Transcript (Random Processes) - faraday - Eastern Mediterranean University

Chapter 6
Random Processes
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Description of Random Processes
Stationarity and ergodicty
Autocorrelation of Random Processes
Properties of autocorrelation
Huseyin Bilgekul
EEE 461 Communication Systems II
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
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Homework Assignments
• Return date: November 8, 2005.
• Assignments:
Problem 6-2
Problem 6-3
Problem 6-6
Problem 6-10
Problem 6-11
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Random Processes
• A RANDOM VARIABLE X, is a rule for
assigning to every outcome, w, of an
experiment a number X(w).
– Note: X denotes a random variable and X(w) denotes
a particular value.
• A RANDOM PROCESS X(t) is a rule for
assigning to every w, a function X(t,w).
– Note: for notational simplicity we often omit the
dependence on w.
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Ensemble of Sample Functions
The set of all possible functions is called the
ENSEMBLE.
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Random Processes
• A general Random or Stochastic
Process can be described as:
– Collection of time functions
(signals) corresponding to various
outcomes of random experiments.
– Collection of random variables
observed at different times.
• Examples of random processes
in communications:
– Channel noise,
– Information generated by a source,
– Interference.
t1
t2
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Collection of Time Functions
• Consider the time-varying function representing a
random process where wi represents an outcome of a
random event.
• Example:
– a box has infinitely many resistors (i=1,2, . . .) of same
resistance R.
– Let wi be event that the ith resistor has been picked up from
the box
– Let v(t, wi) represent the voltage of the thermal noise
measured on this resistor.
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Collection of Random Variables
• For a particular time t=to the value x(to,wi) is a random variable.
• To describe a random process we can use collection of random
variables {x(to,w1) , x(to,w2) , x(to,w3) , . . . }.
• Type: a random processes can be either discrete-time or continuoustime.
• Probability of obtaining a sample function of a RP that passes
through the following set of windows. Probability of a joint event.
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Description of Random Processes
• Analytical description: X(t) =f(t,w) where w is an
outcome of a random event.
• Statistical description: For any integer N and any
choice of (t1, t2, . . ., tN) the joint pdf of {X(t1), X( t2),
. . ., X( tN) } is known. To describe the random
process completely the PDF f(x) is required.
x1  x(t1 ), x  [ x1 , x2 ,... xN ]
f  x )  f {x(t1 ), x(t2 ),.... x(t N )}
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Example: Analytical Description
• Let X t )  A cos  2p f0t  q ) where q is a random variable
uniformly distributed on [0,2p].
• Complete statistical description of X(to) is:
– Introduce Y  2p f 0t0  q
– Then, we need to transform from y to x:
pX  x ) dx  pY  y1 ) dy  pY  y2 ) dy
– We need both y1 and y2 because for a given x the equation
x=A cos (y) has two solutions in [0,2p].
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Analytical (continued)
• Note x and y are actual values of the random
variables X and Y.
• Since dx
2
2
dy
 A sin y 
A x
and pY is uniform in [2pf0t0, 2pf0t0 + 2p], we get
1


p X  x )   p A2  x 2
0

A  x  A
Elsewhere
• Using the analytical description of X(t), we
obtained its statistical description at any time t.
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Example: Statistical Description
• Suppose a random process x(t) has the property that for any
N and (t0,t1, . . .,tN) the joint density function of {x(ti)} is a
jointly distributed Gaussian vector with zero mean and
covariance
 ij   2 min  ti , t j )
• This gives complete statistical description of the random
process x(t).
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Activity: Ensembles
• Consider the random process: x(t)=At+B
• Draw ensembles of the waveforms:
– B is constant, A is uniformly distributed between [-1,1]
– A is constant, B is uniformly distributed between [0,2]
• Does having an “Ensemble” of waveforms give
you a better picture of how the system performs?
x(t)
x(t)
2
B
t
t
Slope Random
B intersect is Random
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Stationarity
• Definition: A random process is STATIONARY to the
order N if for any t1,t2, . . . , tN,
fx{x(t1), x(t2),...x(tN)}=fx{x(t1+t0), x(t2+t0),...,x(tN +t0)}
• This means that the process behaves similarly (follows
the same PDF) regardless of when you measure it.
• A random process is said to be STRICTLY
STATIONARY if it is stationary to the order of N→∞.
• Is the random process from the coin tossing experiment
stationary?
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Illustration of Stationarity
Time functions pass
through the corresponding
windows at different times
with the same probability.
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Example of First-Order Stationarity
RANDOM PROCESS is x  t )  A sin w0t  q0 )
• Assume that A and w0 are constants; q0 is a
uniformly distributed RV from [p,p); t is time.
• From last lecture, recall that the PDF of x(t):
1


