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

466 RANDOM PROCESSES AND SPECTRAL ANALYSIS
and the bar atop represents ensemble average. Note that ensemble averaging is done before
the limiting operation. We shall now show that the PSD as defined in Eq. (9.10a) is the Fourier
transfomi of the autocorrelation function R x ( r) of the process x(t); that is,
(9.1 0b)
This can be proved as follows:
Joo T/2
-oo
!-T/2
Thus, for real x(t),
Xr (f) = x r (t)e - jZrrft dt = x(t)e - jZrrft dt
(9.11)
IXr (/) 1 2 = Xr (-f)XT (f)
T/2
! f T/2
-T/2
T/2 T/2
= / x(t1 )x(t2)e - jZ1rf(tz - ti) dt1dt2
!-T/2 -T/2
= x(t1)rJ 2 rrfti dt1 x(t2)e - jZ1rftz dt2
-T /2
and
S x if) = lim [ IXr (f) l 2 T➔oo T ]
= lim -
[ 1 !
T/2
T/2
!
T➔oo T -T/2 -T/2
x(t1 )x(t2)e-jZ1rf(t 2-1 i l dt1 dt2
]
(9.12)
Interchanging the operation of integration and ensemble averaging,* we get
1 T/2 T/2 -,.......
Sx (/) = lim - x(t1)x(t2)e - 12 rrf (tz - ti) dt1 dt 2
! !
T➔oo T -T/2 -T/2
f T/2 f T/2
1
= lim - R x (t2 - t1)e - 12 rrf (tz - ti) dt1 dt 2
T➔oo T -T/2 -T/2
Here we are assuming that the process x(t) is at least wide-sense stationary, so that x(t1 )x(t2) =
R x (t2 - t1). For convenience, let
Then,
(9.1 3)
I f T/2 T/2
Sx (f) = lim -
(f)(t2 - ti) dt1 dt2
!
T➔oo T -T/2 -T/2
(9.14)
* The operation of ensemble averaging is also an operation of integration. Hence, interchanging integration with
ensemble averaging is equivalent to interchanging the order of integration.