- Email: [email protected]
Variable tap-length non-parametric variable step-size NLMS adaptive filtering algorithm for acoustic echo cancellation
Applied Acoustics 159 (2020) 107074
Contents lists available at ScienceDirect
Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Variable tap-length non-parametric variable step-size NLMS adaptive filtering algorithm for acoustic echo cancellation S. Hannah Pauline a, Dhanalakshmi Samiappan a, R. Kumar a, Ankita Anand b, Asutosh Kar c,⇑ a
Department of Electronics and Communication Engineering, SRM Institute of Science and Technology, Kattankulathur, India Student Member, IEEE, Delhi, India c Department of Electronics and Communication Engineering, Indian Institute of Information Technology, Design and Manufacturing, Kancheepuram, Chennai, India b
a r t i c l e
i n f o
Article history: Received 31 May 2019 Received in revised form 10 September 2019 Accepted 28 September 2019
Keywords: Variable step-size Variable tap-length NLMS Mean square error (MSE)
a b s t r a c t In teleconferencing and communication systems, the use of Loudspeaker Enclosure Microphone device leads to undesired echoes. To reduce these echoes, an acoustic echo canceller is used. In this paper, we propose a Variable Tap-length Non-Parametric Variable Step-Size NLMS (VT-NPVSS-NLMS) algorithm based on adaptive filtering for acoustic echo cancellation. The step-size is adjusted without the need for tuning too many parameters, using only the square of the average autocorrelation of a priori and a posteriori estimates of error. Moreover, the tap-length is varied to facilitate a high convergence speed and a small bias in tap-length from the optimum length. Hence, such a combination of the variable step-size algorithm with a variable tap-length provides faster convergence and reduced steady-state mean square error. The performance of the proposed algorithm is evaluated in terms of steady-state and transient mean square error. The advantages of the proposed algorithm, as compared to other adaptive algorithms, are presented using simulation results. From the results, we infer that for the VT-NPVSSNLMS, the convergence speed is increased and the steady-state mean square error is reduced when compared with the conventional NLMS and the variable-step-size NLMS algorithm with a fixed tap-length. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Acoustic echo cancellation (AEC) has attracted researchers since the advent of advanced communication systems, such as teleconferencing, mobile phones, Voice Over IP, etc. In a loudspeaker enclosure microphone (LEM) device, echoes arise due to the coupling between the loudspeaker and microphone. The quality of the speech signal is highly deteriorated by this acoustic echo. Therefore, it is required to eliminate these undesired echoes using an AEC system. An Adaptive filter is the basic requirement in an AEC system. It estimates the echo path and generates a replica of the echo signal, which is then subtracted from the original echo signal (reference signal) [1]. Several adaptive filtering algorithms have been proposed in literature, the LMS algorithm being the most popular because of its simplicity and robustness. However, it has slow convergence and requires more number of filter coefficients to identify a long echo ⇑ Corresponding author. E-mail addresses: [email protected] (S.H. Pauline), [email protected] (D. Samiappan), [email protected] (R. Kumar), [email protected] (A. Anand), [email protected] (A. Kar). https://doi.org/10.1016/j.apacoust.2019.107074 0003-682X/Ó 2019 Elsevier Ltd. All rights reserved.
path [2]. Hence, a variant of LMS, known as the Normalized LMS (NLMS) algorithm, was proposed [3]. In this algorithm, the stepsize parameter is normalized. However, because the step-size is fixed, it is not possible to achieve fast convergence and low steady-state error at the same time. Hence, the performance is compromised because of the trade-off between fast convergence and low steady-state error. To eliminate this trade-off, various variable-step-Size NLMS (VSS-NLMS) algorithms have been proposed, such as in [4,5]. There are two main classes of the VSS-NLMS algorithms. The first class of such algorithms are derived by considering limit conditions, such as the estimation error, which will decrease depending on the step-size. The other class of VSS-NLMS algorithms are ’heuristic’ algorithms, which are derived on intuitional basis such that an increase in step-size increases the convergence speed, whereas a decrease in the step-size leads to a decrease in the steady-state error. Therefore, by using VSS-NLMS algorithm, the convergence speed is improved and the steady-state error is reduced. However, the step-size can either be varied to reduce the estimation error or to reduce the convergence speed i.e., the step-size is increased to increase the convergence speed and reduced to decrease the convergence speed [6–8]. Thus, it is not easy to obtain a reliable solution for the ‘‘estimation error versus
2
S.H. Pauline et al. / Applied Acoustics 159 (2020) 107074
convergence speed” trade-off, as most of the VSS adaptive algorithms require tuning of several algorithm parameters, thereby making it difficult to control the algorithm. Recent works in variable step-size algorithm can be found in [9–12]. Hence, there arises a need for an elegant variable-step-size algorithm which can be conveniently controlled in practical applications and has reduced sensitivity to the aforementioned trade-off. Therefore, nonparametric VSS adaptive algorithm is preferable over the conventional VSS algorithms. Similar to the variation in step-size, the variation in the taplength of the adaptive filter also affects its performance. Therefore, it is necessary to optimize the tap-length in order to provide efficient performance of the adaptive algorithm. When the taplength is chosen to be large-valued, the convergence speed is reduced at the cost of computational complexity. Whereas if the tap-length is chosen to be small-valued, the steady-state mean square error (MSE) is large [13]. Therefore, it is required to adaptively adjust the tap-length to its optimum length. Several variable tap-length algorithms have been proposed in [14–21] on various strategies to vary the tap-length. The variable tap-length (VT) algorithm should be designed in such a way that the optimum taplength is obtained faster and with lesser steady-state error. The different existing VT algorithms are classified as Segmented Filter (SF) [14], Gradient Descent (GD) [15] and Fractional Tap-length (FT) adaptive algorithms, among which the FT adaptive algorithm [16,17] has gained popularity as it is computationally less complex and more robust, as compared to the other VT algorithms. In the FT algorithm, the error-width parameter influences the convergence speed as well as the bias from the optimal tap-length. When the error-width is large, convergence speed is high and the bias is large, whereas when it is small the convergence is slow and the bias is also small. Therefore, the value of the error-width parameter should be large when the algorithm is not near the optimum value of its tap-length and should be small when the algorithm is near optimum tap-length value [16,17,22]. In this paper, we introduce a new combination of variable taplength NLMS algorithm with a variable step-size. We provide an improved and convenient solution to step-size and tap-length selection by proposing the Variable-Tap-Length Non-Parametric Variable-Step-Size (VT-NPVSS-NLMS) algorithm, wherein the step size as well as the tap length can be varied simultaneously. By adjusting tap length to its optimum length, convergence is faster
and the steady state MSE is reduced to minimum. By adjusting the step size in such a way that the square error is reduced at every time instant the convergence speed can be improved with a smaller steady state MSE.Moreover, we present a detailed analysis of the proposed algorithm’s performance in transient and steady state. This paper has the following sections: Section 2 briefs about the preliminaries of adaptive filtering algorithms, Section 3 explains in detail the proposed VT-NPVSS-NLMS algorithm, Section 4 details the performance analysis of the proposed algorithm, Section 5 provides the simulation results and Section 6 presents the conclusion. 2. Background 2.1. Acoustic echo cancellation In the past, extensive research has been conducted to cancel out the acoustic echo using filtering algorithms. Fig. 1 shows an AEC that uses an adaptive filter, whose function is to converge at high speed and track the echo simultaneously, encurring a low steady-state error. Thus, the echo can be effectively cancelled. The adaptive algorithm used in the filter can adjust the tap weights such that the steady-state error is minimized [23]. Finite impulse response (FIR) filters are used in the design of adaptive filters [24]. The microphone signal is the desired signal and is represented as
dðnÞ ¼ yðnÞ þ v ðnÞ þ sðnÞ;
ð1Þ
where v(n) is the near-end signal, s(n) is the noise signal and
yðnÞ ¼ xT ðnÞ h
ð2Þ
is the acoustic echo signal, where xðnÞ ¼ ½xðnÞ; xðn 1Þ; . . . ; xðn K þ 1Þ is the input signal from far-end and has a length of K and h ¼ ½h0 ; h1 ; . . . ; hK1 is the impulse response of the actual echo path. The adaptive filter estimates the unknown echo path and pro^ðnÞ of the echo signal as vides a replica y
^ðnÞ ¼ xT ðnÞ wðnÞ; y
ð3Þ
where wðnÞ ¼ ½w0 ðnÞ; w1 ðnÞ; . . . ; wK1 ðnÞ is the weight vector of the adaptive filter. The error signal
^ðnÞ eðnÞ ¼ dðnÞ y
Fig. 1. Acoustic echo canceller.
ð4Þ
3
S.H. Pauline et al. / Applied Acoustics 159 (2020) 107074
is obtained by subtracting the estimate of the echo signal from the microphone signal, such that the effect of echo is minimized [25]. 2.2. NLMS algorithm In order to select an optimum step-size l that guarantees better stability, the Normalized LMS is proposed, where the power of the input signal is normalized [25]. It is computationally less complex, has better stability and is more robust with good tracking capabilities [26,27]. The weight update equation for NLMS algorithm is given as
wðn þ 1Þ ¼ wðnÞ þ
l xðnÞeðnÞ ; a þ xT ðnÞxðnÞ
ð5Þ
where l is the step-size parameter, xðnÞ is the input signal vector, wðnÞ is the weight vector and, to avoid division by zero, a small positive value a (known as regularization constant) is used. However, since the step- size is fixed, the NLMS algorithm does not simultaneously provide fast convergence and reduction in steady-state error.
