Jennifer Christensen

1
Errors and Erasures Decoding
Research Objective Evaluate the combination of
MFSK and QPSK communication across an AWGN
channel with Reed-Solomon coding based on
bandwidth efficiency and probability of error
characteristics.
Erasures are an additional feature of a decoder
which label symbols erasures when the validity
of the symbol is called into question, either
due to ambiguity or channel interference. The
diagram displays a visualization of how erasures
are detected. Gamma is an important term
symbols received between this value and the axis
are erasures. The value of gamma may vary, and
the plot below shows the general movement of
the probability curves.
Combination of BFSK, QPSK 4 phases, 2
frequencies-16 signals
Errors and Erasures Probability
Jennifer Christensen South
Dakota School of Mines and Technology SURE
Advisor Dr. J. J. Komo
Simulated M4N signal across an AWGN channel
(µ0, s2 No/2) and through the receiver to
determine probability of bit error.

• Reed-Solomon Coding
• To improve the performance of QPSK-MFSK
  communication, Reed-Solomon (RS) codes are
  incorporated.
• Some characteristics of RS codes
• nonbinary cyclic block codes with m-bit length
  symbols
• capable of correcting any combination of t or
  fewer errors
• described by an (n, k) notation where n total
  number of code symbols in the block and k
  number of data symbols being encoded.
• (n-k) is equal to the number of parity symbols,
  2t
• Can correct up to (n-k)/2 errors

Results show probability is as
expected-curves are equal, independent of the
number of frequencies.
The addition of erasures decoding produces small,
if any, coding gain (lt0.1 dB). For higher
values of gamma, the performance is worse than
without erasures. Therefore, it makes little
difference if erasures are included in the
previous model.
Bandwidth Efficiency Plane The
bandwidth-efficiency curve is an important metric
derived from the Shannon-Hartley Theorem for an
AWGN channel. The ideal curve is where the data
rate (R) equals the channel capacity (C). Eb/No
is the bit energy divided by the noise power
spectral density and R/W is the data rate over
bandwidth. The curve demonstrates the necessary
trade-off between power and bandwidth efficiency.
The above shows different code lengths with
equal code rates (Rc3/4). Clearly, the n256
code length has best results. The improvement
will eventually converge.
Qualitative Results
Next, using n256 and varying the code rates, we
find that the best probability curves for RS
code across an AWGN channel are with code rates
between 0.6 and 0.7. The data collected is then
plotted on our bandwidth efficiency plane.

• Rc is the code rate (k/n)
• Coded Bandwidth WcW/Rc

For coherently detected QPSK the minimum tone
spacing 1/(2Ts) When we extend this to
multiple frequencies the spacing must be
increased to maintain the orthogonality of our
signals (the minimum tone spacing 1/Ts).
Due to this, the bandwidth efficiency
decreases. For all values of N, the
bandwidth efficiency is 2 bits/sec/Hz as shown.

• Conclusions
• MFSK and QPSK combination has same bandwidth as
  QPSK alone
• Bandwidth efficiency decreases with Reed-Solomon
  coding
• Bit error probability decrease reliability
  improves
• Future work
• Soft Decision Decoding
• Evaluation across different channels