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An optimal variable step-size affine projection algorithm for the modified filtered-x active noise control
Signal Processing 114 (2015) 100–111
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An optimal variable step-size affine projection algorithm for the modified filtered-x active noise control Ju-man Song a, PooGyeon Park b,n a b
Division of IT Convergence Engineering, Pohang University of Science and Technology, Pohang, Republic of Korea Department of Electronic and Electrical Engineering, Pohang University of Science and Technology, Pohang, Republic of Korea
a r t i c l e in f o
abstract
Article history: Received 14 July 2014 Received in revised form 5 February 2015 Accepted 7 February 2015 Available online 5 March 2015
This paper introduces an optimal variable step-size affine projection algorithm for the modified filtered-x active noise control systems. First, the recursion form of the error covariance from the tap weight update equation is constructed, not ignoring the dependency between the estimation error and the secondary noise signal. Such consideration has not been concerned previously for the analysis of the modified filtered-x affine projection algorithm. Second, a recursion form of the mean square deviation is derived from that of the error covariance. From the recursion form, an optimal step size is decided to get the fastest convergence rate. Both the recursion forms of the mean square deviation and the optimal step size require scalar additions and multiplications that do not contribute to the overall complexity seriously. The simulation results on the active noise control environments show both fast convergence rate and low steady-state error. & 2015 Elsevier B.V. All rights reserved.
Keywords: Adaptive filters Active noise control (ANC) Affine projection algorithm (APA) Filtered-x affine projection (FxAP) Optimal step size Mean-square deviation (MSD)
1. Introduction The goal of active noise control (ANC) is to cancel out the unwanted disturbance signal using the acoustic signal. The acoustic signal is generated from controllable secondary source. With feedback or feed forward control using electrical signals, the ANC system generates acoustic signals for destructive interference at the sound fields [1]. The unwanted disturbance signals for destructive interference have long wavelengths with low frequency. The low frequency noise is hard to be suppressed by conventional passive ways. Thus, the method named as the ANC is developed to efficiently remove the acoustic noise. In ANC systems, the least mean squares (LMS) algorithm has been used as the form of filtered-x LMS algorithm (FxLMS), for low cost and complexity. To improve the performance, the FxLMS algorithm is modified by various ways: the generalized FxLMS recursion [2], the convex algorithm [3],
n
Corresponding author. Tel.: þ 82 54 279 2238; fax: þ 82 54 279 2903. E-mail address: [email protected] (P. Park).
http://dx.doi.org/10.1016/j.sigpro.2015.02.005 0165-1684/& 2015 Elsevier B.V. All rights reserved.
and variable step-size algorithm [4]. However, a drawback of the FxLMS algorithm is that its convergence rate gets worse in correlated input environments. Typically, due to the secondary path, the ANC system shows the similar effect with correlated inputs. The effect is caused from the pre-filtered input source to the adaptive filter to update coefficient equation, although the input signal to the copy of the adaptive filter is not filtered to the estimate of the secondary path. Thus the FxLMS algorithm shows poor convergence rate at the ANC system. To overcome the weak point of the FxLMS algorithm for the ANC system, the affine projection algorithm (APA) has been used as the filtered-x affine projection (FxAP) algorithm [5] and modified filtered-x affine projection (MFxAP) algorithm [6]. The performance of FxAP algorithm is shown by analyzing the transient and the steady states [7,8]. Each literature analyzes the transient analysis of the conventional FxAP (CFxAP) algorithm, and the steady-state mean square performance of both the CFxAP algorithm and the MFxAP algorithm. Both analyses use the energy conservation relation, which has been already used to analyze the mean-square performa nce of a family of APA [9]. They show good agreement with
J. Song, P. Park / Signal Processing 114 (2015) 100–111
simulations in the ANC system environments. However, those analyses ignore the dependency between the weight-error vector and past noise. Additionally, using the energy conservation relation is too complex to apply at ANC systems for realtime algorithms. On the other hand, a different approach was presented to analyze the mean-square deviation (MSD) of APA, considering the dependency between the weight-error vector and past noise [10]. It constructs the propagation model of the error covariance having lower complexity and showing great simulation result on analysis in better agreement than other analyses. Based on the analysis, the optimal step size is derived to get the best performance [11]. However, there are just few literatures of improving the MFxAP algorithm to get fast convergence rate and low steady state error [12], compared to APA [13– 15]. Thus, to obtain those requirements simultaneously, in this paper, an optimal step size of the MFxAP algorithm is suggested, applying the analysis of APA [10]. This paper uses the optimal step size algorithm of the APA [10]. The proposed algorithm is developed by analyzing the MSD of the basic coefficient update equation of the MFxAP algorithm, which is different from that of the APA. It considered the difference between the APA and the MFxAP algorithm, including the pre-filtered input from the estimate of the secondary path. Full theoretical justification will be provided for the superior performance of the proposed algorithm, with some different proofs from [10] for better way. The nonstationary condition, which is often ignored and is essential for real implementation of the ANC, is also considered. Furthermore, efficiency consideration is also applied to the proposed algorithm to reduce the computational complexity. This paper is organized as follows. Section 2 describes the system of the FxAP algorithm in the ANC environments. Section 3 presents a recursion form of the error covariance considering the dependency between the weight-error vector and the secondary noise vector for the MFxAP algorithm. Based on the recursion form, Section 4 analyzes the MSD and constructs the optimal step size of the MFxAP algorithm. Non-stationary consideration is presented in Section 5, and finally, Section 6 shows some simulation results verifying the performance and properties of the proposed optimal step size algorithm. 2. The filtered-x APA For the ANC systems, each structure of the CFxAP algorithm and the MFxAP algorithm is depicted in Figs. 1 and 2, respectively. First, the primary noise signal, which is unwanted signal, passes through the unknown primary path. After that, the adaptive filter output yi is added to the passed signal and it goes through the secondary path h. The secondary path is considered as a finite impulse response (FIR) filter of different lengths from the primary path. The primary path is determined by assuming that the algorithm converges to the optimal solution wo A Rn1 . Thus, the desired signal di which has to be canceled out is given by di ¼ Vi wo . In practice, however the secondary noise signal should be considered, and the disturbance signal is defined with the projection order M Z 1 as di ¼ VTi wo þ ri ;
ð1Þ
101
where Vi 9 ½vi vi 1 … vi M þ 1 A RnM ;
ð2Þ
vi 9 ½vi vi 1 … vi n þ 1 T A Rn1 ;
ð3Þ
and vi is a signal passed through the estimate of secondary ^ For the estimation of secondary path h^ A RL1 as vi ¼ xi nh. path, it is assumed to be done perfectly in this paper. The secondary noise signal ri A RM1 is a zero mean white 2 Gaussian noise vector with variance σr . The basic coefficient update equation of FxAP can be written with the following recursion formula: 1 ^ iþ1 ¼ w ^ i μi Vi VTi Vi w ei ; ð4Þ ^ i A Rn1 is the impulse response of the adaptive where w filter, and μi A ð0; 1 is a variable step size. For the use of ei , the CFxAP algorithm uses samples of the error signal ei, at the error microphone, because it cannot use the disturbance signal di. However, the MFxAP algorithm can reconstruct the disturbance signal and the error signal by using the output of the estimate of secondary path. Thus, in the MFxAP algorithm, the error vector ei is given by ^ i: ei ¼ di þVTi w
ð5Þ
The plus sign after di means the addition at a sensor like a microphone, and the summation is not an electrical sum but an acoustic sum, so Eq. (5) can be used in the MFxAP algorithm, but not in the CFxAP algorithm. On the other hand, the MFxAP algorithm recovers the desired signal from the error signal using the output from the secondary path estimator after the adaptive filter, so it can use Eq. (5). In this paper, only MFxAP algorithm is handled, because it shows better convergence rate than CFxAP algorithm. 3. Augmented modeling of the modified filtered-x APA From the tap weight update equation, the recursion ~ i ¼ wo w ^ i is formulated as form of the tap error vector w 1 ~ iþ1 ¼ w ~ i þ ri ; ~ i þ μi Vi VTi Vi ð6Þ w VTi w and let the transition matrix be 1 Φðiþ 1; iÞ ¼ In μi Vi VTi Vi VTi ; where Ia is an identity matrix in R matrix has two properties such as
Φði; iÞ ¼ In ;
Φði; jÞ ¼ Φði; kÞΦðk; jÞ
ð7Þ aa
. Then, the transition ð8Þ
With the transition matrix, the recursion form of tap error vector can be rewritten as 3" " # 2 # Z 0 ri þ 1 r r 1 4 5 i þ iþ1 ¼ T ~ iþ1 μi V i V i V i Φði þ 1; iÞ w~ i w 0 h
i
ð9Þ MM
, and 0 is the zero matrix of where Z ¼ IM 1 A R appropriate dimensions. h i ~ Ti w ~i The MSD matrix at each iteration i is defined as E w 0
0 0
¼ TrðPi Þ, where EðÞ is the notation of expectation. Pi is the covariance matrix of tap error vector which can be
102
J. Song, P. Park / Signal Processing 114 (2015) 100–111
Fig. 1. The block diagram of CFxAP.
Fig. 2. The block diagram of MFxAP.
determined as fvj j0 r jr i 1g: "
"
Ri
Si
STi
Pi
Ri
Si
STi
Pi
#
#
follows
with
a
given
0" #" #T 1 ri ri A 9 E@ ~i ~i w w 8 0" 19 #" #T < = ri ri @ jV i 1 A ¼E E ~i ~i w w : ;
set
Vi 1 9
ð10Þ
ð11Þ
"
Ri
Si
STi
Pi
#
02 ¼ E @4
Ri T
Si
Si Pi
31 5A:
ð12Þ
~i Considering the dependency between the estimation error w and the noise signal ri , the recursions of R i , S i and P i with variable step size μi are obtained as Theorem 1. Theorem 1. The recursion forms of the covariance matrices R i , S i , and P i are R i þ 1 ¼ ZR i ZT þ σ 2r a1 aT1
ð13Þ
J. Song, P. Park / Signal Processing 114 (2015) 100–111
1 S i þ 1 ¼ ZS i Φði þ 1; iÞ þ μi ZR i VTi Vi VTi ;
P i þ 1 ¼ Φði þ 1; iÞP i Φ ðiþ 1; iÞ þ μ2i Vi VTi Vi T
ð14Þ 1
R i VTi Vi
1
1 T þ μi Vi VTi Vi S i Φ ðiþ 1; iÞ 1 T VTi ; þ μi Φði þ 1; iÞS i VTi Vi
VTi
ð15Þ
Proof. See Appendix A. If we assume that i Z M, covariance matrices R i and S i can have closed form for any input signal in (13) and (14). In the following corollaries, closed forms are shown.
Ri ¼ σ
2 r IM
vTa vb nEðva vb Þ ¼ ¼ 0: n-1ð J va J J vb J Þ nσ 2v
ð19Þ
Even though the input signal to the adaptive filter of the ANC system is correlated by the estimate of the secondary path, Assumption 1 can be used, because of the self-whitening effect of the APA [16]. To reduce mathematical complexity, from Assumption 1, the following lemma is obtained. Lemma 1. For integers i; jZ 0, Under Assumption 1, it holds that 1 Φðiþ j þ1; i þ1ÞVi VTi Vi ZjT j1
σr is available, we have
¼ ∏
k¼0
ð16Þ
,.
use vTa vb =ð J va J J vb J Þ C 0, because lim
where a1 is vector, whose first element is 1 and the others are 0's in Rn1 .
Corollary 1. For i ZM, if
103
1 1 μi þ j k Vi þ j VTiþ j Vi þ j Zj ZjT
ð20Þ
with probability being one when n-1.
Proof. The proof is omitted.
Proof. See Appendix C.
Corollary 2. For i ZM, it holds that
Thus, for the ANC systems, the recursion form of the covariance matrix P i can be reorganized as the following theorem.
