Widely linear general Kalman filter for stereophonic ... - CiteSeerX
Signal Processing 94 (2014) 570–575
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Widely linear general Kalman filter for stereophonic acoustic echo cancellation Constantin Paleologu a,n, Jacob Benesty b, Silviu Ciochină a a b
University Politehnica of Bucharest, 1-3, Iuliu Maniu Blvd., 061071 Bucharest, Romania INRS-EMT, University of Quebec, Montreal, QC, Canada H5A 1K6
a r t i c l e in f o
abstract
Article history: Received 8 April 2013 Received in revised form 24 June 2013 Accepted 2 August 2013 Available online 9 August 2013
The stereophonic acoustic echo cancellation (SAEC) problem is usually modelled as a twoinput/two-output system with real random variables. Recently, the SAEC scheme was recast as a single-input/single-output system with complex random variables, thanks to the widely linear (WL) model. In this paper, we motivate the use of a more general form of the Kalman filter with the WL model for SAEC. Simulation results indicate that this algorithm outperforms the recursive least-squares (RLS) algorithm, which is usually considered as the benchmark for SAEC. & 2013 Elsevier B.V. All rights reserved.
Keywords: Stereophonic acoustic echo cancellation (SAEC) Widely linear (WL) model Kalman filter Adaptive filters Recursive least-squares (RLS) algorithm
1. Introduction Stereophonic acoustic echo cancellation (SAEC) is a very challenging system identification problem [1]. Usually, an SAEC system consists of four adaptive filters aiming at identifying four echo paths from two loudspeakers to two microphones. The main difficulty comes from the fact that the loudspeaker (input) signals are linearly related, which results in the so-called nonuniqueness problem [2]. This issue can be addressed by manipulating the signals transmitted to the near-end room, e.g., using a preprocessor on the loudspeaker signals to make them less coherent [3], but without affecting much the stereo perception and the sound quality. The adaptive filters used in SAEC should exploit the cross-correlation between the channels [4]. In this context, the most interesting solutions belong to the recursive
n
Corresponding author. Tel.: +40 21 402 4634. E-mail addresses: [email protected] (C. Paleologu), [email protected] (J. Benesty), [email protected] (S. Ciochină). 0165-1684/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2013.08.001
least-squares (RLS) family. Due to their convergence features, these algorithms were preferred in many real-world applications [5,6]. Recently, we proposed a different approach for the SAEC problem [1,7], by using the widely linear (WL) model [8]. Basically, the classical two-input/two-output system with real random variables was recast as a single-input/ single-output system with complex random variables. As a consequence, the four real-valued acoustic impulse responses are converted to one complex-valued impulse response. One advantage of this approach is that instead of handling two (real) output signals separately, we only handle one (complex) output signal, which is convenient for the main challenges of SAEC. In this paper, we derive a general Kalman filter (GKF) with the WL model for SAEC, namely the WL-GKF. The term “general” refers to a different approach we propose, i.e., a block of time samples is considered at each iteration, instead of one time sample (as in the conventional approach). The main motivation behind this work is the appealing performance of the Kalman filter for echo cancellation [9–11]. Also, the WL complex Kalman filters
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Fig. 1. The WL model for SAEC.
[12,13] were found to be attractive for many applications. The proposed algorithm has inherited some similarities with the WL augmented complex Kalman filter presented in [13]. However, the WL-GKF is derived based on a state variable model suitable for the SAEC problem. The proposed WL-GKF joins the advantages of the WL model (as described before) and the good features of the GKF [11]. Simulation results indicate that the developed algorithm outperforms the RLS counterpart. Consequently, it could represent an attractive alternative in SAEC. 2. The WL model for SAEC In this section, we briefly review the WL model for SAEC (Fig. 1) [1,7]. Let us denote the two input (or loudspeaker) signals by xL ðnÞ and xR ðnÞ (i.e., “left” and “right”), and the two output (or microphone signals) by dL ðnÞ and dR ðnÞ, where n is the time index. Therefore, the microphone signals are obtained as dL ðnÞ ¼ yL ðnÞ þ vL ðnÞ;
