Convergence analysis of the conventional filtered-x affine projection algorithm for active noise control

Signal Processing 170 (2020) 107437

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Convergence analysis of the conventional filtered-x affine projection algorithm for active noise control Jianfeng Guo a,c, Feiran Yang a,b,c,∗, Jun Yang a,b,c,∗ a

Key Laboratory of Noise and Vibration Research, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China c School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing 100049, China b

a r t i c l e

i n f o

Article history: Received 19 May 2019 Revised 1 December 2019 Accepted 16 December 2019 Available online 17 December 2019 Keywords: Active noise control Filtered-x affine projection algorithm Transient behavior Steady-state performance

a b s t r a c t The conventional filtered-x affine projection (CFxAP) algorithm has been proposed for active noise control due to its potential good convergence and moderate computationally cost. Although some work has been done to analyze the convergence performance of the CFxAP algorithm, they usually adopted relatively strong approximations and hence came to inaccurate results especially at the steady-state. In this paper, we propose a new theoretical model for the CFxAP algorithm to address this problem. The recursion of an augmented weight-error vector is constructed, which is adopted for the mean and mean-square performance analysis of the CFxAP algorithm. Both the correlation between the past weight-error vectors and the dependency of weight-error vectors on past noise vectors are fully considered in our theoretical model. In addition, the treatment does not impose any restriction to the signal distributions. Simulation results show that our theoretical results are more accurate than the previous methods. © 2019 Elsevier B.V. All rights reserved.

1. Introduction The filtered-x least mean square (FxLMS) algorithm has been widely used for active noise control (ANC) due to its computational simplicity and ease of implementation [1–3]. The main drawback of the FxLMS algorithm is its slow convergence that is governed by the eigenvalue spread of the covariance matrix of the filtered reference signal [1–7]. To address this problem, the filtered-x affine projection (FxAP) algorithm was proposed to improve the convergence speed [8–10]. The complexity of the FxAP algorithm increases as the projection order for a direct implementation, which is still too expensive for certain applications. However, several fast implementation approaches have been proposed [8–15], which reduce the complexity of the FxAP algorithm significantly. There are two versions of the FxAP algorithm depending on the calculation of the error vector. The first one is referred to as the modified filtered-x affine projection (MFxAP) algorithm, which reconstructs the desired signal to generate the estimated error vector internally [8,9]. The second one is the conventional filtered-x affine projection (CFxAP) algorithm, which builds the error vector with past samples of the error signal directly [10]. The CFxAP algorithm is

∗

Corresponding authors. E-mail addresses: [email protected] (F. Yang), [email protected] (J. Yang).

https://doi.org/10.1016/j.sigpro.2019.107437 0165-1684/© 2019 Elsevier B.V. All rights reserved.

computationally more efficient than the MFxAP algorithm, but the former is developed under the slow-convergence assumption. In the literature, many efforts have been made to analyze the convergence performance of the affine projection (AP) algorithm [16–21]. However, the analysis approaches used in the AP algorithm cannot be extended to the FxAP algorithm directly due to the existence of the secondary path. The transient and steady-state performance of the MFxAP algorithm was analyzed under the umbrella of energy conservation arguments [22,23]. The convergence behavior of the CFxAP algorithm is, however, quite different from that of the MFxAP algorithm due to the calculation of the error vector. The evolution of the mean-square error (MSE) of the CFxAP algorithm was derived to predict the transient behavior in [24], and the steady-state theoretical model of the CFxAP algorithm was provided in [25]. However, the covariance matrix of the a priori error vector was approximated by a diagonal matrix in both the transient and steady-state analysis in [24,25]. In addition, the dependency of weight-error vectors on the past noise was neglected in all the performance analysis of the FxAP algorithms [22–25]. The two aforementioned concerns thus result in an inaccurate theoretical prediction especially when the projection order is large. In this paper, we provide a new theoretical model for the CFxAP algorithm in the context of single-channel ANC systems. Specifically, the update equation for an augmented weight-error vector is established to perform the convergence analysis. The mathematical models are then derived for the mean and mean-square behaviors,

