A Diffusion Strategy for the Multichannel Active Noise Control System in Distributed Network

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1 216 International Conference on Computational Science and Computational Intelligence A Diffusion Strategy for the Multichannel Active Noise Control System in Distributed Network Ju-man Song Division of IT Convergence Engineering Pohang University of Science and Technology Pohang, Kyungbuk sjm1924@postech.ac.kr PooGyeon Park Department of Electrical Engineering Pohang University of Science and Technology Pohang, Kyungbuk ppg@postech.ac.kr Abstract This paper introduces a diffusion strategy for the multichannel active noise control. The diffusion strategy is designed to reduce the computational complexity by distributes computations to all nodes of multichannel active noise control system. Thus, the multichannel filtered-x normalized least mean square algorithm, which is the simplest way for real active noise control environments is used as a base algorithm for the diffusion strategy. From the structure of the multichannel active noise control system, the optimal weight vector at a node of the proposed strategy is calculated by considering the weight vector of neighbor nodes. With the proposed strategy, the computational complexity is distributed from one main processor to all nodes. With some simulation results, the performance of the proposed strategy is shown with the reduced computational complexity for one processor. Index Terms Adaptive filters, active noise control (ANC), least mean square (LMS), filtered-x LMS (FxLMS), multichannel, diffusion, distributed network. I. INTRODUCTION In these days, the problems of environmental noises are severe social issues. To remove such noisy acoustic signals, the field of noise control is researched in two domains: passive and active noise control. The low frequency noise, which includes audible frequency is hard to be controlled by a passive way, because the passive way is ineffective on low frequency noise. When the noise control system is built by passive way, the system is costly and bulky to get wanted performance. Thus, an active noise control (ANC) system is developed to overcome the limitation of the passive noise control system. The ANC system is designed to cancel out the unwanted disturbance signal by generating acoustic signals for destructive interference in the acoustic fields. The acoustic signals, which are generated from the electrical system, can be controlled with adaptive filters. The most noise environments are multidimensional spaces, and in such environments, the noise field is too complicated [1]. To remove multidimensional noise, multichannel ANC systems are applied. For the multichannel ANC system, a method of using the filtered-x least mean square (FxLMS) algorithm is suggested [2], and some methods of using the filtered-x affine projection (FxAP) algorithm is also suggested in various ways [3] [5]. However, the computational complexity is too high for real environments, because of the multiple inputs and outputs. Furthermore, if some error microphones have problems to get error signals, the whole environments will be broken. To enhance the multichannel ANC system, some researchers suggest to use the acoustic sensor networks(asn) for cheap, flexible and efficient solution [6]. Using the ASN for ANC system, the computational complexity can be distributed to all nodes. M. Ferrer suggests an ANC system for distributed networks with incremental strategy [7], and C. Antoanazas suggests a diffusion strategy for ANC system on frequency domain [8]. They use the incremental strategy, which is a way of distributed networks. In the incremental strategy, a node communicates with a front node and a next node to modify the weight of multichannel ANC system. In addition to the incremental networks, there are some diffusion strategies to be used for adaptive filter. A LMS type diffusion solution is proposed by Cattivelli, F. S. to get good performance by cooperating with neighbor nodes, not just with the