IEEE SIGNAL PROCESSING LETTERS, VOL. 11, NO. 11, NOVEMBER 2004
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Filtered-X Affine Projection Algorithm for Multichannel Active Noise Control Using Second-Order Volterra Filters Giovanni L. Sicuranza, Senior Member, IEEE, and Alberto Carini, Member, IEEE
Abstract—In this letter, a multichannel controller based on Volterra filters is described. A filtered-X affine projection algorithm is derived in detail for homogeneous quadratic filters. The proposed algorithm can be also extended to higher-order Volterra kernels and includes linear controllers as a particular case. Index Terms—Active noise control, adaptive Volterra filters, affine projection algorithm, multichannel nonlinear controller.
In this letter, a novel filtered-X affine projection (AP) algorithm is derived for a multichannel nonlinear controller modeled by means of quadratic filters. The proposed algorithm exhibits good convergence behavior typical of affine projection methods with a limited increase of computational complexity with respect to filtered-X LMS algorithms.
I. INTRODUCTION
II. MULTICHANNEL NONLINEAR ACTIVE NOISE CONTROLLER
HE principle of active noise control (ANC) consists of the destructive interference in a given location of the noise produced by a primary source with an interefering signal generated by a secondary source driven by an adaptive controller [1]. A commonly used adaptation strategy is based on the so-called feed-forward methods, where reference signals measured in the proximity of the noise source are available. These signals are used together with the error signals captured in the proximity of the zone to be silenced in order to adapt the controller. Singlechannel and multichannel schemes have been demostrated in the literature according to the number of reference sensors, error sensors, and secondary sources used. A single-channel active noise controller gives, in principle, attenuation of the undesired disturbance in the proximity of the point where the error microphone collects the error signal. To spatially extend the silenced region a multichannel approach can be applied at the expense of an increasing implementation complexity. In this respect, fast implementations and fast algorithms have been considered in [2], [3] for multichannel controllers modeled by means of FIR filters. Most of the studies in this field actually refer to linear models, while it is often recognized that nonlinearities can affect practical applications. In fact, nonlinear effects may be present according to the behavior of the noise source and the paths modeling the acoustic systems [4], [5]. Therefore, nonlinear modeling techniques may bring new insights and suggest new developments in the design of active noise controllers.
A multichannel active noise controller scheme is shown in Fig. 1. Here, microphones are used to collect input signals , generated by the noise source and fed to the adaptive controller. The propagation of the original noise up to the region to be silenced is described by suitable transfer characterizing the primary paths. The adapfunctions that are tive controller generates noises propagated through the secondary paths with transfer functions . The signals at the end of the primary and secondary and , destructively interfer. The error mipaths, namely , crophones thus collect errors that are used together with the input signals to adapt the controller. In the linear case, the controller is usually described by means FIR filters connecting any input to any output . Moreof over, preliminary and independent evaluations of the impulse are needed. responses In the nonlinear case, an efficient model is represented by truncated Volterra filters [6]. These filters share with the linear ones the property that their output is linear with respect to the filter coefficients. As a consequence, the adaptation algorithms derived for linear filters can be appropriately extended to the nonlinear case. The model we are proposing here for the controller is based on Volterra filters connecting any input to any output . In order to obtain an efficient realization and also to derive fast AP algorithms it is expedient to resort to the so-called diagonal representation [7]. This representation allows a truncated Volterra filter to be described by the “diagonal” entries of its kernels. In fact, if the th-order kernel is represented as a sampled hypercube of the same order, the diagonal representation implies a change of Cartesian coordinates to coordinates that are aligned along the diagonals of the hypercube. In this way, the Volterra filter can be represented in the form of a filter bank where each filter corresponds to a diagonal of the hypercube. This representation, which naturally leads to a multichannel realization, has been used in [5] to derive a filtered-X LMS algorithm for single-channel
T
Manuscript received May 12, 2004. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Vitor H. Nascimento. G. L. Sicuranza is with Dipartimento di Elettrotecnica Elettronica Informatica (DEEI), University of Trieste, 34127 Trieste, Italy (e-mail:
[email protected]). A. Carini is with Istituto di Scienze e Tecnologie dell’Informazione, Università degli Studi di Urbino, 61029 Urbino, Italy (e-mail:
[email protected]). Digital Object Identifier 10.1109/LSP.2004.836944
1070-9908/04$20.00 © 2004 IEEE
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IEEE SIGNAL PROCESSING LETTERS, VOL. 11, NO. 11, NOVEMBER 2004
is the vector formed with the where coefficients of the -th channel (4) and the corresponding input vector entries, is defined as
, again formed with
.. .
