Alternative Learning Algorithm for Stereophonic Acoustic Echo Canceller without Pre-Processing

Transcription

1 1958 PAPER Special Section on Digital Signal Processing Alternative Learning Algorithm or Stereophonic Acoustic Echo Canceller without Pre-Processing Akihiro HIRANO a),kenjinakayama, Memers, Daisuke SOMEDA, and Masahiko TANAKA, Nonmemers SUMMARY This paper proposes an alternative learning algorithm or a stereophonic acoustic echo canceller without pre-processing which can identiy the correct echo-paths. By dividing the ilter coeicients into the ormer/latter parts and updating them alternatively, conditions oth or unique solution and or perect echo cancellation are satisied. The learning or each part is switched rom one part to the other when that part converges. Convergence analysis clariies the condition or correct echo-path identiication. For ast and stale convergence, a convergence detection and an adaptive step-size are introduced. The modiication amount o the ilter coeicients determines the convergence state and the step-size. Computer simulations show 10 db smaller ilter coeicient error than those o the conventional algorithms without pre-processing. key words: stereophonic acoustic echo canceller, pre-processing 1. Introduction Echo cancellers are used to reduce echoes in a wide range o applications, such as TV conerence systems and handsree telephones. To realistic TV conerencing, multichannel audio, at least stereophonic, is essential. For stereophonic teleconerencing, stereophonic acoustic echo cancellers (SAEC s) [1] [13] have een studied. SAEC s have a undamental prolem in which their ilter coeicients cannot have a unique solution [1] [5]. Though SAEC s with pre-processing [6] [10] are good candidates or solving this prolem, audile sound distortion caused y the pre-processing arises. An SAEC without preprocessing, XM-NLMS algorithm [11], has also een proposed. Though the XM-NLMS converges aster than a standard SAEC [1], its convergence at the optimum coeicient is not conirmed. This paper proposes a stereophonic acoustic echo canceller without pre-processing. Section 2 reviews the SAEC and its undamental prolem. An SAEC without preprocessing, its convergence analysis and adaptation control are presented in Sect. 3. Computer simulation results show the perormance o the proposed algorithm. 2. Uniqueness Condition or Stereophonic Acoustic Echo Canceller Figure 1 shows a teleconerencing using an SAEC. This echo canceller consists o our adaptive ilters corresponding to our echo paths rom two loudspeakers to two microphones. Each adaptive ilter estimates the corresponding echo path. The ar-end signal x i (n)inthei-th channel at time index n is generated rom a talker speech s(n) y passing room A impulse response g i rom the talker to the i-th microphone. x i (n) passes an echo path h i, j rom the i-th loudspeaker to the j-th microphone and ecome an echo d j (n). Similarly, adaptive ilters w i, j (n) generates an echo replica y j (n). w i, j (n) is so updated as to reduce the residual echo e j (n) SAEC s have a undamental prolem in which their ilter coeicients cannot have a unique solution [1]. SAEC s may have ininite numer o solutions other than the optimum solution w i, j (n) = h i, j. Further analyses show that SAEC s may have a unique and optimum solution when the numer o taps N W or SAEC and the impulse response length N A in room A satisy N W < N A [5], [10]. For echo cancellation perormance, N B N W is preerale where N B is the impulse response length in room B. Thereore, i N B N W < N A,SAECin room B achieves oth perect echo cancellation and optimum solution. Such a condition, however, cannot e satisied or SAEC s in oth room A and B. Manuscript received Novemer 28, Manuscript revised March 1, Final manuscript received April 15, The authors are with the Division o Electrical Engineering and Computer Science, Graduate School o Natural Science and Technology, Kanazawa University, Kanazawa-shi, Japan. Presently, with the Designg Department No. 4, Electronics Engineering Division, Electronics Business Group, Tokairtika Co., Ltd. Presently, with the Second Department, Financial Systems Division, Hitachi Sotware Engineering Co., Ltd. a) hirano@t.kanazawa-u.ac.jp Fig. 1 Teleconerencing using SAEC.

