mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2024-11-08 01:56:10 -05:00
ef49f6dbd6
git-svn-id: svn+ssh://svn.code.sf.net/p/wsjt/wsjt/branches/wsjtx@6301 ab8295b8-cf94-4d9e-aec4-7959e3be5d79
1334 lines
30 KiB
Plaintext
1334 lines
30 KiB
Plaintext
#LyX 2.1 created this file. For more info see http://www.lyx.org/
|
|
\lyxformat 474
|
|
\begin_document
|
|
\begin_header
|
|
\textclass paper
|
|
\use_default_options true
|
|
\maintain_unincluded_children false
|
|
\language english
|
|
\language_package default
|
|
\inputencoding auto
|
|
\fontencoding global
|
|
\font_roman default
|
|
\font_sans default
|
|
\font_typewriter default
|
|
\font_math auto
|
|
\font_default_family default
|
|
\use_non_tex_fonts false
|
|
\font_sc false
|
|
\font_osf false
|
|
\font_sf_scale 100
|
|
\font_tt_scale 100
|
|
\graphics default
|
|
\default_output_format default
|
|
\output_sync 0
|
|
\bibtex_command default
|
|
\index_command default
|
|
\float_placement H
|
|
\paperfontsize 12
|
|
\spacing onehalf
|
|
\use_hyperref false
|
|
\papersize default
|
|
\use_geometry true
|
|
\use_package amsmath 1
|
|
\use_package amssymb 1
|
|
\use_package cancel 1
|
|
\use_package esint 1
|
|
\use_package mathdots 1
|
|
\use_package mathtools 1
|
|
\use_package mhchem 1
|
|
\use_package stackrel 1
|
|
\use_package stmaryrd 1
|
|
\use_package undertilde 1
|
|
\cite_engine basic
|
|
\cite_engine_type default
|
|
\biblio_style plain
|
|
\use_bibtopic false
|
|
\use_indices false
|
|
\paperorientation portrait
|
|
\suppress_date false
|
|
\justification true
|
|
\use_refstyle 1
|
|
\index Index
|
|
\shortcut idx
|
|
\color #008000
|
|
\end_index
|
|
\leftmargin 1in
|
|
\topmargin 1in
|
|
\rightmargin 1in
|
|
\bottommargin 1in
|
|
\secnumdepth 3
|
|
\tocdepth 3
|
|
\paragraph_separation indent
|
|
\paragraph_indentation default
|
|
\quotes_language english
|
|
\papercolumns 1
|
|
\papersides 1
|
|
\paperpagestyle default
|
|
\tracking_changes false
|
|
\output_changes false
|
|
\html_math_output 0
|
|
\html_css_as_file 0
|
|
\html_be_strict false
|
|
\end_header
|
|
|
|
\begin_body
|
|
|
|
\begin_layout Title
|
|
A stochastic successive erasures soft-decision decoder for the JT65 (63,12)
|
|
Reed-Solomon code
|
|
\end_layout
|
|
|
|
\begin_layout Author
|
|
Steven J.
|
|
Franke, K9AN and Joseph H.
|
|
Taylor, K1JT
|
|
\end_layout
|
|
|
|
\begin_layout Abstract
|
|
The JT65 protocol has revolutionized amateur-radio weak-signal communication
|
|
by enabling amateur radio operators with small antennas and relatively
|
|
low-power transmitters to communicate over propagation paths not usable
|
|
with traditional technologies.
|
|
A major reason for the success and popularity of JT65 is its use of a strong
|
|
error-correction code: a short block-length, low-rate Reed-Solomon code
|
|
based on a 64-symbol alphabet.
|
|
Since 2004, most programs implementing JT65 have used the patented Koetter-Vard
|
|
y (KV) algebraic soft-decision decoder, licensed to K1JT and implemented
|
|
in a closed-source program for use in amateur radio applications.
|
|
We describe here a new open-source alternative called the Franke-Taylor
|
|
(FT, or K9AN-K1JT) algorithm.
|
|
It is conceptually simple, built around the well-known Berlekamp-Massey
|
|
errors-and-erasures algorithm, and performs even better than the KV decoder.
