Inverse structure for active noise control and ... - Semantic Scholar

IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL. 9, NO. 2, FEBRUARY 2001

141

Inverse Structure for Active Noise Control and Combined Active Noise Control/Sound Reproduction Systems Martin Bouchard, Member, IEEE, and Yu Feng

Abstract—In this paper, a simple inverse structure for multichannel active noise control (ANC) is introduced, based on the use of noncausal models of the (pseudo-)inverse plant between the error sensors (typically microphones in ANC systems) and the actuators (typically loudspeakers). This structure, combined with adaptive FIR filters and filtered-x LMS-based algorithms, can produce both a reduction of the computational load and an increase of the convergence speed, compared to the multichannel filtered-x LMS algorithm or its fast exact realizations. Also, for systems combining active sound cancellation and exact sound reproduction (such as headphones combining both ANC and binaural audio with equalization), the proposed inverse structure can further reduce the computational load compared to common approaches. Simulations using exact and noisy models of a realistic acoustic plant and an inverse plant were performed to verify the performance of the proposed structure. Index Terms—active noise control, adaptive filtering algorithms, inverse modeling.

I. INTRODUCTION

F

OR active noise control (ANC) systems [1]–[3], a common approach is to use adaptive FIR filters trained with the filtered-x LMS algorithm [4]–[6] for both feedforward systems and internal model control (IMC) feedback systems in monochannel and multichannel systems. The widespread use of the filtered-x LMS algorithm is mainly due to the simplicity and the low computational load of the algorithm. Variations of the algorithm sometimes called modified filtered-x LMS algorithms have also been published [7]–[10], which can achieve a faster convergence speed by using a larger step size in the algorithm. Recently, fast exact realizations of the filtered-x LMS and the modified filtered-x LMS algorithms have been published [7]. In most cases, these fast realizations can reduce the computational complexity of the filtered-x LMS and modified filtered-x LMS algorithms for multichannel systems. Many algorithms that can achieve faster convergence than the multichannel filtered-x LMS or the modified filtered-x LMS algorithms for ANC systems have been published over the years. Some of these algorithms work on a sample-by-sample basis, some others on a block-by-block basis. There have been Manuscript received June 29, 1999; revised April 25, 2000. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Michael S. Brandstein. The authors are with the School of Information Technology and Engineering, University of Ottawa, Ottawa, ON, Canada K1N 6N5 (e-mail: [email protected]). Publisher Item Identifier S 1063-6676(01)00848-3.

Fig. 1. ANC system with the controller separated into two parts: a predictor and a delayed noncausal inverse plant model.

Fig. 2. ANC system with a standard controller structure.

numerous approaches used: lattice structures with descent algorithms [11], [12], fast recursive-least-squares (RLS) or fast-transversal-filter (FTF) algorithms [13], [14], transform domain algorithms [15], [16], fast affine projection algorithms [17], etc. However, these algorithms provide increased convergence speed at the cost of increasing the computational load, as compared to the multichannel filtered-x LMS algorithm or its fast exact realizations. In this paper, a simple multichannel algorithm that can both reduce the computational load and increase the convergence speed compared to the multichannel filtered-x LMS algorithm or its fast exact realizations is introduced, using an inverse structure and filtered-x LMS-based algorithms. In the proposed approach, the FIR controller is split into two parts: multichannel predictors and multichannel delayed noncausal models of the (pseudo-)inverse plant between the error sensors (typically microphones in ANC systems) and the actuators (typically loudspeakers). Fig. 1 shows this inverse structure for the simplified monochannel case, while the standard structure for controllers using FIR filters in ANC systems is shown in Fig. 2. It should be noted that in adaptive ANC, it is usually possible to measure the acoustic plant between the actuators and the error sensors by sending a known sequence in the plant and by correlating this sequence with the signals received at the error sensors. This identification can be performed off-line (before the control stage), or on-line (during the control stage). The inverse model of the plant can be identified using the same techniques as for the direct model of the plant. One possibility is to use time-domain adaptive filters for the inverse identification [18], [6]. Another possibility is to use frequency domain techniques with or without regularization [19].

