B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

72 ANALYSIS AND TRANSMISSION OF SIGNALS
Example 3 .2 Find the Fourier transform of g(t)
IT(t/r) (Fig. 3.9a).
Figure 3.9
Rectangular
pulse and its
Fourier spectrum.
-----1-- g(t)
1:
2
(a)
(b)
We have
G(f) = 1_: rr ( ) e - J 2 nftdt
Since IT (t / r) = 1 for ltl < r /2, and since it is zero for ltl > r /2,
Therefore,
r/2
G(f) = l e-J 2 rrft dt
-r/2
- _ _
l _ e -Jrrfr _ Jnfr
2 sin (n/
-
r)
- j2nf
( ) - 2nf
sin (nf r)
= r --- = r sine (n/ r)
(n/ r)
IT ( ) r sine ( r ) = r sine (n/ r) (3.19)
Recall that sine (x) = 0 when x = ±nn. Hence, sine (wr /2) = 0 when wr /2 = ±nn ;
that is, when/ = ±n/r (n = 1, 2, 3, ... ), as shown in Fig. 3.9b. Observe that in this case
G(f) happens to be real. Hence, we may convey the spectral information by a single plot
of G(f) shown in Fig. 3.9b.
Example 3 .3
Find the Fourier transform of the unit impulse signal o(t).
We use the sampling property of the impulse function [Eq. (2.11)] to obtain
F[o(t)] = 1_: 8(t)e-J 2 rrft dt = e-J2rrf-O
= l
(3.20a)
or
o(t) 1
(3.20b)