ð1Þ
dR ðnÞ ¼ yR ðnÞ þ vR ðnÞ;
ð2Þ
where yL ðnÞ and yR ðnÞ denote the stereo echo signals, and vL ðnÞ and vR ðnÞ are the near-end signals (i.e., noise or a combination of noise and near-end speech). The echo signals can be modelled as [2,3] T
T
yL ðnÞ ¼ ht;LL xL ðnÞ þ ht;RL xR ðnÞ;
ð3Þ
T yR ðnÞ ¼ ht;LR xL ðnÞ
ð4Þ
þ
T ht;RR xR ðnÞ;
where ht;LL ; ht;RL ; ht;LR , and ht;RR are L-dimensional vectors of the loudspeaker-to-microphone acoustic impulse responses
(the subscript t stands for “true”), the superscript T denotes transposition, and xL ðnÞ ¼ ½xL ðnÞ xL ðn1Þ ⋯ xL ðnL þ 1ÞT xR ðnÞ ¼ ½xR ðnÞ xR ðn1Þ ⋯ xR ðnL þ 1ÞT
comprise the L most recent loudspeaker signal samples. In this context, the main goal is to estimate the four acoustic impulse responses, ht;LL ; ht;RL ; ht;LR , and ht;RR , from the microphone signals dL ðnÞ and dR ðnÞ. Next, let us form the complex random variable (CRV): ð5Þ dðnÞ ¼ dL ðnÞ þ jdR ðnÞ ¼ yðnÞ þ vðnÞ; pffiffiffiffiffiffiffi where j ¼ 1, yðnÞ ¼ yL ðnÞ þ jyR ðnÞ, and vðnÞ ¼ vL ðnÞþ jvR ðnÞ. Also, let us define the complex random vector: xðnÞ ¼ xL ðnÞ þ jxR ðnÞ:
ð6Þ
Consequently, the (complex) echo signal can be obtained as H
yðnÞ ¼ ht xðnÞ þ ht ′H xn ðnÞ;
ð7Þ
where the superscripts H and n denote conjugate transpose and conjugate, respectively, and ht ¼ ht;1 þ jht;2 ;
ð8Þ
h′t ¼ h′t;1 þ jh′t;2 ;
ð9Þ ′
with ht;1 ¼ ðht;LL þ ht;RR Þ=2, ht;2 ¼ ðht;RL ht;LR Þ=2, ht;1 ¼ ′ ðht;LL ht;RR Þ=2, and ht;2 ¼ ðht;RL þ ht;LR Þ=2. Using the previous notation, we can express (7) as H ~ yðnÞ ¼ h~ t xðnÞ;
ð10Þ
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T ′T ~ where h~ t ¼ ½ht ht T and xðnÞ ¼ ½xT ðnÞ xH ðnÞT . Therefore, the complex observation is H ~ þ vðnÞ: dðnÞ ¼ h~ t xðnÞ
ð11Þ
It can be noticed that we are dealing now with a complex acoustic impulse response of length 2L, i.e., h~ t , whose complex input and output are, respectively, x(n) and d(n). From (7) or (10), we recognize the WL model for CRVs proposed in [8]. Thanks to the WL model, the two-input/ two-output system with real random variables was converted to a single-input/single-output system with CRVs. This approach is in line with the duality principle [14]. In this context, the new goal is to estimate the system ~ h~ t with an adaptive filter, hðnÞ, in such a way that for a reasonable value of n, we have for the (normalized) misalignment: ~ J h~ t hðnÞ J 22 rι; 2 ~ J ht J
ð12Þ
2
where ι is a predetermined small positive number and J J 2 is the ℓ2 norm. As compared to the classical SAEC approach [2,3], which requires four adaptive filters of length L, the WL model involves only one filter of length 2L. On the other hand, we are dealing now with CRVs having both real and imaginary parts. Apparently, the overall complexity of the WL model is similar to the classical approach. However, there are other aspects that should be taken into account in practice (e.g., double-talk detection, echo/noise suppression, implementation issues, etc.), which make more convenient to handle only one adaptive filter instead of four such systems [1]. 3. State variable model for WL SAEC Let us express (11) by considering the P most recent time samples of the microphone signal, i.e., dðnÞ ¼ ½dðnÞ dðn1Þ ⋯ dðnP þ 1ÞT n
~ ðnÞh~ ðnÞ þ vðnÞ; ¼X t T
ð13Þ
where ~ ~ ~ ~ XðnÞ ¼ ½xðnÞ xðn1Þ ⋯ xðnP þ 1Þ
ð14Þ
is the input signal matrix of size 2L P and the noise signal vector, vðnÞ, is defined similar to dðnÞ. We also consider that the system to be identified is time dependent, i.e., h~ t ðnÞ. ~ T ðnÞ is the measurement matrix and In our context, X x(n) is considered as deterministic. Expression (13) is called the observation equation. We assume that h~ t ðnÞ is a zero-mean complex random vector, which follows a simplified first-order Markov model [15], i.e., h~ t ðnÞ ¼ h~ t ðn1Þ þ wðnÞ;
ð15Þ
where wðnÞ is a zero-mean circular complex white Gaussian noise signal vector, which is uncorrelated with h~ t ðn1Þ and vðnÞ. The correlation matrix of wðnÞ is assumed to be R w ðnÞ ¼ s2w ðnÞI2L , where I2L is the 2L 2L identity matrix. The variance, s2w ðnÞ, captures the uncertainties in h~ t ðnÞ. Expression (15) is called the state equation.
