Adaptive Filtering Part 8 docx

A Stereo Acoustic Echo Canceller Using Cross-Channel Correlation 199 () () () () () () () () URi Si Ri URi ULi Si Li ULi kkk k kkk k     xXgx xXgx . (7) In (6), if there are no un-correlated noises, we call the situation as strict single talking. In this chapter, sound source signal( () Si xk), uncorrelated noises ( () URi xk  and () ULi xk  ) are assumed as independent white Gaussian noise with variance xi  and Ni  , respectively. 2.3 Stereo acoustic echo canceller problem For simplification, only one stereo audio echo canceller for the right side microphone’s output signal () i yk  , is explained. This is because the echo canceller for left microphone output is apparently treated as the same way as the right microphone case. As shown in Fig.2, the echo canceller cancels the acoustic echo () i y k as ˆ () () () () iiii ek yk yk nk   (8) where () i ek is acoustic echo canceller’s residual error, () i nkis a independent background noise, ˆ () i yk is an FIR adaptive filter output in the stereo echo canceller, which is given by ˆˆ ˆ () () () () () TT iRiRiLiLi y kkkkkhx hx (9) where ˆ () Ri kh and ˆ () Li kh are N tap FIR adaptive filter coefficient arrays. Error power of the echo canceller for the right channel microphone output, 2 () ei k  , is given as: =( - + 22 ˆ () () () () ()) T ei Ri STi i i k y kkknk  hx (10) where ˆ () STi kh is a stereo echo path model defined as = ˆˆˆ () () () T TT STi Ri Li kkk     hhh . (11) Optimum echo path estimation ˆ OPT h which minimizes the error power 2 () e k  is given by solving the linier programming problem as 1 2 0 () LS N ei k Minimize k             (12) where LS N is a number of samples used for optimization. Then the optimum echo path estimation for the ith LTI period ˆ OPTi h is easily obtained by well known normal equation as =( )) 1 1 0 ˆ (()() LS N OPTi i i NLSi k yk k     hxX  (13) Adaptive Filtering 200 where NLSi X is an auto-correlation matrix of the adaptive filter input signal and is given by (k) (k)) (k) (k)) )= = (k) (k)) (k) (k)) 11 1 00 11 0 00 (( (()() (( LS LS LS LS LS NN TT Ri Ri Ri Li N ii kk T NLSi i i NN ii k TT Li Ri Li Li kk kk                            xx xx AB Xxx CD xx xx . (14) By (14), determinant of NLSi X is given by 1 NLSi i i i i i  XADCAB . (15) In the case of the stereo generation model which is defined by(2), the sub-matrixes in (14) are given by ) ) 1 0 1 0 (() ()2 ()(()) () () ( ( ) ( ) ( )( ( ) ) ( )( ( ) ) ( ) ( ) (() () ()(()) LS LS N TTT i Si RRi Si URi Si Ri URi URi k N TTTT i Si RLi Si URi Si Ri ULi Si Ri URi ULi k TT iSiLRiSiULiSiRiUR kkkk kk k k kk kk kk kkkk          AXGXxXgxx BXGXxXgxXgxx CXGXxXgx ) ) 1 0 1 0 ()( () ) () () (() ()2 ()(()) () () LS LS N TT iSiLiULiURi k N TTT i Si LLi Si ULi Si Li ULi ULi k kk kk kkkk kk         Xg x x DXGXxXgxx .