1. ELEC 303 – Random Signals Lecture 19 – Random processes Dr. Farinaz Koushanfar ECE Dept., Rice University Nov 12, 2009

2. Lecture outline Basic concepts Statistical averages, Autocorrelation function Wide sense stationary (WSS) Multiple random processes

3. 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)

4. 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)

5. Example 2 Randomly choose a phase  ~ U[0,2  ] Generate a sinusoid with fixed amplitude (A) and fixed freq (f 0 ) but a random phase  The RP is X(t)= A cos(2  f 0 t +  )

6. X(t)= A cos(2  f 0 t +  )

7. Example 3 X(t)=X Random variable X~U[-1,1]

8. 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 t 0, the number x(t 0 ;  i ) constitutes a RV X(t 0 ) In other words, at any time instant, the value of a random process is a random variable

9. Example: sample functions of a random process

10. 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

11. 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

12. 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 {X n } n=1 to , X 0 =0, and X n =X n-1 +  i for all n  1

13. Statistical averages m X (t) is the mean, of the random process X(t) At each t=t 0, it is the mean of the RV X(t 0 ) Thus, m X (t)=E[X(t)] for all t The PDF of X(t 0 ) denoted by f X(t0) (x)

14. Mean of a random process

15. Example 7 Randomly choose a phase  ~ U[0,2  ] Generate a sinusoid with fixed amplitude (A) and fixed freq (f 0 ) but a random phase  The RP is X(t)= A cos(2  f 0 t +  ) We can compute the mean For  [1,2  ], f  (  )=1/2 , and zero otherwise E[X(t)]=  {0 to 2  } A cos(2  f 0 t+  )/2 .d  = 0

16. Autocorrelation function The autocorrelation function of the RP X(t) is denoted by R X (t 1,t 2 )=E[X(t 1 )X(t 2 )] R X (t 1,t 2 ) is a deterministic function of t 1 and t 2

17. Example 8 The autocorrelation of the RP in ex.7 is We have used

18. Example 9 X(t)=X Random variable X~U[-1,1] Find the autocorrelation function

19. 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 – m X (t)=E[X(t)] is independent of t – R X (t 1,t 2 ) depends only on the time difference  =t 1 - t 2 and not on the t 1 and t 2 individually From the definition, R X (t 1,t 2 )=R X (t 2,t 1 )  If RP is WSS, then R X (  )=R X (-  )

20. Example 8 (cont’d) The autocorrelation of the RP in ex.7 is Also, we saw that m X (t)=0 Thus, this process is WSS

21. Example 10 Randomly choose a phase  ~ U[0,  ] Generate a sinusoid with fixed amplitude (A) and fixed freq (f 0 ) but a random phase  The new RP is Y(t)= A cos(2  f 0 t +  ) We can compute the mean For  [1,  ], f  (  )=1/ , and zero otherwise M Y (t) = E[Y(t)]=  {0 to  } A cos(2  f 0 t+  )/ .d  = -2A/  sin(2  f 0 t) Since m Y (t) is not independent of t, Y(t) is nonstationary RP

22. Multiple RPs Two RPs X(t) and Y(t) are independent if for all t 1 and t 2, the RVs X(t 1 ) and X(t 2 ) are independent Similarly, the X(t) and Y(t) are uncorrelated if for all t 1 and t 2, the RVs X(t 1 ) and X(t 2 ) 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

23. Cross correlation and joint stationary The cross correlation between two RPs X(t) and Y(t) is defined as R XY (t 1,t 2 ) = E[X(t 1 )X(t 2 )]  clearly, R XY (t 1,t 2 ) = R XY (t 2,t 1 ) Two RPs X(t) and Y(t) are jointly WSS if both are individually stationary and the cross correlation depends on  =t 1 -t 2  for X and Y jointly stationary, R XY (  ) = R XY (-  )