Now, the basic problem may be restated as follows. Given the two fundamental equations, i.e., (13) and (15), our objective is to find the optimal recursive estimator of ~ h~ t ðnÞ denoted by hðnÞ. In this context, the values of s2w ðnÞ play a major role in the performance of the estimator, i.e., small values of s2w ðnÞ imply a good misalignment but a poor tracking, while large values of s2w ðnÞ (meaning that the uncertainties in the echo path are high) imply a good tracking but a high misalignment. Consequently, the values of s2w ðnÞ highly determine the tracking abilities and the convergence of the Kalman filter to be derived for SAEC with the WL model. 4. WL general Kalman filter It is known that, in the context of the linear sequential Bayesian approach, the optimum estimate of the state vector, h~ t ðnÞ, has the form [16] ~ ~ hðnÞ ¼ hðn1Þ þ KðnÞen ðnÞ;
ð16Þ
where KðnÞ is the Kalman gain matrix and ~ T ðnÞh~ n ðn1Þ eðnÞ ¼ dðnÞX
ð17Þ
is the a priori error signal vector, which is obtained using the adaptive filter coefficients at time n1. The a posteriori error signal vector is defined based on the adaptive filter coefficients at time n, i.e., ~ T ðnÞμ~ n ðnÞ þ vðnÞ; ~ T ðnÞh~ n ðnÞ ¼ X ϵðnÞ ¼ dðnÞX
ð18Þ
where ~ ~ μðnÞ ¼ h~ t ðnÞhðnÞ
ð19Þ
is the state estimation error or a posteriori misalignment. ~ The correlation matrix of μðnÞ is ~ μ~ H ðnÞ; R μ~ ðnÞ ¼ E½μðnÞ
ð20Þ
where E½ denotes mathematical expectation. We can also define the a priori misalignment as ~ ~ ~ ¼ μðn1Þ þ wðnÞ; mðnÞ ¼ h~ t ðnÞhðn1Þ
ð21Þ
for which its correlation matrix is ~ ~ ðnÞ ¼ Rμ~ ðn1Þ þ s2w ðnÞI2L : R m~ ðnÞ ¼ E½mðnÞ m H
ð22Þ
The a priori misalignment appears in the a priori error signal vector as ~ ðnÞm ~ ðnÞ þ vðnÞ: eðnÞ ¼ X T
n
ð23Þ
The Kalman gain matrix is obtained by minimizing the criterion [16]: J ðnÞ ¼
1 tr R μ~ ðnÞ 2L
ð24Þ
with respect to KðnÞ, where tr½ denotes the trace of a square matrix. We easily find that ~ ~ H ðnÞR ~ ðnÞXðnÞ ~ X þ s2v IP 1 ; KðnÞ ¼ Rm~ ðnÞXðnÞ½ m where
s2v
ð25Þ
¼ E½jvðnÞj , IP is the P P identity matrix, and 2
~ H ðnÞR m~ ðnÞ: R μ~ ðnÞ ¼ ½I2L KðnÞX
ð26Þ
C. Paleologu et al. / Signal Processing 94 (2014) 570–575
573
gR
gL 0.01
0.01
0
−0.01
0
0
500
1000
1500
2000
−0.01
0
500
1000
ht,LL Amplitude
0.01
2000
0.01
0
0
−0.01
−0.01 0
200
400
0
200
ht,RL
400
ht,RR
0.01
Amplitude
1500
ht,LR
0.01
0
0
−0.01
−0.01 0
200
400
0
Samples
200
400
Samples
Fig. 2. Acoustic impulse responses used in simulations.
Rm~ ðnÞ ¼ R μ~ ðn1Þ þ s2w ðnÞI2L ;
ð28Þ
~ H ðnÞR m~ ðnÞXðnÞ ~ þ s2v IP ; Ren ðnÞ ¼ X
ð29Þ
and do not behave the same way in practice. The two parameters s2w ðnÞ and s2v in the WL-GKF (for which the WL-RLS does not depend on) allow us to better control it. The first (and perhaps the most important) parameter is s2w ðnÞ, which plays a major role in the overall performance of the algorithm, as explained before. Using the ℓ2 norm in both sides of (15), together with the approximation J wðnÞ J 22 2Ls2w ðnÞ (which is valid when 2L b 1), ~ replacing h~ t ðnÞ by its estimate hðnÞ, and also considering the contribution of the model's order, we can evaluate
1 ~ KðnÞ ¼ R m~ ðnÞXðnÞR en ðnÞ;
ð30Þ
s b 2w ðnÞ ¼
eðnÞ ¼ dðnÞXT ðnÞh~ ðn1Þ;
ð31Þ
~ ~ hðnÞ ¼ hðn1Þ þ KðnÞen ðnÞ;
ð32Þ
~ H ðnÞRm~ ðnÞ: Rμ~ ðnÞ ¼ ½I2L KðnÞX
ð33Þ
The estimation from (36) is designed to achieve a proper compromise between good tracking and low misalignment. When the algorithm starts to converge or when there is an abrupt change of the system (e.g., when the ~ echo path changes), the difference between hðnÞ and ~ hðn1Þ is significant, so that the parameter s b 2w ðnÞ takes large values, thus providing fast convergence and tracking. On the other hand, when the algorithm starts to converge ~ ~ to its steady-state, the difference between hðnÞ and hðn1Þ reduces, thus leading to small values of s b 2w ðnÞ and, consequently, to a low misalignment. The second parameter to be found is the noise power, s2v . Usually, it can be estimated during silences of the nearend talker, i.e., in the single-talk scenario [18]. However, this is not always an easy task. The most critical situation in echo cancellation is the double-talk case, when the near-end signal is a combination of the background noise and the near-end speech. However, the variance of the near-end signal naturally appears within the algorithm. This allows us to better control the filter, in terms of its robustness to near-end signal variations. In this scenario,
The correlation matrix of the a priori error signal vector is ~ ðnÞR ~ ðnÞXðnÞ ~ þ Ren ðnÞ ¼ X m H
s2v IP ;
ð27Þ
whose inverse appears explicitly in the Kalman gain matrix. Consequently, the following equations summarize the WL general Kalman filter (WL-GKF):
n
~ The initialization is hðnÞ ¼ 0 and Rμ~ ð0Þ ¼ εI2L , where ε is a small positive constant. When s2w ðnÞ ¼ 0, we have lim Rμ~ ðnÞ ¼ 0;
ð34Þ
lim KðnÞ ¼ 0;
ð35Þ
n-1
n-1
and, obviously, the WL-GKF will never be able to track the changes in h~ t . On the other hand, for large values of s2w ðnÞ, the Kalman gain matrix never goes to zero, which allows the update equation (32) to stay “alert” to any possible random changes of the system. The relation between the Kalman filter and the RLS algorithm is well established [17]. For P ¼1, the WL-GKF has striking resemblances with the WL-RLS algorithm [1,7]. However, the two algorithms are very much different
~ ~ J hðnÞ hðn1Þ J 22 : 2PL
ð36Þ
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the parameter s2v can be estimated as proposed in [19] or [20]. Some recent works, e.g., [11,21], show that the GKF using the s2v estimate proposed in [19] is very robust to large near-end signals (like double-talk).
