Assignment 1
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Assignment 2: Constellation diagrams and the Shannon-Hartley
capacity theorem
Part 1 (12 marks)
For this part of the assignment, you will be working with the 16 QAM constellation diagram from the lecture slides:
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Introduction: Remember that in quadrature amplitude modulation, we transmit a signal with frequency consisting of a sine and a
cosine component. The and coordinates of the constellation points are simply peak amplitude voltages for the cosine and sine
components, respectively. That is, a constellation point (symbol) can be expressed as a tuple of coordinates that puts the point
f I Q (i, q)
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at volts from the -axis in the horizontal direction (positive voltages to the right, negative voltages to the left) and at volts from
the -axis in the vertical direction (positive voltages to the top, negative voltages to the bottom). So the signal at time that
represents the constellation point is given by the equation
where is the carrier frequency. We can also write this in polar coordinates:
Here, is the distance (voltage) between the origin of the - coordinate system and the constellation point, and is the
angle between the positive -axis, the origin, and the constellation point, in radians (that is, has a value between 0 and rather
than 0 and 360 degrees).
We will now assume that the constellation point labelled 1100 has the amplitude coordinates (1, 3) at the transmitter, i.e., 0111 has (-1,
-1), 1010 has (3, -3), and so on.
Your first task is to encode each of the last six digits of your AUID (7 or 9 digit student ID number) into a bit sequence representing
the 8 bit ASCII character for the digit. That is, you should end up with a string of 48 bits. Then convert this string into 16QAM
coordinate pairs (most significant bit first), such that each nibble ("half byte") of each ASCII character becomes one symbol (one
constellation point). E.g., the bit sequence 1011011000101110... would turn into:
(3, -1) // for 1011
(-1, -3) // for 0110
(-3, -3) // for 0010
(1, -3) // for 1110
…
Marking: One mark will be awarded for each correct coordinate pair (you may include comments as above as to which pair relates to
which bit sequence, but this is not required). You may use the ASCII table at http://www.ascii-code.com/ for conversion of your AUID
digits into binary. Part 2 (24 marks)
i Q q I s t (i, q) s(t) = i ? cos(2πft) + q ? sin(2πft) f s(t) = √i2 + q2 ? sin(2πft + ?) √i2 + q2 I Q ? I ? 2π
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In this part, your task is to decode the 16QAM signal, which is given to you as the I- and Q-voltages of the successive constellation
points in the message:
(-0.39, 2.39)
(1.84, 2.50)
(-1.64, -3.78)
(-1.62, 1.92)
(-1.82, -1.75)
(-3.73, -1.64)
(-2.39, -2.41)
(0.17, -1.77)
(-0.22, -0.36)
(-0.23, 3.35)
(-1.78, -2.51)
(3.72, 2.36)
(-2.00, -2.16)
(0.36, -0.57)
(-2.17, -3.75)
(-3.89, 3.85)
(-1.75, -1.89)
(-2.13, 2.28)
(-1.55, -1.69)
(0.49, 2.41)
(-1.76, -3.87)
(-3.99, 1.80)
(-1.69, -0.46)
(3.74, 0.37)
(-3.57, -3.55)
(-3.74, 2.28)
(-0.43, 1.74)
(-3.80, -0.64)
(-0.20, -2.09)
(-1.65, 1.73)
(-1.74, -1.97)
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(1.33, -3.91)
(-0.25, -3.62)
(-0.61, 1.82)
(-0.10, -3.63)
(-0.64, -1.95)
(-0.38, -2.12)
(-2.34, 1.72)
(-1.57, -2.14)
(1.72, 3.80)
(-3.81, -3.51)
(-2.29, 3.42)
(-1.74, -2.21)
(-1.75, -3.90)
(-1.77, -3.99)
(1.75, -1.93)
(-0.29, -1.64)
(-2.23, -2.39)
(-2.42, -2.49)
(-2.29, 1.97)
(-1.88, -1.50)
(-0.16, 2.28)
(-0.27, -2.31)
(3.65, 3.66)
(-1.64, -2.53)
(-1.58, 0.31)
(-3.66, -2.15)
(-3.65, 2.18)
(-1.61, -2.59)
(-0.47, -2.17)
(-1.91, -2.23)
(-2.12, 1.52)
(-0.38, -3.50)
(0.50, -2.35)
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(-1.33, -1.77)
(-2.26, -1.61)
As before in Part 1, each constellation point is a tuple with two voltages. The receiver has amplified the voltages to the original
levels. However there is some added noise on the voltages in the tuples. This noise may sometimes cause you to identify the wrong
constellation point, which results in a symbol error (wrong ASCII character being decoded). Your task is to decode the raw message
and identify and correct the symbol errors. The message contains a (meaningless) sports headline referring to two countries,
something that they do, and something about the circumstances in which this happens / happened / will happen.
