Page 222 - Schaum's Outline of Theory and Problems of Signals and Systems

Chapter 5

Fourier Analysis of Continuous-Time

Signals and Systems

5.1 INTRODUCTION
In previous chapters we introduced the Laplace transform and the z-transform to
convert time-domain signals into the complex s-domain and z-domain representations that
are, for many purposes, more convenient to analyze and process. In addition, greater
insights into the nature and properties of many signals and systems are provided by these
transformations. In this chapter and the following one, we shall introduce other transfor-
mations known as Fourier series and Fourier transform which convert time-domain signals
into frequency-domain (or spectral) representations. In addition to providing spectral
representations of signals, Fourier analysis is also essential for describing certain types of
systems and their properties in the frequency domain. In this chapter we shall introduce
Fourier analysis in the context of continuous-time signals and systems.

5.2 FOURIER SERIES REPRESENTATION OF PERIODIC SIGNALS
A. Periodic Signals:

In Chap. 1 we defined a continuous-time signal x(t) to be periodic if there is a positive
nonzero value of T for which

x(t + T) =x(t) all t (5.1)
The fundamental period To of x(t) is the smallest positive value of T for which Eq. (5.1)
is satisfied, and l/To = fo is referred to as the fundamental frequency.
Two basic examples of periodic signals are the real sinusoidal signal

and the complex exponential signal
X(t) = e~%'
(5.3)
where oo = 2n-/To = 2n- fo is called the fundamental angular frequency.

B. Complex Exponential Fourier Series Representation:

The complex exponential Fourier series representation of a periodic signal x(t) with
fundamental period To is given by