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
Introduction
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
JT65 message frames consist of a short compressed message encoded for transmissi
|
|
on with a Reed-Solomon code.
|
|
Reed-Solomon codes are block codes characterized by
|
|
\begin_inset Formula $n$
|
|
\end_inset
|
|
|
|
, the length of their codewords,
|
|
\begin_inset Formula $k$
|
|
\end_inset
|
|
|
|
, the number of message symbols conveyed by the codeword, and the number
|
|
of possible values for each symbol in the codewords.
|
|
The codeword length and the number of message symbols are specified with
|
|
the notation
|
|
\begin_inset Formula $(n,k)$
|
|
\end_inset
|
|
|
|
.
|
|
JT65 uses a (63,12) Reed-Solomon code with 64 possible values for each
|
|
symbol.
|
|
Each of the 12 message symbols represents
|
|
\begin_inset Formula $\log_{2}64=6$
|
|
\end_inset
|
|
|
|
message bits.
|
|
The source-encoded messages conveyed by a 63-symbol JT65 frame thus consist
|
|
of 72 information bits.
|
|
The JT65 code is systematic, which means that the 12 message symbols are
|
|
embedded in the codeword without modification and another 51 parity symbols
|
|
derived from the message symbols are added to form a codeword of 63 symbols.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The concept of Hamming distance is used as a measure of
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
distance
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
between different codewords, or between a received word and a codeword.
|
|
Hamming distance is the number of code symbols that differ in the two words
|
|
being compared.
|
|
Reed-Solomon codes have minimum Hamming distance
|
|
\begin_inset Formula $d$
|
|
\end_inset
|
|
|
|
, where
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
d=n-k+1.\label{eq:minimum_distance}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
The minimum Hamming distance of the JT65 code is
|
|
\begin_inset Formula $d=52$
|
|
\end_inset
|
|
|
|
, which means that any particular codeword differs from all other codewords
|
|
in at least 52 symbol positions.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Given a received word containing some incorrect symbols (errors), the received
|
|
word can be decoded into the correct codeword using a deterministic, algebraic
|
|
algorithm provided that no more than
|
|
\begin_inset Formula $t$
|
|
\end_inset
|
|
|
|
symbols were received incorrectly, where
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
t=\left\lfloor \frac{n-k}{2}\right\rfloor .\label{eq:t}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
For the JT65 code,
|
|
\begin_inset Formula $t=25$
|
|
\end_inset
|
|
|
|
, so it is always possible to efficiently decode a received word having
|
|
no more than 25 symbol errors.
|
|
Any one of several well-known algebraic algorithms, such as the widely
|
|
used Berlekamp-Massey (BM) algorithm, can carry out the decoding.
|
|
Two steps are necessarily involved in this process, namely
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Determine which symbols were received incorrectly.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Find the correct value of the incorrect symbols.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
If we somehow know that certain symbols are incorrect, this information
|
|
can be used to reduce the work involved in step 1 and allow step 2 to correct
|
|
more than
|
|
\begin_inset Formula $t$
|
|
\end_inset
|
|
|
|
errors.
|
|
In the unlikely event that the location of every error is known and if
|
|
no correct symbols are accidentally labeled as errors, the BM algorithm
|
|
can correct up to
|
|
\begin_inset Formula $d$
|
|
\end_inset
|
|
|
|
errors.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The FT algorithm creates lists of symbols suspected of being incorrect and
|
|
sends them to the BM decoder.
|
|
Symbols flagged in this way are called
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
erasures,
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
while other incorrect symbols will be called
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
errors.