To eliminate the trade-off between fast convergence and good tracking, the step-size should be adjusted [28]. This is done by the VSS-NLMS algorithm. A large step-size should be maintained during initial stages of convergence, thus providing faster convergence. A small step-size is used at the end of the convergence stage, thus giving a low steady-state error. The weight updating of VSS-NLMS is given as
lVN ðnÞ xðnÞeðnÞ ; a þ xT ðnÞxðnÞ
ð6Þ
where lVN ðnÞ is the variable step-size parameter. A variant of the VSS-NLMS is the Non-Parametric VSS-NLMS (NPVSS-NLMS) algorithm. As the number of parameters increase they interact with each other during simulation and are difficult to adjust. In NonParametric VSS, only one parameter (forgetting factor) needs to be adjusted all other parameters are fixed. Therefore it provides better stability while simulating.In this algorithm, the step-size is adjusted in such a way that the squared estimation error is reduced at each time instant. 2.4. Variable tap-length algorithm The Variable-Tap-Length algorithm is required to provide faster convergence and smaller steady-state bias of tap-length in the presence of noise [22]. The weight update equation is given by
wKðnÞ ðn þ 1Þ ¼ wKðnÞ ðnÞ þ lðnÞ
ðKðnÞÞ eKðnÞ xKðnÞ ðnÞ;
ð7Þ
lðnÞ is the variable step-size and KðnÞ is the variable tap-
where
ðKðnÞÞ
length, eKðnÞ is the output error. Let
KðnÞ
and eL KðnÞ
eL
ð8Þ
ðnÞ be the a priori error estimate given as
ðnÞ ¼ dðnÞ xTKðnÞ;1:L ðnÞ wKðnÞ;1:L ðnÞ;
ð9Þ
ð10Þ
ð11Þ
where j is the positive leakage parameter used for overcoming the wandering problem, k is the step-size for adaptation of tap-length, and D is the error width that influences the convergence speed ðKðnÞÞ
and bias from the optimal tap-length, and the errors eKðnÞ ðnÞ and ðKðnÞÞ
eKðnÞD ðnÞ are the output errors calculated using (9). The taplength Kðn þ 1Þ to be used in the NLMS filter coefficient update equation at iteration ðn þ 1Þ is given as
bkf ðnÞc; if jKðnÞ kf ðnÞj > q KðnÞ;
ð12Þ
otherwise;
where b:c represents the floor operator and q is a small integer. The problem with fixed value of D is that the tap-length convergence is faster for large value of D and the bias in tap-length is also large. On the other hand, a small value of D can lead to the under modelling problem and slower convergence. Hence, a variable D can provide a compromise between the bias from the optimum tap-length and quick convergence [29]. The energy of the instantaneous error signal is given as
h i2 ðKðnÞÞ e2i ðnÞ ¼ ce2 ðn 1Þ þ ð1 cÞ eKðnÞ ðnÞ
ð13Þ
and
DðnÞ ¼ min Dmax ; B e2i ðnÞ ;
ð14Þ
where the upper bound of DðnÞ is given by Dmax ; c 1 is called as the smoothing parameter and B is a constant. When the error is large during the initial stages of adaptation, DðnÞ increases and provides faster convergence. As the error starts to decrease, the value of DðnÞ also decreases, thus providing a small bias from the optimum tap-length. However, adaptation in the presence of noise is not well addressed in (13) and (14). Let xðnÞ be the zero-mean Gaussian signal, KðnÞ < K opt at the transient adaptation stage and bðnÞ be stationary with zero mean. It is assumed that bðnÞ is uncorrelated to xðnÞ. The vectors wK opt and xK opt ðnÞ can be written as [30]
T wK opt ¼ ½wKðnÞ ðnÞ w
ð15Þ
and
ðnÞT ; xK opt ¼ ½xKðnÞ ðnÞ x
ð16Þ iT
h
iT
¼ wKðnÞþ1 ; wKðnÞþ2 ; . . . ; wK where wKðnÞ ðnÞ ¼ w1 ; w2 ; . . . ; wKðnÞ ; w and opt
ðnÞ ¼ ðxðn KðnÞÞ; xðn KðnÞ 1Þ; . . . ; xðn K opt þ 1Þ T . Using x (10) in (9), we get ðKðnÞÞ T ðnÞw ðnÞ xTKðnÞ v KðnÞ ðnÞ þ bðnÞ; eKðnÞ ðnÞ ¼ x
where 1 6 L 6 KðnÞ and wKðnÞ;1:L ðnÞ and xKðnÞ;1:L ðnÞ are vectors consisting of first L elements of the vectors wKðnÞ ðnÞ and xKðnÞ ðnÞ, respectively. The desired signal dðnÞ is defined as
dðnÞ ¼ xTK opt ðnÞ wK opt þ bðnÞ;
2 2 ðKðnÞÞ ðKðnÞÞ ; kf ðn þ 1Þ ¼ kf ðnÞ j k eKðnÞ ðnÞ eKðnÞD ðnÞ
h
LKðnÞ ðnÞ be the a posteriori error estimate given as
KðnÞ ðnÞ ¼ dðnÞ xTKðnÞ;1:L ðnÞ wKðnÞ;1:L ðn þ 1Þ L
cient vector and bðnÞ is a stationary zero-mean uncorrelated noise, independent of the input signal xðnÞ. For the tap-length to be variable, the pseudo-tap-length concept is as presented in [16] for the FT NLMS algorithm. Let the pseudotap-length be defined as kf ðnÞ.The tap-length update equation is given as [16]
Kðn þ 1Þ ¼
2.3. Variable step-size NLMS algorithm
wðn þ 1Þ ¼ wðnÞ þ
where K opt is the optimal tap-length, xK opt ¼ ½xðnÞ; xðn 1Þ; . . . ; xðn K opt þ 1ÞT ; wK opt ¼ ½w1 ; w2 ; . . . ; wK opt T is the optimum coeffi-
where
ð17Þ
v KðnÞ ðnÞ is the coefficient error vector defined as
vKðnÞ ðnÞ ¼ wKðnÞ ðnÞ wKðnÞ :
ð18Þ
From (14), we have
E½DðnÞ ¼ B E e2i ðnÞ :
ð19Þ
4
S.H. Pauline et al. / Applied Acoustics 159 (2020) 107074
Considering that the correct estimation of the energy of the output error is given by (13), then, from (17) and assuming that xKðnÞ ðnÞ and wKðnÞ ðnÞ are independent of each other [3], we have
h i h i
2 k2 þ r2x E v TKðnÞ ðnÞv KðnÞ ðnÞ ; E e2i ðnÞ ¼ E b ðnÞ þ r2x kw
ð20Þ
where E½x2 ðnÞ ¼ r2x . Furthermore,
E½DðnÞ ¼ B E e2 ðnÞ h i h i 2 k2 þ r2x E v TKðnÞ ðnÞv KðnÞ ðnÞ : ¼ B E b ðnÞ þ r2x kw
ð21Þ
tive filter is to its optimum solution. However, the efficiency of the adaptation rate of DðnÞ is reduced due to the presence of noise, which causes DðnÞ to be large and produces a large bias. Therefore, update
DðnÞ,the auto-correlation between eðKðnÞÞ and KðnÞ ðnÞ
ðKðn1ÞÞ
ðKðnÞÞ
eKðn1Þ ðn 1Þ is used [31]. The time-average estimate of eKðnÞ ðnÞ and
ðKðn1ÞÞ eKðn1Þ ðn
1Þ is given by [32]
ðKðn1ÞÞ gðnÞ ¼ b gðn 1Þ þ ð1 bÞ eðKðnÞÞ KðnÞ ðnÞ eKðn1Þ ðn 1Þ;
ð22Þ
where b varies from 0 to 1 and influences gðnÞ, and the update equation of DðnÞ is given as
DðnÞ ¼ minðDmax ; AjgðnÞjÞ;
ð23Þ
where A is a constant and j:jis the magnitude selected according to the criteria discussed in detail in Section 4. Hence, by using time average estimate of error gðnÞ, the effect of uncorrelated noise is eliminated from the update equation of DðnÞ and it is possible to measure how close the adaptive filter is to the optimum value. In the initial stages of adaptation, jgðnÞjis large-valued, thus giving large values of DðnÞ. When close to optimum, jgðnÞj 0, thus giving small values of DðnÞ. This achieves fast convergence and small steady-state bias simultaneously, the speed of convergence and steady-state bias being unaffected by the presence of noise.
