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-\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 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 that could
- not be utilized using traditional technologies.
- One reason for the success and popularity of the JT65 mode 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 200?, decoders for the JT65 code have used the 
-\begin_inset Quotes eld
-\end_inset
-
-Koetter-Vardy
-\begin_inset Quotes erd
-\end_inset
-
- (KV) algebraic soft-decision decoder.
- The KV decoder is implemented in a closed-source program that is licensed
- to K1JT for use in amateur applications.
- This note describes a new open-source alternative to the KV decoder called
- the SFRSD decoder.
- The SFRSD decoding algorithm is shown to perform at least as well as the
- KV decoder.
- The SFRSD algorithm is conceptually simple and is built around the well-known
- Berlekamp-Massey errors-and-erasures decoder.
- 
-\end_layout
-
-\begin_layout Section
-Introduction
-\end_layout
-
-\begin_layout Standard
-JT65 message frames consist of a short, compressed, message that is encoded
- for transmission using a Reed-Solomon code.
- Reed-Solomon codes are block codes and, like all block codes, are characterized
- by the length of their codewords, 
-\begin_inset Formula $n$
-\end_inset
-
-, the number of message symbols conveyed by the codeword, 
-\begin_inset Formula $k$
-\end_inset
-
-, and the number of possible values for each symbol in the codewords.
- The codeword length and the number of message symbols are specified as
- a tuple in the form 
-\begin_inset Formula $(n,k)$
-\end_inset
-
-.
- JT65 uses a (63,12) Reed-Solomon code with 64 possible values for each
- symbol, so each symbol represents 
-\begin_inset Formula $\log_{2}64=6$
-\end_inset
-
- message bits.
- The source-encoded messages conveyed by a 63-symbol JT65 frame consist
- of 72 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 the codeword consisting
- of 63 total 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
- that are 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 positions.
- 
-\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 
-\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
-
-, which means that it is always possible to efficiently decode a received
- word that contains no more than 25 symbol errors.
- 
-\end_layout
-
-\begin_layout Standard
-There are a number of well-known algebraic algorithms that can carry out
- the process of decoding a received codeword that contains no more than
- 
-\begin_inset Formula $t$
-\end_inset
-
- errors.
- One such algorithm is the Berlekamp-Massey (BM) decoding algorithm.
-\end_layout
-
-\begin_layout Standard
-A decoder, such as BM, must carry out two tasks: 
-\end_layout
-
-\begin_layout Enumerate
-determine which symbols were received incorrectly 
-\end_layout
-
-\begin_layout Enumerate
-determine the correct value of the incorrect symbols 
-\end_layout
-
-\begin_layout Standard
-If it is somehow known that certain symbols are incorrect, such information
- can be used in the decoding algorithm to reduce the amount of work required
- in step 1 and to allow step 2 to correct more than 
-\begin_inset Formula $t$
-\end_inset
-
- errors.
- In fact, in the unlikely event that the location of each and every error
- is known and is provided to the BM decoder, and if no correct symbols are
- accidentally labeled as errors, then the BM decoder can correct up to 
-\begin_inset Formula $d$
-\end_inset
-
- errors! 
-\end_layout
-
-\begin_layout Standard
-In the decoding algorithm described herein, a list of symbols that are known
- or suspected to be incorrect is sent to the BM decoder.
- Symbols in the received word that are flagged as being incorrect are called
- 
-\begin_inset Quotes eld
-\end_inset
-
-erasures
-\begin_inset Quotes erd
-\end_inset
-
-.
- Symbols that are not erased and that are incorrect will be called 
-\begin_inset Quotes eld
-\end_inset
-
-errors
-\begin_inset Quotes erd
-\end_inset
-
-.
- The BM decoder accepts erasure information in the form of a list of indices
- corresponding to the incorrect, or suspected incorrect, symbols in the
- received word.
- As already noted, if the erasure information is perfect, then up to 51
- errors will be corrected.
- When the erasure information is imperfect, then some of the erased symbols
- will actually be correct, and some of the unerased symbols will be in error.