1063–6676/01$10.00 © 2001 IEEE

142

IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL. 9, NO. 2, FEBRUARY 2001

The approach of splitting the controller into two parts is not new [20], and these two parts are sometimes referred to as the regulator and the compensator. However, the approach reported in [20] does not use adaptive FIR filters as controllers and noncausal models of the (pseudo-)inverse plant. A previously published approach using noncausal models of the inverse plant is known as the filtered- LMS algorithm [21], and uses the noncausal models to filter the signals from the error sensors. However, filtering these error signals by the inverse plant is likely to reduce the magnitude of some energetic components in the spectrum of the error signals, because the spectrum of the disturbance signals (i.e., the error signals when the controller is off) is often correlated with the frequency response of the plant between the actuators and the error sensors. In all cases, the statistics of the filtered error signals will be different from the statistics of the original error signals. Minimizing the filtered error signals in the filtered- LMS algorithm may thus lead to a slow attenuation of the “true” error signals at the error sensors. Indeed, it is reported in [21] that the filtered- LMS algorithm will be even slower than the filtered-x LMS algorithm on average. In the inverse structure proposed in this paper, the signals from the error sensors are directly minimized, thus avoiding this problem of reduced convergence speed. With the proposed inverse structure using adaptive FIR filters, there are two benefits of using noncausal filters modeling the (pseudo-)inverse plant of an ANC system. One benefit is that the combination of the delayed noncausal models and the physical direct plant (or the models of the direct plant) becomes approximately a pure delay (approximately because models are never perfect). Using these pure delay operations can eliminate some costly convolutions when an adaptive filtering algorithm is used to adapt the system of Fig. 1. The second benefit of using noncausal filters modeling the (pseudo-)inverse plant of an ANC system is due to the fact that the convergence speed of filtered-x LMS-based algorithms is related to the eigenvalue spread in the correlation matrix of the filtered reference signals. This eigenvalue spread is caused by the statistics of the reference signals (a nonuniform spectrum in the reference signals will cause an eigenvalue spread in the correlation matrix), and by the frequency response of the direct plant models, used to filter the reference signals in the filtered-x LMS-based algorithms. A nonuniform frequency response in the plant model can thus increase the eigenvalue spread in the multichannel correlation matrix of the filtered reference signals. It is this second source of the eigenvalue spread that can be eliminated using the proposed structure with the noncausal models of the (pseudo)inverse plant, because the resulting combination of the noncausal models and the models of the direct plant will have flat, uniform frequency responses, thus eliminating the eigenvalue spread caused by the models of the direct plant. For systems combining active noise control and exact sound reproduction (such as headphones combining both ANC and binaural audio using equalization), the proposed inverse structure can further reduce the computational load as compared to common approaches. Fig. 3 shows such a system (for the simplified monochannel case), and from this figure it can be seen that exact sound reproduction systems use delayed noncausal models of the (pseudo-)inverse plant. Since this filtering using

Fig. 3. Combined ANC/exact sound reproduction system.

Fig. 4.

Block-diagram of the filtered-x LMS algorithm.

the noncausal models is the same as the filtering performed by the ANC algorithms that use the inverse structure introduced in this paper, exact sound reproduction systems can be combined with those ANC algorithms without a significant extra computational cost. In Section II of this paper, a brief description of the multichannel filtered-x LMS algorithm and a modified filtered-x LMS algorithm will be presented. Section III will then describe the multichannel algorithms that result from the use of the filtered-x LMS and the modified filtered-x LMS algorithms with the proposed inverse structure, i.e., the structure using the delayed noncausal models of the (pseudo-)inverse plant. In Section IV, the optimal solution of the controller in the standard ANC structure and in the proposed inverse structure will be discussed, showing that they have the same causality constraint for broadband disturbance control. In Section V, the computational load of the different algorithms described in this paper for ANC systems will be discussed. Simulations using exact and noisy models of a realistic acoustic plant and an inverse plant are performed in Section VI to show the convergence gain that can be achieved with the proposed inverse structure, while reducing the computational load of the control algorithm. II. MULTICHANNEL FILTERED-X LMS AND MODIFIED FILTERED-X LMS ALGORITHMS A. The Multichannel Filtered-x LMS Algorithm In the multichannel filtered-x LMS algorithm, the gradient of the instantaneous sum of the squared error signals is computed as a function of each coefficient in the adaptive filters. The algorithm simply uses a steepest descent approach, and thus it modifies the coefficients in the opposite direction of the gradient. Fig. 4 shows a block diagram of the algorithm for the monochannel case. In order to briefly describe the multichannel filtered-x LMS algorithm used for ANC systems, the following notation is defined: number of reference sensors in a feedforward ANC system; number of actuators in an ANC system; number of error sensors in an ANC system;

BOUCHARD AND FENG: INVERSE STRUCTURE FOR ACTIVE NOISE CONTROL

length of the FIR adaptive filters; length of the FIR filter models of the direct plant; value at time of the th reference signal in a feedforward ANC system; value at time of the th actuator signal [see (7)]; value at time of the primary sound field (disturbance signal) at the th error sensor; value at time of the signal measured by the th error sensor; value at time of the th coefficient in the adaptive and in the filtered-x FIR filter linking LMS and modified filtered-x LMS algorithms; value of the th coefficient in the FIR filter model and ; of the direct plant between value at time of the filtered reference signals, i.e., signals the signals obtained by filtering the in the filtered-x with the direct plant models LMS and modified filtered-x LMS algorithms [see (8)];

(1) (2) (3) (4) (5) (6) In the notation for ANC systems, feedforward controllers have been implicitly assumed because of the use of reference signals . However, it is well known that feedback ANC systems using the internal model control (IMC) approach behave like feedforward controllers [22], [23], and that the same adaptive filtering algorithms as in feedforward systems can be used for those feedback systems. Using the above notation, the multichannel filtered-x LMS algorithm can be described by (7)–(9):

(7) (8) (9) where is a scalar convergence gain. The error signals measured by the error sensors are the physical sum of the disturbance signals and the contribution of the actuators at the error of the direct sensors, as described by (10) if the models plant are exact:

(10)

143

Fig. 5.

