Stochastic Processes
Definition
()
A random variable is a number X assigned to every outcome
of an experiment.
( )
A stochastic process is the assignment of a function of t X t,
to each outcome of an experiment.
{ ( ) ( )
(
)}
The set of functions X t,1 , X t, 2 ,, X t, N corresponding
to the N outcomes of an experiment is called an ensemble and each
member X t, i is called a sample function of the stochastic
process.
( )
A common convention in the notation describing stochastic
processes is to write the sample functions as functions of t only
and to indicate the stochastic process by X t instead of X t,
and any particular sample function by X i t instead of X t, i .
()
()
( )
( )
Definition
Ensemble
Sample
Function
()
The values of X t at a particular time t1 define a random variable
X t1 or just X 1 .
()
Example of a Stochastic Process
Suppose we place a
temperature sensor at
every airport control
tower in the world
and record the
temperature at noon
every day for a year.
Then we have a
discrete-time,
continuous-value
(DTCV) stochastic
process.
Example of a Stochastic Process
Suppose there is a large number
of people, each flipping a fair
coin every minute. If we assign
the value 1 to a head and the
value 0 to a tail we have a
discrete-time, discrete-value
(DTDV) stochastic process
Continuous-Value vs. Discrete-Value
A continuous-value (CV)
random process has a pdf
with no impulses. A discretevalue (DV) random process
has a pdf consisting only of
impulses. A mixed random
process has a pdf with
impulses, but not just
impulses.
Deterministic vs. NonDeterministic
A random process is deterministic if a sample function
can be described by a mathematical function such that its
future values can be computed. The randomness is in the
ensemble, not in the time functions. For example, let the
sample functions be of the form,
()
(
X t = Acos 2 f0t + )
and let the parameter be random over the ensemble but
constant for any particular sample function.
All other random processes are non-deterministic.
Stationarity
If all the multivariate statistical descriptors of a random
process are not functions of time, the random process is
said to be strict-sense stationary (SSS).
A random process is wide-sense stationary (WSS) if
( ( )) is independent of the choice of t
E X t1
1
and
( ( ) ( ))
E X t1 X t2 depends only on the difference between t1 and t2
Ergodicity
If all of the sample functions of a random process have the
same statistical properties the random process is said to be
ergodic. The most important consequence of ergodicity is that
ensemble moments can be replaced by time moments.
( )
E Xn
T /2
()
1
n
= lim
X
t dt
T T
T /2
Every ergodic random process is also stationary.
Measurement of Process
Parameters
The mean value of an ergodic random process can be estimated
by
T
1
X = X t dt
T 0
()
()
where X t is a sample function of that random process. In
practical situations, this function is usually not known. Instead
samples from it are known. Then the estimate of the mean
value would be
1 N
X = Xi
N i=1
()
where X i is a sample from X t .
Measurement of Process
Parameters
To make a good estimate, the samples from the random process
should be independent.
Correlation Functions
The Correlation Function
If X(t) is a sample function of one stochastic CT process and Y(t) is
a sample function from another stochastic CT process and
()
( )
X 1 = X t1 and Y2 = Y t2
then
( )
(
) R XY t1 ,t2 = E X 1Y2* =
(
)
X 1Y2* f XY x1 , y2 ;t1 ,t2 dx1dy2
is the correlation function relating X and Y. For stationary stochastic
continuous-time processes this can be simplified to
()
( ( ) ( ))
R XY = E X t Y * t + If the stochastic process is also ergodic then the time correlation
function is
T /2
1
*
*
RXY = lim
X
t
Y
t
+
dt
=
X
t
Y
t + = R XY T T T /2
()
() (
)
() (
)
()
Autocorrelation
If X and Y represent the same stochastic CT process then the
correlation function becomes the special case called
autocorrelation.
()
() (
)
R X = E X t X * t + For an ergodic stochastic process,
()
T /2
() (
)
() (
1
*
*
RX = lim
X
t
X
t
+
dt
=
X
t
X
t +
T T
T /2
)
()
= RX Autocorrelation
( ( ))
( )
R X t,t = E X 2 t
Meansquared
value of X
For WSS stochastic continuous-time processes
RX
()
( ( ))
()
T /2
()
()
1
2
2
2
0 = E X t and RX 0 = lim
X
t
dt
=
X
t
T T
T /2
MeanAverage
squared
Signal
value of X
Power of X
The Correlation Function
If X n is a sample function of one stochastic DT process and Y n is a sample function from another stochastic DT process and
X 1 = X n1 and Y2 = Y n2 then
R XY n1 , n2 = E X 1Y2* =
(
) XY
*
1 2
(
)
f XY x1 , y2 ;n1 , n2 dx1dy2
is the correlation function relating X and Y. For stationary stochastic
DT processes this can be simplified to
(
R XY m = E X n Y * n + m )
If the stochastic DT process is also ergodic then the time correlation
function is
1 N 1
*
*
Y
Y
RXY m = lim
X
n
n
+
m
=
X
n
n + m = R XY m N 2N
n= N
Autocorrelation
If X and Y represent the same stochastic DT process then the
correlation function becomes the special case called
autocorrelation.