f x  x )   p A2  x 2
0

xA
x Elsewhere
• Note: there is NO dependence on time, the PDF is
not a function of t.
• The RP is STATIONARY.
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Non-Stationary Example
RANDOM PROCESS is x  t )  A sin w0t  q0 )
• Now assume that A, q0 and w0 are constants; t
is time.
• Value of x(t) is always known for any time
with a probability of 1. Thus the first order
PDF of x(t) is
f  x )    x  A sin w0t  q0 ) )
• Note: The PDF depends on time, so it is
NONSTATIONARY.
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Ergodic Processes
• Definition: A random process is ERGODIC if all time averages of any
sample function are equal to the corresponding ensemble averages
(expectations)
• Example, for ergodic processes, can use ensemble statistics to
compute DC values and RMS values
xDC  x  t )  [ x(t )]  mx
x t )
1
 lim
T  T


T
0
[ x(t )]dt
[ x(t )]   [ x] f ( x)dx mx

xRMS 
Time average
Ensmble average
x 2  t )  x 2   2  mx 2
• Ergodic processes are always stationary; Stationary processes are not
necessarily ergodic
Ergodic  Stationary
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Example: Ergodic Process
RANDOM PROCESS is x  t )  A sin w0t  q0 )
• A and w0 are constants; q0 is a uniformly
distributed RV from [p,p); t is time.
• Mean (Ensemble statistics)

p

p
mx  x   x q ) fq q ) dq   A sin w0t  q )
1
dq  0
2p
• Variance
x 2  
p
p
2
1
A
A2 sin 2 w0t  q )
dq 
2p
2
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Example: Ergodic Process
• Mean (Time Average) T is large
x t )
1
 lim
T  T

T
0
A sin w0t  q ) dt  0
• Variance
x2 t )
1 T 2 2
A2
 lim  A sin w0t  q ) dt 
0
2
T  T
• The ensemble and time averages are the same, so the
process is ERGODIC
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EXERCISE
• Write down the definition of :
– Wide sense stationary
– Ergodic processes
• How do these concepts relate to each other?
• Consider: x(t) = K; K is uniformly distributed
between [-1, 1]
– WSS?
– Ergodic?
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Autocorrelation of Random Process
• The Autocorrelation function of a real random
process x(t) at two times is:
) )
_________________
 
Rx  t1 , t2 )  x t x t    x1 x2 f x  x1 , x2 ) dx1dx2
1
2
 
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Wide-sense Stationary
• A random process that is stationary to order 2 or greater is
Wide-Sense Stationary:
• A random process is Wide-Sense Stationary if:
________
x  t )  constant
Rx  t1 , t2 )  Rx t )
• Usually, t1=t and t2=t+t so that t2- t1 =t.
• Wide-sense stationary process does not DRIFT with time.
• Autocorrelation depends only on the time gap but not where the
time difference is.
• Autocorrelation gives idea about the frequency response of the
RP.
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Autocorrelation Function of RP
• Properties of the autocorrelation function of wide-sense
stationary processes
_________
Rx  0 )  x 2  t )  Second Moment
Rx t )  Rx  t ) , Symmetric
Rx  0 )  Rx t ) , Maximum value at 0
Autocorrelation of slowly and rapidly fluctuating random processes.
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Cross Correlations of RP
• Cross Correlation of two RP x(t) and y(t) is
defined similarly as:
) )
_________________
 
Rxy  t1 , t2 )  x t y t    x1 y2 f xy  x1 , y2 ) dx1dy2
1
2
 
• If x(t) and y(t) are Jointly Stationary processes,
Rxy  t1 , t2 )  Rxy t2  t1 )  Rxy t )
t  t2  t1
• If the RP’s are jointly ERGODIC,
_________________
Rxy t )  x  t ) y  t  t )  x(t ) y (t  t )
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Cross Correlation Properties of Jointly
Stationary RP’s
• Some properties of cross-correlation functions are
Rxy t )  Rxy  t )
Rxy t )  Rx  0 ) Ry  0 )
Rxy t )  12  Rx  0 )  Ry  0 ) 
• Uncorrelated:
• Orthogonal:
__________________
Rxy t )  x  t ) y  t  t )  x y
Rxy t )  0
• Independent: if x(t1) and y(t2) are independent (joint
distribution is product of individual distributions)
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