Acoustic Array Systems

3

Theoretical Preliminaries of Array Signal Processing

In this chapter, a brief review is given of signal processing aspects required in the discussions of microphone arrays. Specifically, we shall review the basics of linear algebra, digital signal processing, array signal processing, and optimization algorithms. The review is not intended to be exhaustive. Only materials relevant to the subsequent development of microphone array techniques will be covered.

3.1 Linear Algebra Basics

This section gives a brief review of linear algebra, including three key elements: the linear system of equations, fundamental vector subspaces, and eigenvalue problems. The presentation will be exemplified in the context of acoustic radiation problems. Many acoustic radiation problems can be formulated into linear systems of equations with appropriate discretization schemes. Or, in matrix form,

(3.1)

where vectors p and q are the pressure output and the source input, respectively, which are related by the propagation matrix which is assumed to be square for simplicity. The associated inverse problems are based on the preceding forward problem, where one attempts to retrieve q, given the information of G and p. It can be shown for some perturbations due to measurement noise, numerical error, and so ...