Multichannel Affine and Fast Affine Projection Algorithms for Active Noise Control and Acoustic Equalization Systems

Transcription

1 54 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL. 11, NO. 1, JANUARY 2003 Multichannel Affine and Fast Affine Projection Algorithms for Active Noise Control and Acoustic Equalization Systems Martin Bouchard, Member, IEEE Abstract In the field of adaptive signal processing, it is well known that affine projection algorithms or their low-computational implementations fast affine projection algorithms can produce a good tradeoff between convergence speed and computational complexity. Although these algorithms typically do not provide the same convergence speed as recursive-least-squares algorithms, they can provide a much improved convergence speed compared to stochastic gradient descent algorithms, without the high increase of the computational load or the instability often found in recursive-least-squares algorithms. In this paper, multichannel affine and fast affine projection algorithms are introduced for active noise control or acoustic equalization. Multichannel fast affine projection algorithms have been previously published for acoustic echo cancellation, but the problem of active noise control or acoustic equalization is a very different one, leading to different structures, as explained in the paper. The computational complexity of the new algorithms is evaluated, and it is shown through simulations that not only can the new algorithms provide the expected tradeoff between convergence performance and computational complexity, they can also provide the best convergence performance (even over recursive-least-squares algorithms) when nonideal noisy acoustic plant models are used in the adaptive systems. Index Terms Acoustic equalization, active noise control, fast affine projection algorithms, multichannel adaptive filtering, sound reproduction. I. INTRODUCTION ACTIVE noise control (ANC) systems [1], [2] work on the principle of destructive interference between an original primary disturbance sound field measured at the location of error sensors (typically microphones), and a secondary sound field that is generated by control actuators (typically loudspeakers). In ANC systems a common approach is to use adaptive FIR filters, in either feedforward or feedback control configurations. A similar problem is the problem of acoustic equalization or deconvolution [3], [4], where the acoustic response of a room between actuators and sensors needs to be inverted and compensated. An application of this is transaural audio or multichannel exact sound reproduction systems, where given waveforms have to be reproduced at some sensor locations. Figs. 1 and 2 show block-diagrams of monochannel im- Manuscript received October 15, 2001; revised August 14, This work was supported in part by an NSERC grant. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Peter Vary. The author is with the School of Information Technology and Engineering, University of Ottawa, Ottawa, ON, K1N 6N5 Canada ( bouchard@site.uottawa.ca). Digital Object Identifier /TSA plementations of feedforward active noise control and acoustic equalization, using adaptive FIR filters. The systems in Figs. 1 and 2 are delay-compensated, i.e., the stabilization time on the error signals caused by updates to the adaptive FIR filter coefficients has been eliminated by minimizing an alternative error signal, with the same steady state statistics as the original error signal. This has also been called the modified filtered- structure [5], and the use of this structure will be assumed in the rest of this paper. Also, the algorithms to be introduced in this paper are for feedforward adaptive active noise control, although it is a simple task to adapt them to either feedback adaptive active noise control [with internal model control (IMC) structures] or acoustic equalization systems [5]. It is well known that affine projection algorithms or their low-computational implementations fast affine projection algorithms can produce a good tradeoff between convergence speed and computational complexity. Although these algorithms typically do not provide the same convergence speed as recursiveleast-squares algorithms, they can provide a much improved convergence speed compared to stochastic gradient descent algorithms, without the high increase of the computational load or the instability often found in recursive-least-squares algorithms, especially for multichannel systems [5] [7]. An adaptation of a fast affine projection algorithm for monochannel active noise control has been previously published [8]. The adaptation is not straightforward, as fast affine projection algorithms compute auxiliary coefficients instead of the normal time-domain coefficients usually computed by adaptive FIR filtering algorithms, and the outputs from those normal coefficients are required for active noise control or for acoustic equalization. In Section II of this paper, the previous work is modified and extended to introduce multichannel affine and fast affine projection algorithms for active noise control or acoustic equalization. Multichannel fast affine projection algorithms have been previously published for acoustic echo cancellation, but the problem of active noise control or acoustic equalization is a very different one. Indeed active noise control and acoustic equalization are obviously control or inverse problems, while acoustic echo cancellation is an identification problem (with of course its own additional constraints such as double-talk, etc.). This leads to different structures (such as the filtered- structure of the filtered- LMS [9] instead of the standard adaptive FIR filter structure, or other structures such as adjoint [5], filtered- [9] or inverse filtered- [10]), to a different number of dimensions for the different signals, and obviously to different multichannel algorithms. In Section III, the computational complexity of the new /03$ IEEE

