Ozyegin Univ. EE503 Spring 2015
Due in class, May 14, 2015
Homework 8
1. Digital modulation using PSK: The data to be modulated, cfw_Xn : n 0 , is modeled by a Bernoulli
process with p = 1/2. Dene the discrete-time phase process cfw_n : n 0 by
n =
+
2
if
clear all,close all,clc
% Part (a)
for i = 1:5
g = randi(2,1);
if (g = 1)
theta_n(i) = -pi/2;
else
theta_n(i) = pi/2;
end
end
theta_t = ones(1,100) * theta_n(1);
for i=2:5
theta_t = [theta_t ones(1,100)*theta_n(i)];
end
t=0:.01:4.99;
X_t=cos(4*pi*t)
clear all,close all,clc
% Part (a)
for i = 1:5
g = randi(2,1);
if (g = 1)
theta_n(i) = -pi/2;
else
theta_n(i) = pi/2;
end
end
theta_t = ones(1,100) * theta_n(1);
for i=2:5
theta_t = [theta_t ones(1,100)*theta_n(i)];
end
t=0:.01:4.99;
X_t=cos(4*pi*t)
clear all,close all,clc
% Part (a)
n = 1:200;
X = randn(1,200);
S = cumsum(X);
S = S./n;
subplot 411, plot(n,S);
xlabel('n'), ylabel('S_n');
title('3(a) Sample average sequence');
% Part (b)
X = randn(5000,200);
S = cumsum(X,2);
S = S./repmat(n,5000,1);
%
clear all,close all,clc
% Part (a)
n = 1:200;
X = randn(1,200);
S = cumsum(X);
S = S./n;
subplot 411, plot(n,S);
xlabel('n'), ylabel('S_n');
title('3(a) Sample average sequence');
% Part (b)
X = randn(5000,200);
S = (cumsum(X')';
S = S./repmat(n,5000,1);
Ozyegin Univ. EE503 Spring 2015
Due in class Apr. 9, 2015
Homework 5
1. Let X U[0, 1] and Y N (0, X 2 ). Find E XY 2 .
Solution:
1
E XY 2 = E E XY 2 |X
= E X E Y 2 |X
x3 dx =
= E XX 2 = E X 3 =
0
1
.
4
2. Let and X be random variables with
f () =
2
5 3
3
Ozyegin Univ. EE503 Spring 2015
Due 5pm Apr. 25, 2015
Homework 7
Please scan and send your homework to my e-mail: ali.ercan@ozyegin.edu.tr
1. In the lecture it was stated that conditionals of a Gaussian random vector are Gaussian. In this
problem you will
Ozyegin Univ. EE503 Spring 2015
Due in class Apr. 2, 2015
Homework 4
1. Function of uniform random variables.
Let X and Y be two independent U[0, 1] random variables. Find the probability density function (pdf)
of Z = [(X + Y ) mod 1] (i.e., Z = X + Y if
Ozyegin Univ. EE503 Spring 2015
Due 5pm Apr. 15, 2015
Homework 6
Please scan and send your homework to my e-mail: ali.ercan@ozyegin.edu.tr
1. Radar signal detection. The received signal S for a radar channel is 0 if there is no target and a random
variabl
Ozyegin Univ. EE503 Spring 2015
Due in class of Mar. 5, 2015
Homework 2
1. Let X be a random variable with the cdf shown below.
F (x)
1
2/3
1/3
1 2
x
3
x
1
2
3
4
Find the probabilities of the following events.
(a) cfw_X = 2.
(b) cfw_X < 2.
(c) cfw_X = 2 c
Ozyegin Univ. EE503 Spring 2015
Due in class Mar. 12, 2015
Homework 3
1. Consider the Laplacian random variable X with pdf f (x) = 1 e|x| .
2
(a) Sketch the cdf of X.
(b) Find Pcfw_|X| 2 or X 0 .
(c) Find Pcfw_|X| + |X 3| 3 .
Solution:
(a) We consider two
Ozyegin Univ. EE503 Spring 2015
Due in class of Feb. 26, 2015
Homework 1
1. Monty Hall. Gold is placed behind one of three curtains. A contestant chooses one of the curtains.
Monty Hall, the game host, opens an unselected empty curtain. The contestant can
Ozyegin Univ. EE503 Spring 2015
Mar. 15, 2015
Practice Problems 1
1. Unions and intersections. A number x is selected at random in the interval [1, +1]. Consider the
events A = cfw_x < 0, B = cfw_|x 0.5| < 1, and C = cfw_x > 0.75.
(a) Find the probabiliti
Ozyegin Univ. EE503 Spring 2015
Due in class Apr. 9, 2015
Homework 5
1. Let X U[0, 1] and Y N (0, X 2 ). Find E XY 2 .
2. Let and X be random variables with
f () =
2
5 3
3
01
0
otherwise
and X|cfw_ = Exp(). Find E(X).
3. Which of the following matrices
1)(contd)
(a) and (c)
Matlab Code
clear all,close all,clc
% Part (a)
for i = 1:5
g = randi(2,1);
if (g = 1)
theta_n(i) = -pi/2;
else
theta_n(i) = pi/2;
end
end
theta_t = ones(1,100) * theta_n(1);
for i=2:5
theta_t = [theta_t ones(1,100)*theta_n(i)];
end
t
EE503 Spring 2010-11
Due in class of May 11, 2011
Homework 10
1. Digital modulation using Phase Shift Keying. The stream of bits to be modulated (and sent over the
communication channel), cfw_Xn : n 0 , is modeled by a Bernoulli process with p = 1/2. Dene
EE503 Spring 2010-11
Due in class of Mar 16, 2011
Homework 5
1. A xed amount a is placed in one envelope and an amount 5a is placed in the other. One of the
envelopes is opened (each envelope is equally probable), and the amount X is observed to be in it.
EE503 Spring 2010-11
Due in class of Mar 16, 2011
Homework 5
1. A xed amount a is placed in one envelope and an amount 5a is placed in the other. One of the
envelopes is opened (each envelope is equally probable), and the amount X is observed to be in it.
Due in class of Mar 2nd , 2011
EE503 Spring 2010-11
Homework 3
1. Let X be a random variable with the cdf shown below.
F (x)
1
2/3
1/3
1 2
x
3
x
1
2
3
4
Find the probabilities of the following events.
(a)
(b)
(c)
(d)
cfw_X
cfw_X
cfw_X
cfw_X
= 2.
< 2.
= 2
EE503 Spring 2010-11
Due in class of Mar 9, 2011
Homework 4
1
1. Let X be a r.v. with Laplacian pdf fX (x) = 2 e|x| , and let Y be dened by the function of X shown
in the gure. Find the cdf, expected value and variance of Y
Y
+a
1
X
+1
a
2. Let X Exp() be