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SIGNAL PROCESSING ELSEVIER
Signal Processing 64 (1998) 87-91
Application of the leaky extended LMS (XLMS) algorithm in stereophonic acoustic echo cancellation T. Hoya*, Y. Loke, J.A. Chambers, P.A. Naylor Department of Electronic and Electrical Engineering, Imperial College of Science, Technology and Medicine, London, SW7 2BT, .!JK
Received 9 October 1997
Abstract Analysis of the leaky least mean square (LMS) adaptive algorithm has justified the use of a leakage factor in many applications. In this work, a similar leakage factor is introduced in the two-channel LMS and the extended LMS (XLMS) algorithms for use in stereophonic acoustic echo cancellation. This is compared with the alternative of adding random white noise to the input stereo signals. Simulations and experimental results indicate that introduction of the leakage factor is superior to the direct addition of random white noise. Performance measures used are based on output error and weight error vector norms. 0 1998 Elsevier Science B.V. All rights reserved. Zusammeufassung Die Analyse des adaptiven ‘leaky least mean square (LMS)’ Algorithmus rechtfertigt fiir viele Anwendungen die Verwendung eines ‘Leakage’-Faktors. In dieser Arbeit wird ein Phnlicher Faktor fiir einen zweikanaligen LMS Algorithmus und einen erweiterten LMS (XLMS) zur zweikanaligen Echokompensation vorgestellt. Dies wird mit der Addition von we&m Rauschen zu den Eingangssignalen verglichen. Simulationen zeigen, da0 die Einfiihrung eines ‘Leakage’-Faktors der einfachen Addition von weisem Rauschen iiberlegen ist. Die verwendeten MaEe zur Beurteilung der Verfahren basieren auf dem Ausgangsfehler und dem Systemabstand. 0 1998 Elsevier Science B.V. All rights reserved. R&urn6 L’analyse de l’algorithme de gradient stochastique avec perte (leaky LMS) a permis de justifier l’usage d’un facteur de perte dans de nombreuses applications. Un facteur de perte similaire est introduit dans cet article pour les algorithmes LMS deux-canaux et LMS etendu (XLMS) dans le contexte d’annulation d’Bcho stiriophonique. Cette approche est cornpark avec celle consistant B additionner un bruit blanc aux signaux stCrBod’entrke. Des simulations et des r&ultats exp&rimentaux indiquent que l’introduction du facteur de perte est prifkrable g l’addition directe de bruit blanc. Les mesures de performance utili&s sont bastes sur l’erreur de sortie et les normes des vecteurs d’erreur sur les coefficients. 0 1998 Elsevier Science B.V. All rights reserved. Keywords:
Leakage factor; Stereophonic acoustic echo cancellation; Leaky XLMS; Two-channel
* Corresponding author. 0165-1684/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved. PI1 SO165-1684(97)00178-3
LMS
88
T. Hoya et al. / Signal Processing 64 (1998) 87-91
1. Introduction Adaptive echo cancellers are widely employed in teleconferencing systems in order to compensate for the impairment in the quality due to undesired ethos resulting from coupling between loudspeaker and microphone. In a stereophonic environment, the fundamental problem of echo cancellation lies in the misalignment of two-channel adaptive filters. In [9], it is also described that conventional acoustic echo cancellation (AEC) approaches give no satisfactory solution. This can be understood by considering the two-channel echo cancellers in Fig. 1. At the remote room on the right, speech is transmitted via two acoustic paths characterised by the impulse responses g1 and g2. In the near room, the signals from each loudspeaker will couple back into the microphone via the impulse responses h1 and hz. Echo cancellation is achieved if the adaptive filters match the receiving room impulse responses. To simplify the diagram, coupling is only shown for one microphone, echo cancellation on the other microphone is essentially an identical process. In the LMS algorithm, the error signal e(n) is written as e(n) = y(n) - Qx, + fi,TX,,
(1)
where c, and gZ are N-dimensional vectors of the adaptive filter coefficients, x1 and x2 are vectors comprising the N most recent samples, with superscript T denoting vector transpose. Similarly, the signal y(n) can be written as y(n) = hTx, + @xx,,
h”z = h2 - g2.
