Signals and Systems - Electrical Engineering

350 CHAPTER 5: Frequency Analysis: The Fourier Transform

The Fourier transform of 2u(t− 1 )is− 2 je−j/+ 2 πδ()so that

Y()=

− 2 jsin()

^2

+j

2 e−j



−j

2 e−j



+ 2 πδ()

=

− 2 jsin()

^2

+ 2 πδ()

To use the integration property we first needX(), which is

X()=

2 sin()



and according to the property,

Y()=

X()

j

+πX( 0 )δ()

=

− 2 jsin()

^2

+ 2 πδ()

sinceX( 0 )=2 (using L’Hopital’s rule). As expected, the two results coincide.ˆ n

5.9 What Have We Accomplished? What Is Next?

You should by now have a very good understanding of the frequency representation of signals and

systems. In this chapter, we have unified the treatment of periodic and nonperiodic signals and their

spectra, and consolidated the concept of frequency response of a linear time-invariant system.

Basic properties of the Fourier transform and important Fourier pairs are given in Tables 5.1 and

5.2. Two significant applications are in filtering and communications. We introduced the basics of

filtering in this chapter and will expand on them in Chapter 6. The fundamentals of modulation

provided in this chapter will be illustrated in Chapter 6 where we will consider their application in

communications.

Certainly the next step is to find out where the Laplace and the Fourier analyses apply, which will be

done in Chapter 6. After that, we will go into discrete-time signals and systems. We will show that

sampling, quantization, and coding bridge the continuous-time and the digital signal processing, and

that transformations similar to the Laplace and the Fourier transforms will permit us to do processing

of discrete–time signals and systems.

Problems............................................................................................

5.1. Fourier series versus Fourier transform—MATLAB

The connection between the Fourier series and the Fourier transform can be seen by considering what

happens when the period of a periodic signal increases to a point at which the periodicity is not clear

as only one period is seen. Consider a train of pulsesx(t)with a periodT 0 = 2 , and a period ofx(t)is

x 1 (t)=u(t+0.5)−u(t−0.5). LetT 0 be increased to4, 8, and 16.