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

108 ANALYSIS AND TRANSMISSION OF SIGNALS
the ESD for this pulse is
\Jl g
(f) = I G(f)l 2 = T 2 sinc 2 (nfI')
This ESD is shown in Fig. 3.34b as a function of wT as well asfI', whereJ is the frequency
in hertz. The energy E B within the band from 0 to B Hz is given by
B
E B = 1 T 2 sinc 2 (nfI') dJ
Setting 2nfI' = x in this integral so that dJ = dx /(2n T), we obtain
-B
T ( 2 nBT
E B
=
; J o sinc 2 (: 2 ) dx
x
Also because E g
= T, we have
E l
2nBT
X
_!!_ = - 1 sinc 2 ( - ) dx
E g
n O 2
The integral on the right-hand side is numerically computed, and the plot of E B /E g vs.
BT is shown in Fig. 3.34c. Note that 90.28% of the total energy of the pulse g (t) is
contained within the band B = l/T Hz. Therefore, by the 90% criterion, the bandwidth
of a rectangular pulse of width T seconds is 1 /T Hz.
3.7.4 Energy of Modulated Signals
We have seen that modulation shifts the signal spectrum G(f) to the left and right by Jo. We
now show that a similar thing happens to the ESD of the modulated signal.
Let g(t) be a baseband signal band-limited to B Hz. The amplitude-modulated signal
<p(t) is
<p(t) = g (t) cos 2nJot
and the spectrum (Fourier transform) of <p(t) is
1
<P(f) = 2
[G(f +Jo) + G(f - Jo)]
The ESD of the modulated signal <p (t) is l <fi (f) 1 2 , that is,
1 2
\Jl rp (f) = 4
1 G(f +Jo) + G(f - Jo) I
If Jo :::: B, then G(f + Jo) and G(f - Jo) are nonoverlapping (see Fig. 3.35), and
\Jl rp (f) = i [IG(f + Jo) 1 2 + IG(f -Jo) l 2 ]
1 1
= 4
\Ji g (f +Jo) + 4 \Jl g (f - Jo) (3.70)