2 HIRANO et al.: ALTERNATIVE LEARNING ALGORITHM FOR STEREOPHONIC ACOUSTIC ECHO CANCELLER Correct Echo-Path Identiication without Pre- Processing 3.1 Algorithm In order to satisy the uniqueness condition or oth SAEC s in room A and room B, the numer o taps or SAEC N W is so chosen as to satisy N W /2 < N A N W and N W /2 < N B N W. I the size o oth rooms are similar, which is usual case, such N W may exist. In adaptation, N W /2 taps are updated at a time; thus the eective numer o taps or SAEC N W /2 is smaller than the impulse response length in the arend room N A. To avoid the perormance degradation caused y the tap shortage, another N W /2 taps will also update at the other time. The ilter coeicient vector w i, j (n) is divided into two su-vectors w i, j, (n) andw i, j, (n) shown y w i, j, (n) = [w i, j,0 (n),,w i, j,nw /2 1(n)] T (1) w i, j, (n) = [w i, j,nw /2(n),,w i, j,nw 1(n)] T. (2) In the irst stage, w i, j, (n) is updated while w i, j, (n) isixed. This stage is repeated until w i, j, (n) converges. As the second stage, w i, j, (n) is updated while w i, j, (n) isixed. This stage is also repeated until w i, j, (n) converges. These two stages are repeated one ater another. 3.2 Convergence Analysis Convergence o the averaged ilter coeicients has een analyzed. First, convergence o w i, j, (n) whenw i, j, (n) isixed as an initial value w i, j, (0) is analyzed. Convergence o w i, j, (n) can e derived in the same manner as w i, j, (n). The ar-end signal on i-th channel x i (n) is derived as x i (n) = g T i s(n) (3) where the talker speech vector s(n) and the impulse response vector g i are deined y g i = [g i,0,g i,1,,g i,na 1] T (4) s(n) = [s(n),, s(n N A + 1)] T. (5) [] T denotes the transpose o []. The echo d j (n) and the echo replica y j (n) are calculated as d j (n) = y j (n) = {h T i, j, x i, (n) + h T i, j, x i,(n)} (6) {w T i, j, (n)x i, (n) + w T i, j, (0)x i,(n)}. (7) h i, j,, h i, j,, x i, (n) andx i, (n), are deined as h i, j, = [h i, j,0,, h i, j,nw /2 1] T (8) h i, j, = [h i, j,nw /2,, h i, j,nw ] T (9) x i, (n) = [x i (n),, x i (n N W /2 + 1)] T (10) x i, (n) = [x i (n N W /2),, x i (n N W )] T, (11) which are su-vectors o h i, j and x j (n). By using (3), the residual echo e j (n) is calculated y e j (n) = {h i, j, w i, j, (n)} T G i, s (n) + {h i, j, w i, j, (0)} T G i, s (n). (12) s (n), s (n N W ) are deined y s i, (n) = [s(n),, s(n N W /2 N A + 1)] (13) s i, (n) = [s(n N W /2),, s(n N W N A + 1)]. (14) G i,p is a matrix deined y (15), which contains g i and perorms convolution etween s i,p (n)andg i,p,wherepis either or. By introducing vectors and matrices deined y h1, d (n) = j, w 1, j, (n) (16) h 2, j, w 2, j, (n) h1, d (n) = j, w 1, j, (n) (17) h 2, j, w 2, j, (n) G1, G = (18) G 2, G1, G =, (19) G 2, simpliied result or e j (n), i.e., e j (n) = d T (n)g s (n) + d T (0)G s (n) (20) is derived. Taking an ensemle average o e j (n) leadsusto where E[e 2 j (n)] = E[e j(n)e T j (n)] = d T (n)q d (n) + d T (0)Q d (0) + 2d T (n)q d (0) (21) R = s (n)s T (n) (22) R = s (n)s T (n) (23) R = s (n)s T (n) (24) Q = G R G T (25) Q = G R G T (26) Q = G R G T (27) and d (n) denotes an average o d (n). From E[e 2 j (n)] = 2Q d (n) + 2Q d (0) = 0, (28) d (n) the averaged ilter coeicient error d which minimizes

