Gujarat Technological University 2010 M.E Statistical Signal Analysis - Question Paper

parameter mean (m) = 0 and s=2. obtain the value of V that provide P[Y<0]=10-6.
06
Q.3 (a) provide ans of subsequent ques.. 06
1.
2.
Show that E[x] for the random variable with CDF FX(x)=1-1/x, for x<1, does not
exist.
describe the joint density function of 2 random variable and state how to get the
marginal density function from joint density.
(b)
1.
2.
provide ans of subsequent ques..
State the Chebyshev's inequality for random variable.
State and prove the central limit theorem.
06
OR
Q.3 (a) provide ans of subsequent ques.. 06
1.
2.
Show how Chebyshev's inequality is useful to decide the narrowness or broadness
of PDFs.
describe characteristics function and Moment generating function of random variable.
(b) provide ans of subsequent ques.. 06
1.
2.
For vectored random variable Xi of dimension of two x one obtain the PDF if random
variables are jointly Gaussian.
If Y=1/X2 , where X and Y both are random variables. obtain the FY(y)in terms of
FX(x)
Q.4 (a) provide ans of subsequent ques.. 06
1.
2.
Let Y= 2X+3, if random variable X is uniformly distributed over [-1, 2], obtain PDF
of random variable Y and Plot PDF of random variable X and Y.
For random variable X and Y Joint density is provided as,
2( )
0 x 1, 0 1
( , )
0 otherwise
XY
ax by
y
f x y a b
? +
? = = = =
= + ?
??
calculate the conditional expectation E[X|Y=1/2]
(b)
1.
2.
provide ans of subsequent ques.
Consider a random process X(t)= U cos(t) + V sin (t) where U and V are
independent random variables, every of which assumes the values -2 and one with the
probabilities ½ and 2/3 , respectively. Show that X(t) is Wide Sence Stationary
random process.
The autocorrelation function of a stochastic process X(t) is Rx(t)=0.5Nod(t). Such
process is called white process. If X(t) is the input to system as shown beneath obtain
the power spectral density at output of the system.
06
OR
Q.4(a) provide ans of subsequent ques.. 06
1.
2.
If W(t) is a Wiener process, the will
(a) a1W(t)+ a2 W(t+T) be a Wiener process? Here T > 0 and a1, a2 are real
constants.
(b) (W(t))2 be a Wiener process?
provide the classification of various kinds of random processes.
(b) A random process x(t) with the PSD shown in figure beneath is passed, through a
band pass filter with frequency response as shown in figure beneath. Determine the
Mean square values of the quadrature components of the output process, presume the
center frequency in the representation to be 0.5MHz.
06
? t
+
-
i/p
x(t)
o/p
y(t) - ? is attenuation
- t is Delay
**********
10-3
0 1MHz f
1
100kHz
0 0.5MHz f
Sx(w)
Q.5 (a) provide ans of subsequent ques.
06
1.
2.
obtain the autocorrelation function of random variable X whose power spectral
density shown in figure beneath.
Let Xn be a sequence of uncorrelated random variable with zero mean and variance
s2. Let
=
=S
1
( ) k
n
jw t
k
k
X t X e
X(t) is a wide sense stationary process?
(b)
1.
provide ans of subsequent ques..
Sketch the ensemble of the random process x(t) = at + b where b is a constant and a
is an RV uniformly distributed in the range (-2,2).
Just by observing the ensemble, determine whether this is a stationary or a non
stationary process.
06
2. discuss the significance of convergence in mean square sense.
OR
Q.5 (a) provide ans of subsequent ques.. 06
1.
2.
Are the subsequent covariance functions of a real stationary process (give reasons).
(a) sin 4t
(b) d(t-2)
(c)
1 |t|<1
( )
0 |t|>1
R t
?
= ?
?
provided a random process x(t) = k, where k is an RV uniformly
distributed in the range (-1,1).
(a) Sketch the ensemble of the random process
(b) Determine x(t)
(c) Determine Rx(t1,t2)
(b)
1.
2.
provide ans of subsequent ques..
provide the condition for which random process will become the Ergodic random
process.
describe the point wise convergence and uniform convergence of random variable.
06
S(f)
B
A
-f2 -f1 0 f1 f2 f

Earning:   Approval pending.