ELEC 303 – Random Signals
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Transcript of ELEC 303 – Random Signals
ELEC 303 – Random Signals
Lecture 19 – Random processesDr. Farinaz Koushanfar
ECE Dept., Rice UniversityNov 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(2f0t + )
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 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
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
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(-)