2

J. Guo, F. Yang and J. Yang / Signal Processing 170 (2020) 107437

The CFxAP algorithm offers a significant convergence improvement over the FxLMS algorithm as K increases. 3. Signal model Consider the desired signal d(n) that arises from the linear model [18,21–25]

d (n ) = −uT (n )wo + v(n )

Fig. 1. Block diagram of a feedforward ANC system using the CFxAP algorithm.

where the dependency of weight-error vectors on past noise vectors [21] and the correlation between the past weight vectors are both considered. The closed-form expressions for the steady-state MSE and mean-square deviation (MSD) are obtained. In addition, we do not impose any restriction to the signal distributions. Computer simulations are carried out to demonstrate the effectiveness of the analytical model. Throughout this paper, small boldface letters denote vectors and bold capital letters denote matrices. The notation 0L and IL are the L × L zero and identity matrices, whereas 0L×L(K−1 ) is an L × L(K − 1 ) all-zero matrix. The operator tr( · ) indicates the trace of a matrix. The Euclidean norm and the mathematical expectation are denoted by  ·  and E[ · ], respectively. The notation σ = vec() creates a column vector obtained by stacking the columns of the matrix  on top of one another, while  = vec−1 (σ ) returns a matrix  from the vector σ . The operator  stands for the Kronecker product. 2. Review of the CFxAP algorithm We now briefly review the CFxAP algorithm [10] for a singlechannel feedforward ANC system as presented in Fig. 1. The reference signal x(n) is picked up by a reference sensor, processed by the control filter to generate the anti-noise signal y(n) that drives a secondary source. The error signal e(n) measured by the error sensor represents the acoustical combination of the desired ˆ accounts for an availsignal d(n) and the control signal. Here h able estimate of the true secondary path h. It can be modeled in the time domain as an M-order finite impulse response function ˆ = [hˆ 0 , hˆ 1 , . . . , hˆ M−1 ]T . The filtered-x signal u(n) is obtained by filh tering the reference signal x(n) with the estimate of the secondary path

ˆ u ( n ) = xT ( n )h

(1) + 1 )] T

where x(n ) = [x(n ), x(n − 1 ), . . . , x(n − M is the reference signal vector. Using the above notation, the update equation of the CFxAP algorithm reads [10]

w(n ) = w(n − 1 ) − μU(n )(UT (n )U(n ) + δ IK )−1e(n )

( n )] T

(2)

where w(n ) = [w0 (n ), w1 (n ), . . . , wL−1 denotes the weight vector of the control filter with length L, μ denotes the step size and δ is the regularization parameter that ensures the numerical stability of matrix inverse. The filteredx signal matrix U(n ) = [u(n ), u(n − 1 ), . . . , u(n − K + 1 )] is constructed by collecting the past K filtered-x signal vectors u(n ) = [ u ( n ) , u ( n − 1 ) , . . . , u ( n − L + 1 )] T . It is noteworthy that the desired signal d(n) is unavailable in practical ANC systems, and thus the MFxAP algorithm adapts the filter using an internally generated error vector. Regarding the CFxAP algorithm, the error vector is directly built with past error signals, i.e., e(n ) = [e(n ), e(n − 1 ), . . . , e(n − K + 1 )]T . For K = 1, the CFxAP algorithm reduces to the normalized FxLMS algorithm.

(3)

where wo is the weight vector of control filter at steady state, and v(n) denotes the measurement noise with variance σv2 = E[v2 (n )]. To simplify the discussion, we consider the case in which the control filter w(n) is changing slowly. The order of w(n) and h in Fig. 1 can be commuted, and an almost equivalent output would be produced under the assumption of slow convergence. Assuming ˆ = h, the signal a perfect modeling of the secondary path, i.e., h at the acoustic summing junction from the output of the adaptive filter in Fig. 1 can then be denoted as uT (n )w(n − 1 ). The error signal can be written as

e(n ) ≈ d (n ) + uT (n )w(n − 1 )  (n − 1 ) + v (n ) = −uT (n )w

(4)