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
As already noted, with perfect erasure information up to 51 incorrect symbols
|
|
can be corrected.
|
|
Imperfect erasure information means that some erased symbols may be correct,
|
|
and some other symbols in error.
|
|
If
|
|
\begin_inset Formula $s$
|
|
\end_inset
|
|
|
|
symbols are erased and the remaining
|
|
\begin_inset Formula $n-s$
|
|
\end_inset
|
|
|
|
symbols contain
|
|
\begin_inset Formula $e$
|
|
\end_inset
|
|
|
|
errors, the BM algorithm can find the correct codeword as long as
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
s+2e\le d-1.\label{eq:erasures_and_errors}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
If
|
|
\begin_inset Formula $s=0$
|
|
\end_inset
|
|
|
|
, the decoder is said to be an
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
errors-only
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
decoder.
|
|
If
|
|
\begin_inset Formula $0<s\le d-1$
|
|
\end_inset
|
|
|
|
(
|
|
\begin_inset Formula $d-1=51$
|
|
\end_inset
|
|
|
|
for JT65), the decoder is called an
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
errors-and-erasures
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
decoder.
|
|
The possibility of doing errors-and-erasures decoding lies at the heart
|
|
of the FT algorithm.
|
|
On that foundation we have built a capability for using
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
soft
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
information on symbol reliability, thereby producing a soft-decision decoder.
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "sec:You've-got-to"
|
|
|
|
\end_inset
|
|
|
|
Do I feel lucky?
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The FT algorithm uses the estimated quality of received symbols to generate
|
|
lists of symbols considered likely to be in error, thus enabling reliable
|
|
decoding of received words with more than 25 errors.
|
|
As a specific example, consider a received JT65 word with 23 correct symbols
|
|
and 40 errors.
|
|
We do not know which symbols are in error.
|
|
Suppose that the decoder randomly selects
|
|
\begin_inset Formula $s=40$
|
|
\end_inset
|
|
|
|
symbols for erasure, leaving 23 unerased symbols.
|
|
According to Eq.
|
|
(
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:erasures_and_errors"
|
|
|
|
\end_inset
|
|
|
|
), the BM decoder can successfully decode this word as long as
|
|
\begin_inset Formula $e$
|
|
\end_inset
|
|
|
|
, the number of errors present in the 23 unerased symbols, is 5 or less.
|
|
The number of errors captured in the set of 40 erased symbols must therefore
|
|
be at least 35.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The probability of selecting some particular number of incorrect symbols
|
|
in a randomly selected subset of received symbols is governed by the hypergeome
|
|
tric probability distribution.
|
|
Let us define
|
|
\begin_inset Formula $N$
|
|
\end_inset
|
|
|
|
as the number of symbols from which erasures will be selected,
|
|
\begin_inset Formula $X$
|
|
\end_inset
|
|
|
|
as the number of incorrect symbols in the set of
|
|
\begin_inset Formula $N$
|
|
\end_inset
|
|
|
|
symbols, and
|
|
\begin_inset Formula $x$
|
|
\end_inset
|
|
|
|
as the number of errors in the erased symbols.
|
|
In an ensemble of many received words,
|
|
\begin_inset Formula $X$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $x$
|
|
\end_inset
|
|
|
|
will be random variables.
|
|
The conditional probability mass function for
|
|
\begin_inset Formula $x$
|
|
\end_inset
|
|
|
|
given stated values of
|
|
\begin_inset Formula $N$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $X$
|
|
\end_inset
|
|
|
|
, and
|
|
\begin_inset Formula $s$
|
|
\end_inset
|
|
|
|
may be written as
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
P(x=\epsilon|N,X,s)=\frac{\binom{X}{x}\binom{N-X}{s-\epsilon}}{\binom{N}{s}}\label{eq:hypergeometric_pdf}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
where
|
|
\begin_inset Formula $\binom{n}{k}=\frac{n!}{k!(n-k)!}$
|
|
\end_inset
|
|
|
|
is the binomial coefficient.
|
|
The binomial coefficient can be calculated using the function
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
nchoosek(
|
|
\begin_inset Formula $n,k$
|
|
\end_inset
|
|
|
|
)
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
in the interpreted language GNU Octave, as well as many free online calculators
|
|
The hypergeometric probability mass function defined in Eq.
|
|
(
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:hypergeometric_pdf"
|
|
|
|
\end_inset
|
|
|
|
) is available in GNU Octave as function
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
hygepdf(
|
|
\begin_inset Formula $x,N,X,s$
|
|
\end_inset
|
|
|
|
)
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
.
