ~~~ {\zeta (n) < \zeta _{th} } \\ 0, ~~~ {\rm else} \\\end{array}\!\!\right.$ $\widehat {\sigma }_e^2 (n)=\alpha \widehat {\sigma }_e^2 (n-1)+(1-\alpha)e^2(n)$	$\alpha = 0.998$, $\beta = 30$, $\zeta _{\rm th} = 0.35$, $\mu_{\min} = 10^{-5}$, and $\mu_{\max} = 1$.
JO-NLMS [30]	$q(n)=\dfrac{p(n)}{L\sigma _v^2 +(L+2)p(n)\widehat {\sigma }_x^2 }$	$\sigma _{v}^{2}$ is known as a priori.
NVSS-LMS [26]	$\mu (n)=\beta \{1/[1+\exp (-\alpha \|e(n)e(n-1)\|)]-0.5\}$	$\alpha = 20$, $\beta = 0.01$.
$l_{2}$-VSS-LMS [29]	$\mu (n)=\beta[1-\exp (-\alpha \|e(n)\|\times \\| \mathit{\boldsymbol{P}}(n)\\|^2)] \\ \mathit{\boldsymbol{P}}(n)=a\mathit{\boldsymbol{P}}(n-1)+be(n)\mathit{\boldsymbol{x}}(n)$	$\alpha = 1$, $\beta=0.001$, $a = 0.98$, and $b = 0.3$.
Proposed algorithm		$K = 4$. $t = 0.5$ for a white input and $t = 0.35$ for an AR(1) process.

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New normalized LMS adaptive filter with a variable regularization factor

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