In the present section, we explain the proposed Variable Taplength Non-Parametric Variable Step-Size NLMS (VT-NPVSSNLMS) algorithm with a variable tap-length and a nonparametric variation of step size for reducing the compromise between reduction in convergence-time and reduction in estimation error of the adaptive filter. The VT-NPVSS-NLMS algorithm weight update equation is given as ðKðnÞÞ
wKðnÞ ðn þ 1Þ ¼ wKðnÞ ðnÞ þ lNP ðnÞ eKðnÞ ðnÞ xKðnÞ ðnÞ;
ð24Þ
where wKðnÞ is the adaptive filter coefficient vector and is defined as wKðnÞ ¼ ½w1 ðnÞ; w2 ðnÞ; . . . ; wKðnÞ ðnÞT ; xKðnÞ ðnÞ is the input signal vector ðKðnÞÞ
given as xKðnÞ ¼ ½xðnÞ; xðn 1Þ; . . . ; xðn KðnÞ þ 1ÞT ; eKðnÞ ðnÞ is the output error and lNP ðnÞ is the non-parametric variable step-size [33,34] which is calculated according to the criterion which reduces the square of error at every instant, forcing the condition
ð25Þ
where E½: is the mathematical expectation. The optimal value of lNP ðnÞ is given by the expression [33,34]
lNP ðnÞ ¼ where
dðnÞ;
re > rs
0;
otherwise;
ð27Þ
where f is a small positive number which is used to avoid division
by zero. Furthermore, r2e ðnÞ ¼ E e2 ðnÞ is the power of the error sig
nal, r2s ðnÞ ¼ E s2 ðnÞ is the power of noise signal of the system which may be known or can be estimated and L is a posteriori error estimate. Then, r2e is calculated using the following expression as
re ðnÞ ¼ m r2e ðn 1Þ þ ð1 mÞ e2 ðnÞ;
ð28Þ
where m is the exponential weighting factor whose value ranges between 0 and 1. 4. Performance analysis of proposed algorithm The mean square performance of the proposed algorithm is analyzed in detail in this section. We make the following assumptions: 1. The input xðnÞ is a zero-mean white Gaussian stationary signal, 2. The noise signal bðnÞ is zero-mean stationary and is independent of the input signal, 3. KðnÞ < K opt (optimum tap-length) during transient behaviour, and 4. KðnÞ P K opt at steady state [17]. 4.1. Mean square performance analysis Substituting (17) in (7) and using (18), we get
vKðnÞ ðn þ 1Þ ¼
h
i I lNPN ðnÞ xKðnÞ ðnÞ xTKðnÞ ðnÞ T
ðnÞ w xKðnÞ ðnÞ þ lNPN ðnÞ x
þ
lNPN ðnÞ xKðnÞ ðnÞ bðnÞ;
ð29Þ
where lNPN ðnÞ is the non parametric variable step size. The MSE is defined as
3. Proposed VT-NPVSS-NLMS algorithm
E e2 ðnÞ ¼ E s2 ðnÞ ;
f þ xTKðnÞ xKðnÞ
#
rs ; 1 KðnÞ L þ re ðnÞ
KðnÞ
h i In (21), the term E v TKðnÞ ðnÞv KðnÞ ðnÞ shows how close the adap-
to
dðnÞ ¼
"
1
ð26Þ
2
ðKðnÞÞ vðKðnÞÞ eKðnÞ ðnÞ KðnÞ ðnÞ ¼ E
:
ð30Þ
Substituting (17) in (30), we can rewrite (30) as
h n
oi
2 2 2 T vðKðnÞÞ ; KðnÞ ðnÞ ¼ P min þ rx kw k þ rx tr E v KðnÞ ðnÞv KðnÞ ðnÞ
where Pmin
ð31Þ
h i 2 ¼ E b ðnÞ is the minimum MSE and tr(.) denotes the
trace operation. On multiplying both the sides of (26) by vTKðnÞ ðn þ 1Þ, calculating the expected value and after using our assumptions, we get
E
h
i
vKðnþ1Þ ðnÞv TKðnþ1Þ ðnÞ
¼E
h
i
vKðnÞ ðnÞvTKðnÞ ðnÞ
h i
2 E lNPN ðnÞ r2x E v KðnÞ ðnÞv TKðnÞ ðnÞ h i
þ 2 E l2NPN ðnÞ r4x E v KðnÞ ðnÞv TKðnÞ ðnÞ n h io
þ E l2NPN ðnÞ r4x tr E v KðnÞ ðnÞv TKðnÞ ðnÞ I
k2 I þ E l2NPN ðnÞ r4x kw 2
2 þ 2 E lNPN ðnÞ rx Pmin I: ð32Þ
On taking expectation of (23), we get
S.H. Pauline et al. / Applied Acoustics 159 (2020) 107074
E½DðnÞ ¼ A E½jgðnÞj:
ð33Þ
To evaluate E½jgðnÞj in (32), we can rewrite gðnÞ in (22) as
gðnÞ ¼ ð1 bÞ
n1 X ðKðniÞÞ ðKðni1ÞÞ bi eKðniÞ ðn iÞ eKðni1Þ ðn i 1Þ:
ð34Þ
i¼0
Using the Central limit theorem, we can assume that gðnÞ is a zero-mean white Gaussian sequence. Therefore,
E½jgðnÞj ¼
rffiffiffiffi 2
p
rgðnÞ ðnÞ;
ð35Þ
where
r2gðnÞ ðnÞ ¼ E jg2 ðnÞj ¼ ð1 bÞ2
n1 X n1 X
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð1 bÞ Pmin ; E½Kð1Þ K opt þ 2 A pð1 þ bÞ 2 l0
5
ð44Þ
where the value of A should be [35]
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi p 1 þ b Dð1Þ ð2 l0 Þ : A¼ 2 Pmin 2 1b
ð45Þ
The optimum value of Dð1Þ 0:1 K opt . The value of A influences DðnÞ. The value of A must be selected such that it causes DðnÞ to be large during the initial convergence stage so that as the adaptive system environment changes, the tap-length converges fast to its
h i ðKðniÞÞ ðKðni1ÞÞ ðKðnjÞÞ ðKðnj1ÞÞ bi bj E eKðniÞ ðn iÞ eKðni1Þ ðn i 1ÞeKðnjÞ ðn jÞ eKðnj1Þ ðn j 1Þ
i¼0 i¼0 n1 h i h i X 2 2 ðKðniÞÞ ðKðni1ÞÞ b2i E ðeKðniÞ ðn iÞÞ E ðeKðniÞ ðn i 1ÞÞ : ¼ ð1 bÞ2
ð36Þ
i¼0
The above equation is derived assuming that the algorithm has h i ðKðniÞÞ ðKðnjÞÞ converged and that E eKðniÞ ðn iÞ eKðnjÞ ðn jÞ ¼ 0 for i – j.
optimal value. Also, the selected value of A must be ensure that DðnÞ is small once the steady state is reached.
4.2. Steady-state performance analysis
5. Simulation results and discussion
At steady-state, the terms that are present in the transient analysis do not appear because Assumption 4. Therefore, (31) can be expressed for steady state as
The simulation results were obtained in the MATLAB environment, using a sampling frequency of 8 kHz. For obtaining the room impulse response (see Fig. 2), a 12 12 8 ft3 enclosure was used. Inside the enclosure, an ordinary loudspeaker and an omnidirectional microphone were kept at a distance of 1 ft. The taplength of the adaptive filter used in the AEC framework is set at 1200. A stationary Gaussian zero-mean stochastic signal with unit variance is given as input to the measured room impulse response to obtain the channel response. The noise signal is considered as a white Gaussian zero-mean sequence with a variance of 0.01. A comparative analysis of the proposed algorithm is carried out with the basic NLMS algorithm and the VT-NLMS algorithm with variable error width, in terms of the MSE incurred, convergence and tracking capabilities, echo return loss enhancement (ERLE) and echo suppression for an SNR of 20 dB. The results obtained are averaged for 250 Monte-Carlo runs. Figs. 3(a) and 3(b) present the MSE performance for SNR values of 10 dB and 30 dB, while Fig. 4 depicts the tracking performances of the NLMS, VT-NLMS with variable error width and the proposed algorithm. From both the figures, it can be observed that the proposed non-parametric algorithm, with variable tap-length and variable step-size, exhibits the fastest convergence (at about 215 iterations), followed by the VT-NLMS with variable error-width (at about 450 iterations). The NLMS algorithm exhibits the slowest convergence (about 1000 iterations) due to its constant tap-length. Moreover, it must be noted from Figs. 3(a) and 3(b) that the proposed algorithm incurs the highest MSE among the algorithms under consideration, while the NLMS algorithm incurs the least MSE. However, it can be observed that the difference in the highest and the lowest MSE incurred is only marginal. Further, Fig. 4 depicts the ability of the considered algorithms to track variations of a non-stationary echo path. An enclosure dislocation is done at 1500 iterations. It can be seen from the figure that the proposed non-parametric algorithm outperforms the rest in its ability to adapt to the changes in the environment, followed by the
vðKð1ÞÞ Kð1Þ ð1Þ ¼ P min þ P ex ð1Þ;
ð37Þ
where P ex ð1Þ is the steady-state Excess MSE (EMSE) and is given as [3]
h i Pex ð1Þ ¼ r2x tr E v Kð1Þ ð1Þ v TKð1Þ ð1Þ :
ð38Þ
By substituting (32) in (38), we get
Pex ð1Þ ¼
E lNPN ð1Þ E½Kð1Þr2x Pmin
; 2 E lNPN ð1Þ ðE½Kð1Þ þ 2Þr2x
ð39Þ
where, using the approximation [35] as
2
E l2NPN ð1Þ l ð1Þ
E NPN E lNPN ð1Þ : lNPN ð1Þ E lNPN ð1Þ
ð40Þ
From (35) and (23), we get
E½Dð1Þ ¼ A
rffiffiffiffi 2
p
rgð1Þ ð1Þ;
ð41Þ
where Dð1Þ is the bias incurred in the tap-length solution. Eq. (36) is evaluated at steady state to get
rgð1Þ ð1Þ ¼
sffiffiffiffiffiffiffiffiffiffiffiffi 1 b ðKð1ÞÞ v ð1Þ: 1 þ b Kð1Þ
ð42Þ
Substituting (37), (39) and (42) in (41), we get
E½Dð1Þ ¼ 2 A where write
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð1 bÞ Pmin : pð1 þ bÞ 2 l0
ð43Þ
l0 is the step size. Assuming that E½Kð1Þ is large, we can
6
S.H. Pauline et al. / Applied Acoustics 159 (2020) 107074
Fig. 2. Room impulse response.
