[N309] Feedforward Active Noise Control Systems with Online Secondary Path Modeling. Muhammad Tahir Akhtar, Masahide Abe, and Masayuki Kawamata

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1 he 32nd International Congre and Expoition on Noie Control Engineering Jeju International Convention Center, Seogwipo, Korea, Augut 25-28, 2003 [N309] Feedforward Active Noie Control Sytem with Online Secondary Path Modeling Muhammad ahir Akhtar, Maahide Abe, and Maayuki Kawamata Graduate School of Engineering, ohoku Univerity, Aoba Yama 05, Sendai , Japan. addre: ABSRAC he baic approach for active noie control (ANC) ytem with online econdary path modeling i called the additive random noie technique. It involve the injection of additional random noie into the ANC ytem and utilize a ytem identification method to model the econdary path. hi technique, propoed by Erikon et el. (1989), comprie two procee; a noie-control proce and a econdary-path modeling proce. Improvement in thi method have been propoed by many reearcher, viz. Bao et el. (1993), Kuo et el. (1997), and Zhang et el. (2001). hee improved method introduce another adaptive filter into the ANC ytem, and hence deign complexity i increaed. In thi paper a new tructure for feedforward ANC ytem with online econdary-path modeling i propoed. he propoed tructure 1) ue the ame error ignal for updating the noie-control proce a ued for the econdary-path modeling proce, and 2) incorporate an averaging baed filtered-reference algorithm in the noie-control proce. In contrat to exiting improved method, no extra adaptive filter i introduced into the ANC ytem. he theoretical conideration are confirmed by computer imulation, which how that the propoed tructure achieve better performance than that of the exiting method. hi improved performance i achieved at the expene of a lightly increaed computational complexity. KEYWORDS: Active noie control, Averaging, Fx algorithm, Online econdary path modeling

2 EXISINGS MEHODS he baic additive random noie technique for online econdary-path modeling in active noie control (ANC) [1] ytem i propoed by Erikon et el. [2]. A hown in Fig. 1, thi ANC ytem comprie two procee; a noie-control proce, and a econdary-path modeling proce. Auming that the control filter W(z) i an FIR filter of order L, the econdary ignal y( i expreed a where y( = w ( x ( (1) 0 ( w1 ( L wl-1( )] w ( = [ w n and L x ( = [ x( x( n 1) L x( n L + 1)] are tap weight vector and reference ignal vector, repectively. An internally generated zero mean white Gauian noie ignal, v(, uncorrelated with the reference noie x(, i injected at the output y( of the control filter. he reidual noie ignal e( i given a e ( = d( y ( + v ( (2) L where d( = p( x( i the primary diturbance ignal at the error microphone, y ( = ( y( i the econdary canceling ignal, v ( = ( v(, denote the convolution operation, and p( and ( are the impule repone of the primary path P(z) and econdary path S(z), repectively. he reidual noie ignal e( i ued a an error ignal for the control proce, and a a deired repone for the modeling proce, i.e., ew ( = d (. he coefficient of the control filter W(z) are updated by the Fx algorithm: w ( n + 1) = w( + µ e ( x ( = w( + µ x ( u( + µ x ( v ( (3) w w w w where µ w i the tep ize for the control proce, x ( i the reference ignal x( filtered through the modeling filter z ), and u ( = d( y ( i a component of the error ignal due to canceling noie only. We ee that the control proce i perturbed by an undeired term µ w x v (. Auming that z ) i repreented by an FIR filter of order M, the filteredreference ignal x ( i obtained a x ( = ˆ ( x M ( (4)