S ¼ σ 2r
M 1 X
μi m Zm VTi m Vi m
1
VTi m
m¼1
Φ ði þ1; i m þ 1Þ: T
ð17Þ
Theorem 2. With Assumption 1 and Lemma 1, the covariance matrix P i has the recursion form for i ZM as P i þ 1 ¼ Φði þ 1; iÞP i Φ ðiþ 1; iÞ 1 1 Gi VTi Vi VTi ; þ σ 2r Vi VTi Vi T
Proof. See Appendix B. Using Corollaries 1 and 2, (15) can be rewritten as 2 T P i þ 1 ¼ Φði þ 1; iÞP i Φ ðiþ 1; iÞ þ σ 2r μ2i Vi VTi Vi VTi þ σ μi 2 r
M 1 X
μ
T i m V i Vi Vi
1
Z
where Gi ¼ μ2i Z0 Z0T
1 m1 MX þ 2 μi μ2i μi m Zm ZmT ∏ 1 μi j :
m
m¼1
1 T VTi m Vi m VTi m Φ ði þ 1; i m þ 1Þ þ σ 2r μi
M 1 X
m¼1
μi m Φði þ1; i m þ 1ÞVi m
m¼1
1 1 VTi m Vi m ZmT VTi Vi VTi :
ð21Þ
ð18Þ
ð22Þ
j¼1
Proof. Lemma 1 , for the ANC systems, (20) is rewritten with different indices as 1 1 Φðiþ 1; i m þ 1ÞVi m VTi m Vi m ZmT VTi Vi VTi m1
¼ ∏
j¼0
1 1 1 μi j Vi VTi Vi Zm ZmT VTi Vi VTi ð23Þ
4. MSD analysis and optimal step size of the modified filtered-x APA The recursion formula of the tap error vector covariance matrix is the most important factor among augmented modeled matrices. However, (18) is too complex to use. Thus, it has to be simplified to analyze MSD of the MFxAP algorithm, and the following assumption is used. The assumption was used in [11], and we modify it for the MFxAP algorithm. Assumption 1. For arbitrary values of a and b such that aab and a; bZ 0, the input vectors satisfy jvTa vb j{ J va J J vb J , where J J denotes the Euclidean norm. Then, n-1, for a ab, it has probability of one to
Substituting (23) into (18) results in (21).
□
To reach our goal, making recursion form of MSD for the MFxAP algorithm can be achieved by taking the trace on both sides of (21): 1 T Tr P i þ 1 CTrfΦ ði þ1; iÞΦði þ 1; iÞP i g þ σ 2r Tr VTi Vi Gi 1 VTi P i ¼ Tr P i 2μi μ2i Tr Vi VTi Vi 1 Gi ; þ σ 2r Tr VTi Vi
ð24Þ
where Tr ðÞ is the notation of tracing matrix. At this point, for easy derivation, following lemma is used.
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J. Song, P. Park / Signal Processing 114 (2015) 100–111
1 1 M m Tr M T m mT g ðmÞ ¼ Tr E Vi Vi ; Z Z C M n 2
Lemma 2. If the projection order is sufficiently high, we can assume that 1 M Tr Vi VTi Vi VTi P i C Tr P i : ð25Þ n
g ¼ ½g ð1Þ g ð2Þ … g ðM 1ÞT ; 3 μi 1 6 7 1 μi 1 μi 2 6 7 7; μi ¼ 6 6 7 ⋮ 4 5 1 μi 1 … 1 μi M þ 2 μi M þ 1
ð32Þ ð33Þ
2
Proof. From [17, Lemma II.1], n X k ¼ M0
M 1 X λk P i rTr Vi VTi Vi VTi P i r λk P i
ð26Þ
k¼1
and M A RMM is the Mth-order autocorrelation matrix of vi. Two matrices g and μi are used for easy manipulation. Additionally, μi can be recursively obtained as follows: μ μi þ 1 ¼ 1 μi ZM 1 μi þ i ; ð35Þ 0 h i where ZM 1 ¼ IM0 2 00 A RðM 1ÞðM 1Þ . To obtain the optimal step size, the partial differential with respect to μi is done to both sides of (30), and the result is obtained as
where M 0 ¼ n M þ 1 and λk ðÞ is the kth largest eigenvalue of the matrix. Each upper and lower average of eigenvalues is defined respectively as
λ Pi 9
M 1 X λ P ; Mk¼1 k i
λ Pi 9
n 1 X λ P : M k ¼ M0 k i
Then, substituting (27) into (26) results in 1 M λ P i r Tr Vi VTi Vi VTi P i r Mλ P i
ð27Þ
M ∂ Tr Pi þ 1 ¼ 2 þ 2μi TrðPi Þ ∂μi n þ2σ 2r fμi g ð0Þ þ 1 2μi αi g:
ð28Þ
Whenthe order is increased, λ P i -Tr P i =n projection and λ P i -Tr P i =n. Thus, by the Squeeze theorem, the lemma is concluded. □
μi ¼
where
m¼1
m1
μi m gðmÞ ∏ 1 μi j ¼ gT μi ;
M TrðPi Þ nσ 2r αi : M TrðPi Þ þnσ 2r g ð0Þ 2nσ 2r αi
ð31Þ
j¼1
Table 1 Algorithm summary. Initialization :
^ 0 ¼ 0; μ0 ¼ ½1 0T , and TrðP0 Þ. w
Update the vectors :
xi ¼ ½xi xi 1 ⋯ xi L þ 1 T , ^ v ¼ xT h, i
i
vi ¼ ½vi vi 1 ⋯ vi n þ 1 T , Vi ¼ ½vi vi 1 ⋯ vi M þ 1 , ei ¼ ½ei ei 1 ⋯ ei M þ 1 T , ^ i, d^ i ¼ ei VTi w ^ . e^ ¼ d^ þ VT w i
Step-size updates :
Filtered-x AP : MSD update :
i
i
i
αi ¼ gT μi , M TrðPi Þ nσ 2r αi , M TrðPi Þ þ nσ 2r gð0Þ 2nσ 2r αi μi μi þ 1 ¼ 1 μi ZM 1 μi þ . 0 1 ^ i μi Vi VTi Vi ^ iþ1 ¼ w ei . w 2 M Tr Pi þ 1 ¼ 1 2μi μi TrðPi Þ þ σ 2r fμ2i gð0Þ þ 2 μi μ2i αi g n μi ¼
ð37Þ
If l o 0, then μl ¼ 1. For the case of unknown initial MSD, it is 2 recommended to use σ 2y =λ M , where σy is the variance of adaptive filter output yi and λ M is defined for the maximum eigenvalue of the filtered input correlation matrix M [18]. The proposed algorithm is summarized in Table 1. From the point of view of complexity aspect, the proposed algorithm could be considered having a highly computational cost, because of its sophisticated procedure to produce and prove. However, from Table 1, additional Oð2MÞ multiplies per iteration are required for step size and MSD updates of proposed algorithm. Compared to the complexity of the MFxAP such that 2L þ2nM þ 2n þ nM2 þ M 2 þO M 3 =2
Taking the expectation on both sides of (29) from (12), the final recursion form of the MSD is derived as M Tr Pi þ 1 C 1 2μi μ2i TrðPi Þ þ σ 2r fμ2i g ð0Þ þ 2 μi μ2i αi g; n ð30Þ
M 1 X
ð36Þ
The optimal step size μi is finally decided for the fastest convergence iteration i. It is obtained by rate at each solving ∂Tr Pi þ 1 = ∂μi ¼ 0 as
With Lemma 2, substituting (25) into (24) leads to 1 M Tr P i þ σ 2r Tr VTi Vi Tr P i þ 1 C 1 2μi μ2i Gi : n ð29Þ
αi ¼
ð34Þ
J. Song, P. Park / Signal Processing 114 (2015) 100–111
multiplies [19], the proposed algorithm makes few contribution to the overall complexity. 5. Analysis on both recursion forms of the step size and the MSD For the proposed algorithm, the convergence can be guaranteed by analyzing both the optimal step size and the MSD recursion form. From the simulation, the tendency of the optimal step size is shown in Fig. 3. From Fig. 3, the optimal step size varies from 1 to 0.0002484, and it shows the boundary as 0 r μi r 1. The boundary of the optimal step size can be proved by analyzing it. To analyze the optimal step size, μi, let us rewrite (37) as
μi ¼
ai ai þ bi
ð38Þ
where
¼ TrðPi 1 Þ
105
a2i 1 ai 1 þbi 1
ð43Þ
a2i 1 n n a þ σ2α M i 1 M r i 1 ai 1 þ bi 1 nn o n ai 1 ðai 1 þ bi 1 Þ ai 1 þ σ 2r αi 1 ¼ M ai 1 þbi 1 M
TrðPi Þ ¼
¼
a2i 1 n M n n þ μ b þ σ2α M ai 1 þ bi 1 M i 1 i 1 M r i 1
¼
a2i 1 n M n n 1 μi 1 σ 2r αi 1 þ μ σ 2 g ð0Þ þ M ai 1 þ bi 1 M i 1 r M
a2i 1 n M n þ σ 2 1 μi 1 αi 1 þ μi 1 g ð0Þ M ai 1 þ bi 1 M r n Z σ 2r αi : ð44Þ M ¼
Thus,
M ai ¼ TrðPi Þ σ 2r αi ; n
ð39Þ
TrðPi Þ
bi ¼ σ 2r g ð0Þ σ 2r αi :
ð40Þ
then by mathematical induction, ai Z0, for i ZM. Consequently, from (38), the theorem is proved. □
Considering both signs of ai and bi, the boundary of the optimal step size can be shown by the following theorem. Theorem 3. If l o M, then μl ¼ 1, then the optimal step size, μi, has boundary as 0 r μi r 1
ð41Þ
Proof. As the first step, let us assume that 0 r μj r1, for j ri 1, then aj Z 0, and bj Z0. Thus, from (39), αi 1 rgð0Þ, and from (31) to (34), αi ¼ gT μi ¼ 1 μi 1 gT Zμi 1 þ μi 1 gT e1 r 1 μi 1 αi 1 þ μi 1 g ð0Þ r 1 μi 1 g ð0Þ þ μi 1 g ð0Þ ¼ g ð0Þ
ð42Þ
Thus, by mathematical induction, bi Z0, for iZ M. From (30) and (38), TrðPi Þ CTrðPi 1 Þ 2μi 1 ai 1 þ μ2i 1 bi 1
n 2 n σ α ¼ a Z0; M r i M i
ð45Þ
From Theorem 3, the stability on the optimal step size is proved, and furthermore, the following property is induced. Property 1. For integers i ZM, the MSD has decreasing property, and the fastest rate is 1 M=n , with the following recursion form:
M Tr Pi þ 1 C 1 TrðPi Þ þ σ 2r g ð0Þ: ð46Þ n when μi ¼ 1. Proof. From (43), and ai Z 0; bi Z 0, a2 Tr Pi þ 1 TrðPi Þ ¼ i r 0: ai þ bi Then the property is proved.
ð47Þ □
From the above property, it guarantees the stability of the proposed algorithm that it will not diverge with the proposed algorithm.
1
6. Efficiency consideration for MFxAP Algorithm
0.9 0.8
step size (µ)
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
1
2
3
4
iteration
Fig. 3. The step size variation.
5
6 4 x 10
Although the proposed algorithm makes few contribution to the overall complexity, the projection order for the proposed algorithm is properly selected according to the degree of the whitening effect. If the correlation property of the input signal is high, then the projection order should be selected high value which increases the overall computational complexity. In the field of ANC environments of the MFxAP algorithm, the secondary path makes the input signal of the adaptive filter to be colored. Thus, when the secondary path gives high correlation property to the input signal, the suitable projection order for the proposed algorithm would be high, and the computational complexity would be increased. To reduce the complexity, in such environments, the white assumption of the MFxAP
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J. Song, P. Park / Signal Processing 114 (2015) 100–111
σ 2e;i is calculated as
in Lemma 2 can be modified as
1 M Tr P i : VTi P i C Tr Vi VTi Vi nQ
ð48Þ
The difference of (25) and (48) is the parameter Q. The role of the parameter Q is suppressing the convergence speed of the proposed algorithm, to compensate Assumption 1. The parameter Q and Assumption 1 together are applied to the proposed algorithm for the white assumption. The white assumption of Lemma 2 is applied to the convergence stage of the MSD learning curves. Thus, as the parameter Q is increased to high value, the prediction accuracy of the convergence speed would be also increased. The effect of the parameter Q is depicted in Fig. 4, with a projection order of 15. When the parameter Q is 1, the performance of the proposed algorithm is poor. However, as the parameter Q is increased from 1, the performance of the proposed algorithm is also increased, and when the value of the parameter Q is equal to and higher than 50, the proposed algorithm shows the best performance. Through the simulation result, we can know that the parameter Q can be chosen with high value.