Block-diagram of the modified filtered-x LMS algorithm.

B. Multichannel Modified Filtered-x LMS Algorithm In the filtered-x LMS algorithm, if there is a delay in the plant, changes to the adaptive filters do not have an immediate impact on the error signals. This causes the additional requirement that a small convergence gain may be required in the algorithm, so that the coefficients of the adaptive filters change slowly (stationary system hypothesis of the filtered-x LMS algorithm [6]). To overcome this problem, modified filtered-x LMS algorithms were developed [7]–[10]. The version found in [7] will be described here. The modified filtered-x LMS algorithm uses the to subtract from the error signals direct plant models the contribution of the actuators, so that an estimate of the is obtained. The modified filtered-x primary field signals LMS algorithm then performs a commutation of the plant and the adaptive filters, and the adaptive filters try to predict the estimated signals instead of the original signals. As a result, are sufficiently accurate, the behavior of the if the models algorithm is identical to a standard LMS algorithm: any change to the adaptive filters has an immediate effect on the new error , no stationary system hypothesis, and, if the plant signals has a delay, a larger convergence gain can be used as compared to the filtered-x LMS algorithm. The structure of the modified filtered-x LMS algorithm is shown in Fig. 5 for the monochannel case. Defining the following variables, the multichannel modified filtered-x LMS algorithm can be described by (7), (8) and (11)–(13): estimate of error computed with the commutation of the plant and the adaptive filters (as in Fig. 5) (11)

(12)

(13) Fast exact realizations of the filtered-x LMS and the modified filtered-x LMS algorithms have also been published [7]. In most cases, these fast realizations can reduce the computational complexity of the filtered-x LMS and modified filtered-x LMS algorithms for multichannel systems. This will be discussed in Section V.

144

IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL. 9, NO. 2, FEBRUARY 2001

III. MULTICHANNEL INVERSE FILTERED-X LMS AND INVERSE MODIFIED FILTERED-X LMS ALGORITHMS A. The Multichannel Inverse Filtered-x LMS Algorithm As mentioned in the introduction, it is possible to both reduce the computational load and increase the convergence speed of the filtered-x LMS-based algorithms used for the training of adaptive FIR filters in ANC systems by using noncausal models of the (pseudo-)inverse plant. This scheme can be applied to the multichannel filtered-x LMS and the modified filtered-x LMS algorithms, and therefore we call the resulting algorithms the multichannel inverse filtered-x LMS (IFX) and multichannel inverse modified filtered-x LMS (IMFX) algorithms. To describe these algorithms, the following notation is defined: length of the FIR filter models of the (pseudo-)inverse plant; value of the th coefficient in the FIR filter modeling and ; the (pseudo-)inverse plant between estimated value at time of the disturbance signal , where is the delay required to make causal [see (14)]. is obtained by filtering with the adaptive filters the reference signals [see (15)]; value at time of the th coefficient in the adapand in the IFX or tive FIR filter linking IMFX algorithms; signal obtained by filtering the reference signal with the filter ; value at time of the filtered reference signals, i.e., sigthe signals obtained by filtering the , in the IFX or nals with the direct plant models IMFX algorithms; (14) (15) (16)

Fig. 6. Block-diagram of the inverse filtered-x LMS (IFX) algorithm.

Fig. 7. Block-diagram of the simplified inverse filtered-x LMS (simplified IFX) algorithm.

(23) (24)

(25) filters in (23) can be convolved with the Note that the filters in (24), and these convolutions only need to be computed once, off-line. Therefore, (23)–(24) can be combined in a single equation, which will significantly reduce the number of computations of the algorithm

(26) (17) (18) (19) (20) The IFX algorithm is described by (21)–(25), and a block diagram of the algorithm appears in Fig. 6 for the monochannel case. In these equations, and are both used as indices for the error sensor signals or their prediction

(21)

