# Random Processes - Robert Marks.org

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24 Νοε 2013 (πριν από 4 χρόνια και 7 μήνες)

95 εμφανίσεις

copyright Robert J. Marks II

EE 505

Random Processes
-

Example Random Processes

copyright Robert J. Marks II

Example RP’s

Example Random Processes

Gaussian

Recall Gaussian pdf

Let
X
k
=X(t
k
) , 1

Then if, for all
n
, the
corresponding pdf’s are Gaussian, then the
RP is Gaussian.

The Gaussian RP is a useful model in signal
processing.

)
(
K
)
(
2
1
2
/
1
2
/
1
|
K
|
2
1
)
(
m
x
m
x
n
X
T
e
x
f

copyright Robert J. Marks II

Flip Theorem

Let A take on values of
+1

and
-
1

with equal
probability

Let X(t) have mean m(t) and
autocorrelation R
X

Let Y(t)=AX(t)

Then Y(t) has mean zero and
autocorrelation R
X

What about the autocovariances?

copyright Robert J. Marks II

Multiple RP’s

X(t) & Y(t)

Independence

(
X(t
1
), X(t
2
), …, X(t
k
)
)

is independent to

(
Y(

1
), Y(

2
), …, Y(

j
)
)

…for All choices of
k

and
j

and

all sample locations

copyright Robert J. Marks II

Multiple RP’s

X(t) & Y(t)

Cross Correlation

R
XY
(t,

⤽䕛堨)⥙(

)

Cross
-
Covariance

C
XY
(t,

⤽)R
XY
(t,

-

䕛E⡴(崠䕛](

)

Orthogonal:

R
XY
(t,

⤠㴠0

Uncorrelated:

C
XY
(t,

⤠㴠=

Note: Independent

the converse.

copyright Robert J. Marks II

Example RP’s

Multiple

Random Process
Examples

Example

X(t) = cos(

)
Ⱐ†⡴(㴠獩s(

)

䉯瑨⁡牥⁺敲漠敡渮

䍲潳猠䍯牲敬慴e潮㴿

p.338

copyright Robert J. Marks II

Example RP’s

Multiple

Random Process Examples

Signal + Noise

X(t) = signal, N(t) = noise

Y(t) = X(t) + N(t)

If X & N are independent,R
XY
=?

p.338

Note: also, var Y = var X + var N

N
var
X
var
SNR

copyright Robert J. Marks II

Example RP’s

Multiple

Random Process Examples (cont)

Discrete time RP’s

X[n]

Mean

Variance

Autocorrelation

Autocovariance

Discrete time i.i.d. RP’s

Bernoulli RP’s

%LQRL53¶
p.340

Binary vs. Bipolar

Random Walk

p.341
-
2

copyright Robert J. Marks II

Autocovariance of Sum Processes

X[k]
’s are iid.

Autocovariance=?

n
k
n
]
k
[
X
S
1
X
n
]
S
[
E
n

)
X
var(
n
]
S
var[
n

copyright Robert J. Marks II

Autocovariance of Sum Processes

When
i=j
, the answer is var(
X
). Otherwise, zero.

How many cases are there where
i = j?

k
j
j
n
i
i
k
n
k
k
n
n
S
X
X
X
X
E
X
k
S
X
n
S
E
S
S
S
S
E
k
n
C
1
1
)
(
)
(
)
)(
(
)
)(
(
)
,
(
)
X
var(
)
k
,
n
min(
)
k
,
n
(
C
S

)
k
,
n
min(
copyright Robert J. Marks II

Autocovariance of Sum Processes

For Bernoulli sum process,

For Bipolar case

pq
)
k
,
n
min(
)
k
,
n
(
C
S

pq
)
X
var(

pq
)
X
var(
4

pq
)
k
,
n
min(
)
k
,
n
(
C
S
4

copyright Robert J. Marks II

Continuous Random Processes

Poisson Random Process

Place n points randomly on line of length T

T
t
p
;
q
p
k
n
]
s
int
po
k
Pr[
k
n
k

T

t

Choose any subinterval of length t.

The probability of finding k points on the
subinterval is

copyright Robert J. Marks II

Continuous Random Processes

Poisson Random Process (cont)

The Poisson approximation: For k big and p small…

!
)
/
(
!
)
(
]
points

Pr[
/
k
T
nt
e
k
np
e
q
p
k
n
k
k
T
nt
k
np
k
n
k

copyright Robert J. Marks II

Continuous Random Processes

The Poisson Approximation…

For n big and p small (implies k << n since p

!
)
(
k
np
e
q
p
k
n
k
np
k
n
k

!
!
)
1
)...(
2
)(
1
(
)!
(
!
!
k
n
k
k
n
n
n
n
k
n
k
n
k
n
k

n
p
n
k
n
k
n
e
p
p
q
)
(
)
1
(
)
1
(

Here’s why…

copyright Robert J. Marks II

Continuous Random Processes

Poisson Random Process (cont)

