Signal Processing and Integrated Circuits

Chapter 2

Analog Continuous-time Signals and Systems

2.1 Introduction

In this chapter the fundamental concepts and mathematical tools of analog signal and system analyses are reviewed. This review can be regarded as a comprehensive summary of the fundamentals of analog signals and systems. It is the distillation of courses on these topics which are usually covered at the junior to senior undergraduate levels. Therefore, the discussion is quite compact, and the material can be used as an easy reference for later chapters and as a short revision course.

2.2 The Fourier Series in Signal Analysis and Function Approximation

2.2.1 Definitions

A signal f(x) defined over an interval [−l, l] and satisfying conditions of considerable generality, can be represented as an infinite series of pure sine and cosine signals as

2.1

where the coefficients are given by

2.2a

2.2b

A simplified notation results by letting

2.3

so that the signal is defined over the range [−π, π] and the series becomes

2.4

with

2.5a

2.5b

or the alternative versions in

2.6

or

2.7

with

2.8

2.9

and

2.10

If the signal ...