# Modern Communication Systems (EEET 4036), SP5-2015

Modern Communication Systems (EEET 4036), SP5-2015
University of South Australia
Homework Assignment 2
September 10, 2015
This homework assignment has 2 questions. There are 29 marks in total. Note that the questions can be
solved independently.
Your answers must be submitted as a single .pdf file not exceeding 4 pages. You may include plots or
drawings to illustrate your answers, but use them sparingly. Matlab code listings are to be submitted as
.m files. All files, combined into a single .zip file, must be submitted via the course website.
In this homework assignment we consider QAM baseband transmission over a noisy channel with signal
attenuation and phase offsets. A block of (L+K) symbols is transmitted at a symbol rate of Rs = 1/T
symbols per second. The symbols uj with j = 1, 2, . . . , L+K are taken from some QAM constellation
alphabet A. The transmitted waveform is
u(t) =
L
X
+K
j=1
ujp(t – jT), (1)
where p(t) is a unit-energy root Nyquist pulse. The propagating signal is both attenuated and experiences
a phase rotation. The scaled, phase-rotated and noisy received signal can be written as
r(t) = a0u(t)e
i?0 + z(t), (2)
where z(t) models the noise, a0 = 0 is the (real-valued) signal amplitude, and ?0 ? [-p, p) is the phase
offset. The received waveform r(t) is passed through a matched filter q(t) to obtain y(t) = r(t)*q(t). The
matched filter output y(t) is sampled at t = kT, where k = 1, 2, . . . , L+K, resulting in the discrete-time
signal model
yk = a0uke
i?0 + wk, (3)
where yk = y(kT) and where wk are the noise samples. In the following, we will exclusively consider the
discrete-time model in (3).
Signal amplitude a0 and phase offset ?0 are unknown to the receiver and need to be estimated before being
able to recover the transmitted information. For the purpose of estimating the unknown parameters, a
fixed length-L sequence {sk} of pilot symbols that are known to the receiver is transmitted at the start
of the burst (also known as preamble). That is, uk = sk for k = 1, . . . , L, and the remaining K symbols
uk for k = L+1, . . . , L+K are random data symbols.
Q1. In this question you will derive the least squares (LS) estimators of the unknown signal amplitude
a0 and the unknown phase offset ?0.
(a) (5 marks)
Give an expression for the LS objective function S(a, ?) and the corresponding minimisation
problem for estimating both a0 and ?0. Based on these expressions, briefly describe the concept
of LS estimation.
(b) (3 marks)
Derive the LS estimator ˆ? of the phase offset by minimising the objective function S(a, ?)
from (a) with respect to ?.
(c) (6 marks)
Substitute ˆ? from (b) into the objective function S(a, ?) from (a) to obtain the objective
function minimised with respect to the phase offset, i.e.
S(a) = min
??[-p,p)
S(a, ?) = S(a, ?ˆ). (4)
Note that the resulting function S(a) only depends on a single variable a. First simplify S(a)
and then derive the LS estimator ˆa of the signal amplitude by minimising with respect to a.
1
Modern Communication Systems (EEET 4036), SP5-2015
University of South Australia
Now consider the case where the signal amplitude is a0 = 1 such that the signal model (3) simplifies to
yk = uke
i?0 + wk. (5)
As shown in the lecture, the corresponding LS phase estimator is
?ˆ = ?
X
L
k=1
yku
*
k
!
= ?
X
L
k=1
yks
*
k
!
, (6)
where ?(·) denotes the complex argument and where it was taken into account that the L symbols used
by the estimator are preamble symbols uk = sk.
Q2. In this question, you will implement and simulate a transmitter, channel with phase offset and
noise, and receiver as shown in Fig. 1. You will implement the shaded components, but will be
provided with a Matlab implementation of the remaining components. Note that Fig. 1 without
the shaded components is the same setup considered in Q2 of Homework Assignment 1.
Adhere to the Matlab implementation guidelines included at the end of this assignment.
(a) (3 marks)
Extend the provided Matlab implementation by adding the blocks for insertion and removal
of the preamble. Generate a sequence {sk} of L = 50 pilot symbols that are used as preamble
and then append the data symbol sequence {uk} provided by the mapper to this preamble.
Use pilot symbols taken from a 2-PAM (also known as BPSK) constellation A = {-1, +1} (the
data symbols are QPSK modulated). You may either choose a fixed or randomly generated
preamble.
Because the preamble is known to the receiver, it does not carry any user data. The purpose
of the block Remove Preamble is simply to discard the preamble symbols such that only the
data symbol sequence is provided to the LS receiver.
Test your code without noise to ensure that insertion and removal of the preamble is working
correctly.
(b) (3 marks)
Extend your implementation from (a) by introducing a fixed phase offset ?0 chosen from the
interval [-p, p). Inspect the scatter diagram at point B for various offsets and noise levels and
(c) (4 marks)
Add code to the top-level script from (b) to step through the entire phase offset range from
-p to p in increments of 0.1p and measure the probability of bit error and the probability of
symbol error for each offset. Use a fixed noise level of sigmaw=0.6, which corresponds to a
signal to noise ratio (SNR) of 4.4 dB. For each phase offset, repeatedly transmit blocks with
K = 500 data symbols until 1000 symbol error events occur.
Plot the measured error probabilities as a function of the phase offset ?0. Discuss your observations.
For reference, the theoretical probabilities of bit and symbol error are 4.78% and
9.33% when no phase offset is present.
Mapper Insert
Preamble ×
+
Preamble
Phase Offset
Correction
tx bits
A
{uk}
{yk}
B
{dk}
C
{wk}
rx bits
e
i?0
{sk}
Figure 1: System model with phase offset and correction. Double arrows indicate complex signals.
2
Modern Communication Systems (EEET 4036), SP5-2015
University of South Australia
(d) (5 marks)
Add the block Phase Offset Correction to your implementation. To correct the phase offset,
you first need to estimate it. Implement the LS estimator ˆ?, using only the preamble symbols
as shown in (6) (recall that the data symbols are unknown at the receiver). Then use the
estimate ˆ? to correct the received symbol sequence {yk}.
Ensure that the phase offset correction operates as intended by comparing the scatter diagrams
of the symbol sequences before and after correction. Then repeat the simulation from (c) and
plot the measured probabilities of bit and symbol error as a function of the phase offset
?0 ? [-p, p). Compare your observations to those in (c).
Implementation Guidelines
• For all questions that require a Matlab implementation, provide code listings as .m file.
• Implement each block in the diagrams as a dedicated Matlab function. Use function names that
resemble the block names, e.g. PhaseOffsetCorrection for the block ‘Phase Offset Correction’.
• Use a separate top-level script for each part of question Q2, e.g. Q2a.m for part (a).
• Include brief comments describing what the major code components are intended to do.
3

