itle Author(s) Convergence Analysis of Stereophoni Relation between PreProcessing a Hirano, Akihiro; Nakayama, Kenji; W Citation Issue Date 9993 ype Conference Paper ext version publisher UR http://hdl.handle.net/97/689 Right *KURA に登録されているコンテンツの著作権は, 執筆者, 出版社 ( 学協会 ) などが有します *KURA に登録されているコンテンツの利用については, 著作権法に規定されている私的使用や引用などの範囲内で行ってください * 著作権法に規定されている私的使用や引用などの範囲を超える利用を行う場合には, 著作権者の許諾を得てください ただし, 著作権者から著作権等管理事業者 ( 学術著作権協会, 日本著作出版権管理システムなど ) に権利委託されているコンテンツの利用手続については, 各著作権等管理事業者に確認してください http://dspace.lib.kanazawau.ac.jp/dspac
CONVERGENCE ANAYSIS OF SEREOPHONIC ECHO CANCEER WIH PREPROCESSING REAION BEWEEN PREPROCESSING AND CONVERGENCE Akihiro HIRANO Kenji NAKAYAMA Kazunobu WAANABE Faculty of Engineering, Kanazawa University 4, Kodatsuno Chome, Kanazawa 98667, Japan Email: hirano@t.kanazawau.ac.jp ABSRAC his paper presents convergence characteristics of stereophonic echo cancellers with preprocessing. he convergence analysis of the averaged tapweights show that the convergence characteristics depends on the relation between the impulse response in the farend room and the changes of the preprocessing filters. Examining the uniqueness of the solution in the frequency domain leads us to the same relation. Computer simulation results show the validity of these analyses.. INRODUCION Echo cancellers are used to reduce echoes in a wide range of applications, such as V conference systems and handsfree telephones. o realistic V conferencing, multichannel audio, at least stereophonic, is essential. For stereophonic teleconferencing, stereophonic acoustic echo cancellers have been studied [9]. In stereophonic echo cancellers, an uniqueness problem is one of the most serious problems [4]. here are infinite number of solutions for tapweights of adaptive filters. Strong crosscorrelation between input signals causes incorrect identification of the echo paths. o overcome this problem, improved echo cancellation algorithms have been proposed [59]. Some of these algorithms introduce a preprocessing which artificially varies the crosscorrelation [5, 6]. he others incorporate independent components into input signals [79]. However, the convergence of these algorithm has not been analyzed. his paper investigates convergence characteristics of stereophonic echo cancellers with preprocessing. he convergence analysis of the averaged tapweights show that the convergence characteristics depends on the relation between the impulse response in the farend room and the changes of the preprocessing filters. Examining the uniqueness of the solution in the frequency domain leads us to the same relation. Computer simulations validate the analyses.. SEREOPHONIC ACOUSIC ECHO CANCEER WIH PREPROCESSING Figure depicts an teleconferencing using stereophonic echo canceller with preprocessing [5, 6]. his echo canceller consists of four adaptive filters corresponding to four echo paths from two loudspeakers to two microphones. Each adaptive filter estimates the corresponding echo path. Preprocessing units F,k Room A echo canceller x () n s () n G F,k x( n) G x () n e () n e () n F,k w,( n ) + w,( n ) w,( n ) s () n w,( n ) + y () n y () n Room B eft h, h, h, Fig.. eleconferencing using stereophonic echo canceller with preprocessing. h, Right and F,k, which are timevarying filters, are introduced for correct echo path identification. he farend signal vector x j (n) inthe jth channel at time index n is generated from a white signal vector x(n) by x j (n) = G j x(n). () G j is a matrix consists of the farend room impulse response g j,k defined by G j = g j, g j, g j,n f +N h g j,k =,k g j, g j, g j,ng,n f +N h k (3) where N g is the length of the farend room impulse response, N f is the number of taps of the preprocessing filters, N g is the number of taps of the adaptive filters, i, j is an i j zero matrix, [ ] denotes the transpose of a matrix [ ]. For colored signals, g j,k contains both a coloring filter and the room impulse response. he preprocessed signal vector s j (n) is calculated by s j (n) = F j,k x j (n) = F j,k G j x(n) where F j,k contains the coefficients of the jth preprocessing filter in the kth state given by F j,k = f j,k, f j,k, f j,k,nh (4) ()
