Nonlinear multichannel active noise control 7 Giovanni L. Sicuranza and Alberto Carini 7.1. Introduction The principle of active noise control (ANC) has been well known for many years, as shown by the seminal contributions in [28, 33]. However, methods for ANC have been intensively studied only in the last three decades as a consequence of the developments in the area of digital technologies. Such studies have already pro- vided a good comprehension of the theory and consequently useful applications in vibration and acoustic noise control tasks. Generally speaking, active control of vibrations and acoustic noises are based on the same principle of the destructive interference in a given location between a disturbance source and an appropriately generated signal with the same amplitude as the disturbance source but with an opposite phase. The main difference in the two situations is due to the nature of the actuators used to generate the interfering signal, which are mainly mechanical in the case of vibration control while they are electroacoustical devices, as loud- speakers, in the case of acoustic noise control. The vibration control is also related to the problem of attenuating the cor- responding sound generated by the vibrations in the auditory spectrum. Studies in this field are currently developed with reference to environmental and legal re- strictions concerning people’s safety and health. Among many examples, we may refer to the application of actively controlling the inputs to vibrating structures, such as airplane cockpits, to minimize the sound radiation. On the other hand, in most cases the acoustic noise can be directly reduced by introducing a secondary noise source, such as a loudspeaker, which produces a sound field that destructively interferes with the original noise in a desired location where the so-called error microphone is placed. The secondary noise source is driven by the active noise controller that is adapted according to the error signal and the input signal collected by a reference microphone in proximity of the noise source. Successful implementations of this approach can be found in systems such as air conditioning ducts, to attenuate the low frequency noise due to the fans, or in transport systems, to reduce the noise generated by the engines inside propeller aircrafts, vehicles, and helicopters. 170 Nonlinear multichannel active noise control Physical limitations generally restrict the frequency range of ANC systems below a few hundred hertz. The main limitation is related to the wavelength of the acoustic waves in connection to the extension of the silenced area. At higher frequencies, it is still possible to cancel the sound in a limited zone around, for example, a listener’s ear, and suitable systems have been already successfully im- plemented for active headsets. To attenuate disturbance frequencies above 1 kH, passive systems, based on the absorption and reflection properties of materials, are widely used and give good performances. The initial activities on vibration and noise control originated in the field of control engineering [24, 32, 53], while in recent years the advantages offered by the signal processing methodologies have been successfully exploited [18, 19, 27]. The latter approach offers great opportunities for real-time implementations as a consequence of recent advances in electroacoustic transducers, digital signal pro- cessors, and adaptation algorithms. Most of the studies presented in the literature refer to linear models and lin- ear adaptive controllers represent the state of the art in the field. However, lin- ear controllers can hardly provide any further improvements in terms of control performance. On the other hand, nonlinear modeling techniques may bring new insights and suggest new developments. In fact, it is often recognized that nonlin- earities may affect actual applications. Nonlinear effects arise from the behavior of the noise source which rather than a stochastic process may be described as a non- linear deterministic process, sometimes of chaotic nature [30, 45]. Moreover, the paths modeling the acoustic system may exhibit a nonlinear behavior, thus further motivating the use of a nonlinear controller [26]. The simplest and most frequently used ANC systems exploit a single-channel configuration where the primary path P consists of the acoustic response from the noise source to an error microphone located at the canceling point. The noise source is sensed by a single reference microphone and an interfering signal is pro- duced by a single loudspeaker. A single-channel active noise controller gives, in principle, attenuation of the undesired disturbance only in the proximity of the error microphone. Even though the silenced region may be relatively large at low frequencies, that is, in a region that is appropriately small with respect to the wave- length of the corresponding acoustical waves, to spatially extend the silenced re- gion a multichannel approach may be preferred. In this case, sets of reference sen- sors, actuators, and