EE 505 - Robert Marks.org
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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
(xm) K (xm)
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 nk
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 nk
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 nk
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
nk
(1 p )
nk
n
(1 p ) ( e
copyright Robert J. Marks II
p n
)
Continuous Random Processes
Poisson Random Process (cont)
Pr[ k
k
n k nk
(
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(
2t
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