2
σ 2e;0 ¼ E e20 ¼ Ef d0 þ vT0 w^ 0 g ¼ σ 2d ;
ð49Þ
σ 2e;i þ 1 ¼ ασ 2e;i þ ð1 αÞe2i ;
ð50Þ
2 d
where σ is the variance of di. The forgetting factor properly selected as
α is
1 ; nK
ð51Þ
α ¼ 1
where K is a positive integer value, because as the length of weight vector n increases, the forgetting factor should also be increased. The positive integer value K is determined by considering the input signal property. If the magnitude of input correlation is high, K should be also high. To check up the system change with an error signal, the covariance of error at steady state is the impact factor. The threshold of an error signal can be determined as the convergence value of ei covariance as n 2 o ~ i þr i lim E e2i ¼ lim E vTi w ð52Þ
i-1
i-1
lim E e2i C σ 2v TrðP1 Þ þ σ 2r 9 e2th ;
i-1
7. Non-stationary consideration For pursuing completely optimal step-size MFxAP algorithm, the non-stationary case of primary path filter should be considered. As the ANC system is designed for acoustic environment, the acoustic path can be interrupted by any obstacles. For example, if anybody passes through between a monitor speaker and a microphone, the acoustic path will be seriously changed. In such an unpredicted sudden change case, suitable algorithms should be constructed. One of them is the reset algorithm aiming instantaneous system change [18]. Through restarting count of iteration number for step size when the unknown weight vector is changed, the reset algorithm can reach the goal. To check the change of the 2 unknown weight vector, there is need of ei , with time average ^ o ¼ 0, value. As the optimal step-size algorithm starts with w
ð53Þ
for a given input signal. From (30), (53) can be rewritten as ð54Þ e2th C 1 þ ζ i σ 2r where
ζi ¼
nσ 2v μ2i g ð0Þ þ 2 μi μ2i αi : M 2μi μ2i
ð55Þ
ζ i seems to be complex for calculation, but almost terms in it can be got from (30) which already has to be calculated. Using (54), a decision point of reset is determined as σ 2e;i 4 γ e2th , when the unknown weight vector is suddenly changed. The design parameter γ has to satisfy
γ 4 E e20 =e2th C
σ 2v TrðP0 Þ : σ 2r 1 þ ζ i
ð56Þ
The reset algorithm for non-stationary consideration is summarized in Table 2.
5 0
8. Simulation
−5
−25
At the environment of ANC systems, some simulations have been carried out. The primary path and the secondary path consist of two ways. First, the primary path is set as unknown 32 tap length FIR filter which is randomly generated and the secondary path is modeled as the following 8-coefficient FIR filter which is given by [7]
−30
H ðzÞ ¼ 0:1 þ 0:2z 1 þ 0:9z 2 0:3z 3
MSD(dB)
−10 −15 −20
þ0:4z 4 0:1z 5 þ 0:1z 6 þ 0:05z 7 :
−35 −40
0
0.5
1
1.5
2 iteration
2.5
3
3.5
4 4
x 10
Fig. 4. MSD learning curves of the proposed algorithm with various parameter Q for a white signal. The projection order is 15. Q is varying from 1 to 50.
ð57Þ
The unknown primary path is generated at each trial with J wo J ¼ 1. The second way of paths is using the models P(z) and S(z), which are similar to the data of the disk included in [20]. The length of the primary path is 145, and that of the secondary path is 51. The secondary path has to filter xi at each iteration to generate pre-filtered input
J. Song, P. Park / Signal Processing 114 (2015) 100–111 5
Table 2 Reset algorithm for system change.
−5
2 e^ th ¼ ð1 þ ζ i Þσ 2r ,
−10
2
MSD(dB)
σ 2e^ ;i þ 1 ¼ ασ 2e^ ;i þ ð1 αÞe^ i if
M=1 M=2 M=4 M=6 M=8 M=10 M=12 M=14 M=16
0
for each i do
2 σ 2e^ ;i 4 γ e^ th then μi ¼ ½1 0T ; TrðPi Þ ¼
107
TrðP0 Þ
endif
−15 −20 −25
1 ^ i μi Vi VTi Vi ^ iþ1 ¼ w e^ i w
−30
endfor
−35 −40 −45 0
2000
4000
6000
8000
10000
12000
iteration
Fig. 5. MSD learning curves of the proposed algorithm with various projection orders and n¼ 32 for a white signal. The step size μ is 0.04.
5 M=1 M=2 M=4 M=6 M=8 M=10 M=12 M=14 M=16
0 −5 −10 MSD(dB)
vector vi . A simulation result of the proposed algorithm with paths of disk model [20] is already shown in Fig. 4. A Gaussian noise with N ð0; 1Þ is employed as the reference signal. The disturbance signal di is generated through (1). For the secondary noise, the variance is set as σ 2r ¼ 10 3 . The simulation results are obtained through 50 independent trials. The initial MSD value is set as TrðP0 Þ ¼ 1, and the initial weight of adaptive filter is set ^ 0 ¼ 0. The Mth-order input autocorrelation matrix of as w vi is numerically estimated from fvi j0 r i r1:2 104 g. Assumption 1 and the approximation (25) can be used, because the APA has self-whitening nature for colored reference signals [16], thus the MFxAP algorithm which uses the APA type can be modified with the proposed algorithm. However, as the MFxAP algorithm makes colored property to the input signal, a suitable projection order M should be chosen, considering the input correlation. Thus, the first simulations have done with various projection orders under the environments of a white signal, an AR signal and an ARMA signal. Each colored input signal is generated through
−15 −20 −25 −30 −35 −40 0
2000
4000
6000
8000
10000
12000
iteration
G1 ðzÞ ¼
1 ; 1 0:95z 1
ð58Þ
G2 ðzÞ ¼
1 þ0:5z 1 þ0:81z 2 : 1 0:59z 1 þ 0:4z 2
ð59Þ
The projection order M is increased from 1 to half length of coefficient of the adaptive filter. The randomly generated 32 length primary path, and the 8 tap secondary path are used for simulations. The suitable projection order is decided by choosing the one which shows the best performance. For example, the MSD learning curves of the proposed algorithm for a white input signal with various projection orders are shown in Fig. 5. The use of the optimal step size APA for a white signal shows the best performance with lowest projection order [11]. As the projection order is increased, the steady state is decreased. Otherwise, the best performance of the proposed algorithm for a white signal is shown with the projection order M¼6. The reason is that the MFxAP algorithm uses the pre-filtered input vector vi from the estimate of secondary path h, so the input vector to adaptive filter has colored property. That is why the proposed algorithm shows the best performance when the projection order sufficiently satisfies the whitening effect for Assumption 1. The signals from (58) and (59) have been used for the simulation of Figs. 6 and 7, respectively. From the
Fig. 6. MSD learning curves of the proposed algorithm with various projection orders and n¼ 32 for an AR input signal (58). The step size μ is 0.04.
simulation, the proposed algorithm shows the best performance for each input signals with the projection orders 8 and 14. For the ARMA signals, the appropriate projection order is almost closed to half length of the adaptive filter. It means that with high projection order, the MFxAP whitens the signal from the estimate of secondary path to establish Assumption 1 properly. As the colored property is increased, higher projection order is needed to show the best performance. The AR signal with first order filter (58) shows a little bit increase of colored property according to Fig. 6, but projection orders closed to one show poor performance. With the paths from [20], white signal is used for the reference signal. To find the best projection order, various projection order simulation has been done. From the simulation result in Fig. 8, the performance of the proposed algorithm is decreased, when the projection order is increased from 1 to 6. However, as the projection order is increased from 6 to 38, both the convergence rate and the steady state error are improved. For the projection order higher than 38, the proposed algorithm is getting better convergence rate, but the steady state error is getting
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J. Song, P. Park / Signal Processing 114 (2015) 100–111
5 M=1 M=2 M=4 M=6 M=8 M=10 M=12 M=14 M=16
0 −5
MSD (dB)
−10 −15 −20 −25 −30 −35 −40 0
2000
4000
6000
8000
10000
12000
iteration
Fig. 7. MSD learning curves of the proposed algorithm with various projection orders and n¼ 32 for an ARMA input signal (59). The step size μ is 0.04.
5 0
With the evolving VSSAP algorithm of the best performance and the proposed algorithm, the simulation results are in Figs. 9 and 10. The evolving VSSAP algorithm uses the projection order of the proposed algorithm as the initial projection order. The proposed algorithm shows far below MSD learning curve, as well as shows the fastest convergence rate compared to other fixed-step-size MFxAP algorithms with different type input signals. Compared to evolving VSSAP algorithm, the convergence speed of both algorithms is alike, but the steady state of proposed algorithm shows lower curves (Fig. 11). In the environment of paths from [20], the proposed algorithm is also compared with the MFxAP algorithms and the evolving VSSAP algorithm. With the white signal, suitable projection order is 38 as the previous simulation result. However, it is too high to use in real-time ANC environments. Thus, the parameter Q is applied to the proposed algorithm, and according to the simulation result in Fig. 4, the parameter Q is set as 50. For the evolving VSSAP algorithm, parameters are determined: μmax ¼ 2; α ¼ 0:9999; μNup ¼ 0:2; μNdown ¼ 0:000001. The simulation result is depicted in Fig. 12, and the proposed algorithm shows the best performance with lower projection order 15 than
−5 5
−10
(a) MFXAP with mu=0.04 (b) MFXAP with mu=0.02 (c) MFXAP with mu=0.005 (d) Evolving VSSAP (e) proposed
−15
−5
−20
−10 MSD(dB)
MSD(dB)
0
−25 −30
−15 (a)
−20 (b)
−25 (d)
(c)
−30
−35
(e)
−35 0
0.5
1
1.5
2 iteration
2.5
3
3.5
4
−40
4
x 10
Fig. 8. MSD learning curves of the proposed algorithm with various projection orders and n¼32 for an white input signal. The step size μ is 0.04, and the secondary path is from [20]. The projection order is varying from 1 to 50.
worse. Thus, the projection order can be chosen, according to the environments and the requirements. The next simulations are done for the test of the proposed algorithm compared with MFxAP algorithms of various step sizes, but same projection order and the evolving variable step-size algorithm [12]. The evolving VSSAP algorithm based on [15] uses the structure of the MFxAP algorithm. The projection orders of the proposed algorithm are determined from the above simulations that compare the proposed algorithm with various projection orders. The initial projection order of the evolving VSSAP algorithm is determined as the projection order of the proposed algorithm, and some tuning parameters are determined through many times simulations. The first comparing simulation is done for the paths of randomly generated 32 length primary path, and the 8 tap secondary path. For the evolving VSSAP algorithm, some tuning parameters are determined: μmax ¼ 2; α ¼ 0:99; μNup ¼ 0:2; μNdown ¼ 0:00001. It was very difficult to get the best performance with the evolving VSSAP algorithm [12].
−45
0
2000
4000
6000
8000
10000
12000
14000
iteration
Fig. 9. MSD learning curves of the proposed algorithm, evolving VSSAP algorithm and standard fixed-step-size MFxAP algorithm with various step sizes for a white signal.
5 (a) MFXAP with mu=0.04 (b) MFXAP with mu=0.02 (c) MFXAP with mu=0.005 (d) Evolving VSSAP (e) proposed
0 −5 −10 MSD(dB)
−40
−15
(a) (b)
−20
(c)
−25
(e)
(d)
−30 −35 −40
0
2000
4000
6000
8000
10000
12000
14000
iteration
Fig. 10. MSD learning curves of the proposed algorithm, evolving VSSAP algorithm and fixed-step-size MFxAP algorithm with various step sizes for an AR input signal (58).
J. Song, P. Park / Signal Processing 114 (2015) 100–111
5
5 (a) MFXAP with mu=0.04 (b) MFXAP with mu=0.02 (c) MFXAP with mu=0.005 (d) Evolving VSSAP (e) proposed
0 −5
−5
(a)
−15
−10
−20
(b)
−25
(c)
(d)
MSD(dB)
MSD(dB)
MFXAP with mu=0.1 MFXAP with mu=0.03 Evolving−VSSAP proposed
0
−10
(e)
−15 −20
−30 −35 −40
109
−25 0
2000
4000
6000
8000
10000
12000
14000
−30
iteration −35
Fig. 11. MSD learning curves of the proposed algorithm, evolving VSSAP algorithm and fixed-step-size MFxAP algorithm with various step sizes for an ARMA input signal (59).
0
0.5
1
1.5
Fig. 13. MSD learning curves of the proposed algorithm, fixed-step-size MFxAP algorithm with various step sizes, and evolving VSSAP algorithm, for an white signal (Q ¼50, α ¼ 0:975, σ 2r ¼ 10 2 ).
