ECEN 661 - Modulation Theory
Homework #9
Assignment Date: 4/23/08
Due Date: 5/1/08 (optional)
This homework assignment is optional. You do not need to turn in solutions. If you choose to turn
in this homework by the above due date, you can use your score
ECEN 661 - Modulation Theory
Homework #8 Solutions
1. Let the BPSK signal be written as s ( t ) = A cos ( c t + ) over the bit interval ( 0, T s ) .
(a) The received will base its decision on a statistic of the form
Ts
z = 2 r ( t ) cos ( c t + ) dt .
0
T
ELEN 661 - Modulation Theory
Homework #8
Assignment Date: 4/14/08
Due Date: 4/22/08
1. Sensitivity of PSK to Phase Errors.
(a) Derive an expression for the probability of error of a coherent BPSK demodulator where the
local oscillator used for demodulatio
ECEN 661 - Modulation Theory
Homework #7 Solutions
1.
(a) First, we start with the trellis diagram for h = 1 3
n =
.
I3 = 1 / e j 2 3 p ( t )
2
n = -3
n = -3
I2 = 1 / ej 3 p ( t ) I = 1 / ej 2 3 p* ( t )
3
I1 = 1 p ( t )
I2 = 1 / ej 3 p * ( t )
I3 = 1
ELEN 661 - Modulation Theory
Homework #7
Assignment Date: 3/26/08
Due Date: 4/2/08
1.
(a) Sketch the phase trellis diagrams for CPFSK with modulation indices of h = 1 3 , h = 1 2
and h = 2 3 .
2
(b) Calculate the asymptotic power efficiency, = d min ( 4 E
ECEN 661 - Modulation Theory
Homework #6 Solutions
1. Let the two FSK waveforms be written as
f
s 0 ( t ) = A cos 2 f c + - t +
2
f
s 1 ( t ) = A cos 2 f c - t +
2
so that the signals are separated in frequency by f . Then the correlation coefficient be
ECEN 661 - Modulation Theory
Homework #6
Assignment Date: 3/17/08
Due Date: 3/25/08
1. (Text 5.12, 4th edition, modified). In Section 4.3.1 in the text it was shown that the minimum
frequency separation for orthogonality of binary FSK signals with coheren
ECEN 661 - Modulation Theory - Spring 2008
Homework #5 Solutions
1.
The decision statistic z will be of the form z = s + n where s cfw_ 0, AT and n is a zero-mean
Gaussian random variable with variance of 2 = N o T 2 .
(a) The decision threshold needs to
ECEN 661 - Modulation Theory
Homework #5
Assignment Date: 2/25/08
Due Date: 3/3/08
1. (Text 5-4, 4th edition - modified) A binary digital communication system employs the signals
s0 ( t ) = 0 , 0 t < T ,
s1 ( t ) = A , 0 t < T .
This is called on-off keyi
ECEN 661 Modulation Theory
Homework #4 Solutions
Spring 2008
1.
(a)
G(f) 2
ss ( f ) = -T
where
T
G ( f ) = - sinc ( fT 2 ) cfw_ e j fT 2 e j 3 fT 2
2
= jT sinc ( fT 2 ) sin ( fT 2 ) e j fT .
Hence,
ss ( f ) = T sinc 2 ( fT 2 ) sin 2 ( fT 2 ) .
(b)
G(f)
ECEN 661 - Modulation Theory
Homework #4
Assignment Date: 2/13/08
Due Date: 2/20/08
g(t)
1. (Text 4.34, 4th edition) The information sequence cfw_ a n is a
sequence of iid random variables, each taking values 1 with
equal probabilities. This sequence is
function z=GMSKphase(BT,t)
% z=GMSKphase(BT,t)
% Creates the phase pulse shape for a GMSK signal with a
% time bandwidth product of BT.
beta=2*pi*BT/sqrt(log(2);
tu=beta*(t+1/2);
tl=beta*(t-1/2);
z=(exp(-(tu.^2)/2)-exp(-(tl.^2)/2)/sqrt(2*pi);
z=z-tu.*Q(tu
function z=GMSKfreq(BT,t)
% z=GMSKfreq(BT,t)
% Creates the frequency pulse shape for a GMSK signal with a
% time bandwidth product of BT.
beta=2*pi*BT/sqrt(log(2);
tu=beta*(t+1/2);
tl=beta*(t-1/2);
z=Q(tl)-Q(tu);
ECEN 661 - Modulation Theory
Homework #3
Assignment Date: 2/4/08
Due Date: 2/11/08
1. Sketch the phase trellis diagram and the phase state diagram for each of the following modulation formats:
2
(a) Full response binary CPFSK with h = - .
3
1
(b) Partial
ECEN 661 - Modulation Theory
Homework #2 Solutions
1.
(a). Start with (4.1-38):
n ( t ) = x ( t ) cos ( c t ) y ( t ) sin ( c t ) .
Next compute the autocorrelation of the bandpass process,
E [ n ( t ) n ( t + ) ] = E [ x ( t ) x ( t + ) ] cos ( c t ) cos
ECEN 661 - Modulation Theory
Homework #2
Assignment Date: 1/28/08,
Due Date: 2/4/08
1. Text 4.3 (4th ed.) Suppose that N ( t ) is a zero-mean, stationary, narrowband process represented by
N ( t ) = X ( t ) cos ( c t ) Y ( t ) sin ( c t ) .
The autocorrel
function [S, f]=Psd(x,dx)
% This function estimates the PSD of the input signal.
N=length(x);
cor=conv(x,fliplr(conj(x)/N;
Nr=length(cor);
S=fft(cor);
S=fftshift(dx*abs(S);
Nf=(Nr-1)/2;
df=1/(dx*Nr);
f=[-Nf:Nf]*df;
ECEN 661 - Modulation Theory
Homework #1
Assignment Date: 1/16/08
Due Date: 1/23/08
1. An audio signal m ( t ) is FM modulated to produce
s ( t ) = A cos 2 f c t + 2 k f
t
m ( ) d ,
where f c is the carrier frequency and k f is the frequency deviation co
clear all;
N=20; 0umber of wave periods
Ns=50; 0umber of samples per period
dt=10^-5; %time between samples
T=dt*50; %time of the square wave period
t=[dt:dt:T*N];
%create the square wave
m=zeros(1,N*Ns);
for i=1:N
m(i-1)*Ns+1:(i-1)*Ns+Ns/2)=1;
m(i-1)*Ns+
ECEN 661 - Modulation Theory
Take Home Final
Assignment Date: 4/28/08
Due Date: 5/5/08
Preface: The final exam in this course will be a take-home exam. The exercise described here is
to be turned in by noon on Tuesday May 6th. This assignment is to be don
ECEN 661 -Modulation Theory
Spring 2008
Midterm Exam #2
Problem 1. (40 points)
A certain 8-ary quadrature amplitude modulation scheme sends one of the following signals:
s i ( t ) = A i cos ( c t ) + B i sin ( c t ) , 0 t < T s , i = 1, 2, , 8 ,
where A i
EE661 -Modulation Theory
Spring 2008
Practice Midterm Exam #2
The following problems were taken from previous exams. They were not given all in the same
year so do not interpret the set of problems as representative of a complete exam, but rather take
eac
ECEN 661
Exam 1 Solutions
Spring 2008
Problem 1. (25 points)
H(f)
Zero-mean, white Gaussian noise with
a PSD of N 0 2 is passed through a
BPF whose transfer function is as
described in the figure.
B
Half cycle of
a sine wave
A
f
fc
(a) Find the variance o