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Edits to ftrsd paper.

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Steven Franke 2016-01-18 03:19:08 +00:00
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lib/ftrsd/ftrsd_paper/ftrsd.lyx
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@ -138,7 +138,7 @@ key "kv2001"
Since 2001 the KV decoder has been considered the best available soft-decision
decoder for Reed Solomon codes.
We describe here a new open-source alternative called the Franke-Taylor
(FT, or K9AN-K1JT) algorithm.
(FT, or K9AN-K1JT) soft-decision decoding algorithm.
It is conceptually simple, built around the well-known Berlekamp-Massey
errors-and-erasures algorithm, and in this application it performs even
better than the KV decoder.
@ -298,7 +298,15 @@ d=n-k+1.\label{eq:minimum_distance}
\end_inset
The minimum Hamming distance of the JT65 code is
With
\begin_inset Formula $n=63$
\end_inset
and
\begin_inset Formula $k=12$
\end_inset
the minimum Hamming distance of the JT65 code is
\begin_inset Formula $d=52$
\end_inset
@ -361,8 +369,11 @@ erasures.
\begin_inset Quotes erd
\end_inset
With perfect erasure information up to 51 incorrect symbols can be corrected
for the JT65 code.
With perfect erasure information up to
\begin_inset Formula $n-k=51$
\end_inset
incorrect symbols can be corrected for the JT65 code.
Imperfect erasure information means that some erased symbols may be correct,
and some other symbols in error.
If
@ -701,8 +712,31 @@ How might we best choose the number of symbols to erase, in order to maximize
\end_inset
symbols.
Decoding will then be assured if the set of erased symbols contains at
least 37 errors.
According to equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:erasures_and_errors"
\end_inset
, with
\begin_inset Formula $s=45$
\end_inset
and
\begin_inset Formula $d=52$
\end_inset
then
\begin_inset Formula $e\le3$
\end_inset
, so decoding will be assured if the set of erased symbols contains at least
\begin_inset Formula $40-3=37$
\end_inset
errors.
With
\begin_inset Formula $N=63$
\end_inset
@ -826,19 +860,32 @@ The Franke-Taylor Decoding Algorithm
\begin_layout Standard
Example 3 shows how statistical information about symbol quality should
make it possible to decode received frames having a large number of errors.
In practice the number of errors in the received word is unknown, so we
use a stochastic algorithm to assign high erasure probability to low-quality
symbols and relatively low probability to high-quality symbols.
In practice the number of errors in the received word is unknown, so our
algorithm simply assigns a high erasure probability to low-quality symbols
and relatively low probability to high-quality symbols.
As illustrated by Example 3, a good choice of erasure probabilities can
increase by many orders of magnitude the chance of producing a codeword.
Note that at this stage we must treat any codeword obtained by errors-and-erasu
res decoding as no more than a
Once erasure probabilities have been assigned to each of the 63 received
symbols, the FT algorithm uses a random number generator to decide whether
or not to erase each symbol according to its assigned erasure probability.
The list of erased symbols is then submitted to the BM decoder which either
produces a codeword or fails to decode.
\end_layout
\begin_layout Standard
The process of selecting the list of symbols to erase and calling the BM
decoder comprises one cycle of the FT algorithm.
The next cycle proceeds with a new selection of erased symbols.
At this stage we must treat any codeword obtained by errors-and-erasures
decoding as no more than a
\emph on
candidate
\emph default
.
Our next task is to find a metric that can reliably select one of many
proffered candidates as the codeword actually transmitted.
\end_layout
\begin_layout Standard
@ -898,7 +945,7 @@ The FT algorithm uses quality indices made available by a noncoherent 64-FSK
\end_inset
of the symbol's fractional power
\begin_inset Formula $p_{1,\,j}$
\begin_inset Formula $p_{1,\, j}$
\end_inset
in a sorted list of
@ -923,7 +970,8 @@ The FT algorithm uses quality indices made available by a noncoherent 64-FSK
\end_layout
\begin_layout Standard
We use an empirical table of symbol error probabilities derived from a large
We use a 3-bit quantization of these two metrics to index the entries in
an 8x8 table of symbol error probabilities derived empirically from a large
dataset of received words that were successfully decoded.
The table provides an estimate of the
\emph on
@ -938,10 +986,22 @@ a priori
\end_inset
metrics.
These probabilities are close to 1 for low-quality symbols and close to
0 for high-quality symbols.
This table is a key element of the algorithm, as it will define which symbols
are effectively
\begin_inset Quotes eld
\end_inset
protected
\begin_inset Quotes erd
\end_inset
from erasure.
The a priori symbol error probabilities are close to 1 for low-quality
symbols and close to 0 for high-quality symbols.
