Random Process Calculus
Random Process Calculus
Random Process Calculus
and we write
l.i. m.
n
n
X X
( random process ( ) { }
X t
is said to be continuous at a point
'
t t
in the mean%square
sense if
( ) ( )
'
'
l.i. m.
t t
X t X t
or equivalently
( ) ( )
'
&
'
lim '
t t
E X t X t
1
]
Mean-square continuity and autocorrelation function
)$* ( random process ( ) { }
X t
is +, continuous at
'
t
if its auto correlation function
( )
$ &
,
X
R t t
is continuous at
' '
) , *. t t
Proof:
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
&
& &
' ' '
' ' '
&
, & , ,
X X X
E X t X t E X t X t X t X t
R t t R t t R t t
+ 1
]
+
-f ( )
$ &
,
X
R t t
is continuous at
' '
) , *, t t
then
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
' '
&
' ' ' '
' ' ' ' ' '
lim lim , & , ,
, & , ,
'
X X X
t t t t
X X X
E X t X t R t t R t t R t t
R t t R t t R t t
+ 1
]
+
)&* -f ( ) { }
X t
is +, continuous at
'
t
its mean is continuous at
'
. t
#his follows from the fact that
( ) ( ) ( ) ( ) ( ) ( )
& &
' '
E X t X t E X t X t 1 1
] ]
( ) ( ) ( ) ( )
' '
& &
' '
lim lim '
t t t t
E X t X t E X t X t
1 1 1
] ] ]
( ) EX t
is continuous at
'
. t
Example
Consider the random binary wave
. ) */ X t
discussed in xample . ( typical reali!ation
of the process is shown in Fig. below. #he reali!ation is a discontinuous function.
#he process has the autocorrelation function given by
$
) *
' otherwise
p
p X
T
T R
'
We observe that
) *
X
R
is continuous at '. #herefore,
) *
X
R
is continuous at all
.
p
T
$
$
'
) * X t
t
K
K
p
T
Example For a Wiener process ( ) { }
, X t
( ) ( )
( ) ( )
$ & $ &
, min ,
where is a constant.
, min ,
X
X
R t t t t
R t t t t t
#hus the autocorrelation function of a Wiener process is continuous everywhere implying
that a Wiener process is m.s. continuous everywhere. We can similarly show that the
0oisson process is m.s. continuous everywhere.
Mean-square differentiability
#he random process ( ) { }
X t
is said to have the mean%square derivative ( ) 1 X t
at a point
, t
provided
( ) ( ) X t t X t
t
+
approaches ( ) 1 X t
in the mean square sense as
' t . -n other words, the random process ( ) { }
X t
has a m%s derivative ( ) 1 X t
if
( ) ( )
( )
&
'
lim 1 '
t
X t t X t
E X t
t
+ 1
1
]
Remark
)$* -f all the sample functions of a random process ( ) X t
are differentiable, then the
above condition is satisfied and the m%s derivative exists.
Example
Consider the random%phase sinusoid { } ) * X t
given by
'
) * cos) * X t A w t +
where
'
and A w
are constants and
2 3', & 4.
#hen for each
,
) * X t
is differentiable. #herefore, the m.s. derivative is
' '
) * sin) * X t Aw w t +
M.S. Deriatie and !utocorrelation functions
#he m%s derivative of a random process ( ) X t
at a point t exists if
( )
&
$ &
$ &
,
X
R t t
t t
exists at the point
) , *. t t
(pplying the Cauchy criterion, the condition for existence of m%s derivative is
( ) ( ) ( ) ( )
$ &
&
$ &
, '
$ &
lim '
t t
X t t X t X t t X t
E
t t
+ + 1
1
]
xpanding the square and taking expectation results,
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
$ &
$ &
&
$ &
, '
$ &
$ $ $ & & &
& &
, '
$ &
$ & $ &
$ &
lim
, , & , , , & ,
lim
, , , ,
&
t t
X X X X X X
t t
X X X X
X t t X t X t t X t
E
t t
R t t t t R t t R t t t R t t t t R t t R t t t
t t
R t t t t R t t t R t t t R t t
t t
+ + 1
1
]
+ + + + + + + + 1 1
+
1 1
] ]
+ + + + + 1
1
]
ach of the above terms within square bracket converges to
( )
$ &
&
$ &
$ &
,
,
X
t t t t
R t t
t t
1
1
]
if the
second partial derivative exists.
