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

104 ANALYSIS AND TRANSMISSION OF SIGNALS
Example 3. 16 Verify Parseval's theorem for the signal g(t) = e-at u(t) (a > 0).
We have
E g = 1 00 g 2 (t) dt = f 00 e - 2a 1 dt = ]_
-oo lo 2a
We now determine E g
from the signal spectrum G(j) given by
(3.66)
1
G(j) = j2nf + a
and from Eq. (3.65),
100 1
00
1 1 2nf 1
00
E g
= I G(f)l 2 df =
2 2 df = - tan- 1 -
-oo -oo (2nf) + a 2na a _ 00
1
2a
which verifies Parseval's theorem.
3.7.2 Energy Spectral Density (ESD)
Equation (3.65) can be interpreted to mean that the energy of a signal g(t) is the result of
energies contributed by all the spectral components of the signal g(t). The contribution of a
spectral component of frequency f is proportional to I G(j) 1 2 . To elaborate this further, consider
a signal g (t) applied at the input of an ideal bandpass filter, whose transfer function H(j) is
shown in Fig. 3.32a. This filter suppresses all frequencies except a narrow band !).j (!).f -+ 0)
centered at angular frequency wo (Fig. 3.32b). If the filter output is y(t), then its Fourier
transform Y (j) = G(j)H (j), and E y
, the energy of the output y(t), is
(3.67)
Because H (j) = 1 over the passband !).j , and zero everywhere else, the integral on the
right-hand side is the sum of the two shaded areas in Fig. 3.32b, and we have (for !).j -+ 0)
E y
= 2 I G(Jo) 1 2 df
Thus, 21G(J) 1 2 df is the energy contributed by the spectral components within the two narrow
bands, each of width !).j Hz, centered at ±Jo. Therefore, we can interpret I G(f) 1 2 as the energy
per unit bandwidth (in hertz) of the spectral components of g(t) centered at frequency f.
In other words, I G(J)l 2 is the energy spectral density (per unit bandwidth in hertz) of g(t).
Actually, since both the positive- and the negative-frequency components combine to form the
components in the band !).j , the energy contributed per unit bandwidth is 21 G(j) 1 2 . However,
for the sake of convenience we consider the positive- and negative-frequency components to
be independent. The energy spectral density (ESD) W g
(t) is thus defined as
(3.68)