(16) where ,,, TTTT RRi Ri Ri RLi Ri Li LRi Li Ri LLi Li Li GggGggGggGgg . (17) In the cease of strict single talking where () URi kx and () ULi kx do not exist, (16) becomes very simple as ) ) ) ) 1 0 1 0 1 0 1 0 (() () (() () (() () (() () LS LS LS LS N T iSiRRiSi k N T iSiRLiSi k N T iSiLRiSi k N T iSiLLiSi k kk kk kk kk                 AXGX BXGX CXGX DXGX . (18) To check the determinant NLSi X , we calculate NLSi i XCconsidering T ii BC as A Stereo Acoustic Echo Canceller Using Cross-Channel Correlation 201 1 1 ( ( NLSi i i ii ii ii iiiiiii     XCADCCABC ADCCBCA . (19) Then 1 ii iiii  DC CBCA becomes zero as ) 1 1 1 0 1 21 0 ( ( )( ) ( ) ( ) ( )) (()( ( ( ) )()) 0 LS LS ii i i ii N TT Si LLi LRi RRi RLi Si Si LRi Si k N TTTTTT xi Si Li Li Li Ri Li Ri Ri Ri Ri Ri Si k kkkk Nk k               DC CA BC XGGGGXXGX XggggggggggX . (20) Hence no unique solution can be found by solving the normal equation in the case of strict single talking where un-correlated components do not exist. This is a well known stereo adaptive filter cross-channel correlation problem. 3. Stereo acoustic echo canceller methods To improve problems addressed above, many approaches have been proposed. One widely accepted approach is de-correlation of stereo sound. To avoid the rank drop of the normal equation(13), small distortion such as non-linear processing or modification of phase is added to stereo sound. This approach is simple and effective to endorse convergence of the multi-channel adaptive filter, however it may degrade the stereo sound by the distortion. In the case of entertainment applications, such as conversational DTV, the problem may be serious because customer’s requirement for sound quality is usually very high and therefore even small modification to the speaker output sound cannot be accepted. From this view point, approaches which do not need to add any modification or artifacts to the speaker output sound are desirable for the entertainment use. In this section, least square (LS), stereo affine projection (AP), stereo normalized least mean square (NLMS) and WARP methods are reviewed as methods which do not need to change stereo sound itself. 3.1 Gradient method Gradient method is widely used for solving the quadratic problem iteratively. As a generalized gradient method, let denote M sample orthogonalized error array (k) Mi ε based on original error array (k) Mi e as (k) (k)() Mi i Mi kε Re (21) where (k) Mi e is an M sample error array which is defined as (k) [ ( ), ( 1), ( 1)] T Mi i i i ekek ek Me  (22) Adaptive Filtering 202 and ( ) i kR is a M M  matrix which orthogonalizes the auto-correlation matrix (k) (k) T Mi Mi ee . The orthogonalized error array is expressed using difference between adaptive filter coefficient array (k) ˆ STi h and target stereo echo path 2N sample response ST h as (k) ( - (k) 2 ˆ () () ) T Mi i M Ni ST STi kkε RX hh (23) where 2 () MNi kX is a Mx2N matrix which is composed of adaptive filter stereo input array as defined by 2 ( ) [ ( ), ( 1), ( 1)] MNi i i i kkk kMXxxx . (24) By defining an echo path estimation error array ( ) STi kd which is defined as =- (k) ˆ () STi ST STi kdhh (25) estimation error power (k) 2 i   is obtained by (k) (k) (k)= (k) (k) 2 22 () TT iMiMiSTiNNiSTi k    εε dQ d (26) where 22 2 2 () () () () () TT NNi MNi i i MNi kkkkkQXRRX. (27) Then, (26) is regarded as a quadratic function of ˆ () STi kh as (k) (k) (k) 22 22 1 ˆˆ ˆˆ (()) () 2 TTT STi STi N Ni STi STi N Ni ST fk khhQhhQh . (28) For the quadratic function, gradient () i kΔ is given by =- (k) 22 () () iNNiSTi kkΔ Qd. (29) Iteration of ˆ () STi kh which minimizes (k) 2 i   is given by = (k) =(k) 22 2 ˆˆ (1) () () ˆ () () ˆ () () () () STi STi i STi N Ni STi T STi M Ni i i Mi kkk kk kkkk       hhΔ hQd hXRRe (30) where  is a constant to determine step size. Above equation is very generic expression of the gradient method and following approaches are regarded as deviations of this iteration. 