f j,k,l =,l f j,k, f j,k, f j,k,n f,nh l. (5) For a recursive preprocessing filters, an FIR approximation can be used. In order to overcome the uniqueness problem, multiple F j,k s are used. Each F j,k will be selected in the order of k, orin a random order. he echo y i (n) intheith channel is generated by y i (n) = Σ s j (n)h j,i (6) j= where h j,i is the impulse response vector of the echo path from the jth loudspeaker to the ith microphone. he echo replica ŷ i (n) is calculated by ŷ i (n) = Σ s j (n)w j,i (n) (7) j= where w j,i (n) is the tapweight vector of the adaptive filter which estimates h j,i. he residual echo e i (n) isgiv enby e i (n) = y i (n) ŷ i (n) = x (n) Σ G j F j,k(h j,i w j,i (n)). (8) i= Assuming an MS (east Mean Squares) algorithm, the tapweight vector is updated by w j,i (n + ) = w j,i (n) + µx j (n)e i (n) (9) where the step size µ is a positive constant. 3. CONVERGENCE OF AVERAGED APWEIGHS Ensemble average of the tapweight error defined by m j,i (n) = E[h j,i w j,i (n)] () will be analyzed. he simultaneous difference equation for m j,i (n) are m,i (n + ) = (I µf,k G G F,k)m,i (n) µf,k G G F,km,i (n) () m,i (n + ) = (I µf,k G G F,k)m,i (n) µf,k G G F,km,i (n) () where I is an unit matrix. hese equations can be rewritten as M(n + ) = I µf kgg F k M(n) (3) where M(n) = m,i (n), G = G, F m,i (n) G k = F,k. (4) F,k et us assume that there are two preprocessing filters F and F and that they are periodically switched; using F for iterations and then F for iterations. Extension of this results to more general case is strait forward. he av eraged tapweight error matrix M(n) becomes M((m + )) = I µf GG F I µf GG F M(m) (5) where m is an integer. By introducing the difference of two filters F = F F, (6) M((m + )) is derived as M((m + )) = I µ(f + F)GG (F + F ) I µf GG F M(m) = I µf GG F µ FGG F + F GG F + FGG F I µf GG F M(m). (7) Using a matrix D defined by D = µ FGG F + F GG F + FGG F, (8) M((m + )) is derived as M((m + )) = Σ C(i, ) i= I µf GG F D i M(m) (9) where C(i, ) is number of combinations selecting i from. If D =, i.e., the preprocessor is a fixed filter, the right hand side of (9) becomes (I µf GG F ) M(m). his corresponds to the convergence of the traditional stereophonic echo canceller []. Because of the uniqueness problem, (I µf GG F ) i do not converge to zero. Some of the eigenvalues of F GG F are zero and components corresponding to zero eigenvalues do not converge to the optimum values[4]. For convergence of the tapweights to the echo paths, D should not be zero. Nonzero D will change the minimum eigenvalue from zero to nonzero. arger D may results in smaller eigenvalue spread, and therefore, faster convergence will be achieved. In D, FG denotes a combined characteristics of G and F F, i.e., filtering signals by G and then by F F. hus, relation between G and F F has a large influence on the convergence of the echo canceller. If some frequency components are removed by either G or F F, the frequency response of the adaptive filter corresponding to these frequencies is not determined. Since users cannot modify the farend room characteristics G, F F should be so designed as not to reduce any frequency components which G passes. Otherwise, the convergence speed becomes slow. Similarly, large FGG F requires that the passband of F should cover that of G. 4. ANAYSIS IN FREQUENCY DOMAIN he influence of the relation between the farend room characteristics and the preprocessing filters on the convergence will further be analyzed in the frequency domain. For kth pre +i
processing filter, the residual echo E i,k (z) intheith channel is calculated by E i,k (z) = Σ j= H j,i(z) W j,i (z) F j,k(z)g j (z)x(z) () where G j (z) is the farend room transfer function, F j,k (z) isthe preprocessing filter, H j,i (z) is the echo path, W j,i (z) is the adaptive filter, X(z) is the signal source. For a simple case where F, (z) = F, (z) =, solving E i, (z) = E i, (z) = results in H,i(z) W,i (z) F,(z) F, (z) G (z)x(z) =. () In order to have an optimum solution W,i (z) = H,i (z), (F, (z) F, (z))g (z)x(z) should not be zero. herefore, the passband of F, (z) F, (z) should cover all the passband of G (z). Similarly, the passband of F, (z) F, (z) should also cover all the passband of H,i (z) W,i (z). For more general case, the convergence condition becomes F,(z)F, (z) F, (z)f, (z) G j(z). () Introducing F j, (z) = F j, (z) F j (z) (3) leads to F, (z)f, (z)( F (z) F (z))g j (z). (4) his suggests that the passbands of F, (z), F, (z) and F (z) F (z) should cover the passband of G j (z) for good convergence. hese results from frequency domain analysis agree with the convergence analysis of the averaged tapweights shown in previous section. 