(5)
Let us now define two vectors of elements (6)
Fig. 1. Multichannel active noise control.
(7) controllers equipped with Volterra filters. A similar representation has been used in [8] to derive a filtered-X AP algorithm for single-channel quadratic controllers. In this letter, we propose a filtered-X AP algorithm for multichannel quadratic controllers. What is peculiar in this case is that two multichannel representations, one relative to the structure of the controller and one relative to the realization of the Volterra filters, are efficiently nested together. In other words, each filter connecting any input to any output is assumed to be a quadratic filter implemented in the form of a filter bank. Each channel of the filter bank contains an FIR filter formed with the coefficients of the diagonals of the quadratic kernel. Moreover, to reduce the computational complexity, the quadratic kernel is described by means of the so-called triangular representation ([6, p. 35]). The essential steps needed to derive the filtered-X AP algorithm for homogeneous quadratic filters are reported in the next section. This algorithm also includes linear filters as a particular case.
formed, respectively, with the vectors and related to the single channels of the filter bank. Thus the output of the homogeneous quadratic filter can be written as (8) While the filtered-X LMS algorithm minimizes the error at time , according to the stochastic gradient approximation, the filtered-X AP algorithm of order minimizes the coefficient variations within the constraint that the last a posteriori errors are set to zero. In what follows, we consider the cost function to be minimized, given by (9) where (10)
III. UPDATING ALGORITHM In the filter bank representation, any output from the multichannel quadratic controller of Fig. 1 can be written, according to the diagonal representation, as (1) is the number of channels actually used, with is the output from the generic channel of the FIR filter connecting the input to the output , given by the following equation
and the -th element is the signal measured by the -th error , at time , given microphone, by (11) where and
is the sound generated by the noise source
where , and
(2) where is the memory length of the quadratic filter and denotes the kernel coefficients. Using vector notation, (2) becomes (3)
(12) is the interfering sound generated through the secondary paths by the actuator sources. The symbol indicates the linear convolution operator. We first derive the LMS algorithm for this cost function. According to the stochastic gradient approximation, the components of the gradient vector of the error function are given by (13)
SICURANZA AND CARINI: FILTERED-X AFFINE PROJECTION ALGORITHM
is a
matrix defined as
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in [10]. Therefore, we formulate the final updating relationship derived by (16) as (14)
Therefore, the updating relationship is (17) (15) are the parameters that control both the conThe step sizes vergence rate and the stability of the algorithm. This rule is es[9]. sentially a sliding-window LMS algorithm with Even though it has better convergence properties compared to the standard memoryless LMS algorithm, it still suffers from convergence difficulties due to the correlations existing in the input vectors to the multichannel filter banks implementing the Volterra filters in the controller. In fact, since the entries in these vectors are given in form of products of input samples, correlations exist among them even when the input signals are white ([6, p. 253]). Even though the study of the convergence properties is beyond the scope of this short contribution, it is worth pointing out some relevant aspects of the problem. To improve the performance of the algorithm, it is convenient to employ a decorrelation matrix in the updating rule as done in the RLS and matrix AP algorithms. We thus introduce the following which can be seen as an estimate of the correlation matrix of the filtered-X signals, i.e., of the input signals filtered by the impulse responses of the secondary paths. Moreover, the input signals are formed with products of couples of input samples, obtained using the last input vectors. The adaptation rule then becomes
