PART 3 Random Processes

Document Sample
PART 3 Random Processes Powered By Docstoc
					    PART 3
Random Processes




                   Huseyin Bilgekul
     Eeng571 Probability and astochastic Processes
   Department of Electrical and Electronic Engineering
           Eastern Mediterranean University              EE571   1
Random Processes




                   EE571   2
Kinds of Random Processes




                            EE571   3
              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.




                                                   EE571   4
Conceptual Representation of RP




                              EE571   5
Ensemble of Sample Functions
           The set of all possible functions is called the
           ENSEMBLE.




                                                  EE571      6
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,                       t1           t2
   – Information generated by a source,
   – Interference.




                                               EE571        7
                  Random Processes
Let  denote the random outcome of an experiment. To every such
outcome suppose a waveform                     X (t,  )
 X (t , ) is assigned.
                                                    
The collection of such            X (t,  )
                                  n
waveforms form a                  X (t,  )         
                                  k
stochastic process. The
set of { k } and the time        X (t,  )
                                  2
                                                    
index t can be continuous
                                  X (t,  )
or discrete (countably            1
                                                                   t
                                             0           t t
infinite or finite) as well.                   1        2



For fixed  i  S (the set of
all experimental outcomes), X (t , ) is a specific time function.
For fixed t,
                          X 1  X (t1 , i )
is a random variable. The ensemble of all such realizations
X (t , ) over time represents the stochastic                 EE571    8
Random Process for a Continuous Sample Space




                                        EE571   9
Random Processes




                   EE571   10
Wiener Process Sample Function




                                 EE571   11
EE571   12
Sample Sequence for Random Walk




                                  EE571   13
Sample Function of the Poisson Process




                                         EE571   14
Random Binary Waveform




                         EE571   15
Autocorrelation Function of the Random Binary Signal




                                             EE571   16
Example




          EE571   17
EE571   18
Random Processes
 Introduction (1)




                    EE571   19
                  Introduction

• A random process is a process (i.e., variation in
  time or one dimensional space) whose behavior
  is not completely predictable and can be
  characterized by statistical laws.
• Examples of random processes
  – Daily stream flow
  – Hourly rainfall of storm events
  – Stock index




                                              EE571   20
                      Random Variable
• A random variable is a mapping function which assigns outcomes of a
  random experiment to real numbers. Occurrence of the outcome
  follows certain probability distribution. Therefore, a random variable
  is completely characterized by its probability density function (PDF).




                                                              EE571   21
STOCHASTIC PROCESS




                     EE571   22
STOCHASTIC PROCESS




                     EE571   23
STOCHASTIC PROCESS




                     EE571   24
           STOCHASTIC PROCESS

• The term “stochastic processes” appears mostly in
  statistical textbooks; however, the term “random
  processes” are frequently used in books of many
  engineering applications.




                                              EE571   25
STOCHASTIC PROC ESS




                      EE571   26
  DENSITY OF STOCHASTIC PROCESSES
• First-order densities of a random process
     A stochastic process is defined to be completely or
     totally characterized if the joint densities for the
     random variables X (t1 ), X (t2 ),  X (tn ) are known for all
     times t1 , t2 ,, tn and all n.
     In general, a complete characterization is practically
     impossible, except in rare cases. As a result, it is
     desirable to define and work with various partial
     characterizations. Depending on the objectives of
     applications, a partial characterization often suffices to
     ensure the desired outputs.



                                                            EE571     27
DENSITY OF STOCHASTIC PROCESSES


• For a specific t, X(t) is a random variable with
  distribution F ( x, t )  p[ X (t ) .x]
• The function F ( x, t ) is defined as the first-order
  distribution of the random variable X(t). Its
  derivative with respect to x
                                F ( x, t )
                   f ( x, t ) 
                                  x

  is the first-order density of X(t).