2 BOUCHARD: MULTICHANNEL AFFINE AND FAST AFFINE PROJECTION ALGORITHMS 55 Fig. 1. Delay compensated modified-filtered-x structure for active noise control. Fig. 2. Delay compensated modified-filtered-x structure for acoustic equalization or exact sound reproduction. introduced algorithms is evaluated. In Section IV, it is shown through simulations that not only can the new algorithms provide the expected tradeoff between convergence performance and computational complexity, they can also provide the best convergence performance (even over recursive-least-squares algorithms) when nonideal noisy acoustic plant models are used in the adaptive systems. II. MULTICHANNEL AFFINE AND FAST AFFINE PROJECTION ALGORITHMS FOR ACTIVE NOISE CONTROL A. Multichannel Affine Projection Algorithm for Active Noise Control An affine projection algorithm for multichannel active noise control is described in this section. Even though a fast affine projection algorithm for multichannel active noise control with a lower complexity will be developed in the next subsection, it may be useful to use the nonsimplified affine projection version because for small affine projection orders the complexity of the affine projection algorithm may be similar to the complexity of the fast affine projection algorithm, and its implementation is much simpler; the affine projection algorithm does not require the use of any sliding window recursive-least-squares algorithm to compute parameters, therefore in some implementations its numerical robustness may be greater than for fast affine projection algorithms. The mathematical foundation of the affine projection algorithm will not be described here since it is available in the literature [11], but an emphasis will be put on how the different signals in multichannel ANC must be structured so that the affine projection algorithm can be used. In particular, the dimensions of the different resulting signals will be emphad. To describe the multichannel delay-compensated modified filtered- affine projection algorithm (MFXAP), the following notation is defined (refer to Fig. 1): number of reference sensors in an ANC system; number of actuators in an ANC system; number of error sensors in an ANC system; length of the adaptive FIR filters; affine projection order; length of (fixed) FIR filters modeling the plant (transfer functions between the actuators and the error sensors) in an ANC system; value at time of the th reference signal; value at time of the th actuator signal; value at time of the primary sound field at the th error sensor; value at time of the th error sensor; estimate of, computed in delay-compensated modified filtered- structures; value at time of the alternative error signal for the th sensor, computed in delay-compensated modified filtered- structures; value at time of the th coefficient in the adaptive FIR filter linking and ; value of the th coefficient in the (fixed) FIR filter modeling the plant between and ; value at time of the filtered reference signal, i.e., the signal obtained by filtering the signal with the plant model filter;

3 56 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL. 11, NO. 1, JANUARY 2003 The interlaced notation used for or is not the only possible notation, but it is required in order to make a block time series for which it will be possible to develop fast algorithms such as fast recursive-least-squares algorithms for multichannel ANC systems [5] [7] or the fast affine projection algorithm for multichannel ANC systems to be introduced later in this section. Using the above notation, the multichannel MFXAP algorithm for active noise control can be described by (1) (5) (1) (2) (5) where is a normalized convergence gain, and is a regularization factor that may be used to help with eventual numerical instability. It should be noted that the performance of the MFXAP algorithm is sensitive to the value of this regularization factor, so it should be tuned carefully (tradeoff between slow convergence and numerical instability). Also, for a proper initialization of the MFXAP algorithm, at the first iteration of the algorithm it was found that better performance was obtained if the first components (upper components) of in (4) and (5) were nonzero [some other components of will also be nonzero]. B. Multichannel Fast Affine Projection Algorithm for Active Noise Control Derivations of the original fast affine projection algorithm can be found in [12], [13], and the modifications required in order to adapt the algorithm to the problem of active noise control or (3) (4) acoustic equalization were discussed in [8]. These modifications are required because of the computation of auxiliary coefficients by fast affine projection algorithms, instead of normal coefficients. In this paper, the fast affine projection (FAP) algorithm selected for an extension to the problem of multichannel active noise control will use a built-in sliding window recursive-least-squares (RLS) algorithm instead of a sliding window fast-rls or fast-transversal-filter (FTF) algorithm. It is therefore a multichannel FAP-RLS algorithm [14] for active noise control, using the delay-compensated modified filtered- structure: the MFXFAP-RLS algorithm. The rationale for using the sliding window RLS is that simulation results of multichannel active noise control using fast affine projection algorithms with sliding window fast-rls or FTF algorithms showed severe numerical instability, even with double precision floating point number representation. Also, the selected projection order will be typically 10 or less, therefore the computational gain of using fast realizations is not very large in this case. To help with the stability of the sliding window RLS algorithm, the inverse correlations matrices in (12) and in (15) were forced to be symmetric by computing only the upper triangular part and copying the values to the lower triangular part. In the description of the multichannel fast affine projection algorithm for ANC, the emphasis again will be put on the structure and dimensions of the different signals. The following additional notation is defined for the MFXFAP-RLS algorithm: value at time of the th auxiliary coefficient in the adaptive FIR filter linking and. These auxiliary coefficients are the coefficients computed by the fast affine projection algorithm. They are different from the coefficients, and extra equations are required in active noise control systems to compute and from instead of [see (6) (9)]. inverse correlation matrices used by the sliding window RLS algorithm. is initialized as an identity matrix multiplied by the scalar, where is a regularization factor to be adjusted. the first columns from the matrix. correlation vector of associated with the th actuator, initialized with zero values. correlation matrix of, initialized with zero values. vector of sparsely filled with the values of. The rows to be filled with the values are the same rows associated with the th actuator in the signal. error vector. the first columns of. the last columns of. the last rows of.