|
|
The cumulative probability that
|
|
\emph on
|
|
at least
|
|
\emph default
|
|
|
|
\begin_inset Formula $\epsilon$
|
|
\end_inset
|
|
|
|
errors are captured in a subset of
|
|
\begin_inset Formula $s$
|
|
\end_inset
|
|
|
|
erased symbols selected from a group of
|
|
\begin_inset Formula $N$
|
|
\end_inset
|
|
|
|
symbols containing
|
|
\begin_inset Formula $X$
|
|
\end_inset
|
|
|
|
errors is
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
P(x\ge\epsilon|N,X,s)=\sum_{j=\epsilon}^{N}P(x=j|N,X,s).\label{eq:cumulative_prob}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Paragraph
|
|
Example 1:
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Suppose a received word contains
|
|
\begin_inset Formula $X=40$
|
|
\end_inset
|
|
|
|
incorrect symbols.
|
|
In an attempt to decode using an errors-and-erasures decoder,
|
|
\begin_inset Formula $s=40$
|
|
\end_inset
|
|
|
|
symbols are randomly selected for erasure from the full set of
|
|
\begin_inset Formula $N=n=63$
|
|
\end_inset
|
|
|
|
symbols.
|
|
The probability that
|
|
\begin_inset Formula $x=35$
|
|
\end_inset
|
|
|
|
of the erased symbols are actually incorrect is then
|
|
\begin_inset Formula
|
|
\[
|
|
P(x=35)=\frac{\binom{40}{35}\binom{63-40}{40-35}}{\binom{63}{40}}\simeq2.4\times10^{-7}.
|
|
\]
|
|
|
|
\end_inset
|
|
|
|
Similarly, the probability that
|
|
\begin_inset Formula $x=36$
|
|
\end_inset
|
|
|
|
of the erased symbols are incorrect is
|
|
\begin_inset Formula
|
|
\[
|
|
P(x=36)\simeq8.6\times10^{-9}.
|
|
\]
|
|
|
|
\end_inset
|
|
|
|
Since the probability of erasing 36 errors is so much smaller than the probabili
|
|
ty of erasing 35 errors, we may safely conclude that the probability of
|
|
randomly choosing an erasure vector that can decode the received word is
|
|
approximately
|
|
\begin_inset Formula $P(x=35)\simeq2.4\times10^{-7}$
|
|
\end_inset
|
|
|
|
.
|
|
The odds of successfully decoding the word on the first try are very poor,
|
|
about 1 in 4 million.
|
|
\end_layout
|
|
|
|
\begin_layout Paragraph
|
|
Example 2:
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
How might we best choose the number of symbols to erase, in order to maximize
|
|
the probability of successful decoding? By exhaustive search over all possible
|
|
values up to
|
|
\begin_inset Formula $s=51$
|
|
\end_inset
|
|
|
|
, it turns out that for
|
|
\begin_inset Formula $X=40$
|
|
\end_inset
|
|
|
|
the best strategy is to erase
|
|
\begin_inset Formula $s=45$
|
|
\end_inset
|
|
|
|
symbols.
|
|
Decoding will then be assured if the set of erased symbols contains at
|
|
least 37 errors.
|
|
With
|
|
\begin_inset Formula $N=63$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $X=40$
|
|
\end_inset
|
|
|
|
, and
|
|
\begin_inset Formula $s=45$
|
|
\end_inset
|
|
|
|
, the probability of successful decode in a single try is
|
|
\begin_inset Formula
|
|
\[
|
|
P(x\ge37)\simeq1.9\times10^{-6}.
|
|
\]
|
|
|
|
\end_inset
|
|
|
|
This probability is about 8 times higher than the probability of success
|
|
when only 40 symbols were erased.
|
|
Nevertheless, the odds of successfully decoding on the first try are still
|
|
only about 1 in 500,000.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Paragraph
|
|
Example 3:
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Examples 1 and 2 show that a random strategy for selecting symbols to erase
|
|
is unlikely to be successful unless we are prepared to wait a long time
|
|
for an answer.
