Multi-Channel Adaptive Feedforward Control of Noise in an Acoustic Duct

Although considerable progress has been made in the applications of active noise and vibration control since the pioneering work of 2, successful implementations of these techniques have recently been made. Such success can be attributed to the rapid progress of the technology in three major areas: 1) novel implementation of the electro-acoustic transducers, 2) development of the advanced adaptive control algorithms, and 3) inexpensive and reliable applications of the digital signal processing (DSP) hardware. In order to compensate for the fact that majority of the models are inaccurate most active noise controllers are made adaptive. These controllers can operate with limited modeling information and their major advantage is that controllers can be changed if the system response varies. Implementations of the adaptive feedforward ANC systems are now a common practice. The feedforward algorithm uses both a reference signal, that is ideally unaffected by the controller, and an error signal, to tune the controller which creates the closed-loop action or the feedback dynamics. In the feedforward ANC systems, the acoustical feedback from the secondary source(s) to the reference point(s) is one of the major reasons for instability, which makes the reference signal non-stationary during the adaptation 3. There exist various techniques to either completely eliminate or compensate for the effect of the acoustical feedback 4. A considerable global noise reduction has been achieved with a simple single channel controller only under certain circumstances 5. However, to obtain global noise reduction in general, a multi-channel controller is needed. Whenever the geometry of the sound field is complicated or the input primary noise is broadband, then it is not just enough to adjust the single secondary source to cancel out the acoustic field. Bao et al. 6 showed that global noise reductions could not be achieved by the use of a single channel system at resonant frequencies, dominated by several coupled acoustical modes. The optimum location of the sensor/actuator is an important feature in the design of multi-channel active noise controllers. A good number of studies have been reported in this interesting area, 789. One control strategy, which has been successfully adapted in controlling the sound in a complicated acoustic field, is to tune the outputs of a number of secondary sources in order to minimize the sum of the squares of the outputs from the error microphones. This strategy, with an adaptive finite impulse response (FIR) filtering scheme, can lead to the multiple error filtered-x least mean squares (MEFXLMS) algorithm, which was proposed by Elliott et al. (1992) 10. This is based on a multi-channel generalization of the filtered-x LMS algorithm, developed earlier by Widrow and Stearns, 11. The stability and convergence of the MEFXLMS algorithm with the effect of the secondary paths and the system variables on the convergence speed and the stability range, in both time and frequency domains, were investigated by Elliott, 10 and others 121314 A new implementation of the multi-channel filtered-x LMS and the LMS algorithms for the feedforward active noise control was described by Douglas, 15. Since 1990s, much attention has been paid on the multi-channel ANC systems Bai and Lin, 61617 investigated the multiple-channel ANC systems based on the FXLMS and the H∞ algorithms 18. They carried out tests on the feedforward and the feedback ANC structures and showed that the adaptive and H∞ methods exhibit comparable performance for both random and engine noise irrespective of the control structure. However, they found that the multi-input/multi-output (MIMO) feedforward H∞ technique could only be effective in suppressing the transient noise. Recently, an adaptive feedforward algorithm has been de-veloped that minimizes a more general cost function, which is the p-norm 1⩽p⩽∞ of a vector composed of the error signals. For the limiting case of very large value of p, the Minimax algorithm leads to a uniform residual field and hence, reduces the computational load in comparison with the MEFXLMS algorithm 19. In this paper, the transfer functions of the primary, secondary and the acoustical feedback paths for the multi-channel ANC system of an acoustic duct are presented. The physical models were based on the closed-form solutions of a one-dimensional dynamic equation of sound with accurate models for the loudspeakers and the microphones, developed earlier by the authors 1. Here, the feedforward control algorithm is used to minimize a general cost function, which is the p-norm of a vector composed of the error signals. Simulations are used to compare the MEFXLMS and Minimax algorithms. The system of a hard-walled finite-length duct with a point dipole source, located at xp, is shown in Fig. 