front node and the next node [9]. Based on this diffusion strategy, in this paper, a new version of diffusion algorithm for ANC system in distributed networks is proposed. The proposed strategy uses the multichannel FxNLMS algorithm for low computational complexity. All node in the proposed strategy consist of an actuator, an error microphone, and a computational capacity. All nodes communicate with each of neighbor nodes to get robust and good performance. Some simulation results are shown with the multichannel ANC system. II. MULTICHANNEL FXNLMS The structure of the multichannel ANC system is depicted on Fig.1. The block diagram of Fig.1 includes two reconstruction blocks. This scheme is called as the modified structure [1]. It is designed to reconstruct the error signal and the disturbance signal. The error signal in the acoustic field cannot be used directly, because the summation of error signal is not an electrical sum, but an acoustic sum. To reconstruct the error signal and the disturbance signal, the output signal from the adaptive filter is used after passing through the estimate of the secondary paths. If the estimation of secondary paths is perfect, the reconstructed signals are same with the original signals. The case of /16 $ IEEE DOI 1.119/CSCI

2 Fig. 1. The block diagram of MFxAP for multichannel ANC systems. imperfect estimation is described in [11]. The authors of [11] says that imperfect estimation does not affect the steady state error. Even though there are some noises to the estimation of the secondary path, the change of the convergence rate is very low. The multichannel ANC system has I reference microphones, J actuators, and K error microphones. I primary noise signals are measured at I reference microphones, and each primary noise signal is defined as x i (n)=[x i (n), x i (n 1),,x i (n H +1)] T R H 1. (1) Each primary noise signal passes through primary paths to K error microphones. At each error microphones, there is a secondary noise signal, r k (n). The secondary noise signal is generated as an white Gaussian noise of zero mean and σ 2 r variance. With this secondary noise signal, the disturbance signal is generated as where d k (n) = u T k (n) w o + r k (n) (2) u k (n) =[u T k,1 (n),, u T k,j (n)] T R L I J 1 (3) u k,j (n) =[u T k,j,1 (n),, u T k,j,i (n)] T R L I 1, (4) and u k,j,i (n) is generated from the ith reference signal by passing through the estimate of the secondary path, ĥk,j. There are J actuators and K error microphones in the multichannel ANC system, and J output signals from J actuators arrive at K error microphones. Thus, the total number of secondary paths is J K. With the estimate of the secondary path, ĥk,j, the filtered input signal is defined as u k,j,i (n) =[u k,j,i (n), u k,j,i (n 1),,u k,j,i (n L+1)] T (5) u k,j,i (n) =ĥk,j x i (n). (6) With the filtered input signals, the weight of adaptive filter can be updated iteratively. The most used algorithm for the ANC system is the FxNLMS algorithm, and the weight update equation is defined as ŵ j,i (n +1)=ŵ j,i (n) μ where K k=1 u k,j,i (n) u k,j,i (n) 2 e k (n), (7) ŵ j (n) =[ŵj,1 T (n),, ŵj,i T (n)] T R L I 1 (8) ŵ (n) =[ŵ1 T (n),, ŵj T (n)] T R L I J 1. (9) The weight update equation (7) can be extended to the whole multichannel system as ŵ (n +1)=ŵ (n) μ K k=1 u k (n) u k (n) 2 e k (n), (1) Notation of the multichannel FxNLMS algorithm is summarized in Table I. The multichannel ANC algorithm with FxNLMS algorithm does not use any network process among sensors. Just a main processor calculates the optimal weight to remove the unwanted disturbance signals. All computations