                                                  EE571   28
  DENSITY OF STOCHASTIC PROCESSES


• If the first-order densities defined for all time t, i.e. f(x,t),
  are all the same, then f(x,t) does not depend on t and we call
  the resulting density the first-order density of the random
  process X (t ); otherwise, we have a family of first-order
  densities.
• The first-order densities (or distributions) are only a partial
  characterization of the random process as they do not
  contain information that specifies the joint densities of the
  random variables defined at two or more different times.



                                                          EE571   29
       MEAN AND VARIANCE OF RP
• Mean and variance of a random process
      The first-order density of a random process, f(x,t), gives the
      probability density of the random variables X(t) defined for all time
      t. The mean of a random process, mX(t), is thus a function of time
      specified by
                                              
       m X (t )  E[ X (t )]  E[ X t ]   xt f ( xt , t )dxt
                                              
• For the case where the mean of X(t) does not depend on t, we have

             mX (t )  E[ X (t )]  mX (a constant).
• The variance of a random process, also a function of time, is defined
  by
       X (t )  E  X (t )  m X (t )]2   E[ X t2 ]  [m X (t )] 2
        2
                   [

                                                                        EE571   30
     HIGHER ORDER DENSITY OF RP

• Second-order densities of a random process
    For any pair of two random variables X(t1) and X(t2),
    we define the second-order densities of a random
    process as f ( x1 , x2 ; t1 , t2 ) or f ( x1 , x2 ) .
• Nth-order densities of a random process
      The nth order density functions for X (t )at times
     t1 , t2 ,, tn are given by
   f ( x , x ,, x ; t , t ,, t ) or f ( x1 , x2 ,, xn ) .
      1   2      n   1   2   n




                                                        EE571   31
           Autocorrelation function of RP

• Given two random variables X(t1) and X(t2), a
  measure of linear relationship between them is
  specified by E[X(t1)X(t2)]. For a random process,
  t1 and t2 go through all possible values, and
  therefore, E[X(t1)X(t2)] can change and is a
  function of t1 and t2. The autocorrelation function
  of a random process is thus defined by

     R(t1 , t2 )  EX (t1 ) X (t2 )  R(t2 , t1 )

                                                  EE571   32
Autocovariance Functions of RP




                                 EE571   33
            Stationarity of Random Processes




  f x1 , x2 ,, xn ; t1 , t2 ,, tn )  f x1 , x2 ,, xn ; t1  , t2  ,, tn   )
• Strict-sense stationarity seldom holds for random
  processes, except for some Gaussian processes.
  Therefore, weaker forms of stationarity are needed.

                                                                             EE571    34
       Stationarity of Random Processes

               PDF of X(t)
X(t)




                                    Time, t




                                        EE571   35
Wide Sense Stationarity (WSS) of Random Processes




  EX (t )  m (constant) for all t.
  R(t1, t2 )  Rt2  t1 )  Rt2  t1 ), for all t1 and t2 .
                                                     EE571   36
          Equality and Continuity of RP

• Equality




• Note that “x(t, wi) = y(t, wi) for every wi” is not
  the same as “x(t, wi) = y(t, wi) with probability
  1”.



                                                 EE571   37
Equality and Continuity of RP




                                EE571   38
        Mean Square Equality of RP
• Mean square equality




                                     EE571   39
Equality and Continuity of RP




                                EE571   40
EE571   41
Random Processes
 Introduction (2)




                    EE571   42
Stochastic Continuity




                        EE571   43
Stochastic Continuity




                        EE571   44
Stochastic Continuity




                        EE571   45
Stochastic Continuity




                        EE571   46
Stochastic Continuity




                        EE571   47
Stochastic Continuity




                        EE571   48
         Stochastic Convergence

• A random sequence or a discrete-time random
  process is a sequence of random variables
  {X1(w), X2(w), …, Xn(w),…} = {Xn(w)}, w  .
• For a specific w, {Xn(w)} is a sequence of
  numbers that might or might not converge. The
  notion of convergence of a random sequence
  can be given several interpretations.




