A Signal Theoretic Introduction to Random Processes

3BACKGROUND: SIGNAL THEORY

3.1 INTRODUCTION

This chapter details relevant signal theory and associated results that are important for subsequent chapters. Material on orthogonality, Dirichlet theory, and the Dirac Delta provide the basis for a discussion of Fourier theory and signal power. This is followed by a discussion of the power spectral density, and autocorrelation, functions. Finally, an introduction to the Cramer representation of a signal, and the Cramer transform, is given.

3.2 SIGNAL ORTHOGONALITY

Orthogonality and signal decomposition underpin a significant level of modern analysis, particularly, in the communications field. The following definitions are fundamental:

Definition: Orthogonality

Two Lebesgue integrable signals and are orthogonal on an interval [α, β] if

(3.1)

Definition: Orthogonal Set

A set of functions is an orthogonal set for [α, β] if

(3.2)

Definition: Orthonormal Set

A set of functions is an orthonormal set for [α, β] if

(3.3)

Definition: Orthogonal/Orthonormal ...