5 MFXAP with mu=0.02 MFXAP with mu=0.005 Evolving−VSSAP proposed
0
−5
5 MFXAP with mu=0.04 MFXAP with mu=0.02 MFXAP with mu=0.005 proposed
0 −10 −5 −15 −10
−20
MSD(dB)
MSD(dB)
2 5
x 10
iteration
−25
−15
−30
−20
−35
−25
−40 0
0.5
1
1.5 iteration
2
2.5
3
−30
4
x 10
−35
Fig. 12. MSD learning curves of the proposed algorithm, fixed-step-size MFxAP algorithm with various step sizes, and evolving VSSAP algorithm, for an white signal (Q ¼50, α ¼ 0:9999, σ 2r ¼ 10 3 ).
the optimal projection order 38. As the projection order is decreased, the computational complexity is also decreased, maintaining the best performance among other algorithms. Additional simulation has been carried out, in order to evaluate the proposed algorithm with low SNR. The variance of the secondary noise is set as σ 2r ¼ 10 3 , and the parameter α of the evolving VSSAP algorithm is set as 0.975, but the other factors of the simulation environments are same as Fig. 4. The parameter α is changed, because the noise variance is also changed. The simulation result is depicted in Fig. 12. The proposed algorithm shows the best performance with the fastest convergence rate and the lowest steady-state error, even though the variance of the secondary noise is increased. In these simulation environments, the optimal step size for one trial is depicted in Fig. 13. It shows that the optimal step size is higher than 0 and lower than 1. The last simulation is done to test the reset algorithm on non-stationary consideration. To check up the biggest change of the unknown primary path, the sign of it is
0
2000
4000
6000
8000
10000
iteration
Fig. 14. MSD learning curves of the proposed algorithm and fixed-stepsize MFxAP algorithm with various step sizes for an ARMA signal in a non-stationary environment in which the sign of the unknown primary path is changed in the middle of the test interval.
assumed to change at iteration 5000. The other environmental conditions are same with the simulation of Fig. 10. With the ARMA signal, (59), a positive integer K ¼2 is used. The design parameter, γ ¼ 1700, is determined empirically on the consideration of (56). When the parameter is smaller than this value, the detection of weight vector change is barely done, while it is larger, much more resets including undesirable ones is done. With the proper parameter, the wanted reset is done successfully, and the proposed algorithm traces the changed weight vector as it did before reset (Fig. 14). 9. Conclusion This paper proposed an MFxAP algorithm with the optimal step size for ANC systems. To improve the MFxAP algorithm, it analyzes the MSD learning curve of the MFxAP
110
J. Song, P. Park / Signal Processing 114 (2015) 100–111
algorithm and derives the step size at each iteration for ANC systems. With the pre-filtered input, it considers the crosscorrelation between the secondary noise vector and the current weight error vector to achieve more accurate prediction of the learning curve of the MFxAP algorithm. With numerically estimated Mth-order input autocorrelation matrix of vi, but without any parameters to be tuned in the basic algorithm, the proposed algorithm achieves optimal performance in two senses as the simulation results show. One is the lowest steady-state error and the other is the fastest convergence rate.
ΦB ði; kÞ 9 Bi 1 Bi 2 ⋯Bk ;
ΦB ði; iÞ 9 In :
ðB:4Þ
S , Ai ¼ Z, Bi ¼ Φðiþ 1; iÞ, and Ci ¼ σ 2r μi Z Let X i¼ 1 i T Vi Vi VTi , then from (14) and (16), we can obtain 1 X
S ¼ σ 2r
μi m Zm VTi m Vi m
1
VTi m Φ ði þ 1; i mþ 1Þ: T
ðB:5Þ
m¼1
From the fact that Z is nilpotent (i.e., Zj ¼ 0) if j ZM, the corollary is concluded. Appendix C. Proof of Lemma 1 Under Assumption 1, the term of Vi is
Acknowledgment
Vi VTi Vi This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF2014R1A1A2055122).
4
Riþ1 T
Si þ 1
Pi þ 1
2
T 6Z 4 0
2 4
Riþ1 T
Si þ 1
3
Si þ 1
Si þ 1 Pi þ 1
2 T 6Z 4 0
02
5 ¼ E@4
1 μi Vi VTi Vi
k ¼ k0 þ 1 0 X
2
5¼4
ðA:1Þ
3" #" #T r ri 5 i Φði þ 1; iÞ w~ i w~ i
ðA:2Þ
3 0" #" #T 1 r ri 5E@ i jV i A ~i ~i Φði þ 1; iÞ w w
1
ðA:3Þ
~ Ti T and for given Vi by V i , where D is the cross term of ½rTi w ~ i with vi , Eð½rTi r i þ 1 . From the independence of ri and w ~ Ti T ½rTi w ~ Ti jV i Þ is equivalent to Eð½rTi w ~ Ti T ½rTi w ~ Ti jV i 1 Þ, and w the theorem is proved.
From the recursive equation ðB:1Þ
the closed form can be obtained as Xi ¼ ¼
T B ði; kÞ
1 ¼ Zj þ VTiþ j Vi VTi Vi
ΦA ði; i j þ1ÞCi j Φ
0
k X
J vi þ k J 2
vi þ k aT1 k
ðC:1Þ
J vi þ k J 2
vTiþ j ⋯vTiþ j þ 1 M
k ¼ 1M
iv aT iþk 1k J vi þ k J 2 ðC:2Þ
With (C.1) and (C.2), the lemma can be proved by mathematical induction. As the first step, let us take j¼0, then (20) is satisfied as follows: 1 left hand side ðLHSÞ ¼ Φði þ 1; i þ 1ÞVi VTi Vi 1 ¼ Vi VTi Vi ¼ right hand side ðRHSÞ:
ðB:2Þ
j0 1
¼ ∏
1 0 0 1 μi þ j0 k Vi þ j0 VTiþ j0 Vi þ j0 Zj Zj T :
ðC:4Þ
Then, from the assumption with (7) and (8), for the case of j ¼ j0 þ 1, the left-hand side (LHS) of (20) can be rewritten as 1 0 Φ iþ j0 þ 2; i þ 1 Vi VTi Vi Zðj þ 1ÞT ¼ Φ i þ j0 þ2; i þj0 þ 1
0
j 1
¼ ∏
k¼0
ðB:3Þ
ðC:3Þ
The next step is assuming that (20) holds for unspecified values of j ¼ j0 1 Φ iþ j0 þ 1; i þ 1 Vi VTi Vi ZjT
where, for i4 k, and
ΦA ði; iÞ 9 IM
k ¼ 1M
k0 h X
Φ iþ j0 þ 2; i þ 1 Vi VTi Vi T B ði; i j þ 1Þ
vi þ k aT1 k
0
j¼1
ΦA ði; kÞ 9Ai 1 Ai 2 ⋯Ak ;
J vi þ k þ j J
2
0
k X
þ
Φ i þ j0 þ 1; i þ1 Vi VTi Vi
ΦA ði; kÞCk 1 Φ
k ¼ 1 1 X
J vi þ k J 2
k ¼ 1M
where k ¼ minðM; jÞ M. From (C.1), pre-multiplying VTiþ j leads to
k¼0
Appendix B. Proof of Corollary 2
i X
vi þ k aT1 k
¼ Zj :
0
Xi þ 1 ¼ Ai Xi BTi þ Ci ;
J vi þ k J
2
0
k X
þ
k ¼ 1M
0
3 VTi 7 5 þ σ 2r a1 aT1 ; ΦT ði þ 1; iÞ
μi VTi Vi
J vi þ k J 2
1 ¼ Vi þ j VTiþ j Vi þ j Zj þ
1
Z 1 μi Vi VTi Vi
vi þ k aT1 k
vi þ k þ j aT1 k Zj
k ¼ 1M
3 1 VTi 7 2 C 5 þr i þ 1 a1 aT1 þ DjV i A ΦT ðiþ 1; iÞ
μi VTi Vi 3
Z
0 X
0 X vi þ k aT1 k
¼
Appendix A. Proof of Theorem 1 From (11) and (12), it leads to 2 3 0" #" #T 1 R i þ 1 Si þ 1 ri þ 1 ri þ 1 4 T 5 ¼ E@ A ~ iþ1 ~ i þ 1 jV i w w Si þ 1 Pi þ 1
¼
k ¼ 1M
¼
2
1
1 μ i þ j0 k
1
1
Zðj
0
0
Zðj
þ 1ÞT
ðC:5Þ
þ 1ÞT
1 In μi þ j0 þ 1 Vi þ j0 þ 1 VTiþ j0 þ 1 Vi þ j0 þ 1
o 1 0 0 Zj Zðj þ 1ÞT VTiþ j0 þ 1 Vi þ j0 VTiþ j0 Vi þ j0
ðC:6Þ
J. Song, P. Park / Signal Processing 114 (2015) 100–111
Φ i þ j0 þ 2; i þ 1 Vi VTi Vi j0 1
¼ ∏
k¼0
1 μi þ j0 k
1
0
Zðj
þ 1ÞT
1 0 0 Vi þ j0 VTiþ j0 Vi þ j0 Zj Zðj þ 1ÞT
1 VTiþ j0 þ 1 Vi þ j0 μi þ j0 þ 1 Vi þ j0 þ 1 VTiþ j0 þ 1 Vi þ j0 þ 1 1 0 0 VTiþ j0 Vi þ j0 Zj Zðj þ 1ÞT :
ðC:7Þ
Then, for 1 M r k r0 and j k Z M, note that aT1 k Zj ¼ 0, and from (C.1), we can say that 1 0 1 Zj ¼ Vi þ j0 þ 1 VTiþ j0 þ 1 Vi þ j0 þ 1 Vi þ j0 VTiþ j0 Vi þ j0 0
0
ZZj þ
k X
vi þ j0 þ k aT1 k j0 Z ‖vi þ j0 þ k ‖2 k ¼ 1M
ðC:8Þ
1 0 1 0 Zj ¼ Vi þ j0 þ 1 VTiþ j0 þ 1 Vi þ j0 þ 1 Zðj þ 1Þ Vi þ j0 VTiþ j0 Vi þ j0 ðC:9Þ Thus, with (C.2) and (C.9), (C.7) leads to 1 0 0 1 μi þ j0 k Vi þ j0 þ 1 VTiþ j0 þ 1 Vi þ j0 þ 1 Zðj þ 1Þ Zðj þ 1ÞT
j0 1
¼ ∏
k¼0
1 0 0 μi þ j0 þ 1 Vi þ j0 þ 1 VTiþ j0 þ 1 Vi þ j0 þ 1 Zðj þ 1Þ Zðj þ 1ÞT
ðC:10Þ 0
j
¼ ∏
k¼0
1 0 0 1 μi þ j0 þ 1 k Vi þ j0 þ 1 VTiþ j0 þ 1 Vi þ j0 þ 1 Zðj þ 1Þ Zðj þ 1ÞT
ðC:11Þ Then, by mathematical induction, it holds for all i; jZ 0, and the lemma is concluded. References [1] S. Elliott, P. Nelson, Active noise control, IEEE Signal Process. Mag. 10 (4) (1993) 12–35. [2] M. Rupp, A. Sayed, Robust fxlms algorithms with improved convergence performance, IEEE Trans. Speech Audio Process. 6 (1) (1998) 78–85.