1. The dimensions of the cross-section of the duct are assumed sufficiently small compared to its length, and hence, a one-dimensional wave-guide with the spatial coordinate x could be modeled. Based on the Green’s function in the Laplace domain, the closed-form solution of the acoustic pressure would be obtained. Detailed analysis of this was reported by the authors in a recent article 1. The transfer functions of the acoustic field are obtained for four different cases:• The acoustic transfer function for the sound pressure at the reference microphone point, xr and primary source, Qps is (1)Hprps=Pxr,sQps=−e−asxr−xp/1+bs−γ0se−asxr+xp/1+bs+γ1se−as2−xr−xp/1+bs+γ0sγ1se−as2−xr+xp/1+bs21+bs1−γ0sγ1se−2as/1+bs• The acoustic transfer function for the sound pressure at the error microphone point, xe and the sound pressure at the reference microphone point, xr is (2)Hpers=Pxe,sPxr,s=−e−asxe−xp/1+bs−γ0se−asxe+xp/1+bs+γ1se−as2−xe−xp/1+bs+γ0sγ1se−as2−xe+xp/1+bs−e−asxr−xp/1+bs−γ0se−asxr+xp/1+bs+γ1se−as2−xr−xp/1+bs+γ0sγ1se−as2−xr+xp/1+bs• The acoustic transfer function for the sound pressure at the reference microphone point, xr and secondary source, Qss is (3)Hsrss=Pxr,sQss=e−asxs−xr/1+bs−γ0se−asxr+xs/1+bs+γ1se−as2−xr−xs/1+bs−γ0sγ1se−as2−xs+xr/1+bs21+bs1−γ0sγ1se−2as/1+bs• The acoustic transfer function for the sound pressure at the error microphone point, xe and the secondary source, Qss is (4)Hsess=Pxe,sQss=−e−asxe−xs/1+bs−γ0se−asxe+xs/1+bs+γ1se−as2−xe−xs/1+bs+γ0sγ1se−as2−xe+xs/1+bs21+bs1−γ0sγ1se−2as/1+bsIn the above equation, a=L/c,b=2β/c,L is the length of the duct, c is the speed of the acoustic wave, β is the air damping being proportional to the kinematic viscosity of the air and γ0s and γ1s are the reflection coefficients at either end. The reflection coefficient γs satisfies the relation −1⩽γs⩽1. When γs=0, the boundary transmits (or absorbs) all the waves, and hence no waves are reflected i.e., equivalent to a semi-infinite duct. One extreme value of γs=1, leads to p=0 is the classical open-end condition where the displacement wave is totally reflected. For another extreme case, γs=−1, a close-end condition of total reflective is obtained. The intermediate values of γs provide a more realistic boundary condition and Morris showed that γs strongly affects the achievable noise reduction 20. Two major components of the physical model of the duct are microphones and loudspeakers. It is necessary to consider the dynamics of these components in order to obtain an accurate simulation model. The transfer function of the condenser microphone and the preamplifier is presented by Clark 21. The transfer function of the loudspeaker, which relates the speaker’s strength to the applied voltage, can be written as 121: (5)Hspks=QssVs=H1s1−H2sH3s(6)H1s=SsB1sAs,H2s=−Ss2Rs+Ls2As(7)H3s=−1−γ0se−2asxs/1+bs+γ1se−2as1−xs/1+bs+γ0sγ1se−2as/1+bs21+bs1−γ0sγ1se−2as/1+bsx=xs+H3s=1−γ0se−2asxs/1+bs+γ1se−2as1−xs/1+bs−γ0sγ1se−2as/1+bs21+bs1−γ0sγ1se−2as/1+bsx=xs−where H1s is the dynamics of the loudspeaker, H2s is the mechano-acoustic coupling term, and H3s is the dynamics of the acoustic duct at the loudspeaker position. L and R are the respective inductance and resistance of the armature, Bl is the product of the field strength and the conductor length, and Ss is the surface area of the loudspeaker. (8)As=[LMss3+RMs+LDss2+RDs+LKs+Bl2s+RKs]where Ms,Ds, and Ks are the respective mechanical mass, damping and stiffness of the loudspeaker. It must be noted that the presented model for the loudspeaker includes both the electro-mechanical and the mechano-acoustical couplings. The filtered-x least-mean-square (FXLMS) algorithm is the most popular adaptive control algorithm used in DSP implementations of active vibration and noise control. This is found to be quite suitable for both broadband and narrowband active noise control, and robust when physical modeling errors are present. Due to the algorithm’s extensive use of the multiply/accumulate (MAC) operation, its structure and the operation are suitable for the DSP chips. To obtain the global noise reduction under general circumstances, a multi-channel controller must be sought. Elliott developed the multiple error FXLMS (MEFXLMS) algorithm 10, which is a multi-channel generalization of the filtered-x LMS algorithm. This, with an adaptive finite impulse response, FIR, filtering scheme, would minimize a cost function that is defined as the sum of the mean squares of the measured error signals. In a more general case, it is possible to define a family of algorithms, where each of them minimizing a different measure of the error vector composed of the error signals 19. Therefore, the error criterion, also known as the cost function, is defined as the p-norm of the error vector. The single-reference/multiple-output feedforward ANC system, shown in the block diagram form of Fig. 2, uses one reference sensor, K secondary sources and M error