Let n

㵮⽔=㴠晲敱略湣礠潦⁰潩湴猠

!
)
(
]

interval
on

points

Pr[
k
t
e
t
k
k
t

!
)
/
(
!
)
(
]
points

Pr[
/
k
T
nt
e
k
np
e
q
p
k
n
k
k
T
nt
k
np
k
n
k

copyright Robert J. Marks II

Continuous Random Processes

Poisson Random Process (cont)

This is a Poisson process with parameter

Examples: Modeling

Popcorn

Rain (Both in space and time)

Passing cars

Shot noise

Packet arrival times

!
k
)
t
(
e
]
t
k
Pr[
k
t

interval
on

points

t

copyright Robert J. Marks II

Continuous Random Processes

Poisson Counting Process

Poisson Points

!
k
)
t
(
e
]
k
)
t
(
X
Pr[
k
t

)
t
(
X
copyright Robert J. Marks II

Continuous Random Processes

Recall for Poisson RV with parameter a

Poisson Counting Process Expected Value is
thus

t
)]
t
(
X
[
E

!
k
)
a
(
e
]
k
X
Pr[
k
a

a
)
X
var(
X

copyright Robert J. Marks II

Continuous Random Processes

The Poisson Counting Process is independent
increment process. Thus, for

)!
(
)
(
!
]
)
(
)
(
Pr[
]
)
(
Pr[
]
)
(
)
(
,
)
(
Pr[
]
)
(
,
)
(
Pr[
)
(
i
j
e
t
i
e
t
i
j
t
X
X
i
t
X
i
j
t
X
X
i
t
X
j
X
i
t
X
t
i
j
t
i

copyright Robert J. Marks II

Continuous Random Processes

Autocorrelation: If

㸠

t
t
t
t
)
t
(
t
)
t
(
X
E
)
t
(
X
)
(
X
E
)
t
(
X
E
)
t
(
X
E
)
t
(
X
)
(
X
)
t
(
X
E
)
t
(
X
)
t
(
X
)
(
X
)
t
(
X
E
)
(
X
)
t
(
X
E
)
,
t
(
R
X

2
2
2
2
2
)
,
t
min(
t
)
,
t
(
R
X

2
t

copyright Robert J. Marks II

Continuous Random Processes

Autocovariance of a Poisson sum process

)
,
t
min(
t
)
,
t
min(
t
)
(
X
E
)
t
(
X
E
)
,
t
(
R
)
,
t
(
C
X
X



2
copyright Robert J. Marks II

Continuous Random Processes

Other RP’s related to the Poisson process

Random telegraph signal

)
t
(
X
Poisson Points

copyright Robert J. Marks II

Poisson Random Processes

Random telegraph signal

|
t
|
X
e
)
,
t
(
C

2
|
|
2
)]
(
[
t
e
t
X
E

PROOF…

copyright Robert J. Marks II

Poisson Random Processes

Random telegraph signal. For t>0,

odd

is

0
on

points

of
number
Pr

even

is

0
on

points

of
number
Pr
]
1
)
(
Pr[
)
1
(
1
)
(
Pr
1
)]
(
[
,t)
(
,t)
(
t
X
t
X
t
X
E

copyright Robert J. Marks II

Poisson Random Processes

Random telegraph signal. For t>0,

t
e
t
t
e
,t)
(
t
t

cosh
...
!
4
)
(
!
2
)
(
1
even

is

0
on

points

of
number
Pr
4
2

copyright Robert J. Marks II

Poisson Random Processes

Random telegraph signal. For t>0.

Similarly…

)
sinh(
...
!
5
)
(
!
3
)
(
odd

is

0
on

points

of
number
Pr
5
3
t
e
t
t
t
e
,t)
(
t
t

copyright Robert J. Marks II

Poisson Random Processes

Random telegraph signal. For t>0.

Thus

0
;
)
sinh(
)
cosh(
odd

is

0
on

points

of
number
Pr

even

is

0
on

points

of
number
Pr
]
1
)
(
Pr[
)
1
(
1
)
(
Pr
1
)]
(
[
2

t
e
t
t
e
,t)
(
,t)
(
t
X
t
X
t
X
E
t
t

|
|
2
)]
(
[
t
e
t
X
E

For all t…

copyright Robert J. Marks II

Poisson Random Processes

Random telegraph signal. For t >

,

X(t)

X(

)

