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

44
SIGNALS AND SIGNAL SPACE
Example 2 .4 Find the exponential Fourier series for the periodic square wave w(t) shown in Fig. 2.16.
Figure 2. 16
A square pulse
periodic signal.
w(t)
-37t
-27t
-7t
-zi; 2
7t
2
7t 21t 37t 1
(a)
where
w(t) = L D n
J n 2nfot
n=-oo
Do = _!__ f w(t) dt = !
To lr 0 2
D n = _!__ { w(t)e-Jn2nfiit dt,
To lr 0
1 1
To/4
= _ e-jn2nfot dt
To -To/4
= l
[e-jn2nfoTo/4 _ J n 2nfoTo/4 ]
-jn2nfoTo
2
= sin ( n2 nfoTo ) = _I_ sin (nn)
n2nfoTo 4 nn 2
In this case D n is real. Consequently, we can do without the phase or angle plot if we plot
D n vs.f instead of the amplitude spectrum (ID n l vs.!) as shown in Fig. 2.17.
Figure 2. 17
Exponential
Fourier spectrum
of the square
pulse periodic
signal.
0.5
1
lt
-7 .,.. . .-- - - ... .. . -3
-9 ·--... .. ·· -5
-1\
31t
3 .. . . .. . - ···-... _ 7
5 ··- __ ./ 9