3.2 Least Square (LS) method (M=2N) From(30), the estimation error power between estimated adaptive filter coefficients and stereo echo path response , ( ) ( ) T ii kkddis given by A Stereo Acoustic Echo Canceller Using Cross-Channel Correlation 203 =)) 222222 ( 1) ( 1) ()( ()( () () TT T ii iNNNiNNNii kk k k kk    dd dIQ IQ d (31) where 2N I is a 2 2NN  identity matrix. Then the fastest convergence is obtained by finding ( ) i kR which orthogonalizes and minimizes eigenvalue variance in 22 () NNi kQ . If M=2N, 22 () MNi kX is symmetric square matrix as = 22 () () T MNi MNi kkXX (32) and if (= ) 22 22 () () () () TT MNiMNi MNiMNi kk kkXX XX is a regular matrix so that inverse matrix exists, () () T ii kkRR which orthogonalizes 22 () NNi kQ is given by ) 1 22 22 () () ( () () TT ii NNi NNi kk k k  RR X X (33) By substituting (33) for (30) =)(k) 1 22 22 22 ˆˆ (1) () ()( () () T STi STi NNi NNi NNi Ni kk kkk   hhXXX e (34) Assuming initial tap coefficient array as zero vector and 0   during 0 to 2N-1th samples and 1   at 2Nth sample , (34) can be re-written as = (2N-1) (2N-1) (2N-1)) (2N-1) 1 22 22 22 ˆ (2 ) ( T STi N Ni N Ni N Ni i N  hX X X y (35) where (k) i y is 2 N sample echo path output array and is defined as (k)=[ ( ), ( 1), ( 2 1)] T iii i ykyk yk Ny  (36) This iteration is done only once at 2 1Nth  sample. If 2 LS NN , inverse matrix term in (35) is written as =)= 1 22 22 0 () () ( () () LS N TT N Ni N Ni i i NLSi k kk kk    XX xxX (37) Comparing (13) and (35) with (37), it is found that LS method is a special case of gradient method when M equals to 2N. 3.3 Stereo Affine Projection (AP) method (M=P  N) Stereo affine projection method is assumed as a case when M is chosen as FIR response length P in the LTI system. This approach is very effective to reduce 2Nx2N inverse matrix operations in LS method to PxP operations when the stereo generation model is assumed to be LTI system outputs from single WGN signal source with right and left channel independent noises as shown in Fig.2. For the sake of explanation, we define stereo sound signal matrix 2 () PNi kX which is composed of right and left signal matrix ( ) Ri kX and ( ) Li kX for P samples as 2 2 2 () () () () () () () T T Si Ri URi TT PNi Ri Li T Si Li ULi kk kkk kk              XGX XXX XGX (38) Adaptive Filtering 204 where 2 ( ) [ ( ), ( 1), ( 2 2)] Si Si Si Si kkk kP  Xxx x (39) () URi kX and () ULi kX are un-correlated signal matrix defined as ( ) [ ( ), ( 1), ( 1)] ( ) [ ( ), ( 1), ( 1)] URi URi URi URi ULi ULi ULi ULi kkk kP kkk kP   Xxx x Xxx x   (40) Ri G and Li G are source to microphones response (2P-1)xP matrixes and are defined as 2 ,0, 2 ,0, 2 ,1, 2 ,1, 2, 1, 2, 1, 00 00 00 00 , 00 0 00 0 00 00 TT TT Ri RLi Ri Li TT TT Ri Li Ri Li Ri Li TT TT RP i LP i Ri Li                                gg gg gg gg GG gg gg      . (41) As explained by(31), 22 () NNi kQ determines