|
|
So let's modify the strategy to tip the odds in our favor.
|
|
Let the received word contain
|
|
\begin_inset Formula $X=40$
|
|
\end_inset
|
|
|
|
incorrect symbols, as before, but suppose we know that 10 received symbols
|
|
are significantly more reliable than the other 53.
|
|
We might therefore protect the 10 most reliable symbols from erasure, selecting
|
|
erasures from the smaller set of
|
|
\begin_inset Formula $N=53$
|
|
\end_inset
|
|
|
|
less reliable symbols.
|
|
If
|
|
\begin_inset Formula $s=45$
|
|
\end_inset
|
|
|
|
symbols are chosen randomly for erasure in this way, it is still necessary
|
|
for the erased symbols to include at least 37 errors, as in Example 2.
|
|
However, the probabilities are now much more favorable: with
|
|
\begin_inset Formula $N=53$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $X=40$
|
|
\end_inset
|
|
|
|
, and
|
|
\begin_inset Formula $s=45$
|
|
\end_inset
|
|
|
|
, Eq.
|
|
(
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:hypergeometric_pdf"
|
|
|
|
\end_inset
|
|
|
|
) yields
|
|
\begin_inset Formula $P(x\ge37)=0.016$
|
|
\end_inset
|
|
|
|
.
|
|
Even better odds are obtained by choosing
|
|
\begin_inset Formula $s=47$
|
|
\end_inset
|
|
|
|
, which requires
|
|
\begin_inset Formula $x\ge38$
|
|
\end_inset
|
|
|
|
.
|
|
With
|
|
\begin_inset Formula $N=53$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $X=40$
|
|
\end_inset
|
|
|
|
, and
|
|
\begin_inset Formula $s=47$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $P(x\ge38)=0.027$
|
|
\end_inset
|
|
|
|
.
|
|
The odds for successful decoding on the first try are now about 1 in 38.
|
|
A few hundred independently randomized tries would be enough to all-but-guarant
|
|
ee production of a valid codeword by the BM decoder.
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "sec:The-decoding-algorithm"
|
|
|
|
\end_inset
|
|
|
|
The Franke-Taylor decoding algorithm
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Example 3 shows how reliable 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.
|
|
As illustrated by Example 3, a good choice of erasure probabilities can
|
|
increase the chance of a successful decode by many orders of magnitude.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The FT algorithm uses two quality indices made available by a noncoherent
|
|
64-FSK demodulator.
|
|
The demodulator computes the power spectrum for each symbol and identifies
|
|
the most likely symbol value based on the largest signal-plus-noise power
|
|
in 64 frequency bins.
|
|
The fractions of total power in the two bins containing the largest and
|
|
second-largest powers (denoted by
|
|
\begin_inset Formula $p_{1}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $p_{2}$
|
|
\end_inset
|
|
|
|
, respectively) are passed to the decoder from the demodulator as
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
soft-symbol
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
information.
|
|
The decoder derives two metrics from
|
|
\begin_inset Formula $\{p_{1},p_{2}\}:$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
\begin_inset Formula $p_{1}$
|
|
\end_inset
|
|
|
|
-rank: the rank
|
|
\begin_inset Formula $\{1,2,\ldots,63\}$
|
|
\end_inset
|
|
|
|
of the symbol's fractional power,
|
|
\begin_inset Formula $p_{1}$
|
|
\end_inset
|
|
|
|
in the sorted list of
|
|
\begin_inset Formula $p_{1}$
|
|
\end_inset
|
|
|
|
values.
|
|
High ranking symbols have larger signal-to-noise ratio than those with
|
|
lower rank.