5 WL−GKF, P=1 WL−GKF, P=2 WL−GKF, P=4
Misalignment (dB)
0 −5 −10 −15 −20
Misalignment (dB)
Simulations are performed in the context of SAEC, as described in Fig. 1. The acoustic impulse responses used for the far-end and near-end locations are shown in Fig. 2 [1]. Impulse responses in the far-end [i.e., g L ðnÞ and g R ðnÞ] have 2048 coefficients, while the length of the impulse responses in the near-end [i.e., ht;LL ðnÞ, ht;RL ðnÞ, ht;LR ðnÞ, and ht;RR ðnÞ] is L ¼512. The length of the WL adaptive filters used in the experiments is 2L ¼ 1024; sample rate is 8 kHz. Two source signals are used: a white Gaussian signal and a speech sequence. The input signals are preprocessed with positive and negative half-wave rectifiers, using a distortion parameter αr ¼ 0:3 [1]. All simulations are performed in the single-talk scenario. We can define the stereo echo-to-noise ratio (SENR) [which is equivalent to the signal-to-noise ratio (SNR)] as SENR ¼ s2y =s2v , where s2y ¼ E½jyðnÞj2 is the variance of y(n). In our simulations, the background noise is independent white Gaussian distributed, whose level is set such that SENR ¼30 dB. We assume that the variance of the noise, s2v , is available in all the simulations. The performance measures is the normalized misalignment (in dB), which is computed based on (12). In order to evaluate the tracking capabilities of the algorithms, the impulse responses in the near-end location are shifted to the right by 12 samples in the middle of each experiment. The performance of the WL-GKF using different model orders is presented in Fig. 3, where the source signal is white and Gaussian. We can see that the performance of the WL-GKF improves when the value of P increases. However, this improvement is not very significant for P 42. In Fig. 4, the WL-GKF with P ¼1 is compared to the WL-RLS algorithm using different values of the forgetting factor ð0 oλ r 1Þ. This specific parameter of the WL-RLS
WL−RLS, λ=1−1/(2L) WL−RLS, λ=1−1/(40L) WL−GKF, P=1
0
−5
−10
−15
−20
−25 0
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Time (seconds) Fig. 4. Misalignment of the WL-RLS algorithm with two different values of the forgetting factor, λ, and misalignment of the WL-GKF with P ¼1. The source signal is white Gaussian.
10 WL−RLS, λ=1−1/(40L) WL−GKF, P=1 WL−GKF, P=2
5
Misalignment (dB)
5. Simulation results
5
0
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−20 0
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Time (seconds) Fig. 5. Misalignment of the WL-RLS algorithm with λ ¼ 11=ð40LÞ, and misalignment of the WL-GKF with P¼ 1 and P¼ 2. The source signal is speech.
algorithm addresses the compromise between convergence rate/tracking capabilities on the one hand and misadjustment/stability on the other hand [22]. It can be noticed that the WL-GKF compromises better between the tracking capability and steady-state misalignment level, as compared to the WL-RLS algorithm. In Fig. 5, the source signal is a speech sequence and the WL-GKF (with P ¼1 and P ¼2) is compared to the WL-RLS algorithm using λ ¼ 11=ð40LÞ. Overall, this result also proves that the WL-GKF performs better as compared to the WL-RLS algorithm. 6. Conclusions
−25 −30 0
5
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15
20
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30
Time (seconds) Fig. 3. Misalignment of the WL-GKF with different values of P. The source signal is white Gaussian.
Due to their convergence features, RLS-based algorithms are frequently involved in SAEC. In this paper, we have motivated the use of the Kalman filter in this application. Indeed, we have developed a general Kalman filter (by considering, at each iteration, a block of time samples instead of one time sample as it is the case in the
C. Paleologu et al. / Signal Processing 94 (2014) 570–575
conventional approach) with the WL model for SAEC. As compared to the WL-RLS algorithm, the proposed WL-GKF compromises better between the tracking capability and steady-state misalignment level. Of course, the main issue remains the computational complexity, which will be addressed in future work by developing fast versions of the WL-GKF.
Acknowledgment This work was supported by the UEFISCDI Romania under Grants PN-II-RU-TE no. 7/5.08.2010 and PN-II-IDPCE-2011-3-0097. The authors would like to thank the Handling Editor and the reviewers for the valuable comments and suggestions. References [1] J. Benesty, C. Paleologu, T. Gänsler, S. Ciochină, A Perspective on Stereophonic Acoustic Echo Cancellation, Springer-Verlag, Berlin, Germany, 2011. [2] M.M. Sondhi, D.R. Morgan, J.L. Hall, Stereophonic acoustic echo cancellation—an overview of the fundamental problem, IEEE Signal Processing Letters 2 (8) (1995) 148–151. [3] J. Benesty, D.R. Morgan, M.M. Sondhi, A better understanding and an improved solution to the specific problems of stereophonic acoustic echo cancellation, IEEE Transactions on Speech and Audio Processing 6 (3) (1998) 156–165. [4] S. Emura, Y. Haneda, A. Kataoka, S. Makino, Stereo echo cancellation algorithm using adaptive update on the basis of enhanced inputsignal vector, Signal Processing 86 (2006) 1157–1167. [5] P. Eneroth, S.L. Gay, T. Gänsler, J. Benesty, A real-time implementation of a stereophonic acoustic echo canceler, IEEE Transactions on Speech and Audio Processing 9 (5) (2001) 513–523. [6] T. Gänsler, J. Benesty, New insights into the stereophonic acoustic echo cancellation problem and an adaptive nonlinearity solution, IEEE Transactions on Speech and Audio Processing 10 (5) (2002) 257–267.