(22)

and where “*” is the offline convolution operator. Since are models of the direct and (pseudo-)inverse plant, their combination can also be estimated by pure delays (the same causal). Therefore, it is possible delay required to make to simplify (23)–(26) and use (27) instead, to have a simplified IFX algorithm (27) is the delay required to make causal. A block where diagram of the simplified IFX algorithm is shown in Fig. 7 for the monochannel case. B. Multichannel Inverse Modified Filtered-x LMS Algorithm The inverse modified filtered-x LMS (IMFX) algorithm combines a modified filtered-x LMS algorithm with the inverse structure using the models of the (pseudo-)inverse plant. The algorithm can be described by (21)–(22), (26) and (28)–(30),

BOUCHARD AND FENG: INVERSE STRUCTURE FOR ACTIVE NOISE CONTROL

145

IV. OPTIMAL SOLUTION AND CAUSALITY CONSTRAINT OF THE ANC STRUCTURES

Fig. 8. Block-diagram of the inverse modified filtered-x LMS (IMFX) algorithm.

Fig. 9. Block-diagram of the simplified inverse modified filtered-x LMS (simplified IMFX) algorithm.

and a block diagram of the algorithm is shown in Fig. 8 for the monochannel case

The optimal broadband solutions for the controllers in the two ANC structures of Fig. 10 (direct and inverse structures) are not the same. To describe the two solutions, a simple monochannel ANC system will be used, with the following definitions (as in Fig. 10): causal impulse response between the reference signal and the error sensor. This impulse response samples; includes a delay of causal impulse response (plant) between the actuator and the error sensor. This impulse response insamples; cludes a delay of truncated and delayed (causal) version of the noncausal and possibly two-sided infinite impulse rebetween the error sponse of the inverse plant is required sensor and the actuator. A delay of causal. The truncation of is to make is not formally defined here. It is assumed that (a performance a good delayed estimate of includes 99% of the encriteria could be that ), and it is assumed that is the ergy in smallest value that meets this performance criteria kept in (i.e., that the noncausal part of is as short as possible); optimal broadband solution of a controller in Fig. 10. does not mean here that The notation are time-varying. This notation is used instead so that the time shifting operation becomes easy to write in the rest of this section. For the standard (noninverse) structure, it can be is seen from Fig. 10(a) that the optimal solution for (34)

(28)

(29)

where “*” is the convolution operator once again. The delay in is , and the criteria the resulting impulse response , or . For the for a causal solution is thus inverse structure, it can be seen from Fig. 10(b) that the optimal is broadband solution

(30) (35) and are models of the direct and (pseudo-) Again, since inverse plant, their combination in (26) can be estimated by pure and in (22), (28) delays. Moreover, the combination of can also be simplified to pure delays. The resulting simplified IMFX algorithm is described by (21), (22) and by (31)–(33) (31) (32) (33) A block diagram of the simplified IMFX algorithm is shown in Fig. 9, for the monochannel case.

, the delay To make sure that a causal solution exists for must be sufficiently high so that still remains in a causal impulse response. Therefore, the following condition . In principle, the causality must again be respected: constraint is thus the same for both structures. However, if the is not respected, the shapes of the optimal solucriteria will determine the performance that can be achieved tions is not respected by causal controllers. For example, if for and the noncausal coefficients of the optimal solution a particular structure are very energetic, then the causal solution would produce a performance with a severe degradation compared to the performance of the optimal controller. In the case of narrowband or periodic disturbance signals, it that is not necessary to find a broadband optimal solution

146

IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL. 9, NO. 2, FEBRUARY 2001

(a) Fig. 10.

(b)

Block-diagram of the system used in the simulations for (a) the standard ANC structure and (b) the inverse structure. TABLE I COMPUTATIONAL COMPLEXITY (NUMBER OF MULTIPLIES) OF THE MULTICHANNEL ALGORITHMS

is valid at all frequencies. The main factor that limits the performance of causal FIR filter controllers in that case is the ability of the controller to predict the disturbance signals. For the inverse structure, it can be seen from Fig. 7 that it is a -samples-ahead prediction of the disturbance signals from the delayed predicted from reference signals, i.e., . In the case of the standard ANC struc-samples-ahead predicture, Fig. 4 shows that it is instead a tion of the disturbance signals from the filtered reference signals, predicted from , i.e., is produced by filtering with the plant model where that includes a delay of . The delays and thus have a direct effect on the optimal narrowband steady-state performance that can be achieved by the adaptive FIR filter algorithms used with the two control structures, and they also have an effect on the tracking capabilities of the algorithms. A low value (standard structure) means that the of (inverse structure) or tracking capabilities of the corresponding structure is increased. is different in Because the broadband optimal solution the structures of Fig. 10(a) and (b), the number of coefficients required in the standard algorithms of Section II will typically be different from the number of coefficients required in the inverse algorithms of Section III for broadband control. It can be expected that on average, the number of coefficients required in the adaptive FIR filters for the inverse structure (Section III) will be lower than for the standard structure, because only is part of the optimal solution in the inverse structure (35), and for the standard structure (34). as opposed to V. COMPUTATIONAL LOAD OF THE ADAPTIVE FIR FILTER ALGORITHMS FOR ANC The computational load of the multichannel filtered-x LMS algorithm, the modified filtered-x LMS algorithm, their fast