3 where ˆ( = [ˆ 0 ( ˆ 1( L ˆ M -1( ] i the impule repone of the modeling filter z ) and x ( = [ x( x( n 1) L x( n M + 1)]. he update equation for z ) i given a M ˆ( n + 1) = ˆ( + µ e ( v( = ˆ( + µ v( [ d ( ] = ˆ( + µ v( [ v ( ] + µ v( u( where µ i the tep ize of the modeling proce, v ˆ ( i an etimate of v ( obtained from the modeling filter, and v ( = [ v( v( n 1) L v( n M + 1)] (5). Eq. (5) how that the performance of the modeling proce i degraded by an undeired term µ v ( u( and in thort cae the modeling proce may diverge. Bao method i an attempt to improve the convergence of z ) in the preence of u(, by introduction of adaptive noie cancellation (ADNC) filter (Fig. 2(a)) [3]. he ADNC filter x( Noie Control Proce (Fx Algorithm) Copy P(z) W(z) z ) x White Noie Generator y( ( v( d( e ( = u( + v ( S(z) y - v ( d ( v ˆ ( z ) Secondary-path Modeling Proce e ( Fig. 1. Block diagram of active noie control ytem with online econdary path modeling (Erikon method). e( ( d ( ADNC filter x( B(z) ( d ( e( Prediction Error Filter z - C(z) e( n- ) v ˆ ( ( d ( e( ADNC filter with Cro-updating z( H(z) x( When B( z) converge then d ( v ( When C( z) converge then d ( v ( When Sˆ( z) and H( z) convergethen e w (, z( u( and d ( v ( (a) (b) (c) Fig. 2. Improved method for online econdary-path modeling in ANC ytem of Fig. 1: (a) Bao method (b) Kuo method, (c) Zhang method

4 B(z), excited by x(, cancel the interference introduced by u( in the etimation of S(z). In the method propoed by Kuo et el. (Fig. 2(b)), the adaptive prediction-error filter excited by delayed verion of e(, predict and hence cancel the interference caued by u( [4]. In Zhang method the ADNC filter, W(z) and z ) are cro-updated to reduce the mutual coupling between the modeling proce and the control proce (Fig. 2(c)) [5]. he imulation conducted by Zhang et el. how that thi tructure give the bet performance of all exiting method. Hereafter thee three method are referred a improved method. PROPOSED MEHOD From above dicuion we ee that all improved method concentrate on the modeling proce and control proce i ame a in the Erikon method. In preent tudy we conider improving the performance of the control proce. he baic idea i that if control proce i efficient in it noie-reduction performance, then modeling procill in turn converge fat. A hown in Fig. 1, the error ignal for the control proce and the modeling proce are given a ( = u( + v (. (6) e ( = u( + [ v ( ]. (7) From thee expreion we oberve that: Both ( and e ( contain u(, the error ignal required for the noie control proce. In (, u( i corrupted by the component v ( and in e ( it i corrupted by the term [ v ]. A compared with e(, e ( i a better error ignal for the control proce, becaue v < vˆ and when z ) converge then (ideally) v vˆ [ v ] 0. Since v( i a white Gauian noie of zero mean hence both v ( and [ v ] are random in nature and can be averaged out. On the bai of the above analyi two modification are uggeted to the Erikon method

5 Firt, uing e ( a an error ignal for both the control proce and the modeling proce, i.e., e ( = e ( (. Second, replacing Fx algorithm with the adaptive w filtering with averaging (AFA) [6] baed filtered-x (FxAFA) algorithm. he propoed ANC ytem i hown Fig. 3. he FxAFA algorithm [7] i a modified verion of the Fx algorithm, where averaging i incorporated with both the iteration vector and the obervation vector. Due to averaging the random fluctuation in the learning curve are reduced and hence iteration move toward the optimal olution at a fater rate. Referring to the Fx algorithm in Eq. (3), the FxAFA algorithm can be formulated a given in Eq. (8). γ n k w w ( n + 1) = w( + (1/ n ) = 1 µ w e ( k) x ( k); 1/ 2 < γ < 1 (8) n wher ( = (1/ k = 1 w( k). It i important to note that computing the running average of the data doe not put o much computational burden ince average can be calculated recurively [7]. Now better noie reduction performance i expected and ANC will reduce the reidual noie component u( at fat convergence rate. hi mean that the modeling proce i expected to converge fat now. In comparion to exiting improved method, the main feature of the propoed ANC ytem are ummarized below: An effort i made to improve the control proce, o that better noie reduction i achieved and hence modeling proce converge fat. he propoed tructure ue two adaptive filter, W(z) and z ), to perform noiecontrol and econdary-path modeling imultaneouly. hi i in contrat to exiting improved method, which ue three adaptive filter. COMPUAIONAL COMPLEXIY able 1 preent the computational complexity (multiplication per iteratio comparion of the propoed method with the exiting method. It i aumed that three adaptive filter, B(z), C(z) and H(z) in Bao' method, Kuo' method and Zhang' method, repectively, are elected of order N. Hence thee improved method have the ame computational complexity. he order of the third filter, N, i uually elected le than L (a thumb rule i N < L/2). If a large value for N i elected then the convergence of the third filter i low, and overall performance