- If a total of 
-\begin_inset Formula $n_{e}$
-\end_inset
-
- symbols are erased and the remaining unerased symbols contain 
-\begin_inset Formula $x$
-\end_inset
-
- errors, then the BM algorithm can find the correct codeword as long as
- 
-\begin_inset Formula 
-\begin{equation}
-n_{e}+2x\le d-1\label{eq:erasures_and_errors}
-\end{equation}
-
-\end_inset
-
-If 
-\begin_inset Formula $n_{e}=0$
-\end_inset
-
-, then the decoder is said to be an 
-\begin_inset Quotes eld
-\end_inset
-
-errors-only
-\begin_inset Quotes erd
-\end_inset
-
- decoder and it can correct up to 
-\begin_inset Formula $t$
-\end_inset
-
- errors (
-\begin_inset Formula $t$
-\end_inset
-
-=25 for JT65).
- If 
-\begin_inset Formula $0<n_{e}\le d-1$
-\end_inset
-
- (
-\begin_inset Formula $d-1=51$
-\end_inset
-
- for JT65), then the decoder is said to be an 
-\begin_inset Quotes eld
-\end_inset
-
-errors-and-erasures
-\begin_inset Quotes erd
-\end_inset
-
- decoder.
- 
-\end_layout
-
-\begin_layout Standard
-For the JT65 code, (
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:erasures_and_errors"
-
-\end_inset
-
-) says that if 
-\begin_inset Formula $n_{e}$
-\end_inset
-
- symbols are declared to be erased, then the BM decoder will find the correct
- codeword as long as the remaining un-erased symbols contain no more than
- 
-\begin_inset Formula $\left\lfloor \frac{51-n_{e}}{2}\right\rfloor $
-\end_inset
-
- errors.
- The errors-and-erasures capability of the BM decoder is a very powerful
- feature that serves as the core of the new soft-decision decoder described
- herein.
- 
-\end_layout
-
-\begin_layout Standard
-It will be helpful to have some understanding of the errors and erasures
- tradeoff described by (
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:erasures_and_errors"
-
-\end_inset
-
-) to appreciate how the new decoder algorithm works.
- Section NN describes some examples that illustrate ho w the errors-and-erasures
- capability can be combined with some information about the quality of the
- received symbols to enable a decoding algorithm to reliably decode received
- words that contain many more than 25 errors.
- Section NN describes the SFRSD decoding algorithm.
-\end_layout
-
-\begin_layout Section
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:You've-got-to"
-
-\end_inset
-
-You've got to ask yourself.
- Do I feel lucky?
-\end_layout
-
-\begin_layout Standard
-Consider a particular received codeword that contains 40 incorrect symbols
- and 23 correct symbols.
- It is not known which 40 symbols are in error
-\begin_inset Foot
-status open
-
-\begin_layout Plain Layout
-In practice the number of errors will not be known either, but this is not
- a serious problem.
-\end_layout
-
-\end_inset
-
-.
- Suppose that the decoder randomly chooses 40 symbols to erase (
-\begin_inset Formula $n_{e}=40$
-\end_inset
-
-), leaving 23 unerased symbols.
- According to (
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:erasures_and_errors"
-
-\end_inset
-
-), the BM decoder can successfully decode this word as long as the number
- of errors, 
-\begin_inset Formula $x$
-\end_inset
-
-, present in the 23 unerased symbols is 5 or less.
- This means that the number of errors captured in the set of 40 erased symbols
- must be at least 35.
- 
-\end_layout
-
-\begin_layout Standard
-The probability of selecting some particular number of bad symbols in a
- randomly selected subset of the codeword symbols is governed by the hypergeomet
-ric probability distribution.
-\end_layout
-
-\begin_layout Standard
-Define:
-\end_layout
-
-\begin_layout Itemize
-\begin_inset Formula $n$
-\end_inset
-
-= number of symbols in a codeword (63 for JT65),
-\end_layout
-
-\begin_layout Itemize
-\begin_inset Formula $X$
-\end_inset
-
-= number of incorrect symbols in a codeword,
-\end_layout
-
-\begin_layout Itemize
-\begin_inset Formula $n_{e}$
-\end_inset
-
-= number of symbols erased for errors-and-erasures decoding,
-\end_layout
-
-\begin_layout Itemize
-\begin_inset Formula $x$
-\end_inset
-
-= number of incorrect symbols in the set of erased symbols.
