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

3.8 Signal Power and Power Spectral Density 111
ESD of the Input and the Output
If x(t) and y(t) are the input and the corresponding output of a linear time-invariant (LTI)
system, then
Therefore,
Y (J) = H (J)X (J)
This shows that
\J!y (J ) = IHlf) l 2 \J!x lf)
Thus, the output signal ESD is 1Hlf)l 2 times the input signal ESD.
(3.75)
3.8 SIGNAL POWER AND POWER
SPECTRAL DENSITY
For a power signal, a meaningful measure of its size is its power [defined in Eq. (2.4)] as the
time average of the signal energy averaged over the infinite time interval. The power Pg of a
real-valued signal g(t) is given by
l T/2
Pg = lim - l g 2 (t) dt
T➔oo T -T/2
(3.76)
The signal power and the related concepts can be readily understood by defining a truncated
signal g T (t) as
g T (t) - { g (t) -
0
ltl :s; T/2
ltl > T/2
The truncated signal is shown in Fig. 3.36. The integral on the right-hand side of Eq. (3.76)
yields E gr
, which is the energy of the truncateq signal g T (t). Thus,
. E gT
Pg = hm -
T➔oo T
(3.77)
This equation describes the relationship between power and energy of nonperiodic signals.
Understanding this relationship will be very helpful in understanding and relating all the
power concepts to the energy concepts. Because the signal power is just the time average of
energy, all the concepts and results of signal energy also apply to signal power if we modify
the concepts properly by taking their time averages.
3. 8. 1 Power Spectral Density (PSD)
If the signal g (t) is a power signal, then its power is finite, and the truncated signal g T (t) is an
energy signal as long as T is finite. If g T (t) G T lf), then from Parseval's theorem,