                                          EE571   49
Sure Convergence (Convergence Everywhere)

• The sequence of random variables {Xn(w)}
  converges surely to the random variable X(w) if the
  sequence of functions Xn(w) converges to X(w) as n
    for all w  , i.e.,
  Xn(w)  X(w) as n   for all w  .




                                               EE571   50
Stochastic Convergence




                         EE571   51
Stochastic Convergence




                         EE571   52
Almost-sure convergence (Convergence with
              probability 1)




                                    EE571   53
Almost-sure Convergence (Convergence with
              probability 1)




                                    EE571   54
Mean-square Convergence




                          EE571   55
Convergence in Probability




                             EE571   56
Convergence in Distribution




                              EE571   57
                        Remarks
•   Convergence with probability one applies to the
    individual realizations of the random process.
    Convergence in probability does not.
•   The weak law of large numbers is an example of
    convergence in probability.
•   The strong law of large numbers is an example of
    convergence with probability 1.
•   The central limit theorem is an example of convergence
    in distribution.




                                                    EE571    58
Weak Law of Large Numbers (WLLN)




                              EE571   59
Strong Law of Large Numbers (SLLN)




                                EE571   60
The Central Limit Theorem




                            EE571   61
Venn Diagram of Relation of Types of Convergence



                                 Note that even sure
                                 convergence may not
                                 imply mean square
                                 convergence.




                                             EE571   62
Example




          EE571   63
Example




          EE571   64
Example




          EE571   65
Example




          EE571   66
Ergodic Theorem




                  EE571   67
Ergodic Theorem




                  EE571   68
The Mean-Square Ergodic Theorem




                             EE571   69
   The Mean-Square Ergodic Theorem

The above theorem shows that one can expect
a sample average to converge to a constant in
mean square sense if and only if the average of
the means converges and if the memory dies
out asymptotically, that is , if the covariance
decreases as the lag increases.




                                           EE571   70
Mean-Ergodic Process




                       EE571   71
Strong or Individual Ergodic Theorem




                                 EE571   72
Strong or Individual Ergodic Theorem




                                 EE571   73
Strong or Individual Ergodic Theorem




                                 EE571   74
    Examples of Stochastic Processes



• iid random process
  A discrete time random process {X(t), t = 1, 2,
  …} is said to be independent and identically
  distributed (iid) if any finite number, say k, of
  random variables X(t1), X(t2), …, X(tk) are
  mutually independent and have a common
  cumulative distribution function FX() .



                                               EE571   75
       iid Random Stochastic Processes

• The joint cdf for X(t1), X(t2), …, X(tk) is given by
     FX 1 , X 2 ,, X k ( x1 , x2 ,, xk )  P X 1  x1 , X 2  x2 ,, X k  xk )
     FX ( x1 ) FX ( x2 ) FX ( xk )

• It also yields
   p X 1 , X 2 ,, X k ( x1 , x2 ,, xk )  p X ( x1 ) p X ( x2 ) p X ( xk )
  where p(x) represents the common probability
  mass function.



                                                                                EE571   76
Bernoulli Random Process




                           EE571   77
Random walk process




                      EE571   78
                Random walk process

• Let 0 denote the probability mass function of
  X0. The joint probability of X0, X1,  Xn is

  P( X 0  x0 , X 1  x1 ,, X n  xn )
   P X 0  x0 , 1  x1  x0 ,,  n  xn  xn 1 )
   P( X 0  x0 ) P(1  x1  x0 ) P( n  xn  xn 1 )
    0 ( x0 ) f ( x1  x0 ) f ( xn  xn 1 )
    0 ( x0 ) P( x1 | x0 ) P( xn | xn 1 )



                                                        EE571   79
         Random walk process

P( X n 1  xn 1 | X 0  x0 , X 1  x1 ,, X n  xn )
  P( X 0  x0 , X 1  x1 ,, X n  xn , X n 1  xn 1 )

         P( X 0  x0 , X 1  x1 ,, X n  xn )
   0 ( x0 ) P( x1 | x0 ) P( xn | xn 1 )  P( xn 1 | xn )

             0 ( x0 ) P( x1 | x0 ) P( xn | xn 1 )
 P( xn 1 | xn )



                                                      EE571    80
                     Random walk process


     The property
P( X n 1  xn 1 | X 0  x0 , X 1  x1 ,, X n  xn )  P( X n  xn 1 | X n  xn )
     is known as the Markov property.
     A special case of random walk: the Brownian
     motion.