= −ea (n ) + v(n ) ˜ (n ) is the difference between the where the weight-error vector w estimated and the steady-state weight vector

 ( n ) = wo − w ( n ) w

(5)

˜ 0 (n ), w ˜ 1 (n ), . . . , w ˜ L−1 (n )]T , = [w and ea (n) is the a priori error

˜ ( n − 1 ). ea ( n ) = uT ( n )w

(6)

This approach in deriving (4) has been already used for the convergence analysis of FxLMS algorithm in [1,26], and also applied to the transient and steady-state analysis of the CFxAP algorithm in [24,25], respectively. Using (4), the error vector used in the CFxAP algorithm can be expanded as

e(n ) ≈ −ea (n ) + v(n )

⎡

⎤

˜ (n − 1 ) + v (n ) −uT (n )w ˜ (n − 2 ) + v (n − 1 ) −u (n − 1 )w ⎢ ⎥ ⎥ =⎢ .. ⎣ ⎦ . T ˜ (n − K ) + v (n − K + 1 ) −u (n − K + 1 )w T

(7)

where ea (n ) = [ea (n ), ea (n − 1 ), . . . , ea (n − K + 1 )]T denotes the a priori error vector and v(n ) = [v(n ), v(n − 1 ), . . . , v(n − K + 1 )]T is the noise vector. As seen from (7), the a priori error vector of the CFxAP algo˜ (n − k ) for rithm depends on the K past weight-error vectors w k = 1, . . . , K, which makes the analysis of the CFxAP much more difficult than that of, for instance, the standard AP and MFxAP algorithms. To well handle this problem, we hence define an augmented weight-error vector which consists of the K past weighterror vectors with length KL

⎡

⎤

˜ (n ) w ˜ (n − 1 ) ⎥ ⎢ w ⎥, ˜ (n ) = ⎢ W .. ⎣ ⎦ . ˜ (n − K + 1 ) w

(8)

and the corresponding filtered-x signal matrix with size K × KL

⎡

uT ( n ) ⎢ 01×L Us ( n ) = ⎢ . ⎣ .. 01×L

01×L uT ( n − 1 ) .. . 01×L

01×L 01×L .. . 01×L

⎤

01×L 01×L ⎥ ⎥. .. ⎦ . T u (n − K + 1 )

(9)

J. Guo, F. Yang and J. Yang / Signal Processing 170 (2020) 107437

Using the definitions in (8) and (9), we rewrite (7) as

˜ (n − 1 ) + v (n ) e(n ) = −Us (n )W

(10)

Subtracting wo from both sides of (2) and using (10), we obtain



˜ (n − 1 ) + v (n ) ˜ (n ) = w ˜ (n − 1 ) + μU(n )R−1 (n ) −Us (n )W w



(11)

where R(n ) = UT (n )U(n ) + δ IK . Note that the left-hand side of ˜ (n ), while the right-hand (11) is the current weight-error vector w ˜ (n − 1 ) and side of (11) involves the previous weight-error vector w ˜ (n − 1 ). We have found that it is more the augmented version W convenient to use the augmented weight vector for the conver˜ (n ) gence analysis. To this end, we present a relation between W ˜ (n − 1 ) and W

˜ ( n ) = QW ˜ (n − 1 ) + P (w ˜ (n − 1 ) ) ˜ (n ) − w W

(12)

of (17) are ignored. This is true for K = 1, but it may not hold for K > 1. To see this, we obtain using (15)

˜ (n − 1 )vT (n )] = E[A(n − 1 )W ˜ ( n − 2 ) v T ( n )] E[W +E[B(n − 1 )]E[v(n − 1 )vT (n )].