111
[3] M. Ferrer, A. Gonzalez, M. De Diego, G. Pinero, Convex combination filtered-x algorithms for active noise control systems, IEEE Trans. Audio Speech Lang. Process. 21 (1) (2013) 156–167. [4] B. Huang, Y. Xiao, J. Sun, G. Wei, A variable step-size fxlms algorithm for narrowband active noise control, IEEE Trans. Audio Speech Lang. Process. 21 (2) (2013) 301–312. [5] S. Elliott, P. Nelson, Active noise control, IEEE Signal Process. Mag. 10 (4) (1993) 12–35. [6] E. Bjarnason, Active noise cancellation using a modified form of the filtered-x lms algorithm, in: Proceedings of 6th European Signal Processing Conference, vol. 2, 1992, pp. 1053–1056. [7] M. Ferrer, A. Gonzalez, M. De Diego, G. Pinero, Transient analysis of the conventional filtered-x affine projection algorithm for active noise control, IEEE Trans. Audio Speech Lang. Process. 19 (3) (2011) 652–657. [8] M. Ferrer, M. De Diego, A. Gonzalez, G. Piero, Steady-state mean square performance of the multichannel filtered-x affine projection algorithm, IEEE Trans. Signal Process. 60 (6) (2012) 2771–2785. [9] H.-C. Shin, A. Sayed, Mean-square performance of a family of affine projection algorithms, IEEE Trans. Signal Process. 52 (1) (2004) 90–102. [10] P. Park, C.H. Lee, J.-W. Ko, Mean-square deviation analysis of affine projection algorithm, IEEE Trans. Signal Process. 59 (12) (2011) 5789–5799. [11] C. Lee, P. Park, Optimal step-size affine projection algorithm, IEEE Signal Process. Lett. 19 (7) (2012) 431–434. [12] A. Gonzalez, F. Albu, M. Ferrer, M. Diego, Evolutionary and variable step size strategies for multichannel filtered-x affine projection algorithms, IET Signal Process. 7 (6) (2013) 471–476. [13] P. Park, J.-H. Seo, N. Kong, Variable matrix-type step-size affine projection algorithm with orthogonalized input vectors, Signal Process. 98 (2014) 135–142. [14] I. Song, P. Park, A variable step-size affine projection algorithm with a step-size scaler against impulsive measurement noise, Signal Process. 96 (Part B) (2014) 321–324. [15] A. Gonzalez, M. Ferrer, M. de Diego, G. Piero, An affine projection algorithm with variable step size and projection order, Digit. Signal Process. 22 (4) (2012) 586–592. [16] M. Rupp, A family of adaptive filter algorithms with decorrelating properties, IEEE Trans. Signal Process. 46 (3) (1998) 771–775. [17] J.-B. Lasserre, A trace inequality for matrix product, IEEE Trans. Autom. Control 40 (8) (1995) 1500–1501. [18] C.H. Lee, P. Park, Scheduled-step-size affine projection algorithm, IEEE Trans. Circuits Syst. I: Regul. Pap. 59 (9) (2012) 2034–2043. [19] M. Ferrer, A. Gonzalez, M. De Diego, G. Pinero, Fast affine projection algorithms for filtered-x multichannel active noise control, IEEE Trans. Audio Speech Lang. Process. 16 (8) (2008) 1396–1408. [20] S.M. Kuo, D. Morgan, Active Noise Control Systems: Algorithms and DSP Implementations, Wiley Series in Telecommunications and Signal Processing, Wiley, New York, 1996.
Contents lists available at ScienceDirect
Signal Processing journal homepage: www.elsevier.com/locate/sigpro
An optimal variable step-size affine projection algorithm for the modified filtered-x active noise control Ju-man Song a, PooGyeon Park b,n a b
Division of IT Convergence Engineering, Pohang University of Science and Technology, Pohang, Republic of Korea Department of Electronic and Electrical Engineering, Pohang University of Science and Technology, Pohang, Republic of Korea
a r t i c l e in f o
abstract
Article history: Received 14 July 2014 Received in revised form 5 February 2015 Accepted 7 February 2015 Available online 5 March 2015
This paper introduces an optimal variable step-size affine projection algorithm for the modified filtered-x active noise control systems. First, the recursion form of the error covariance from the tap weight update equation is constructed, not ignoring the dependency between the estimation error and the secondary noise signal. Such consideration has not been concerned previously for the analysis of the modified filtered-x affine projection algorithm. Second, a recursion form of the mean square deviation is derived from that of the error covariance. From the recursion form, an optimal step size is decided to get the fastest convergence rate. Both the recursion forms of the mean square deviation and the optimal step size require scalar additions and multiplications that do not contribute to the overall complexity seriously. The simulation results on the active noise control environments show both fast convergence rate and low steady-state error. & 2015 Elsevier B.V. All rights reserved.
Keywords: Adaptive filters Active noise control (ANC) Affine projection algorithm (APA) Filtered-x affine projection (FxAP) Optimal step size Mean-square deviation (MSD)
1. Introduction The goal of active noise control (ANC) is to cancel out the unwanted disturbance signal using the acoustic signal. The acoustic signal is generated from controllable secondary source. With feedback or feed forward control using electrical signals, the ANC system generates acoustic signals for destructive interference at the sound fields [1]. The unwanted disturbance signals for destructive interference have long wavelengths with low frequency. The low frequency noise is hard to be suppressed by conventional passive ways. Thus, the method named as the ANC is developed to efficiently remove the acoustic noise. In ANC systems, the least mean squares (LMS) algorithm has been used as the form of filtered-x LMS algorithm (FxLMS), for low cost and complexity. To improve the performance, the FxLMS algorithm is modified by various ways: the generalized FxLMS recursion [2], the convex algorithm [3],
n
Corresponding author. Tel.: þ 82 54 279 2238; fax: þ 82 54 279 2903. E-mail address: [email protected] (P. Park).
http://dx.doi.org/10.1016/j.sigpro.2015.02.005 0165-1684/& 2015 Elsevier B.V. All rights reserved.
and variable step-size algorithm [4]. However, a drawback of the FxLMS algorithm is that its convergence rate gets worse in correlated input environments. Typically, due to the secondary path, the ANC system shows the similar effect with correlated inputs. The effect is caused from the pre-filtered input source to the adaptive filter to update coefficient equation, although the input signal to the copy of the adaptive filter is not filtered to the estimate of the secondary path. Thus the FxLMS algorithm shows poor convergence rate at the ANC system. To overcome the weak point of the FxLMS algorithm for the ANC system, the affine projection algorithm (APA) has been used as the filtered-x affine projection (FxAP) algorithm [5] and modified filtered-x affine projection (MFxAP) algorithm [6]. The performance of FxAP algorithm is shown by analyzing the transient and the steady states [7,8]. Each literature analyzes the transient analysis of the conventional FxAP (CFxAP) algorithm, and the steady-state mean square performance of both the CFxAP algorithm and the MFxAP algorithm. Both analyses use the energy conservation relation, which has been already used to analyze the mean-square performa nce of a family of APA [9]. They show good agreement with
J. Song, P. Park / Signal Processing 114 (2015) 100–111
simulations in the ANC system environments. However, those analyses ignore the dependency between the weight-error vector and past noise. Additionally, using the energy conservation relation is too complex to apply at ANC systems for realtime algorithms. On the other hand, a different approach was presented to analyze the mean-square deviation (MSD) of APA, considering the dependency between the weight-error vector and past noise [10]. It constructs the propagation model of the error covariance having lower complexity and showing great simulation result on analysis in better agreement than other analyses. Based on the analysis, the optimal step size is derived to get the best performance [11]. However, there are just few literatures of improving the MFxAP algorithm to get fast convergence rate and low steady state error [12], compared to APA [13– 15]. Thus, to obtain those requirements simultaneously, in this paper, an optimal step size of the MFxAP algorithm is suggested, applying the analysis of APA [10]. This paper uses the optimal step size algorithm of the APA [10]. The proposed algorithm is developed by analyzing the MSD of the basic coefficient update equation of the MFxAP algorithm, which is different from that of the APA. It considered the difference between the APA and the MFxAP algorithm, including the pre-filtered input from the estimate of the secondary path. Full theoretical justification will be provided for the superior performance of the proposed algorithm, with some different proofs from [10] for better way. The nonstationary condition, which is often ignored and is essential for real implementation of the ANC, is also considered. Furthermore, efficiency consideration is also applied to the proposed algorithm to reduce the computational complexity. This paper is organized as follows. Section 2 describes the system of the FxAP algorithm in the ANC environments. Section 3 presents a recursion form of the error covariance considering the dependency between the weight-error vector and the secondary noise vector for the MFxAP algorithm. Based on the recursion form, Section 4 analyzes the MSD and constructs the optimal step size of the MFxAP algorithm. Non-stationary consideration is presented in Section 5, and finally, Section 6 shows some simulation results verifying the performance and properties of the proposed optimal step size algorithm. 2. The filtered-x APA For the ANC systems, each structure of the CFxAP algorithm and the MFxAP algorithm is depicted in Figs. 1 and 2, respectively. First, the primary noise signal, which is unwanted signal, passes through the unknown primary path. After that, the adaptive filter output yi is added to the passed signal and it goes through the secondary path h. The secondary path is considered as a finite impulse response (FIR) filter of different lengths from the primary path. The primary path is determined by assuming that the algorithm converges to the optimal solution wo A Rn1 . Thus, the desired signal di which has to be canceled out is given by di ¼ Vi wo . In practice, however the secondary noise signal should be considered, and the disturbance signal is defined with the projection order M Z 1 as di ¼ VTi wo þ ri ;
ð1Þ
101
where Vi 9 ½vi vi 1 … vi M þ 1 A RnM ;
ð2Þ
vi 9 ½vi vi 1 … vi n þ 1 T A Rn1 ;
ð3Þ
and vi is a signal passed through the estimate of secondary ^ For the estimation of secondary path h^ A RL1 as vi ¼ xi nh. path, it is assumed to be done perfectly in this paper. The secondary noise signal ri A RM1 is a zero mean white 2 Gaussian noise vector with variance σr . The basic coefficient update equation of FxAP can be written with the following recursion formula: 1 ^ iþ1 ¼ w ^ i μi Vi VTi Vi w ei ; ð4Þ ^ i A Rn1 is the impulse response of the adaptive where w filter, and μi A ð0; 1 is a variable step size. For the use of ei , the CFxAP algorithm uses samples of the error signal ei, at the error microphone, because it cannot use the disturbance signal di. However, the MFxAP algorithm can reconstruct the disturbance signal and the error signal by using the output of the estimate of secondary path. Thus, in the MFxAP algorithm, the error vector ei is given by ^ i: ei ¼ di þVTi w
ð5Þ
The plus sign after di means the addition at a sensor like a microphone, and the summation is not an electrical sum but an acoustic sum, so Eq. (5) can be used in the MFxAP algorithm, but not in the CFxAP algorithm. On the other hand, the MFxAP algorithm recovers the desired signal from the error signal using the output from the secondary path estimator after the adaptive filter, so it can use Eq. (5). In this paper, only MFxAP algorithm is handled, because it shows better convergence rate than CFxAP algorithm. 3. Augmented modeling of the modified filtered-x APA From the tap weight update equation, the recursion ~ i ¼ wo w ^ i is formulated as form of the tap error vector w 1 ~ iþ1 ¼ w ~ i þ ri ; ~ i þ μi Vi VTi Vi ð6Þ w VTi w and let the transition matrix be 1 Φðiþ 1; iÞ ¼ In μi Vi VTi Vi VTi ; where Ia is an identity matrix in R matrix has two properties such as
Φði; iÞ ¼ In ;
Φði; jÞ ¼ Φði; kÞΦðk; jÞ
ð7Þ aa
. Then, the transition ð8Þ
With the transition matrix, the recursion form of tap error vector can be rewritten as 3" " # 2 # Z 0 ri þ 1 r r 1 4 5 i þ iþ1 ¼ T ~ iþ1 μi V i V i V i Φði þ 1; iÞ w~ i w 0 h
i
ð9Þ MM
, and 0 is the zero matrix of where Z ¼ IM 1 A R appropriate dimensions. h i ~ Ti w ~i The MSD matrix at each iteration i is defined as E w 0
0 0
¼ TrðPi Þ, where EðÞ is the notation of expectation. Pi is the covariance matrix of tap error vector which can be
102
J. Song, P. Park / Signal Processing 114 (2015) 100–111
Fig. 1. The block diagram of CFxAP.
Fig. 2. The block diagram of MFxAP.
determined as fvj j0 r jr i 1g: "
"
Ri
Si
STi
Pi
Ri
Si
STi
Pi
#
#
follows
with
a
given
0" #" #T 1 ri ri A 9 E@ ~i ~i w w 8 0" 19 #" #T < = ri ri @ jV i 1 A ¼E E ~i ~i w w : ;
set
Vi 1 9
ð10Þ
ð11Þ
"
Ri
Si
STi
Pi
#
02 ¼ E @4
Ri T
Si
Si Pi
31 5A:
ð12Þ
~i Considering the dependency between the estimation error w and the noise signal ri , the recursions of R i , S i and P i with variable step size μi are obtained as Theorem 1. Theorem 1. The recursion forms of the covariance matrices R i , S i , and P i are R i þ 1 ¼ ZR i ZT þ σ 2r a1 aT1
ð13Þ
J. Song, P. Park / Signal Processing 114 (2015) 100–111
1 S i þ 1 ¼ ZS i Φði þ 1; iÞ þ μi ZR i VTi Vi VTi ;
P i þ 1 ¼ Φði þ 1; iÞP i Φ ðiþ 1; iÞ þ μ2i Vi VTi Vi T
ð14Þ 1
R i VTi Vi
1
1 T þ μi Vi VTi Vi S i Φ ðiþ 1; iÞ 1 T VTi ; þ μi Φði þ 1; iÞS i VTi Vi
VTi
ð15Þ
Proof. See Appendix A. If we assume that i Z M, covariance matrices R i and S i can have closed form for any input signal in (13) and (14). In the following corollaries, closed forms are shown.