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
\begin_inset Formula $p_{2}/p_{1}$
|
|
\end_inset
|
|
|
|
: when
|
|
\begin_inset Formula $p_{2}/p_{1}$
|
|
\end_inset
|
|
|
|
is not small compared to 1, the most likely symbol value is only slightly
|
|
more reliable than the second most likely one.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The FT decoder uses a table of symbol error probabilities derived from a
|
|
large dataset of received words that have been successfully decoded.
|
|
The table provides an estimate of the
|
|
\emph on
|
|
a-priori
|
|
\emph default
|
|
probability of symbol error based on a given symbol's
|
|
\begin_inset Formula $p_{1}$
|
|
\end_inset
|
|
|
|
-rank and
|
|
\begin_inset Formula $p_{2}/p_{1}$
|
|
\end_inset
|
|
|
|
metrics.
|
|
These probabilities are close to 1 for low-quality symbols and close to
|
|
0 for high-quality symbols.
|
|
Recall from Examples 2 and 3 that best performance was obtained with
|
|
\begin_inset Formula $s>X$
|
|
\end_inset
|
|
|
|
.
|
|
Correspondingly, the FT algorithm works best when the probability of erasing
|
|
a symbol is somewhat larger than the probability that the symbol is incorrect.
|
|
We found empirically that good decoding performance is obtained when the
|
|
symbol erasure probability is about 1.3 times the symbol error probability.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The FT algorithm tries successively to decode the received word using independen
|
|
t
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
educated guesses
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
to select symbols for erasure.
|
|
For each iteration a stochastic erasure vector is generated based on the
|
|
symbol erasure probabilities.
|
|
The erasure vector is sent to the BM decoder along with the full set of
|
|
63 received symbols.
|
|
When the BM decoder finds a candidate codeword it is assigned a quality
|
|
metric
|
|
\begin_inset Formula $d_{s}$
|
|
\end_inset
|
|
|
|
defined as the soft distance between the received word and the codeword,
|
|
where
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
d_{s}=\sum_{i=1}^{n}\alpha_{i}\,(1+p_{1,i}).\label{eq:soft_distance}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
Here
|
|
\begin_inset Formula $\alpha_{i}=0$
|
|
\end_inset
|
|
|
|
if received symbol
|
|
\begin_inset Formula $i$
|
|
\end_inset
|
|
|
|
is the same as the corresponding symbol in the codeword,
|
|
\begin_inset Formula $\alpha_{i}=1$
|
|
\end_inset
|
|
|
|
if the received symbol and codeword symbol are different, and
|
|
\begin_inset Formula $p_{1,i}$
|
|
\end_inset
|
|
|
|
is the fractional power associated with received symbol
|
|
\begin_inset Formula $i$
|
|
\end_inset
|
|
|
|
.
|
|
Think of the soft distance as made up of two terms: the first is the Hamming
|
|
distance between the received word and the codeword, and the second ensures
|
|
that if two candidate codewords have the same Hamming distance from the
|
|
received word, a smaller soft distance will be assigned to the one where
|
|
differences occur in symbols of lower estimated reliability.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Technically the FT algorithm is a list decoder, potentially generating a
|
|
list of candidate codewords.
|
|
Among the list of candidate codewords found by the stochastic search algorithm,
|
|
only the one with the smallest soft distance from the received word is
|
|
retained.
|
|
As with all such algorithms, a stopping criterion is necessary.
|
|
FT accepts a codeword unconditionally if its soft distance is smaller than
|
|
an empirically determined acceptance threshold,
|
|
\begin_inset Formula $d_{a}$
|
|
\end_inset
|
|
|
|
.
|
|
A timeout is used to limit the algorithm's execution time if no codewords
|
|
within soft distance
|
|
\begin_inset Formula $d_{a}$
|
|
\end_inset
|
|
|
|
of the received word are found in a reasonable number of trials.
|
|
\end_layout
|
|
|
|
\begin_layout Paragraph
|
|
Algorithm pseudo-code:
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
For each received symbol, define the erasure probability as 1.3 times the
|
|
|
|
\emph on
|
|
a priori
|
|
\emph default
|
|
symbol-error probability determined from soft-symbol information
|
|
\begin_inset Formula $\{p_{1}\textrm{-rank},\,p_{2}/p_{1}\}$
|
|
\end_inset
|
|
|
|
.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Make independent stochastic decisions about whether to erase each symbol
|
|
by using the symbol's erasure probability, allowing a maximum of 51 erasures.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Attempt errors-and-erasures decoding by using the BM algorithm and the set
|
|
of erasures determined in step 2.