3 1960 g i,p,0 g i,p,1 g i,p,na g i,p,0 g i,p,1 g.... i,p,na 1.. G i,p = N W /2 (15) g i,p,0 g i,p,1 g i,p,na 1 } {{ } N W /2 + N A 1 p = or E[e 2 j (n)] is determined as d = Q 1 Q d (0). (29) Thereore, the ilter coeicients will converge on w1, j, ( ) h1, = j, w 2, j, ( ) h 2, j, + Q 1 h1, Q j, w 1, j, (0). (30) h 2, j, w 2, j, (0) The irst term in the right hand side is the optimum solution. The second term is the error caused y the tap shortage. Results or w i, j, (n) ecomes d = Q 1 QT d (0) (31) which can e derived y the same manner as or w i, j, (n). error d (1) By updating w i, j, (n) until convergence, the coeicient or w i, j, (n) converges at d (1) = K d (0) (32) where K is deined y K = Q 1 Q. (33) Then updating w i, j, (n) with the initial coeicient error d (1) or w i, j, (n), the coeicient error d (1) or w i, j, (n) converges at d (1) = K d (1) = K K d (0) (34) where K is deined y K = Q 1 QT. (35) By repeating these processes once more, d (2) and d (2) are calculated as d (2) = K d (1) = K K K d (0) (36) d (2) = K d (2) = (K K ) 2 d (0). (37) Finally, d (m) = K (K K ) m 1 d (0) (38) d (m) = (K K ) m d (0) (39) are otained. The conditions in which the ilter coeicients converge at the optimum value are the ollowings. 1. The inverse matrices Q 1 and Q 1 exist. 2. The maximum asolute eigenvalue o K K is less than 1. The irst condition is similar to the uniqueness condition discussed in [5], [10]. To satisy this, the matrices R and R should have their inverse. Also, G and G should have pseudo-inverse. R 1 and R 1 always exist ecause R and R are the auto-correlation matrices. A neccesary condition or existence o pseudo-inverse is N W /2 < N A, which is assumed in this algorithm. The neccesary and suicient condition is rankg = rankg = N W /2. This condition might hold i g 1 and g 2 suiciently dier. In order to examine the condition on eigenvalue o K K, let up rewite the matrices Q, Q and Q. By introducing vectors x1, x (n) = (40) x 2, x1, x (n) =, (41) x 2, Q, Q and Q ecomes Q = G R G T = E[x (n)x T (n)] (42) Q = G R G T = E[x (n)x T (n)] (43) Q = G R G T = E[x (n)x T (n)]. (44) The matrices Q and Q are the auto-correlation o x (n)and x (n), respectively. Similarly, Q is the cross-correlation etween x (n) andx (n). Note that these matrices include inter-channel correlation. Using (42), (43) and (44), K K ecomes K K = Q 1 Q Q 1 Q = E[x (n)x T (n)] 1 E[x (n)x T (n)] E[x (n)x T (n)] 1 E[x (n)x T (n)]. (45) Thus, K K is square o the cross-correlation Q multiplied y the inverse o the auto-correlation Q and Q. Intuitively, the eigenvalues o K K might not e so large acause the auto-correlation is larger than the cross-correlation.