3 TABLE I NOTATION AND SUMMARY FOR THE CMFXNLMS Notation Dimensions Description I 1 Number of reference sensors J 1 Number of actuators K 1 Number of error sensors L 1 Length of the adaptive filter H 1 Length of the secondary path estimation ĥ w o L I J 1 Optimal Solution of the adaptive filter ŵ j,i (n) L 1 Weight Vector of the adaptive filter from the input i to the output j ŵ j (n) =[ŵj,1 T (n),, ŵt j,i (n)]t L I 1 Weight Vector of the adaptive filter related the output j ŵ (n) =[ŵ1 T (n),, ŵt J (n)]t L I J 1 Full Weight Vector of the adaptive filter x i (n) =[x i (n),x i (n 1),,x i (n H +1)] T H 1 Primary noise signal vector from i-th primary input source x (n) =[x T 1 (n),, xt I (n)]t H I 1 Full Primary noise signal vector u k,j,i (n) =x T i (n) ĥk,j 1 Filtered primary noise signal from i-th input signal to secondary path ĥk,j, u k,j,i (n) =[u k,j,i (n),u k,j,i (n 1),,u k,j,i (n L +1)] T L 1 Filtered primary noise signal vector from i-th input signal to secondary path ĥ k,j, u k,j (n) =[u T k,j,1 (n),, ut k,j,i (n)]t L I 1 Filtered primary noise signal vector related to j-th output and secondary path ĥk,j, u k (n) =[u T k,1 (n),, ut k,j (n)]t L I J 1 Filtered primary noise signal vector related to k-th error sensor, r k (n) 1 Secondary noise signal at time n d k (n) = u T k (n) wo + r k (n) 1 Disturbance signal to k-th error sensor e k (n) =d k (n)+u T k (n) ŵ (n) 1 k-th error sensor signal at time n ŵ (n +1)=ŵ (n) μ K k=1 u k (n) ( u T k (n) u k (n) ) 1 ek (n) L I J 1 Weight update equation at time n are centralized to the main processor. Thus, in this paper, we will call this algorithm as the centralized multichannel FxNLMS (CMFxNLMS) algorithm. III. A DIFFUSION STRATEGY FOR ANC In this section, we will develop a diffusion strategy for ANC system to reduce the computational complexity in distributed networks. The CMFxNLMS algorithm needs the whole information from all nodes, and the optimal weight is calculated at just the central processor. However, when the calculation of the optimal weight is distributed to all nodes, the computational complexity should be also distributed from one processor to other processors. There are some distributed strategies, such as the incremental network, and the diffusion strategy. The diffusion strategy will be dealt in this paper, because it can get good performance by communicate with neighbor nodes. Above all, the diffusion strategy is suitable for the multichannel ANC system, because all nodes are in one environment. Thus, the communication with neighbor nodes is a valuable thing. To define the diffusion system for the multichannel ANC system, at first, a node of the diffusion strategy should be defined. In this paper, a node has an actuator, an error microphone, and a processor. A node has its own weight which will be used at diffusion step, and it is defined as ŵ ka (n) =[ŵ T k a,1 (n),, ŵ T k a,i (n)] T R L I 1, (11) where 1 k a K, and K is the number of node. ŵ kai (n) is the weight vector of node k a from input i. The notation k a means for the perspective of actuators, and the notation k e means for that of error sensors. Of course, a node has both an error sensor and an actuator, but k a, k e are used for easy comparison of the CMFxNLMS algorithm with diffusion algorithm. With these weights, the weight vector for the hole ANC system can be defined as ŵ (n) =[ŵ T 1 (n),, ŵ T K (n)] T R L I K 1. (12) At the diffusion environments, the reference signal vector from ith reference sensor is defined as x i (n)=[x i (n), x i (n 1),,x i (n H +1)] T R H 1. (13) This signal is filtered through the estimate of the secondary path ĥk e,k a, and the filtered signal is defined as u ke,k a,i (n) =ĥk e,k a x i (n) R L 1, 1 k e K (14) u ke,k a (n)= ĥk e,k a x (n) =[u T k e,k a,1(n),, u T k e,k a,i(n)] T R L I 1, (15)

4 Fig. 2. The block diagram of MFxAP for multichannel ANC systems. u ke (n) =[u T k e,1 (n),, u T k e,k (n)] T R L I K 1 (16) With this filtered signal, the unwanted disturbance signal can be defined as d ke (n) = u ke (n) T w o + r ke (n), (17) where w o is the optimal weight of the whole ANC system, and r ke (n) is the secondary noise signal for node k e. Then, the error signal at node k can be defined as e ke (n) =d ke (n)+u T k e (n) w ke (n) = u T k e (n) w ke (n)+r ke (n), (18) where w ke (n) is the weight at node k e, and w ke (n) is the weight error at node k e. With this error signal, the intermediate weight at node k is determined as Ψ ke (n +1)=w ke (n) μ ke (n) u ke (n) u