4 BOUCHARD: MULTICHANNEL AFFINE AND FAST AFFINE PROJECTION ALGORITHMS 57 The MFXFAP-RLS algorithm is then described by (2), (3) and (6) (18) (6) The performance of the MFXFAP-RLS algorithm is sensitive to the value of the regularization factor, so it should be tuned carefully (again, tradeoff between slow convergence and numerical instability). Also, for a proper initialization of the MFXFAP-RLS algorithm, at the first iteration of the algorithm only the first components of in (6), (8), (10), (11), (13), and (14) should be nonzero, all the other components should be zero. This requirement is caused by the sliding window RLS algorithm which does not have a forgetting factor. (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) III. COMPUTATIONAL COMPLEXITY OF THE ALGORITHMS Table I lists the computational complexity of the two algorithms introduced in Section II, estimated by the number of multiplies for one iteration of the algorithms. Matrix inversions were assumed to be performed with standard LU decomposition: multiplies, where is the of a square matrix. In Table II, the number of multiplies required by the two algorithms of Section II is compared to other previously published multichannnel adaptive FIR filtering algorithms for active noise control or acoustic equalization, based on least-mean-squares (LMS) or recursive-least-squares algorithms [5] [7]. Two cases are considered in Table II: a monochannel system with,,,, and a multichannel system with,,,,. It can be found from Table II that except for the multichannel modified filtered- LMS algorithm, the algorithm with the lowest computational load is the MFXFAP-RLS algorithm introduced in Section II (for a projection order ). This is true even though the MFXFAP-RLS of Section II uses a sliding window RLS algorithm, and not a sliding window fast-rls or FTF algorithm. Since the MFXFAP-RLS can also provide a good improvement of the convergence speed over the multichannel modified filtered- LMS algorithm (as the simulations of the next section will show), the MFXFAP-RLS is therefore an attractive algorithm for practical real-time implementations. Table II also shows that the computational load of the MFXAP algorithm is significantly higher: typically between the fast recursive-least-squares algorithms and the recursive-least-squares algorithms. But the use of the MFXAP may still be a good solution for low projection orders, or for systems where the other algorithms show numerical instability. IV. SIMULATION OF THE MULTICHANNEL ANC ALGORITHMS In order to compare the convergence of the MFXAP and MFXFAP-RLS algorithms with other algorithms for multichannel ANC, ANC simulations were performed using Matlab with acoustic transfer functions experimentally measured in a duct. The algorithms that were implemented for comparison with the algorithms of Section II are the multichannel modified filtered- LMS and RLS algorithms [5]. The RLS algorithm was modified to force the symmetry of the inverted correlation matrix, since this greatly helps the numerical stability of the algorithm [6], [7]. The simulated system had the dimensions, and. This is to reflect the well known principle that an additional actuator can greatly help to find a causal solution for an acoustic control system or

5 58 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL. 11, NO. 1, JANUARY 2003 TABLE I COMPUTATIONAL LOAD OF THE MFXAP AND MFXFAP-RLS ALGORITHMS, ESTIMATED BY THE NUMBER OF MULTIPLIES FOR ONE ITERATION OF THE ALGORITHMS TABLE II COMPARISON OF THE COMPUTATIONAL LOAD OF THE MFXAP AND MFXFAP-RLS ALGORITHMS WITH OTHER MULTICHANNEL DELAY-COMPENSATED MODIFIED FILTERED-x ALGORITHMS FOR ANC an inverse acoustic system (MINT algorithm [15]). However a system with, and is typically underdetermined, therefore the global correlation matrix to be recurrently inverted by the modified filtered- RLS algorithm is singular, and in order to avoid instability some noise (1%) was added to the signals used to compute this inverted correlation matrix. The impulse responses used for the multichannel acoustic plant had 64 samples each ( ), while the adaptive filters had 150 coefficients ( ). The forgetting factor coefficient in the recursive-least-squares algorithm was set to. The step and