|
|
If the BM decoder is successful go to step 5.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
If decoding is not successful, go to step 2.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Calculate the soft distance
|
|
\begin_inset Formula $d_{s}$
|
|
\end_inset
|
|
|
|
between the candidate codeword and the received symbols.
|
|
Set
|
|
\begin_inset Formula $d_{s,min}=d_{s}$
|
|
\end_inset
|
|
|
|
if the soft distance is the smallest one encountered so far.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
If
|
|
\begin_inset Formula $d_{s,min}\le d_{a}$
|
|
\end_inset
|
|
|
|
, go to 8.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
If the number of trials is less than the maximum allowed number, go to 2.
|
|
Otherwise, declare decoding failure and exit.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
A
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
best
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
codeword with
|
|
\begin_inset Formula $d_{s,min}\le d_{a}$
|
|
\end_inset
|
|
|
|
has been found.
|
|
Declare a successful decode and return this codeword .
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
Comparison with Berlekamp-Massey and Koetter-Vardy
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Comparisons of decoding performance are usually presented in the professional
|
|
literature as plots of word error rate as a function of
|
|
\begin_inset Formula $E_{b}/N_{0}$
|
|
\end_inset
|
|
|
|
, the signal-to-noise ratio per information bit.
|
|
Results for the Berlekamp-Massey, Koetter-Vardy, and Franke-Taylor decoding
|
|
algorithms on the (63,12) code are shown in Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:WER"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
For these initial tests we generated 1000 signals at each signal-to-noise
|
|
ratio, assuming the additive white gaussian noise (AWGN) channel, and processed
|
|
the data using each algorithm.
|
|
It's easy to see that, as expected, the soft-decision algorithms FT and
|
|
KV are about 2 dB better than the hard-decision BM algorithm, and that
|
|
FT has a slight edge (about 0.2 dB) over KV.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename fig_wer.pdf
|
|
lyxscale 120
|
|
scale 120
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "fig:WER"
|
|
|
|
\end_inset
|
|
|
|
Word error rate (WER) as a function of
|
|
\begin_inset Formula $E_{b}/N_{0}$
|
|
\end_inset
|
|
|
|
for non-fading signals in AWGN.
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
In the professional literature plots like Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:WER"
|
|
|
|
\end_inset
|
|
|
|
usually extend downward to even smaller error rates, say
|
|
\begin_inset Formula $10^{-6}$
|
|
\end_inset
|
|
|
|
or less, because of the importance of error-free transmission.
|
|
The circumstances for minimal amateur-radio QSOs are very different, however:
|
|
error rates on the order of 0.1, or ever higher, may be acceptable.
|
|
In this case the essential information is better presented in a plot like
|
|
Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:Psuccess"
|
|
|
|
\end_inset
|
|
|
|
, which shows the percentage of transmissions copied correctly as a function
|
|
of signal-to-noise ratio in a standard bandwidth.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
In Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:Psuccess"
|
|
|
|
\end_inset
|
|
|
|
we have plotted the results of simulations for signal-to-noise ratios
|
|
\begin_inset Formula $-30\leq SNR\leq-18$
|
|
\end_inset
|
|
|
|
dB, again using 1000 simulated signals for each point.
|
|
For each decoding algorithm we include three curves: one for the AWGN channel
|
|
and no fading, and two more for Doppler spreads of 0.2 and 1.0 Hz.
|
|
(Note that the JT65 symbol rate is about 2.69 Hz; the simulated Doppler
|
|
spreads are comparable to those encountered on HF ionospheric paths and
|
|
for EME at VHF and lower UHF bands.)
