mirror of https://github.com/saitohirga/WSJT-X.git
More massaging of draft paper on the FT dedoder.
git-svn-id: svn+ssh://svn.code.sf.net/p/wsjt/wsjt/branches/wsjtx@6205 ab8295b8-cf94-4d9e-aec4-7959e3be5d79
This commit is contained in:
Joe Taylor
parent
546da13f9e
commit
b8f772d0b9
|
|
@ -89,23 +89,14 @@ The JT65 mode 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.
|
||||
One reason for the success and popularity of JT65 is its use of strong
|
||||
error-correction coding.
|
||||
The JT65 code is a short block-length, low-rate, Reed-Solomon code based
|
||||
on a 64-symbol alphabet.
|
||||
Since 2004, most JT65 decoders have used the patented
|
||||
\begin_inset Quotes eld
|
||||
\end_inset
|
||||
|
||||
Koetter-Vardy
|
||||
\begin_inset Quotes erd
|
||||
\end_inset
|
||||
|
||||
(KV) algebraic soft-decision decoder.
|
||||
A major reason for the success and popularity of JT65 is its use of strong
|
||||
error-correction coding: a short block-length, low-rate, Reed-Solomon code
|
||||
based on a 64-symbol alphabet.
|
||||
Since 2004, most JT65 decoders have used the patented Koetter-Vardy (KV)
|
||||
algebraic soft-decision decoder.
|
||||
The KV decoder is implemented in a closed-source program licensed to K1JT
|
||||
for use in amateur radio applications.
|
||||
We describe here a new open-source alternative called the FTRSD (or FT)
|
||||
algotithm.
|
||||
We describe here a new open-source alternative called the FT algotithm.
|
||||
It is conceptually simple, is built around the well-known Berlekamp-Massey
|
||||
errors-and-erasures algorithm, and perform at least as well as the KV decoder.
|
||||
\end_layout
|
||||
|
|
@ -116,7 +107,7 @@ Introduction
|
|||
|
||||
\begin_layout Standard
|
||||
JT65 message frames consist of a short, compressed message encoded for transmiss
|
||||
ion using a Reed-Solomon code.
|
||||
ion with a Reed-Solomon code.
|
||||
Reed-Solomon codes are block codes; as such they are characterized by the
|
||||
length of their codewords,
|
||||
\begin_inset Formula $n$
|
||||
|
|
@ -134,7 +125,8 @@ ion using a Reed-Solomon code.
|
|||
|
||||
.
|
||||
JT65 uses a (63,12) Reed-Solomon code with 64 possible values for each
|
||||
symbol, so each symbol represents
|
||||
symbol.
|
||||
Each symbol represents
|
||||
\begin_inset Formula $\log_{2}64=6$
|
||||
\end_inset
|
||||
|
||||
|
|
@ -181,9 +173,9 @@ The minimum Hamming distance of the JT65 code is
|
|||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Given only 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
|
||||
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
|
||||
|
||||
|
|
@ -199,11 +191,11 @@ For the JT65 code,
|
|||
\begin_inset Formula $t=25$
|
||||
\end_inset
|
||||
|
||||
, which means that it is always possible to efficiently decode a received
|
||||
word having no more than 25 symbol errors.
|
||||
: 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 ncessarily involved, namely
|
||||
Two steps are ncessarily involved in this process, namely
|
||||
\end_layout
|
||||
|
||||
\begin_layout Enumerate
|
||||
|
|
@ -215,16 +207,16 @@ determine the correct value of the incorrect symbols
|
|||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
If it is somehow known that certain symbols are incorrect, this information
|
||||
can be used to reduce the amount of work in step 1 and to allow step 2
|
||||
to correct more than
|
||||
If we somehow know that certain symbols are incorrect, this information
|
||||
can be used to reduce the work 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 can be provided
|
||||
to the BM decoder, and if no correct symbols are accidentally labeled as
|
||||
errors, the BM algorithm can correct up to
|
||||
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
|
||||
|
||||
|
|
@ -233,34 +225,33 @@ If it is somehow known that certain symbols are incorrect, this information
|
|||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
The FT algorithm creates a list of symbols suspected of being incorrect
|
||||
and sends it to the BM decoder.
|
||||
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
|
||||
erasures,
|
||||
\begin_inset Quotes erd
|
||||
\end_inset
|
||||
|
||||
, while other incorrect symbols will be called
|
||||
while other incorrect symbols will be called
|
||||
\begin_inset Quotes eld
|
||||
\end_inset
|
||||
|
||||
errors
|
||||
errors.
|
||||
\begin_inset Quotes erd
|
||||
\end_inset
|
||||
|
||||
.
|
||||
As already noted, with perfect erasure information up to 51 errors can
|
||||
be corrected.