˜ ( n − 1 ) = A ( n − 1 )W ˜ ( n − 2 ) + B ( n − 1 )v ( n − 1 ) W ˜ (n − 3 ) = A ( n − 1 )A ( n − 2 )W +A ( n − 1 )B ( n − 2 )v ( n − 2 )

where P = [IL , 0L×L(K−1 ) is the matrix with size LK × L and Q is the LK × LK matrix defined as

IL ⎢ IL ⎢0 L Q=⎢ ⎢. ⎣ .. 0L

0L 0L IL .. . 0L

... ... ... .. . ...

+B ( n − 1 )v ( n − 1 )



⎤

0L 0L ⎥ ⎥ 0 L ⎥. .. ⎥ .⎦ IL

(18)

For K = 1, the noise vector v(n) becomes the scalar v(n), and hence E[v(n − 1 )vT (n )] = 0. For K > 1, because v(n) and v(n − 1 ) have K − 1 elements in common, and then E[v(n − 1 )vT (n )] = 0K . In this case, the correlation between the augmented weight vector ˜ (n − 1 ) and the noise vector v(n) cannot be neglected. AccordW ˜ (n − 1 ) as follows ing to (15), we get the recursion relation for W

]T

⎡

3

=

K−1

˜ (n − K ) A (n − k ) W

k=1

(13)

+B ( n − 1 )v ( n − 1 ) K−1 

+

Substituting (11) into (12), the recursion of the augmented vector readily reads

˜ (n ) = (Q − μPU(n )R−1 (n )Us (n ))W ˜ (n − 1 ) W +μPU(n )R−1 (n )v(n ).

(14)

μPU(n )R−1 (n )

Defining two matrices B(n ) = and A(n ) = Q − B(n )Us (n ), Eq. (14) can be rewritten in a more compact form

˜ ( n ) = A ( n )W ˜ ( n − 1 ) + B ( n )v ( n ). W

(15)

4. Convergence analysis Using the aforementioned signal model, we then investigate the transient and steady-state behaviors of the CFxAP algorithm in this section. To facilitate the theoretical analysis, we introduce two assumptions. A1). The reference signal x(n) and the desired signal d(n) are stationary processes, and the noise v(n) is statistically independent ˜ (n ) is indeof any other signals. A2). The weight-error vector w pendent of the filter-x signal vector u(n). The assumption A2) is referred to as the independence assumption [27,28].

k=2

˜ (n )] = E[A(n )]E[W ˜ ( n − 1 )] , E[W

(16)



A ( n − l ) B ( n − k )v ( n − k ) .

(19)

l=1

˜ ( n − 1 ) v T ( n ) B T ( n )] E[A(n )W





  T T ˜ = E A (n ) A ( n − k ) W ( n − K )v ( n )B ( n ) k=1   +E A ( n )B ( n − 1 )v ( n − 1 )vT ( n )BT ( n )      K−1 k −1  +E A ( n ) A ( n − l ) B ( n − k ) v ( n − k ) vT ( n ) BT ( n )   k=2 l=1    K−1 k −1  =E A ( n − l ) B ( n − k ) v ( n − k ) vT ( n ) BT ( n )  k=1  l=0   K−1 k −1  =E A(n − l ) B(n − k )E[v(n − k )vT (n )]BT (n ) K−1 

k=1

l=0

= σv2 γ ,

(20) where



K−1  k=1

Taking expectation on both sides of (15) and using the two assumptions, we obtain the update equation of the augmented weight-error vector in the mean sense

k −1

Substituting (19) into the second term of the right-hand side of (17) and using the assumption A1), we obtain

γ =E

4.1. Mean convergence





k −1





A(n − l ) B(n − k )T(k )B (n ) T

(21)

l=0

and

T(k ) = E[v(n − k )vT (n )]/σv2 .

(22)

Similarly, the third term of the right-hand side of (17) can be rewritten as

where we have used the relation E[B(n )v(n )] = 0LK×1 .