c41. 2. The leaky extended LMS (XLMS) algorithm The leaky LMS adaptive algorithm has played a significant role in channel equalisation [7] and ADPCM coders [S]. Leakage is successful both in stabilising the systems and in alleviating ‘stalling’ of adaptive coefficients due to very low input signal and quantisation effects. Similarly, the leakage factor can be incorporated into the XLMS algorithm [3], to yield the leaky XLMS algorithm. The leaky XLMS algorithm is a direct approximation of the two-channel RLS algorithm with a leakage factor y. Using the same notation as above, the main update in the algorithm is as follows: ~:~~~:i]=(1-~)~~6i]+~~~-l(~+l)
(3)
XI@ +1) 1
X
(2)
where h1 and h2 are the true impulse response vectors in the receiving room. We can denote the misalignments as h”, = hl - RI,
the input signals are correlated, because x1 and x2 are the convolution of the same signal s(n) with impulse responses g1 and g2. Thus the main strategy with stereophonic echo cancellation is to find a method to decorrelate x1 and x2, and to do this without affecting stereophonic perception. In other work, the problem of stereophonic AEC has been investigated, and several projection-based algorithms for multichannel AEC [1,3,10,1 l] and the extended LMS (XLMS) have been proposed
[ x2@ + 1)
4n
prl(n + 1Y
M(n + 1) =
[
P-12@
+
1)1
P22h
h”;x, + @x2 = 0.
P11@ + 1) = XT@+ lF1(n + l), P22b
+
r12.(n +
1) =xTc,
+
1)x2@
1
Prr2(n + 1)1
where
From here, we can immediately see that unless x1 and x2 are linearly independent, this does not imply that h”, = k2 = 0. This illustrates the fundamental problem with stereophonic signals, that is
(5)
11,
where c+ is the adaptation gain. The matrix M(n + 1) is defined as
Now assuming e(n) has been driven to be zero. It follows that (4)
+
+
I),
1) = xT(n + 1)x2@ + l),
and I denotes the identity matrix.
+
I)1
'
T. Hoya et al. / Signal Processing 64 (1998) 87-91
M(n + 1) can be interpreted as a simplified twochannel correlation matrix which takes into account the cross correlation between x1 and x2. pii and p22 are the sum squared data in the two channel input taps, and r 12 is the sum squared cross channel coefficient. p is a correlation coefficient that scales the cross-correlation by a fixed amount. The stability conditions of XLMS are
89
whereas the addition of random white noise only decreases the weight error vector norm.
3. Simulations and experimental results
O
Signal Processing 64 (1998) 87-91
Application of the leaky extended LMS (XLMS) algorithm in stereophonic acoustic echo cancellation T. Hoya*, Y. Loke, J.A. Chambers, P.A. Naylor Department of Electronic and Electrical Engineering, Imperial College of Science, Technology and Medicine, London, SW7 2BT, .!JK
Received 9 October 1997
Abstract Analysis of the leaky least mean square (LMS) adaptive algorithm has justified the use of a leakage factor in many applications. In this work, a similar leakage factor is introduced in the two-channel LMS and the extended LMS (XLMS) algorithms for use in stereophonic acoustic echo cancellation. This is compared with the alternative of adding random white noise to the input stereo signals. Simulations and experimental results indicate that introduction of the leakage factor is superior to the direct addition of random white noise. Performance measures used are based on output error and weight error vector norms. 0 1998 Elsevier Science B.V. All rights reserved. Zusammeufassung Die Analyse des adaptiven ‘leaky least mean square (LMS)’ Algorithmus rechtfertigt fiir viele Anwendungen die Verwendung eines ‘Leakage’-Faktors. In dieser Arbeit wird ein Phnlicher Faktor fiir einen zweikanaligen LMS Algorithmus und einen erweiterten LMS (XLMS) zur zweikanaligen Echokompensation vorgestellt. Dies wird mit der Addition von we&m Rauschen zu den Eingangssignalen verglichen. Simulationen zeigen, da0 die Einfiihrung eines ‘Leakage’-Faktors der einfachen Addition von weisem Rauschen iiberlegen ist. Die verwendeten MaEe zur Beurteilung der Verfahren basieren auf dem Ausgangsfehler und dem Systemabstand. 0 1998 Elsevier Science B.V. All rights reserved. R&urn6 L’analyse de l’algorithme de gradient stochastique avec perte (leaky LMS) a permis de justifier l’usage d’un facteur de perte dans de nombreuses applications. Un facteur de perte similaire est introduit dans cet article pour les algorithmes LMS deux-canaux et LMS etendu (XLMS) dans le contexte d’annulation d’Bcho stiriophonique. Cette approche est cornpark avec celle consistant B additionner un bruit blanc aux signaux stCrBod’entrke. Des simulations et des r&ultats exp&rimentaux indiquent que l’introduction du facteur de perte est prifkrable g l’addition directe de bruit blanc. Les mesures de performance utili&s sont bastes sur l’erreur de sortie et les normes des vecteurs d’erreur sur les coefficients. 0 1998 Elsevier Science B.V. All rights reserved. Keywords:
Leakage factor; Stereophonic acoustic echo cancellation; Leaky XLMS; Two-channel
* Corresponding author. 0165-1684/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved. PI1 SO165-1684(97)00178-3
LMS
88
T. Hoya et al. / Signal Processing 64 (1998) 87-91
1. Introduction Adaptive echo cancellers are widely employed in teleconferencing systems in order to compensate for the impairment in the quality due to undesired ethos resulting from coupling between loudspeaker and microphone. In a stereophonic environment, the fundamental problem of echo cancellation lies in the misalignment of two-channel adaptive filters. In [9], it is also described that conventional acoustic echo cancellation (AEC) approaches give no satisfactory solution. This can be understood by considering the two-channel echo cancellers in Fig. 1. At the remote room on the right, speech is transmitted via two acoustic paths characterised by the impulse responses g1 and g2. In the near room, the signals from each loudspeaker will couple back into the microphone via the impulse responses h1 and hz. Echo cancellation is achieved if the adaptive filters match the receiving room impulse responses. To simplify the diagram, coupling is only shown for one microphone, echo cancellation on the other microphone is essentially an identical process. In the LMS algorithm, the error signal e(n) is written as e(n) = y(n) - Qx, + fi,TX,,
(1)
where c, and gZ are N-dimensional vectors of the adaptive filter coefficients, x1 and x2 are vectors comprising the N most recent samples, with superscript T denoting vector transpose. Similarly, the signal y(n) can be written as y(n) = hTx, + @xx,,
h”z = h2 - g2.
c41. 2. The leaky extended LMS (XLMS) algorithm The leaky LMS adaptive algorithm has played a significant role in channel equalisation [7] and ADPCM coders [S]. Leakage is successful both in stabilising the systems and in alleviating ‘stalling’ of adaptive coefficients due to very low input signal and quantisation effects. Similarly, the leakage factor can be incorporated into the XLMS algorithm [3], to yield the leaky XLMS algorithm. The leaky XLMS algorithm is a direct approximation of the two-channel RLS algorithm with a leakage factor y. Using the same notation as above, the main update in the algorithm is as follows: ~:~~~:i]=(1-~)~~6i]+~~~-l(~+l)
(3)
XI@ +1) 1
X
(2)
where h1 and h2 are the true impulse response vectors in the receiving room. We can denote the misalignments as h”, = hl - RI,
the input signals are correlated, because x1 and x2 are the convolution of the same signal s(n) with impulse responses g1 and g2. Thus the main strategy with stereophonic echo cancellation is to find a method to decorrelate x1 and x2, and to do this without affecting stereophonic perception. In other work, the problem of stereophonic AEC has been investigated, and several projection-based algorithms for multichannel AEC [1,3,10,1 l] and the extended LMS (XLMS) have been proposed
[ x2@ + 1)
4n
prl(n + 1Y
M(n + 1) =
[
P-12@
+
1)1
P22h
h”;x, + @x2 = 0.
P11@ + 1) = XT@+ lF1(n + l), P22b
+
r12.(n +
1) =xTc,
+
1)x2@
1
Prr2(n + 1)1
where
From here, we can immediately see that unless x1 and x2 are linearly independent, this does not imply that h”, = k2 = 0. This illustrates the fundamental problem with stereophonic signals, that is
(5)
11,
where c+ is the adaptation gain. The matrix M(n + 1) is defined as
Now assuming e(n) has been driven to be zero. It follows that (4)
+
+
I),
1) = xT(n + 1)x2@ + l),
and I denotes the identity matrix.
+
I)1
'
T. Hoya et al. / Signal Processing 64 (1998) 87-91
M(n + 1) can be interpreted as a simplified twochannel correlation matrix which takes into account the cross correlation between x1 and x2. pii and p22 are the sum squared data in the two channel input taps, and r 12 is the sum squared cross channel coefficient. p is a correlation coefficient that scales the cross-correlation by a fixed amount. The stability conditions of XLMS are
89
whereas the addition of random white noise only decreases the weight error vector norm.
3. Simulations and experimental results
O
