Multi-Channel Active Noise Control for All Uncertain Primary and Secondary Paths

Multi-channel Active Noise Control for All Uncertain Primary and Secondary Paths

Yuhsuke Ohta and Akira Sano Department of System Design Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

Abstract filter matrices in an on-line manner, which enables the noise cancellation at the objective points. Unlike the pre- Fully adaptive feedforward control algorithm is proposed vious indirect approaches based on the explicit on-line for general multi-channel active noise control (ANC) identification, neither dither sounds nor the PE property when all the noise transmission channels are uncertain. of the source noises are required in the proposed scheme. To reduce the actual canceling error, two kinds of virtual errors are introduced and are forced into zero by adjusting 2. Feedforward Adaptive Active Noise Control three adaptive FIR filter matrices in an on-line manner, which can result in the canceling at the objective points. Primary Reference Secondary Error Noise Microphones Sources Microphones Unlike other conventional approaches, the proposed algo- Sources G (z) rithm does not need exact identification of the secondary 1

e1 (k) paths, and so requires neither any dither signals nor the v1(k) s1 (k) PE property of the source noises, which is a great advan- r1(k) G (z) tage of the proposed adaptive approach. 3 G2 (z) G4 (z) s(k)

1. Introduction vN (k) c e (k) Nc s (k) Ns r (k) Active noise control (ANC) is efficiently used to sup- Nr press unwanted low frequency noises generated from primary sources by emitting artificial secondary sounds to u(k) r(k) Adaptice Active objective points [1][2]. Adaptive feedforward control Noise Controller Cˆ(z,k) schemes using the primary noises measured by reference e(k) microphones are effective to ANC, since the noise path channels cannot be precisely modeled and are uncertainly changeable. A variety of filtered-x algorithms have been Figure 1: Schematic diagram of multi-channel adaptive adopted to attain the feedforward adaptation [2][3][4], on feedforward active noise control system. the assumption that all the secondary channels are known a priori. To deal with a general case when the secondary The setup of multi-channel feedforward active noise path channels are also unknown, almost previous works control system is illustrated in Fig.1. The primary noises N were based on indirect adaptive approaches which employ s(k) ∈ R s are generated from the Ns sources, and de- on-line identification of the secondary channels. There- tected by the Nr reference microphones. The detected N fore, dither noise should be added to artificial control signals r(k) ∈ R r are the input to the Nc × Nr adap- sound to assure the PE condition. By taking an exten- tive feedforward controller matrix Cˆ (z,k), where Nc is sion of the filtered-x type of approaches, the identified the number of the secondary loudspeakers which pro- models of the secondary channels are used correspond- duce artificial sounds to cancel the primary noises at ingly in the filtered-x algorithms [3][6][7][8] or the feed- the Nc objective points. The canceling errors are de- N forward controller [4][5]. Recently different approaches tected as ec(k) ∈ R c by the Nc error microphones. N ×N N ×N without secondary path identification have also been pro- G1(z) ∈Z c s and G2(z) ∈Z r s represent the posed [9][10]. primary channel matrices from the primary noise s(k) The aim of this paper is to propose a fully direct adap- to the reference microphones and error microphones, re- N ×N N ×N tive control approach which does not need explicit identi- spectively. G3(z) ∈Z r c and G4(z) ∈Z c c are fication of the secondary channels. To reduce the actual the secondary channel matrices from the secondary con- canceling errors, two kinds of virtual errors are introduced trol sounds u(k) to the reference and error microphone and are forced into zero by adjusting three adaptive FIR sets, respectively, where G3(z) is referred to as the cou- 3ULPDU\ 3DWK ea k eb k → k →∞ &KDQQHO Thus, if ( ) and ( ) 0 for is satisfied and *] \N ˆ ˆ the FIR parameters of C(z,k) and K(z,k) converge to HN . any constants, then the second and third terms in the VN R T *] 6HFRQGDU\ 3DWK right hand side of (4) can be cancelled, then by the rela- &KDQQHO ea k eb k e k R UN XN tion ( )+ ( )= c( ), thus it can also be attained *] R &]N *] \N that ec(k) → 0 [11]. $GDSWLYH )LOWHU , , Dˆ z,k Kˆ z,k R HN It seems that ( ) and ( ) are the identified .]N R T models for G1(z) and G4(z) respectively and the adaptive $GDSWLYH )LOWHU ,, Cˆ z,k ']N controller ( ) is adjusted according to the identified

