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

534 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
If the n variables are uncorrelated, aij = 0 (i -I j), and K x reduces to a diagonal
matrix. Thus, Eq. (10.62) becomes
n
I
;-1 2n:a?-
2 2a;
Px1X2···X n
(x1 , x2, . .. , Xn ) = n
exp [ - (x; -x;) 2 ]
-
l
(10.64a)
(10.64b)
As we observed earlier, independent variables are always uncorrelated, but uncorrelated
variables are not necessarily independent. For the case of jointly Gaussian RVs, however,
uncorrelatedness implies independence.
P-3: When x 1 , x 2 , ... , Xn are jointly Gaussian, all the marginal densities, such as Px; (x;), and
all the conditional densities, such as Px;xjlxkxi•••x/x;, Xj lXk , xz, . .. , x p
), are Gaussian.
This property can be readily verified (Prob. 8.2-9).
P-4: Linear combinations of jointly Gaussian variables are also jointly Gaussian. Thus, if we
form m variables YI , Y2, . .. , Y m (m :S n) obtained from
n
y; = I: a;kXk
k=l
(10.65)
then YI , Y 2 , . .. , Y m are also jointly Gaussian variables.
10.5.4 Properties of Gaussian Random Process
A random process x(t) is said to be Gaussian if the RVs x(t1), x(t 2 ), ... , x(t n ) are jointly
Gaussian [Eq. (10.62)] for every n and for every set (t1, t2 , ... , t n )- Hence, the joint PDF of
RVs x(t1), x(t 2 ), ... , x(t n ) of a Gaussian random process is given by Eq. (10.62) in which the
mean and the covariance matrix K x are specified by
and (10.66)
This shows that a Gaussian random process is completely specified by its autocorrelation
function R x Cti, f j
) and its mean value x(t).
As discussed in Chapter 9, if the Gaussian random process satisfies two additional
conditions:
(10.67a)
and
x(t) = constant for all t
(10.67b)
then it is a wide-sense stationary process. Moreover, Eqs. (I 0.67) also mean that the joint PDF
of the Gaussian RVs x(t,), x(t2), ... , x(t n ) is also invariant to a shift of time origin. Hence,