exact realizations [7], and the inverse algorithms described in Section III (IFX, IMFX and their simplified versions) was estimated by counting the number of multiplies required for each algorithm. The result is shown in Table I. As it was discussed in Section IV, the number of coefficients required in the adaptive FIR filters for the algorithms using the inverse structure of Fig. 1 can be expected to be lower than the number of coefficients required in the algorithms using the standard structure of Fig. 2. This is why the variable was split into two for the algorithms using the standard variables in Table I: for the algorithms using the inverse structure of Fig. 2, and structure of Fig. 1. To have a better view of the computational complexity of the different algorithms, the number of required multiplies was of ANC systems and evaluated for several dimensions several ratios of adaptive filter coefficients to plant model coefor ficients or (pseudo-)inverse plant model coefficients ( ). Values and were used, as in the simulations of Section VI. The values of the dimensions and filter lengths that were used and the results for the different algorithms are shown in Table II. From this table, the following observations can be made: • the simplified form of the IFX algorithm always produces a lower computational load than the filtered-x LMS algorithm or its fast exact realization, for monochannel or multichannel systems; • the simplified form of the IMFX algorithm always produces a lower computational load than the modified filtered-x LMS algorithm or its fast exact realization, for monochannel or multichannel systems. The reduction of the computation load is even greater in this case then the one achieved by the IFX algorithm;

BOUCHARD AND FENG: INVERSE STRUCTURE FOR ACTIVE NOISE CONTROL

147

TABLE II EVALUATION OF THE COMPUTATIONAL COMPLEXITY (NUMBER OF MULTIPLIES) OF THE FILTERED-x LMS (FXLMS), FAST-EXACT FXLMS, MODIFIED FXLMS, FAST-EXACT MODIFIED FXLMS, IFX, SIMPLIFIED IFX, IMFX AND SIMPLIFIED IMFX ALGORITHMS, FOR SEVERAL DIMENSIONS AND RATIOS OF COEFFICIENTS IN ANC SYSTEMS

• nonsimplified forms of the IFX algorithm or the IMFX algorithm typically produce higher computational loads than the filtered-x LMS algorithm, the modified filtered-x LMS algorithm or their fast exact realizations, for monochannel or multichannel systems; • in multichannel systems, fast exact realizations of the filtered-x LMS or the modified filtered-x LMS algorithms do not always reduce the number of computations: it deratio and the dimensions ; pends on the • for monochannel systems, fast exact realizations of the filtered-x LMS or the modified filtered-x LMS algorithms typically increase the number of computations. The approach used in [7] to develop fast exact realizations of the filtered-x LMS and the modified filtered-x LMS algorithms could also be used for the IFX and the IMFX algorithms. It remains to be seen if there is a need for such fast exact realizations of these inverse algorithms, because the simplifications that were described in Section III (i.e., using pure delays) provide a greater reduction of the computational load than the fast exact realizations would. However, when the models of the plant and the (pseudo-)inverse plant are inaccurate, the simplified versions of the inverse algorithms may not perform as well as the nonsimplified inverse algorithms, as will be seen in Section VI. Therefore, in this case, the use of fast

Fig. 11. An exact sound reproduction system combined with the simplified IFX ANC algorithm.

realizations of the nonsimplified inverse algorithms could be an alternative. As mentioned in the introduction, for ANC systems combined with exact sound reproduction systems (such as binaural or transaural audio systems with equalization), the algorithms using the inverse structure provide an extra reduction of the computational load because there is no additional multiply operation to perform in the system with the introduction of the sound reproduction equalization component. This is because the inverse filtering required by the exact sound reproduction component is already performed in the inverse structure algorithms. Fig. 11 illustrates this for the simplified version of the IFX algorithm.

148

IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL. 9, NO. 2, FEBRUARY 2001

(a) Fig. 12.

(a) Impulse response and (b) frequency response of the plant used in the simulations.

(a) Fig. 13.

(b)

(b)

(a) Impulse response and (b) frequency response of the inverse plant used in the simulations.

VI. SIMULATION RESULTS

(a)

(b) Fig. 14. (a) Impulse response and (b) frequency response of the transfer function between the reference signal and the disturbance signal.

