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

3.3 Some Properties of the Fourier Transform 75
Example 3 .7
Find the Fourier transform of the sign function sgn (t) (pronounced signum t), shown in
Fig. 3.13. Its value is + 1 or - 1, depending on whether t is positive or negative:
sgn (t) = {
- 1
t > 0
t=O
t <O
(3.24)
We cannot use integration to find the transform of sgn (t) directly. This is because sgn (t)
violates the Dirichlet condition [see E.g. (3.14) and the associated footnote]. Specifically,
sgn (t) is not absolutely integrable. However, the transform can be obtained by considering
sgn t as a sum of two exponentials, as shown in Fig. 3.13, in the limit as a ---+ 0:
Figure 3. 13
Sign function.
I''" (,)
I ................:----------------------------------- ,-"'u(, I
-e a1 u(-t ) ·······················································
1-
Therefore,
I
.F[sgn(t)] = lim {.F[e- a1 u(t)] - .F[e a1 u(-t)] }
a➔O
= (--1- lim - __ 1_)
(see pairs 1 and 2 in Table 3. 1)
a-->0 a+ j2nf a - j2nf
. -j4nf 1
= ( ) = (3.25)
l a 2 + 4n 2 f2 jnf
3.3 SOME PROPERTIES OF THE
FOURIER TRANSFORM
We now study some of the important properties of the Fourier transform and their implications
as well as their applications. Before embarking on this study, it is important to point out a
pervasive aspect of the Fourier transform-the time-frequency duality.