sensors. The feedback neutralization technique has been used to overcome the acoustical feedback effects. Pz is the transfer vector M×1 of the primary paths Pmz, which relates the reference microphone signal un, to every error sensor emn.Fz is the transfer vector 1×K of the acoustic feedback paths Fkz, from the K secondary sources to the reference sensor. The transfer matrix Sz,M×K represents the secondary paths Smkz, formed by the actuators, acoustic error paths, and the error sensors from the K secondary sources to the M error sensors. The error signal vector en, measured by the M error sensors, can be written as: (9)en≡[e1ne2n…eMn]T(10)en=dn−y′nwhere (11)dn≡[d1nd2n…dMn]Tis the primary noise vector, with dmn representing the noise at the mth error sensor, and (12)y′n≡[y1′ny2′n…ym′n]Tis the canceling noise vector at the error sensors, in which ym′n is the sum of the canceling noise from the K secondary sources to the mth error sensor. The vector y′n can be expressed as: (13)y′n=Sn*ynwhere the symbol * denotes the linear convolution, and (14)Sn=s11ns12n…s1Kns21ns22n…s2kn............sM1nsM2n…sMKncontains the impulse response functions, where smkn is the response from the kth secondary source to the mth error sensor. The output signal vector yn:(15)yn≡[y1ny2n…yKn]Tis then used to drive the K secondary sources and could be obtained from (16)yn=XTnwnThe vector wn, in Eq. (16), represents the adaptive weights associated with all the K adaptive filters. (17)wn≡[w1Tnw2Tn…wKTn]Twhere (18)wkn≡[wk,0nwk,1n…wk,L−1n]T,k=1,2,…,Kare the weight vectors of the K adaptive filters, each of which is assumed to be of the order L. Moreover, Xn is a KL×K block-diagonal matrix defined as: (19)Xn=xn0…00xn0…...0.0..0…0xnwhere (20)xn≡[xnxn−1…xn−L+1]Tis the common reference signal vector for all the adaptive filters. By combining Eqs. 13 and 16, and substituting in Eq. 10, the error signal vector could be obtained. (21)en=dn−Sn*yn=dn−Sn*[XTnwn]In the feedforward ANC system, a positive feedback loop exists between the canceling loudspeaker and the reference microphone, which tends to destabilize the ANC system. The acoustic feedback introduces closed-loop poles to the system, which makes the system unstable if the feedback loop gain becomes too large. In addition, it changes the direction of the adaptive control filter updating. The effects of the acoustic feedback on the performance of the feedforward ANC system were reported by the authors 1. The simple approach to solving the feedback problem is to use a separate feedback cancellation, or “neutralization” filter within the controller, which is exactly the same technique as used in acoustic In this a feedback neutralization technique has been to overcome the acoustic feedback This model of the feedback is by the secondary signal, and its output is from the reference sensor The feedback of the reference microphone signal is a feedback neutralization which models the feedback The reference signal is as: is the of the feedback and is a of the primary source noise and the feedback from the secondary Since the primary noise is with the the adaptation of the feedback neutralization filter must be when the ANC system is in operation, to adaptive during of feedback neutralization is in by an adaptive for the transfer function of the feedback 4. The cost function of an adaptive filter is defined as the p-norm of the vector composed of the measured error signals functions, being are used in the The solution be the algorithm to adjust the coefficients of the adaptive in the direction of of the cost This lead to a global of the cost function since are no extreme The general algorithm for the p-norm cost function is a convergence that the stability and the convergence of the algorithm. The of is with to the kth weight vector at the time which can be expressed from Eqs. and weight can be obtained by the as: denotes the product convolution, obtained by each of with to form the and is a vector defined as: is not in an value of by can be obtained by an modeling Eq. Eq. leads the above a matrix of the reference signal vectors with the general of is the reference signal vector, formed by filtering and the from the kth secondary source to the mth error sensor. Eq. can be K as: values of the lead to the of the sum of the squares for the of the measured signal as tends to In this important for and have been The multiple error FXLMS algorithm is a case of the family of the algorithms when and in this case, to Eq. the cost function of an adaptive filter is In when minimizing a cost of then the be in the vector by Eq. This in the cost function when the vector its The of Eq. is more uniform acoustical can be obtained by the value of the in the cost function of Eq. For the limiting case, as tends to the is Minimax algorithm. The this is to the acoustical field the control in an active control system. To this it is necessary to minimize the values