6 able 1. Computational Complexity Comparion of the Propoed Method with the Exiting Method NUMBER OF MULIPLICAIONS/IERAION Analytical Expreion L=M=64, N=32 L=M=128, N=64 Erikon Method 2L+3M Improved Method 2L+3M+2N Propoed Method 6L+3M x( Copy P(z) Noie Control Proce (FxAFA Algorithm) W(z) z ) AFA x White Noie Generator v( y( d( e ( = u( + v ( S(z) y - v ( ( v ˆ ( z ) Secondary-path Modeling Proce e ( Fig. 3. Propoed method for active noie control ytem with online econdary path modeling. of the ANC ytem i degraded. With thi contraint, the analytical expreion in able 1 how that the computational complexity of the propoed method i greater than the exiting cheme. o get a clear picture, a few numerical example are alo preented in able 1. SIMULAION RESULS In thi ection we compare the performance of the propoed method with thoe of the Erikon' method and Zhang' method. he performance comparion i done on the bai of the relative modeling error, RME = 10log[ ( ˆ( / ( ]. For the acoutical path P(z) and S(z) the experimental data provided by [1] i ued, where both are modeled by IIR filter of order 25. We truncate the impule repone of the acoutic path to get FIR (order 64) repreentation. hee truncated impule repone are hown in Fig. 4. he modeling filter z ) and control filter W(z), repectively, are FIR filter of order 64. he ADNC filter H(z) in Zhang' method i elected a an FIR filter of order 32. All initial valu(0), ŝ (0), and h(0) are et to zero. he ampling frequency of 4kHz i ued

7 he reference noie ignal i a inuoid of frequency 300 Hz and variance 2. o have low teady tate reidual noie, a zero-mean white Gauian noie of variance 0.05 i ued in the modeling proce. he parameter are adjuted for fat and table convergence. In Fig. 5, are hown the curve of the relative modeling error, RME, for the propoed method in comparion to the exiting method. Here Modified Erikon method i Erikon' method with e ( = e ( (. hi tructure i obtained by replacing the FxAFA w algorithm with the Fx algorithm in the ANC ytem of Fig. 3. We ee that the modified Erikon method beat the Erikon' method, and the propoed method give the bet reult of all. REFERENCES 1. S.M. Kuo and D.R. Morgan, Active Noie Control Sytem Algorithm and DSP Implementation (New York: Wiley, 1996) 2. L. J. Erikon, and M. C. Allie, Ue of random noie for on-line tranducer modeling in an adaptive active attenuation ytem, J. Acout. Soc. Am., 85, , (1989) 3. C. Bao, P. Sa, and H. V. Bruel, Adaptive active control of noie in 3-D reverberant encloure, J. Sound Vibr., 161, (1993) 1.5 Amplitude Sample Number 1.5 (a) Amplitude Sample Number (b) Fig. 4. Impule repone of acoutic path: (a) Impule repone of primary path P(z), (b) Impule repone of econdary path S(z)

8 4. S.M. Kuo, and D. Vijayan, A econdary path modeling technique for active noie control ytem, IEEE ran. Speech Audio Proceing, 5, (1997) 5. M. Zhang, H. Lan, and W. Ser, Cro-updated active noie control ytem with online econdary path modeling, IEEE ran. Speech, Audio Proc., 9, (2001) 6. G. Yin, Adaptive filtering with averaging, in Adaptive Control, Filtering, and Signal Proceing, K.J. Åtröm, G.C. Goodwin, and P.R. Kumar, Ed. (New York: Springer-Verlog, 1995) 7. A.M. ahir, M. Abe, and M. Kawamata, On ue of averaging in Fx algorithm for ingle-channel feedforward ANC ytem, Proc. IEEE ISCAS2003, 4, IV389 IV392, (hailand, 2003) 0 (a) (b) -10 (c) RME -20 (d) Iteration time x 10 4 Fig. 5. Relative modeling error RME veru iteration time n: (a) Erikon method (µ w =1 10-5, µ = ), (b) Modified Erikon method (µ w =2 10-5, µ = ), (c) Zhang method (µ w =2 10-5, µ =2 10-3, µ h = ), (d) Propoed method (µ w =2 10-5, µ =2 10-3, γ=0.5)