˜ T (n − 1 )AT (n )] = σv2 γ T . E[B(n )v(n )W

4.2. Mean-square performance

Substituting (20) and (23) into (17) and considering the assumptions A1) and A2), we get

Multiplying both sides of (15) from the right by their respective transposes and taking statistical expectation, we obtain

˜ ( n )W ˜ T (n )] = E A(n )E[W ˜ ( n − 1 )W ˜ T ( n − 1 )] A T ( n ) E[W

˜ ( n )W ˜ (n )] = E[A(n )W ˜ ( n − 1 )W ˜ ( n − 1 ) A ( n )] E[W ˜ ( n − 1 ) v T ( n ) B T ( n )] + E[A(n )W T

T

(17) − 1 ) v T ( n )]

˜ (n In the previous analysis, it is assumed that E[W = 0KL×K , and hence the second and third terms in the right-hand side



+ σv2

T

˜ T ( n − 1 ) A T ( n )] + E[B(n )v(n )W + E[B(n )v(n )vT (n )BT (n )].



(23)



γ + γ T + E[B(n )BT (n )] .

(24)

The vectorization operation is employed for the evaluation of (24), which does not restrict the input to be Gaussian or white [18,22– 25,29,30]. We introduce the following property of the vectorization operator [31], for matrices M,  and N with compatible dimensions





vec(MN ) = NT  M vec().

(25)

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J. Guo, F. Yang and J. Yang / Signal Processing 170 (2020) 107437

We define a K2 L2 × 1 vector z(n) that is obtained by stacking the ˜ ( n )W ˜ T (n )] on top of one another columns of the matrix E[W

T

 ( n ) W ( n )] ) z(n ) = vec(E[W =

˜ L−1 (n − K + 1 )w ˜ L−1 (n − K + 1 )]} . . . , E[w =

(26)

˜ (n ) ] = lim E[w 2

n→∞

{ z0,0 (n ), z1,0 (n ), . . . , zL−1,L−1 (n − K + 1 )]}T

˜ i ( n )w ˜ j (n )]. Applying the vectorization operator where zi, j (n ) = E[w to both sides of (24) and using the property (25), we have

z ( n ) = z ( n − 1 ) + 

z(∞ ) = (IK 2 L2 − )−1 

(36)

The steady-state MSD can be computed using

{E[w˜ 0 (n )w˜ 0 (n )], E[w˜ 1 (n )w˜ 0 (n )], T



Using (35), we readily obtain

(27)

L−1 

zi,i (∞ )

(37)

i=0

Using (32), we could work out the steady-state EMSE of the CFxAP algorithm





lim E[ea (n ) ] = tr Ru PT vec−1 (z(∞ ) )P . 2

n→∞

(38)

5. Simulation results

where

 = E[A(n )  A(n )],

(28)

 = σv2 vec(γ + γ T + E[B(n )BT (n )] ).

(29)

The MSD is defined as the distance between the estimated and the steady-state weight vector, which can be readily computed by

˜ (n ) ] = E[w 2

L−1 

zi,i (n )

(30)

i=0

Notice from (27) that the proposed theoretical model could provide a more accurate prediction than the previous approaches because of two reasons. First, the terms γ and γ T represent the effect of the past noise on the weight-error vector, which was neglected in [22–25]. Also, the covariance matrix of the augmented weight˜ (n ) instead of that of the current weight-error vecerror vector W ˜ (n ) is adopted to fully consider the correlation between past tor w weight-error vectors. We continue to examine the MSE performance of the CFxAP al2 gorithm. Using (4), the MSE ε (n ) = E [e(n ) ] can be expressed as

ε (n ) = εex (n ) + σv

2

(31)

where εex (n ) = E [ea (n ) ] is the excess mean-square error (EMSE). Using assumption A2) and the property tr(MN ) = tr(NM ) [31], the EMSE learning curve of the CFxAP algorithm can be evaluated by 2

