1. Mathematical Background

This chapter provides the mathematical background necessary for studying spherical array processing. Spherical arrays typically sample functions on a sphere (e.g. sound pressure); therefore, this chapter begins by presenting the spherical coordinate system, as well as some examples of functions on the sphere. Spherical harmonics are a central theme of this book as they form a basis for representing functions on the sphere. Therefore, spherical harmonics are first defined and illustrated, and then an introduction to the spherical Fourier transform and a description of functions on the sphere in Hilbert space follows. The chapter concludes with a presentation of the topics of rotation, convolution, and correlation defined for functions on the sphere.