 Open Access
 Total Downloads : 4
 Authors : Gopalaiah, Dr.K. Suresh
 Paper ID : IJERTCONV2IS13010
 Volume & Issue : NCRTS – 2014 (Volume 2 – Issue 13)
 Published (First Online): 30072018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Acoustic Echo Cancellation using Variable Step Size NLMS Algorithms
GOPALAIAH
Research Scholar, Dayananda Sagar College of Engineering
Bangalore, India gopaliah@gmail.com
Dr.K. SURESH
Professor
Sri Darmasthala Manjunatheswara Institute of Technology Ujire, India
ksece1@gmail.com
AbstractThe purpose of a variable stepsize normalized
LMS filter is to solve the dilemma of fast convergence rate and low excess MSE. In the past two decades, many VSSNLMS algorithms have been presented and have claimed that they have good convergence and tracking properties. This paper summarizes several promising algorithms and gives a performance comparison via extensive simulation. Simulation results demonstrate that Benestys NPVSS and our GSER have the best performance in both timeinvariant and timevarying systems.
Index TermsAdaptive filters, normalized least mean square (NLMS), variable stepsize NLMS, regularization parameter.

INTRODUCTION
Adaptive filtering algorithms have been widely employed in many signal processing applications. Among them, the normalized least mean square (NLMS) adaptive filter is most popular due to its simplicity. The stability of the basic NLMS is controlled by a fixed stepsize constant , which also governs the rate of convergence, speed of tracking ability and the amount of steadystate excess meansquare error (MSE).
In practice, the NLMS is usually implemented by adding the squared norm of input vector with a small positive number commonly called the regularization parameter. For the basic NLMS algorithm, the role of is to prevent the associated denominator from getting too close to zero, so as to keep the filter from divergence. Since the performance of NLMS is affected by the overall stepsize parameter, the regularization parameter has an effect on convergence properties and the excess MSE as well, i.e., a too big may slow down the adaptation of the filter in certain applications.
There are conflicting objectives between fast convergence and low excess MSE for NLMS with fixed regularization parameter. In the past two decades, many variable stepsize NLMS (VSSNMS) algorithms have been proposed to solve this dilemma of the conventional NLMS. For example, Kwong used the power of instantaneous error to introduce a variable stepsize LMS (VSSLMS) filter [6]. This VSSLMS has a larger step size when the error is large, and has a smaller step size when the error is small. Later Aboulnasr pointed out that VSSLMS algorithm is fairly sensitive to the accompanying noise, and presented a modified VSSLMS
(MVSS) algorithm to alleviate the influence of uncorrelated disturbance. The stepsize update of MVSS is adjusted by utilizing an estimate of the autocorrelation of errors at adjacent time samples. Recently Shin, Sayed, and Song used the norm of filter coefficient error vector as a criterion for optimal variable stepsize, and proposed a variable stepsize affine projection algorithm (VSAPA), and a variable stepsize NLMS (VSNLMS) as well. Lately Benesty proposed a non parametric VSS NLMS algorithm (NPVSS), which need not tune many parameters as that of many variable step size algorithms.
Another type of VSS algorithms has timevarying regularization parameter that is fixed in the conventional – NLMS filters. By making the regularization parameter gradientadaptively, Mandic presented a generalized normalized gradient descent (GNGD) algorithm. Mandic claimed that the GNGD adapts its learning rate according to the dynamics of the input signals, and its performance is bounded from below by the performance of the NLMS. Very recently, Mandic introduced another scheme with hybrid filters structure to further improve the steadystate misadjustment of the GNGD. Choi, Shin, and Song then proposed a robust regularized NLMS (RRNLMS) filter, which uses a normalized gradient to update the regularization parameter. While most variable stepsize algorithms need to tune several parameters for better performance, we presented an almost tuningfree generalized squareerrorregularized NLMS algorithm (GSER) recently. Our GSER exhibits very good performance with fast convergence, quick tracking
and low steadystate MSE.
The purpose of this paper is to provide a fair comparison among these VSS algorithms. In Section II, we summarize the algorithms. Section III illustrates the simulation results. Conclusions are given in Section IV.

VARIABLE STEPSIZE ALGORITHMS
In this section, we summarize several variable stepsize adaptive filtering algorithms including VSSLMS , MVSS , VS APA , VSNLMS, NPVSS, GNGD, RRNLMS, and GSER
algorithm .
Let d(n) be the desired response signal of the adaptive filter
d(n) = xT (n)h(n) + v(n) , (1)
where h(n) denotes the coefficient vector of the unknown system with length M ,
h(n )= [h0(n ), p(n ),.. .,hM 1(n)]T, (2)
x(n) is the input vector
x(n) = [x(n), x(n 1),, x(n M +1)]T , (3)
and v(n) is the system noise that is independent of x(n) .
Let the adaptive filter have same structure and same order as that of the unknown system. Denoting the coefficient vector of the filter at iteration n as w(n) . We express the a priori estimation error as
e(n) = d(n) xT (n)w(n) . (4)

