Jacob Benesty - Array Beamforming with Linear Difference Equations (Springer Topics in Signal Processing, 20)

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Voice communication and human-machine speech interface systems have to face the problem of sound acquisition in complex acoustic environments, where noise, reverberation, and competing sources may coexist. To deal with this challenging issue, microphone arrays, which consist of multiple sensors arranged into a certain one-, two-, or three-dimensional topology, have to be used. The central component of an array system is signal processing, which operates on the signals observed by the array of sensors to restore a signal of interest while suppressing the unwanted signal components. Ideally, one would wish to simply improve the signal-to-noise ratio (SNR) without adding any distortion to either the signal of interest or the unwanted signals, thereby preserving the fidelity of both the signal and ambience. However, this seemingly easy objective turns out to be extremely difficult to achieve. With all the methods and techniques developed in the literature, signal enhancement is generally achieved by adding some degree of distortion to either the signal of interest or the ambience or even both at the same time. There is one exception: differential beamforming, whose root can be traced back to the principle of directional microphones invented in the 1940s; this principle was then introduced into the regime of microphone arrays in the 1990s to form differential microphone arrays (DMAs), which are designed to measure the spatial derivatives of the sound pressure field. Basically, a first-order DMA consists of two closely spaced omnidirectional pressure microphones, and its output is the subtraction of one sensors output from that of the other. The two omnidirectional microphones measure the sound pressure field, each from its own viewpoint. Since they are closely spaced, the difference between their outputs approximates very well the spatial derivative of the sound pressure field along the axis that connects the two sensors. So, this simple subtraction operation yields a differential beamformer that is responsive to the first-order differential of the acoustic pressure field. Similarly, a general Nth-order DMA can be formed by subtractively combining the outputs of two DMAs of order N1. This simple way of constructing microphone arrays has laid out the foundation for differential beamforming. The resulting beamformers feature a prominent property, that is, their beampatterns are frequency independent by theory, and have a great potential for use in high fidelity sound acquisition and enhancement as they do not add spectral distortion to the source signals regardless of the source incidence angle. A great deal of effort has been devoted to the study and design of differential beamformers over the last two decades. Many methods have been developed, most of which attempt to design the beamforming filter from the beampattern perspective, that is, the parameters of the beamformers are determined in such a way that the resulting beampattern matches well a target frequency-independent beampattern. Interestingly, few works have attempted to address the problem from the difference equations perspective, though this view is the foundation for differential beamforming. This book is intended to study the link between differential beamforming and differential equations, which is missed in the literature. This link enables to study the fundamental theory and methods of beamforming from a rather different perspective, leading to new insights into the problem and new methods to solve the problem. The book first presents a brief overview of the problems and methods for beamforming and some performance measures popularly used either to evaluate beamformers or to derive optimal beamformers. Then, first-order, second-order, and general high-order linear difference equations are discussed, based on which we show how to formulate the beamforming problem and derive different beamforming methods, including fixed and adaptive ones. Furthermore, we show how to apply the theory of difference equations to the general problem of speech enhancement and deduce a number of noise reduction filters, including the maximum SNR filter, the Wiener filter, the MVDR filter, etc. Also covered in the book are the difference equations and differential beamforming from the spectral graph perspective.