-\end_layout
-
-\begin_layout Standard
-In an ensemble of received words, 
-\begin_inset Formula $X$
-\end_inset
-
- and 
-\begin_inset Formula $x$
-\end_inset
-
- will be random variables.
- Let 
-\begin_inset Formula $P(x|(X,n_{e}))$
-\end_inset
-
- denote the conditional probability mass function for the number of incorrect
- symbols, 
-\begin_inset Formula $x$
-\end_inset
-
-, given that the number of incorrect symbols in the codeword is X and the
- number of erased symbols is 
-\begin_inset Formula $n_{e}$
-\end_inset
-
-.
- Then
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula 
-\begin{equation}
-P(x|(X,n_{e}))=\frac{\binom{X}{x}\binom{n-X}{n_{e}-x}}{\binom{n}{n_{e}}}\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 
-\begin_inset Quotes eld
-\end_inset
-
-nchoosek(
-\begin_inset Formula $n,k$
-\end_inset
-
-)
-\begin_inset Quotes erd
-\end_inset
-
- function in Gnu Octave.
- The hypergeometric probability mass function defined in (
-\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 $x,n,X,n_{e}$
-\end_inset
-
-)
-\begin_inset Quotes erd
-\end_inset
-
-.
- 
-\end_layout
-
-\begin_layout Paragraph
-Case 1
-\end_layout
-
-\begin_layout Case
-A codeword contains 
-\begin_inset Formula $X=40$
-\end_inset
-
- incorrect symbols.
- In an attempt to decode using an errors-and-erasures decoder, 
-\begin_inset Formula $n_{e}=40$
-\end_inset
-
- symbols are randomly selected for erasure.
- The probability that 
-\begin_inset Formula $35$
-\end_inset
-
- of the erased symbols are incorrect is:
-\begin_inset Formula 
-\[
-P(x=35)=\frac{\binom{40}{35}\binom{63-40}{40-35}}{\binom{63}{40}}=2.356\times10^{-7}.
-\]
-
-\end_inset
-
-Similarly:
-\begin_inset Formula 
-\[
-P(x=36)=8.610\times10^{-9}.
-\]
-
-\end_inset
-
-Since the probability of catching 36 errors is so much smaller than the
- probability of catching 35 errors, it is safe to say that the probability
- of randomly selecting an erasure vector that can decode the received word
- is essentially equal to 
-\begin_inset Formula $P(X=35)\simeq2.4\times10^{-7}$
-\end_inset
-
-.
- The odds of successfully decoding the word on the first try are about 1
- in 4 million.
-\end_layout
-
-\begin_layout Paragraph
-Case 2
-\end_layout
-
-\begin_layout Case
-It is interesting to work out the best choice for the number of symbols
- that should be selected at random for erasure if the goal is to maximize
- the probability of successfully decoding the word.
- By exhaustive search, it turns out that if 
-\begin_inset Formula $X=40$
-\end_inset
-
-, then the best strategy is to erase 
-\begin_inset Formula $n=45$
-\end_inset
-
- symbols, in which case the word will be decoded 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
-
-, 
-\begin_inset Formula $n_{e}=45$
-\end_inset
-
-, then 
-\begin_inset Formula 
-\[
-P(x\ge37)\simeq2\times10^{-6}.
-\]
-
-\end_inset
-
-This probability is about 8 times higher than the probability of success
- when only 
-\begin_inset Formula $40$
-\end_inset
-
- symbols were erased, and the odds of successfully decoding on the first
- try are roughly 1 in 500,000.
- 
-\end_layout
-
-\begin_layout Paragraph
-Case 3
-\end_layout
-
-\begin_layout Case
-Cases 1 and 2 illustrate the fact that a strategy that tries to guess which
- symbols to erase is not going to be very successful unless we are prepared
- to wait all day for an answer.
- Consider a slight modification to the strategy that can tip the odds in
- our favor.
- Suppose that the codeword contains 
-\begin_inset Formula $X=40$
-\end_inset
-
- incorrect symbols, as before.
- In this case it is known that 10 of the symbols are much more reliable
- than the other 53 symbols.
- The 10 most reliable symbols are all correct and these 10 symbols are protected
- from erasure, i.e.