VSSLMS algorithm
Kwong used the squared instantaneous a priori estimation error to update the step size as
(n +1) = (n) + e2 (n) , (5)
where 0 < < 1 , > 0 , and (n +1) is restricted in some pre decided[min ,max ] . The filter coefficient vector update recursion is given by
w(n +1) = w(n) + (n)e(n)x(n) . (6)
where 2 is a positive number proportional to K, max < 2 ,and
p(n) is an M Ã—1 vector recursively given by
p(n) = p(n 1)+(1)X(n)(XT(n)X(n)+1I)1 (13)
A variable step size NLMS (VSNLMS) was obtained as a special case of VSAPA by choosing K = 1 as follows.
D.NPVSS algorithm
Benesty argued that many variable stepsize algorithms may not work reliably because they need to set several parameters which are not easy to tune in practice, and proposed a nonparametric variable stepsize NLMS algorithm (NPVSS).
The filter coefficient vector update recursion is given as that of (6), and the variable step size is updated as
v
Where 3, 4 are positive numbers, 2 is the power of the

MVSS algorithm
Aboulnasr utilized an estimate of the autocorrelation of e(n) at adjacent time samples to update the variable step size
system noise, and the power of the error signal is estimated as
E.GNGD algorithm.
as
where
(n +1) = (n) + p2
(n) , (7)
The GNGD belongs to the family of timevarying regularized VSS algorithm. The filter coefficient vector is updated as
p(n) = p(n 1) + (1 )e(n)e(n 1) , (8) and 0 < < 1 .

VSAPA and VSNLMS
Shin, Sayed, and Song proposed a variable stepsize affine projection algorithm (VSAPA), which employed an error vector, instead of a scalar error as used in VSSLMS [6] and MVSS, to adjust the variable step size. The coefficient vector update recursion is given by
w(n +1) = w(n)+Âµ(n)X(n)(XT(n)X(n)+1I)1 (9)
where 1 is a small positive number, I is an unit matrix of size K Ã— K , X(n) is an M Ã— K input matrix defined as X(n) = [x(n),x(n 1),, x(n K +1)] , (10)
and
e(n) = [e(n), e(n 1),, e(n K +1)]T . (11)
The variable step size (n) is obtained by
,
where c is a fixed step size, and the regularization parameter
(n) is recursively calculated as
where is an adaptation parameter needs tuning, and the initial value (0) has to be set as well.
F.RRNLMS Algorithm
Chois RRNLMS algorithm is a modified version of GNGD. The regularization parameter is updated as
where sgn(x) represents the sign function, and min is a parameter needs tuning.
G. GSER Algorithm. The GSER updates w(n) as follows,
where is a positive parameter that makes the filter more general, and the power of the error signal is estimated.


SIMULATION RESULTS
In this section, we present the comparison results of several experiments of VSSLMS, MVSS , VSAPA, VSNLMS, NPVSS, GNGD, RRNLMS, and GSER algorithm. The adaptive filter is used to identify a 128tap acoustic echo system ho(n ) .We have used the normalized squared coefficient error (NSCE) to evaluate the performance of the algorithms. The NSCE is defined as
We have run extensive simulations. The results are reasonably consistent. In this section, we show some simulation results with the following parameters setup: = = 0.99, = 5Ã—105, 1 = 0.1, 2=104, 3=20, 4=103, min=104 min=10
3,max=1,c= 1, = 0.15, and K = 4 . We assume the power of
slightly NSCE advantage than that of GSER in 20dB SNR case. It should be noted that NPVSS assume 2
v is available in the simulation. Figures 5 and 6 are the results of AR process input with 30dB SNR and 20dB SNR, respectively. RRNLMS has worst NSCE and shows slower tracking behavior compare to white Gaussian signal input case. VSAPA still has problem in low SNR situation: the NSCE of VSAPA is 10 dB worse than that of its competing algorithms. GSER has fastest tracking and convergence speed in 30dB SNR case.
v
system noise, 2
, is available for NPVSS algorithm.

TimeInvariant System
The reference input, x(n) , is either a zero mean, unit variance white Gaussian signal or a secondorder AR process. The power of the echo system is about 1. An independent white Gaussian signal with zero mean and variance 0.001 is added to the system output. Figures 1 and 2 are the results of white Gaussian signal input and AR process input, respectively. The NSCE curves are ensemble averages over 20 independent runs. As can be seen, VSNLMS has the worst performance. GNGD and RRNLMS have similar convergence speed in the early period, and GNGD exhibits very limited performance in later phase while RRNLMS keeps adaptation to a lower NSCE. However, we notice that RRNLMS is outperformed by the rest algorithms in this category. VSSLMS and MVSS have the same performance in the simulation. VSAPA, NPVSS and GSER are among the best group that has fast convergence speed and low NSCE.

TimeVarying System
Tracking the timevarying system is an important issue in adaptive signal processing. We compare RRNLMS, VAAPA, NPVSS and GSER in a scenario that the acoustic echo system h (n) is changed to its negative value at sample 35,000. The additive zero mean white Gaussian noise, v(n) , is either with variance 0.01 or 0.001. Figures 3 and 4 are the results of white Gaussian signal input with 30dB signaltonoise ration (SNR) and 20dB SNR, respectively. All algorithms have fast tracking performance. RRNLMS has worst NSCE. VSAPA achieves the lowest NSCE when SNR is 30dB. However, the NSCE of VSAPA is 5dB worse than that of NPVSS and GSER. Notice that VSAPA exhibits slow convergence rate. NPVSS has


CONCLUSIONS
Many variable stepsize NLMS algorithms have been proposed to achieve fast convergence rate, rapid tracking, and low misalignment in the past two decades. This paper summarized several promising algorithms and presented a performance comparison by means of extensive simulation. According to the simulation, Benestys NPVSS and our GSER have the best performance in both timeinvariant and time varying systems.
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