A random or stochastic process is the
assignment of a function of a real variable to each sample point
ωω in a sample space. Thus,
the process
Xωt
X
ω
t
can be considered a function of two variables. For
each ωω, the time function
must be well-behaved and may or may not look random to the eye.
Each time function of the process is called a sample
function and must be defined over the entire domain of
interest. For each tt, we have a
function of ωω, which is
precisely the definition of a random variable. Hence the
amplitude of a random process is a random variable.
The amplitude distribution of a process refers to
the probability density function of the amplitude:
pXtx
p
X
t
x
. By examining the process's amplitude at several
instants, the joint amplitude distribution can also be defined.
For the purposes of this module, a process is said to be
stationary when the joint amplitude distribution
depends on the differences between the selected time instants.
The expected value or mean of a
process is the expected value of the amplitude at each
tt.
EXt=
m
X
t=∫-∞∞xpXtxdx
X
t
m
X
t
x
x
p
X
t
x
For the most part, we take the mean to be zero. The
correlation function is the first-order joint
moment between the process's amplitudes at two times.
R
X
t
1
t
2
=∫-∞∞∫-∞∞
x
1
x
2
pX
t
1
X
t
2
x
1
x
2
d
x
1
d
x
2
R
X
t
1
t
2
x
2
x
1
x
1
x
2
p
X
t
1
X
t
2
x
1
x
2
Since the joint distribution for stationary processes depends
only on the time difference, correlation functions of stationary
processes depend only on
|
t
1
-
t
2
|
t
1
t
2
. In this case, correlation functions are really
functions of a single variable (the time difference) and are
usually written as
R
X
τ
R
X
τ
where
τ=
t
1
-
t
2
τ
t
1
t
2
. Related to the correlation function is the
covariance function
K
X
τ
K
X
τ
, which equals the correlation function minus the
square of the mean.
K
X
τ=
R
X
τ-
m
X
2
K
X
τ
R
X
τ
m
X
2
The variance of the process equals the covariance function
evaluated as the origin. The power spectrum of a
stationary process is the Fourier Transform of the correlation
function.
𝒮
X
ω=∫-∞∞
R
X
τⅇ-ⅈωτdτ
𝒮
X
ω
τ
R
X
τ
ω
τ
A particularly important example of a random process is
white noise. The process
Xt
X
t
is said to be white if it has zero mean and a
correlation function proportional to an impulse.
EXt=0
X
t
0
R
X
τ=
N
0
2δτ
R
X
τ
N
0
2
δ
τ
The power spectrum of white noise is constant for all
frequencies, equaling
N
0
2
N
0
2
which is known as the spectral height.
When a stationary process
Xt
X
t
is passed through a stable linear, time-invariant
filter, the resulting output
Yt
Y
t
is also a stationary process
having power density spectrum
𝒮
Y
ω=|Hⅈω|2
𝒮
X
ω
𝒮
Y
ω
H
ω
2
𝒮
X
ω
where
Hⅈω
H
ω
is the filter's transfer function.