$GDSWLYH models of the secondary path dynamics. However, even )LOWHU ,,, - R HN when the source noise does not satisfy the PE property, .]N &]N T $GDSWLYH )HHGIRUZDUG [N &RQWUROOHU IRU $1& $GDSWLYH the proposed algorithm does not require the convergence )LOWHU ,, $GDSWLYH )LOWHU , of the adjusted parameters to their true values, but only the convergence of their parameters to any constants such ea k eb k Figure 2: Direct fully adaptive algorithm for single chan- that the errors ( ) and ( ) can converge to zero (see nel ANC Fig.9). Therefore, any dither sounds are not needed unlike the conventional indirect adaptive algorithm. The pling channel matrix. All the channels contain model degradation and complexity caused by the dither signals uncertainty and parameter changeability. can also be overcome by the proposed direct adaptive al- It follows from Fig.1 that gorithm. e k G z s k − G z u k c( )= 1( ) ( ) 4( ) ( ) (1a) 3.2 New ANC algorithm in two-channel case r k G z s k G z u k ( )= 2( ) ( )+ 3( ) ( ) (1b) In a multi-channel case, the exchange of two matrices u(k)=Cˆ (z,k)r(k) (1c) gives a different result, so the algorithm should be mod- ified. To mitigate the problem, we employ a diagonal Cˆ z,k where ( ) is an FIR type of adaptive feedforward con- matrix as Kˆ (z,k) to derive a fully adaptive algorithm for troller matrix, and the coefficients in Cˆ (z,k) can be up- adjusting the controller parameters. Here for the sim- dated by various filtered-x adaptive algorithms using the plicity of notation, we show the adaptive algorithm in given secondary path models. two-channel case. Similar to the single-channel case, we define two kinds 3. New Adaptive Algorithm for ea k eb k Uncertain Secondary Channels of virtual error vectors ( ) and ( ) which can be de- scribed by 3.1 Key idea for new adaptation algorithm in a e1(k) ec1(k) Kˆ11(z)0 u1(k) single channel case a = + e2(k) ec2(k) 0 Kˆ22(z) u2(k) We give a new direct adaptive algorithm which does not Dˆ11(z) Dˆ12(z) r1(k) need explicit identification of the secondary path chan- − (5) Dˆ21(z) Dˆ22(z) r2(k) nels, unlike the filtered-x algorithms using the identified eb k Dˆ z Dˆ z r k model of G¯4(z). The basic structure of the proposed 1( ) 11( ) 12( ) 1( ) b = e2(k) Dˆ 21(z) Dˆ 22(z) r2(k) adaptive feedforward control algorithm is illustrated in   e k ea k eb k Fig.2, where c( ), ( ), ( ) can be expressed as: x11(k) Cˆ z Cˆ z  x k  ec(k)=G1(z)r(k) − G4(z)u(k) (2a) 11( ) 12( )0 0  12( )  −   (6) a 00Cˆ21(z) Cˆ22(z) x21(k) e (k)=ec(k)+Kˆ (z,k)u(k) − Dˆ(z,k)r(k) (2b) x22(k) eb(k)=Dˆ(z,k)r(k) − Cˆ(z,k)x(k) (2c) where    where G1(z) and G4(z) are defined in the previous sec- Kˆ z x11(k) 11( )0 tion, and the control input u(k) and the auxiliary signal      x12(k)   0 Kˆ11(z)  r1(k) x(k) are also defined as   =   (7) x21(k) Kˆ22(z)0 r2(k) u k Cˆ z,k r k ,xk Kˆ z,k r k x k ( )= ( ) ( ) ( )= ( ) ( ) (3) 22( ) 0 Kˆ22(z) It follows from Fig.2 and the definitions that u1(k) Cˆ11(z) Cˆ12(z) r1(k) = (8) a b u k Cˆ z Cˆ z r k e (k)+e (k)=[ec(k)+Kˆ (z,k)u(k) − Dˆ(z,k)r(k)] 2( ) 21( ) 22( ) 2( ) Dˆ z,k r k − Cˆ z,k x k +[ ( ) ( ) ( ) ( )] In the proposed adaptive algorithm, the coefficient pa- = ec(k)+Kˆ (z,k)Cˆ(z,k)r(k) − Cˆ(z,k)Kˆ (z,k)r(k) (4) rameters of Kîi(z) and Dˆ ij(z) are updated so that the a a first virtual errors e1(k) and e2(k) can be eliminated, where i = 1 and 2. while the parameters of Cîj(z) are updated so that the Other adaptive algorithms can also be derived, for in- b b 2 second virtual errors e1(k) and e2(k) can be eliminated. stance, the norm e(k) of the augmented error e(k)= If these parameters converge to constant and the two vir- (eaT (k), ebT (k))T can also minimized, which gives an al- tual errors can be reduced almost to zero, then it follows thernative algorithm which requries more computation. from (5) ∼ (8) that Therefore, we adopted the proposed algorithm given by (10) to (14) in the experiment. a b ei (k)+ei (k)=eci(k)+Kîi(z)Cî1(z)r1(k) +Kîi(z)Cî2(z)r2(k) − Cî1(z)Kîi(z)r1(k) 4. Simulation Results −Cî2(z)Kˆ ii(z)r2(k) for i = 1 and 2 (9) 0.04 0.04 ea k eb k 0.02 0.02 Thus, if the virtual errors i ( ) and i ( ) can be reduced 0 0 to zero and all the adjustable parameters converge to con- Amplitude -0.02 Amplitude -0.02 -0.04 -0.04 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 stant, then other four terms in the right hand side can Time [s] Time [s] be cancelled and finally it follows that the actual errors (a) G111(z) (b) G411(z) eci(k) can also eliminated. Let the adjustable parameters be the coefficients of the Figure 3: Example of identified FIR channels FIR filters defined as: Fig.3 depicts examples of FIR models of the primary (1) (mc ) c −1 ij −mij Cîj (z,k)=ˆcij (k)z + · +ˆcij (k)z and secondary channels which were idenitifed by exper- (1) −1 (mk ) −mk iments for the active noise control in a room. These Kˆ z,k kˆ k z · kˆ ii k z ii ii( )= ii ( ) + + ii ( ) models are used in the simulations. Two primary loud- (1) (md ) d −1 ij −mij speakers, two reference microphones, two secondary loud- Dˆ ij(z,k)=dîj (k)z + · + dîj (k)z speakers, and two error microphones are placed initially (m) where i, j = 1 and 2. The adaptive parameters {kîi (k)} at specific locations, which configures a two-channel ac- (m) tive noise control system (Ns = Nr = Nc = Ne = 2). The and {dîj (k)} are updated so that the instantaneous error norm ea(k)2 may be minimized, while the pa- sampling period is chosen 1 ms. The power spectra of the (m) primary source noises s1(k) and s2(k) are limited in low rameters {dˆ (k)} are updated so that the error norm ij frequency range from 50 to 400 Hz, or have sometimes ea k 2 ( ) may be minimized. sinusoids with unknown frequencies which do not satisfy Let the parameter vectors be defined as the PE condition. Therefore in the conventional indirect k (1) (mii) T approaches need dither sounds which are additively gen- θˆKii(k)=(kîi (k), ··· , kîi (k)) d erated from the secondary loudspeakers to identify the (1) (mij ) T θˆDij(k)=(dîj (k), ··· , dîj (k)) secondary channels to assure the PE condition. On the (mc ) other hand, the proposed approach does not need any θˆ k c(1) k , ··· , c ij k T Cij( )=(îj ( ) îj ( )) dither sounds and the PE condition, but it can attain the and let the regressor vectors be denoted as noise attenuation even in the presence of uncertainties of the secondary channels. k T e k ζi(k)=[ui(k − 1), ··· ,ui(k − mii)] Figs.4 to 6 show the actual canceling errors c1( ) d T and ec2(k). In the numerical simulations, the locations ξj(k)=[rj (k − 1), ··· ,rj(k − mij)] of the two error microphones are changed by 34[cm] in- ϕ k x k − , ··· ,x k − mc T ij( )=[ ij ( 1) ij ( ij)] stantaneously far from the original positions by using the switches at 10[s] after the start of control. This causes the where i = 1 and 2, and xij (k)=Kîi(z)rj(k). uncertain changes of the secondary path channels, and Thus the adaptive algorithm for updating these param- the extended error based filtered-x algorithm [5] could eters is summarized as follows: not keep the stable attenuation performance as shown in a Fig.5, since it needs precise knowledge on the secondary θˆKii(k +1)=θˆKii(k) − γK ζi(k)εi (k) (10) channels. On the other hand, the proposed method can θˆ k θˆ k γ ξ k εa k Dij( +1)= Dij( )+ D j( ) i ( ) (11) attain the stable control performance even if the sec- a a 2 2 εi (k)=ei (k)/[1 + γD(ξ1(k) + ξ2(k) ) ondary channels change very rapidly. Thus, the proposed 2 algorithm does not need on-line identification of the sec- +γK ζi(k) ] (12) ondary channels. θˆ k θˆ k γ ϕ k εb k Cij( +1)= Cij( )+ C ij( ) i ( ) (13) Figs.7 and 8 show the control results and Fig.9 shows eb k Dˆ z,k Kˆ z,k εb k i ( ) one of parameters of 11( ) and 11( ), when the i ( )= 2 2 (14) 1+γC (ϕi1(k) + ϕi2(k) ) primary source noises consist of sinusoids with unknown 0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2

