Gaussian Processes
2.6
2001/08/06
2004/01/07 09:58:43 US/Central
Behnaam
Aazhang
aaz@ece.rice.edu
Dinesh
Rajan
dinesh@ece.rice.edu
Mohammad
Jaber
Borran
mohammad@ece.rice.edu
Roy
Ha
rha@rice.edu
Shawn
Stewart
mrshawn@alumni.rice.edu
Behnaam
Aazhang
aaz@ece.rice.edu
Gaussian process
random process
stochastic process
Describes signals that cannot be precisely characterized.
Gaussian Random Processes
Gaussian process
A process with mean
μ
X
t
and covariance function
C
X
t
2
t
1
is said to be a Gaussian process if any
X
X
t
1
X
t
2
…
X
t
N
formed by any sampling of the process is a
Gaussian random vector, that is,
f
X
x
1
2
N
2
Σ
X
1
2
1
2
x
μ
X
Σ
X
x
μ
X
for all
x
n
where
μ
X
μ
X
t
1
⋮
μ
X
t
N
and
Σ
X
C
X
t
1
t
1
…
C
X
t
1
t
N
⋮
⋱
C
X
t
N
t
1
…
C
X
t
N
t
N
.
The complete statistical properties of
X
t
can be obtained from the second-order statistics.
Properties
- If a Gaussian process is WSS, then it is strictly stationary.
- If two Gaussian processes are uncorrelated, then they are also
statistically independent.
- Any linear processing of a Gaussian process results in a
Gaussian process.
X and
Y are Gaussian and zero mean
and independent.
Z
X
Y
is also Gaussian.
φ
X
u
u
X
u
2
2
σ
X
2
for all
u
φ
Z
u
u
X
Y
u
2
2
σ
X
2
u
2
2
σ
Y
2
u
2
2
σ
X
2
σ
Y
2
therefore Z is also Gaussian.