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 wellbehaved 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:
p
Xt
x
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=∫−∞∞xp
Xt
xdx
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 firstorder joint
moment between the process's amplitudes at two times.
R
X
t
1
t
2
=∫−∞∞∫−∞∞
x
1
x
2
p
X
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−(iωτ)dτ
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, timeinvariant
filter, the resulting output
Yt
Y
t
is also a stationary process
having power density spectrum
𝒮
Y
ω=Hiω2
𝒮
X
ω
𝒮
Y
ω
H
ω
2
𝒮
X
ω
where
Hiω
H
ω
is the filter's transfer function.