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.
Combined QPSK and MFSK
Communication Across an
AWGN Channel
QPSK Detection Decision Regions for RS Coding
Original
Signal
γ
γ
Correct Signal
Detected
Erasure Region
Error Region
Combination of BFSK, QPSK:
4 phases, 2 frequencies-16 signals
si (t ) 
Es
Ts
 
 (2i1  1) 
 (2i2  1) 

  cos  2 t 

cos 1t 
4
4



 
Jennifer Christensen
South Dakota School of Mines
and Technology
SURE Advisor: Dr. J. J. Komo
0  t  Ts , 0  i1 , i2  3, i  4i2  i
Simulated M=4N signal
across an AWGN channel
(μ=0, σ2 = No/2) and
through the receiver to
determine probability of
bit error.
Transmitted
signal plus
noise
Correlator
Receiver
Output - Gaussian
random variable
 dt
r(t)=si(t)+n(t)
n (t )
.
.
.
0
T
PCo  PC
where   1 
Pers
z(T)=ai(T)+n0 Decision
Stage:
.
.
.
Determine
closest
match
si(t)
 dt
Reed-Solomon Coding
1  (1  Pb ) 2 N
0
Results show
probability is as
expected-curves are
equal, independent
of the number of
frequencies.
At Pb=10-5
Eb/No = 9.52 dB
for M=4, 16, 64
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.
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.
n
j
1
 Psd   j ( nj )Psc (1  Psc ) n j
n j d 1
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
Bandwidth Efficiency Plane
M=4N, Pb=10-5
n v
1 n  n  cv
 n  v  c n v  w c w
P     PE
(v  w)
Pers
 PC

n v 0  v  w max( 0,d 2v )
 w 
d
s
1 (t )
T
Eb W  C W 
  2  1

NO C 
Errors and Erasures Probability
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.
The above shows different code lengths
with equal code rates (Rc=3/4). Clearly,
the n=256 code length has best results.
The improvement will eventually converge.
The addition of erasures decoding produces small, if any,
coding gain (<0.1 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.
Qualitative Results
M=4N
Next, using n=256 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.
Code Rate
Uncoded
Rc=7/8
(16,14)
(64,56)
(256, 224)
Rc=3/4
(16,12)
(64,48)
(256, 192)
Rc=5/8
(16,10)
(64, 40)
(256, 160)
Rc is the code rate (k/n)
Coded Bandwidth: Wc=W/Rc
n=256, Pb
=10-5
k=240
k=224
k=192
k=160
Pb=10-5
R/W
2.00
1.75
Pb =10 -6
Coding Gain (dB)
0 (9.58 dB) 0 (10.53 dB)
1.52
2.76
3.42
1.71
3.26
4.15
2.06
3.34
3.87
2.36
3.92
4.65
2.21
3.41
3.87
2.52
4.03
4.66
1.50
1.25
Conclusions
M=16, N=2 (no coding)
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