convergence speed of the gradient method. In this section, we derive affine projection method by minimizing the max-min eigenvalue variance in 22 () NNi kQ . Firstly, the auto-correlation matrix is expressed by sub-matrixes for each stereo channel as = 2 () () () () () ANNi BNNi NNi CNNi DNNi kk k kk         QQ Q QQ (42) where ( ) ANNi kQ and ( ) DNNi kQ are right and left channel auto-correlation matrixes, () BNNi kQ and () CNNi kQ are cross channel-correlation matrixes. These sub-matrixes are given by +2 +2 22 2 22 () () () () () () () () () () () () () () () () () () () () () () () () () () ( TT T T T ANNi Si Ri i i Ri Si URi i i URi TT Si i i URi TT T T T BNNi Si Ri i i Li Si URi i i ULi TT URi i i ULi CNNi kkkk kkkkk kkk k kkkk kkkkk kkk k   QXGRRGXXRRX XRRX QXGRRGXXRRX XRRX Q +2 +2 22 22 2 )()()() ()()()()() () () () () () () () () () () () () () () () () () TT T T T Si Li i i Ri Si ULi i i URi TT ULi i i UTi TT T T T DNNi Si Li i i Li Si ULi i i ULi TT Si i i ULi kkkk kkkkk kkk k kkkk kkkkk kkk k   XGRRGX XRRX XRRX QXGRRGXXRRX XRRX (43) Since the iteration process in (30) is an averaging process, the auto-correlation matrix 22 () NNi kQ is approximated by using expectation value of it, 22 22 () () NNi NNi kkQQ  . Then expectation values for sub-matrixes in (42) are simplified applying statistical independency between sound source signal and noises and Tlz function defined in Appendix as A Stereo Acoustic Echo Canceller Using Cross-Channel Correlation 205 22 22 22 2 ( () ()() ()) ( ()()() ()) (() ()() ()) (() ()() ()) (() TT T T ANNi Si Ri i i Ri Si URi i i URi TT T BNNi Si Ri i i Li Si TT T CNNi Si Li i i Ri Si TT DNNi Si Li i Tlz k k k k Tlz k k k k Tlz k k k k Tlz k k k k Tlz k     QXGRRGX XRRX QXGRRGX QXGRRGX QXGR          2 () () ()) ( () () () ()) T iLiSi ULiiiULi kk kTlz kkk kRGX X R RX   (44) where = = = 222 2 ( ) [ ( ), ( 1), ( 1)] ( ) [ ( ), ( 1), ( 1)] ( ) [ ( ), ( 1), ( 1)] T Si Si Si Si URi URi URi URi ULi ULi ULi ULi kkk kP kkk kP kkk kP    Xxx x Xxx x Xxx x             (45) with 2 ( ) [ ( ), ( 1), ( 2 2)] ( ) [ ( ), ( 1), ( 1)] ( ) [ ( ), ( 1), ( 1)] T Si Si Si Si T URi URi URi URi T ULi ULi ULi ULi kxkxk xkp kxkxk xkp kxkxk xkp    x x x       . (46) Applying matrix operations to 22NNi Q , a new matrix 22NNi  Q which has same determinant as 22NNi Q is given by = 22 () () () ANNi NNi DNNi k k k            Q0 Q 0Q (47) where (), () A NNi ANNi DNNi DNNi Tlz Tlz  QQQQ. (48) Since both 2 () T Si Ri kXG  and 2 () T Si Li kXG  are symmetric PxP square matrixes, A NNi  Q and BNNi  Q are re-written as ( 2222 22 2 () () () () () () () () () () () () () )()()() () ()()() (( TT T TT T ANNi Si Ri i i Ri Si Si Li i i Li Si TT URi i i URi TTTT TT Si Ri Ri Li Li Si i i URi URi i i T Xi Ri Ri kkk kkkk k kkk k k k kk k k kk N        Q X GR RGX X GRRGX XRRX X GGGGX RR X X RR GG G    2 2 ))()() () () TT Li Li Ni P i i T DNNi Ni P i i Nkk Nkk      GIRR QIRR . (49) As evident by(47), (48) and(49), 22 () NNi k  Q is composed of major matrix ( ) ANNi k  Q and noise matrix ( ) DNNi k  Q . In the case of single talking where sound source signal power 2 X  is much Adaptive Filtering 206 larger than un-correlated signal power 2 Ni  , ( ) ( ) T ii kkRR which minimizes eigenvalue