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Chapter 6Chapter 6Random ProcessesRandom Processes
Huseyin BilgekulEEE 461 Communication Systems II
Department of Electrical and Electronic Engineering Eastern Mediterranean University
Description of Random Processes Stationarity and ergodicty Autocorrelation of Random Processes Properties of autocorrelation
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Homework Homework AssignmentsAssignments
• 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 ProcessesRandom Processes• A RANDOM VARIABLE X, is a rule for
assigning to every outcome, of an experiment a number X(. – Note: X denotes a random variable and X( denotes
a particular value. • A RANDOM PROCESS X(t) is a rule for
assigning to every a function X(t, – Note: for notational simplicity we often omit the
dependence on .
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Ensemble of Sample Ensemble of Sample FunctionsFunctions
The set of all possible functions is called the ENSEMBLE.
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• 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
Random ProcessesRandom Processes
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Collection of Time FunctionsCollection of Time Functions• Consider the time-varying function representing a
random process where i represents an outcome of a random event.
• Example: – a box has infinitely many resistors (i=1,2, . . .) of same
resistance R. – Let i be event that the ith resistor has been picked up from
the box– Let v(t, i) represent the voltage of the thermal noise
measured on this resistor.
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Collection of Random Collection of Random VariablesVariables
• For a particular time t=to the value x(to,i is a random variable. • To describe a random process we can use collection of random
variables {x(to,1 , x(to,2 , x(to,3 , . . . }. • Type: a random processes can be either discrete-time or continuous-
time. • 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 Description of Random ProcessesProcesses
• Analytical description: X(t) =f(t,) where 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.
1 1 1 2
1 2
( ), [ , ,... ]
{ ( ), ( ),.... ( )}N
N
x x t x x x
f f x t x t x t
xx
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Example: Analytical Example: Analytical DescriptionDescription
• Let where is a random variable uniformly distributed on [0,2.
• Complete statistical description of X(to) is: – Introduce – Then, we need to transform from y to x:
– We need both y1 and y2 because for a given x the equation x=A cos (y) has two solutions in [0,2].
0cos 2X t A f t
0 02Y f t
1 2X Y Yp x dx p y dy p y dy
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Analytical (continued)Analytical (continued)• Note x and y are actual values of the random
variables X and Y. • Since
and pY is uniform in [2f0t0, 2f0t0 + 2we get
• Using the analytical description of X(t), we obtained its statistical description at any time t.
2 2sindx A y A xdy
2 2
1
0 ElsewhereX
A x Ap x A x
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Example: Statistical Example: Statistical DescriptionDescription
• 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
• This gives complete statistical description of the random
process x(t).
2 min ,ij i jt t
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Activity: EnsemblesActivity: 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?
B
x(t)
t
Slope Random
2x(t)
t
B intersect is Random
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StationarityStationarity• 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 StationarityIllustration of Stationarity
Time functions pass through the corresponding windows at different times with the same probability.
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Example of First-Order Example of First-Order StationarityStationarity
• Assume that A and 0 are constants; 0 is a uniformly distributed RV from ) t is time.
• From last lecture, recall that the PDF of x(t):
• Note: there is NO dependence on time, the PDF is not a function of t.
• The RP is STATIONARY.
0 0RANDOM PROCESS is sinx t A t
2 2x
1
0 Elsewhere
x Af x A x
x
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Non-Stationary ExampleNon-Stationary Example
• Now assume that A, 0 and 0 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
• Note: The PDF depends on time, so it is NONSTATIONARY.
0 0sinf x x A t
0 0RANDOM PROCESS is sinx t A t
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Ergodic ProcessesErgodic 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
• Ergodic processes are always stationary; Stationary processes are not necessarily ergodic
Ergodic tationaryS
0
2 2 2 2
[ ( )]
1 [ ( )] Time average
[ ( )] [ ] ( ) Ensmble average
limDC x
T
T
x
RMS x
x x t x t m
x t x t dtT
x t x f x dx m
x x t x m
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Example: Ergodic ProcessExample: Ergodic Process
• A and 0 are constants; 0 is a uniformly distributed RV from ) t is time.
• Mean (Ensemble statistics)
• Variance
01sin 0
2xm x x f d A t d
2
2 2 20
1sin2 2x
AA t d
0 0RANDOM PROCESS is sinx t A t
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Example: Ergodic ProcessExample: Ergodic Process• Mean (Time Average) T is large
• Variance
• The ensemble and time averages are the same, so the process is ERGODIC
00
1 sin 0limT
Tx t A t dt
T
2
2 2 200
1 sin2lim
T
T
Ax t A t dtT
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EXERCISEEXERCISE• 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 Autocorrelation of Random ProcessProcess
• The Autocorrelation function of a real random process x(t) at two times is:
_________________
1 2 1 2 1 2 1 2, ,1 2x xR t t x t x t x x f x x dx dx
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Wide-sense StationaryWide-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:
• Usually, t1=t and t2=t+ so that t2- t1 =• 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 Autocorrelation Function of RPRP• Properties of the autocorrelation function of wide-sense
stationary processes
Autocorrelation of slowly and rapidly fluctuating random processes.Autocorrelation of slowly and rapidly fluctuating random processes.
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Cross Correlations of RPCross Correlations of RP• Cross Correlation of two RP x(t) and y(t) is
defined similarly as:
• If x(t) and y(t) are Jointly Stationary processes,
• If the RP’s are jointly ERGODIC,
_________________
xy 1 2 1 2 xy 1 2 1 2, ,1 2R t t x t y t x y f x y dx dy
xy 1 2 xy 2 1 xy 2 1, R t t R t t R t t
_________________
xy ( ) ( )R x t y t x t y t
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Cross Correlation Properties of Cross Correlation Properties of Jointly Stationary RP’sJointly Stationary RP’s
• Some properties of cross-correlation functions are
• Uncorrelated:
• Orthogonal:
• Independent: if x(t1) and y(t2) are independent (joint distribution is product of individual distributions)
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