dit View Go Tools Window Help Final Exam_Review Questions.pdf (page 9 of 10) Lec

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dit View Go Tools Window Help Final Exam_Review Questions.pdf (page 9 of 10) Lecture 10: Introduction to Random Processes 31. An ergodic random process X(0) has the autocorrelation function Rxx (r) = 12+36r' 1 +4z2 Determine the following: a. The mean of X(t) b. The mean-square value of X(t) c. Assume that X(0) has a direct-current component K and a noise component N(o). That is, X() K+ N(t). Then we know that we can express the autocorrelation function as follows: Rx (t) KRw (t). Find the expressions for K and Ry (t)

Explanation / Answer

First of all,In econometrics and signal processing,a stochastic process is said to be ergodic if its statistical properties can be deduced from a single,sufficiently long,random sample of the process.if process is not ergodic means erratic.

Auto correlation function Rxx(T)=(12+36T2)/(1+4T2).

Auto correlation means correlation of same signal with it self.

Auto correlation gives the product of signal with its shifted version.that means

Rxx(T)=| -&->&x(t)x(t-T)dt=(12+36T2)/(1+4T2). I am taking integral symbol as | and infinity as &.

From this relation

x(t)x(t-T)=d/dT(12+36T2)/(1+4T2)

Apply derivative formula of u/v,we will get

x(t)x(t-T)=((1+4T2)72T-(12+36T2)(8T))/(1+4T2)2. substitute t in place of T.

x(t)x(t-T)=((72t+288T3)-(96T+288T3))/(1+4T2)2.

Finally

x(t)x(t-T)=-24T/(1+4T2)2

From the above expression X(t) maybe =(1/(1+4T2).

In other way,Rxx(T) directly equal to E(x(t)x(t)).

(a).mean of x(t)=|-&->&xX(t)dx=x(t)=(1/1+4T2)(x2/2)-&->&=&.

Normally maximum value of Rxx(T) gives the total power spectral densty of the given signal when Rxx(T)=Rxx(0).

Substitute 0 in place of T in the given autocorrelation function

Rxx(T)=Rxx(0)=12.

This is the maximum mean for the signal X(t).

(b).Mean square value means square root of the mean of X(t).

Mean square value of X(t)=&=&.

Maximum Mean square value of signal =12=23.

(c).Now,assume that the signal X(t) has both direct component and noise component that is X(t)=k+N(t).

Then we can express autocorrelation function as Rxx(T)=k2+RNN(T).

Now,from given data in the problem

k2+RNN(T)=(12+36T2)/(1+4T2)=(12/(1+4T2))+(36T2/(1+4T2)).

After equating above expressions on both sides,we will get the expressions for K2 and RNN(T) as below.

K2=(12/(1+4T2)).

RNN(T)=(36T2/(1+4T2)).

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