This document provides instructions for a computer assignment on calculating the bit error rate (BER) of 16QAM modulation with Gray encoding. Students are asked to: 1) Consider the 16QAM constellation and Gray encoding, 2) Calculate the average symbol energy, 3) Derive the theoretical symbol error rate (SER) formula, 4) Approximate the theoretical BER from the SER, and 5) Simulate the transmission of 16QAM symbols over an AWGN channel and calculate the simulated BER to plot against theoretical BER curves.

1. University of Isfahan - Department of Electrical Engineering
Digital Communications Due: 94/10/30
Computer Assignment II: BER of 16QAM with Gray Encoding
Instructor: M. F. Sabahi TA: I. Taghavi
1. Consider a 16QAM modulation scheme with Gray encoding and its constellation.
2. Calculate the average energy of the 16QAM symbols.
3. Theoretically, one can show that the symbol error rate (SER) of a 16QAM in presence
of AWGN is equal to
Ps = 3Q
4Eb
5N0
−
9
4
Q
4Eb
5N0
2
.
*Note that, if we denote average energy of bits and symbols with Eb and Es, respectively,
signal to noise ratio can be written as
SNR =
Eb
N0
=
Es
N0 log2 M
where M = 16 is the alphabet size and N0 denotes noise power.
4. In a 16QAM symbol constellation with Gray encoding, adjacent symbols differ only
by one bit. So, for high enough SNRs, bit error rate (BER) of the received signal can be
calculate using the following approximation, theoretically.
Pb ≈
Ps
log2 M
.
5. (Numerical Simulation)
Generate a stream of 16QAM symbols, pass it through the complex AWGN channel and
calculate the bit error rate after demodulation/detection.
*Don’t use MATLAB built-in functions or objects such as “modem” for modulation and
demodulation.
6. Plot bit error rate (BER) obtained from simulations versus different SNRs and
compare it with the theoretical derivation (Fig.1).
Report:
Write your conclusions and depict the results of this experiment. Send only one zip file (or
any other compressed file) including your short report (PDF format) and simulation file(s)
to ui.homeworks@gmail.com before the assignment deadline.
Remark: Please use the following name for the email subject and attached file
“CA2 your student number”
* If you have any questions or comments regarding the steps outlined in this document,
please contact me by sending an email to ui.homeworks@gmail.com.
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2. University of Isfahan - Department of Electrical Engineering
SNR [dB]
0 2 4 6 8 10 12 14
BitErrorRate(BER)
10
-6
10
-5
10
-4
10
-3
10-2
10-1
100
Simulation
Theory
Figure 1: BER of 16QAM versus different SNRs.
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