- the set of erasures is chosen from the smaller set of 53 less reliable
- symbols.
- If 
-\begin_inset Formula $n_{e}=45$
-\end_inset
-
- symbols are chosen randomly from the set of 
-\begin_inset Formula $n=53$
-\end_inset
-
- least reliable symbols, it is still necessary for the erased symbols to
- include at least 37 errors (as in Case 2).
- In this case, with 
-\begin_inset Formula $n=53$
-\end_inset
-
-, 
-\begin_inset Formula $X=40$
-\end_inset
-
-, 
-\begin_inset Formula $n_{e}=45$
-\end_inset
-
-, 
-\begin_inset Formula $P(x\ge37)=0.016$
-\end_inset
-
-! Now, the situation is much better.
- The odds of decoding the word on the first try are approximately 1 in 62.5!
- 
-\end_layout
-
-\begin_layout Standard
-Even better odds are obtained with 
-\begin_inset Formula $n_{e}=47$
-\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
-
-, 
-\begin_inset Formula $n_{e}=47$
-\end_inset
-
-, 
-\begin_inset Formula $P(x\ge38)=0.0266$
-\end_inset
-
-, which makes the odds the best so far; about 1 in 38.
- 
-\end_layout
-
-\begin_layout Section
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:The-decoding-algorithm"
-
-\end_inset
-
-The SFRSD decoding algorithm
-\end_layout
-
-\begin_layout Standard
-Case 3 illustrates how, with the addition of some reliable information about
- the quality of just 10 of the 63 symbols, it is possible to devise an algorithm
- that can decode received words containing a relatively large number of
- errors using only the BM errors-and-erasures decoder.
- The key to improving the odds enough to make the strategy of 
-\begin_inset Quotes eld
-\end_inset
-
-guessing
-\begin_inset Quotes erd
-\end_inset
-
- at the erasure vector useful for practical implementation is to use information
- about the quality of the received symbols to decide which ones are most
- likely to be in error.
- In practice, because the number of errors in the received word is unknown,
- rather than erase a fixed number of symbols, it is better use a stochastic
- algorithm which assigns a relatively high probability of erasure to the
- lowest quality symbols and a relatively low probability of erasure to the
- highest quality symbols.
- As illustrated by case 3, a good choice of the erasure probabilities can
- increase the probability of a successful decode by many orders of magnitude
- relative to a bad choice.
-\end_layout
-
-\begin_layout Standard
-The SFRSD algorithm uses two quality indices available from the JT65 noncoherent
- 64-FSK demodulator to assign a variable probability of erasure to each
- received symbol.
- The demodulator identifies the most likely received symbol based on which
- of 64 frequency bins contains the the largest signal plus noise power.
- The percentage of the total signal plus noise 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 power percentage, 
-\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 lower ranked
- symbols.
- 
-\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 is not much better than
- the second most likely symbol
-\end_layout
-
-\begin_layout Standard
-The decoder has a built-in 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 that is expected 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 
-\emph on
-a-priori
-\emph default
- symbol error probabilities will be close to 1 for lower-quality symbols
- and closer to 0 for high-quality symbols.
- Recall, from Cases 2 and 3, that the best performance was obtained when
- 
-\begin_inset Formula $n_{e}>X$
-\end_inset
-
-.
- Correspondingly, the SFRSD algorithm works best when the probability of
- erasing a symbol is somewhat larger than the probability that the symbol
- is incorrect.
- Empirically, it was determined that good performance of the SFRSD algorithm
- is obtained when the symbol erasure probability is a factor of 
-\begin_inset Formula $1.3$
-\end_inset
-
- larger than the symbol error probability.
-\end_layout
-
-\begin_layout Standard
-The SFRSD algorithm successively tries to decode the received word using
- educated guesses at the symbols that should be erased.
- In each iteration, an independent stochastic erasure vector is generated
- based on the symbol erasure probabilities.
- The guessed erasure vector is provided to the BM decoder along with the
- received word.