the regularization factor used by some algorithms were adjusted by trial and error, and the values producing the fastest convergence speed were selected. For all the affine projection algorithms a step close to unity was used, even under noisy plant model conditions. Figs. 3 and 4 show the performance of the MFXAP and MFXFAP-RLS algorithms for different affine projection orders, from to. In these figures the convergence is defined as the ratio of the sum of the error signals power over the sum of primary field (i.e., the disturbance signals) power. As can be seen from these figures, in both cases a projection order of is sufficient to get a significantly improved convergence performance over a projection order of (the case is a normalized stochastic gradient descent algorithm [11]). Except for (which is an unrealistic high value), the performance of the MFXAP and MFXFAP-RLS algorithms is similar. Fig. 5 compares the performance of the MFXFAP-RLS algorithm for projection order with the multichannel modified filtered- LMS and RLS algorithms. As expected, the convergence performance of the MFXFAP-RLS Fig. 3. Convergence curves of the MFXAP algorithm, for different affine projection orders. From top to bottom: N =1, N =5, N =10 and N = 100. Fig. 4. Convergence curves of the MFXFAP-RLS algorithm, for different affine projection orders. From top to bottom: N =1, N =5, N =10and N = 100. algorithm is found between the convergence performance of the LMS-based algorithm and the RLS-based algorithm. The

6 BOUCHARD: MULTICHANNEL AFFINE AND FAST AFFINE PROJECTION ALGORITHMS 59 Fig. 5. Convergence curves of the MFXFAP-RLS algorithm and multichannel delay-compensated modified filtered-x LMS and RLS algorithms, for ideal plant models. From top to bottom: LMS, FAP-RLS, RLS. Fig. 7. Convergence curves of the MFXFAP-RLS algorithm and multichannel delay-compensated modified filtered-x LMS and RLS algorithms, for 10 db SNR plant models. From top to bottom: RLS, LMS, FAP-RLS. multichannel modified filtered- RLS, on top of also having a much lower computational load. This is shown in Fig. 7. Since in practice it may not always be possible to have plant models that are very accurate (due to time-variance of the plant or a slow on-line identification process), the fact that the MFXFAP-RLS seems more robust to plant model noise (at least for the acoustic system considered here) is another reason to consider this algorithm for practical implementations. Results similar to Figs. 5 7 were found when the MFXAP algorithm was used instead of the MFXFAP-RLS algorithm. Fig. 6. Convergence curves of the MFXFAP-RLS algorithm and multichannel delay-compensated modified filtered-x LMS and RLS algorithms, for 20 db SNR plant models. From top to bottom: LMS, FAP-RLS, RLS. convergence speed gain of the MFXFAP-RLS over the multichannel modified filtered- LMS is considerable, and since the computational load of the MFXFAP-RLS with is of the same order (although higher) than the LMS-based algorithm, the use of the MFXFAP-RLS algorithm is an interesting option for practical implementations. This is particularly true since the multichannel recursive-least-squares algorithms for ANC listed in Table II either have a much higher computational load, or have serious numerical instability problems [5] [7]. With the regularization factor, the MFXFAP-RLS did not show any numerical instability in the simulations. Simulations with noisy acoustic plant models ( model in Figs. 1 and 2) were then performed to compare the robustness of the different algorithms to plant models inaccuracy. So far ideal plant models had been assumed. The noise added to the ideal plant models was added on a frequency by frequency basis, where a random complex value with a magnitude of 20 or 10 db less that the original magnitude was added to each frequency in the frequency response. Fig. 6 first shows the performance when plant models with a 20 db SNR were used. In this case the performance of all algorithms was similar to the case when ideal plant models were used (Fig. 5), except for the initial convergence of the RLS algorithm which become slower. However, when 10 db SNR models were used, the multichannel modified filtered- RLS required a much smaller step in order to