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

3.8 Signal Power and Power Spectral Density 113
3.8.2 Time Autocorrelation Function of Power Signals
The (time) autocorrelation function R g
( r) of a real power signal g(t) is defined as*
1 T/2
R g
(r) = lim - ! g(t)g(t - r) dt
T➔oo T -T/2
(3.82a)
We can use the same argument as that used for energy signals [Eqs. (3.72b) and (3.72c)] to
show that R g
(r) is an even function of r. This means that for a real g(t),
1 T/2
R g
(r) = lim - 1 g(t)g(t + r) dt
T➔oo T -T/2
(3.82b)
and
R g
(r) = R g
(-r)
(3.83)
For energy signals, the ESD \II g
(f) is the Fourier transform of the autocorrelation function
1/f g
( r). A similar result applies to power signals. We now show that for a power signal, the
PSD S g
(f) is the Fourier transform of the autocorrelation function R g
(r). From Eq. (3.82b)
and Fig. 3.36,
1
R g
(r) = lim -loo 1/f
gy(t)gy(t + r) dt = lim _ g _ T (r)
_
T➔oo T
T➔oo T _ 00
(3.84)
Recall from the Wiener-Khintchine theorem that 1/f gT (r) {=::=> 1Gy (f)l 2 • Hence, the Fourier
transform of the preceding equation yields
R g
(r) {=::=>
. I Gy (f)l 2
hm --- = S g
(f)
T
T➔oo
(3.85)
Although we have proved these results for a real g(t), Eqs. (3.80), (3.81a), (3.81b), and (3.85)
are equally valid for a complex g(t).
The concept and relationships for signal power are parallel to those for signal energy. This
is brought out in Table 3.3.
Signal Power Is Its Mean Square Value
A glance at Eq. (3.76) shows that the signal power is the time average or mean of its squared
value. In other words P g
is the mean square value of g(t). We must remember, however, that
this is a time mean, not a statistical mean (to be discussed in later chapters). Statistical means
are denoted by overbars. Thus, the (statistical) mean square of a variable x is denoted by x 2 .
To distinguish from this kind of mean, we shall use a wavy overbar to denote a time average.
Thus, the time mean square value of g(t) will be denoted by g 2 (t). The time averages are
conventionally denoted by angle brackets, written as (g 2 (t)). We shall, however, use the wavy
* For a complex g(t), we define
I /2 I T /2
R g
(r) = lim - !T
g(t)g* (t -r)dt = lim - ! g* (t)g(t +r)dt
T-HX) T -T/2 T-HX) T -T/2