- If the BM decoder finds a candidate codeword, then the codeword is assigned
- a quality metric, defined to be the soft distance, 
-\begin_inset Formula $d_{s}$
-\end_inset
-
-, between the received word and the codeword, where
-\begin_inset Formula 
-\begin{equation}
-d_{s}=\sum_{i=1}^{n}(1+p_{1,i})\alpha_{i}.\label{eq:soft_distance}
-\end{equation}
-
-\end_inset
-
-and 
-\begin_inset Formula $p_{1,i}$
-\end_inset
-
- is the fractional power associated with the i'th received symbol and 
-\begin_inset Formula $\alpha_{i}=0$
-\end_inset
-
- if the i'th received symbol is the same as the corresponding symbol in
- the codeword, and 
-\begin_inset Formula $\alpha_{i}=1$
-\end_inset
-
- if the i'th symbol in the received word and the codeword 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.
- 
-\end_layout
-
-\begin_layout Standard
-Technically, the 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
- kept.
- As with all such algorithms, a stopping criterion is necessary.
- SFRSD 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 employed to limit the execution time of the algorithm in cases
- where 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
-\end_layout
-
-\begin_layout Enumerate
-For each symbol in the received word, define the erasure probability to
- be 1.3 times the a priori symbol-error probability determined by the soft-symbol
- information 
-\begin_inset Formula $\{p_{1}\textrm{-rank},p_{2}/p_{1}\}$
-\end_inset
-
-.
- 
-\end_layout
-
-\begin_layout Enumerate
-Make independent decisions about whether or not to erase each symbol in
- the word using the symbol's erasure probability.
- Allow a total of up to 51 symbols to be erased.
- 
-\end_layout
-
-\begin_layout Enumerate
-Attempt BM errors-and-erasures decoding with the set of erased symbols that
- was 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 word.
- 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 codeword with 
-\begin_inset Formula $d_{s}\le d_{a}$
-\end_inset
-
- has been found.
- Declare a successful decode.
- Return the best codeword found so far.
-\end_layout
-
-\begin_layout Section
-Results
-\end_layout
-
-\begin_layout Section
-Summary 
-\end_layout
-
-\begin_layout Bibliography
-\begin_inset CommandInset bibitem
-LatexCommand bibitem
-key "key-1"
-
-\end_inset
-
-
-\end_layout
-
-\end_body
-\end_document
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+
+\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 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.
+ 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.
+ 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
+
+\begin_layout Section
+Introduction
+\end_layout
+
+\begin_layout Standard
+JT65 message frames consist of a short, compressed message encoded for transmiss
+ion using a Reed-Solomon code.
+ Reed-Solomon codes are block codes; as such they are characterized by the
+ length of their codewords, 
+\begin_inset Formula $n$
+\end_inset
+
+, the number of message symbols conveyed by the codeword, 
+\begin_inset Formula $k$
+\end_inset
+
+, and the number of possible values for each symbol in the codewords.
+ The codeword length and the number of message symbols are specified as
+ a tuple in the form 
+\begin_inset Formula $(n,k)$
+\end_inset
+
+.
+ JT65 uses a (63,12) Reed-Solomon code with 64 possible values for each
+ symbol, so each symbol represents 
+\begin_inset Formula $\log_{2}64=6$
+\end_inset
+
+ message bits.
+ The source-encoded messages conveyed by a 63-symbol JT65 frame consist
+ of 72 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 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 
+\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
+
+, which means that 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
+\end_layout
+
+\begin_layout Enumerate
+determine which symbols were received incorrectly 
+\end_layout
+
+\begin_layout Enumerate
+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 
+\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 
+\begin_inset Formula $d$
+\end_inset
+
+ errors.
+ 
+\end_layout
+
+\begin_layout Standard
+The FT algorithm creates a list of symbols suspected of being incorrect
+ and sends it 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 errors can
+ be corrected.
+ When the erasure information is imperfect, some of the 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 
+\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 and can correct up to 
+\begin_inset Formula $t$
+\end_inset
+
+ errors (
+\begin_inset Formula $t$
+\end_inset
+
+=25 for JT65).
+ If 
+\begin_inset Formula $0<s\le d-1$
+\end_inset
+
+ (
+\begin_inset Formula $d-1=51$
+\end_inset
+
+ for JT65), the decoder is said to be an 
+\begin_inset Quotes eld
+\end_inset
+
+errors-and-erasures
+\begin_inset Quotes erd
+\end_inset
+
+ decoder.
