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

80 ANALYSIS AND TRANSMISSION OF SIGNALS
Figure 3.16
Scaling property
of the Fourier
transform.
1
g(t)
-----
_ ..1..
2
( a )
2 f --
,:
1
g(t)
-
-T
0
t ---
Reciprocity of Signal Duration and Its Bandwidth
The time-scaling property implies that if g(t) is wider, its spectrum is narrower, and vice
versa. Doubling the signal duration halves its bandwidth, and vice versa. This suggests that the
bandwidth of a signal is inversely proportional to the signal duration or width (in seconds). We
have already verified this fact for the rectangular pulse, where we found that the bandwidth
of a gate pulse of width r seconds is 1 / r Hz. More discussion of this interesting topic can be
found in the literature. 2
Example 3 .9
Show that
g(-t) {=> G(-J) (3.30)
Use this result and the fact that e-at u(t) {=> 1/(a + j2nf), to find the Fourier transforms of
e at u(-t) and e-altl .
Equation (3.30) follows from Eq. (3.29) by letting a = -1. Application of Eq. (3.30) to
pair 1 of Table 3.1 yields
Also
1
e a1 u(-t) {=> --­
a -j2nf
Therefore,
-altl
1 1 2a
e {=> --- + --- = -----
a + j2nf a - j2nf a 2 + (2nf) 2 (3.31)