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

10.5 Vector Decomposition of White Noise Random Processes 535
we can conclude that a wide-sense stationary Gaussian random process is also strict-sense
stationary.
Another significant property of the Gaussian process is that the response of a linear system
to a Gaussian process is also a Gaussian process. This arises from property P-4 of the Gaussian
RVs. Let x(t) be a Gaussian process applied to the input of a linear system whose unit impulse
response is h(t). If y(t) is the output (response) process, then
y(t) = L: x(t - r)h(r) dr
00
= lim L x(t - kf..r)h(kf..r) f.. r
L'> r-+0 k=- oo
is a weighted sum of Gaussian RVs. Because x(t) is a Gaussian process, all the variables
x(t - kf..r) are jointly Gaussian (by definition). Hence, the variables y(t i ), y(t 2 ), ... , y(t n )
for all n and every set (t1, t2, ... , t n ) are linear combinations of variables that are jointly
Gaussian. Therefore, the variables y(t1 ), y(t 2 ), . .. , y(t n ) must be jointly Gaussian, according
to the earlier discussion. It follows that the process y(t) is a Gaussian process.
To summarize, the Gaussian random process has the following properties:
1. A Gaussian random process is completely specified by its autocorrelation function and mean
value.
2. If a Gaussian random process is wide-sense stationary, then it is stationary in the strict
sense.
3. The response of a linear system to a Gaussian random process is also a Gaussian random
process.
Consider a white noise process nw (t) with PSD N /2. Then any complete set of orthonormal
basis signals <p1 (t), ({J2 (t), . .. can decompose nw(t) into
nw(t) = n1<p1 (t) + n2<p2 (t) + · · ·
= 'E nk({Jk (t)
k
White noise has infinite bandwidth. Consequently, the dimensionality of the signal space is
infinity.
We shall now show that RVs n1 , n 2 , ... are independent, with variance N/2 each. First,
we have