ELEC 303 – Random Signals

Lecture 19 – Random processesDr. Farinaz Koushanfar

ECE Dept., Rice UniversityNov 12, 2009

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(2f0t + )

X(t)= A cos(2f0t + )

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 n1

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)

dx)x(xf)t(m)]t(X[E )t(x0X0 0

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(2f0t + )• We can compute the mean• For [1,2], f()=1/2, and zero otherwise

• E[X(t)]= {0 to 2} A cos(2f0t+)/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(2f0t + )• We can compute the mean• For [1,], f()=1/, and zero otherwise

• MY(t) = E[Y(t)]= {0 to } A cos(2f0t+)/.d = -2A/ sin(2f0t)

• 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(-)