# Our Service Charter

1. ### Excellent Quality / 100% Plagiarism-Free

We employ a number of measures to ensure top quality essays. The papers go through a system of quality control prior to delivery. We run plagiarism checks on each paper to ensure that they will be 100% plagiarism-free. So, only clean copies hit customers’ emails. We also never resell the papers completed by our writers. So, once it is checked using a plagiarism checker, the paper will be unique. Speaking of the academic writing standards, we will stick to the assignment brief given by the customer and assign the perfect writer. By saying “the perfect writer” we mean the one having an academic degree in the customer’s study field and positive feedback from other customers.
2. ### Free Revisions

We keep the quality bar of all papers high. But in case you need some extra brilliance to the paper, here’s what to do. First of all, you can choose a top writer. It means that we will assign an expert with a degree in your subject. And secondly, you can rely on our editing services. Our editors will revise your papers, checking whether or not they comply with high standards of academic writing. In addition, editing entails adjusting content if it’s off the topic, adding more sources, refining the language style, and making sure the referencing style is followed.
3. ### Confidentiality / 100% No Disclosure

We make sure that clients’ personal data remains confidential and is not exploited for any purposes beyond those related to our services. We only ask you to provide us with the information that is required to produce the paper according to your writing needs. Please note that the payment info is protected as well. Feel free to refer to the support team for more information about our payment methods. The fact that you used our service is kept secret due to the advanced security standards. So, you can be sure that no one will find out that you got a paper from our writing service.
4. ### Money Back Guarantee

If the writer doesn’t address all the questions on your assignment brief or the delivered paper appears to be off the topic, you can ask for a refund. Or, if it is applicable, you can opt in for free revision within 14-30 days, depending on your paper’s length. The revision or refund request should be sent within 14 days after delivery. The customer gets 100% money-back in case they haven't downloaded the paper. All approved refunds will be returned to the customer’s credit card or Bonus Balance in a form of store credit. Take a note that we will send an extra compensation if the customers goes with a store credit.

We have a support team working 24/7 ready to give your issue concerning the order their immediate attention. If you have any questions about the ordering process, communication with the writer, payment options, feel free to join live chat. Be sure to get a fast response. They can also give you the exact price quote, taking into account the timing, desired academic level of the paper, and the number of pages.

Excellent Quality
Zero Plagiarism
Expert Writers

or

Instant Quote