+ The errors-and-erasures capability of Reed-Solomon codes lies at the core
+ of the FTRSD algorithm.
+ 
+\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 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
+ symbols and 40 errors.
+ We do not know which symbols are in error.
+ Suppose that the decoder randomly chooses 
+\begin_inset Formula $s=40$
+\end_inset
+
+ symbols to erase, 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 received set, 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.
+ Let 
+\begin_inset Formula $P(x|N,X,s)$
+\end_inset
+
+ denote the conditional probability mass function for 
+\begin_inset Formula $x$
+\end_inset
+
+, the number of erased incorrect symbols, given the stated values of 
+\begin_inset Formula $N$
+\end_inset
+
+, 
+\begin_inset Formula $X$
+\end_inset
+
+, and 
+\begin_inset Formula $s$
+\end_inset
+
+.
+ Then
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+P(x|N,X,s)=\frac{\binom{X}{x}\binom{N-X}{s-x}}{\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 
+\begin_inset Quotes eld
+\end_inset
+
+nchoosek(
+\begin_inset Formula $n,k$
+\end_inset
+
+)
+\begin_inset Quotes erd
+\end_inset
+
+ function in Gnu Octave.
+ 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
+
+.]
+\end_layout
+
+\begin_layout Paragraph
+Example 1:
+\end_layout
+
+\begin_layout Case
+A codeword 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 incorrect is then
+\begin_inset Formula 
+\[
+P(x=35)=\frac{\binom{40}{35}\binom{63-40}{40-35}}{\binom{63}{40}}=2.356\times10^{-7}.
+\]
+
+\end_inset
+
+Similarly, the probability that 
+\begin_inset Formula $x=36$
+\end_inset
+
+ of the erased symbols are incorrect is
+\begin_inset Formula 
+\[
+P(x=36)=8.610\times10^{-9}.
+\]
+
+\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 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_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, and 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 
+\[
+P(x\ge37)\simeq2\times10^{-6}.
+\]
+
+\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.
+ 
+\end_layout
+
+\begin_layout Paragraph
+Example 3:
+\end_layout
+
+\begin_layout Case
+Examples 1 and 2 show that a strategy of randomly 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 
+\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 
+\begin_inset Formula $N=53$
+\end_inset
+
+ less reliable symbols.
+ If 
+\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 
+\begin_inset Formula $N=53$
+\end_inset
+
+, 
+\begin_inset Formula $X=40$
+\end_inset
+
+, 
+\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 with 
+\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 $P(x\ge38)=0.0266$
+\end_inset
+
+.
+ These odds are the best so far, about 1 in 38.
+ 
+\end_layout
+
+\begin_layout Section
+\begin_inset CommandInset label
+LatexCommand label
+name "sec:The-decoding-algorithm"
+
+\end_inset
+
+The FTRSD 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.
+\end_layout
+
+\begin_layout Standard
+The FTRSD 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
+ power.
+ The fraction 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 lower ranked
+ symbols.
+ 
+\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 FTRSD 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 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
+ 
+\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.
+ 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, 
+\begin_inset Formula $d_{s}$
+\end_inset
+
+, 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}
+\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
+
+ if received symbol 
+\begin_inset Formula $i$
+\end_inset
+
+ is the same as the corresponding symbol in the codeword, and 
+\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.
+ 
+\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 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, 
+\begin_inset Formula $d_{a}$
+\end_inset
+
+.
+ A timeout is employed 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 to be 1.3 times
+ the 
+\emph on
+a priori
+\emph default
+ symbol-error probability determined by the 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 with the BM algorithm and the set of
+ eraseures 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 codeword with 
+\begin_inset Formula $d_{s}\le d_{a}$
+\end_inset
+
+ has been found.
+ Declare a successful decode and return the codeword .
+\end_layout
+
+\begin_layout Section
+Results and Comparison with KVASD
+\end_layout
+
+\begin_layout Standard
+Possible figures:
+\end_layout
+
+\begin_layout Itemize
+histogram of 
+\begin_inset Formula $s$
+\end_inset
+
+ (number of erasures) for succeffsul 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
+
+
+\end_layout
+
+\end_body
+\end_document