Random Processes - Random Signals and Processes Primer with MATLAB - page 374

Note that the only difference in the previously considered n -dimensional variable
is that in this case the distribution as well as the density function are functions of the
time instants, t 1 , , t n .

If we know an n -dimensional density then all densities of a lower order are also
known, as demonstrated below:

...

1

f X 1 X 2 ... X n 1 ðx 1 ; ... ; x n 1 ; t 1 ; ... t n 1 Þ ¼ f X 1 X 2 ... X n ðx 1 ; ... ; x n ; t 1 ; ... t n Þ d x n :

1

f X 1 X 2 ... X n 2 ðx 1 ; ... ; x n 2 ; t 1 ; ... ; t n 2 Þ ¼ Ð 1
1

f X 1 X 2 ... X n 1 ðx 1 ; ... ; x n 1 ; t 1 ; ... t n 1 Þ d x n 1

:

f X 1 ðx 1 ; t 1 Þ ¼ Ð 1
1

f X 1 X 2 ðx 1 ; x 2 ; t 1 ; t 2 Þ d x 2

(6.10)

The calculation of n -dimensional distribution and density functions is a very
complex task. Fortunately, in many practical situations, it is enough to observe a
process in one or two time instants and then deal with a one-dimensional PDF and a
two-dimensional PDFs.

6.3 Stationary Random Processes

A process is called stationary if none of its statistical characteristics change with
time [PEE93, p. 168].
We can define different types of stationary processes according to observed
statistical characteristics [PEE93, pp. 169-170].
A process is said to be a first-order stationary process if its first density function
is independent of a time shifting D , that is:

f X ðx; tÞ ¼ f X ðx; t þ DÞ; (6.11)

for any t and D .

Therefore,

f X ðx; tÞ ¼ f X ðxÞ: (6.12)

Similarly, a process is called a second-order stationary process if its second
order density has the following property for any value of t 1 , t 2 , and

D

:

f X 1 X 2 ðx 1 ; x 2 ; t 1 ; t 2 Þ ¼ f X 1 X 2 ðx 1 ; x 2 ; t 1 þ D; t 2 þ DÞ: (6.13)