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

78 ANALYSIS AND TRANSMISSION OF SIGNALS
The duality property states that if the Fourier transform of g(t) is G(f) then the Fourier
transform of G(t), withf replaced by t, is the g(-f) which is the original time domain signal
with t replaced by -f .
Proof From Eq. (3.9b),
Hence,
g (t) = I: G(x)J 2rrxt dx
Changing t tof yields Eq. (3.26).
g(-t) = I: G(x)e-j 2 :rc xt dx
■
Example 3.8
In this example we shall apply the duality property [Eq. (3.26)] to the pair in Fig. 3.15a.
Figure 3. 15 g(t)
Duality property
of the Fourier
transform.
__ 1_
----
(a)
g(t)
(b)
t--
From Eq. (3. 19) we have
TI ( f) r sine (n:f r)
n ( f )
'-,.-'
g (t)
G(f)
(3.27a)
(3.27b)
Also G(t) is the same as G(f) withf replaced by t, and g(-f) is the same as g(t) with t
replaced by -f . Therefore, the duality property (3.26) yields
n(-f) = n(f)
G(t)
'-.-'
g (-f)
(3.28a)