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Signal Processing journal homepage: www.elsevier.com/locate/sigpro
Fast communication
Widely linear general Kalman filter for stereophonic acoustic echo cancellation Constantin Paleologu a,n, Jacob Benesty b, Silviu Ciochină a a b
University Politehnica of Bucharest, 1-3, Iuliu Maniu Blvd., 061071 Bucharest, Romania INRS-EMT, University of Quebec, Montreal, QC, Canada H5A 1K6
a r t i c l e in f o
abstract
Article history: Received 8 April 2013 Received in revised form 24 June 2013 Accepted 2 August 2013 Available online 9 August 2013
The stereophonic acoustic echo cancellation (SAEC) problem is usually modelled as a twoinput/two-output system with real random variables. Recently, the SAEC scheme was recast as a single-input/single-output system with complex random variables, thanks to the widely linear (WL) model. In this paper, we motivate the use of a more general form of the Kalman filter with the WL model for SAEC. Simulation results indicate that this algorithm outperforms the recursive least-squares (RLS) algorithm, which is usually considered as the benchmark for SAEC. & 2013 Elsevier B.V. All rights reserved.
Keywords: Stereophonic acoustic echo cancellation (SAEC) Widely linear (WL) model Kalman filter Adaptive filters Recursive least-squares (RLS) algorithm
1. Introduction Stereophonic acoustic echo cancellation (SAEC) is a very challenging system identification problem [1]. Usually, an SAEC system consists of four adaptive filters aiming at identifying four echo paths from two loudspeakers to two microphones. The main difficulty comes from the fact that the loudspeaker (input) signals are linearly related, which results in the so-called nonuniqueness problem [2]. This issue can be addressed by manipulating the signals transmitted to the near-end room, e.g., using a preprocessor on the loudspeaker signals to make them less coherent [3], but without affecting much the stereo perception and the sound quality. The adaptive filters used in SAEC should exploit the cross-correlation between the channels [4]. In this context, the most interesting solutions belong to the recursive
n
Corresponding author. Tel.: +40 21 402 4634. E-mail addresses: [email protected] (C. Paleologu), [email protected] (J. Benesty), [email protected] (S. Ciochină). 0165-1684/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2013.08.001
least-squares (RLS) family. Due to their convergence features, these algorithms were preferred in many real-world applications [5,6]. Recently, we proposed a different approach for the SAEC problem [1,7], by using the widely linear (WL) model [8]. Basically, the classical two-input/two-output system with real random variables was recast as a single-input/ single-output system with complex random variables. As a consequence, the four real-valued acoustic impulse responses are converted to one complex-valued impulse response. One advantage of this approach is that instead of handling two (real) output signals separately, we only handle one (complex) output signal, which is convenient for the main challenges of SAEC. In this paper, we derive a general Kalman filter (GKF) with the WL model for SAEC, namely the WL-GKF. The term “general” refers to a different approach we propose, i.e., a block of time samples is considered at each iteration, instead of one time sample (as in the conventional approach). The main motivation behind this work is the appealing performance of the Kalman filter for echo cancellation [9–11]. Also, the WL complex Kalman filters
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Fig. 1. The WL model for SAEC.
[12,13] were found to be attractive for many applications. The proposed algorithm has inherited some similarities with the WL augmented complex Kalman filter presented in [13]. However, the WL-GKF is derived based on a state variable model suitable for the SAEC problem. The proposed WL-GKF joins the advantages of the WL model (as described before) and the good features of the GKF [11]. Simulation results indicate that the developed algorithm outperforms the RLS counterpart. Consequently, it could represent an attractive alternative in SAEC. 2. The WL model for SAEC In this section, we briefly review the WL model for SAEC (Fig. 1) [1,7]. Let us denote the two input (or loudspeaker) signals by xL ðnÞ and xR ðnÞ (i.e., “left” and “right”), and the two output (or microphone signals) by dL ðnÞ and dR ðnÞ, where n is the time index. Therefore, the microphone signals are obtained as dL ðnÞ ¼ yL ðnÞ þ vL ðnÞ;
ð1Þ
dR ðnÞ ¼ yR ðnÞ þ vR ðnÞ;
ð2Þ
where yL ðnÞ and yR ðnÞ denote the stereo echo signals, and vL ðnÞ and vR ðnÞ are the near-end signals (i.e., noise or a combination of noise and near-end speech). The echo signals can be modelled as [2,3] T
