Information Theory Project
Channel Capacity and Capacities of Complex
Constellations
Deadline [Wednesday, 12 April 2017]
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Short Description:
Theoretically Channel Capacity for two dimensional
continuous-continuous [complex] signals c, is bounded by Shannon limit:
c ≤ log2 (1 + 𝑆𝑁𝑅)
(1)
Where c here is considered the practical data rate which is bounded by the channel capacity.
However, to study the effect of M-ary modulation for different schemes with M=2𝑏 symbols,
the channel capacity of any discrete input, continuous output complex constellation is limited
by a maximum of b bits/symbol (This saturation happens at high SNRs depending on the
modulation type). Thus, the capacity in this case assuming equi-probable input signals P(s)
=1/𝑀𝑠 is obtained as:
(2)
Where s is the complex constellation set, which is composed of 𝑠𝑟 + 𝑗𝑠𝑖. The received symbol
y𝑟 + 𝑗y𝑖. In the following, we see the capacity for different QAM modulation order.
Project Background:
To calculate the Channel Capacity for M-ary Modulation schemes in equation 1, it can be
implemented using one of the following three techniques (MATLAB Integration function,
Monte-Carlo, Gauss Hermite). Where MATLAB integration function technique calculates
the exact equation using the integration function to evaluate the expectation. And both MonteCarlo and Gauss Hermite techniques approximate the integration operation in calculating the
capacity.
1) Implementing the Integration without any approximations gives the best and most
accurate results. However, it is a complex integration which needs to be split into two
integrations (real and imaginary).
2) Using Monte-Carlo simulation as an approximation to the discrete-input continuousoutput capacity for QAM, PSK, and PAM gives the worst results when compared to other
techniques. It is implemented using the mutual information formula I(X;Y)= H(X)-H(X|Y).
3) Using Gauss Hermite approximation, gives a better approximation compared to the
Monte-Carlo. The code can be found in the uploaded Gauss Hermite compressed file.
Gauss Hermite Folder Notes:
The uploaded Gauss Hermite folder contains two m files QAMCapacity, and PlotAndSave
QAMCapacity.m contains the technique implementation with an example on how to calculate
the capacity of BPSK, 4, 16, 32, 64 QAM, and 8 PSK
PlotAndSave.m calls the QAMCapacity function and plot all the previous modulation
examples compared with the two-dimensional Shannon limit in the same figure
Project Procedure:
1) Using Gauss Hermite technique that approximates equation 1, Modify
QAMCapacity.m to calculate the channel capacity curves for all the following three
modulation schemes then update PlotAndSave.m to plot the following three figures
over SNR range [-10:44]
a) Figure 1: 2,4,16,32 and 64 QAM (with two dimensional Shannon capacity curve)
b) Figure 2: 2,4,8,16,32 and 64 PSK (with two dimensional Shannon capacity curve)
c) Figure 3: 2,4,8,16,32 and 64 PAM (with one dimensional Shannon capacity curve)
2) For each capacity curve, compute the Operation Point which is the exact SNR
value at which the rate of each modulation curve saturates to its maximum number
of bits (for example at which SNR BPSK reaches 2 bits/symbol? And so on)
3) For each capacity curve, compute the Rate Loss which is the difference between
the Operation Point and the SNR value at which the Shannon curve reaches the
saturation value of the curve.
4) Three more Figures comparing (16 PAM with 16 QAM), (16 PSK with 16 QAM)
and (16 QAM with Shannon limit)
Deliverables:
1) A hard copy of three sections computerized report including:
a) A short Introduction section describing the derivation for the Shannon AWGN
Channel capacity (Continuous - Continuous), and the derivation for the (Discrete –
Continuous) channel capacity after adding the modulation and demodulation blocks
and clarifying the objective of modulation which is enhancing the capacity to
approach the Shannon limit. (Derivations can be attached handwritten within the
report)
b) A short Background section about Adaptive Modulation and the types of
modulation (single sided – double sided) and explaining the tradeoff between the
capacity and the bit error rate happening due to the increase in the order of
modulation. (for example increasing the PAM order from 2 PAM to 8 PAM
increases the capacity, however the BER increases also)
c) Results and Conclusion section including the 3 capacity graphs for the 3
modulation schemes (QAM, PSK, PAM) each with the corresponding Shannon
limit and commenting on the graphs according to the operation point and the rate
loss of each modulation curve, the 3 more comparing graphs with comments.
2) A working MATLAB code that prints out all the required output should be submitted
to [email protected]
Team Work Hints:
Groups of up to THREE students MAX are allowed. However, full understanding of the
project by all the group members is important for the evaluation.