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

112 ANALYSIS AND TRANSMISSION OF SIGNALS
Figure 3.36
Limiting process
in derivation of
PSD.
g(I)
, ___
Hence, P g
, the power of g(t), is given by
P g
=
E g r = lim _!_ [1 00 I GrCf) l 2 df ]
lim
(3.78)
T➔oo T T➔ oo T -oo
As T increases, the duration of gy(t) increases, and its energy E g1
also increases proportionately.
This means that IGr(/) 1 2 also increases with T, and as T ➔ oo, 1 Gr(f) l 2 also approaches
oo. However, 1 Gr(f) l 2 must approach oo at the same rate as T because for a power signal, the
right-hand side of Eq. (3.78) must converge. This convergence permits us to interchange the
order of the limiting process and integration in Eq. (3.78), and we have
P g
= l oo lim
_00 T--+oo
IGr (f) l 2 df
T
(3.79)
We define the power spectral density (PSD) S g
(w) as
Consequent! y, *
S (f) _ 1 . 1 Gr(f) l 2
g - 1m
T--+oo T
(3.80)
P g
=
1_:
S rs lf)df
= 2 fo 00 S g (f)df
(3.81a)
(3.81b)
This result is parallel to the result [Eq. (3.69a)] for energy signals. The power is the area under
the PSD. Observe that the PSD is the time average of the ESD of gy (t) [Eq. (3.80)].
As is the case with ESD, the PSD is also a positive, real, and even function off. If g (t) is
a voltage signal, the units of PSD are volts squared per hertz.
* One should be cautious in using a unilateral expression such as P g = 2 Jo"'" S 8
(f) df when S g (f) contains an
impulse at the origin (a de component). The impulse part should not be multiplied by the factor 2.