Ri ¼ σ
2 r IM
vTa vb nEðva vb Þ ¼ ¼ 0: n-1ð J va J J vb J Þ nσ 2v
ð19Þ
Even though the input signal to the adaptive filter of the ANC system is correlated by the estimate of the secondary path, Assumption 1 can be used, because of the self-whitening effect of the APA [16]. To reduce mathematical complexity, from Assumption 1, the following lemma is obtained. Lemma 1. For integers i; jZ 0, Under Assumption 1, it holds that 1 Φðiþ j þ1; i þ1ÞVi VTi Vi ZjT j1
σr is available, we have
¼ ∏
k¼0
ð16Þ
,.
use vTa vb =ð J va J J vb J Þ C 0, because lim
where a1 is vector, whose first element is 1 and the others are 0's in Rn1 .
Corollary 1. For i ZM, if
103
1 1 μi þ j k Vi þ j VTiþ j Vi þ j Zj ZjT
ð20Þ
with probability being one when n-1.
Proof. The proof is omitted.
Proof. See Appendix C.
Corollary 2. For i ZM, it holds that
Thus, for the ANC systems, the recursion form of the covariance matrix P i can be reorganized as the following theorem.
S ¼ σ 2r
M 1 X
μi m Zm VTi m Vi m
1
VTi m
m¼1
Φ ði þ1; i m þ 1Þ: T
ð17Þ
Theorem 2. With Assumption 1 and Lemma 1, the covariance matrix P i has the recursion form for i ZM as P i þ 1 ¼ Φði þ 1; iÞP i Φ ðiþ 1; iÞ 1 1 Gi VTi Vi VTi ; þ σ 2r Vi VTi Vi T
Proof. See Appendix B. Using Corollaries 1 and 2, (15) can be rewritten as 2 T P i þ 1 ¼ Φði þ 1; iÞP i Φ ðiþ 1; iÞ þ σ 2r μ2i Vi VTi Vi VTi þ σ μi 2 r
M 1 X
μ
T i m V i Vi Vi
1
Z
where Gi ¼ μ2i Z0 Z0T
1 m1 MX þ 2 μi μ2i μi m Zm ZmT ∏ 1 μi j :
m
m¼1
1 T VTi m Vi m VTi m Φ ði þ 1; i m þ 1Þ þ σ 2r μi
M 1 X
m¼1
μi m Φði þ1; i m þ 1ÞVi m
m¼1
1 1 VTi m Vi m ZmT VTi Vi VTi :
ð21Þ
ð18Þ
ð22Þ
j¼1
Proof. Lemma 1 , for the ANC systems, (20) is rewritten with different indices as 1 1 Φðiþ 1; i m þ 1ÞVi m VTi m Vi m ZmT VTi Vi VTi m1
¼ ∏
j¼0
1 1 1 μi j Vi VTi Vi Zm ZmT VTi Vi VTi ð23Þ
4. MSD analysis and optimal step size of the modified filtered-x APA The recursion formula of the tap error vector covariance matrix is the most important factor among augmented modeled matrices. However, (18) is too complex to use. Thus, it has to be simplified to analyze MSD of the MFxAP algorithm, and the following assumption is used. The assumption was used in [11], and we modify it for the MFxAP algorithm. Assumption 1. For arbitrary values of a and b such that aab and a; bZ 0, the input vectors satisfy jvTa vb j{ J va J J vb J , where J J denotes the Euclidean norm. Then, n-1, for a ab, it has probability of one to
Substituting (23) into (18) results in (21).
□
To reach our goal, making recursion form of MSD for the MFxAP algorithm can be achieved by taking the trace on both sides of (21): 1 T Tr P i þ 1 CTrfΦ ði þ1; iÞΦði þ 1; iÞP i g þ σ 2r Tr VTi Vi Gi 1 VTi P i ¼ Tr P i 2μi μ2i Tr Vi VTi Vi 1 Gi ; þ σ 2r Tr VTi Vi
ð24Þ
where Tr ðÞ is the notation of tracing matrix. At this point, for easy derivation, following lemma is used.
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J. Song, P. Park / Signal Processing 114 (2015) 100–111
1 1 M m Tr M T m mT g ðmÞ ¼ Tr E Vi Vi ; Z Z C M n 2
Lemma 2. If the projection order is sufficiently high, we can assume that 1 M Tr Vi VTi Vi VTi P i C Tr P i : ð25Þ n
g ¼ ½g ð1Þ g ð2Þ … g ðM 1ÞT ; 3 μi 1 6 7 1 μi 1 μi 2 6 7 7; μi ¼ 6 6 7 ⋮ 4 5 1 μi 1 … 1 μi M þ 2 μi M þ 1
ð32Þ ð33Þ
2
Proof. From [17, Lemma II.1], n X k ¼ M0
M 1 X λk P i rTr Vi VTi Vi VTi P i r λk P i
ð26Þ
k¼1
and M A RMM is the Mth-order autocorrelation matrix of vi. Two matrices g and μi are used for easy manipulation. Additionally, μi can be recursively obtained as follows: μ μi þ 1 ¼ 1 μi ZM 1 μi þ i ; ð35Þ 0 h i where ZM 1 ¼ IM0 2 00 A RðM 1ÞðM 1Þ . To obtain the optimal step size, the partial differential with respect to μi is done to both sides of (30), and the result is obtained as
where M 0 ¼ n M þ 1 and λk ðÞ is the kth largest eigenvalue of the matrix. Each upper and lower average of eigenvalues is defined respectively as
λ Pi 9
M 1 X λ P ; Mk¼1 k i
λ Pi 9
n 1 X λ P : M k ¼ M0 k i
Then, substituting (27) into (26) results in 1 M λ P i r Tr Vi VTi Vi VTi P i r Mλ P i
ð27Þ
M ∂ Tr Pi þ 1 ¼ 2 þ 2μi TrðPi Þ ∂μi n þ2σ 2r fμi g ð0Þ þ 1 2μi αi g:
ð28Þ
Whenthe order is increased, λ P i -Tr P i =n projection and λ P i -Tr P i =n. Thus, by the Squeeze theorem, the lemma is concluded. □
μi ¼
where
m¼1
m1
μi m gðmÞ ∏ 1 μi j ¼ gT μi ;
M TrðPi Þ nσ 2r αi : M TrðPi Þ þnσ 2r g ð0Þ 2nσ 2r αi
ð31Þ
j¼1
Table 1 Algorithm summary. Initialization :
^ 0 ¼ 0; μ0 ¼ ½1 0T , and TrðP0 Þ. w
Update the vectors :
xi ¼ ½xi xi 1 ⋯ xi L þ 1 T , ^ v ¼ xT h, i
i
vi ¼ ½vi vi 1 ⋯ vi n þ 1 T , Vi ¼ ½vi vi 1 ⋯ vi M þ 1 , ei ¼ ½ei ei 1 ⋯ ei M þ 1 T , ^ i, d^ i ¼ ei VTi w ^ . e^ ¼ d^ þ VT w i
Step-size updates :
Filtered-x AP : MSD update :
i
i
i
αi ¼ gT μi , M TrðPi Þ nσ 2r αi , M TrðPi Þ þ nσ 2r gð0Þ 2nσ 2r αi μi μi þ 1 ¼ 1 μi ZM 1 μi þ . 0 1 ^ i μi Vi VTi Vi ^ iþ1 ¼ w ei . w 2 M Tr Pi þ 1 ¼ 1 2μi μi TrðPi Þ þ σ 2r fμ2i gð0Þ þ 2 μi μ2i αi g n μi ¼
ð37Þ
If l o 0, then μl ¼ 1. For the case of unknown initial MSD, it is 2 recommended to use σ 2y =λ M , where σy is the variance of adaptive filter output yi and λ M is defined for the maximum eigenvalue of the filtered input correlation matrix M [18]. The proposed algorithm is summarized in Table 1. From the point of view of complexity aspect, the proposed algorithm could be considered having a highly computational cost, because of its sophisticated procedure to produce and prove. However, from Table 1, additional Oð2MÞ multiplies per iteration are required for step size and MSD updates of proposed algorithm. Compared to the complexity of the MFxAP such that 2L þ2nM þ 2n þ nM2 þ M 2 þO M 3 =2
Taking the expectation on both sides of (29) from (12), the final recursion form of the MSD is derived as M Tr Pi þ 1 C 1 2μi μ2i TrðPi Þ þ σ 2r fμ2i g ð0Þ þ 2 μi μ2i αi g; n ð30Þ
M 1 X
ð36Þ
The optimal step size μi is finally decided for the fastest convergence iteration i. It is obtained by rate at each solving ∂Tr Pi þ 1 = ∂μi ¼ 0 as
With Lemma 2, substituting (25) into (24) leads to 1 M Tr P i þ σ 2r Tr VTi Vi Tr P i þ 1 C 1 2μi μ2i Gi : n ð29Þ
αi ¼
ð34Þ
J. Song, P. Park / Signal Processing 114 (2015) 100–111
multiplies [19], the proposed algorithm makes few contribution to the overall complexity. 5. Analysis on both recursion forms of the step size and the MSD For the proposed algorithm, the convergence can be guaranteed by analyzing both the optimal step size and the MSD recursion form. From the simulation, the tendency of the optimal step size is shown in Fig. 3. From Fig. 3, the optimal step size varies from 1 to 0.0002484, and it shows the boundary as 0 r μi r 1. The boundary of the optimal step size can be proved by analyzing it. To analyze the optimal step size, μi, let us rewrite (37) as
μi ¼
ai ai þ bi
ð38Þ
where
¼ TrðPi 1 Þ
105
a2i 1 ai 1 þbi 1
ð43Þ
a2i 1 n n a þ σ2α M i 1 M r i 1 ai 1 þ bi 1 nn o n ai 1 ðai 1 þ bi 1 Þ ai 1 þ σ 2r αi 1 ¼ M ai 1 þbi 1 M
TrðPi Þ ¼
¼
a2i 1 n M n n þ μ b þ σ2α M ai 1 þ bi 1 M i 1 i 1 M r i 1
¼
a2i 1 n M n n 1 μi 1 σ 2r αi 1 þ μ σ 2 g ð0Þ þ M ai 1 þ bi 1 M i 1 r M
a2i 1 n M n þ σ 2 1 μi 1 αi 1 þ μi 1 g ð0Þ M ai 1 þ bi 1 M r n Z σ 2r αi : ð44Þ M ¼
Thus,
M ai ¼ TrðPi Þ σ 2r αi ; n
ð39Þ
TrðPi Þ
bi ¼ σ 2r g ð0Þ σ 2r αi :
ð40Þ
then by mathematical induction, ai Z0, for i ZM. Consequently, from (38), the theorem is proved. □
Considering both signs of ai and bi, the boundary of the optimal step size can be shown by the following theorem. Theorem 3. If l o M, then μl ¼ 1, then the optimal step size, μi, has boundary as 0 r μi r 1
ð41Þ
Proof. As the first step, let us assume that 0 r μj r1, for j ri 1, then aj Z 0, and bj Z0. Thus, from (39), αi 1 rgð0Þ, and from (31) to (34), αi ¼ gT μi ¼ 1 μi 1 gT Zμi 1 þ μi 1 gT e1 r 1 μi 1 αi 1 þ μi 1 g ð0Þ r 1 μi 1 g ð0Þ þ μi 1 g ð0Þ ¼ g ð0Þ
ð42Þ
Thus, by mathematical induction, bi Z0, for iZ M. From (30) and (38), TrðPi Þ CTrðPi 1 Þ 2μi 1 ai 1 þ μ2i 1 bi 1
n 2 n σ α ¼ a Z0; M r i M i
ð45Þ
From Theorem 3, the stability on the optimal step size is proved, and furthermore, the following property is induced. Property 1. For integers i ZM, the MSD has decreasing property, and the fastest rate is 1 M=n , with the following recursion form:
M Tr Pi þ 1 C 1 TrðPi Þ þ σ 2r g ð0Þ: ð46Þ n when μi ¼ 1. Proof. From (43), and ai Z 0; bi Z 0, a2 Tr Pi þ 1 TrðPi Þ ¼ i r 0: ai þ bi Then the property is proved.
ð47Þ □
From the above property, it guarantees the stability of the proposed algorithm that it will not diverge with the proposed algorithm.
1
6. Efficiency consideration for MFxAP Algorithm
0.9 0.8
step size (µ)
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
1
2
3
4
iteration
Fig. 3. The step size variation.