Fig. 3. MSE performance of the algorithms under consideration for SNR (a) 10 dB. (b) 30 dB.
S.H. Pauline et al. / Applied Acoustics 159 (2020) 107074
7
Fig. 4. Tracking performance of the algorithms under consideration.
VT-NLMS and the NLMS. The improved performance of the proposed and the VT-NLMS algorithms is due to their ability to dynamically adapt the tap-length to the varying room impulse response. In a non-stationary environment, it is desirable that the adaptive algorithm tracks the statistical environment variations quickly to effectively suppress the echo from the output and a slight increment in the MSE is a small price to pay for the same. Thus, the proposed VT-NPVSS-NLMS algorithm is more preferable as compared to the rest. Fig. 5 depicts the ERLE performance of the considered algorithms. The measure of the amount of echo (in dB) attenuated by the AEC is the ERLE. It can be defined as the ratio of the power of the far-end signal to the power of the residual echo signal after the cancellation process as [3]
ERLE ¼ 10 log10
power of far-end signal : power of residual echo signal
ð46Þ
It can be seen from Fig. 5 that the proposed algorithm results in the highest ERLE much faster than the NLMS and the VS-NLMS algorithm. It must be noted that even though all three of the algorithms attain similar ERLE, the proposed algorithm incurs the least
time to attain the highest ERLE, as compared to the NLMS and the VS-NLMS algorithm. This shows that the proposed algorithm suppresses the effect of echo more efficiently and faster than the algorithms under consideration. Fig. 6 depicts the testing of the proposed algorithm with an acoustic signal. The effect of echo on the desired signal can be clearly seen in Fig. 6(c). On comparing Fig. 6(a) and 6(d), the effectiveness of the proposed algorithm in removing echo from the acoustic signal can be conveniently observed. The enhanced signal waveform in Fig. 6(d) is similar to that of the desired signal waveform in Fig. 6(a), and the echo-effect is attenuated well. 6. Conclusion In this paper, we introduced a novel adaptive algorithm which combines variation in tap-length as well as in step-size. The proposed Variable-Tap-Length Non-Parametric Variable-Step-Size (VT-NPVSS-NLMS) algorithm offered an improved and convenient solution to simultaneous selection of step-size and tap-length selection to obtain fast convergence and a small steady-state MSE, without the need for adjusting too many parameters in an
Fig. 5. ERLE performance of the algorithms under consideration.
8
S.H. Pauline et al. / Applied Acoustics 159 (2020) 107074
Fig. 6. Sound signal waveforms. (a) Desired signal, (b) Echo, (c) Microphone signal, and (d) processed signal from the proposed algorithm.
AEC framework. A comprehensive analysis of the transient and steady-state performance of the proposed algorithm was done. From the computer simulations, it was concluded that the proposed VT-NPVSS-NLMS algorithm outperforms the conventional NLMS and the variable-step-size NLMS algorithm with a fixed tap-length in terms of an increased convergence speed and faster tracking and improved ERLE performance. Further, MSE incurred by the proposed algorithm is comparable to the NLMS and the VS-NLMS algorithm with a fixed tap-length. As a result, it was concluded that the proposed algorithm is more preferable for AEC systems, as compared to the basic NLMS and the VS-NLMS algorithm with a fixed tap-length.
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
References [1] Paleologu C, Benesty J, Ciochina S. Sparse adaptive filters for echo cancellation. Georgia: Morgan and Claypool Publishers; 2010. [2] Tingchan W, Chutchavong V, Benjangkaprasert C. Performance of a robust variable step-size LMS adaptive algorithm for multiple echo cancellation in telephone network. In: Proceeding of SICE-ICASE International Joint Conference, Korea. p. 3173–6. [3] Haykin S. Adaptive filter theory. 3rd ed. New York: Prentice Hall Inc.; 1996. [4] Casco-Sñchez FM, Medina-Ramírez RC, Lopez-Guerrero M. A new variable step-size NLMS algorithm and its performance evaluation in echo cancelling applications. J Appl Res Technol 2011;9(3):302–13.
[5] Kun S, Xiaoli M. A variable step-size NLMS algorithm using statistics of channel response. Signal Process 2010;90(6):2107–11. [6] Zhua YG, Lia YG, Guana SY, Chenb QS. A novel variable step-size NLMS algorithm and its analysis. Proc Eng 2012;29:1181–5. [7] Tianhui W, Liyuan M, Yongjun L, Yonggang D. An improved variable step-size LMS adaptive spectral-line enhancement algorithm and its simulation. In: Proceeding of First International Conference on Pervasive Computing Signal Processing and Applications, PCSPA 2010. China: IEEE; 2010. p. 727–30. [8] Kar A, Swamy M. Convergence and steady state analysis of a tap-length optimization algorithm for linear adaptive filters. AEU – Int J Electron Commun 2016;70:1114–21. [9] Bendoumia Rédha, Djendi M. Variable step-size subband backward bss algorithms for speech quality enhancement. Appl Acoust 2014;86:25–41. [10] Niu Qun, Chen T. A new variable step size LMS adaptive algorithm. Proceedings of IEEE Chinese Control and Decision Conference 2018. [11] Resende L, Haddad D, Petroglia MR. A variable step size NLMS algorithm with adaptive coefficient vector reusing. In: Proceedings of IEEE international conference on electro information technology. [12] Yuan Z, Tao XS. New LMS adaptive filtering algorithm with variable step size. In: Proceedings of IEEE conference on vision, image and signal processing. [13] Kar A, Chandra M. An improved variable structure adaptive filter design and analysis for acoustic echo cancellation. Radio Eng 2015;24:252–61. [14] Riera-Palou F, Noras JM, Cruickshank DGM. Linear equalizers with dynamic and automatic length selection. Electron Lett 2001;37(25):1553–4. [15] Gu Y, Tang J, Cui H, Du W. LMS algorithm with gradient descent filter length. IEEE Signal Process Lett 2004;11(3):305–7. [16] Kar A, Chandra M. Performance evaluation of a new variable tap-length learning algorithm for automatic structure adaptation in linear adaptive filters. AEU – Int J Electron Commun 2015;69:253–61. [17] Zhang Y, Chambers JA, Li Ning, Sayed AH. Steady-state performance analysis of a variable tap-length LMS algorithm. IEEE Trans Signal Process 2008;56 (2):839–45. [18] Kar A, Swamy M. Tap-length optimization of adaptive filters used in stereophonic acoustic echo cancellation. Signal Process 2017;131:422–33. [19] Kar A, Padhi T, Majhi B, Swamy MNS. Analysing the impact of system dimension on the performance of a variable-tap-length adaptive algorithm. Appl Acoust 2019;150:207–15. [20] Wei Y, Yan Z. Variable tap length LMS algorithm with adaptive step size. Circuits Syst Signal Process 2017;16:2815–27.
S.H. Pauline et al. / Applied Acoustics 159 (2020) 107074 [21] Wei Y, Yan Z. Simple variable tap length algorithm for high noise environment. IET Electron Lett 2017;53(5):320–2. [22] Li N, Zhang Y, Zhao Y, Hao Y. An improved variable tap-length LMS algorithm. Signal Process 2009;89(5):908–12. [23] Widrow B. Adaptive signal processing. Upper Saddle River, New Jersey: Prentice Hall Inc.; 1985. [24] Mader A, Puder H, Schmidt GU. Step-size control for acoustic echo cancellation filters – an overview. Signal Process 2000;80(9):1697–719. [25] Kuech F, Kellermann W. Orthogonalized power filters for nonlinear acoustic echo cancellation. Signal Process 2006;86(6):1168–81. [26] Kuhn EV, Kolodziej JE, Seara R. Stochastic modeling of the NLMS algorithm for complex gaussian input data and non-stationary environment. Digit Signal Process 2014;30:55–66. [27] Jeong JJ, Koo K, Koo G, Kim SW. Steady-state mean-square deviation analysis of improved normalized subband adaptive filter. Signal Process 2015;106:49–54.
9
[28] Shi K, Ma X. A variable step-size NLMS algorithm using statistics of channel response. Signal Process 2010;90(6):2107–11. [29] Feuer A, Weinstein E. Convergence analysis of LMS filters with uncorrelated gaussian data. IEEE Trans Acoust Speech Signal Process ASSP-33 1985:222–30. [30] Mayyas K. Performance analysis of the deficient length LMS adaptive algorithm. IEEE Trans Signal Process 2005;53(8):2727–34. [31] Aboulnasr T, Mayyas K. A robust variable step-size LMS-type algorithm: analysis and simulations. IEEE Trans Signal Process 1997;45(3):631–9. [32] Mayyas K. A variable step-size selective partial update LMS algorithm. Digit Signal Process 2013;23(1):75–85. [33] Benesty J, Rey H, Rey Vega L, Tressens S. A non-parametric VSS-NLMS algorithm. IEEE Signal Process Lett 2006;13(10):581–4. [34] Cohen I, Benesty J, Gannot S. Speech processing in modern communication challenges and perspectives. Springer Sci Bus Media 2009;3. [35] Mayyas K, AbuSeba HA. A new variable length nlms adaptive algorithm. Digital Signal Process 2014;34:82–91.