ke (n) 2 e k e (n). (19) At the diffusion step, the weight for the k e -th node is determined as K w ke (n +1)= a l,ke Ψ l (n +1), (2) l=1 where a l,ke is a combination coefficient, which is the weight connecting node k e to its neighbor node l N ke. a l,ke can be chosen such that K l=1 a l,k =1for all k. According to determining the combination coefficient, this proposed strategy can describe all ASN for the multichannel ANC system. This proposed strategy uses the adapt-then-combine (ATC) diffusion LMS algorithm of [9]. At first, each node calculates its intermediate weight, and as the next step, a weighted sum is done to get the weight of estimate w o at each node. We will call this diffusion strategy for the multichannel ANC system as the diffusion multichannel FxNLMS (DMFxNLMS), and it is summarized in Table II. IV. SIMULATION RESULTS In the environments of multichannel ANC system, the length of weight vector of the adaptive filter from the input i to the output k a, ŵ ka,i (n), is set as 32, and the secondary path ĥk,j is modeled as 8-coefficient FIR filter, which is generated randomly. Pre-filtered input vectors from i-th input signal to secondary path ĥk e,k a, u ke,k a,i (n) are generated. The unknown primary path, w o, is generated with separated unit weight vector, ŵ j,i (n). A Gaussian noise with N (, 1) is employed as the i-th primary input source, x i (n). With the variance of σr 2 k =1 3, the secondary noise at the k-th error sensor, r k (n) is also generated as a Gaussian distribution. For the initialization, the initial weight of adaptive filter is set as ŵ =. All simulations have done in 5 independent trials in common. In order to evaluate the proposed strategy, we uses the mean square deviation (MSD), which is defined as MSD(n) 1 K E { w k T K e (n) w ke (n) }, (21) k=1 where w ke = w o w ke is the weight error of node k. The simulations have done under the environments of a white signal. The simulation results are depicted on Fig. 3-5, with K=4, K=6, K=8. The combination coefficient of Fig. 3-5 uses belows: A = [ ] a l,k = (22) The above combination coefficient means that a node uses the intermediate weight from its neighbor nodes by weighted

5 TABLE II NOTATION AND SUMMARY FOR THE DMFXNLMS Notation Dimensions Description I 1 Number of reference sensors K 1 Number of nodes L 1 Length of the adaptive filter H 1 Length of the secondary path estimation ĥ w o L I K 1 Optimal Solution of the adaptive filter ŵ ka,i (n) L 1 Weight Vector of the adaptive filter from the input i to the output k a ŵ ka (n) =[ŵk T (n),, a,1 ŵt k a,i (n)]t L I 1 Weight Vector of the adaptive filter related the output k a ŵ (n) =[ŵ1 T (n),, ŵt K (n)]t L I K 1 Full Weight Vector of the adaptive filter x i (n) =[x i (n),x i (n 1),,x i (n H +1)] T H 1 Primary noise signal vector from i-th primary input source x (n) =[x T 1 (n),, xt I (n)]t H I 1 Full Primary noise signal vector u ke,ka,i (n) =x T i (n) ĥk,k a 1 Filtered primary noise signal from i-th input signal to secondary path ĥk,k a, u ke,ka,i(n)=[u ke,ka,i(n),u ke,ka,i(n 1),,u ke,ka,i(n L+1)] T L 1 Filtered primary noise signal vector from i-th input signal to secondary path ĥk e,k a, u ke,ka (n) =[u T k (n),, e,k a,1 ut k e,k a,i (n)]t L I 1 Filtered primary noise signal vector related to k a-th output and secondary path ĥ ke,ka, u ke (n) =[u T k (n),, e,1 ut k e,k (n)]t L I K 1 Filtered primary noise signal vector related to k e-th error sensor, r ke (n) 1 Secondary noise signal at time n d ke (n) = u T k e (n) w o + r ke (n) 1 Disturbance siganl to k e-th error sensor e ke (n) =d ke (n)+u T k e (n) ŵ (n) 1 k e-th error sensor signal at time n ( Ψ ke (n+1)=w ke (n) μ ke (n) u ke (n) u T 1eke k e (n) u ke (n)) (n) L I K 1 Intermediate weight update equation at time n and node k e w ke (n +1)= K l=1 a l,k e Ψ l (n +1) L I K 1 Diffusion step at time n and node k e -5-1 Diffusion FxNLMS, μ=.5 Diffusion FxNLMS, μ=.3 Diffusion FxNLMS, μ= Diffusion FxNLMS, μ=.5 Diffusion FxNLMS, μ=.3 Diffusion FxNLMS, μ= MSD(dB) MSD(dB) iteration 1 5 Fig. 3. MSD learning curves of the proposed DMFxNLMS algorithm with various step size for a white signal and randomly generated secondary path. The number of node is iteration 1 5 Fig. 4. MSD learning curves of the proposed DMFxNLMS algorithm with various step size for a white signal and randomly generated secondary path. The number of node is