|
||||
When the erasure information is imperfect, some of the erased symbols may
|
||||
be correct and some other symbols in error.
|
||||
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 unerased symbols contain
|
||||
symbols are erased and the remaining (unerased) symbols contain
|
||||
\begin_inset Formula $e$
|
||||
\end_inset
|
||||
|
||||
|
|
@ -301,7 +292,7 @@ errors-only
|
|||
\begin_inset Formula $d-1=51$
|
||||
\end_inset
|
||||
|
||||
for JT65), the decoder is said to be an
|
||||
for JT65), the decoder is called an
|
||||
\begin_inset Quotes eld
|
||||
\end_inset
|
||||
|
||||
|
|
@ -310,8 +301,8 @@ errors-and-erasures
|
|||
\end_inset
|
||||
|
||||
decoder.
|
||||
The errors-and-erasures capability of Reed-Solomon codes lies at the core
|
||||
of the FTRSD algorithm.
|
||||
The possibility of doing errors-and-erasures decoding lies at the heart
|
||||
of the FT algorithm.
|
||||
|
||||
\end_layout
|
||||
|
||||
|
|
@ -326,17 +317,17 @@ Do I feel lucky?
|
|||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
The FTRSD algorithm uses a statistical argument based on the quality of
|
||||
received symbols to generate lists of symbols likely to be in error, thereby
|
||||
enabling reliable decoding of received codewords with more than 25 errors.
|
||||
As a specific example, consider a received JT65 codeword with 23 correct
|
||||
The FT algorithm uses the estimated quality of received symbols to generate
|
||||
lists of symbols considered likely to be in error, thereby enabling reliable
|
||||
decoding of received words with more than 25 errors.
|
||||
As a specific example, consider a received JT65 signal producing 23 correct
|
||||
symbols and 40 errors.
|
||||
We do not know which symbols are in error.
|
||||
Suppose that the decoder randomly chooses
|
||||
Suppose that the decoder randomly selects
|
||||
\begin_inset Formula $s=40$
|
||||
\end_inset
|
||||
|
||||
symbols to erase, leaving 23 unerased symbols.
|
||||
symbols for erasure, leaving 23 unerased symbols.
|
||||
According to Eq.
|
||||
(
|
||||
\begin_inset CommandInset ref
|
||||
|
|
@ -381,15 +372,11 @@ tric probability distribution.
|
|||
\end_inset
|
||||
|
||||
will be random variables.
|
||||
Let
|
||||
\begin_inset Formula $P(x|N,X,s)$
|
||||
\end_inset
|
||||
|
||||
denote the conditional probability mass function for
|
||||
The conditional probability mass function for
|
||||
\begin_inset Formula $x$
|
||||
\end_inset
|
||||
|
||||
, the number of erased incorrect symbols, given the stated values of
|
||||
given stated values of
|
||||
\begin_inset Formula $N$
|
||||
\end_inset
|
||||
|
||||
|
|
@ -401,8 +388,7 @@ tric probability distribution.
|
|||
\begin_inset Formula $s$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
Then
|
||||
may be written as
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
|
@ -418,7 +404,7 @@ where
|
|||
\end_inset
|
||||
|
||||
is the binomial coefficient.
|
||||
[The binomial coefficient can be calculated using the
|
||||
The binomial coefficient can be calculated using the function
|
||||
\begin_inset Quotes eld
|
||||
\end_inset
|
||||
|
||||
|
|
@ -430,7 +416,7 @@ nchoosek(
|
|||
\begin_inset Quotes erd
|
||||
\end_inset
|
||||
|
||||
function in Gnu Octave.
|
||||
in the interpreted language GNU Octave.
|
||||
The hypergeometric probability mass function defined in Eq.
|
||||
(
|
||||
\begin_inset CommandInset ref
|
||||
|
|
@ -439,7 +425,7 @@ reference "eq:hypergeometric_pdf"
|
|||
|
||||
\end_inset
|
||||
|
||||
) is available in Gnu Octave as function
|
||||
) is available in GNU Octave as function
|
||||
\begin_inset Quotes eld
|
||||
\end_inset
|
||||
|
||||
|
|
@ -451,15 +437,15 @@ hygepdf(
|
|||
\begin_inset Quotes erd
|
||||
\end_inset
|
||||
|
||||
.]
|
||||
.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Paragraph
|
||||
Example 1:
|
||||
\end_layout
|
||||
|
||||
\begin_layout Case
|
||||
A codeword contains
|
||||
\begin_layout Standard
|
||||
Suppose a codeword contains
|
||||
\begin_inset Formula $X=40$
|
||||
\end_inset
|
||||
|
||||
|
|
@ -477,7 +463,7 @@ A codeword contains
|
|||
\begin_inset Formula $x=35$
|
||||
\end_inset
|
||||
|
||||
of the erased symbols are incorrect is then
|
||||
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}}=2.356\times10^{-7}.