4 HIRANO et al.: ALTERNATIVE LEARNING ALGORITHM FOR STEREOPHONIC ACOUSTIC ECHO CANCELLER Adaptation Control In this approach, the adaptation control is important. To identiy the echo path, each stage should e terminated when the ilter coeicients converge. This causes two prolems: selection o the switching interval etween two stages and a random walk around the convergence value. An adaptive step-size and a convergence detection are introduced or ast convergence with a small computational cost. The adaptive step-size and the convergence detection are carried out asedonthe coeicient modiication amount deined y 2 w i, j,p (mk) w i, j,p ((m 1)K) 2 D j (m) = 2 (46) w i, j,p (mk) 2 where i and j are the channel index or the loudspeaker and the microphone in the near-end room, respectively. p is either or. To avoid the increase o the computational cost, (46) is calculated once in a K iterations. Coeicient adaptation is stopped when (46) is calculated. Figure 2 demonstrates an example o coeicient modiication amount D j (m). D j (m) decreases until the ilter coeicients converge. Ater convergence, D j (m) walks up and down caused y a random walk around the convergence value. Thereore, the ilter coeicients are considered to e converged i D j (m 1) < D j (m) is satisied. The step-size is controlled y ( ) D j (m) µ j (m) = µ max (47) D j,max where D j,max is the maximum value o D j (m) inthesame stage. Usually, D j (1)isusedastheD j,max. µ j (n) isused within mk < n < (m+1)k. (x) is a monotonically increasing unction o x. An example is ( ) D j (m) α µ j (m) = µ max (48) D j,max where α is a constant. The step-size µ j (m) in (48) is used or oth w i, j, and w i, j,. An overview o the adaptation control is as ollows: 1. Update ilter coeicients with µ j (0) = µ max or the irst K iterations. 2. Calculate D j (1). D j.max = D j (1). 3. Update ilter coeicients with µ j (m) controlled y (48) or the next K iterations. 4. Calculate D j (m). 5. I D j (m 1) < D j (m), then proceed to the next stage. 6. I D j,max < D j (m), then D j,max = D j (m). 7. Goto Computer Simulations Simulations have een carried out to show the perormance o the proposed algorithm. The simulation conditions and parameters in Tale 1 are used, except or specially noted cases. Far-end room impulse responses g i are 60-tap FIR ilters (N A = 60) while those or near-end room h i, j are 64- tap FIR ilters (N B = 64). In this case, SAEC s do not have an unique solution. Adaptive ilters are 64-tap FIR ilters (N W = 64). In this case, the condition or echo-path identiication N W /2 < N A N B is satisied. White Gaussian signals are used as a talker signal and an additive noise. The echo-to-noise ratio (ENR) is 60 db. As an adaptation algorithm, Normalized Least Mean Squares (NLMS) algorithm [14] is used. The proposed algorithm is compared with the standard SAEC [1] and the XM- NLMS algorithm [11]. For XM-NLMS, smaller step-size is also used ecause o the staility. As a perormance measure, the normalized coeicient error (NCE) and the echo return loss enhancement (ERLE) deined y 2 w i, j (n) h i, j 2 NCE(n) = 2 (49) h i, j 2 ERLE(n) = 9999 k=0 d2 j (n k) 9999 k=0 e2 j (n k) (50) are used. The ERLE is time-averaged over samples. To show the eect o the adaptive step-size and adaptive interval, several cominations are compared in addition Fig. 2 An example o coeicient modiication amount. Tale 1 Parameters Simulation conditions. N A 60 N B, N W 64 Fixed switching interval Averaging interval K 4000 Adaptation algorithm NLMS Fixed step-size 1.0 Variale step-size µ max ( D j (m) D j,max ) 1/4 µ max 1.0 Far-end talker signal s(n) White Gaussian Additive noise White Gaussian, ENR=60 db