5. COMPUER SIMUAIONS Simulations have been carried out to examine the relation between the characteristics of the preprocessor and that of the room impulse response. th order Butterworth filters are used as the transfer functions in the farend room G j (z) s and the echo paths H j,i (z) s. Both highpass filters (F s) and lowpass filters (PF s) are used. Example of the transfer functions are shown in Fig.. For simplicity, F, (z) = F, (z) = F, (z) = are assumed. hus, F, (z) = F(z) is periodically inserted and then removed in the first channel. As a preprocessing filter, four F(z) s are used: () F(z) = z. F(z) is F. his can be considered as a simplified version of [5]. () F(z) = a + z. F(z) is a F. his can be considered as a simplified version of [6]. + az (3) F(z) = z. F(z) isapf. ( a) z (4) F(z) =. F(z) is an allpass filter (APF). + az Figure 3 demonstrates the transfer function F(z) for each F(z). 64tap adaptive FIR filters with the normalized MS (least mean squares) algorithm, which is identical to the MS algorithm for stationary inputs, are used. he step size is µ =. 5..5.5.8.6.4...4.6.8.8.6.4...4.6.8 Fig.. Amplitude response of echo path. ()..4.6.8.5.5 (3)..4.6.8.5.5 ()..4.6.8.5.5 (4)..4.6.8 Fig. 3 Amplitude response of F(z). Figure 4 () depicts the convergence characteristics of the tapweight error vector norm for F(z) = z. A curve with "P" label denotes the characteristics when both the echo path and the farend room transfer functions are PF s, and vice versa. As expected by the analyses, convergence speed for the PF case is slower than that for the F case. For the PF case, ( F(z))G j (z) becomes too small for a low frequency range. he convergence becomes slow for the low frequency range. he results in Fig. 4 () is similar to that in () because F(z) is also a PF. he opposite situation occurs when F(z) isa PF. As shown in Fig. 4 (3), the convergence is faster when the echo path and the farend room transfer functions are PF s. Figure 4 (4) shows that the convergence speed is almost independent of the G j (z) and H j,i (z) if F(z) is an APF. his is because the amplitude of ( F(z))G j (z) is same as that of G j (z). 6. CONCUSION he convergence characteristics of the stereophonic echo cancellers with preprocessing have been examined. he convergence analysis of the averaged tapweights show that the
P P.5.5.5 3 3.5 4 4.5 5 IERAION [sample] x 4 () F(z) = z.5.5.5 3 3.5 4 4.5 5 IERAION [sample] x 4 (3) F(z) is allpass filter. P P.5.5.5 3 3.5 4 4.5 5 IERAION [sample] x 4 () F(z) = z Fig. 4 Convergence of tapweight error vector norm. convergence characteristics depends on the relation between the impulse response in the farend room and the changes of the preprocessing filters. he analysis in the frequency domain results in a same relation. Computer simulation results show the validity of these analyses. REFERENCES []. Fujii et al., "A Note on MultiChannel Echo Cancellers," echnical Reports of IEICE on CS, pp. 74, January 984 (in Japanese). [] A. Hirano et al., "Convergence Characteristics of a Multi Channel Echo Canceller with Strongly Crosscorrelated Input Signals Analytical Results," Proc. of 6th DSP Symposium, pp. 4449, November 99. [3] M. M. Sondhi et al., "Stereophonic Acoustic Echo Cancellation An Overview of the Fundamental Problem," IEEE SP etters, vol., no. 8, pp. 485, August 995. [4] A. Hirano et al., "Convergence Analysis of a Stereophonic Acoustic Echo Canceller Part I: Convergence Characteristics of ap Weights," Proc. of th DSP Symposium, pp..5.5.5 3 3.5 4 4.5 5 IERAION [sample] x 4 (4) F(z) is allpass filter. Fig. 4 Convergence of tapweight error vector norm (Conti.). 569574, November 996. [5] Y. Joncour et al., "A Stereo Echo Canceler with PreProcessing for Correct Echo Path Identification," Proc. of ICASSP 98, pp. 3677368, May 998. [6] M. Ali, "Stereophonic Acoustic Echo Cancellation System Using imevarying AllPass Filtering for Signal Decorrelation," Proc. of ICASSP 98, pp. 3689369, May 998. [7] J. Benesty et al., "Stereophonic Acoustic Echo Cancellation Using Nonlinear ransformations and Comb Filtering," Proc. of ICASSP 98, pp. 36373676, May 998. [8] A. Gilloire et al., "Using Auditory Properties to Improve he Behaviour of Stereophonic Acoustic Echo Cancellers," Proc. of ICASSP 98, pp. 3683684, May 998. [9] S. Shimauchi et al., "New Configuration for A Stereo Echo Canceller with Nonlinear PreProcessing," Proc. of ICASSP 98, pp. 36853688, May 998.