|
||||
|
|
@ -513,10 +499,10 @@ ty of erasing 35 errors, we may safely conclude that the probability of
|
|||
Example 2:
|
||||
\end_layout
|
||||
|
||||
\begin_layout Case
|
||||
How might we best choose the number of symbols to be chosen for erasure,
|
||||
so as to maximize the probability of successful decoding? By exhaustive
|
||||
search over all possible values up to
|
||||
\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
|
||||
|
||||
|
|
@ -551,8 +537,9 @@ P(x\ge37)\simeq2\times10^{-6}.
|
|||
\end_inset
|
||||
|
||||
This probability is about 8 times higher than the probability of success
|
||||
when only 40 symbols were erased, but the odds of successfully decoding
|
||||
on the first try are still only about 1 in 500,000.
|
||||
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
|
||||
|
||||
|
|
@ -560,19 +547,19 @@ This probability is about 8 times higher than the probability of success
|
|||
Example 3:
|
||||
\end_layout
|
||||
|
||||
\begin_layout Case
|
||||
Examples 1 and 2 show that a strategy of randomly selecting symbols to erase
|
||||
\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 symbol set contain
|
||||
Let the received word contain
|
||||
\begin_inset Formula $X=40$
|
||||
\end_inset
|
||||
|
||||
incorrect symbols, as before, but suppose it is known that 10 symbols are
|
||||
much more reliable than the other 53.
|
||||
The 10 most reliable symbols are therefore protected from erasure, and
|
||||
erasures chosen from the smaller set of
|
||||
incorrect symbols, as before, but suppose we know that 10 symbols are significa
|
||||
ntly more reliable than the other 53.
|
||||
We might therefore protect the 10 most reliable symbols from erasure, and
|
||||
choose erasures from the smaller set of
|
||||
\begin_inset Formula $N=53$
|
||||
\end_inset
|
||||
|
||||
|
|
@ -581,11 +568,9 @@ Examples 1 and 2 show that a strategy of randomly selecting symbols to erase
|
|||
\begin_inset Formula $s=45$
|
||||
\end_inset
|
||||
|
||||
symbols are now chosen randomly from the set of 53 least reliable symbols,
|
||||
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
|
||||
symbols are chosen randomly 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
|
||||
|
||||
|
|
@ -593,7 +578,7 @@ Examples 1 and 2 show that a strategy of randomly selecting symbols to erase
|
|||
\begin_inset Formula $X=40$
|
||||
\end_inset
|
||||
|
||||
,
|
||||
, and
|
||||
\begin_inset Formula $s=45$
|
||||
\end_inset
|
||||
|
||||
|
|
@ -610,7 +595,7 @@ reference "eq:hypergeometric_pdf"
|
|||
\end_inset
|
||||
|
||||
.
|
||||
Even better odds are obtained with
|
||||
Even better odds are obtained by choosing
|
||||
\begin_inset Formula $s=47$
|
||||
\end_inset
|
||||
|
||||
|
|
@ -627,7 +612,7 @@ reference "eq:hypergeometric_pdf"
|
|||
\begin_inset Formula $X=40$
|
||||
\end_inset
|
||||
|
||||
, and
|
||||
, and
|
||||
\begin_inset Formula $s=47$
|
||||
\end_inset
|
||||
|
||||
|
|
@ -636,8 +621,9 @@ reference "eq:hypergeometric_pdf"
|
|||
\end_inset
|
||||
|
||||
.
|
||||
These odds are the best so far, about 1 in 38.
|
||||
|
||||
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 from the BM decoder.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Section
|
||||
|
|
@ -647,23 +633,21 @@ name "sec:The-decoding-algorithm"
|
|||
|
||||
\end_inset
|
||||
|
||||
The FTRSD decoding algorithm
|
||||
The FT decoding algorithm
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Example 3 shows how reliable information about symbol quality might lead
|
||||
to an algorithm capable of decoding received frames with a large number
|
||||
of errors.
|
||||
In practice the number of errors in the received word is unknown, so it
|
||||
is better use a stochastic algorithm to assign a high probability of erasure
|
||||
to low-quality symbols and a 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.
|
||||
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 a high erasure probability to low-quality
|
||||
symbols and a relatively low probability to high-quality symbols.
|
||||
As illustrated by Example 3, a good choice of these probabilities can increase
|
||||
the chance of a successful decode by many orders of magnitude.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
The FTRSD algorithm uses two quality indices made available by a noncoherent
|
||||
The FT algorithm uses two quality indices made available by a noncoherent
|
||||
64-FSK demodulator.
