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

Upon carrying out this convolution (integration), we have
p y
(y) = --
00
I
1 (
x2
(y _ x) 2
exp -- - ---)
dx
2na1a2 -oo 2af 20-2
y2
(
8.6 Sum of Random Va riables 445
[ 2 ] 2 )
1 - 00
2
I I a
= e <0-1 +0-2
>
exp ---- x -
1
I 2 a f +a f af +a}
Jzn(a 2 +a 2 ) .j'f;;if;;f J_oo 2 •/•l af +a}
By a simple change of variable
y
(8.94)
dx
we can rewrite the integral of Eq. (8.94) as
(8.95)
By examining Eq. (8.95), it can be seen that y is a Gaussian RV with zero mean and variance:
In fact, because x1 and x2 are independent, they must be uncorrelated. This relationship can
be obtained from Eq. (8.81).
More generally, 5 if x1 and x2 are jointly Gaussian but not necessarily independent, then
y = x1 + x2 is Gaussian RV with mean
and variance
Based on induction, the sum of any number of jointly Gaussian distributed RV's is still
Gaussian. More importantly, for any fixed constants { ai, i = 1, ... , m} and jointly Gaussian
RVs {xi, i = I, ... , m},
remains Gaussian. This result has important practical implications. For example, if Xk is a
sequence of jointly Gaussian signal samples passing through a discrete time filter with impulse
response {hd, then the filter output
00
y = LhiXk-i
i=O
(8.96)