In your solution, identify the 4-bit string (nibble) that each tuple decodes to, and the ASCII symbol that each byte resulting from two
nibbles corresponds to. If the ASCII symbol is not printable, mark it as “[NP]” instead, or if it is a space character, mark it as [ ]. The
original message consists of spaces and printable letters only. If the decoded character is in error, give the intended character as well,
and identify the bit in error. Examples:
(-0.85, -1.17) decodes as 0111
(2.91, -3.02) decodes as 1010 - together, 01111010 gives “z”
(-2.81, - 2.94) decodes as 0010
(-3.18, 3.05) decodes as 0000 - together, 00100000 gives “[ ]”
(-2.02, -2.94) decodes as 0010
(-1.02, 2.79) decodes as 0100 - together, 00100100 gives “$”
Probable error in most significant nibble, likely intended bit sequence: 01100100 “d”
...
Finally, state the decoded message:
With symbol errors (characters in error) included, and
with symbol errors corrected.
Marking: 12 marks for correct decoding of the constellation points (as received) into bytes and conversion into ASCII, 6 marks for
correct identification of the bit errors, and 6 marks for the correction of the resulting symbol errors.
Part 3 (36 marks)
In this part, your task is to apply the Shannon-Hartley capacity theorem.
(i, q)
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There are two versions of the theorem:
1. (precise version)
2. (approximate version for much larger than 1)
Here, is the capacity of the channel (its theoretically achievable maximal bit rate in bits/second), is the bandwidth in hertz, is
the signal power in watts and is the noise power, also in watts. is also called the signal-to-noise ratio (with being the more
accurate version that takes into account that we can't separate the noise from the signal at the receiver).
You may use the approximate version of the theorem for .
The problems that you must solve in this part are as follows:
1. You work for an international consultancy implementing communication solutions for governments all over the world. The
government of country name censored for national security reasons wishes to establish a radio-based data link between two of its police bases. It has invited your company to tender of the implementation of a coding and modulation scheme for the link.
In its tender documents, the CENSORED Communications Commission has assigned the frequency band between 19.1 MHz and
19.14 MHz for the project.
Based on the specifications for the transmitter and antennas to be used, as well as the distance between the bases, you calculate
that the received power will be around 20 pW. You submit a tender and the government offers you the contract. The small print of the project says that contractors will have to
stay in a secure guest room in one of the police bases until the project is finished to specification and handed over in working
order. Also, the government states that the noise power measured in the band at the receiving sites is 0.15456 fW. The bit rate to
be achieved is just 1 Mb/s. You estimate that your company's technology will be able to achieve half of the capacity theoretically
possible under the Shannon-Hartley theorem. Should you sign the contract? Justify your answer. 2. The government of CENSORED has bought equipment from your company that operates a data link at 100 kb/s in a band from 4 GHz to 4.000042 GHz.
The goverment is happy and reports that it is operating quite satisfactorily. Given that your equipment can only use half the
channel capacity, can you give an estimate for the signal-to-noise ratio (SNR) that is achieved or exceeded at the receiver (to the
closest whole dB)? Show your working.
