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Random Processes and Spectral Analysis Chap. 6
Properties of Gaussian Processes
Some properties of Gaussian processes are as follows:
1. f x (x) depends only on C and m, which is another way of saying that the N-dimensional
Gaussian PDF is completely specified by the first- and second-order moments
(i.e., means, variances, and covariances).
2. Since the {x i = x(t i )} are jointly Gaussian, the x i = x(t i ) are individually Gaussian.
3. When C is a diagonal matrix, the random variables are uncorrelated. Furthermore, the
Gaussian random variables are independent when they are uncorrelated.
4. A linear transformation of a set of Gaussian random variables produces another set of
Gaussian random variables.
5. A wide-sense stationary Gaussian process is also strict-sense stationary † [Papoulis,
1984, p. 222; Shanmugan and Breipohl, 1988, p. 141].
Property 4 is very useful in the analysis of linear systems. This property, as well as the
following theorem, will be proved subsequently.
THEOREM. If the input to a linear system is a Gaussian random process, the system
output is also a Gaussian process.
Proof. The output of a linear network having an impulse response h(t) is
This can be approximated by
q
y(t) = h(t) * x(t) = h(t - l)x(l) dl
L
y(t) = a
N
j = 1
-q
h(t - l j )x(l j ) ¢l
which becomes exact as N gets large and ¢ → 0.
The output random variables for the output random process are
In matrix notation, these equations become
D
y 1
y 2
T
o
y N
y(t 1 ) = a
N
o
j = 1
y(t 2 ) = a
N
j =1
N
y(t N ) = a [h(t N - l j ) ¢l]x(l j )
j=1
[h(t 1 - l j ) ¢l]x(l j )
[h(t 2 - l j ) ¢l] x(l j )
h 11 h 12
Á h1N
h 21 h 22
Á h2N
= D
T
o o ∞ o
h N1 h N2
Á hNN
D T
o
x N
(6–114)
† This follows directly from Eq. (6–112), because the N-dimensional Gaussian PDF is a function only of t and
not the absolute times.
x 1
x 2