In the previous section, the low computational load of some algorithms using the inverse structure has been described. In this section, simulations will show that the convergence speed of those algorithms can also be faster than for the standard filtered-x LMS and modified filtered-x LMS algorithms. The feedforward ANC system shown in Fig. 10 was simulated here. The simulated plant was the measured impulse response between a loudspeaker and a microphone in an active headset. The inverse plant was identified offline from the impulse response of the (direct) plant, using an adaptive fast-transversal-filter (FTF) algorithm. The inverse plant impulse response was identified with dB modeling error precision over the normalized frea dB modeling error quency bandwidth of 0.075 to 0.45. This precision was evaluated by performing the convolution of the identified inverse impulse response with the impulse response of the plant, subtracting the ideal Dirac impulse from the result, computing the magnitude of the Fourier transform of the resulting signal, and finally evaluating the magnitude between normalized frequency 0.075 to 0.45. Because the accuracy of the inverse model is only good between these two normalized frequencies, the broadband reference signal that was used in the simulations was band-limited between these two normalized frequencies. The impulse response and the frequency response of the plant are shown in Fig. 12. The delay of this plant is small (8–10

BOUCHARD AND FENG: INVERSE STRUCTURE FOR ACTIVE NOISE CONTROL

(a) Fig. 15.

(b)

(a) Impulse response and (b) frequency response of the adaptive filter after convergence, for the standard ANC structure.

(a) Fig. 16.

149

(b)

(a) Impulse response and (b) frequency response of the adaptive filter after convergence, for the inverse ANC structure.

samples), so it can be expected that the gain of the modified filtered-x LMS algorithm over the filtered-x LMS algorithm will not be large. Nevertheless, some convergence gain was observed. The noncausal impulse response and the frequency response of the inverse plant are shown in Fig. 13. It can be seen that a delay of 10–15 samples is required to make the inverse impulse response causal. A delay of 15 samples was used in the simulations. The length of the impulse response of the plant (and its noisy models) was 256 samples, and the length of the delayed impulse response of the inverse plant (and its noisy models) was also 256 samples, including 15 noncausal samples that were delayed. For the transfer function between the reference signal and the disturbance signal at the error sensor, an experimentally measured acoustic impulse response was used, again with a length of 256 samples. Fig. 14 shows the impulse response and the frequency response of that transfer function. The length of the adaptive filter was chosen to be coefficients for the algorithms using the standard structure of coefficients for the algorithms using the Fig. 2, and inverse structure of Fig. 1. These lengths were chosen such that the performance of the Wiener solutions (estimated by a RLS algorithm with a forgetting factor close to unity, i.e., a long memory) of the adaptive filters for the two structures would be approximately the same in the case where exact plant model and/or inverse plant model were used. Therefore, to obtain algorithms producing similar misadjustments, it was only required to obtain the same steady-state error levels with the different algorithms. In the simulations, the scalar convergence gain that produced the fastest convergence speed was found by trial and error

for each algorithm. When exact models or models with dB modeling error were used, it was verified that all algorithms produced nearly the same steady state residual error level (within a or dB mod1 dB margin). However, for models with eling errors, it was not possible to adjust the gain so that the steady-state residual error would be the same. This is because the Wiener solutions of the different algorithms are not the same in this case: each algorithm uses a different structure, and this changes the statistics of the filtered (or sometimes delayed) reference signals and the (sometimes estimated) disturbance signals. The first simulations that were performed were with exact models of the plant and the inverse plant. In this case, the FIR filter solutions that were found by the adaptive filtering algorithms after 25 dB of convergence are shown in Figs. 15 and 16, for the standard structure and the inverse structure, respectively. As expected, it can be seen from Fig. 16 that the solution for the inverse structure is a negated shifted version of the impulse response of Fig. 14. Fig. 17 shows the learning curves for the different algorithms and structures, where the convergence is defined by the average energy of the error signal divided by the average energy of the disturbance signal, in decibels. It is clear from Fig. 17 that with exact models, the algorithms using the inverse structure outperform the algorithms using the standard structure. There is no significant difference of performance between the different versions of the algorithms using the inverse structure: the IFX algorithm, the IMFX algorithm and their simplified versions. For the standard structure, the modified filtered-x LMS outperforms the filtered-x LMS, as previously reported in the literature.

150

IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL. 9, NO. 2, FEBRUARY 2001

Fig. 17. Learning curves of the different algorithms with the use of exact models: IFX, simplified IFX (SIFX), IMFX, simplified IMFX (SIMFX), modified filtered-x LMS (MFXLMS) and filtered-x LMS (FXLMS) algorithms.

0

Fig. 18. Learning curves of the different algorithms using models with 20 dB modeling error level: IFX, simplified IFX (SIFX), IMFX, simplified IMFX (SIMFX), modified filtered-x LMS (MFXLMS) and filtered-x LMS (FXLMS) algorithms.