-
1

1

1

-
1

1
)
(
Pr
1
)
(
|
1
)
(
Pr

1
)
(
Pr
1
)
(
|
1
)
(
Pr
1
)
(
,
1
)
(
Pr

1
)
(
,
1
)
(
Pr
]
1
)
(
)
(
Pr[

X
X
t
X
X
X
t
X
X
t
X
X
t
X
X
t
X
copyright Robert J. Marks II

Poisson Random Processes

Random telegraph signal. For t >

,

)
(
cosh
even

is

)
,
(
on

points

of
number
Pr
1
)
(
|
1
)
(
Pr
1
)
(
|
1
)
(
Pr
)
(

t
e
t
X
t
X
X
t
X
t





e
e
t
X
X
t
X
X
t
X
t
)
cosh(
)
(
cosh
1
)
(
Pr
1
)
(
|
1
)
(
Pr
1
)
(
,
1
)
(
Pr
)
(
Thus…

copyright Robert J. Marks II

Poisson Random Processes

Random telegraph signal. For t >

,

)
cosh(
)
(
cosh
1
)
(
Pr
1
)
(
|
1
)
(
Pr
1
)
(
,
1
)
(
Pr


t
e
t
X
X
t
X
X
t
X

And…

)
sinh(
)
(
cosh
1
)
(
Pr
1
)
(
|
1
)
(
Pr
1
)
(
,
1
)
(
Pr


t
e
t
X
X
t
X
X
t
X

X(t)

X(

)

-
1

1

1

-
1

copyright Robert J. Marks II

Poisson Random Processes

Random telegraph signal. For t >

.

Onward…

)
(
sinh
odd

is

)
,
(
on

points

of
number
Pr
1
)
(
|
1
)
(
Pr
1
)
(
|
1
)
(
Pr
)
(

t
e
t
X
t
X
X
t
X
t
copyright Robert J. Marks II

Poisson Random Processes

Random telegraph signal. For t >

.

)
cosh(
)
(
sinh
1
)
(
Pr
1
)
(
|
1
)
(
Pr
1
)
(
,
1
)
(
Pr


t
e
t
X
X
t
X
X
t
X

And…

)
sinh(
)
(
sinh
1
)
(
Pr
1
)
(
|
1
)
(
Pr
1
)
(
,
1
)
(
Pr


t
e
t
X
X
t
X
X
t
X

X(t)

X(

)

-
1

1

1

-
1

copyright Robert J. Marks II

Poisson Random Processes

Random telegraph signal. For t >

.

)
cosh(
)
(
sinh
1
)
(
Pr
1
)
(
|
1
)
(
Pr
1
)
(
,
1
)
(
Pr


t
e
t
X
X
t
X
X
t
X

And…

)
sinh(
)
(
sinh
1
)
(
Pr
1
)
(
|
1
)
(
Pr
1
)
(
,
1
)
(
Pr


t
e
t
X
X
t
X
X
t
X

X(t)

X(

)

-
1

1

1

-
1

copyright Robert J. Marks II

Poisson Random Processes

Random telegraph signal. For t >

.

)
sinh(
)
(
cosh
)
cosh(
)
(
sinh
)
sinh(
)
(
sinh
)
cosh(
)
(
cosh
1
)
(
)
(
Pr
1
1
)
(
)
(
Pr
1
)
(
)
(
)
,
(








t
t
t
t
X
e
t
e
t
e
t
e
t
X
t
X
X
t
X
X
t
X
E
t
R

X(t)

X(

)

-
1

1

1

-
1

In general…

|
|
2
)
(
)
(
)
,
(
)
,
(

t
X
X
e
X
t
X
t
R
t
C
copyright Robert J. Marks II

Continuous Random Processes

Other RP’s related to the Poisson process

Poisson point process, Z(t)

Let X(t) be a Poisson sum process. Then

pp.352

)
S
t
(
)
t
(
X
dt
d
)
t
(
Z
n
n

)
t
(
Z
Poisson Points

copyright Robert J. Marks II

Continuous Random Processes

Other RP’s related to the Poisson process

Shot Noise, V(t)

Z(t)

V(t)

pp.352

)
S
t
(
h
)
t
(
V
n
n

Poisson Points

h(t)

)
t
(
V
copyright Robert J. Marks II

Continuous Random Processes

Wiener Process

Assume bipolar Bernoulli sum process with jump
bilateral height h and time interval

E[X(t)]=0; Var X(n

⤠㴠㑮灱栠
2

= nh
2

Take limit as h

〠慮搠

〠步数0湧n

㴠栠
2

/

constant and t = n

.

Then
Var X(t)

By the central limit theorem, X(t) is Gaussian with zero
mean and
Var X(t)
=

We could use any zero mean process to generate the
Wiener process.

p.355

copyright Robert J. Marks II

Continuous Random Processes

Wiener Processes:

=
1

copyright Robert J. Marks II

Continuous Random Processes

Wiener processes in finance

S= Price of a Security.

㴠楮晬慴楯湡特a景牣攮†

If there is no risk…interest earned is proportional to investment.

Solution is

With “volatility”

, we have the most commonly used model in
finance for a security:

V(t) is a Wiener process.

S
dt
dS
dt
)
t
(
S
)
t
(
dS

t
e
S
)
t
(
S

0

)
t
(
dV
)
t
(
S
dt
)
t
(
S
)
t
(
dS