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

3.3 Some Properties of the Fourier Transform 89
then (time differentiation)*
and (time integration)
dg (t)
dt
j2nfG(f)
(3.47)
G(f) I
11
g(r)dr -. - + - G(0)8(f)
-oo J27rf 2
(3.48)
Proof Differentiation of both sides of Eq. (3.9b) yields
dg (t)
dt
= l oo j2nfG(f)ei 2 n:ft df
_ 00
This shows that
Repeated application of this property yields
dg (t) j2nfG(f)
dt
d n g(t) (j2nftG(f) (3.49)
dt n
The time integration property [Eq. (3.48)] already has been proved in Example 3.12.
Example 3. l 3 Use the time differentiation property to find the Fourier transform of the triangular pulse ,0,.(t /r)
shown in Fig. 3.23a.
■
Figure 3.23
Using the time
differentiation
property to
finding the
Fourier transform
of a
piecewise-linear
signal.
-'t
- 2 -
0
2
't
dg
dt
't
2
t--
(a)
-'t
- 2 -
-2
't
0
't
2
t-'>
(b)
(d) f -
2
't
d 2 g
dt 2
2
't
-'t
--
2
0
't
2
t---'>
(c)
-4
't
* Valid only if the transform of dg(t)/dt exists.