Simulations with models of the plant and the inverse plant dB, dB and having modeling error levels of dB were then performed. The modeling error was added on a frequency-by-frequency basis, by adding for each frequency a random complex value whose magnitude was for instance dB less than the magnitude of the frequency response at that frequency. These levels of modeling error are all achievable in practice, and for stationary system models, a better accuracy than those error levels can often be achieved. Note that these simulations with noisy models can only provide some indication of the algorithms’ behavior with inaccurate models, and that further theoretical analysis would be an interesting topic. dB modeling error level. Fig. 18 shows the result for the In this case, it can be seen that the algorithms using the inverse structure are still more than twice as fast as the other algorithms for the simulated system, while also having a lower computational load. However, it can be seen that the convergence gain is reduced as compared to the use of exact models. Again, there is no significant difference of performance between the different versions of the algorithms using the inverse structure, and for the standard structure the modified filtered-x LMS algorithm outperforms the filtered-x LMS algorithm. dB modeling Fig. 19 shows the simulation results for a dB error level appeared to be the turning error level. This point where the algorithms using the inverse structure lose their performance gain as compared to the filtered-x LMS and modified filtered-x LMS algorithms (but the inverse structure algorithms still keep their low computational load advantage). It can be seen from Fig. 19 that all the algorithms using the inverse

0

Fig. 19. Learning curves of the different algorithms using models with 10 dB modeling error level: IFX, simplified IFX (SIFX), IMFX, simplified IMFX (SIMFX), modified filtered-x LMS (MFXLMS) and filtered-x LMS (FXLMS) algorithms.

0

Fig. 20. Learning curves of the different algorithms using models with 5 dB modeling error level: IFX, simplified IFX (SIFX), IMFX, simplified IMFX (SIMFX), modified filtered-x LMS (MFXLMS) and filtered-x LMS (FXLMS) algorithms.

structure achieve a steady-state performance of approximately 15 dB (i.e., sound reduction) only, while the other algorithms can achieve more than 20 dB. Thus in this case, the designer’s choice for the algorithm would be to choose between a low comdB putational load and a greater noise attenuation. For the modeling error level, as shown in Fig. 20, the algorithms using the inverse structure performed poorly, producing only 6–8 dB of sound attenuation. The inverse algorithms using the simplification (the pure delay combination of the plant model and the inverse plant model) produced the worst performance. The performance of the filtered-x LMS and the modified filtered-x LMS algorithms was better and almost identical: more than 20 dB of sound attenuation. In the simulations where the algorithms using the inverse structure produced a good performance (simulations with moddB), all the versions of these eling error levels of less than algorithms produced very similar performance. Therefore, the use of the inverse algorithm with the lowest computational load (i.e., the simplified IFX algorithm) may be the appropriate choice among the four inverse algorithms. VII. CONCLUSION In this paper, an inverse structure was introduced for the use of adaptive FIR filters in ANC systems. Multichannel adaptive FIR filter learning algorithms based on the filtered-x LMS algorithm were introduced for this inverse structure. For systems using this inverse structure for ANC, the extra computational load required to convert these ANC systems to combined ANC/exact

BOUCHARD AND FENG: INVERSE STRUCTURE FOR ACTIVE NOISE CONTROL

151

sound reproduction systems is also reduced. It was shown in simulations with a realistic acoustical plant that some versions of the introduced algorithms can achieve both a reduction of the computational load and an increase of the convergence speed as compared to standard algorithms for ANC such as the multichannel filtered-x LMS algorithm, the modified filtered-x LMS algorithm or their fast exact realizations. However, it was found in the simulations that the modeling error level on the models of the plant and the inverse plant was required to be less than dB in order to achieve a performance gain with the inverse structure. Some theoretical work on the effect of plant modeling error and inverse plant modeling error would be of interest.

[15] M. Bouchard and B. Paillard, “A transform domain optimization to increase the convergence of the multichannel filtered-x LMS algorithm,” J. Acoust. Soc. Amer., vol. 100, pp. 3203–3214, Oct. 1996. [16] T. Kosaka, S. J. Elliott, and C. C. Boucher, “Novel frequency domain filtered-X LMS algorithm for active noise reduction,” Proc. ICASSP-97, vol. 1, pp. 403–406, 1997. [17] S. C. Douglas, “A fast affine projection algorithm for active noise control,” in Proc. 29th Asilomar Cong. Signals, Systems, Computers, vol. 2, 1996, pp. 1245–1249. [18] P. A. Nelson, F. Orduna-Bustamante, and H. Hamada, “Inverse filter design and equalization zones in multi-channel sound reproduction,” IEEE Trans. Speech Audio Processing, vol. 3, pp. 1–8, Jan. 1995. [19] O. Kirkeby, P. A. Nelson, H. Hamada, and F. Orduna-Bustamante, “Fast deconvolution of multichannel systems using regularization,” IEEE Trans. Speech Audio Processing, vol. 6, pp. 189–195, Mar. 1998. [20] E. F. Berkman and E. K. Bender, “Perspectives on active noise and vibration control,” Sound Vibr., vol. 31, pp. 80–94, Jan. 1997. [21] B. Widrow and E. Walach, Adaptive Inverse Control. Englewood Cliffs, NJ: Prentice-Hall, 1996. [22] S. E. Forsythe, M. D. McCollum, and A. D. McCleary, “Stabilization of a digitally controlled active-isolation system,” in Proc. Recent Advances Active Control Sound Vibration, 1991, pp. 879–889. [23] S. J. Elliott, T. J. Sutton, B. Rafaely, and M. Johnson, “Design of feedback controllers using a feedforward approach,” in Proc. ACTIVE 95, 1995, pp. 863–874.