spread in 22 () NNi kQ so as to attain the fastest convergence is given by making A NNi  Q as a identity matrix by setting ( ) ( ) T ii kkRR as 21 () () ( ( )) TTT i i Xi Ri Ri Li Li kkN   RR GGGG (50) In other cases such as double talking or no talk situations, where we assume 2 X  is almost zero, () () T ii kkRR which orthogonalizes A NNi   Q is given by 21 () () ( ) T ii NiP kkN   RR I (51) Summarizing the above discussions, the fastest convergence is attained by setting () () T ii kkRR as   1 22 () () () () TT ii PNiPNi kk k k  RR X X . (52) Since 22 2 22 2 22 22 2 () () () () () () () () () () () () () () () () () () ( T PNi PNi T Si Ri URi TT TT Ri Si URi Li Si ULi T Si Li ULi TTTTT T Ri Si Si Ri Li Si Si Li URi URi ULi ULi T Xi Ri Ri kk kk kk kk kk kk kk k k kk N             XX XGX GX X GX X XGX GXXGGXXGX X X X GG  )+2 2T Li Li Ni P N   GG I . (53) By substituting (52) for (30), we obtain following affine projection iteration : =(k) 1 22 ˆˆ (1) () ()( () ()) T STi STi i P Ni P Ni Pi kkkkk   hhXXXe . (54) In an actual implementation  is replaced by μ for forgetting factor and I  is added to the inverse matrix to avoid zero division as shown bellow. 1 222 ˆˆ ( 1) ( )+ (k)[ (k) (k) I] ( ) T ST ST P Ni P Ni P Ni Pi kk k    hhXXX μe (55) where (1)   is very small positive value and 1 [1,(1 ), ,(1 ) ] p diag   μ  . (56) The method can be intuitively understood using geometrical explanation in Fig. 3. As seen here, from a estimated coefficients in a k-1th plane a new direction is created by finding the nearest point on the i th plane in the case of traditional NLMS approach. On the other hand, affine projection creates the best direction which targets a location included in the both i-1 and i th plane. A Stereo Acoustic Echo Canceller Using Cross-Channel Correlation 207   (1),(1) RL kkxx Space   (1),(1) RL kk  xx   () , () RL kkxx   () , () RL kkxx Space NLMS Iteration Affine Projection Goal Fig. 3. Very Simple Example for Affine Method 3.4 Stereo Normalized Least Mean Square (NLMS) method (M=1) Stereo NLMS method is a case when M=1 of the gradient method. Equation (54) is re-written when M =1 as =(k) 1 ˆˆ ( 1) () ()( () () () ()) TT STi STi i Ri Ri Li Li i kkkkkkke    hhxxxxx (57) It is well known that convergence speed of (57) depends on the smallest and largest eigen- value of the matrix 22NNi Q . In the case of the stereo generation model in Fig.2 for single talking with small right and left noises, we obtain following determinant of 22NNi Q for M=1 as (( 1 22 12 () ()( () ()) () )) TT NNi i i i i TT TT Ri Ri Li Li Ri Ri Li Li N N kkkkk      Qxxxx gg gg gg gg I (58) If eigenvalue of TT Ri Ri Li Li gg gg are given as ( 22 min max ) TT Ri Ri Li Li i i  gg gg  (59) where 2 mini  and 2 maxi  are the smallest and largest eigenvalues, respectively. Adaptive Filtering 208 22 () NNi kQ is given by assuming un-correlated noise power 2 Ni  is very small ( 22 minNi i   ) as 12 2 2 2 22 min max () ( ) TT NNi Ri Ri Li Li Ni Ni i i k      Qgggg (60) Hence, it is shown that stereo NLMS echo-canceller’s convergence speed is largely affected by the ratio between the largest eigenvalue of TT Ri Ri Li Li gg gg and non-correlated signal power 2 Ni  . If the un-correlated sound