converge, which greatly slowed down its convergence speed. In this case the MFXFAP-RLS algorithm greatly outperformed the V. CONCLUSION In this paper, multichannel affine projection and fast affine projection algorithms were introduced for active noise control or acoustic equalization systems using adaptive FIR filters. It was shown through simulations that the proposed algorithms can provide a great increase of convergence speed over a multichannel least-mean-squares algorithm, while the increase of the computational load introduced by the new MFXFAP-RLS algorithm may be acceptable for many applications. The new algorithms were found to be numerically robust. In the realistic case of noisy plant models, it was found that the proposed algorithms could even outperform more expensive recursive-least-squares algorithms. Therefore they seem to be an interesting option for practical multichannel ANC or acoustic equalization systems. REFERENCES [1] S. Elliott, Signal Processing for Active Control. London: Academic, [2] S. M. Kuo and D. R. Morgan, Active noise control: A tutorial review, Proc. IEEE, vol. 87, pp , June [3] J. Bauck and D. H. Cooper, Generalized transaural stereo and applications, J. Audio Eng. Soc., vol. 44, pp , Sept [4] P. A. Nelson, F. Orduna-Bustamante, and H. Hamada, Inverse filter design and equalization zones in multi-channel sound reproduction, IEEE Trans. Speech Audio Process., vol. 3, pp. 1 8, Jan [5] 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, no. 5, pp , Sept [6] F. Yu and M. Bouchard, Recursive least-squares algorithms with good numerical stability for multichannel active noise control, in Proc. ICASSP 2001, vol. 5, Salt Lake City, UT, May 2001, pp [7] M. Bouchard, Numerically stable fast convergence least-squares algorithms for multichannel active sound cancellation systems and sound deconvolution systems, Signal Process., vol. 82, no. 5, pp , May 2002.

7 60 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL. 11, NO. 1, JANUARY 2003 [8] S. C. Douglas, The fast affine projection algorithm for active noise control, in Proc. 29th Asilomar Conf. Sign., Syst., Comp., vol. 2, Pacific Grove, CA, Oct. 1995, pp [9] B. Widrow and E. Walach, Adaptive Inverse Control. Upper Saddle River, NJ: Prentice-Hall, [10] M. Bouchard and F. Yu, Inverse structure for active noise control and combined active noise control/sound reproduction systems, IEEE Trans. Speech Audio Processing, vol. 9, pp , Feb [11] K. Ozeki and T. Umeda, An adaptive filtering algorithm using an orthogonal projection to an affine subspace and its properties, Elec. Comm. Japan, vol. J67-A, pp , Feb [12] S. L. Gay and S. Tavathia, The fast affine projection algorithm, in Proc. ICASSP 1995, vol. 5, Detroit, MI, May 1995, pp [13] M. Tanaka, Y. Kaneda, S. Makino, and J. Kojima, Fast projection algorithm and its step control, in Proc. ICASSP 1995, vol. 2, Detroit, MI, May 1995, pp [14] M. Ghanassi and B. Champagne, Acoustic Signal Processing for Telecommunication, S. L. Gay and J. Benesty, Eds. Norwell, MA: Kluwer, 2000, pp [15] M. Miyoshi and Y. Kaneda, Inverse filtering of room acoustics, IEEE Trans. Acoust., Speech, Signal Processing, vol. 36, pp , Feb Martin Bouchard (M 96) received the B.Eng., M.App.Sc., 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, for the construction of the Lauralco alumine plant, and at CAE Electronics in , for the development of a visual system in a flight simulator. From 1993 to 1997, he worked as a Research Engineer at Sherbrooke University, where he was responsible for the real-time implementation of DSP-based adaptive digital controllers. From 1995 to 1997, he also worked at SoftDb Active Noise Control Systems that he co-founded. In January 1998, he joined the School of Information Technology and Engineering (SITE) at the University of Ottawa, initially as an Assistant Professor and later as an Associate Professor (2002). He has conducted consulting activities with Lumic Electronics, Ottawa, ON, Canada, and with the Communications Research Center (CRC), Ottawa. His current research interests are signal processing applied to telecommunications, speech/audio processing and acoustics. Dr. Bouchard is a member of the Ordre des Ingénieurs du Québec and the Audio Engineering Society.