Transcript EE 505 - Robert Marks.org

ECE 5345
Random Processes - Example Random Processes
copyright Robert J. Marks II
Example RP’s
Example Random Processes
 Gaussian
Recall Gaussian pdf
f X

(x) 
1
2 
n/2
1/ 2
1   T 1  
 (xm) K (xm)
e 2
|K |
Let Xk=X(tk) , 1  k  n. 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.
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 RX
Let Y(t)=AX(t)
Then Y(t) has mean zero and
autocorrelation RX
What about the autocovariances?
copyright Robert J. Marks II
Multiple RP’s
X(t) & Y(t)

Independence
(X(t1), X(t2), …, X(tk ))
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
RXY(t,  )=E[X(t)Y()]
Cross-Covariance
CXY(t,  )= RXY(t,  ) - E[X(t)] E[Y()]
Orthogonal: RXY(t,  ) = 0
Uncorrelated: CXY(t,  ) = 0
Note: Independent Uncorrelated, but not
the converse.
copyright Robert J. Marks II
Example RP’s
Multiple Random Process
Examples
 Example
X(t) = cos(t+), Y(t) = sin(t+),
Both are zero mean.
Cross Correlation=?
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,RXY=?
Note: also, var Y = var X + var N
SNR 
var X
var N
copyright Robert J. Marks II
p.338
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 Binomial RP’s

Binary vs. Bipolar
 Random Walk
p.341-2
copyright Robert J. Marks II
p.340
Autocovariance of Sum Processes
n
Sn   X [ k ]
X[k]’s are iid.
k 1
E [ S n ]  nX
var[ S n ]  n var( X )
Autocovariance=?
copyright Robert J. Marks II
Autocovariance of Sum Processes


 E ( S n  n X )( S k  k X ) 
C S ( n , k )  E ( S n  S n )( S k  S k )
 n
  k

 E  ( X i  X ) ( X
  j 1
  i 1


j
 
 X )
 
When i=j, the answer is var(X). Otherwise, zero.
How many cases are there where i = j?
min( n , k )  C S ( n , k )  min( n , k ) var( X )
copyright Robert J. Marks II
Autocovariance of Sum Processes
For Bernoulli sum process,
var( X )  pq
C S ( n , k )  min( n , k ) pq
For Bipolar case
var( X )  4 pq
C S ( n , k )  4 min( n , k ) pq
copyright Robert J. Marks II
Continuous Random Processes
Poisson Random Process

Place n points randomly on line of length T
t
T

Choose any subinterval of length t.

The probability of finding k points on the
subinterval is
Pr[ k
 n  k nk
t
po int s ]    p q
;p
T
k
copyright Robert J. Marks II
Continuous Random Processes
Poisson Random Process (cont)

The Poisson approximation: For k big and p small…
Pr[ k
k
 n  k nk
 np ( np )
points ]    p q
e
k!
k 
 nt / T ( nt / T )
e
k!
copyright Robert J. Marks II
k
Continuous Random Processes
The Poisson Approximation…

For n big and p small (implies k << n since p k/n<<1)
k
n
  k nk
 np ( np )
  p q
e
k!
k 
Here’s why…
k
n
 
n!
n ( n  1)( n  2 )...( n  k  1) n
  


k!
k!
 k  k ! ( n  k )!
q
nk
 (1  p )
nk
n
 (1  p )  ( e
copyright Robert J. Marks II
p n
)
Continuous Random Processes
Poisson Random Process (cont)
Pr[ k
k
 n  k nk
(
np
)
 np
points ]    p q
e
k!
k 
e
 nt / T
( nt / T )
k
k!

Let n  such that =n/T = frequency of points
remains constant.
Pr[ k
points on interval
t]  e
 t (t )
k!
copyright Robert J. Marks II
k
Continuous Random Processes
Poisson Random Process (cont)
Pr[ k
points on interval
t]  e
 t ( t )
k!