5
6 4 x 10
Although the proposed algorithm makes few contribution to the overall complexity, the projection order for the proposed algorithm is properly selected according to the degree of the whitening effect. If the correlation property of the input signal is high, then the projection order should be selected high value which increases the overall computational complexity. In the field of ANC environments of the MFxAP algorithm, the secondary path makes the input signal of the adaptive filter to be colored. Thus, when the secondary path gives high correlation property to the input signal, the suitable projection order for the proposed algorithm would be high, and the computational complexity would be increased. To reduce the complexity, in such environments, the white assumption of the MFxAP
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J. Song, P. Park / Signal Processing 114 (2015) 100–111
σ 2e;i is calculated as
in Lemma 2 can be modified as
1 M Tr P i : VTi P i C Tr Vi VTi Vi nQ
ð48Þ
The difference of (25) and (48) is the parameter Q. The role of the parameter Q is suppressing the convergence speed of the proposed algorithm, to compensate Assumption 1. The parameter Q and Assumption 1 together are applied to the proposed algorithm for the white assumption. The white assumption of Lemma 2 is applied to the convergence stage of the MSD learning curves. Thus, as the parameter Q is increased to high value, the prediction accuracy of the convergence speed would be also increased. The effect of the parameter Q is depicted in Fig. 4, with a projection order of 15. When the parameter Q is 1, the performance of the proposed algorithm is poor. However, as the parameter Q is increased from 1, the performance of the proposed algorithm is also increased, and when the value of the parameter Q is equal to and higher than 50, the proposed algorithm shows the best performance. Through the simulation result, we can know that the parameter Q can be chosen with high value.
2
σ 2e;0 ¼ E e20 ¼ Ef d0 þ vT0 w^ 0 g ¼ σ 2d ;
ð49Þ
σ 2e;i þ 1 ¼ ασ 2e;i þ ð1 αÞe2i ;
ð50Þ
2 d
where σ is the variance of di. The forgetting factor properly selected as
α is
1 ; nK
ð51Þ
α ¼ 1
where K is a positive integer value, because as the length of weight vector n increases, the forgetting factor should also be increased. The positive integer value K is determined by considering the input signal property. If the magnitude of input correlation is high, K should be also high. To check up the system change with an error signal, the covariance of error at steady state is the impact factor. The threshold of an error signal can be determined as the convergence value of ei covariance as n 2 o ~ i þr i lim E e2i ¼ lim E vTi w ð52Þ
i-1
i-1
lim E e2i C σ 2v TrðP1 Þ þ σ 2r 9 e2th ;
i-1
7. Non-stationary consideration For pursuing completely optimal step-size MFxAP algorithm, the non-stationary case of primary path filter should be considered. As the ANC system is designed for acoustic environment, the acoustic path can be interrupted by any obstacles. For example, if anybody passes through between a monitor speaker and a microphone, the acoustic path will be seriously changed. In such an unpredicted sudden change case, suitable algorithms should be constructed. One of them is the reset algorithm aiming instantaneous system change [18]. Through restarting count of iteration number for step size when the unknown weight vector is changed, the reset algorithm can reach the goal. To check the change of the 2 unknown weight vector, there is need of ei , with time average ^ o ¼ 0, value. As the optimal step-size algorithm starts with w
ð53Þ
for a given input signal. From (30), (53) can be rewritten as ð54Þ e2th C 1 þ ζ i σ 2r where
ζi ¼
nσ 2v μ2i g ð0Þ þ 2 μi μ2i αi : M 2μi μ2i
ð55Þ
ζ i seems to be complex for calculation, but almost terms in it can be got from (30) which already has to be calculated. Using (54), a decision point of reset is determined as σ 2e;i 4 γ e2th , when the unknown weight vector is suddenly changed. The design parameter γ has to satisfy
γ 4 E e20 =e2th C
σ 2v TrðP0 Þ : σ 2r 1 þ ζ i
ð56Þ
The reset algorithm for non-stationary consideration is summarized in Table 2.
5 0
8. Simulation
−5
−25
At the environment of ANC systems, some simulations have been carried out. The primary path and the secondary path consist of two ways. First, the primary path is set as unknown 32 tap length FIR filter which is randomly generated and the secondary path is modeled as the following 8-coefficient FIR filter which is given by [7]
−30
H ðzÞ ¼ 0:1 þ 0:2z 1 þ 0:9z 2 0:3z 3
MSD(dB)
−10 −15 −20
þ0:4z 4 0:1z 5 þ 0:1z 6 þ 0:05z 7 :
−35 −40
0
0.5
1
1.5
2 iteration
2.5
3
3.5
4 4
x 10
Fig. 4. MSD learning curves of the proposed algorithm with various parameter Q for a white signal. The projection order is 15. Q is varying from 1 to 50.
ð57Þ
The unknown primary path is generated at each trial with J wo J ¼ 1. The second way of paths is using the models P(z) and S(z), which are similar to the data of the disk included in [20]. The length of the primary path is 145, and that of the secondary path is 51. The secondary path has to filter xi at each iteration to generate pre-filtered input
J. Song, P. Park / Signal Processing 114 (2015) 100–111 5
Table 2 Reset algorithm for system change.
−5
2 e^ th ¼ ð1 þ ζ i Þσ 2r ,
−10
2
MSD(dB)
σ 2e^ ;i þ 1 ¼ ασ 2e^ ;i þ ð1 αÞe^ i if
M=1 M=2 M=4 M=6 M=8 M=10 M=12 M=14 M=16
0
for each i do
2 σ 2e^ ;i 4 γ e^ th then μi ¼ ½1 0T ; TrðPi Þ ¼
107
TrðP0 Þ
endif
−15 −20 −25
1 ^ i μi Vi VTi Vi ^ iþ1 ¼ w e^ i w
−30
endfor
−35 −40 −45 0
2000
4000
6000
8000
10000
12000
iteration
Fig. 5. MSD learning curves of the proposed algorithm with various projection orders and n¼ 32 for a white signal. The step size μ is 0.04.
5 M=1 M=2 M=4 M=6 M=8 M=10 M=12 M=14 M=16
0 −5 −10 MSD(dB)
vector vi . A simulation result of the proposed algorithm with paths of disk model [20] is already shown in Fig. 4. A Gaussian noise with N ð0; 1Þ is employed as the reference signal. The disturbance signal di is generated through (1). For the secondary noise, the variance is set as σ 2r ¼ 10 3 . The simulation results are obtained through 50 independent trials. The initial MSD value is set as TrðP0 Þ ¼ 1, and the initial weight of adaptive filter is set ^ 0 ¼ 0. The Mth-order input autocorrelation matrix of as w vi is numerically estimated from fvi j0 r i r1:2 104 g. Assumption 1 and the approximation (25) can be used, because the APA has self-whitening nature for colored reference signals [16], thus the MFxAP algorithm which uses the APA type can be modified with the proposed algorithm. However, as the MFxAP algorithm makes colored property to the input signal, a suitable projection order M should be chosen, considering the input correlation. Thus, the first simulations have done with various projection orders under the environments of a white signal, an AR signal and an ARMA signal. Each colored input signal is generated through
−15 −20 −25 −30 −35 −40 0
2000
4000
6000
8000
10000
12000
iteration
G1 ðzÞ ¼
1 ; 1 0:95z 1
ð58Þ
G2 ðzÞ ¼
1 þ0:5z 1 þ0:81z 2 : 1 0:59z 1 þ 0:4z 2
ð59Þ
The projection order M is increased from 1 to half length of coefficient of the adaptive filter. The randomly generated 32 length primary path, and the 8 tap secondary path are used for simulations. The suitable projection order is decided by choosing the one which shows the best performance. For example, the MSD learning curves of the proposed algorithm for a white input signal with various projection orders are shown in Fig. 5. The use of the optimal step size APA for a white signal shows the best performance with lowest projection order [11]. As the projection order is increased, the steady state is decreased. Otherwise, the best performance of the proposed algorithm for a white signal is shown with the projection order M¼6. The reason is that the MFxAP algorithm uses the pre-filtered input vector vi from the estimate of secondary path h, so the input vector to adaptive filter has colored property. That is why the proposed algorithm shows the best performance when the projection order sufficiently satisfies the whitening effect for Assumption 1. The signals from (58) and (59) have been used for the simulation of Figs. 6 and 7, respectively. From the
Fig. 6. MSD learning curves of the proposed algorithm with various projection orders and n¼ 32 for an AR input signal (58). The step size μ is 0.04.
simulation, the proposed algorithm shows the best performance for each input signals with the projection orders 8 and 14. For the ARMA signals, the appropriate projection order is almost closed to half length of the adaptive filter. It means that with high projection order, the MFxAP whitens the signal from the estimate of secondary path to establish Assumption 1 properly. As the colored property is increased, higher projection order is needed to show the best performance. The AR signal with first order filter (58) shows a little bit increase of colored property according to Fig. 6, but projection orders closed to one show poor performance. With the paths from [20], white signal is used for the reference signal. To find the best projection order, various projection order simulation has been done. From the simulation result in Fig. 8, the performance of the proposed algorithm is decreased, when the projection order is increased from 1 to 6. However, as the projection order is increased from 6 to 38, both the convergence rate and the steady state error are improved. For the projection order higher than 38, the proposed algorithm is getting better convergence rate, but the steady state error is getting
108
J. Song, P. Park / Signal Processing 114 (2015) 100–111
5 M=1 M=2 M=4 M=6 M=8 M=10 M=12 M=14 M=16
0 −5
MSD (dB)
−10 −15 −20 −25 −30 −35 −40 0
2000
4000
6000
8000
10000
12000
iteration
Fig. 7. MSD learning curves of the proposed algorithm with various projection orders and n¼ 32 for an ARMA input signal (59). The step size μ is 0.04.
5 0
With the evolving VSSAP algorithm of the best performance and the proposed algorithm, the simulation results are in Figs. 9 and 10. The evolving VSSAP algorithm uses the projection order of the proposed algorithm as the initial projection order. The proposed algorithm shows far below MSD learning curve, as well as shows the fastest convergence rate compared to other fixed-step-size MFxAP algorithms with different type input signals. Compared to evolving VSSAP algorithm, the convergence speed of both algorithms is alike, but the steady state of proposed algorithm shows lower curves (Fig. 11). In the environment of paths from [20], the proposed algorithm is also compared with the MFxAP algorithms and the evolving VSSAP algorithm. With the white signal, suitable projection order is 38 as the previous simulation result. However, it is too high to use in real-time ANC environments. Thus, the parameter Q is applied to the proposed algorithm, and according to the simulation result in Fig. 4, the parameter Q is set as 50. For the evolving VSSAP algorithm, parameters are determined: μmax ¼ 2; α ¼ 0:9999; μNup ¼ 0:2; μNdown ¼ 0:000001. The simulation result is depicted in Fig. 12, and the proposed algorithm shows the best performance with lower projection order 15 than
−5 5
−10
(a) MFXAP with mu=0.04 (b) MFXAP with mu=0.02 (c) MFXAP with mu=0.005 (d) Evolving VSSAP (e) proposed
−15
−5
−20
−10 MSD(dB)
MSD(dB)
0
−25 −30
−15 (a)
−20 (b)
−25 (d)
(c)
−30
−35
(e)
−35 0
0.5
1
1.5
2 iteration
2.5
3
3.5
4
−40
4
x 10
Fig. 8. MSD learning curves of the proposed algorithm with various projection orders and n¼32 for an white input signal. The step size μ is 0.04, and the secondary path is from [20]. The projection order is varying from 1 to 50.
worse. Thus, the projection order can be chosen, according to the environments and the requirements. The next simulations are done for the test of the proposed algorithm compared with MFxAP algorithms of various step sizes, but same projection order and the evolving variable step-size algorithm [12]. The evolving VSSAP algorithm based on [15] uses the structure of the MFxAP algorithm. The projection orders of the proposed algorithm are determined from the above simulations that compare the proposed algorithm with various projection orders. The initial projection order of the evolving VSSAP algorithm is determined as the projection order of the proposed algorithm, and some tuning parameters are determined through many times simulations. The first comparing simulation is done for the paths of randomly generated 32 length primary path, and the 8 tap secondary path. For the evolving VSSAP algorithm, some tuning parameters are determined: μmax ¼ 2; α ¼ 0:99; μNup ¼ 0:2; μNdown ¼ 0:00001. It was very difficult to get the best performance with the evolving VSSAP algorithm [12].
−45
0
2000
4000
6000
8000
10000
12000
14000
iteration
Fig. 9. MSD learning curves of the proposed algorithm, evolving VSSAP algorithm and standard fixed-step-size MFxAP algorithm with various step sizes for a white signal.
5 (a) MFXAP with mu=0.04 (b) MFXAP with mu=0.02 (c) MFXAP with mu=0.005 (d) Evolving VSSAP (e) proposed
0 −5 −10 MSD(dB)
−40
−15
(a) (b)
−20
(c)
−25
(e)
(d)
−30 −35 −40
0
2000
4000
6000
8000
10000
12000
14000
iteration
Fig. 10. MSD learning curves of the proposed algorithm, evolving VSSAP algorithm and fixed-step-size MFxAP algorithm with various step sizes for an AR input signal (58).
