8.1 INTRODUCTION

In this chapter, we study some mathematical principles that will be very useful to us in order to understand the next two chapters. First, we define the meaning of orthogonal functions, which are used to represent deterministic signals in a series expansion known as the generalized Fourier series. We use the Gram-Schmidt procedure to transform a set of M linear dependent or independent functions into a set of K, K ? M, orthogonal functions. We also discuss geometric representation of signals in the signal space, which can be used to determine decision regions in M-ary detection of signals in noise, as be will be seen later. Then, integral equations are studied. The relation between integral equations and their corresponding linear differential equations are established through Green's function or the kernel. In solving integral equations, we present an approach by which we obtain the eigenfunctions and eigenvalues from the linear differential equation. In Section 8.4, we discuss the series representation of random processes by orthogonal functions known as Karhunen-Lo ve expansion. Specifically, we consider processes with rational power spectral densities, the Wiener process, and the white Gaussian noise process.

8.2 ORTHOGONAL FUNCTIONS

From vector analysis, we say that two vectors X and Y are orthogonal (perpendicular) if their dot or inner product is zero. That is,

Let X and Y be two vectors in , such that X = [ x ₁ x ₂ x _K] ^T and