This is a Poisson process with parameter
 occurrences per unit time
Examples: Modeling
 Popcorn
 Rain (Both in space and time)
 Passing cars
 Shot noise
 Packet arrival times
copyright Robert J. Marks II
t
k
Continuous Random Processes
Poisson Counting Process
X(t )
Poisson Points
Pr[ X ( t )  k ]  e
copyright Robert J. Marks II
 t ( t )
k!
k
Continuous Random Processes
Recall for Poisson RV with parameter a
Pr[ X  k ]  e
a ( a )
k!
k
X  var( X )  a
Poisson Counting Process Expected Value is
thus
E [ X ( t )]   t
copyright Robert J. Marks II
Continuous Random Processes
The Poisson Counting Process is independent
increment process. Thus, for   t and j  i,
Pr[ X ( t )  i , X ( )  j ]
 Pr[ X ( t )  i , X ( )  X ( t )  j  i ]
 Pr[ X ( t )  i ] Pr[ X ( )  X ( t )  j  i ]

 t i e   t
i!

 (
 t)
j  i   (  t )
e
( j  i )!
copyright Robert J. Marks II
Continuous Random Processes

t
Autocorrelation: If  > t
R X ( t , )  E  X ( t ) X (  ) 

 E X ( t ) X (  )  X ( t )   X ( t )
2


 E  X ( t ) X (  )  X ( t )   E X ( t )
2


 E  X ( t ) E  X (  )  X ( t )   E X ( t )

  t   (   t )   t   t 
2
  t   t
2

2
2

R X ( t , )   t    min( t , )
copyright Robert J. Marks II
Continuous Random Processes
Autocovariance of a Poisson sum process
C X ( t , )  R X ( t , )  E  X ( t )  E  X (  ) 
  t    min( t , )   t 
2
  min( t , )
copyright Robert J. Marks II

Continuous Random Processes
Other RP’s related to the Poisson process

Random telegraph signal
X(t )
Poisson Points
copyright Robert J. Marks II
Poisson Random Processes

Random telegraph signal
E [ X ( t )]  e
 2 | t |
C X ( t , )  e
 2 |t  |
PROOF…
copyright Robert J. Marks II
Poisson Random Processes

Random telegraph signal. For t>0,
E [ X ( t )]   1  Pr  X ( t )  1  (  1)  Pr[ X ( t )   1]
 Pr number
of points on ( 0 ,t) is even
 Pr number

of points on ( 0 ,t) is odd
copyright Robert J. Marks II

Poisson Random Processes

Random telegraph signal. For t>0,
Pr number
of points on ( 0 ,t) is even
2
4



(

t
)
(

t
)
 t
 e   1 

 ...  
2!
4!


e
 t
cosh  t 
copyright Robert J. Marks II

Poisson Random Processes
Random telegraph signal. For t>0.
Similarly…

Pr number
of points on ( 0 ,t) is odd

3
5


(

t
)
(

t
)
 t 
e
t 

 ... 


3!
5!


e
 t
sinh(  t )
copyright Robert J. Marks II
Poisson Random Processes
Random telegraph signal. For t>0.
Thus

E [ X ( t )]   1  Pr  X ( t )  1  (  1)  Pr[ X ( t )   1]
 Pr number
of points on ( 0 ,t) is even
 Pr number
e
e
 t
cosh(
 2t

of points on ( 0 ,t) is odd

 t )  sinh(  t ) 
;t  0
For all t…
E [ X ( t )]  e
copyright Robert J. Marks II
 2 | t |
Poisson Random Processes

Random telegraph signal. For t > ,
1
X()
-1
1
X(t)
-1
Pr[ X ( t ) X ( )  1]  Pr  X ( t )  1, X ( )  1
 Pr  X ( t )   1, X ( )   1
 Pr  X ( t )  1 | X ( )  1 Pr  X ( )  1 
 Pr  X ( t )   1 | X ( )   1 Pr  X ( )   1 
copyright Robert J. Marks II
Poisson Random Processes