                                                                         EE571    81
              Gaussian process

• A random process {X(t)} is said to be a
  Gaussian random process if all finite
  collections of the random process, X1=X(t1),
  X2=X(t2), …, Xk=X(tk), are jointly Gaussian
  random variables for all k, and all choices of t1,
  t2, …, tk.
• Joint pdf of jointly Gaussian random variables
  X1, X2, …, Xk:




                                               EE571   82
Gaussian process




                   EE571   83
Time series – AR random process




                                  EE571   84
                 The Brownian motion
      (one-dimensional, also known as random walk)


• Consider a particle randomly moves on a real line.
• Suppose at small time intervals  the particle jumps a small
  distance  randomly and equally likely to the left or to the
  right.
• Let X  (t ) be the position of the particle on the real line at
  time t.




                                                          EE571   85
                 The Brownian motion
• Assume the initial position of the particle is at the
  origin, i.e. X  (0)  0
• Position of the particle at time t can be expressed as
  X  (t )   Y1  Y2    Y[ t /  ] ) where Y1 , Y2 ,
  are independent random variables, each having
  probability 1/2 of equating 1 and 1.
  ( t /   represents the largest integer not exceeding
   t /  .)


                                                    EE571   86
                  Distribution of X(t)

• Let the step length  equal  , then
           X (t )   Y1  Y2    Y[t /  ] )
• For fixed t, if  is small then the distribution of X  (t )
  is approximately normal with mean 0 and variance t,
  i.e., X  (t ) ~ N 0, t ) .




                                                     EE571   87
Graphical illustration of Distribution of X(t)

             PDF of X(t)
X(t)




                                     Time, t




                                           EE571   88
• If t and h are fixed and  is sufficiently small
  then
     X  (t  h)  X  (t )
       Y1  Y2 
                              Y[(t  h ) / ] )  Y1  Y2     Y[ t /  ] ) 
                                                                                
       Y[t / ]1  Y[t / ] 2      Y[(t  h ) / ] )

        Yt    Yt  2  
                                         Yt  h  
                                                     
                                           




                                                                                    EE571   89
Graphical Distribution of the displacement of
                 X  (t  h)  X  (t )


• The random variable X  (t  h)  X  (t ) is normally
  distributed with mean 0 and variance h, i.e.
                                                u2 
P X  (t  h)  X  (t ) )  x 
                                    1    x

                                    2h   2h 
                                           exp
                                              
                                                     du
                                                     




                                                  EE571    90
• Variance of X  (t ) is dependent on t, while variance
  of X  (t  h)  X  (t ) is not.
• If 0  t1  t2    t2 m , then X  (t2 )  X  (t1 ) ,
  X  (t4 )  X  (t3 ), , X  (t2 m )  X  (t2 m 1 )
  are independent random variables.




                                                       EE571   91
X




     t




    EE571   92
Covariance and Correlation functions of                                                          X  (t )

Cov X  (t ), X  (t  h)  E X  (t ) X  (t  h)

 E  Y1  Y2    Y t     Y1  Y2    Yt  h  
     
     
                              
                                                       
                                                             
                                                               
                            2
                               Y  Y    Y   Y                                                    
 E  Y1  Y2    Y t    1 2                    t     t  1  Yt   2    Yt  h  
                                                                                                   
                            
                              2
 E  Y1  Y2    Y t   
                          

t

                                                       Cov  X  (t ), X  (t  h) 
              Correl  X  (t ), X  (t  h)  
                                                                 t  t  h )
                                                             t
                                                   =
                                                         t  t  h)                           EE571       93

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:10
posted:12/14/2011
language:English
pages:93