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-compu- tational implementations fast affine projection algorithms can produce a good tradeoff between convergence speed and compu- tational 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, multi- channel 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 com- putational 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 A CTIVE 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 re- sponse 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 loca- tions. Figs. 1 and 2 show block-diagrams of monochannel im- Manuscript received October 15, 2001; revised August 14, 2002. 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 Engi- neering, University of Ottawa, Ottawa, ON, K1N 6N5 Canada (e-mail: bouchard@site.uottawa.ca). Digital Object Identifier 10.1109/TSA.2002.805642 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 coef- ficients 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- ” struc- ture [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 algo- rithms can produce a good tradeoff between convergence speed and computational complexity. Although these algorithms typ- ically 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 al- gorithms, without the high increase of the computational load or the instability often found in recursive-least-squares algo- rithms, especially for multichannel systems [5]–[7]. An adapta- tion 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 com- pute auxiliary coefficients instead of the “normal” time-domain coefficients usually computed by adaptive FIR filtering algo- rithms, and the outputs from those “normal” coefficients are re- quired for active noise control or for acoustic equalization. In Section II of this paper, the previous work is modified and ex- tended to introduce multichannel affine and fast affine projec- tion algorithms for active noise control or acoustic equalization. Multichannel fast affine projection algorithms have been previ- ously 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 can- cellation is an identification problem (with of course its own ad- ditional constraints such as double-talk, etc.). This leads to dif- ferent 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 algo- rithms. In Section III, the computational complexity of the new 1063-6676/03$17.00 © 2003 IEEE