Recall from Examples 2 and 3 that candidate codewords are produced with
higher probability when
higher probability when the number of erased symbols is larger than the
number of symbols that are in error, i.e.
when
\begin_inset Formula $s>X$
\end_inset
@ -968,7 +1028,7 @@ t educated guesses to select symbols for erasure.
, the soft distance between the received word and the codeword:
\begin_inset Formula
\begin{equation}
d_{s}=\sum_{j=1}^{n}\alpha_{j}\,(1+p_{1,\,j}).\label{eq:soft_distance}
d_{s}=\sum_{j=1}^{n}\alpha_{j}\,(1+p_{1,\, j}).\label{eq:soft_distance}
\end{equation}
\end_inset
@ -986,7 +1046,7 @@ Here
\end_inset
if the received symbol and codeword symbol are different, and
\begin_inset Formula $p_{1,\,j}$
\begin_inset Formula $p_{1,\, j}$
\end_inset
is the fractional power associated with received symbol
@ -1030,7 +1090,7 @@ In practice we find that
\begin_layout Standard
\begin_inset Formula
\begin{equation}
u=\frac{1}{n}\sum_{j=1}^{n}S(c_{j},\,j).\label{eq:u-metric}
u=\frac{1}{n}\sum_{j=1}^{n}S(c_{j},\, j).\label{eq:u-metric}
\end{equation}
\end_inset
@ -1063,7 +1123,7 @@ The correct JT65 codeword produces a value for
bins containing noise only.
Thus, if the spectral array
\begin_inset Formula $S(i,\,j)$
\begin_inset Formula $S(i,\, j)$
\end_inset
has been normalized so that the average value of the noise-only bins is
@ -1078,7 +1138,7 @@ The correct JT65 codeword produces a value for
\begin_layout Standard
\begin_inset Formula
\begin{equation}
\bar{u}_{1}=1+y,\label{eq:u1-exp}
\bar{u}_{c}=1+y,\label{eq:u1-exp}
\end{equation}
\end_inset
@ -1090,7 +1150,7 @@ where
is the signal-to-noise ratio in linear power units.
If we assume Gaussian statistics and a large number of trials, the standard
deviation of measured values of
\begin_inset Formula $u_{1}$
\begin_inset Formula $u$
\end_inset
is
@ -1099,7 +1159,7 @@ where
\begin_layout Standard
\begin_inset Formula
\begin{equation}
\sigma_{1}=\left(\frac{1+2y}{n}\right)^{1/2}.\label{eq:sigma1}
\sigma_{c}=\left(\frac{1+2y}{n}\right)^{1/2}.\label{eq:sigma1}
\end{equation}
\end_inset
@ -1143,12 +1203,28 @@ i.e.
\begin_layout Standard
\begin_inset Formula
\begin{equation}
\sigma_{i}=\frac{1}{n}\left[n+2y(k-1)\right]^{1/2}.\label{eq:sigma2}
\sigma_{i}=\frac{1}{n}\left[n+2y(k-1)\right]^{1/2},\label{eq:sigma2}
\end{equation}
\end_inset
where the subscript
\begin_inset Quotes eld
\end_inset
i
\begin_inset Quotes erd
\end_inset
is an abbreviation for
\begin_inset Quotes eld
\end_inset
incorrect
\begin_inset Quotes erd
\end_inset
.
\end_layout
\begin_layout Standard
@ -1162,11 +1238,11 @@ If
\end_inset
to be drawn from a population with statistics described by
\begin_inset Formula $\bar{u}_{1}$
\begin_inset Formula $\bar{u}_{c}$
\end_inset
and
\begin_inset Formula $\sigma_{1}.$
\begin_inset Formula $\sigma_{c}.$
\end_inset
If no tested codeword is correct,
@ -1235,14 +1311,8 @@ reference "sec:Theory,-Simulation,-and"
\end_layout
\begin_layout Standard
Technically the FT algorithm is a list decoder.
Among the list of candidate codewords found by the stochastic search algorithm,
only the one with the largest
\begin_inset Formula $u$
\end_inset
is retained.
As with all such algorithms, a stopping criterion is necessary.
As with all decoding algorithms that generate a list of possible codewords,
a stopping criterion is necessary.
FT accepts a codeword unconditionally if the Hamming distance
\begin_inset Formula $X$
\end_inset
@ -1312,7 +1382,7 @@ For each received symbol, define the erasure probability as 1.3 times the
a priori
\emph default
symbol-error probability determined from soft-symbol information
\begin_inset Formula $\{p_{1}\textrm{-rank},\,p_{2}/p_{1}\}$
\begin_inset Formula $\{p_{1}\textrm{-rank},\, p_{2}/p_{1}\}$
\end_inset
.