0 0 0 0 y_2 y_1 y_1 y_2

-0.2 -0.2 -0.2 -0.2

-0.4 -0.4 -0.4 -0.4 0 5 10 15 20 0 5 10 15 20 0 2 4 6 8 10 0 2 4 6 8 10 Time [s] Time [s] Time [s] Time [s]

Figure 4: ec1(k) and ec2(k) in case without control. Figure 7: ec1(k) and ec1(k) in case without control

0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2

0 0 0 0 e_c1 e_c2 e_C1 e_C2 -0.2 -0.2 -0.2 -0.2

-0.4 -0.4 -0.4 -0.4 0 5 10 15 20 0 5 10 15 20 0 2 4 6 8 10 0 2 4 6 8 10 Time [s] Time [s] Time [s] Time [s]

Figure 5: ec1(k) and ec2(k) obtained by the extended Figure 8: ec1(k) and ec2(k) obtained by the proposed fully error based ﬁltered-x algorithm (Shimizu et al., 2002). adaptive algorithm

(16) d k(4) 11 11 0.4 0.4 0.04 0.04

0.2 0.03 0.2 0.02 0 0 0.02 e_C1 e_C2

Amplitude 0 Amplitude 0.01 -0.2 -0.2 −0.02 0 -0.4 -0.4 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 5 10 15 20 0 5 10 15 20 Time [s] Time [s] Time [s] Time [s]

d(16) k k(4) k Dˆ z,k Figure 6: ec1(k) and ec2(k) obtained by the direct fully Figure 9: Parameters 11 ( ) and 11 ( )in 11( ) adaptive algorithm. and Kˆ11(z,k), respectively. frequencies (actually about 150 Hz and 250 Hz in the in- References terval (0s, 4s), and 100 Hz in the interval (8s, 10s)). In [1] S. J. Elliot and P. A. Nelson: Active Noise Control, Aca- Fig.9, dotted lines indicate 16th coefficient of G111(z) (left G z demic Press, 1993. figure) and 4th coefficient of 411( ) (right figure). Even [2] S. M. Kuo and D. R. Morgan: Active Noise Control Sys- when the primary source noises have an insufficient PE tems, John Wiley and Sons, 1996. condition of the primary source noises like sinusoids, the [3] L. J. Eriksson, ”Development of the filtered-U algorithm proposed algorithm can give very nice attenuation perfor- for active noise control”, J. Acoust. Soc. Am., Vol. 89, 16 4 mance in the interval (0s, 4s), and d11(k) and k11(k) con- pp.257-265, 1991. verged to any constants. During the interval, the adaptive [4] F. Jiang, H. Tsuji, H. Ohmori and A. Sano, ”Adaptation algorithm updates only small number of parameters re- for active noise control”, IEEE Control Systems, Vol.17, quired for reducing the canceling errors. During the inter- No.6, pp.36-47, 1997. [5] T. Shimizu, T. Kohno, H. Ohmori and A. Sano, ”Mul- val (4s, 8s), the primary source noises have rather much tichannel active noise control”, in Active Sound and Vi- PE property and then the many parameters of the adap- bration Control, IEE, 2002. tive filters should be updated, so the convergence time [6] N. Saito, T. Sone, T. Ise and M. Akiho,”Optimal on-line is need to achieve nice attenuation performance. During modeling of primary and secondary paths in active noise the interval (8s, 10s) the primary source noises are sinu- control systems”, J. Acoust. Soc. Jpn (E), Vol.17, No.6, soids again, however, since the adjustment of almost all pp.275-283, 1996. adaptive parameters has been completed , then no param- [7] S. Kim and Y. J. Park, ”Unified-error filtered-x LMS al- eters are required to update. Thus, the proposed control gorithm for on-line active control of noise in time-varying scheme is very robust to the insufficiency of the primary environment”, Proc. Active 97, pp.801-810, 1997. source noises unlike the conventional approaches. [8] M. Zhang, H. Lan and W. Ser,”Cross-updated active noise control system with online secondary path modeling”, IEEE Trans. Speech and Audio Processing, Vol.9, 5. Conclusion No.5, pp.598-602, 2001. [9] Y. Kajikawa and Y. Nomura, ”An active noise control The proposed fully direct adaptive control approach system without secondary path model”, Trans. IEICE, can work effectively even when all of the primary and Vol.J82-A, No.2, pp.209-217, 1999. secondary channel dynamics are uncertainly changeable. [10] K. Fujii, M. Muneyasu and J. Ohga, ”Active noise con- To reduce the canceling error, two virtual errors are in- trol systems by using the simultaneous equations method without estimation of the error path filter coefficients”, troduced and are forced into zero by adjusting the three Trans. IECIE, Vol.J82-A, No.3, pp.299-305, 1999. adaptive FIR filter matrices in an on-line manner, which [11] Y. Ohta, T. Kouno and A. Sano,” Adaptive active enables the noise cancellation at the objective points. Un- noise control for uncertain secondary path,” Proc. IEEE like the previous methods, neither dither noises nor the ICASSP, Hong Kong, China, 2003. PE property of the source noises are required.