REFERENCES [1] S. J. Elliott, “Down with noise,” IEEE Spectrum, vol. 36, pp. 54–61, June 1999. [2] P. A. Nelson and S. J. Elliott, Active Control of Sound. New York: Academic, 1992. [3] M. O. Tokhi and R. R. Leitch, Active Noise Control. Oxford, U.K.: Clarendon, 1992. [4] S. M. Kuo and D. R. Morgan, Active Noise Control Systems: Algorithms and DSP Implementations. New York: Wiley, 1996. [5] S. J. Elliott, I. M. Stothers, and P. A. Nelson, “A multiple error LMS algorithm and its application to the active control of sound and vibration,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 35, pp. 1423–1434, Oct. 1987. [6] B. Widrow and S. D. Stearns, Adaptive Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1985. [7] S. C. Douglas, “Fast implementations of the filtered-x LMS and LMS algorithms for multichannel active noise control,” IEEE Trans. Speech Audio Processing, pp. 454–465, July 1999. [8] I. S. Kim, H. S. Na, K. J. Kim, and Y. Park, “Constraint filtered-x and filtered-u least-mean-square algorithms for the active control of noise in ducts,” J. Acoust. Soc. Amer., vol. 95, pp. 3379–3389, June 1994. [9] M. Rupp and R. Frenzel, “The behavior of LMS and NLMS algorithms with delayed coefficient update in the presence of spherically invariant processes,” IEEE Trans. Signal Processing, vol. 42, pp. 668–672, Mar. 1994. [10] M. Rupp and A. H. Sayed, “Robust FxLMS algorithms with improved convergence performance,” IEEE Trans. Signal Processing, vol. 6, pp. 78–85, Jan. 1998. [11] Y. C. Park and S. D. Sommerfeldt, “A fast adaptive noise control algorithm based on the lattice structure,” Appl. Acoust., vol. 47, no. 1, pp. 1–25, 1996. [12] J. H. Yun, Y. C. Park, D. H. Youn, and I. W. Cha, “Efficient active noise control algorithm based on the lattice-transversal joint (LTJ) filter structure,” IEICE Trans. Fundamentals Electron. Commun. Comput., vol. 81, pp. 1755–1757, Aug. 1998. [13] M. Bouchard and S. Quednau, “Multichannel recursive-least-squares algorithms and fast-transversal-filter algorithms for active noise control and sound reproduction systems,” IEEE Trans. Speech Audio Processing, vol. 8, pp. 606–618, Sept. 2000. [14] T. Auspitzer, D. Guicking, and S. J. Elliott, “Using a fast-recursive-leastsquared algorithm in a feedback controller,” in Proc. IEEE ASSP Workshop Applications Signal Procesing Audio Acoustics, 1995.

Martin Bouchard (M’98) received the B.Ing., M.Sc.A., and Ph.D. degrees in electrical engineering from Sherbrooke University, Sherbrooke, QC, Canada, in 1993, 1995, and 1997 respectively. He worked in an instrumentation group at Bechtel-Lavalin in 1991, and at CAE Electronics, Inc., in 1992–1993 in a real-time software group. From 1993 to 1997, he was with the Groupe d’Acoustique et de Vibrations de l’Université de Sherbrooke, working as a Signal Processing/Control Research Engineer. From 1995 to 1997, he also was with SoftDb Active Noise Control Systems, Inc., which he cofounded. In January 1998, he joined the University of Ottawa, Ottawa, ON, Canada, where he is currently an Assistant Professor with the School of Information Technology and Engineering. His current research interests are signal processing, adaptive filtering and neural networks, applied to speech coding, audio, and acoustics. Dr. Bouchard is a member of the Ordre des Ingénieurs du Québec, the Audio Engineering Society, and the Acoustical Society of America.

Yu Feng was born in Kunming, Yunnan Province, China, in 1971. He received the B.A. degree from Tsinghua University, Beijing, China, in 1995, and the M.S. degree from Ottawa University, Ottawa, ON, Canada in 2000, respectively, all in electrical engineering. From 1995 to 1998, he was an Engineer with the Management Information System, Civil Administrator of Aviation, China. He is interested in signal processing (acoustics, adaptive filtering, speech, etc.) and virtual environments.