EE 536 Digital Communications Systems
Midterm Exam (Take-Home) Due: 4-June-2014, 17:00
20 May 2015
Surname
Name
ID
I declare that I have solved all the problems on my own and
indicated any reference I have used during the work clearly, and
did not supply information about my own solutions to any other
student, and finally, I will accept any grade for this exam if the
instructor determines that I have violated the exam rules.
Signature:
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Note: Supply printout of source codes you have used and whenever
feasible submit computer generated plots.
1) (30 pts) We consider a simplified analysis of the Gardners
symbol synchronizer (see class notes) in this problem.
Let
=k
kj kTtgIetx )()(
be a lowpass equivalent PAM signal that appears at the output of
the matched filters (following the I/Q demodulator) in the
receiver. denotes the carrier phase error. We omit noise. kI are
independent equally likely quaternary, j ,1 symbols. is the unknown
delay which is to be estimated by the synchronizer.
a) Find the expected value, P, of the phase detector signal ( nv
:class notes) as a function of the actual delay, , the current
delay estimate used by the synchronizer, , and other relevant
parametes.
b) Find and plot the phase detector characteristics, P, as a
function of estimated delay, ,for the pulse shape TtT
TtT
Atg
=)( . Explain qualitatively how this synchronizer works based on
your findings.
2) (35 pts) Let = + 0.8 + 0.3 +
where are 1,3 independent, equally likely symbols. are real,
white, Gaussian. SNR of (average power of all signal components
divided by average noise power) is 30 dB.
a) Compute the taps of the MMSE equalizer of length=11. Give all
important intermediate results leading to your solution.
b) Find the resulting MMSE and the SINR (desired signals power
to power of all interference at the output) in dB.
c) Write a Matlab code to operate an LMS equalizer for the
channel described in this problem (do not use stock programs in
Matlab tool boxes). Equalizer length is 11. Simulate the equalizer
for 2000 iterations. Initial value of the equalizer tap vector is
all zero, except the center tap, which is set to 1. Obtain curves
(learning curves) for 50 independent runs and plot the average of
them. Try step size () values: [0.0005, 0.001, 0.002, 0.005, 0.01].
Comment on the convergence rate and MSE attained at the end of 2000
iterations. What is the maximum value of which you observe for the
algorithm to converge?
3) (35 pts) The discrete time equivalent channel filter in a
communication system is 15.08.0)( = zzF
with independent equally likely 1,3 sequence input, white
Gaussian noise sequence corruption at the output. We make 7
observations from the channel as:
1v
2v
3v
4v
5v 6v 7v
-1.78 5.48 -2.77 -0.64 2.23 0.59 0.03