6 MSD(dB) Diffusion FxNLMS, μ=.5 Diffusion FxNLMS, μ=.3 Diffusion FxNLMS, μ= iteration 1 6 Fig. 5. MSD learning curves of the proposed DMFxNLMS algorithm with various step size for a white signal and randomly generated secondary path. The number of node is 8. sum according to their connection states. The proposed algorithm converges to steady state, although connections between nodes are perfectly connected, and the number of nodes is increased from 4 to 8. It means that the proposed algorithm is robust for the multichannel ANC environments of various connection environments. [2] S. Elliott, I. Stothers, and P. Nelson, A multiple error lms algorithm and its application to the active control of sound and vibration, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 35, no. 1, pp , Oct [3] M. Bouchard, Multichannel affine and fast affine projection algorithms for active noise control and acoustic equalization systems, Speech and Audio Processing, IEEE Transactions on, vol. 11, no. 1, pp. 54 6, 23. [4] G. Sicuranza and A. Carini, Filtered-x affine projection algorithm for multichannel active noise control using second-order volterra filters, Signal Processing Letters, IEEE, vol. 11, no. 11, pp , 24. [5] A. Carini and G. Sicuranza, Steady-state and transient analysis of multichannel filtered-x affine projection algorithms, in Acoustics, Speech, and Signal Processing, 25. Proceedings. (ICASSP 5). IEEE International Conference on, vol. 4, 25, pp. iv/345 iv/348 Vol. 4. [6] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, A survey on sensor networks, IEEE Communications Magazine, vol. 4, no. 8, pp , Aug 22. [7] M. Ferrer, M. de Diego, G. Piero, and A. Gonzalez, Active noise control over adaptive distributed networks, Signal Processing, vol. 17, pp , 215. [8] C. Antoñanzas, M. Ferrer, A. Gonzalez, M. de Diego, and G. Piñero, Diffusion algorithm for active noise control in distributed networks, in 22nd International Congress on Sound and Vibration, 215. [9] F. S. Cattivelli and A. H. Sayed, Diffusion lms strategies for distributed estimation, IEEE Transactions on Signal Processing, vol. 58, no. 3, pp , March 21. [1] E. Bjarnason, Active noise cancellation using a modified form of the filtered-x lms algorithm, in Proc. 6th Eur. Signal Processing Conf, vol. 2, 1992, pp [11] I. Ardekani and W. Abdulla, Effects of imperfect secondary path modeling on adaptive active noise control systems, Control Systems Technology, IEEE Transactions on, vol. 2, no. 5, pp , 212. V. CONCLUSION This paper proposed a diffusion strategy for the multichannel ANC systems. The diffusion strategy is designed to get robust performance in ASN, and reduce the computational complexity from a centralized processor to distributed nodes. Especially, the FxNLMS algorithm is adopted, because of its stability and less complexity than other algorithms. The proposed strategy estimate the optimal weight vector at each node by considering the weight vector of its neighbor nodes. By this strategy, computations to estimate the optimal weight vector is distributed to all nodes from one node. The proposed algorithm achieves stable performance with various number of nodes as the simulation result shows. Still, the proposed algorithm has to be modified to get better performance, and it can be done by get the optimal step size or other various adaptive filter techniques. VI. ACKNOWLEDGMENT This research was supported by the MSIP (Ministry of Science, ICT and Future Planning), Korea, under the ICT Consilience Creative Program (IITP-R ) supervised by the IITP (Institute for Information & communications Technology Promotion) REFERENCES [1] S.M. Kuo, D.R. Morgan, Active noise control systems Algorithms and DSP implementation. John-Wiley and sons,