ECE300 Digital Modulation Simulation Project solution

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Original Work ?

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Introduction

As you know from class, many digital communication systems can be described by a two-dimensional
constellation of symbols with basis functions proportional to p(t) cos ωct and −p(t) sin ωct, defined
from time 0 to T, for some pulse p(t).

We usually represent the symbol as A + Bj where A2
is
the energy in the in-phase component and B2
is the energy in the quadrature component.

You
should also recall from class that we can analyze the impact of noise entirely through this I/Q
representation of the symbol. This simplifies the framing of the least squares decision metric from
an L2 problem (requiring integration) to finding the minimum distance between two points in the
plane.

Using this representation, we can simulate millions of symbol transmissions in MATLAB quite
quickly – this is necessary if we want to explore systems with error rates.

You will explore QAM, PSK and DPSK systems using this framework, with the goal, largely,
of producing SNR per bit vs.probability of symbol error plots, and comparing system performance.
Many such plots are available in your book for reference. A large emphasis will be placed on the
writing of efficient MATLAB code.

MATLAB Problems

1. My constellation consists of M symbols, c1, . . . , cM ∈ C. I transmit N symbols, a1, . . . , aN ,
from this constellation. They are each corrupted by AWGN, so that at the receiver ai + ni
is seen. The least squares decision ˆai
is the constellation point nearest ai + ni
in the usual
distance sense over C.

Write a function which takes a constellation (a vector of M complex
values) and a noisy symbol vector (a vector containing N complex values) as inputs, and
outputs the length N estimated symbol vector (all values of which are valid constellation
points) based on the least squares decision method.

The code for this function should not be
long, should contain no loops and should be fast. To ensure it works, run a few tests with,
for example, N = 5 randomly generated signals and a constellation such as {1, −1, j, −j}
(don’t present this in your report, just do it to make sure your whole project is built on solid
ground).

2. Write a function which takes as an input a base constellation, a list of N noise-free transmitted
symbols (these symbols will be drawn from the base constellation symbols), a value for the
noise power N0 and a desired value for the SNR per bit in dB.

The function first should find
the desired average symbol energy (from the SNR per bit, the noise power and the number
of symbols in the constellation). It will then scale the base constellation to form the true
constellation, which has the correct average energy.

The transmitted symbols will be scaled
by the same amount, as they must come from the true constellation. Produce N realizations
from a noise process with the correct noise power (note that the noise for each symbol is
drawn from a complex Gaussian Z = X + jY where X and Y are also zero-mean Gaussian
with half of the variance of Z – think about the theorem for linear combinations of Gaussian
random variables) and add this to the scaled transmitted symbol vector.

Then apply your
function from the previous problem to determine an estimate given the true constellation and
the noisy signal. The outputs of the function should be the true noise-free transmitted vector
(scaled) and the estimated received vector.

I know this sounds long, but it corresponds closely to a MATLAB class homework – I suggest
writing and testing it in modular parts to ensure it is working as desired.

3. Write a function that, given the true and estimated symbol sequence determines the number
of errors assuming a nondifferential scheme. Divided by N, this estimates probability of error.
Do not use any for loops. You might want to make use of the MATLAB function nnz. You
also will probably want to avoid near-zero elements being counted as nonzero – use a proper
threshold.

4. Repeat the above for a differential scheme (where N transmissions corresponds to N − 1
symbols).

You are done with the hard part! Give yourself a pat on the back! Make sure to drink
water. From this point forward, you will simply use the functions you’ve created to generate
nice plots. You should use N = 106
, and generate your symbols from the constellation by
using a MATLAB function that samples from a given vector (equiprobably).

You are not
locked in to using N = 106 or the given resolution/range in SNR per bit – you can lower or
increase either to generate smooth plots in a resaonable amount of time.

5. Note that binary antipodal and binary orthogonal signaling are special cases, with easily
defined two-element constellations. In class, we found theroretical formulas for the probability
of error as a function of SNR per bit, making this a natural place to start.

From -4 dB to 20
dB SNR per bit with 1 dB resolution, use your code to generate a plot of the probability of
error for each of these signalling methods, and compare it to a graph of the theoretical SNR
per bit.

6. For M = 4, 8, 16 and 32, from -4 dB to 20 dB SNR per bit with 1 dB resolution, use your code
to generate a plot of the probability of error for M-ary PSK. Provide a legend, and include
the M = 2 case which you computed above. This corresponds directly to a figure in your
text.

7. Perform the above for DPSK (but omit the M = 2 case).

8. Repeat for M-ary QAM with M = 4, 16, 32 and 64 (you have already computed the M = 4
case).

Report Guidelines
I would like you to present your findings in a short report, containing at least four sections:

1. Introduction: Give an overview of the project and the theory behind it. What did you set
out to explore? Which modulation schemes are you comparing? How are they defined?

2. Methods: Describe your MATLAB functions, how they work, and which vector methods
you used to avoid loops and speed things up. This can be a short section, but I want to know
how your decision function works and how you determine the number of errors.

3. Results: The bulk of your report – Each figure generated in the MATLAB code should appear
here, with appropriate axis labels, titles and legends. This should be the binary antipodal
and orthogonal experimental and theoretical SNR per bit vs probability of error curves, as
well as the PSK, DPSK and QAM plots as described above.

4. Discussion: How do the schemes compare to each other? How do differential and nondifferential methods compare? What is the effect of raising M? Tell me about how long it
takes to make these graphs (you can approximate, or use tic and toc).

I would suggest using a LaTeX editor if you can to write this report – it is useful to start learning
this skill early – but I will not require it. There is no upper or lower page limit on the report. You
must send me the MATLAB files used to generate the results.

1. Report Content (70%): The majority of your grade will be from the content of your report,
including your responses to the questions and your ability to produce correct graphs (whether
or not they are pretty).
2. Report Style (15%): It is important that you present your results with labeled graphs and
grammatically correct sentences. Your report should be organized well enough that someone
(for example, yourself in the future) might be able to look at it as a reference on the topic it
is covering.
3. MATLAB Style (15%): Just as in your homeworks – don’t write bad MATLAB code!