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					ELEC 303 – Random Signals

 Lecture 19 – Random processes
       Dr. Farinaz Koushanfar
      ECE Dept., Rice University
             Nov 9, 2010
               Lecture outline
•   Basic concepts
•   Statistical averages,
•   Autocorrelation function
•   Wide sense stationary (WSS)
•   Multiple random processes
            Random processes
• A random process (RP) is an extension of a RV
• Applied to random time varying signals
• Example: “thermal noise” in circuits caused by
  the random movement of electrons
• RP is a natural way to model info sources
• RP is a set of possible realizations of signal
  waveforms governed by probabilistic laws
• RP instance is a signal (and not just one number
  like the case of RV)
                 Example 1
• A signal generator generates six possible
  sinusoids with amplitude one and phase zero.
• We throw a die, corresponding to the value F,
  the sinusoid frequency = 100F
• Thus, each of the possible six signals would be
  realized with equal probability
• The random process is X(t)=cos(2  100F t)
                 Example 2
• Randomly choose a phase  ~ U[0,2]
• Generate a sinusoid with fixed amplitude (A)
  and fixed freq (f0) but a random phase 
• The RP is X(t)= A cos(2f0t + )
X(t)= A cos(2f0t + )
               Example 3
• X(t)=X
• Random variable
  X~U[-1,1]
           Random processes
• Corresponding to each i in the sample space
  , there is a signal x(t; i) called a sample
  function or a realization of the RP
• For the different I’s at a fixed time t0, the
  number x(t0; i) constitutes a RV X(t0)
• In other words, at any time instant, the value
  of a random process is a random variable
Example: sample functions of a
      random process
                 Example 4
• We throw a die, corresponding to the value F,
  the sinusoid frequency = 100F
• Thus, each of the possible six signals would be
  realized with equal probability
• The random process is X(t)=cos(2  100F t)
• Determine the values of the RV X(0.001)
• The possible values are cos(0.2), cos(0.4),
  …, cos(1.2) each with probability 1/6
                   Example 5

•  is the sample space
  for throwing a die
• For all i let x(t; i)=
  i e-1
• X is a RV taking values
  e-1, 2e-1, …, 6e-1, each
  with probability 1/6
                  Example 6
• Example of a discrete-time random process
• Let i denote the outcome of a random
  experiment of independent drawings from
  N(0,1)
• The discrete–time RP is {Xn}n=1 to , X0=0, and
  Xn=Xn-1+ i for all n1
              Statistical averages
•   mX(t) is the mean, of the random process X(t)
•   At each t=t0, it is the mean of the RV X(t0)
•   Thus, mX(t)=E[X(t)] for all t
•   The PDF of X(t0) denoted by fX(t0)(x)

                                     
      E [ X ( t0 )]  m X ( t0 )   xf x ( t0 ) ( x )dx
                                     
Mean of a random process
                Example 7
• Randomly choose a phase  ~ U[0,2]
• Generate a sinusoid with fixed amplitude (A)
  and fixed freq (f0) but a random phase 
• The RP is X(t)= A cos(2f0t + )
• We can compute the mean
• For [1,2], f()=1/2, and zero otherwise
• E[X(t)]= {0 to 2} A cos(2f0t+)/2.d = 0
        Autocorrelation function
• The autocorrelation function of the RP X(t) is
  denoted by RX(t1,t2)=E[X(t1)X(t2)]
• RX(t1,t2) is a deterministic function of t1 and t2
                 Example 8
• The autocorrelation of the RP in ex.7 is




• We have used
                  Example 9
• X(t)=X
• Random variable X~U[-1,1]
• Find the autocorrelation function
   Wide sense stationary process
• A process is wide sense stationary (WSS) if its
  mean and autocorrelation do not depend on
  the choice of the time origin
• WSS RP: the following two conditions hold
  – mX(t)=E[X(t)] is independent of t
  – RX(t1,t2) depends only on the time difference =t1-
    t2 and not on the t1 and t2 individually
• From the definition, RX(t1,t2)=RX(t2,t1)  If RP
  is WSS, then RX()=RX(-)
           Example 8 (cont’d)
• The autocorrelation of the RP in ex.7 is




• Also, we saw that mX(t)=0
• Thus, this process is WSS
                 Example 10
• Randomly choose a phase  ~ U[0,]
• Generate a sinusoid with fixed amplitude (A) and
  fixed freq (f0) but a random phase 
• The new RP is Y(t)= A cos(2f0t + )
• We can compute the mean
• For [1,], f()=1/, and zero otherwise
• MY(t) = E[Y(t)]= {0 to } A cos(2f0t+)/.d
         = -2A/ sin(2f0t)
• Since mY(t) is not independent of t, Y(t) is
  nonstationary RP
                 Multiple RPs
• Two RPs X(t) and Y(t) are independent if for all t1
  and t2, the RVs X(t1) and X(t2) are independent
• Similarly, the X(t) and Y(t) are uncorrelated if for
  all t1 and t2, the RVs X(t1) and X(t2) are
  uncorrelated
• Recall that independence  uncorrelation, but
  the reverse relationship is not generally true
• The only exception is the Gaussian processes
  (TBD next time) were the two are equivalent
 Cross correlation and joint stationary
• The cross correlation between two RPs X(t)
  and Y(t) is defined as
            RXY(t1,t2) = E[X(t1)X(t2)]
  clearly, RXY(t1,t2) = RXY(t2,t1)
• Two RPs X(t) and Y(t) are jointly WSS if both
  are individually stationary and the cross
  correlation depends on =t1-t2
   for X and Y jointly stationary, RXY() = RXY(-)

				
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