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(ELEC530)[2007](f)midterm~2250^_10331.pdf
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29 October 2007 Bertram Shi
ELEC 530
Probability, Random Variables and Stochastic Processes
Midterm Exam

Name: Student ID:


show all your work



unless otherwise specified, numerical answers are not required. You can specify answers in a format that can be easily evaluated on a standard handheld calculator For example, an answer specified as


55 22

AeCB + C where A = ------. B = ---------------
-. and C = 18.5
38 38
+
is OK.

Question Total Score
1 10
2 10
3 10
4 10
5 10
6 10
60

1. Suppose that Xt() is a continuous-time wide sense stationary Gaussian random process with non-zero mean and autocorrelation given by
t1C t2
mX()t= 1 and RX(t1. t2) = 6e C4
+1 Let Z =2X 2C 0.5X(1.9) . Write a single or a set of closed form expression(s) for fZ z
( ) () over the entire range of z and give the numerical value of () .
fZ 0
2. Suppose that St() and Nt() are independent continuous-time wide sense stationary processes with power spectral densities given by
32
S() = ----------------S() = -------------------
Sj.Nj.
.2+5 .2 + 10
Let Zt() = St()+ Nt() . Find the transfer function of the minimum mean squared error filter for estimatingSt( + 0.1) from Z. for
() .. t.
3. Suppose that Xt() is a zero mean continuous-time wide sense stationary random process with autocorrela-tion
R()= e C .
X.
and that t .0.2 0 ... 5
() ()XtC . ()
Yt= h.()d. where h.=
..
0 0 otherwise
.
Find EYt. ()2. for t. 0.
4. Suppose that a random process Zt() is
Zt() = St()+ Nt()
where St() and Nt() are continuous time zero-mean independent wide sense stationary random processes with power spectral densities
24 50
S() = ------------------and () ---------------------
Sj.SNj.= .2+ 16 .2 + 100
Suppose we try to remove the noise by using a low pass filter with transfer function
K
Hs() = -----------
s+5 Let the output of the filter be Yt() . Find the value ofK that minimizes the mean squared error between Yt() and St(), ESt.. ()C Yt().2. .
5. Suppose that Xand Yare jointly distributed with probability density function which is given by
.
x
. 6xy .0 . x. 2. and .0 . y. 1 C --
.
f( . )= ..
XYxy. 2 .
. 0 otherwise where ux is the unit step function. Find .
() EYX= 0.5. . Be careful with your limits of integration here.
6. Suppose that S and N are independent discrete-time wide sense stationary random process with autocor-
nn
relation functions
( ) = 3.15. k and
() ()
RSkRNk
=

.....

2.05 if k =0
1 if k = .1
0 otherwise
Find the impulse response of the non-causal Wiener filter for estimating from
+

for all
m
C.. m .. .