Random processes-3
Docsity.com
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Random Processes: Gaussian Random Process and Poisson Process, Slides of Structural Analysis
The concepts of gaussian random processes and poisson processes in the context of random events and their probability distributions. It covers topics such as stationary increments, independent arrivals, and the stationary arrival rule. The document also includes equations and examples to illustrate these concepts.
Typology: Slides
2012/2013
1 / 34
This page cannot be seen from the preview
Don't miss anything!
Related documents
Partial preview of the text
Download Random Processes: Gaussian Random Process and Poisson Process and more Slides Structural Analysis in PDF only on Docsity!
- Random processes-
22
^
^
^
^
^
^
^
^
exp
exp
XX^
XX
XX^
XX
S^
R^
i^
d
R^
S^
i^
d
^
^
Recall
44
^
^ ^
^
^
^
^
^
^
1 1
1
1 2
1 2
1
1 2 2 Continuing further, consider
time instants
and
associated random variables
.
Let the jpdf of
be given by
,^ , ,^
;^ ,^
,^ ,
1
1 exp^
;^
1,
2
2
n i i
n i i
n i i
XX^ X^
n^
n t
i
n i
n^
t
X^ t
X^ t
p^
x^ x
x^ t^
t^
t x^
S^
x^
x^
i^
n
S S
^
^
^
^
^
^
^
^
^ ^
^ ^
^ ^
^
^ ^
^
^
^
1
2
1
2 :^
&^
is positive definite.
is said to be a Gaussian random process if the above form of pdf is true for any
and for any cho
j^
i^
X^ i
j^
X^ j
t
t
X^
X^
X^ n n
X^ t
m^
t^
X^ t
m^
t
Note
S^
S^
S m^
t^
m^
t^
m^
t
x^
x^
x^
x
X^ t
n
^
^
^
^
^
^
^
^
Definition
^
1 ice of
n. t^ i^ i
55
^
^
^
^
^
^
^
^
^
1 2
1 2
1
2
1 2
1
2
1 2 1
2
(a) A Gaussian random process is completely specified through its mean
and covariance
,^.
(b)^
( ) is stationary
&^
,
,^ ; ,^
,^ ;
X^ ( ) is 2nd order
XX X^
X^
XX^
XX
XX^
XX
m^
t^
C^
t^ t
X t^
m^
t^
m^
C^
t^ t^
C^
t^ t
p^
x^ x
t^
t^
p^
x^ x
t^
t
X t
^
^
^
^
^
Remarks
SSS
is SSS.
(c) A stationary Gaussian random process with zero mean iscompletely described by its autocovariance function or itspdf function.(d) Linear transformation of Gaussian random processes p
X^ t
reserve the
Gaussian nature. Gaussian distributed loads on linear systems produceGaussian distributed responses.
77
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
1
1
1 1
(^21)
cos^
sin^
cos^
sin
cos^
sin^
cos^
sin
cos
n^
n^
n^
n^
n^
n^
n^
n
n^
n
n^
n^
n^
n^
n^
n^
n^
n
n^ m XX
n^
n n
X^ t X^
t^
a^
t^ b
t^
a^
t^
b^
t
a^
t^ b
t^ a
t^
b^
t
R
^
^
^
^
^
^
^
^
^ ^
^
^
^
^
^
^
^
^
^
^
is a WSS random process.is Gaussian. is a SSS process.
X^
t X^
t X^
t
Docsity.com
88
^
^
^
^
^ ^
^
^
^
^
^
^ ^
^
^
^
^
1
(^11) 2
Consider the psd function
cos
cos
Compare this with
cos
XX^
n^
n^
n
n XX^
n^
n^
n
n
XX^
n^
n^
n
n
XX^
n
S^
S
R^
S
R^
S
R
^
^
^
^
^
^ ^
^
^
^
^
^
^
Fourier representation of a Gaussian random process (continued)
^
1 2
By choosing
, we see that the two ACF-s 2
coincide.