(
R X m = E X n X * n + m )
For an ergodic stochastic DT process,
1
RX m = lim
N 2N
N 1
n= N
X n X * n + m = X n X * n + m = R X m Autocorrelation
(
R X n, n = E X 2 n )
Meansquared
value of X
For WSS stochastic discrete-time processes
N 1
1
2
2
n R X 0 = E X 2 n and RX 0 = lim
X
n
=
X
N 2N
n= N
MeanAverage
squared
Signal
value of X
Power of X
(
)
Autocorrelation
Example
()
Let X t be ergodic and let
each sample function be a
sequence of pulses of width
T whose amplitudes are A and
zero with equal probability and
with uniformly-distributed
pulse positions. Let the pulse
amplitudes be independent of
each other. Find the
autocorrelation.
Autocorrelation
Example
()
( ( ) ( )) = E ( X ( t ) X ( t + ))
R X = E X t X* t + For delays whose magnitude is greater than the pulse width,
the pulse amplitudes are uncorrelated and therefore
()
( ( )) ( (
RX = E X t E X t +
)) = ( A / 2 )
2
= A2 / 4 , > T
Autocorrelation
For delays whose magnitude is less than the pulse width it
is possible that the two times, separated by , are in the same
pulse. When is zero that probability is one and as approaches
the pulse width the probability approaches zero, linearly.
T
P t and t+ are in the same pulse interval =
, < ta
T
When the two times are in the same pulse there are two possible
cases of X t X t + A A and 0 0 both with probability 1/2.
(
)
() ( )
E ( X ( t ) X ( t + ) ) = A P ( A) + 0 P ( 0 ) = A
2
2
2
/2
When the two times are not in the same pulse there are four
possible cases of X t X t + A A , 0 0 , 0 0 , 0 0 each
with probability 1/4.
() (
)
( ( ) ( ) ) = A P ( A) P ( A) + 0 P ( A) P ( 0 ) + 0 P ( 0 ) P ( A) + 0 P ( 0 ) P ( 0 ) = A
E X t X t +
2
2
2
2
2
/4
Autocorrelation
For delays of magnitude less than the pulse width,
RX
times are in
= P
same pulse
()
times are not 2
2
A
/
2
+
P
A
/4
in same pulse
A2 T A2 T A2 2T =
+
1
=
2 T
4 T 4 T and the overall autocorrelation function is
RX
2T , T
A =
T
4 1
,
>
T
A2 T + 1
=
tri 4 T ()
2
Properties of Autocorrelation
Autocorrelation is an even function
()
( )
R X = R X or R X m = R X m The magnitude of the autocorrelation value is never greater
than at zero delay.
()
()
R X R X 0 or R X m R X 0 ()
If X has a non-zero expected value then R X or R X m will also and it will be the square of the expected value of X.
()
If X has a periodic component then R X or R X m will
also, with the same period.
Properties of Autocorrelation
{ ( )} is ergodic with zero mean and no periodic components
If X t
then
()
lim R X = 0 or lim R X m = 0
m Only autocorrelation functions for which
{ ( )} 0 for all f or F {R
F RX X
}
m 0 for all are possible
A time shift of a function does not affect its autocorrelation
Autocovariance
Autocovariance is similar to autocorrelation. Autocovariance
is the autocorrelation of the time-varying part of a signal.
()
()
( )
( )
C X = R X E 2 X or C X m = R X m E 2 X
Measurement of Autocorrelation
Autocorrelation of a real-valued WSS stochastic CT process can
be estimated by
T 1
R̂ X =
X t X t + dt , 0 << T
T 0
()
() ( )
But because the function X ( t ) is usually unknown it is much
more likely to be estimated from samples.
1 N n1
x k x k + n , n = 0,1,2,3,…, M << N
R̂ x nTs =
N n k =0
( )
where the x’s are samples taken from the stochastic process X
and the time between samples is Ts .
Measurement of Autocorrelation
The expected value of the autocorrelation estimate is
( ( ))
E R̂ X nTs
1 N n1
= E
X k X k + n N n
k =0
(
1 N n1
=
E X k X k + n N n k =0
( ( ))
E R̂ X nTs
)
1 N n1
=
R X nTs = R X nTs
N n k =0
This is an unbiased estimator.
( )
( )
Crosscorrelation
Properties
R XY = R YX ()
()
R XY R X
( )
(0) R (0)
Y
or R XY m = R YX m or R XY m R X 0 R Y 0 If two stochastic processes X and Y are statistically independent
()
( ) ( )
()
R XY = E X E Y * = R YX ( ) ( )
or R XY m = E X E Y * = R YX m If X is stationary CT and X is its time derivative
( ( ))
( ) ( ( )) R ( )
If Z ( t ) = X ( t ) ± Y ( t ) and X and Y are independent and at least
R XX d
=
RX d
one of them has a zero mean
()
X
()
d2
= 2 RX d
()
R Z = R X + RY