error microphones are used increasing the complexity of the ANC system. In this chapter, the main aspects of the active control of acoustic noises are reviewed from a signal processing point of view with particular attention to the nonlinear multichannel framework. While different nonlinear models for the con- trollers have been considered recently, polynomial or Volterra filters [29]havebeen shown to be particularly useful. Thus in this chapter we will refer to a class of non- linear controllers that contains all the filters that are characterized by a linear rela- tionship of the output with respect to the filter coefficients. Such a class includes in particular truncated Volterra filters [29] as well as functional-link artificial neural networks [35]. G. L. Sicuranza and A. Carini 171 The main problems arising in the nonlinear controllers are the following. (i) The increase in the computational load caused by the complexity of both the multichannel approach and the representation of nonlinearities. (ii) The correlations existing among signals in multichannel schemes due to the similarities in the acoustic paths and their interaction. (iii) The correlations existing, even when the input signals are white, between the elements of the input vectors to the Volterra filters which are given by products of samples [29, page 253]. All these problems have an impact on the efficiency and the accuracy of the adap- tation algorithms used for the controllers. As a consequence, the main aspect to study is the derivation of fast and reliable adaptation algorithms for nonlinear multichannel controllers. It is worth noting that this research area represents a challenge for modern nonlinear signal processing. In this chapter, a description of the ANC scenario is given first with partic- ular reference to the so-called filtered-X adaptation algorithms for linear single- channel and multichannel schemes. Then, the nonlinear effects that may influence the behavior of an ANC system are reviewed. It is also shown how the filtered-X updating algorithms previously described can be profitably applied to the class of nonlinear active noise controllers whose filters are characterized by a linear rela- tionship of the output with respect to the filter coefficients. A few experimental results for nonlinear multichannel active noise controllers belonging to this class are presented and commented. Finally, current work and future research lines are exposed. 7.2. The active noise control scenario Actual ANC systems are essentially mixed analog-digital systems that require A/D and D/A converters, power amplifiers, loudspeakers, and transducers. In this sec- tion, the digital core of single-channel and multichannel ANC systems is briefly described, together with the corresponding algorithms used for adapting the con- trollers in the linear case. The feedforward adaptation algorithms are considered in more detail since they can be profitably exploited in nonlinear situations. The derivations that lead to the filtered-X least mean square (LMS) and the filtered- Xaffine projection (AP) algorithms for single-channel and multichannel schemes are summarized. All the derivations make use of a vector-matrix notation. Due to space limitations, in some cases only the rational of the adaptation algorithms is reported while the detailed derivations can be found in the specific papers referred to. 7.2.1. Single-channel and multichannel schemes for active noise control A typical single-channel acoustic noise control scheme is shown in Figure 7.1. The primary path P consists of the acoustic response from the noise source to the error microphone located at the canceling point. The noise source is sensed by a reference microphone that collects the samples of the noise source ns(n)and 172 Nonlinear multichannel active noise control Primary path Error microphone Noise source e(n) ns(n) u(n) Reference microphone Secondary path Adaptive controller Figure 7.1. Principle of single-channel active noise control. ns(n) P nm(n) dp(n) e(n) ++ F ds(n) x(n) Adaptive D + controller S ( ) H u n Adaptive algorithm Figure 7.2. Block diagram of a single-channel active noise control system. feeds them as input to the adaptive controller. The controller is adapted using ns(n) and the error signal e(n) sent back by the error microphone. The signal u(n), generated by the controller through a loudspeaker and traveling through the secondary path S, gives a signal which destructively interferes with the undesired noise at the error microphone location. A block diagram depicting this situation is shown in Figure 7.2. In this scheme, P and S represent the transfer functions of the primary and secondary paths, respectively, and dp(n)andds(n) are the signals that destructively interfere at the location where the error microphone is placed. D is the transfer function of the detector path, that is, the path between the noise source and the reference microphone, while F is a transfer function modeling the acoustical feedback between the loudspeaker and the reference microphone.