Page 60 - Fundamentals of Communications Systems

2.12 Chapter Two

are J/Hz. Because of this characteristic, two important properties, of the energy
spectral density are

G x (f ) ≥ 0 ∀ f (Energy in a signal cannot be negative valued)

∞ ∞
2
E x = |x(t)| dt = G x (f )df (2.49)
−∞ −∞
This last result is a restatement of Rayleigh’s energy theorem and the analogy
to Parseval’s theorem should be noted.

EXAMPLE 2.20
Example 2.1(cont.). For the Fourier transform pair of

1 0 ≤ t ≤ T p
x(t) = X(f ) = T p exp[− j π fT p ]sinc( fT p ) (2.50)
0 elsewhere
the energy spectrum is
2
G x (f ) = T (sinc( fT p )) 2 (2.51)
p
The energy spectrum of the pulse is shown in Figure 2.3 (a).

EXAMPLE 2.21
Example 2.2(cont.). For the Fourier transform pair of

1 | f |≤ W
x(t) = 2Wsinc(2Wt) X(f ) = (2.52)
0 elsewhere
the energy spectrum is

1 | f |≤ W
G x (f ) = (2.53)
0 elsewhere

EXAMPLE 2.22
Example 2.3(cont.). The energy spectrum of the computer-generated voice signal is
shown in Figure 2.3 (b). The two characteristics that stand out in examining this spec-
trum are that the energy in the signal starts to significantly drop off after about 2.5 kHz
and that the DC content of this voice signal is small.