Signals and Systems - Electrical Engineering

580 C H A P T E R 10: Fourier Analysis of Discrete-Time Signals and Systems

FIGURE 10.2

DTFT of (a) a sampled

signal, and (b) the

magnitude of DTFT and

(c) the phase of DTFT as

functions of(rad/sec).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.2

0.4

0.6

0.8

1

nTs (sec)

x

(nT

)s

− 300 − 200 − 100 0 100 200 300

0

20

|X

(e

jΩ

Ts

)|

− 300 − 200 − 100 0 100 200 300

− 1

0

1

<X

(e

jΩ

Ts

)

Ω (rad /sec)

(a)

(b)

(c)

The results are shown in Figure 10.2. Given that the signal is very smooth, most of the frequency

components have low frequencies.

10.2.4 Time and Frequency Supports

The Fourier representation of a discrete-time signal gives a complementary characterization to its

time representation, just like in the continuous-time case. The following examples illustrate the

complementary nature of the DTFT of discrete-time signals.

Just like with analog signals, the frequency support of the DTFT of a discrete-time signal is inversely

proportional to the time support of the signal.

nExample 10.3

Consider a discrete pulse

p[n]=u[n]−u[n−N]

Find its DTFTP(ejω)and discuss the relation between its frequency support and the time support

ofp[n] whenN=1 and whenN→∞.