(16) which is the adaptation rule of the multichannel filtered-X AP algorithm [9]. As a matter of fact, for , i.e., in the single-channel case, it coincides with the filtered-X AP the algorithm derived in [8]. On the other hand, for becomes a simple normalization matrix factor and the updating rule coincides with the filtered-X (16) NLMS algorithm. It can also be noted that for tends to behave as the RLS algorithm. These considerations suggest a first explanation for the faster convergence behaviors experimentally verified for increasing values of . However, a further modification to (16) is still necessary. In fact, correlations also exist among the filtered-X signals, due to the similarities among the signals collected by the reference microphones and among the impulse responses of the secondary paths. This situation corresponds to that encountered in the problem of stereophonic acoustic echo cancellation [10]. As a may consequence, the correlation matrix be ill-conditioned and the adaptive behavior may be degraded. A first remedy to avoid possible numerical instabilities consists a diagonal matrix of adding to each matrix , where is a small positive constant. A second possibility is to resort to the use of a small leakage factor as suggested
The
computation
of
the inverse of matrices is required at each time . Even though this step is often a critical one, it should be noted that for filtered-X AP algorithms of low orders, i.e., , direct matrix inversion is still an affordable with task. Within these conditions, the complexity of the filtered-X per sample for each quadratic AP algorithm is filter in the controller. It can also be noted that while the filtered-X AP algorithm can be applied to quadratic filters characterized by full triangular representations by simply , using a smaller number of channels often setting permits one to obtain good adaptation performance with a remarkably reduced computational complexity. Moreover, in case of a nonhomogeneous quadratic filter, the linear term can and be considered as an additional channel. The vectors are modified by inserting on the top the coefficients of the linear filter and the samples of the input signal, respectively. As a consequence, the updating rule of (17) still applies. IV. SIMULATION RESULTS In this section, we show some simulation results that validate the proposed multichannel adaptation method. As shown in [4] for the single-channel case, a nonlinear controller is required if the secondary path is modeled as a nonminimum-phase FIR filter and the input signal is a nonlinear and deterministic process of chaotic rather than stochastic nature. Such a noise can be efficiently modeled by a second-order white and predictable nonlinear process as the logistic noise generated by the equation , with and assumed here equal to 0.9. In the multichannel case we are considering, and two outputs . the controller has one input are used. The nonlinear Four error microphones process has been normalized in order to have unit signal . The primary and secondary paths power are modeled with FIR and nonminimum-phase FIR filters, respectively, as shown in Table I. The system is identified using two second-order Volterra filters with linear and quadratic . The quadratic kernels contain parts of memory length corresponding to the principal only two channels diagonal and the adjacent one in the triangular representation. In the experiment the step sizes of the linear and quadratic parts and , respectively, the have been fixed equal to and the constant leakage factor has been set equal to has been set equal to . Fig. 2 plots the ensemble average of the mean attenuation at the error microphones for 50 runs of the simulation system using the updating rule of (17). The four curves refer to different values of the affine corresponds to the NLMS projection order . The order adaptation algorithm. For higher orders of affine projections the improvement in the convergence behavior is evident.
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IEEE SIGNAL PROCESSING LETTERS, VOL. 11, NO. 11, NOVEMBER 2004
TABLE I TRANSFER FUNCTIONS OF PRIMARY AND SECONDARY PATHS USED IN THE EXPERIMENT
with ten coefficients and the same experimental conditions above reported are used, the convergence rate is quite low and the final steady state errors are about ten times higher than for the nonlinear case. If we resort to FIR filters with 29 coefficients, i.e., the same total number of coefficients used in the nonlinear case, the diagrams shown in Fig. 3 are obtained. These results are derived by assuming the same experimental conditions as before, but with optimized step sizes set equal . The comparison of the diagrams in Fig. 2 to with those in Fig. 3 demonstrates that second-order Volterra filters offer better performance than FIR filters of equivalent complexity. V. CONCLUDING REMARKS
Fig. 2. Mean attenuation at the error microphones in a multichannel active noise controller using quadratic filters.