power is very small in single talking, the stereo NLMS echo canceller’s convergence speed becomes very slow. 3.5 Double adaptive filters for Rapid Projection (WARP) method Naming of the WARP is that this algorithm projects the optimum solution between monaural space and stereo space. Since this algorithm dynamically changes the types of adaptive filters between monaural and stereo observing sound source characteristics, we do not need to suffer from rank drop problem caused by strong cross-channel correlation in stereo sound. The algorithm was originally developed for the acoustic echo canceller in a pseudo-stereo system which creates artificial stereo effect by adding delay and/or loss to a monaural sound. The algorithm has been extended to real stereo sound by introducing residual signal after removing the cross-channel correlation. In this section, it is shown that WARP method is derived as an extension of affine projection which has been shown in 3.3. By introducing error matrix ( ) i kE which is defined by )()() () ( -1 - 1 iPiPi Pi kkk kp      Eee e (61) iteration of the stereo affine projection method in (54) is re-written as = 1 222 ˆˆ (1) () ()( () ()) () T STi STi P Ni P Ni P Ni i kkkkkk   HHXXXE (62) where )() ˆˆ ˆ ˆ () () ( -1 - 1 STi STi STi STi kkk kp      Hhh h (63) In the case of strict single talking, following assumption is possible in the ith LTI period by (53) () () T PNi PNi RRLLi kkXX G (64) where RRLLi G is a PxP symmetric matrix as ) 2 ( TT RRLLi Xi Ri Ri Li Li N  GGGGG (65) By assuming RRLLi G as a regular matrix, (62) can be re-written as = 2 ˆˆ (1) () ()() STi RRLLi STi RRLLi P Ni i kkkk  HGHGXE (66) [...]... simulations 222 Adaptive Filtering (b) Short tap adaptive filter(AF2)) Gsub1 1.0 CCTF Responce (a) Long tap adaptive filter(AF1)) 1.0 0 .8 response 0.6 0 .8 0.4 Response (AF1 Coefficinets) 0.2 0.0 0.6 -0.2 -0.4 0.4 0 1 2 3 4 5 6 7 8 sample 0.2 0.0 -0.2 -0.4 0 20 40 60 80 100 120 140 Sample Fig 11 Impulse Response Estimation Results in CCTF Block Fig 12 Estimated Tap Coefficients by Short Tap Adaptive Filter... Loc(2)=(-0 .8, 0.5), C Loc(3)=(-0 .8, 0.0), D Loc(4)=(-0 .8, -0.5) and D Loc(5)=(-0 .8, -1.0) are used and R/L microphone locations are set to R-Mic=(0,0.5) and L-Mic=(0,-0.5), respectively Delay is calculated assuming voice wave speed as 300m/sec In this set-up, talker’s position change for WGN is assumed to be from location A to location B and finally to location D, in which each talker stable period is set to 80 ... )z( dRLi  dRLi1 )  RR  Transition (1 08)  dRLi  dRLi 1 ˆ 1 ˆ h Monoi  1 ( z) ˆ ( z )  h Monoi ( z )  lRLi lRLi  1φ( RLi , z)φ(  RLi  1, z)z hLi 1 1  (lRLi lRLi  1 )φ(  RLi  1, z )φ( RLi , z)z ( dRLi  dRLi1 ) 2 18 Adaptive Filtering These functions are assumed to be digital filters for the echo path estimation results as shown in Fig .8 ˆ Hi ( z) + ˆ l 1RLi1 RLi1 ( z 1 )... and 130 frames As evident from the results in Fig 18, WARP method shows better performances for the stereo echo path estimation regardless far-end double talking existence Even in the case 10% far end signal level shit, WARP method attains more than 20% NORM comparing affine method (P=3) with 10% absolute non-linear result 2 28 Adaptive Filtering