of the mean values of all the error signals. This algorithm, its has good convergence and has stability the MEFXLMS algorithm. Moreover, the Minimax algorithm to be robust changes in the of the transfer functions 19. For the Minimax algorithm the cost function and its vector are denotes the of the error vector with the at each and is the impulse response from the kth secondary source to the error sensor. For the Minimax algorithm, based on Eq. the Eq. can be that the adaptive weights are only one of the error which be at each and this the computational load when compared with the MEFXLMS algorithm. The in Eq. has not been since it has no effect on the The of the MEFXLMS algorithm during the convergence the adaptation is or can be by the of the mean The mean error is that of the sum of the signals from the M error transducers, which the mean The stability of the MEFXLMS algorithm, in the of time in the secondary transfer function, only on the of the matrix of the reference the of the error sensors and the control sources in the system the value of the convergence coefficient by an proportional to this the if the of the secondary transfer function are by either of the control and error sources or by a in the of the it an proportional in the stability of the algorithm. This effect is to that obtained for the system. However, the only for the implementation is that the effect of a in the transfer function of one is to by the transfer functions of the For this the systems have to changes in every components of the system. The physical system the of the MEFXLMS algorithm as it to a to its The speed of convergence is a function of both the geometry of the control system and the resonant of the of the physical system. resonant the speed of the and hence, the geometry of the control source and the error sensor would both the speed of the and the between the and 10. The convergence of the Minimax algorithm can be in a to that of the MEFXLMS algorithm. The Minimax algorithm the error surface defined by the single error signal with the Since the error surface changes with the algorithm must for all which the convergence time and the for the convergence For this algorithm, the stability condition for the was obtained 19. the system response between every secondary source and the error sensor. is the of the of the reference signal were carried out to the dynamic of the and the multiple-channel ANC systems in an acoustic duct and to compare the performance of the MEFXLMS algorithm with that of the all the were in the time domain, their were in both the time and frequency The acoustic duct used for the simulation is a one-dimensional hard-walled finite-length duct with a cross-section and the equation of the duct system the are from are not located on the since one of the boundary is Moreover, all the of the duct system have and hence, the system is For the finite-length duct, the dynamics of the acoustic system are reflected by the poles and and not by the time as for the case of the The ANC system, shown in Fig. has one primary source located at secondary sources at and one reference microphone at and error microphones at the microphones are of the K which has one and poles at and these poles not with the system response the frequency to transfer function of the condenser microphone by Clark of a gain made for the transfer function of microphone is However, for the microphones, this is not and transfer function can be from the to the and be obtained as the In this for the of the transfer function of the microphone was as i.e., with the have been used as the secondary transfer functions of these loudspeakers have been obtained from Eq. which includes the mechano-acoustical and the electro-mechanical couplings. The block diagram of the single-reference/multiple-output feedforward ANC system for the system, is shown in Fig. 4. It can be that the feedback neutralization technique must be used in order to overcome the effects of the acoustical the and the the transfer function of the primary, secondary and the acoustic feedback paths can be obtained in the form of the filter the acoustic field, microphone and the loudspeaker models an for a of the transfer function between the secondary source and the error microphone is its frequency response is shown in Fig. 5. It is possible to the order of the model by the technique In order to compare the dynamic performance of different ANC algorithms, the have been the performance of the single-reference/multiple-output ANC systems, consider the dynamic of different of the FXLMS ANC system, i.e., the effects of both the secondary source and the error microphone under is of no to that the of the sensors and the would the performance of the ANC system. Therefore, in general it is to the physical of the control and the error sources for a single or multiple control system. There are one the achievable global reduction in