The mean function of a random process
is defined as the expected value of
for all
's.
Autocorrelation
The autocorrelation function of the random process
is defined as
Fact
If
is second-order stationary, then
only depends on
.
If
depends on
only, then we will represent the autocorrelation with only one variable
Properties
and
is uniformly distributed between
and
.
The mean function
The autocorrelation function
Not a function of
since the
second term in the right hand side of the equality in
[link] is zero.
Toss a fair coin every
seconds. Since
is a discrete valued random process, the statistical characteristics
can be captured by the pmf and the mean function is written as
when
and
when
and
with
A function of
and
.
Wide Sense Stationary
A process is said to be wide sense stationary if
is constant and
is only a function of
.
Fact
If
is strictly stationary, then it is wide sense stationary. The
converse is not necessarily true.
Autocovariance
Autocovariance of a random process is defined as
The variance of
is
Two processes defined on one experiment (
[link] ).
Crosscorrelation
The crosscorrelation function of a pair of random processes
is defined as
Jointly Wide Sense Stationary
The random processes
and
are said to be jointly wide sense stationary if
is a function of
only and
and
are constant.