T
yL ðnÞ ¼ ht;LL xL ðnÞ þ ht;RL xR ðnÞ;
ð3Þ
T yR ðnÞ ¼ ht;LR xL ðnÞ
ð4Þ
þ
T ht;RR xR ðnÞ;
where ht;LL ; ht;RL ; ht;LR , and ht;RR are L-dimensional vectors of the loudspeaker-to-microphone acoustic impulse responses
(the subscript t stands for “true”), the superscript T denotes transposition, and xL ðnÞ ¼ ½xL ðnÞ xL ðn1Þ ⋯ xL ðnL þ 1ÞT xR ðnÞ ¼ ½xR ðnÞ xR ðn1Þ ⋯ xR ðnL þ 1ÞT
comprise the L most recent loudspeaker signal samples. In this context, the main goal is to estimate the four acoustic impulse responses, ht;LL ; ht;RL ; ht;LR , and ht;RR , from the microphone signals dL ðnÞ and dR ðnÞ. Next, let us form the complex random variable (CRV): ð5Þ dðnÞ ¼ dL ðnÞ þ jdR ðnÞ ¼ yðnÞ þ vðnÞ; pffiffiffiffiffiffiffi where j ¼ 1, yðnÞ ¼ yL ðnÞ þ jyR ðnÞ, and vðnÞ ¼ vL ðnÞþ jvR ðnÞ. Also, let us define the complex random vector: xðnÞ ¼ xL ðnÞ þ jxR ðnÞ:
ð6Þ
Consequently, the (complex) echo signal can be obtained as H
yðnÞ ¼ ht xðnÞ þ ht ′H xn ðnÞ;
ð7Þ
where the superscripts H and n denote conjugate transpose and conjugate, respectively, and ht ¼ ht;1 þ jht;2 ;
ð8Þ
h′t ¼ h′t;1 þ jh′t;2 ;
ð9Þ ′
with ht;1 ¼ ðht;LL þ ht;RR Þ=2, ht;2 ¼ ðht;RL ht;LR Þ=2, ht;1 ¼ ′ ðht;LL ht;RR Þ=2, and ht;2 ¼ ðht;RL þ ht;LR Þ=2. Using the previous notation, we can express (7) as H ~ yðnÞ ¼ h~ t xðnÞ;
ð10Þ
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T ′T ~ where h~ t ¼ ½ht ht T and xðnÞ ¼ ½xT ðnÞ xH ðnÞT . Therefore, the complex observation is H ~ þ vðnÞ: dðnÞ ¼ h~ t xðnÞ
ð11Þ
It can be noticed that we are dealing now with a complex acoustic impulse response of length 2L, i.e., h~ t , whose complex input and output are, respectively, x(n) and d(n). From (7) or (10), we recognize the WL model for CRVs proposed in [8]. Thanks to the WL model, the two-input/ two-output system with real random variables was converted to a single-input/single-output system with CRVs. This approach is in line with the duality principle [14]. In this context, the new goal is to estimate the system ~ h~ t with an adaptive filter, hðnÞ, in such a way that for a reasonable value of n, we have for the (normalized) misalignment: ~ J h~ t hðnÞ J 22 rι; 2 ~ J ht J
ð12Þ
2
where ι is a predetermined small positive number and J J 2 is the ℓ2 norm. As compared to the classical SAEC approach [2,3], which requires four adaptive filters of length L, the WL model involves only one filter of length 2L. On the other hand, we are dealing now with CRVs having both real and imaginary parts. Apparently, the overall complexity of the WL model is similar to the classical approach. However, there are other aspects that should be taken into account in practice (e.g., double-talk detection, echo/noise suppression, implementation issues, etc.), which make more convenient to handle only one adaptive filter instead of four such systems [1]. 3. State variable model for WL SAEC Let us express (11) by considering the P most recent time samples of the microphone signal, i.e., dðnÞ ¼ ½dðnÞ dðn1Þ ⋯ dðnP þ 1ÞT n
~ ðnÞh~ ðnÞ þ vðnÞ; ¼X t T
ð13Þ
where ~ ~ ~ ~ XðnÞ ¼ ½xðnÞ xðn1Þ ⋯ xðnP þ 1Þ
ð14Þ
is the input signal matrix of size 2L P and the noise signal vector, vðnÞ, is defined similar to dðnÞ. We also consider that the system to be identified is time dependent, i.e., h~ t ðnÞ. ~ T ðnÞ is the measurement matrix and In our context, X x(n) is considered as deterministic. Expression (13) is called the observation equation. We assume that h~ t ðnÞ is a zero-mean complex random vector, which follows a simplified first-order Markov model [15], i.e., h~ t ðnÞ ¼ h~ t ðn1Þ þ wðnÞ;
ð15Þ
where wðnÞ is a zero-mean circular complex white Gaussian noise signal vector, which is uncorrelated with h~ t ðn1Þ and vðnÞ. The correlation matrix of wðnÞ is assumed to be R w ðnÞ ¼ s2w ðnÞI2L , where I2L is the 2L 2L identity matrix. The variance, s2w ðnÞ, captures the uncertainties in h~ t ðnÞ. Expression (15) is called the state equation.
Now, the basic problem may be restated as follows. Given the two fundamental equations, i.e., (13) and (15), our objective is to find the optimal recursive estimator of ~ h~ t ðnÞ denoted by hðnÞ. In this context, the values of s2w ðnÞ play a major role in the performance of the estimator, i.e., small values of s2w ðnÞ imply a good misalignment but a poor tracking, while large values of s2w ðnÞ (meaning that the uncertainties in the echo path are high) imply a good tracking but a high misalignment. Consequently, the values of s2w ðnÞ highly determine the tracking abilities and the convergence of the Kalman filter to be derived for SAEC with the WL model. 4. WL general Kalman filter It is known that, in the context of the linear sequential Bayesian approach, the optimum estimate of the state vector, h~ t ðnÞ, has the form [16] ~ ~ hðnÞ ¼ hðn1Þ þ KðnÞen ðnÞ;