the sound pressure is not a linear function of the control source The the location of the optimum error sensor on the location of the control a general Bai and the and sensors be at the pressure of the acoustic in the or duct noise were obtained from the possible of the FXLMS ANC system, which are in 1. It must be noted that the length of the filters, in all these has been sufficiently in order to obtain accurate Moreover, for the convergence the value of the convergence that a their optimum values have been It can be that the value of the noise reduction strongly on the of the primary the error microphone location and the secondary source position. an for with the secondary source located at and the error microphone at a noise reduction of has been By the of the secondary source from to much be obtained The effect of the secondary source location on the performance of the ANC system for is shown in Fig. In this the sound pressure at the error for three different of the secondary source, are also shown It can be it is not possible to have a considerable noise at and irrespective of the location of the error microphone, if the secondary source is at The effect of the error microphone on the of the ANC system is shown in Fig. The secondary source is located at three different for error microphone are 6 and and that the performance of FXLMS ANC system is more to the of the secondary source the of error for the single-reference/multiple-output ANC systems, presented in 2, the obtained convergence of the algorithms either the of the error vector or the It can be that the performance of the Minimax algorithm is that of MEFXLMS for 2, and the performance of the algorithms is the same for the MEFXLMS algorithm for The comparison between and that much noise can be achieved by the single-reference/multiple-output ANC system, for of the FXLMS ANC system. it is possible to have a of the ANC system with a performance the multi-channel either the Minimax or MEFXLMS algorithm for one of the input be obtained by the ANC system when the frequency is changed and would and Moreover, it can be from that the total noise reduction of the Minimax algorithm, compared with that of the MEFXLMS algorithm, is much on the frequency of the input The sound pressure from the MEFXLMS and the Minimax algorithms and the FXLMS ANC for are shown in and It can be that for the multi-channel ANC systems the Minimax and the MEFXLMS algorithms, is achieved at all at for the MEFXLMS algorithm. Moreover, exists a at in the transient response of the Minimax algorithm for the primary which be out a considerable effects of the feedback neutralization technique on the sound pressure from the MEFXLMS algorithm, are shown in Fig. 10. For acoustical feedback and the of the acoustical feedback neutralization technique then be this case a described by the equation, was for the primary noise from to at a uniform of The sound pressure response of the error microphone point the ANC system was is shown in Fig. However, with the control system being the response for the ANC system is shown in Fig. and the response of the multi-channel system the Minimax algorithm is in Fig. It can be that the Minimax algorithm a much for this of the primary simulation of a multi-channel active noise control system, based on a suitable model of a one-dimensional acoustic duct system the dynamics of the microphones and were A developed multi-channel adaptive filtering algorithm is applied to minimize the p-norm 1⩽p⩽∞ of a vector composed of the error signals. The dynamic of the MEFXLMS algorithm that minimizes the sum of the squares of the error signals and the Minimax algorithm, which minimizes the of the measured were The of a of the adaptive feedforward ANC system with the FXLMS algorithm, for the and the input were compared with of the multi-channel adaptive feedforward ANC systems the MEFXLMS and the Minimax algorithms. obtained for different of the ANC system that the noise reduction is not a linear function of the control source and the optimum value of the error sensor location on the control source Moreover, the value of the noise reduction strongly on the of the primary and the of the error microphone and the secondary that the performance of the multi-channel feedforward ANC system the Minimax algorithm is that of the same system the MEFXLMS algorithm. Since the Minimax algorithm uses only one single error signal for the the computational load is much that of the MEFXLMS algorithm. However, the total noise reduction of the Minimax algorithm, compared with that of the MEFXLMS algorithm, is much on the frequency of the input obtained for the multi-channel ANC system that for the performance of the control system and to overcome the acoustical feedback it is necessary to use the acoustical feedback neutralization technique for systems reflective boundary

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