Contents lists available at ScienceDirect
Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Variable tap-length non-parametric variable step-size NLMS adaptive filtering algorithm for acoustic echo cancellation S. Hannah Pauline a, Dhanalakshmi Samiappan a, R. Kumar a, Ankita Anand b, Asutosh Kar c,⇑ a
Department of Electronics and Communication Engineering, SRM Institute of Science and Technology, Kattankulathur, India Student Member, IEEE, Delhi, India c Department of Electronics and Communication Engineering, Indian Institute of Information Technology, Design and Manufacturing, Kancheepuram, Chennai, India b
a r t i c l e
i n f o
Article history: Received 31 May 2019 Received in revised form 10 September 2019 Accepted 28 September 2019
Keywords: Variable step-size Variable tap-length NLMS Mean square error (MSE)
a b s t r a c t In teleconferencing and communication systems, the use of Loudspeaker Enclosure Microphone device leads to undesired echoes. To reduce these echoes, an acoustic echo canceller is used. In this paper, we propose a Variable Tap-length Non-Parametric Variable Step-Size NLMS (VT-NPVSS-NLMS) algorithm based on adaptive filtering for acoustic echo cancellation. The step-size is adjusted without the need for tuning too many parameters, using only the square of the average autocorrelation of a priori and a posteriori estimates of error. Moreover, the tap-length is varied to facilitate a high convergence speed and a small bias in tap-length from the optimum length. Hence, such a combination of the variable step-size algorithm with a variable tap-length provides faster convergence and reduced steady-state mean square error. The performance of the proposed algorithm is evaluated in terms of steady-state and transient mean square error. The advantages of the proposed algorithm, as compared to other adaptive algorithms, are presented using simulation results. From the results, we infer that for the VT-NPVSSNLMS, the convergence speed is increased and the steady-state mean square error is reduced when compared with the conventional NLMS and the variable-step-size NLMS algorithm with a fixed tap-length. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Acoustic echo cancellation (AEC) has attracted researchers since the advent of advanced communication systems, such as teleconferencing, mobile phones, Voice Over IP, etc. In a loudspeaker enclosure microphone (LEM) device, echoes arise due to the coupling between the loudspeaker and microphone. The quality of the speech signal is highly deteriorated by this acoustic echo. Therefore, it is required to eliminate these undesired echoes using an AEC system. An Adaptive filter is the basic requirement in an AEC system. It estimates the echo path and generates a replica of the echo signal, which is then subtracted from the original echo signal (reference signal) [1]. Several adaptive filtering algorithms have been proposed in literature, the LMS algorithm being the most popular because of its simplicity and robustness. However, it has slow convergence and requires more number of filter coefficients to identify a long echo ⇑ Corresponding author. E-mail addresses: [email protected] (S.H. Pauline), [email protected] (D. Samiappan), [email protected] (R. Kumar), [email protected] (A. Anand), [email protected] (A. Kar). https://doi.org/10.1016/j.apacoust.2019.107074 0003-682X/Ó 2019 Elsevier Ltd. All rights reserved.
path [2]. Hence, a variant of LMS, known as the Normalized LMS (NLMS) algorithm, was proposed [3]. In this algorithm, the stepsize parameter is normalized. However, because the step-size is fixed, it is not possible to achieve fast convergence and low steady-state error at the same time. Hence, the performance is compromised because of the trade-off between fast convergence and low steady-state error. To eliminate this trade-off, various variable-step-Size NLMS (VSS-NLMS) algorithms have been proposed, such as in [4,5]. There are two main classes of the VSS-NLMS algorithms. The first class of such algorithms are derived by considering limit conditions, such as the estimation error, which will decrease depending on the step-size. The other class of VSS-NLMS algorithms are ’heuristic’ algorithms, which are derived on intuitional basis such that an increase in step-size increases the convergence speed, whereas a decrease in the step-size leads to a decrease in the steady-state error. Therefore, by using VSS-NLMS algorithm, the convergence speed is improved and the steady-state error is reduced. However, the step-size can either be varied to reduce the estimation error or to reduce the convergence speed i.e., the step-size is increased to increase the convergence speed and reduced to decrease the convergence speed [6–8]. Thus, it is not easy to obtain a reliable solution for the ‘‘estimation error versus
2
S.H. Pauline et al. / Applied Acoustics 159 (2020) 107074
convergence speed” trade-off, as most of the VSS adaptive algorithms require tuning of several algorithm parameters, thereby making it difficult to control the algorithm. Recent works in variable step-size algorithm can be found in [9–12]. Hence, there arises a need for an elegant variable-step-size algorithm which can be conveniently controlled in practical applications and has reduced sensitivity to the aforementioned trade-off. Therefore, nonparametric VSS adaptive algorithm is preferable over the conventional VSS algorithms. Similar to the variation in step-size, the variation in the taplength of the adaptive filter also affects its performance. Therefore, it is necessary to optimize the tap-length in order to provide efficient performance of the adaptive algorithm. When the taplength is chosen to be large-valued, the convergence speed is reduced at the cost of computational complexity. Whereas if the tap-length is chosen to be small-valued, the steady-state mean square error (MSE) is large [13]. Therefore, it is required to adaptively adjust the tap-length to its optimum length. Several variable tap-length algorithms have been proposed in [14–21] on various strategies to vary the tap-length. The variable tap-length (VT) algorithm should be designed in such a way that the optimum taplength is obtained faster and with lesser steady-state error. The different existing VT algorithms are classified as Segmented Filter (SF) [14], Gradient Descent (GD) [15] and Fractional Tap-length (FT) adaptive algorithms, among which the FT adaptive algorithm [16,17] has gained popularity as it is computationally less complex and more robust, as compared to the other VT algorithms. In the FT algorithm, the error-width parameter influences the convergence speed as well as the bias from the optimal tap-length. When the error-width is large, convergence speed is high and the bias is large, whereas when it is small the convergence is slow and the bias is also small. Therefore, the value of the error-width parameter should be large when the algorithm is not near the optimum value of its tap-length and should be small when the algorithm is near optimum tap-length value [16,17,22]. In this paper, we introduce a new combination of variable taplength NLMS algorithm with a variable step-size. We provide an improved and convenient solution to step-size and tap-length selection by proposing the Variable-Tap-Length Non-Parametric Variable-Step-Size (VT-NPVSS-NLMS) algorithm, wherein the step size as well as the tap length can be varied simultaneously. By adjusting tap length to its optimum length, convergence is faster
and the steady state MSE is reduced to minimum. By adjusting the step size in such a way that the square error is reduced at every time instant the convergence speed can be improved with a smaller steady state MSE.Moreover, we present a detailed analysis of the proposed algorithm’s performance in transient and steady state. This paper has the following sections: Section 2 briefs about the preliminaries of adaptive filtering algorithms, Section 3 explains in detail the proposed VT-NPVSS-NLMS algorithm, Section 4 details the performance analysis of the proposed algorithm, Section 5 provides the simulation results and Section 6 presents the conclusion. 2. Background 2.1. Acoustic echo cancellation In the past, extensive research has been conducted to cancel out the acoustic echo using filtering algorithms. Fig. 1 shows an AEC that uses an adaptive filter, whose function is to converge at high speed and track the echo simultaneously, encurring a low steady-state error. Thus, the echo can be effectively cancelled. The adaptive algorithm used in the filter can adjust the tap weights such that the steady-state error is minimized [23]. Finite impulse response (FIR) filters are used in the design of adaptive filters [24]. The microphone signal is the desired signal and is represented as
dðnÞ ¼ yðnÞ þ v ðnÞ þ sðnÞ;
ð1Þ
where v(n) is the near-end signal, s(n) is the noise signal and
yðnÞ ¼ xT ðnÞ h
ð2Þ
is the acoustic echo signal, where xðnÞ ¼ ½xðnÞ; xðn 1Þ; . . . ; xðn K þ 1Þ is the input signal from far-end and has a length of K and h ¼ ½h0 ; h1 ; . . . ; hK1 is the impulse response of the actual echo path. The adaptive filter estimates the unknown echo path and pro^ðnÞ of the echo signal as vides a replica y
^ðnÞ ¼ xT ðnÞ wðnÞ; y
ð3Þ
where wðnÞ ¼ ½w0 ðnÞ; w1 ðnÞ; . . . ; wK1 ðnÞ is the weight vector of the adaptive filter. The error signal
^ðnÞ eðnÞ ¼ dðnÞ y
Fig. 1. Acoustic echo canceller.
ð4Þ
3
S.H. Pauline et al. / Applied Acoustics 159 (2020) 107074
is obtained by subtracting the estimate of the echo signal from the microphone signal, such that the effect of echo is minimized [25]. 2.2. NLMS algorithm In order to select an optimum step-size l that guarantees better stability, the Normalized LMS is proposed, where the power of the input signal is normalized [25]. It is computationally less complex, has better stability and is more robust with good tracking capabilities [26,27]. The weight update equation for NLMS algorithm is given as
wðn þ 1Þ ¼ wðnÞ þ
l xðnÞeðnÞ ; a þ xT ðnÞxðnÞ
ð5Þ
where l is the step-size parameter, xðnÞ is the input signal vector, wðnÞ is the weight vector and, to avoid division by zero, a small positive value a (known as regularization constant) is used. However, since the step- size is fixed, the NLMS algorithm does not simultaneously provide fast convergence and reduction in steady-state error.
