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 pri- mary 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 un- like 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 Kˆii(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 Cˆij(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.