5 1962 Tale 2 Adaptation control methods. Name Step-size Interval Averaging K Proposed Adaptive Adaptive No 4000 Reerence 1 Fixed Fixed Yes 5000 Reerence 2 Adaptive Adaptive Yes 5000 Reerence 3 Fixed Adaptive Yes 5000 Reerence 4 Fixed Adaptive No 4000 Fig. 4 () ERLE NCE and ERLE or some algorithms witout pre-processing. Fig. 3 () ERLE NCE and ERLE or some adaptation control methods. to the proposed method. Adaptation control methods are shown in Tale 2. In some cominations, time-averaging o the ilter coeicients is introduced in order to supress an inluence o a random walk around the convergence value. The NCE is depicted in Fig. 3(a). The proposed algorithm converges to 20 db o the NCE almost 10 times aster than Reerence 1 with ixed step-size and ixed interval. Since the convergence characteristics o the proposed algorithm is etter than Reerence 2, the averaging is not required. Comparison o the proposed algorithm with Reerence 3 and Reerence 4 shows the advantage o the adaptive step-size. Without the adaptive step-size, the NCE ecomes large even i the averaging is used. Though the NCE or the proposed method reaches 16 db within samples, its convergence to the inal value requires very long time. The ERLE is compared y 3(). Higher ERLE is achieved y the proposed and Reerence 2 algorithms with the adaptive step-size, compared with those with the ixed step-size. Reerence 1 with ixed interval converges slowest. Figure 4(a) compares the NCE or some adaptation algorithms without pre-processing. The proposed algorithm achieves 28 db o the NCE which is almost 10 db smaller than the standard SAEC. The XM-NLMS ailed to converge at the optimum value or larger step-size. Though the XM- NLMS with µ = 0.1 converges, the NCE is almost same as that or the standard SAEC. The convergence speed o XM- NLMS with µ = 0.1 is slow ecause o a small step-size. Figure 4() shows the ERLE or some adaptation algorithms without pre-processing. The proposed algorithm, the standard SAEC, and the XM-NLMS with µ = 0.5 achieve almost the same ERLE, almost 60 db. The ELRE or the XM-NLMS with µ = 0.1 is slightly larger ecause o the smaller step-size. Figure 5 compares the perormance when the correct echo-path identiication condition N W /2 < N A N B is not satisied. N A is selected as 30, which is less than N W /2 = 32. In this case, neither Q 1 nor Q 1 exists. The NCE or the proposed algorithm is almost same as those or the standard

6 HIRANO et al.: ALTERNATIVE LEARNING ALGORITHM FOR STEREOPHONIC ACOUSTIC ECHO CANCELLER 1963 NCE and ERLE when correct identiication condition is not satis- Fig. 5 ied. () ERLE Fig. 6 () ERLE NCE and ERLE or longer impulse response. SAEC and the XM-NLMS. The ERLE or the proposed algorithm is almost same as that or the standard SAEC. The ELRE or the XM-NLMS is slightly larger ecause o the smaller step-size. These results show that the perormance o the proposed algorithm is comparale to those or the conventional algorithms even when the correct echo-path identiication condition is not satisied. Simulations or a longer impulse response case have also een carried out. N A = 800, N B = N W = 1024 are used. For the proposed algorithm, K = 2000 is used. The NCE and the ERLE are shown in Fig. 6. The proposed algorithm improves the NCE, while almost the same ERLE is achieved. 5. Conclusions This paper proposes an alternative learning algorithm or a stereophonic acoustic echo canceller without pre-processing which can identiy the correct echo-paths. Convergence analysis clariies the condition or correct echo-path identiication. A convergence detection and an adaptive step-size ased on the modiication amount o the ilter coeicients are also introduced. Simulation results show 10 db smaller coeicient error than those o the conventional algorithms without pre-processing. Detailed analyses on the convergence condition, perormance evaluation or more noisy case such as doule-talk, and or real speech signals should e uture study. Reerences [1] M.M. Sondhi and D.R. Morgan, Stereophonic acoustic echo cancellation An overview o the undamental prolem, IEEE Signal Process. Lett., vol.2, no.8, pp , Aug [2] A. Hirano and A. Sugiyama, Convergence characteristics o a multi-channel echo canceller with strongly cross-correlated input signals Analytical results, Proc. 6th DSP Symposium, pp , Nov [3] A. Hirano and S. Koike, Convergence