Fig 18 Echo Path Estimation Performance Comparison for... ( z)x i ( z) i y i ( z)  hT ( z)x i ( z)  n i ( z) (87 ) ˆ ˆ where n i ( z ) is a room noise, hi ( z) and hi ( z) are stereo adaptive filter and stereo echo path characteristics at the end of ith LTI period respectively and which are defined as ˆ  h ( z)  h ( z ) ˆ HSTi ( z)   Ri  , HST ( z)   R  ˆ ( z)   hLi   hL ( z)     (88 ) Then cancellation error is given neglecting near end... (CCTF) The estimators are prepared for right microphone side sound source case and left microphone side sound source case, respectively Each estimator has two NLMS adaptive filters, longer (1 28) tap one and shorter (8) tap one The longer tap adaptive filter (AF1) is used to find a main tap and shorter one (AF2) is used to estimate the transfer function precisely as an impulse response Figure 11 shows... expressed as shown in Fig 7 (85 ) 214 Adaptive Filtering xSi ( z) xURi (z)  xULi (z) Cross Channel Correlation Generation Matrix W (z) i 0 xRi (z) xLi (z) Cross Channel Correlation Cancellation Matrix STi i Monoi xRi (z) Cross Channel Correlation Recovery Matrix Wi1(z) WARP Matrix ˆ ˆ HMonoi (z)  Wi1(z)HST (z) ˆ ˆ H (z)  W (z)H (z) xSi (z) xLi (z) Wi (z) Multi-Channel Adaptive Filter Multi-Channel... sampled at f S  8KHz after 3.4kHz cut-off low-pass filtering Frame length is set to 100 samples Since the stereo sound generation model is essentially a continuous time signal system, over-sampling (x6, f A  48KHz ) is applied to simulate it In the stereo sound 221 A Stereo Acoustic Echo Canceller Using Cross-Channel Correlation generation model, three far-end talker’s locations, A Loc(1)=(-0 .8, 1.0), B... the S/N is set to 20dB ~ 40dB y 0 .8 A FAR ( z) 0.5 E X Ri , j ( z ) FAL ( z) FB L ( z  ) D LPF Right microphone side N R ( z) 0.5 C X MicRi , j ( z ) FB R ( z  ) B x Left microphone side N L ( z) LPF X Li , j ( z ) X MicLi , j ( z ) d Over-sampling (x6) area to simulate analog delay Simulation set-up for Stereo Sound Generation AF1 1 28 tap N-LMS AF － + D2 AF2 8 tap N-LMS AF － + CL(dB) Calculation... noise xURi ( ) and xULi ( ) as x Ri ( )  gSRi ( )xSi ( )  xURi ( ) xLi ( )  gSLi ( )xSi ( )  xULi ( ) (83 ) In the case of simple direct-wave systems, (83 ) can be re-written as xRi ( )  lRi e  j Ri xSi ( )  xURi ( ) xLi ( )  lLi e  j Li xSi ( )  xULi ( ) (84 ) where lRi and lLi are attenuation of the transfer functions and  Ri and  Li are analog delay values Since the right . Transition zzz       φφ  (1 08) Adaptive Filtering 2 18 These functions are assumed to be digital filters for the echo path estimation results as shown in Fig .8. ˆ H( ) i z 1 ˆ H() i z  + + ˆ H() LRi z 1 ˆ H() i z  + + ˆ H() RRi z ˆ H(.  -domain as exp[2 / ] s zj    . (85 ) In z-domain, the system in Fig.4 is expressed as shown in Fig. 7. Adaptive Filtering 214 Multi-Channel Adaptive Filter + () i zy Cross Channel Correlation Cancellation Matrix Cross Channel Correlation Recovery Matrix Cross Channel Correlation Generation Matrix () i zW () Si zx () () 0   URi ULi z z x x 1 ˆ ˆˆ (). 2 mini  and 2 maxi  are the smallest and largest eigenvalues, respectively. Adaptive Filtering 2 08 22 () NNi kQ is given by assuming un-correlated noise power 2 Ni  is very small