J. Song, P. Park / Signal Processing 114 (2015) 100–111
5
5 (a) MFXAP with mu=0.04 (b) MFXAP with mu=0.02 (c) MFXAP with mu=0.005 (d) Evolving VSSAP (e) proposed
0 −5
−5
(a)
−15
−10
−20
(b)
−25
(c)
(d)
MSD(dB)
MSD(dB)
MFXAP with mu=0.1 MFXAP with mu=0.03 Evolving−VSSAP proposed
0
−10
(e)
−15 −20
−30 −35 −40
109
−25 0
2000
4000
6000
8000
10000
12000
14000
−30
iteration −35
Fig. 11. MSD learning curves of the proposed algorithm, evolving VSSAP algorithm and fixed-step-size MFxAP algorithm with various step sizes for an ARMA input signal (59).
0
0.5
1
1.5
Fig. 13. MSD learning curves of the proposed algorithm, fixed-step-size MFxAP algorithm with various step sizes, and evolving VSSAP algorithm, for an white signal (Q ¼50, α ¼ 0:975, σ 2r ¼ 10 2 ).
5 MFXAP with mu=0.02 MFXAP with mu=0.005 Evolving−VSSAP proposed
0
−5
5 MFXAP with mu=0.04 MFXAP with mu=0.02 MFXAP with mu=0.005 proposed
0 −10 −5 −15 −10
−20
MSD(dB)
MSD(dB)
2 5
x 10
iteration
−25
−15
−30
−20
−35
−25
−40 0
0.5
1
1.5 iteration
2
2.5
3
−30
4
x 10
−35
Fig. 12. MSD learning curves of the proposed algorithm, fixed-step-size MFxAP algorithm with various step sizes, and evolving VSSAP algorithm, for an white signal (Q ¼50, α ¼ 0:9999, σ 2r ¼ 10 3 ).
the optimal projection order 38. As the projection order is decreased, the computational complexity is also decreased, maintaining the best performance among other algorithms. Additional simulation has been carried out, in order to evaluate the proposed algorithm with low SNR. The variance of the secondary noise is set as σ 2r ¼ 10 3 , and the parameter α of the evolving VSSAP algorithm is set as 0.975, but the other factors of the simulation environments are same as Fig. 4. The parameter α is changed, because the noise variance is also changed. The simulation result is depicted in Fig. 12. The proposed algorithm shows the best performance with the fastest convergence rate and the lowest steady-state error, even though the variance of the secondary noise is increased. In these simulation environments, the optimal step size for one trial is depicted in Fig. 13. It shows that the optimal step size is higher than 0 and lower than 1. The last simulation is done to test the reset algorithm on non-stationary consideration. To check up the biggest change of the unknown primary path, the sign of it is
0
2000
4000
6000
8000
10000
iteration
Fig. 14. MSD learning curves of the proposed algorithm and fixed-stepsize MFxAP algorithm with various step sizes for an ARMA signal in a non-stationary environment in which the sign of the unknown primary path is changed in the middle of the test interval.
assumed to change at iteration 5000. The other environmental conditions are same with the simulation of Fig. 10. With the ARMA signal, (59), a positive integer K ¼2 is used. The design parameter, γ ¼ 1700, is determined empirically on the consideration of (56). When the parameter is smaller than this value, the detection of weight vector change is barely done, while it is larger, much more resets including undesirable ones is done. With the proper parameter, the wanted reset is done successfully, and the proposed algorithm traces the changed weight vector as it did before reset (Fig. 14). 9. Conclusion This paper proposed an MFxAP algorithm with the optimal step size for ANC systems. To improve the MFxAP algorithm, it analyzes the MSD learning curve of the MFxAP
110
J. Song, P. Park / Signal Processing 114 (2015) 100–111
algorithm and derives the step size at each iteration for ANC systems. With the pre-filtered input, it considers the crosscorrelation between the secondary noise vector and the current weight error vector to achieve more accurate prediction of the learning curve of the MFxAP algorithm. With numerically estimated Mth-order input autocorrelation matrix of vi, but without any parameters to be tuned in the basic algorithm, the proposed algorithm achieves optimal performance in two senses as the simulation results show. One is the lowest steady-state error and the other is the fastest convergence rate.
ΦB ði; kÞ 9 Bi 1 Bi 2 ⋯Bk ;
ΦB ði; iÞ 9 In :
ðB:4Þ
S , Ai ¼ Z, Bi ¼ Φðiþ 1; iÞ, and Ci ¼ σ 2r μi Z Let X i¼ 1 i T Vi Vi VTi , then from (14) and (16), we can obtain 1 X
S ¼ σ 2r
μi m Zm VTi m Vi m
1
VTi m Φ ði þ 1; i mþ 1Þ: T
ðB:5Þ
m¼1
From the fact that Z is nilpotent (i.e., Zj ¼ 0) if j ZM, the corollary is concluded. Appendix C. Proof of Lemma 1 Under Assumption 1, the term of Vi is
Acknowledgment
Vi VTi Vi This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF2014R1A1A2055122).
4
Riþ1 T
Si þ 1
Pi þ 1
2
T 6Z 4 0
2 4
Riþ1 T
Si þ 1
3
Si þ 1
Si þ 1 Pi þ 1
2 T 6Z 4 0
02
5 ¼ E@4
1 μi Vi VTi Vi
k ¼ k0 þ 1 0 X
2
5¼4
ðA:1Þ
3" #" #T r ri 5 i Φði þ 1; iÞ w~ i w~ i
ðA:2Þ
3 0" #" #T 1 r ri 5E@ i jV i A ~i ~i Φði þ 1; iÞ w w
1
ðA:3Þ
~ Ti T and for given Vi by V i , where D is the cross term of ½rTi w ~ i with vi , Eð½rTi r i þ 1 . From the independence of ri and w ~ Ti T ½rTi w ~ Ti jV i Þ is equivalent to Eð½rTi w ~ Ti T ½rTi w ~ Ti jV i 1 Þ, and w the theorem is proved.
From the recursive equation ðB:1Þ
the closed form can be obtained as Xi ¼ ¼
T B ði; kÞ
1 ¼ Zj þ VTiþ j Vi VTi Vi
ΦA ði; i j þ1ÞCi j Φ
0
k X
J vi þ k J 2
vi þ k aT1 k
ðC:1Þ
J vi þ k J 2
vTiþ j ⋯vTiþ j þ 1 M
k ¼ 1M
iv aT iþk 1k J vi þ k J 2 ðC:2Þ
With (C.1) and (C.2), the lemma can be proved by mathematical induction. As the first step, let us take j¼0, then (20) is satisfied as follows: 1 left hand side ðLHSÞ ¼ Φði þ 1; i þ 1ÞVi VTi Vi 1 ¼ Vi VTi Vi ¼ right hand side ðRHSÞ:
ðB:2Þ
j0 1
¼ ∏
1 0 0 1 μi þ j0 k Vi þ j0 VTiþ j0 Vi þ j0 Zj Zj T :
ðC:4Þ
Then, from the assumption with (7) and (8), for the case of j ¼ j0 þ 1, the left-hand side (LHS) of (20) can be rewritten as 1 0 Φ iþ j0 þ 2; i þ 1 Vi VTi Vi Zðj þ 1ÞT ¼ Φ i þ j0 þ2; i þj0 þ 1
0
j 1
¼ ∏
k¼0
ðB:3Þ
ðC:3Þ
The next step is assuming that (20) holds for unspecified values of j ¼ j0 1 Φ iþ j0 þ 1; i þ 1 Vi VTi Vi ZjT
where, for i4 k, and
ΦA ði; iÞ 9 IM
k ¼ 1M
k0 h X
Φ iþ j0 þ 2; i þ 1 Vi VTi Vi T B ði; i j þ 1Þ
vi þ k aT1 k
0
j¼1
ΦA ði; kÞ 9Ai 1 Ai 2 ⋯Ak ;
J vi þ k þ j J
2
0
k X
þ
Φ i þ j0 þ 1; i þ1 Vi VTi Vi
ΦA ði; kÞCk 1 Φ
k ¼ 1 1 X
J vi þ k J 2
k ¼ 1M
where k ¼ minðM; jÞ M. From (C.1), pre-multiplying VTiþ j leads to
k¼0
Appendix B. Proof of Corollary 2
i X
vi þ k aT1 k
¼ Zj :
0
Xi þ 1 ¼ Ai Xi BTi þ Ci ;
J vi þ k J
2
0
k X
þ
k ¼ 1M
0
3 VTi 7 5 þ σ 2r a1 aT1 ; ΦT ði þ 1; iÞ
μi VTi Vi
J vi þ k J 2
1 ¼ Vi þ j VTiþ j Vi þ j Zj þ
1
Z 1 μi Vi VTi Vi
vi þ k aT1 k
vi þ k þ j aT1 k Zj
k ¼ 1M
3 1 VTi 7 2 C 5 þr i þ 1 a1 aT1 þ DjV i A ΦT ðiþ 1; iÞ
μi VTi Vi 3
Z
0 X
0 X vi þ k aT1 k
¼
Appendix A. Proof of Theorem 1 From (11) and (12), it leads to 2 3 0" #" #T 1 R i þ 1 Si þ 1 ri þ 1 ri þ 1 4 T 5 ¼ E@ A ~ iþ1 ~ i þ 1 jV i w w Si þ 1 Pi þ 1
¼
k ¼ 1M
¼
2
1
1 μ i þ j0 k
1
1
Zðj
0
0
Zðj
þ 1ÞT
ðC:5Þ
þ 1ÞT
1 In μi þ j0 þ 1 Vi þ j0 þ 1 VTiþ j0 þ 1 Vi þ j0 þ 1
o 1 0 0 Zj Zðj þ 1ÞT VTiþ j0 þ 1 Vi þ j0 VTiþ j0 Vi þ j0
ðC:6Þ
J. Song, P. Park / Signal Processing 114 (2015) 100–111
Φ i þ j0 þ 2; i þ 1 Vi VTi Vi j0 1
¼ ∏
k¼0
1 μi þ j0 k
1
0
Zðj
þ 1ÞT
1 0 0 Vi þ j0 VTiþ j0 Vi þ j0 Zj Zðj þ 1ÞT
1 VTiþ j0 þ 1 Vi þ j0 μi þ j0 þ 1 Vi þ j0 þ 1 VTiþ j0 þ 1 Vi þ j0 þ 1 1 0 0 VTiþ j0 Vi þ j0 Zj Zðj þ 1ÞT :
ðC:7Þ
Then, for 1 M r k r0 and j k Z M, note that aT1 k Zj ¼ 0, and from (C.1), we can say that 1 0 1 Zj ¼ Vi þ j0 þ 1 VTiþ j0 þ 1 Vi þ j0 þ 1 Vi þ j0 VTiþ j0 Vi þ j0 0
0
ZZj þ
k X
vi þ j0 þ k aT1 k j0 Z ‖vi þ j0 þ k ‖2 k ¼ 1M
ðC:8Þ
1 0 1 0 Zj ¼ Vi þ j0 þ 1 VTiþ j0 þ 1 Vi þ j0 þ 1 Zðj þ 1Þ Vi þ j0 VTiþ j0 Vi þ j0 ðC:9Þ Thus, with (C.2) and (C.9), (C.7) leads to 1 0 0 1 μi þ j0 k Vi þ j0 þ 1 VTiþ j0 þ 1 Vi þ j0 þ 1 Zðj þ 1Þ Zðj þ 1ÞT
j0 1
¼ ∏
k¼0
1 0 0 μi þ j0 þ 1 Vi þ j0 þ 1 VTiþ j0 þ 1 Vi þ j0 þ 1 Zðj þ 1Þ Zðj þ 1ÞT
ðC:10Þ 0
j
¼ ∏
k¼0
1 0 0 1 μi þ j0 þ 1 k Vi þ j0 þ 1 VTiþ j0 þ 1 Vi þ j0 þ 1 Zðj þ 1Þ Zðj þ 1ÞT
ðC:11Þ Then, by mathematical induction, it holds for all i; jZ 0, and the lemma is concluded. References [1] S. Elliott, P. Nelson, Active noise control, IEEE Signal Process. Mag. 10 (4) (1993) 12–35. [2] M. Rupp, A. Sayed, Robust fxlms algorithms with improved convergence performance, IEEE Trans. Speech Audio Process. 6 (1) (1998) 78–85.
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