To eliminate the trade-off between fast convergence and good tracking, the step-size should be adjusted [28]. This is done by the VSS-NLMS algorithm. A large step-size should be maintained during initial stages of convergence, thus providing faster convergence. A small step-size is used at the end of the convergence stage, thus giving a low steady-state error. The weight updating of VSS-NLMS is given as
lVN ðnÞ xðnÞeðnÞ ; a þ xT ðnÞxðnÞ
ð6Þ
where lVN ðnÞ is the variable step-size parameter. A variant of the VSS-NLMS is the Non-Parametric VSS-NLMS (NPVSS-NLMS) algorithm. As the number of parameters increase they interact with each other during simulation and are difficult to adjust. In NonParametric VSS, only one parameter (forgetting factor) needs to be adjusted all other parameters are fixed. Therefore it provides better stability while simulating.In this algorithm, the step-size is adjusted in such a way that the squared estimation error is reduced at each time instant. 2.4. Variable tap-length algorithm The Variable-Tap-Length algorithm is required to provide faster convergence and smaller steady-state bias of tap-length in the presence of noise [22]. The weight update equation is given by
wKðnÞ ðn þ 1Þ ¼ wKðnÞ ðnÞ þ lðnÞ
ðKðnÞÞ eKðnÞ xKðnÞ ðnÞ;
ð7Þ
lðnÞ is the variable step-size and KðnÞ is the variable tap-
where
ðKðnÞÞ
length, eKðnÞ is the output error. Let
KðnÞ
and eL KðnÞ
eL
ð8Þ
ðnÞ be the a priori error estimate given as
ðnÞ ¼ dðnÞ xTKðnÞ;1:L ðnÞ wKðnÞ;1:L ðnÞ;
ð9Þ
ð10Þ
ð11Þ
where j is the positive leakage parameter used for overcoming the wandering problem, k is the step-size for adaptation of tap-length, and D is the error width that influences the convergence speed ðKðnÞÞ
and bias from the optimal tap-length, and the errors eKðnÞ ðnÞ and ðKðnÞÞ
eKðnÞD ðnÞ are the output errors calculated using (9). The taplength Kðn þ 1Þ to be used in the NLMS filter coefficient update equation at iteration ðn þ 1Þ is given as
bkf ðnÞc; if jKðnÞ kf ðnÞj > q KðnÞ;
ð12Þ
otherwise;
where b:c represents the floor operator and q is a small integer. The problem with fixed value of D is that the tap-length convergence is faster for large value of D and the bias in tap-length is also large. On the other hand, a small value of D can lead to the under modelling problem and slower convergence. Hence, a variable D can provide a compromise between the bias from the optimum tap-length and quick convergence [29]. The energy of the instantaneous error signal is given as
h i2 ðKðnÞÞ e2i ðnÞ ¼ ce2 ðn 1Þ þ ð1 cÞ eKðnÞ ðnÞ
ð13Þ
and
DðnÞ ¼ min Dmax ; B e2i ðnÞ ;
ð14Þ
where the upper bound of DðnÞ is given by Dmax ; c 1 is called as the smoothing parameter and B is a constant. When the error is large during the initial stages of adaptation, DðnÞ increases and provides faster convergence. As the error starts to decrease, the value of DðnÞ also decreases, thus providing a small bias from the optimum tap-length. However, adaptation in the presence of noise is not well addressed in (13) and (14). Let xðnÞ be the zero-mean Gaussian signal, KðnÞ < K opt at the transient adaptation stage and bðnÞ be stationary with zero mean. It is assumed that bðnÞ is uncorrelated to xðnÞ. The vectors wK opt and xK opt ðnÞ can be written as [30]
T wK opt ¼ ½wKðnÞ ðnÞ w
ð15Þ
and
ðnÞT ; xK opt ¼ ½xKðnÞ ðnÞ x
ð16Þ iT
h
iT
¼ wKðnÞþ1 ; wKðnÞþ2 ; . . . ; wK where wKðnÞ ðnÞ ¼ w1 ; w2 ; . . . ; wKðnÞ ; w and opt
ðnÞ ¼ ðxðn KðnÞÞ; xðn KðnÞ 1Þ; . . . ; xðn K opt þ 1Þ T . Using x (10) in (9), we get ðKðnÞÞ T ðnÞw ðnÞ xTKðnÞ v KðnÞ ðnÞ þ bðnÞ; eKðnÞ ðnÞ ¼ x
where 1 6 L 6 KðnÞ and wKðnÞ;1:L ðnÞ and xKðnÞ;1:L ðnÞ are vectors consisting of first L elements of the vectors wKðnÞ ðnÞ and xKðnÞ ðnÞ, respectively. The desired signal dðnÞ is defined as
dðnÞ ¼ xTK opt ðnÞ wK opt þ bðnÞ;
2 2 ðKðnÞÞ ðKðnÞÞ ; kf ðn þ 1Þ ¼ kf ðnÞ j k eKðnÞ ðnÞ eKðnÞD ðnÞ
h
LKðnÞ ðnÞ be the a posteriori error estimate given as
KðnÞ ðnÞ ¼ dðnÞ xTKðnÞ;1:L ðnÞ wKðnÞ;1:L ðn þ 1Þ L
cient vector and bðnÞ is a stationary zero-mean uncorrelated noise, independent of the input signal xðnÞ. For the tap-length to be variable, the pseudo-tap-length concept is as presented in [16] for the FT NLMS algorithm. Let the pseudotap-length be defined as kf ðnÞ.The tap-length update equation is given as [16]
Kðn þ 1Þ ¼
2.3. Variable step-size NLMS algorithm
wðn þ 1Þ ¼ wðnÞ þ
where K opt is the optimal tap-length, xK opt ¼ ½xðnÞ; xðn 1Þ; . . . ; xðn K opt þ 1ÞT ; wK opt ¼ ½w1 ; w2 ; . . . ; wK opt T is the optimum coeffi-
where
ð17Þ
v KðnÞ ðnÞ is the coefficient error vector defined as
vKðnÞ ðnÞ ¼ wKðnÞ ðnÞ wKðnÞ :
ð18Þ
From (14), we have
E½DðnÞ ¼ B E e2i ðnÞ :
ð19Þ
4
S.H. Pauline et al. / Applied Acoustics 159 (2020) 107074
Considering that the correct estimation of the energy of the output error is given by (13), then, from (17) and assuming that xKðnÞ ðnÞ and wKðnÞ ðnÞ are independent of each other [3], we have
h i h i
2 k2 þ r2x E v TKðnÞ ðnÞv KðnÞ ðnÞ ; E e2i ðnÞ ¼ E b ðnÞ þ r2x kw
ð20Þ
where E½x2 ðnÞ ¼ r2x . Furthermore,
E½DðnÞ ¼ B E e2 ðnÞ h i h i 2 k2 þ r2x E v TKðnÞ ðnÞv KðnÞ ðnÞ : ¼ B E b ðnÞ þ r2x kw
ð21Þ
tive filter is to its optimum solution. However, the efficiency of the adaptation rate of DðnÞ is reduced due to the presence of noise, which causes DðnÞ to be large and produces a large bias. Therefore, update
DðnÞ,the auto-correlation between eðKðnÞÞ and KðnÞ ðnÞ
ðKðn1ÞÞ
ðKðnÞÞ
eKðn1Þ ðn 1Þ is used [31]. The time-average estimate of eKðnÞ ðnÞ and
ðKðn1ÞÞ eKðn1Þ ðn
1Þ is given by [32]
ðKðn1ÞÞ gðnÞ ¼ b gðn 1Þ þ ð1 bÞ eðKðnÞÞ KðnÞ ðnÞ eKðn1Þ ðn 1Þ;
ð22Þ
where b varies from 0 to 1 and influences gðnÞ, and the update equation of DðnÞ is given as
DðnÞ ¼ minðDmax ; AjgðnÞjÞ;
ð23Þ
where A is a constant and j:jis the magnitude selected according to the criteria discussed in detail in Section 4. Hence, by using time average estimate of error gðnÞ, the effect of uncorrelated noise is eliminated from the update equation of DðnÞ and it is possible to measure how close the adaptive filter is to the optimum value. In the initial stages of adaptation, jgðnÞjis large-valued, thus giving large values of DðnÞ. When close to optimum, jgðnÞj 0, thus giving small values of DðnÞ. This achieves fast convergence and small steady-state bias simultaneously, the speed of convergence and steady-state bias being unaffected by the presence of noise.