   (n − 1 )2 εex (n ) = E  uT (n )w    (n − 1 )w  T (n − 1 )u(n ) = E uT (n )w 

  (n − 1 )w  T (n − 1 ) = tr Ru E w

Computer simulations in the context of a single channel ANC are carried out to verify the effectiveness of the proposed model. A modified version of the secondary path in [24] is used in the following experiments by introducing eight-sample delay, which could model the acoustic delay in practice. The secondary path is modeled by an 16-tap FIR filter h = [0 0 0 0 0 0 0 0 − 0.1 0.2 0.9 − 0.3 0.4 − 0.1 0.1 0.05]T . The steady-state weight vector is randomly generated with 16 taps by a zero-mean white noise sequence. The weight vector is designed to have the same length as the steady-state weight vector, i.e., L = 16. The reference signal x(n) is obtained by filtering a white  Gaussian sequence with a first order autoregressive (AR) system 1 − 0.92 /(1 − 0.9z−1 ). The disturbance signal d(n) is computed according to model (3) with Gaussian noise of the variance σv2 = 0.001. The expectations  and  in (27) are estimated via ensemble averaging. The regularization parameter is set to δ = 10−5 . 5.1. Transient behavior Figs. 2 and 3 show the learning curves of the CFxAP algorithm with the projection order K = 3 and K = 6, respectively. The step size is μ = 0.03. The simulation results are obtained by averaging 200 independent experiments, and the theoretical MSD and EMSE learning curves of the proposed approach are obtained by (30) and (32). The simulation results match the predicted EMSE

(32)

where Ru = E[u(n )uT (n )] denotes the covariance matrix of the filtered reference signal. Using (8), the current weight-error vector ˜ (n ) can be extracted from the augmented one by w

˜ ( n ). ˜ ( n ) = PT W w

(33)

Substituting (33) into (32) yields



  (n )W  T (n ) P εex (n ) = tr Ru PT E W   = tr Ru PT vec−1 (z(n ) )P .

(34)

4.3. Steady-state solution In the previous section, the state recursion (27) has been used to characterize the transient behavior of the CFxAP algorithm. We can further derive the steady-state solution through this relation. Assuming the step-size is chosen to guarantee the stability of the CFxAP, the matrix  should satisfy the relation ρ () < 1, where ρ () denotes the spectral radius of the matrix . At the steady state, the recursion (27) becomes

z ( ∞ ) = z ( ∞ ) +  .

(35)

Fig. 2. Learning curves of the CFxAP algorithm with the projection order K = 3 and step size μ = 0.03. (a) EMSE curves. (b) MSD curves.

J. Guo, F. Yang and J. Yang / Signal Processing 170 (2020) 107437

Fig. 3. Learning curves of the CFxAP algorithm with the projection order K = 6 and step size μ = 0.03. (a) EMSE curves. (b) MSD curves.

MSD (dB)

Simulation Proposed

-30 -40 -50

10-3

MSD (dB)

-10 -20

10-2

10-1

10-2

10-1

10-2

10-1

Simulation Proposed

-40

-10

MSD (dB)

(b)

-30 -50

-20

10-3

(c)

Simulation Proposed

-30 -40 -50

Fig. 5. Steady-state EMSE of the CFxAP algorithm as a function of the step size μ. (a) K = 2. (b) K = 4. (c) K = 6.

(a)

-10 -20

5

10-3

to a level close to a nearly constant EMSE value, e.g., reaching to its steady-state. As a result, the chosen length could ensure that the algorithm can achieve steady state for other step sizes. The experimental results are obtained by averaging 20 independent runs of 10 0 0 samples at the steady state. In Fig. 5, the theoretical steadystate EMSE computed using (47) in [25] is involved for the comparisons. It is observed that the theoretical results from [25] match the simulated ones when the projection order is small, while the theoretical results deviate from the simulated ones as the projection order increases. However, there is a very good agreement between the simulation results and the proposed theoretical values in terms of both the steady-state MSD and EMSE as shown in Figs. 4 and 5. This performance improvement is attributed to two aspects. First, the correlation between the weight-error vector and the noise vector is well considered. As shown in (21), the larger the projection order is, the higher steady-state MSD becomes. Second, the consideration of the cross-correlation between the past weight-error vectors results in a better prediction. We also repeated the experiments in Figs. 2–5 using the acoustic paths measured in a typical one-dimensional duct (not shown here), and obtained a good agreement between simulation results and the proposed theoretical model.