C = B log2 S+NN C = B log2 SN SN C B S N SN S+NN SN > 30
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3. A measurement of the actual SNR at the receiver in question 2 reveals that it is 63 dB. His Excellency Field Marshall Professor
Doctor Embetz Elment (NOT THEIR REAL NAME), the brother-in-law of the prime minister of CENSORED, would like run a
private 500 kb/s data link for his game server between the same sites and using the same transmitter equipment, but using twice
the bandwidth of the existing link. You know that this means that its receiver would receive approximately the same power from
the transmitter, but that the SNR would decrease by the same factor by which the bandwidth would increase, i.e. by half, because
you are now capturing twice as much noise. Would this work? Explain why (or why not). Show your working. Note: Again, assume
that your technology will only be able to reach half the channel capacity that the Shannon-Hartley theorem allows for.
Note: Units in the question set above may have the usual multiplier prefixes attached, e.g., mW = milliwatts or Mb/s =
megabits/second. For test and exam, we expect you to be familiar with them and how to convert between them (e.g., between mm
and km):
Prefix Prefix name Multiplier Example
f “femto…” 10-15 fW (femtowatts)
p “pico…” 10-12 pW (picowatts)
n “nano…” 10-9 ns (nanoseconds)
μ “micro…” 10-6 μW (microwatts)
m “milli…” 10-3 mm (millimetres)
k “kilo…” 103 kHz (kilohertz)
M “mega…” 106 Mb (megabits)
G “giga…” 109 GHz (gigahertz)
T “tera…” 1012 TB (terabytes)
You may use a calculator where this is appropriate, however you are expected to:
1. Show your working. We don’t just want to see the result, but also how you got there.
2. Use appropriate rounding: The result should not be accurate to more significant digits than any of the input quantities, and do
keep in mind the behaviour of the log function please. E.g., for the purposes of this assignment:
Marks will be awarded for proper numerical presentation!
12.3 log2 345.6789 = 12.3 log2 345.1234 = 103.7
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Marking: Each question carries up to 10 marks for technical correctness and up to 2 marks for proper numerical presentation.
Part 4 (28 marks)
In this part, your task is to check the correctness of a message based on a CRC checksum.
A receiver receives the binary message 001011110001000100001010. Its last 4 bits contain a CRC checksum obtained with the 5 bit
generator polynomial 11001. Has the message been received correctly? Show your working.
Marking: 4 marks for figuring out which polynomial you need to divide by what, 20 marks for correct working and 4 marks for coming
to the correct conclusion based on your working.
Submission
Submit your answers to this assignment in a PDF file via the assignment dropbox . Your submission must not be larger than 300 kB
(i.e., don't scan anything handwritten).
? The expected learning outcomes of this assignment are as follows.
Being able to encode data as, and decode data from, I-Q-coordinates in a constellation diagram.
Being able to use the Shannon-Hartley capacity theorem to determine the workability of communication solutions, achievable
data rates, required bandwidths and signal-to-noise ratios.
Being able to verify a given binary message with CRC checksum for correctness, given the generator polynomial.
Please do talk to your tutor if you think you have not met these outcomes.
This assignment is personalised for 461934529. Academic Integrity
Assignment 1
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Sharing assignment solutions and source code does not help learning. Consequently, our academic integrity policy does not permit
sharing of solutions or source code leading to solutions, nor does it allow sourcing solutions or source code from any third party.
Violation of this will result in your assignment submission attracting no marks, and you may face further disciplinary action. Therefore,
please do not share assignments, assignment solutions and/or source code leading to assignment solutions, or use material from
others in your assignments. You must not publish assignments or solutions in any form online at any time. You will be liable if
someone copies your solution. There are also copyright and IP issues. Please come talk to us if you have any doubt over what is
legitimate and what is not.
You can refer to online tutorials and resources about QAM and the Shannon-Hartley capacity theorem. However, you must not post
this assignment or parts of it online in order to obtain answers. You must implement your own solutions yourself. Do not blindly copy
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