ð16Þ
where KðnÞ is the Kalman gain matrix and ~ T ðnÞh~ n ðn1Þ eðnÞ ¼ dðnÞX
ð17Þ
is the a priori error signal vector, which is obtained using the adaptive filter coefficients at time n1. The a posteriori error signal vector is defined based on the adaptive filter coefficients at time n, i.e., ~ T ðnÞμ~ n ðnÞ þ vðnÞ; ~ T ðnÞh~ n ðnÞ ¼ X ϵðnÞ ¼ dðnÞX
ð18Þ
where ~ ~ μðnÞ ¼ h~ t ðnÞhðnÞ
ð19Þ
is the state estimation error or a posteriori misalignment. ~ The correlation matrix of μðnÞ is ~ μ~ H ðnÞ; R μ~ ðnÞ ¼ E½μðnÞ
ð20Þ
where E½ denotes mathematical expectation. We can also define the a priori misalignment as ~ ~ ~ ¼ μðn1Þ þ wðnÞ; mðnÞ ¼ h~ t ðnÞhðn1Þ
ð21Þ
for which its correlation matrix is ~ ~ ðnÞ ¼ Rμ~ ðn1Þ þ s2w ðnÞI2L : R m~ ðnÞ ¼ E½mðnÞ m H
ð22Þ
The a priori misalignment appears in the a priori error signal vector as ~ ðnÞm ~ ðnÞ þ vðnÞ: eðnÞ ¼ X T
n
ð23Þ
The Kalman gain matrix is obtained by minimizing the criterion [16]: J ðnÞ ¼
1 tr R μ~ ðnÞ 2L
ð24Þ
with respect to KðnÞ, where tr½ denotes the trace of a square matrix. We easily find that ~ ~ H ðnÞR ~ ðnÞXðnÞ ~ X þ s2v IP 1 ; KðnÞ ¼ Rm~ ðnÞXðnÞ½ m where
s2v
ð25Þ
¼ E½jvðnÞj , IP is the P P identity matrix, and 2
~ H ðnÞR m~ ðnÞ: R μ~ ðnÞ ¼ ½I2L KðnÞX
ð26Þ
C. Paleologu et al. / Signal Processing 94 (2014) 570–575
573
gR
gL 0.01
0.01
0
−0.01
0
0
500
1000
1500
2000
−0.01
0
500
1000
ht,LL Amplitude
0.01
2000
0.01
0
0
−0.01
−0.01 0
200
400
0
200
ht,RL
400
ht,RR
0.01
Amplitude
1500
ht,LR
0.01
0
0
−0.01
−0.01 0
200
400
0
Samples
200
400
Samples
Fig. 2. Acoustic impulse responses used in simulations.
Rm~ ðnÞ ¼ R μ~ ðn1Þ þ s2w ðnÞI2L ;
ð28Þ
~ H ðnÞR m~ ðnÞXðnÞ ~ þ s2v IP ; Ren ðnÞ ¼ X
ð29Þ
and do not behave the same way in practice. The two parameters s2w ðnÞ and s2v in the WL-GKF (for which the WL-RLS does not depend on) allow us to better control it. The first (and perhaps the most important) parameter is s2w ðnÞ, which plays a major role in the overall performance of the algorithm, as explained before. Using the ℓ2 norm in both sides of (15), together with the approximation J wðnÞ J 22 2Ls2w ðnÞ (which is valid when 2L b 1), ~ replacing h~ t ðnÞ by its estimate hðnÞ, and also considering the contribution of the model's order, we can evaluate
1 ~ KðnÞ ¼ R m~ ðnÞXðnÞR en ðnÞ;
ð30Þ
s b 2w ðnÞ ¼
eðnÞ ¼ dðnÞXT ðnÞh~ ðn1Þ;
ð31Þ
~ ~ hðnÞ ¼ hðn1Þ þ KðnÞen ðnÞ;
ð32Þ
~ H ðnÞRm~ ðnÞ: Rμ~ ðnÞ ¼ ½I2L KðnÞX
ð33Þ
The estimation from (36) is designed to achieve a proper compromise between good tracking and low misalignment. When the algorithm starts to converge or when there is an abrupt change of the system (e.g., when the ~ echo path changes), the difference between hðnÞ and ~ hðn1Þ is significant, so that the parameter s b 2w ðnÞ takes large values, thus providing fast convergence and tracking. On the other hand, when the algorithm starts to converge ~ ~ to its steady-state, the difference between hðnÞ and hðn1Þ reduces, thus leading to small values of s b 2w ðnÞ and, consequently, to a low misalignment. The second parameter to be found is the noise power, s2v . Usually, it can be estimated during silences of the nearend talker, i.e., in the single-talk scenario [18]. However, this is not always an easy task. The most critical situation in echo cancellation is the double-talk case, when the near-end signal is a combination of the background noise and the near-end speech. However, the variance of the near-end signal naturally appears within the algorithm. This allows us to better control the filter, in terms of its robustness to near-end signal variations. In this scenario,
The correlation matrix of the a priori error signal vector is ~ ðnÞR ~ ðnÞXðnÞ ~ þ Ren ðnÞ ¼ X m H
s2v IP ;
ð27Þ
whose inverse appears explicitly in the Kalman gain matrix. Consequently, the following equations summarize the WL general Kalman filter (WL-GKF):
n
~ The initialization is hðnÞ ¼ 0 and Rμ~ ð0Þ ¼ εI2L , where ε is a small positive constant. When s2w ðnÞ ¼ 0, we have lim Rμ~ ðnÞ ¼ 0;
ð34Þ
lim KðnÞ ¼ 0;
ð35Þ
n-1
n-1
and, obviously, the WL-GKF will never be able to track the changes in h~ t . On the other hand, for large values of s2w ðnÞ, the Kalman gain matrix never goes to zero, which allows the update equation (32) to stay “alert” to any possible random changes of the system. The relation between the Kalman filter and the RLS algorithm is well established [17]. For P ¼1, the WL-GKF has striking resemblances with the WL-RLS algorithm [1,7]. However, the two algorithms are very much different
~ ~ J hðnÞ hðn1Þ J 22 : 2PL
ð36Þ
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the parameter s2v can be estimated as proposed in [19] or [20]. Some recent works, e.g., [11,21], show that the GKF using the s2v estimate proposed in [19] is very robust to large near-end signals (like double-talk).