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename fig_psuccess.pdf
|
|
lyxscale 90
|
|
scale 90
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "fig:Psuccess"
|
|
|
|
\end_inset
|
|
|
|
Percentage of JT65 messages successfully decoded as a function of SNR in
|
|
2500 Hz bandwidth.
|
|
Results are shown for the hard-decision Berlekamp-Massey (BM) and soft-decision
|
|
Franke-Taylor (FT) decoding algorithms.
|
|
Curves labeled DS correspond to the hinted-decode (
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
Deep Search
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
) matched-filter algorithm.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename fig_ntrials_vs_nhard.pdf
|
|
lyxscale 120
|
|
scale 120
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
The number of trials needed to decode a received word vs the Hamming distance
|
|
between the received word and the decoded codeword plotted for 1000 simulated
|
|
frames with no fading.
|
|
The SNR in 2500 Hz bandwidth is -24 dB (
|
|
\begin_inset Formula $E_{s}/N_{o}=5.7$
|
|
\end_inset
|
|
|
|
dB).
|
|
Execution time will be roughly proportional to the number of trials.
|
|
The mean and variance of the number of trials (and execution time) increase
|
|
with the number of errors in the received word.
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename fig_wer2.pdf
|
|
lyxscale 120
|
|
scale 120
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
Word error rate (WER) as a function of
|
|
\begin_inset Formula $E_{s}/N_{o}$
|
|
\end_inset
|
|
|
|
for Rayleigh-fading with Doppler-spread
|
|
\begin_inset Formula $\sigma_{f}=0.2$
|
|
\end_inset
|
|
|
|
Hz.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Possible figures:
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
histogram of
|
|
\begin_inset Formula $s$
|
|
\end_inset
|
|
|
|
(number of erasures) for successful decodes with HF and EME data
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
histogram of
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
ntrials
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
(or execution time)
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
Number of decodes vs.
|
|
ntrials
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
Probability of successful decode vs.
|
|
Es/No or S/N in 2500 Hz BW
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
other...
|
|
?
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
Summary
|
|
\end_layout
|
|
|
|
\begin_layout Bibliography
|
|
\begin_inset CommandInset bibitem
|
|
LatexCommand bibitem
|
|
key "key-1"
|
|
|
|
\end_inset
|
|
|
|
"Stochastic Chase Decoding of Reed-Solomon Codes", Camille Leroux, Saied
|
|
Hemati, Shie Mannor, Warren J.
|
|
Gross, IEEE Communications Letters, Vol.
|
|
14, No.
|
|
9, September 2010.
|
|
\end_layout
|
|
|
|
\begin_layout Bibliography
|
|
\begin_inset CommandInset bibitem
|
|
LatexCommand bibitem
|
|
key "key-2"
|
|
|
|
\end_inset
|
|
|
|
"Soft-Decision Decoding of Reed-Solomon Codes Using Successive Error-and-Erasure
|
|
Decoding," Soo-Woong Lee and B.
|
|
V.
|
|
K.
|
|
Vijaya Kumar, IEEE
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
GLOBECOM
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
2008 proceedings.
|
|
\end_layout
|
|
|
|
\begin_layout Bibliography
|
|
\begin_inset CommandInset bibitem
|
|
LatexCommand bibitem
|
|
key "key-3"
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
Stochastic Erasure-Only List Decoding Algorithms for Reed-Solomon Codes,
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
Chang-Ming Lee and Yu T.
|
|
Su, IEEE Signal Processing Letters, Vol.
|
|
16, No.
|
|
8, August 2009.
|
|
\end_layout
|
|
|
|
\begin_layout Bibliography
|
|
\begin_inset CommandInset bibitem
|
|
LatexCommand bibitem
|
|
key "key-4"
|
|
|
|
\end_inset
|
|
|
|
“Algebraic soft-decision decoding of Reed-Solomon codes,” R.
|
|
Köetter and A.
|
|
Vardy, IEEE Trans.
|
|
Inform.
|
|
Theory, Vol.
|
|
49, Nov.
|
|
2003.
|
|
\end_layout
|
|
|
|
\begin_layout Bibliography
|
|
\begin_inset CommandInset bibitem
|
|
LatexCommand bibitem
|
|
key "key-5"
|
|
|
|
\end_inset
|
|
|
|
Berlekamp-Massey decoder written by Phil Karn, http://www.ka9q.net/code/fec/
|
|
\end_layout
|
|
|
|
\end_body
|
|
\end_document
|