Random telegraph signal. For t > ,
Pr  X ( t )  1 | X ( )  1
 Pr  X ( t )   1 | X ( )   1
 Pr number
e
Thus…
of points on ( , t ) is even
 ( t  )

cosh   ( t   ) 
Pr  X ( t )  1, X ( )  1
 Pr  X ( t )  1 | X ( )  1 Pr  X ( )  1 
 cosh  ( t   ) e
  ( t  )
cosh(  ) e
copyright Robert J. Marks II
 
Poisson Random Processes

Random telegraph signal. For t > ,
Pr  X ( t )  1, X ( )  1
 Pr  X ( t )  1 | X ( )  1 Pr  X ( )  1 
 cosh  ( t   ) e
 t
1
cosh(  )
And…
Pr  X ( t )   1, X ( )   1
-1
 Pr  X ( t )   1 | X ( )   1 Pr  X ( )   1 
 cosh  ( t   ) e
 t
sinh(  )
copyright Robert J. Marks II
X()
1
-1
X(t)
Poisson Random Processes

Random telegraph signal. For t > .
Onward…
Pr  X ( t )  1 | X ( )   1
 Pr  X ( t )   1 | X ( )  1
 Pr number
e
  ( t  )
of points on ( , t ) is odd
sinh  ( t   ) 
copyright Robert J. Marks II

Poisson Random Processes

Random telegraph signal. For t > .
Pr  X ( t )   1, X ( )  1
 Pr  X ( t )   1 | X ( )  1 Pr  X ( )  1 
 sinh  ( t   ) e
 t
1
cosh(  )
X()
And…
Pr  X ( t )  1, X ( )   1
-1
 Pr  X ( t )  1 | X ( )   1 Pr  X ( )   1 
 sinh  ( t   ) e
 t
sinh(  )
copyright Robert J. Marks II
1
-1
X(t)
Poisson Random Processes

Random telegraph signal. For t > .
Pr  X ( t )   1, X ( )  1
 Pr  X ( t )   1 | X ( )  1 Pr  X ( )  1 
 sinh  ( t   ) e
 t
1
cosh(  )
X()
And…
Pr  X ( t )  1, X ( )   1
-1
 Pr  X ( t )  1 | X ( )   1 Pr  X ( )   1 
 sinh  ( t   ) e
 t
sinh(  )
copyright Robert J. Marks II
1
-1
X(t)
Poisson Random Processes

Random telegraph signal. For t > .
1
X()
-1
1
-1
R X ( t ,  )  E  X ( t ) X ( ) 
 1  Pr  X ( t ) X ( )  1  1  Pr  X ( t ) X ( )   1

 sinh
 cosh  ( t   ) e
 t
cosh(  )  sinh  ( t   ) e
 ( t   ) e   t cosh(
X(t)
 )  cosh  ( t   ) e
 t
 t
In general…C X ( t ,  )  R X ( t ,  )  X ( t ) X ( )  e
copyright Robert J. Marks II

 ) 
sinh(  )
sinh(
 2 |t   |
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

Z(t ) 
d
dt
X ( t )    ( t  Sn )
Z(t )
n
pp.352
Poisson Points
copyright Robert J. Marks II
Continuous Random Processes
Other RP’s related to the Poisson process
Shot Noise, V(t)
Z(t)

h(t)
V(t)
V ( t )   h( t  S n )
n
V(t )
pp.352
Poisson Points
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) = 4npqh 2 = nh 2
Take limit as h 0 and   0 keeping  = h 2 / 
constant and t = n .
Then Var X(t)   t
By the central limit theorem, X(t) is Gaussian with zero
mean and Var X(t) =  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.  = inflationary force.
If there is no risk…interest earned is proportional to investment.
dS ( t )   S ( t ) dt 
dS
t
dt
 S
Solution is S ( t )  S 0 e
With “volatility” , we have the most commonly used model in
finance for a security:
dS ( t )   S ( t ) dt   S ( t ) dV ( t )
V(t) is a Wiener process.
copyright Robert J. Marks II