n
n
n^
n
n
S
^
10
Ensemble ofrealizations of randomprocesses can bedigitally simulated
10 Docsity.com
11
^
^
^
^
^
^
^
^
^
^
^
^
^ ^
^
^
^
^
^
1
2
2
2 2
2 2
2
2
2
2 2
2
2
2
Let^
be an iid sequence of random variables withP P such that
P^
P
P^
P
Var
X^ i^ i X^
x^
p X^
x^
q p^ q X^
X^
x^
x^
X^
x^
x
x^ p q X^
X^
x^
x^
X^
x^
x
x^ p
q X X
X
x^ p
q^
x^ p
q
x^ p
q^
x^ p
q
^ ^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
Simple random walk
^
^
^ ^
2
2
2
2
1
4
p^ q
x^
p^ q
p^ q
pq^
^ x
^
^
^
^
^
^
^
13
^
0
0 0
0 (^00)
is known as a simple random walk.
( ) is a discrete state, discrete parameter random process.Consider the limit of
0 as
lim^
lim
and lim Var
lim
x^
x t^
t
S t S t x t
x^
t
x
S^
t^ p
q
t
S t
^
^
^
^
^
^
^
^
Remarks
2
In the limit of
0 as
0,^
( ) becomes
a deterministic function.This is not an interesting limit from probabilisticpoint of view.
x t
x
t^ pq
t
x^
t^
S t
^
^
14
2
2
Consider the following limit of the simple random walk
0 as
with
;^
;^
x^ Var This is an interesting limit!
t
t^
t
x^
t^
p^
q
S t
t
S t
t
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
Wiener Process
1616
Random events and Poisson process
^
^
^
^
^
^
^ ^
^
1 1
2 2
1
1
2
2
Let^
( ) be the number of events occuring randomly in the interval
If there exists probability functions
,^
,^
,^ ;^
then we say that
( ) is a counting proc
N^
NN
N t^
t
P^
n t^
P^ N t
n^
P^
n^ t^
n^ t
P^ N t
n^
N t
n
N t
^
^
^
^
^
^
^
^
^
^
^
ess (discrete state, continuous
parameter random process).
^
^
^
^
^
Inter-arrival time
time
t
t^^1
t^2
s^2
s^1
1717
^ ^
^
^
^
^ ^
^
^
^
1
1
2
2
1
1
1 1
2 2
( ) is said to be a Poisson process with stationary increments if thefollowing conditions are satisfied(a) That is,
where
,^ &
are
N t
P^ N t
N^ s
n N t
N^ s
m^
P^ N t
N^ s
n
s^ t^
s^ t ^
^
^
^
^
^
^
^
^
^
^
^
Independent arrivals :^ ^
^
^
^
^
^
1
1
2
2
mutually exclusive and
&^
( )^
( )^
(^
)^1
;^
(c)^
( )^
&^
( )^
s^
t^
s^
t
b P^ N t
dt
N t
P^ N t
dt
h N t
h
dt
P^ N t
dt
N t
dt^
P^ N t
dt
N t
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
Stationary arrival ruleNegligible probability for simultaneous arrivals
^
^
Under these conditions it can be shown that
( )^
exp^
;^
,^
k t!
P N t
k
t^ k
^ k
^
^
^
^
^
^
^
^
s^^1
s^2 t^^1
t^2
1919
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^ ^
^
^
^
^
0
0 0 0 1 Thus with
=0, we have 0,^
exp^
exp^
1,
Clearly,
1,^
1
0
0,^
exp
We have
0,^
0
0
1
counting begins after
0
1
0,^
exp
Consider now
1,
t
N^
N
N N
N N N
n
P^
t^
A^
t^
t^
P^
d
P^
P^ N
P^
t^
A^
t
P^
P^ N
t
A^
P^
t^
t
n
P^
t^
A
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
0
1
0
1 1
exp^
exp^
0,
exp^
exp^
exp
exp^
exp
We have
1,^
0
1
0
0=^
1,^
t exp
N
t N N
t^
t^
P^
d
A^
t^
t^
d
A^
t^
t^
t
P^
P^ N
A^
P^
t^
t^
t
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
2020
^
^
^
^
^
^
^
^ ^
^
0
0
Repeating this process for
we get
,^
exp
;^
If the stationary arrival rule is relaxed, the above model can bemodified to read as
,^
exp
n!
N
n
t^
t
N
n
t
P^
n t^
t^ n
n
P^
n t^
d^
d
^ n
^
^
^
^
^
^
^
^
^
^
^
^
^