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

3.7 Signal Energy and Energy Spectral Density 109
Figure 3.35
Energy spectral
densities of
modulating and
modulated
signals.
0
l (a)
f-
-Jo
K/4
r-,n
0
(b)
The ESDs of both g(t) and the modulated signal <p(t) are shown in Fig. 3.35. It is clear that
modulation shifts the ESD of g(t) by ±Jo. Observe that the area under W rp
(f) is half the area
under W g
(f). Because the energy of a signal is proportional to the area under its ESD, it follows
that the energy of <p(t) is half the energy of g(t), that is,
Jo ?:. B (3.71)
It may seem surprising that a signal <p(t), which appears so energetic in comparison to g(t),
should have only half the energy of g(t). Appearances are deceiving, as usual. The energy of
a signal is proportional to the square of its amplitude, and higher amplitudes contribute more
energy. Signal g(t) remains at higher amplitude levels most of the time. On the other hand,
<p(t), because of the factor cos 2nJot, dips to zero amplitude levels many times, which reduces
its energy.
3.7.5 Time Autocorrelation Function and
the Energy Spectral Density
In Chapter 2, we showed that a good measure of comparing two signals g(t) and z(t) is the
cross-correlation function 1/r g
z(r) defined in Eq. (2.46). We also defined the correlation of a
signal g(t) with itself [the autocorrelation function 1/r g
( r)] in Eq. (2.47). For a real signal g(t),
the autocorrelation function 1/r g ( r) is given by*
1/r g
(r) = 1_: g(t)g(t + r:)dt
(3.72a)
* For a complex signal g(t), we define
1/fg (T) = L: g(t)g* (t -,)dt = L: g* (t)g(t + ,)dt