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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.
- 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 JT65 decoders have used the patented Koetter-Vardy (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 FT algotithm.
- It is conceptually simple, built around the well-known Berlekamp-Massey
- errors-and-erasures algorithm, and performs 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 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 using
- 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 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 ncessarily 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 errors 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 (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.
- 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 build a capability for using 
-\begin_inset Quotes eld
-\end_inset
-
-soft
-\begin_inset Quotes erd
-\end_inset
-
- information on symbol reliability.
-\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, thereby 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 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.
- 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|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 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.
- 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 Standard
-Suppose 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 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, 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.
- 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 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
-
- less reliable symbols.
- If 
-\begin_inset Formula $s=45$
-\end_inset
-
- 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
-
-, 
-\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 $P(x\ge38)=0.0266$
-\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 FT 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 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 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
- 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 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 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 
-\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
-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
-\begin_inset Float figure
-wide false
-sideways false
-status open
-
-\begin_layout Plain Layout
-\align center
-\begin_inset Graphics
-	filename fig_psuccess.pdf
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset Caption Standard
-
-\begin_layout Plain Layout
-Percentage of JT65 messages successfully decoded as a function of SNR in
- 2.5 kHz bandwidth.
- Results are shown for the hard-decision Berlekamp-Massey (BM) and the Franke-Ta
-ylor (FT) decoding algorithms.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Itemize
-other...
- ?
-\end_layout
-
-\begin_layout Section
-Summary 
-\end_layout
-
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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.
+ 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 JT65 decoders have used the patented Koetter-Vardy (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 FT algotithm.
+ It is conceptually simple, built around the well-known Berlekamp-Massey
+ errors-and-erasures algorithm, and performs 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 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 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 ncessarily 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 errors 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 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.
+ 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|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 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.
+ 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 Standard
+Suppose 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 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, 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.
+ 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 symbols are significa
+ntly 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 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 $P(x\ge38)=0.0266$
+\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 FT 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 these 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 identifies the most likely value for each symbol based
+ on the largest signal-plus-noise power in 64 frequency bins.
+ 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 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 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 
+\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
+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 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 120
+	scale 120
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Percentage of JT65 messages successfully decoded as a function of SNR in
+ 2.5 kHz bandwidth.
+ Results are shown for the hard-decision Berlekamp-Massey (BM) and the sofft-dec
+ision Franke-Taylor (FT) decoding algorithms.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\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