analysis o a stereophonic acoustic echo canceller part 1: Convergence characteristics o tap weights, Proc. 11th DSP Symposium, pp , Nov [4] A. Hirano, Convergence analysis o a multi-channel acoustic echo canceller, Proc. 12th DSP Symposium, pp , Nov [5] M. Tanaka, A. Hirano, and K. Nakayama, On uniqueness o ilter coeicients in stereophonic acoustic echo canceller, Proc. 14th DSP Symposium, pp.31 36, Nov [6] A. Sugiyama, Y. Joncour, and A. Hirano, A stereo echo canceler with correct echo-path identiication ased on an input-sliding technique, IEEE Trans. Signal Process., vol.49, no.11, pp , Nov

7 1964 [7] M. Ali, Stereophonic acoustic echo cancellation system using timevarying all-path iltering or signal decorrelation, Proc. ICASSP 98, pp , May [8] A. Hirano, K. Nakayama, and K. Watanae, Convergence analysis o stereophonic echo canceller with pre-processing Relation etween pre-processing and convergence, Proc. ICASSP 99, pp , March [9] A. Hirano, K. Nakayama, and K. Takee, Stereophonic acoustic echo canceller with pre-processing Second-order pre-processing ilter and its convergence, Proc. 8th IWAENC, pp.63 66, Sept [10] J. Benesty, D. Morgan, and M. Sondhi, A etter understanding and an improved solution to the speciic prolems o stereophonic acoustic echo cancellation, IEEE Trans. Speech Audio Process., vol.6, no.2, pp , March [11] A.W.H. Khong and P. Naylor, The use o partial update schemes to reduce inter-channel coherence in adaptive stereophonic acoustic echo cancellation, Proc. IWAENC 2003, pp.59 62, Sept [12] A. Hirano and A. Sugiyama, A compact multi-channel echo canceller with a single adaptive ilter per channel, Proc. IEEE 1992 ISCAS, pp , May [13] A. Hirano, A. Sugiyama, Y. Arasawa, and N. Kawayachi, Dsp implementation and perormance evaluation o a compact stereo echo canceller, Proc IEEE ICASSP, vol.2, pp , April [14] J. Nagumo and A. Noda, A learning method or system identiication, IEEE Trans. Autom. Control, vol.12, no.3, pp , March Daisuke Someda entered Department o Electrical and Computer Engineering, Faculty o Engineering, Knazawa University in He elonged to the adaptative systems laoratory in 2002, and studied a stereo echo canceller. Ater graduation, he joined Tokairtika Co., Ltd. and engaged in design o car electronics devices. Masahiko Tanaka received the B.Eng. and M.Eng. degrees rom Kanazawa University, Kanazawa, Japan in 1999, 2001, respectively. He has een engaged in researches on adaptive signal processing. He was involved in Hitachi SotWare Engineering Corporation, Japan rom 2000, where he had een a System Engineer in Second Department, Financial Systems Division. Akihiro Hirano received the B.Eng., M.Eng. and Dr. Eng. degrees rom Kanazawa University, Kanazawa, Japan in 1987, 1989 and 2000, respectively. He was involved in NEC Corporation, Kawasaski, Japan rom 1989 to 1998, where he had een a Research Engineer in Research and Development Group. He joined Faculty o Engineering, Kanazawa University in He is currently an Assistant Proessor in Division o Electrical Engineering and Computer Science, Graduate School o Natural Science and Technology, Kanazawa University. He has een engaged in researches on adaptive signal processing and neural networks. He was awarded the 1995 Academic Encouragement Award y IEICE. Dr. Hirano is a memer o IEEE. Kenji Nakayama received the B.E. and Dr. degrees in electronics engineering rom Tokyo Institute o Technology (TIT), Tokyo, Japan, in 1971 and 1983, respectively. From 1971 to 1972 he was engaged in the research on classical network theory in TIT. He was involved in NEC Corporation rom 1972 to 1988, where his research sujects were ilter design methodology and digital signal processing. He joined the Department o Electrical and Computer Engineering at Kanazawa University, in Aug He is currently a Proessor o Graduate School o Natural Science and Technology. His current research interests include adaptive signal processing and neural networks. He is a senior memer o IEEE and a memer o INNS.