Fig. 4. Steady-state MSD of the CFxAP algorithm as a function of the step size μ. (a) K = 2. (b) K = 4. (c) K = 6.

learning curves well in [24] during the transient regime, but there is a deviation up to 12 dB at the steady state for a high project order K = 6. However, it is observed that the proposed transient model fits almost perfectly to the simulation results for different projection orders, as illustrated in Figs. 2 and 3. 5.2. Steady state behavior Figs. 4 and 5 present the steady-state MSD and EMSE of the CFxAP algorithm as a function of the step size. The step size is chosen from 0.0 0 05 to 0.1, and the projection order is set to K = 2, 4, 6. The length of the reference signal is chosen such that the CFxAP algorithm for the smallest step size μ = 0.0 0 05 converges

6. Conclusion This paper has developed an analytical model for the meansquare performance of the CFxAP algorithm under the assumption of slow convergence, including both the transient and steady-state behaviors. The error vector of the CFxAP algorithm depends on the past K weight vectors, which is different from that of the standard AP algorithm. The covariance matrix of the augmented weight vector was employed to investigate the MSD and EMSE performance. In addition, the dependency of weight-error vectors on past noise vectors was considered to further improve the theoretical results in steady state. Extensive computer simulations demonstrated that the proposed theoretical results match experimental results much better than the previous schemes.

6

J. Guo, F. Yang and J. Yang / Signal Processing 170 (2020) 107437

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. CRediT authorship contribution statement Jianfeng Guo: Methodology, Software, Validation, Investigation, Visualization, Writing - original draft, Writing - review & editing. Feiran Yang: Conceptualization, Data curation, Project administration, Visualization, Writing - original draft, Writing - review & editing. Jun Yang: Data curation, Resources, Supervision, Funding acquisition, Project administration, Writing - review & editing. Acknowledgments We would like to thank the Associate Editor and two anonymous reviewers for their careful review and useful comments, which considerably improved the presentation of the paper. This work was supported by Youth Innovation Promotion Association of Chinese Academy of Sciences under Grant 2018027, the Strategic Priority Research Program of Chinese Academy of Sciences under Grant XDC02020400, IACAS Young Elite Researcher Projects QNYC201812 and QNYC201722, National Key R&D Program of China under Grant 2017YFC0804900, and National Natural Science Foundation of China under Grants 61501449, 11674348, and 11804368. References [1] S.M. Kuo, D.R. Morgan, Active Noise Control Systems: Algorithms and DSP Implementations, Wiley, New York, 1996. [2] S.J. Elliott, P.A. Nelson, Active noise control, IEEE Signal Process. Mag. 10 (4) (1993) 12–35. [3] Y. Kajikawa, W.S. Gan, S.M. Kuo, Recent advances on active noise control: open issues and innovative applications, APSIPA Transa. Signal Inf. Process. 1 (2) (2012) e3. [4] E. Bjarnason, Analysis of the filtered-x LMS algorithm, in: IEEE International Conference on Acoustics Speech and Signal Processing Proceedings, 3, 1993, pp. 511–514. [5] I.T. Ardekani, W.H. Abdulla, Theoretical convergence analysis of FxLMS algorithm, Signal Process. 90 (12) (2010) 3046–3055. [6] C. Hansen, S. Snyder, X. Qiu, L. Brooks, D. Moreau, Active Control of Noise and Vibration, second ed., CRC Press, 2012. [7] J. Lorente, M. Ferrer, M. de Diego, A. Gonzalez, L. Fuster, The frequency partitioned block modified filtered-x NLMS with orthogonal correction factors for multichannel active noise control, Digit. Signal Process. 43 (2015) 47–58. [8] S.C. Douglas, The fast affine projection algorithm for active noise control, in: Asilomar Conference on Signals, 2, 1995, pp. 1245–1249. [9] M. Bouchard, Multichannel affine and fast affine projection algorithms for active noise control and acoustic equalization systems, IEEE Trans. Speech Audio Process. 16 (4) (1999) 13–41.

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