9.

a.

A WSS process X(n) is to be generated with RXX(O) = 0? and RXX(I) = pd by
passing white noise process with unit variance through a system described by a
stochastic differential equation X(n) = aX(n-l ) +bW(n). Find a and b. (4)

i(t) = ‏اہر‎ 0൭൪ converges to X(t) in mean square sense where the set of
orthonormal function (0൩൬) are the solution of the integral equation[-T/2.
+T/2]

இ
-7/2 Rxx(tl 2)on(t2)dt2= in $n(tl)
(5)

Also prove the coefficient of the random variable Xn are statistically orthogonal

(2)

Derive the sufficient condition for the random process X(t) to be ergodic in mean

(5)

If the WSS process X(t) is mean square periodic, then X(t) = X(t+T) in the mean

square sense. (3)

b.

Prove that

Anexp(jnwot)

in the mean square sense (4)

What is convergence in distribution and convergence in probability? Consider a
random sequence {Xn}, where = I - (I/n) and 1] =I/n
Check whether the random sequence Xn converges to zero in

1) Distribution 2) probability 3) mean square (5)