In the present section, we explain the proposed Variable Taplength Non-Parametric Variable Step-Size NLMS (VT-NPVSSNLMS) algorithm with a variable tap-length and a nonparametric variation of step size for reducing the compromise between reduction in convergence-time and reduction in estimation error of the adaptive filter. The VT-NPVSS-NLMS algorithm weight update equation is given as ðKðnÞÞ
wKðnÞ ðn þ 1Þ ¼ wKðnÞ ðnÞ þ lNP ðnÞ eKðnÞ ðnÞ xKðnÞ ðnÞ;
ð24Þ
where wKðnÞ is the adaptive filter coefficient vector and is defined as wKðnÞ ¼ ½w1 ðnÞ; w2 ðnÞ; . . . ; wKðnÞ ðnÞT ; xKðnÞ ðnÞ is the input signal vector ðKðnÞÞ
given as xKðnÞ ¼ ½xðnÞ; xðn 1Þ; . . . ; xðn KðnÞ þ 1ÞT ; eKðnÞ ðnÞ is the output error and lNP ðnÞ is the non-parametric variable step-size [33,34] which is calculated according to the criterion which reduces the square of error at every instant, forcing the condition
ð25Þ
where E½: is the mathematical expectation. The optimal value of lNP ðnÞ is given by the expression [33,34]
lNP ðnÞ ¼ where
dðnÞ;
re > rs
0;
otherwise;
ð27Þ
where f is a small positive number which is used to avoid division
by zero. Furthermore, r2e ðnÞ ¼ E e2 ðnÞ is the power of the error sig
nal, r2s ðnÞ ¼ E s2 ðnÞ is the power of noise signal of the system which may be known or can be estimated and L is a posteriori error estimate. Then, r2e is calculated using the following expression as
re ðnÞ ¼ m r2e ðn 1Þ þ ð1 mÞ e2 ðnÞ;
ð28Þ
where m is the exponential weighting factor whose value ranges between 0 and 1. 4. Performance analysis of proposed algorithm The mean square performance of the proposed algorithm is analyzed in detail in this section. We make the following assumptions: 1. The input xðnÞ is a zero-mean white Gaussian stationary signal, 2. The noise signal bðnÞ is zero-mean stationary and is independent of the input signal, 3. KðnÞ < K opt (optimum tap-length) during transient behaviour, and 4. KðnÞ P K opt at steady state [17]. 4.1. Mean square performance analysis Substituting (17) in (7) and using (18), we get
vKðnÞ ðn þ 1Þ ¼
h
i I lNPN ðnÞ xKðnÞ ðnÞ xTKðnÞ ðnÞ T
ðnÞ w xKðnÞ ðnÞ þ lNPN ðnÞ x
þ
lNPN ðnÞ xKðnÞ ðnÞ bðnÞ;
ð29Þ
where lNPN ðnÞ is the non parametric variable step size. The MSE is defined as
3. Proposed VT-NPVSS-NLMS algorithm
E e2 ðnÞ ¼ E s2 ðnÞ ;
f þ xTKðnÞ xKðnÞ
#
rs ; 1 KðnÞ L þ re ðnÞ
KðnÞ
h i In (21), the term E v TKðnÞ ðnÞv KðnÞ ðnÞ shows how close the adap-
to
dðnÞ ¼
"
1
ð26Þ
2
ðKðnÞÞ vðKðnÞÞ eKðnÞ ðnÞ KðnÞ ðnÞ ¼ E
:
ð30Þ
Substituting (17) in (30), we can rewrite (30) as
h n
oi
2 2 2 T vðKðnÞÞ ; KðnÞ ðnÞ ¼ P min þ rx kw k þ rx tr E v KðnÞ ðnÞv KðnÞ ðnÞ
where Pmin
ð31Þ
h i 2 ¼ E b ðnÞ is the minimum MSE and tr(.) denotes the
trace operation. On multiplying both the sides of (26) by vTKðnÞ ðn þ 1Þ, calculating the expected value and after using our assumptions, we get
E
h
i
vKðnþ1Þ ðnÞv TKðnþ1Þ ðnÞ
¼E
h
i
vKðnÞ ðnÞvTKðnÞ ðnÞ
h i
2 E lNPN ðnÞ r2x E v KðnÞ ðnÞv TKðnÞ ðnÞ h i
þ 2 E l2NPN ðnÞ r4x E v KðnÞ ðnÞv TKðnÞ ðnÞ n h io
þ E l2NPN ðnÞ r4x tr E v KðnÞ ðnÞv TKðnÞ ðnÞ I
k2 I þ E l2NPN ðnÞ r4x kw 2
2 þ 2 E lNPN ðnÞ rx Pmin I: ð32Þ
On taking expectation of (23), we get
S.H. Pauline et al. / Applied Acoustics 159 (2020) 107074
E½DðnÞ ¼ A E½jgðnÞj:
ð33Þ
To evaluate E½jgðnÞj in (32), we can rewrite gðnÞ in (22) as
gðnÞ ¼ ð1 bÞ
n1 X ðKðniÞÞ ðKðni1ÞÞ bi eKðniÞ ðn iÞ eKðni1Þ ðn i 1Þ:
ð34Þ
i¼0
Using the Central limit theorem, we can assume that gðnÞ is a zero-mean white Gaussian sequence. Therefore,
E½jgðnÞj ¼
rffiffiffiffi 2
p
rgðnÞ ðnÞ;
ð35Þ
where
r2gðnÞ ðnÞ ¼ E jg2 ðnÞj ¼ ð1 bÞ2
n1 X n1 X
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð1 bÞ Pmin ; E½Kð1Þ K opt þ 2 A pð1 þ bÞ 2 l0
5
ð44Þ
where the value of A should be [35]
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi p 1 þ b Dð1Þ ð2 l0 Þ : A¼ 2 Pmin 2 1b
ð45Þ
The optimum value of Dð1Þ 0:1 K opt . The value of A influences DðnÞ. The value of A must be selected such that it causes DðnÞ to be large during the initial convergence stage so that as the adaptive system environment changes, the tap-length converges fast to its
h i ðKðniÞÞ ðKðni1ÞÞ ðKðnjÞÞ ðKðnj1ÞÞ bi bj E eKðniÞ ðn iÞ eKðni1Þ ðn i 1ÞeKðnjÞ ðn jÞ eKðnj1Þ ðn j 1Þ
i¼0 i¼0 n1 h i h i X 2 2 ðKðniÞÞ ðKðni1ÞÞ b2i E ðeKðniÞ ðn iÞÞ E ðeKðniÞ ðn i 1ÞÞ : ¼ ð1 bÞ2
ð36Þ
i¼0
The above equation is derived assuming that the algorithm has h i ðKðniÞÞ ðKðnjÞÞ converged and that E eKðniÞ ðn iÞ eKðnjÞ ðn jÞ ¼ 0 for i – j.
optimal value. Also, the selected value of A must be ensure that DðnÞ is small once the steady state is reached.
4.2. Steady-state performance analysis
5. Simulation results and discussion
At steady-state, the terms that are present in the transient analysis do not appear because Assumption 4. Therefore, (31) can be expressed for steady state as
The simulation results were obtained in the MATLAB environment, using a sampling frequency of 8 kHz. For obtaining the room impulse response (see Fig. 2), a 12 12 8 ft3 enclosure was used. Inside the enclosure, an ordinary loudspeaker and an omnidirectional microphone were kept at a distance of 1 ft. The taplength of the adaptive filter used in the AEC framework is set at 1200. A stationary Gaussian zero-mean stochastic signal with unit variance is given as input to the measured room impulse response to obtain the channel response. The noise signal is considered as a white Gaussian zero-mean sequence with a variance of 0.01. A comparative analysis of the proposed algorithm is carried out with the basic NLMS algorithm and the VT-NLMS algorithm with variable error width, in terms of the MSE incurred, convergence and tracking capabilities, echo return loss enhancement (ERLE) and echo suppression for an SNR of 20 dB. The results obtained are averaged for 250 Monte-Carlo runs. Figs. 3(a) and 3(b) present the MSE performance for SNR values of 10 dB and 30 dB, while Fig. 4 depicts the tracking performances of the NLMS, VT-NLMS with variable error width and the proposed algorithm. From both the figures, it can be observed that the proposed non-parametric algorithm, with variable tap-length and variable step-size, exhibits the fastest convergence (at about 215 iterations), followed by the VT-NLMS with variable error-width (at about 450 iterations). The NLMS algorithm exhibits the slowest convergence (about 1000 iterations) due to its constant tap-length. Moreover, it must be noted from Figs. 3(a) and 3(b) that the proposed algorithm incurs the highest MSE among the algorithms under consideration, while the NLMS algorithm incurs the least MSE. However, it can be observed that the difference in the highest and the lowest MSE incurred is only marginal. Further, Fig. 4 depicts the ability of the considered algorithms to track variations of a non-stationary echo path. An enclosure dislocation is done at 1500 iterations. It can be seen from the figure that the proposed non-parametric algorithm outperforms the rest in its ability to adapt to the changes in the environment, followed by the
vðKð1ÞÞ Kð1Þ ð1Þ ¼ P min þ P ex ð1Þ;
ð37Þ
where P ex ð1Þ is the steady-state Excess MSE (EMSE) and is given as [3]
h i Pex ð1Þ ¼ r2x tr E v Kð1Þ ð1Þ v TKð1Þ ð1Þ :
ð38Þ
By substituting (32) in (38), we get
Pex ð1Þ ¼
E lNPN ð1Þ E½Kð1Þr2x Pmin
; 2 E lNPN ð1Þ ðE½Kð1Þ þ 2Þr2x
ð39Þ
where, using the approximation [35] as
2
E l2NPN ð1Þ l ð1Þ
E NPN E lNPN ð1Þ : lNPN ð1Þ E lNPN ð1Þ
ð40Þ
From (35) and (23), we get
E½Dð1Þ ¼ A
rffiffiffiffi 2
p
rgð1Þ ð1Þ;
ð41Þ
where Dð1Þ is the bias incurred in the tap-length solution. Eq. (36) is evaluated at steady state to get
rgð1Þ ð1Þ ¼
sffiffiffiffiffiffiffiffiffiffiffiffi 1 b ðKð1ÞÞ v ð1Þ: 1 þ b Kð1Þ
ð42Þ
Substituting (37), (39) and (42) in (41), we get
E½Dð1Þ ¼ 2 A where write
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð1 bÞ Pmin : pð1 þ bÞ 2 l0
ð43Þ
l0 is the step size. Assuming that E½Kð1Þ is large, we can
6
S.H. Pauline et al. / Applied Acoustics 159 (2020) 107074
Fig. 2. Room impulse response.