( ) ( ) ( ) ( ) ( ) ( ) ( )
$ &
$ &
&
& & &
$ & $ & $ & $ &
, '
$ & $ & $ & $ &
,
, , ,
lim &
'
X X X
t t
t t t t
X t t X t X t t X t R t t R t t R t t
E
t t t t t t t t
1 + + 1
+
1 1
] ]
#hus, ( ) { }
X t
is m%s differentiable at t if
( )
&
$ &
$ &
,
X
R t t
t t
exists at
) , * . t t
0articularly, if ( ) X t
is W,,,
( ) ( )
$ & $ &
,
X X
R t t R t t
,ubstituting
$ &
t t
, we get
( ) ( )
( ) ( )
( )
( )
& &
$ & $ &
$ & $ &
$ &
$ &
&
&
$
&
&
,
.
X X
X
X
X
R t t R t t
t t t t
dR t t
t d t
d R
d t
d R
d
#herefore, a W,, process ( ) X t
is m%s differentiable if ( )
X
R
has second derivative at
' .
Example
Consider a W,, process ( ) { }
X t
with autocorrelation function
( ) ( )
exp
X
R a
( )
X
R
does not have the first and second derivative at '. ( ) { }
X t
is not mean%square
differentiable.
Example #he random binary wave ( ) { }
X t
has the autocorrelation function
$
) *
' otherwise
p
p X
T
T R
'
( )
X
R
does not have the first and second derivative at '. #herefore, ( ) { }
X t
is not
mean%square differentiable.
Example For a Wiener process ( ) { }
, X t
( ) ( )
( )
( )
( )
$ & $ &
& &
&
&
&
&
$
&
&
$ &
$ &
$ &
, min ,
where is a constant.
if '
',
' other wise
if '
',
' if '
does not exist if if '
,
does not exist at ) ', '*
X
X
X
X
R t t t t
t t
R t
t
R t
t
t
t
R t t
t t
t t
<
'
<
>
'
#hus a Wiener process is m.s. differentiable nowhere.
Mean and !utocorrelation of t"e Deriatie process
We have,
( )
( ) ( )
( ) ( )
( ) ( )
( )
'
'
'
1 lim
lim
lim
1
t
t
X X
t
X
X t t X t
EX t E
t
EX t t EX t
t
t t t
t
t
+ 1
]
For a W,, process
( ) ( ) ( )
( )
$ & $ &
1
X
X
dR
EX t X t R t t
t d
and
( ) ( )
( )
( )
( ) ( )
( )
&
$ &
$ &
$ &
&
&
&
&
'
var
X
X
X
R t t
EX t X t
t t
d R
d
d R
X t
d
Mean Square #nte$ral
Recall that the definite integral )Riemannian integral* of a function ( ) ! t
over the interval
[ ]
'
, t t
is defined as the limiting sum given by
( ) ( )
'
$
'
'
lim
t
n
k k
n k
k
t
! d !
Where
' $ $
................
n n
t t t t t
.
For a random process ( ) . / X t
, the m%s integral can be similarly defined as the process
( ) . / " t
given by
( ) ( ) ( )
'
$
'
'
l.i .m.
t
n
k k
n k
k
t
" t X d X
Existence of M.S. #nte$ral
-t can be shown that the sufficient condition for the m%s integral ( )
'
t
t
X d
to
exist is that the double integral ( )
' '
$ & $ &
,
t t
X
t t
R d d
exists.
-f ( ) . / X t
is +.,. continuous, then the above condition is satisfied and the
process is +.,. integrable.
Mean and !utocorrelation of t"e #nte$ral of a %SS process
We have
( ) ( )
( )
'
'
'
'
) *
t
t
t
t
t
X
t
X
E" t E X d
EX d
d
t t
#herefore, if
',
X
( ) { }
" t
is necessarily non%stationary.
( )
( ) ( )
( ) ( )
( )
$ &
' '
$ &
' '
$ &
' '
$ & $ &
$ & $ &
$ & $ &
$ & $ &
, ) * ) *
"
t t
t t
t t
t t
t t
X
t t
R t t E" t " t
E X X d d
EX X d d
R d d
which is a function of
$ &
and . t t
Thus the integral of a WSS process is always non-stationary.
Remark #he nonstatinarity of the +.,. integral of a random process has physical
importance 6 the output of an integrator due to stationary noise rises unboundedly.
Example #he random binary wave ( ) { }
X t
has the autocorrelation function
Fig. )a* Reali!ation of a W,, process
) * X t
)b* corresponding integral
) * " t
$
) *
' otherwise
p
p X
T
T R
'
( )
X
R
is continuous at ' implying that ( ) { }
X t
is +.,. continuous. #herefore,
( ) { }
X t
is mean%square integrable.