|
||||
The demodulator identifies the most likely value for each symbol based
|
||||
on which of 64 frequency bins contains the the largest signal-plus-noise
|
||||
|
|
@ -710,9 +694,8 @@ soft-symbol
|
|||
\end_inset
|
||||
|
||||
values.
|
||||
High ranking symbols have larger signal-to-noise ratio than lower ranked
|
||||
symbols.
|
||||
|
||||
High ranking symbols have larger signal-to-noise ratio than those with
|
||||
lower rank.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
|
|
@ -728,8 +711,8 @@ soft-symbol
|
|||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
The FTRSD decoder uses a table of symbol error probabilities derived from
|
||||
a large dataset of received words that have been successfully decoded.
|
||||
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
|
||||
|
|
@ -743,50 +726,40 @@ a-priori
|
|||
\end_inset
|
||||
|
||||
metrics.
|
||||
These probabilities will be close to 1 for low-quality symbols and close
|
||||
to 0 for high-quality symbols.
|
||||
Recall from Examples 2 and 3 that the best performance was obtained when
|
||||
|
||||
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 FTRSD algorithm works best when the probability of
|
||||
erasing a symbol is somewhat larger than the probability that the symbol
|
||||
is incorrect.
|
||||
Correspondingly, the FT algorithm works best when the probability of erasing
|
||||
a symbol is somewhat larger than the probability that the symbol is incorrect.
|
||||
Empirically, we found good decoding performance when the symbol erasure
|
||||
probability is about 1.3 times the symbol error probability.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
The FTRSD algorithm tries successively to decode the received word using
|
||||
educated guesses to select symbols for erasure.
|
||||
For each iteration an independent stochastic erasure vector is generated
|
||||
based on the symbol erasure probabilities.
|
||||
The erasure vector is provided to the BM decoder along with the full set
|
||||
of 63 received symbols.
|
||||
If the BM decoder finds a candidate codeword it is assigned a quality metric,
|
||||
defined as the soft distance,
|
||||
The FT algorithm tries successively to decode the received word using independen
|
||||
t educated guesses to select symbols for erasure.
|
||||
For each iteration an 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
|
||||
|
||||
, between the received word and the codeword:
|
||||
defined as the soft distance between the received word and the codeword:
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
d_{s}=\sum_{i=1}^{n}(1+p_{1,i})\alpha_{i}.\label{eq:soft_distance}
|
||||
d_{s}=\sum_{i=1}^{n}\alpha_{i}\,(1+p_{1,i}).\label{eq:soft_distance}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
Here
|
||||
\begin_inset Formula $p_{1,i}$
|
||||
\end_inset
|
||||
|
||||
is the fractional power associated with received symbol
|
||||
\begin_inset Formula $i$
|
||||
\end_inset
|
||||
|
||||
;
|
||||
\begin_inset Formula $\alpha_{i}=0$
|
||||
\end_inset
|
||||
|
||||
|
|
@ -794,28 +767,36 @@ Here
|
|||
\begin_inset Formula $i$
|
||||
\end_inset
|
||||
|
||||
is the same as the corresponding symbol in the codeword, and
|
||||
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.
|
||||
This soft distance can be written as two terms, the first of which is just
|
||||
the Hamming distance between the received word and the codeword.
|
||||
The second term ensures that if two candidate codewords have the same Hamming
|
||||
distance from the received word, a smaller distance will be assigned to
|
||||
the one where the different symbols occurred in lower quality symbols.
|
||||
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 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 distance will be assigned to the one where differences
|
||||
occur in symbols of lower quality.
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Technically the FT algorithm is a list-decoder, potentially generating a
|
||||
Technically the FT algorithm is a list decoder, potentially generating a
|
||||
list of candidate codewords.
|
||||
Among the list of candidate codewords found by this 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.
|
||||
FTRSD accepts a codeword unconditionally if its soft distance is smaller
|
||||
than an empirically determined acceptance threshold,
|
||||
FT accepts a codeword unconditionally if its soft distance is smaller than
|
||||
an empirically determined acceptance threshold,
|
||||
\begin_inset Formula $d_{a}$
|
||||
\end_inset
|
||||
|
||||
|
|
|
|||
|
|
@ -2348,8 +2348,8 @@ void MainWindow::guiUpdate()
|
|||
m_ntx=7;
|
||||
ui->rbGenMsg->setChecked(true);
|
||||
} else {
|
||||
m_ntx=6;
|
||||
ui->txrb6->setChecked(true);
|
||||
//JHT 11/29/2015 m_ntx=6;
|
||||
// ui->txrb6->setChecked(true);
|
||||
}
|
||||
}
|
||||
}
|
||||
|
|
|
|||
Loading…
Reference in New Issue