Fig. 3. Mean attenuation at the error microphones in a multichannel active noise controller using linear filters.
To further validate our results, a few tests have been performed using only linear filters in the controller. When the FIR filters
In this letter, a quadratic multichannel structure for ANC has been presented and a novel filtered-X AP algorithm has been described using a heuristic approach. It is worth noting, however, that the proposed adaptation algorithm can be considered as an approximate implementation of an exact AP-based algorithm requiring the simultaneous identification of the coefficients of all the quadratic filters in the controller. The approximation introduced here actually consists of using separate updating rules for the filter coefficients of each quadratic filter, without considering the coupling effects existing among them. From a practical point of view, the consequence of this matrices are assumptions is that involved in the proposed updating rules in place of whole matrices. The analysis and comparison of convergence behavior, performance, and complexity of the approximate and exact AP methods will be part of a future extended paper. Another part of our research plans will be the extension of the proposed algorithm to schemes including in the secondary paths nonlinearities modeling the distortions due to loudspeakers and power amplifiers. The structure and the filtered-X AP adaptation algorithm proand posed in this letter include as particular cases: for , the single-channel filtered-X LMS algorithm for linear controllers given in [2], and that for quadratic controllers given in [5], apart from the normalization factor used and , the single-channel filhere; for tered-X AP algorithm for quadratic controllers proposed in [8]; and or or , the multichannel filfor tered-X LMS algorithm for linear controllers given in [2], again and apart from the normalization factor used here; for or or , the multichannel filtered-X AP algorithm for quadratic controllers, which actually constitutes the novelty of the present contribution. As a final remark, it can be noted that the proposed updating rule can be suitably extended to higher-order Volterra filters by still resorting to the diagonal representation [7]. REFERENCES [1] P. A. Nelson and S. J. Elliott, Active Control of Sound. London, U.K.: Academic, 1995. [2] S. C. Douglas, “Fast implementations of the filtered-X LMS and LMS algorithms for multichannel active noise control,” IEEE Trans. Speech Audio Processing, vol. 7, pp. 454–465, July 1999.
SICURANZA AND CARINI: FILTERED-X AFFINE PROJECTION ALGORITHM
[3] M. Bouchard, “Multichannel affine and fast affine projection algorithms for active noise control and acoustic equalization systems,” IEEE Trans. Speech Audio Processing, vol. 11, pp. 54–60, Jan. 2003. [4] P. Strauch and B. Mulgrew, “Active control of nonlinear noise processes in a linear duct,” IEEE Trans. Signal Processing, vol. 46, no. 9, pp. 2404–2412, Sept. 1998. [5] L. Tan and J. Jiang, “Adaptive Volterra filters for active noise control of nonlinear processes,” IEEE Trans. Signal Processing, vol. 49, pp. 1667–1676, Aug. 2001. [6] V. J. Mathews and G. L. Sicuranza, Polynomial Signal Processing. New York: Wiley, 2000. [7] G. M. Raz and B. D. Van Veen, “Baseband Volterra filters for implementing carrier based nonlinearities,” IEEE Trans. Signal Processing, vol. 46, pp. 103–114, Jan. 1998.
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[8] A. Carini and G. L. Sicuranza, “Filtered-X affine projection algorithm for nonlinear active noise controllers,” in Proc. 2003 Sixth Workshop on Nonlinear Signal and Image Processing, Grado, Italy, June 8–11, 2003. [9] G.-O. Glentis, K. Berberidis, and S. Theodoridis, “Efficient least squares adaptive algorithms for FIR transversal filtering,” Signal Processing Mag., vol. 16, pp. 13–41, July 1999. [10] C. Breining, P. Dreiseitel, E. Hänsler, A. Mader, B. Nitsch, H. Puder, T. Schertler, G. Schmidt, and J. Tilp, “Acoustic echo control: An application of very-high-order adaptive filters,” Signal Processing Mag., vol. 16, pp. 42–69, July 1999.