5 WL−GKF, P=1 WL−GKF, P=2 WL−GKF, P=4
Misalignment (dB)
0 −5 −10 −15 −20
Misalignment (dB)
Simulations are performed in the context of SAEC, as described in Fig. 1. The acoustic impulse responses used for the far-end and near-end locations are shown in Fig. 2 [1]. Impulse responses in the far-end [i.e., g L ðnÞ and g R ðnÞ] have 2048 coefficients, while the length of the impulse responses in the near-end [i.e., ht;LL ðnÞ, ht;RL ðnÞ, ht;LR ðnÞ, and ht;RR ðnÞ] is L ¼512. The length of the WL adaptive filters used in the experiments is 2L ¼ 1024; sample rate is 8 kHz. Two source signals are used: a white Gaussian signal and a speech sequence. The input signals are preprocessed with positive and negative half-wave rectifiers, using a distortion parameter αr ¼ 0:3 [1]. All simulations are performed in the single-talk scenario. We can define the stereo echo-to-noise ratio (SENR) [which is equivalent to the signal-to-noise ratio (SNR)] as SENR ¼ s2y =s2v , where s2y ¼ E½jyðnÞj2 is the variance of y(n). In our simulations, the background noise is independent white Gaussian distributed, whose level is set such that SENR ¼30 dB. We assume that the variance of the noise, s2v , is available in all the simulations. The performance measures is the normalized misalignment (in dB), which is computed based on (12). In order to evaluate the tracking capabilities of the algorithms, the impulse responses in the near-end location are shifted to the right by 12 samples in the middle of each experiment. The performance of the WL-GKF using different model orders is presented in Fig. 3, where the source signal is white and Gaussian. We can see that the performance of the WL-GKF improves when the value of P increases. However, this improvement is not very significant for P 42. In Fig. 4, the WL-GKF with P ¼1 is compared to the WL-RLS algorithm using different values of the forgetting factor ð0 oλ r 1Þ. This specific parameter of the WL-RLS
WL−RLS, λ=1−1/(2L) WL−RLS, λ=1−1/(40L) WL−GKF, P=1
0
−5
−10
−15
−20
−25 0
5
10
15
20
25
30
Time (seconds) Fig. 4. Misalignment of the WL-RLS algorithm with two different values of the forgetting factor, λ, and misalignment of the WL-GKF with P ¼1. The source signal is white Gaussian.
10 WL−RLS, λ=1−1/(40L) WL−GKF, P=1 WL−GKF, P=2
5
Misalignment (dB)
5. Simulation results
5
0
−5
−10
−15
−20 0
5
10
15
20
25
30
Time (seconds) Fig. 5. Misalignment of the WL-RLS algorithm with λ ¼ 11=ð40LÞ, and misalignment of the WL-GKF with P¼ 1 and P¼ 2. The source signal is speech.
algorithm addresses the compromise between convergence rate/tracking capabilities on the one hand and misadjustment/stability on the other hand [22]. It can be noticed that the WL-GKF compromises better between the tracking capability and steady-state misalignment level, as compared to the WL-RLS algorithm. In Fig. 5, the source signal is a speech sequence and the WL-GKF (with P ¼1 and P ¼2) is compared to the WL-RLS algorithm using λ ¼ 11=ð40LÞ. Overall, this result also proves that the WL-GKF performs better as compared to the WL-RLS algorithm. 6. Conclusions
−25 −30 0
5
10
15
20
25
30
Time (seconds) Fig. 3. Misalignment of the WL-GKF with different values of P. The source signal is white Gaussian.
Due to their convergence features, RLS-based algorithms are frequently involved in SAEC. In this paper, we have motivated the use of the Kalman filter in this application. Indeed, we have developed a general Kalman filter (by considering, at each iteration, a block of time samples instead of one time sample as it is the case in the
C. Paleologu et al. / Signal Processing 94 (2014) 570–575
conventional approach) with the WL model for SAEC. As compared to the WL-RLS algorithm, the proposed WL-GKF compromises better between the tracking capability and steady-state misalignment level. Of course, the main issue remains the computational complexity, which will be addressed in future work by developing fast versions of the WL-GKF.
Acknowledgment This work was supported by the UEFISCDI Romania under Grants PN-II-RU-TE no. 7/5.08.2010 and PN-II-IDPCE-2011-3-0097. The authors would like to thank the Handling Editor and the reviewers for the valuable comments and suggestions. References [1] J. Benesty, C. Paleologu, T. Gänsler, S. Ciochină, A Perspective on Stereophonic Acoustic Echo Cancellation, Springer-Verlag, Berlin, Germany, 2011. [2] M.M. Sondhi, D.R. Morgan, J.L. Hall, Stereophonic acoustic echo cancellation—an overview of the fundamental problem, IEEE Signal Processing Letters 2 (8) (1995) 148–151. [3] J. Benesty, D.R. Morgan, M.M. Sondhi, A better understanding and an improved solution to the specific problems of stereophonic acoustic echo cancellation, IEEE Transactions on Speech and Audio Processing 6 (3) (1998) 156–165. [4] S. Emura, Y. Haneda, A. Kataoka, S. Makino, Stereo echo cancellation algorithm using adaptive update on the basis of enhanced inputsignal vector, Signal Processing 86 (2006) 1157–1167. [5] P. Eneroth, S.L. Gay, T. Gänsler, J. Benesty, A real-time implementation of a stereophonic acoustic echo canceler, IEEE Transactions on Speech and Audio Processing 9 (5) (2001) 513–523. [6] T. Gänsler, J. Benesty, New insights into the stereophonic acoustic echo cancellation problem and an adaptive nonlinearity solution, IEEE Transactions on Speech and Audio Processing 10 (5) (2002) 257–267.
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