Fig. 3. MSE performance of the algorithms under consideration for SNR (a) 10 dB. (b) 30 dB.
S.H. Pauline et al. / Applied Acoustics 159 (2020) 107074
7
Fig. 4. Tracking performance of the algorithms under consideration.
VT-NLMS and the NLMS. The improved performance of the proposed and the VT-NLMS algorithms is due to their ability to dynamically adapt the tap-length to the varying room impulse response. In a non-stationary environment, it is desirable that the adaptive algorithm tracks the statistical environment variations quickly to effectively suppress the echo from the output and a slight increment in the MSE is a small price to pay for the same. Thus, the proposed VT-NPVSS-NLMS algorithm is more preferable as compared to the rest. Fig. 5 depicts the ERLE performance of the considered algorithms. The measure of the amount of echo (in dB) attenuated by the AEC is the ERLE. It can be defined as the ratio of the power of the far-end signal to the power of the residual echo signal after the cancellation process as [3]
ERLE ¼ 10 log10
power of far-end signal : power of residual echo signal
ð46Þ
It can be seen from Fig. 5 that the proposed algorithm results in the highest ERLE much faster than the NLMS and the VS-NLMS algorithm. It must be noted that even though all three of the algorithms attain similar ERLE, the proposed algorithm incurs the least
time to attain the highest ERLE, as compared to the NLMS and the VS-NLMS algorithm. This shows that the proposed algorithm suppresses the effect of echo more efficiently and faster than the algorithms under consideration. Fig. 6 depicts the testing of the proposed algorithm with an acoustic signal. The effect of echo on the desired signal can be clearly seen in Fig. 6(c). On comparing Fig. 6(a) and 6(d), the effectiveness of the proposed algorithm in removing echo from the acoustic signal can be conveniently observed. The enhanced signal waveform in Fig. 6(d) is similar to that of the desired signal waveform in Fig. 6(a), and the echo-effect is attenuated well. 6. Conclusion In this paper, we introduced a novel adaptive algorithm which combines variation in tap-length as well as in step-size. The proposed Variable-Tap-Length Non-Parametric Variable-Step-Size (VT-NPVSS-NLMS) algorithm offered an improved and convenient solution to simultaneous selection of step-size and tap-length selection to obtain fast convergence and a small steady-state MSE, without the need for adjusting too many parameters in an
Fig. 5. ERLE performance of the algorithms under consideration.
8
S.H. Pauline et al. / Applied Acoustics 159 (2020) 107074
Fig. 6. Sound signal waveforms. (a) Desired signal, (b) Echo, (c) Microphone signal, and (d) processed signal from the proposed algorithm.
AEC framework. A comprehensive analysis of the transient and steady-state performance of the proposed algorithm was done. From the computer simulations, it was concluded that the proposed VT-NPVSS-NLMS algorithm outperforms the conventional NLMS and the variable-step-size NLMS algorithm with a fixed tap-length in terms of an increased convergence speed and faster tracking and improved ERLE performance. Further, MSE incurred by the proposed algorithm is comparable to the NLMS and the VS-NLMS algorithm with a fixed tap-length. As a result, it was concluded that the proposed algorithm is more preferable for AEC systems, as compared to the basic NLMS and the VS-NLMS algorithm with a fixed tap-length.
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
References [1] Paleologu C, Benesty J, Ciochina S. Sparse adaptive filters for echo cancellation. Georgia: Morgan and Claypool Publishers; 2010. [2] Tingchan W, Chutchavong V, Benjangkaprasert C. Performance of a robust variable step-size LMS adaptive algorithm for multiple echo cancellation in telephone network. In: Proceeding of SICE-ICASE International Joint Conference, Korea. p. 3173–6. [3] Haykin S. Adaptive filter theory. 3rd ed. New York: Prentice Hall Inc.; 1996. [4] Casco-Sñchez FM, Medina-Ramírez RC, Lopez-Guerrero M. A new variable step-size NLMS algorithm and its performance evaluation in echo cancelling applications. J Appl Res Technol 2011;9(3):302–13.
[5] Kun S, Xiaoli M. A variable step-size NLMS algorithm using statistics of channel response. Signal Process 2010;90(6):2107–11. [6] Zhua YG, Lia YG, Guana SY, Chenb QS. A novel variable step-size NLMS algorithm and its analysis. Proc Eng 2012;29:1181–5. [7] Tianhui W, Liyuan M, Yongjun L, Yonggang D. An improved variable step-size LMS adaptive spectral-line enhancement algorithm and its simulation. In: Proceeding of First International Conference on Pervasive Computing Signal Processing and Applications, PCSPA 2010. China: IEEE; 2010. p. 727–30. [8] Kar A, Swamy M. Convergence and steady state analysis of a tap-length optimization algorithm for linear adaptive filters. AEU – Int J Electron Commun 2016;70:1114–21. [9] Bendoumia Rédha, Djendi M. Variable step-size subband backward bss algorithms for speech quality enhancement. Appl Acoust 2014;86:25–41. [10] Niu Qun, Chen T. A new variable step size LMS adaptive algorithm. Proceedings of IEEE Chinese Control and Decision Conference 2018. [11] Resende L, Haddad D, Petroglia MR. A variable step size NLMS algorithm with adaptive coefficient vector reusing. In: Proceedings of IEEE international conference on electro information technology. [12] Yuan Z, Tao XS. New LMS adaptive filtering algorithm with variable step size. In: Proceedings of IEEE conference on vision, image and signal processing. [13] Kar A, Chandra M. An improved variable structure adaptive filter design and analysis for acoustic echo cancellation. Radio Eng 2015;24:252–61. [14] Riera-Palou F, Noras JM, Cruickshank DGM. Linear equalizers with dynamic and automatic length selection. Electron Lett 2001;37(25):1553–4. [15] Gu Y, Tang J, Cui H, Du W. LMS algorithm with gradient descent filter length. IEEE Signal Process Lett 2004;11(3):305–7. [16] Kar A, Chandra M. Performance evaluation of a new variable tap-length learning algorithm for automatic structure adaptation in linear adaptive filters. AEU – Int J Electron Commun 2015;69:253–61. [17] Zhang Y, Chambers JA, Li Ning, Sayed AH. Steady-state performance analysis of a variable tap-length LMS algorithm. IEEE Trans Signal Process 2008;56 (2):839–45. [18] Kar A, Swamy M. Tap-length optimization of adaptive filters used in stereophonic acoustic echo cancellation. Signal Process 2017;131:422–33. [19] Kar A, Padhi T, Majhi B, Swamy MNS. Analysing the impact of system dimension on the performance of a variable-tap-length adaptive algorithm. Appl Acoust 2019;150:207–15. [20] Wei Y, Yan Z. Variable tap length LMS algorithm with adaptive step size. Circuits Syst Signal Process 2017;16:2815–27.
S.H. Pauline et al. / Applied Acoustics 159 (2020) 107074 [21] Wei Y, Yan Z. Simple variable tap length algorithm for high noise environment. IET Electron Lett 2017;53(5):320–2. [22] Li N, Zhang Y, Zhao Y, Hao Y. An improved variable tap-length LMS algorithm. Signal Process 2009;89(5):908–12. [23] Widrow B. Adaptive signal processing. Upper Saddle River, New Jersey: Prentice Hall Inc.; 1985. [24] Mader A, Puder H, Schmidt GU. Step-size control for acoustic echo cancellation filters – an overview. Signal Process 2000;80(9):1697–719. [25] Kuech F, Kellermann W. Orthogonalized power filters for nonlinear acoustic echo cancellation. Signal Process 2006;86(6):1168–81. [26] Kuhn EV, Kolodziej JE, Seara R. Stochastic modeling of the NLMS algorithm for complex gaussian input data and non-stationary environment. Digit Signal Process 2014;30:55–66. [27] Jeong JJ, Koo K, Koo G, Kim SW. Steady-state mean-square deviation analysis of improved normalized subband adaptive filter. Signal Process 2015;106:49–54.
9
[28] Shi K, Ma X. A variable step-size NLMS algorithm using statistics of channel response. Signal Process 2010;90(6):2107–11. [29] Feuer A, Weinstein E. Convergence analysis of LMS filters with uncorrelated gaussian data. IEEE Trans Acoust Speech Signal Process ASSP-33 1985:222–30. [30] Mayyas K. Performance analysis of the deficient length LMS adaptive algorithm. IEEE Trans Signal Process 2005;53(8):2727–34. [31] Aboulnasr T, Mayyas K. A robust variable step-size LMS-type algorithm: analysis and simulations. IEEE Trans Signal Process 1997;45(3):631–9. [32] Mayyas K. A variable step-size selective partial update LMS algorithm. Digit Signal Process 2013;23(1):75–85. [33] Benesty J, Rey H, Rey Vega L, Tressens S. A non-parametric VSS-NLMS algorithm. IEEE Signal Process Lett 2006;13(10):581–4. [34] Cohen I, Benesty J, Gannot S. Speech processing in modern communication challenges and perspectives. Springer Sci Bus Media 2009;3. [35] Mayyas K, AbuSeba HA. A new variable length nlms adaptive algorithm. Digital Signal Process 2014;34:82–91.