WSJT-X/lib/ftrsd/ftrsd_paper/ftrsd.lyx

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\begin_body

\begin_layout Title
Open Source 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 Section
\begin_inset CommandInset label
LatexCommand label
name "sec:Introduction-and-Motivation"

\end_inset

Introduction and Motivation
\end_layout

\begin_layout Standard
The JT65 protocol has revolutionized amateur-radio weak-signal communication
 by enabling operators with small or compromise antennas and relatively
 low-power transmitters to communicate over propagation paths not usable
 with traditional technologies.
 The protocol was developed in 2003 for Earth-Moon-Earth (EME, or 
\begin_inset Quotes eld
\end_inset

moonbounce
\begin_inset Quotes erd
\end_inset

) communication, where the scattered return signals are always weak.
 It was soon found that JT65 also enables worldwide communication on the
 HF bands with low power, modest antennas, and efficient spectral usage.
\end_layout

\begin_layout Standard
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.
 Until now, nearly all programs implementing JT65 have used the patented
 Koetter-Vardy (KV) algebraic soft-decision decoder 
\begin_inset CommandInset citation
LatexCommand cite
key "kv2001"

\end_inset

, licensed to and implemented by K1JT as a closed-source executable for
 use only in amateur radio applications.
 Since 2001 the KV decoder has been considered the best available soft-decision
 decoder for Reed Solomon codes.
 We describe here a new open-source alternative called the Franke-Taylor
 (FT, or K9AN-K1JT) algorithm.
 It is conceptually simple, built around the well-known Berlekamp-Massey
 errors-and-erasures algorithm, and in this application it performs even
 better than the KV decoder.
 The FT algorithm is implemented in the popular program 
\emph on
WSJT-X
\emph default
, widely used for amateur weak-signal communication with JT65 and other
 specialized digital modes.
 The program is freely available and licensed under the GNU General Public
 License.
\end_layout

\begin_layout Standard
The JT65 protocol specifies transmissions that normally start one second
 into a UTC minute and last for 46.8 seconds.
 Receiving software therefore has up to several seconds to decode a message,
 before the operator sends a reply at the start of the next minute.
 With today's personal computers, this relatively long time available for
 decoding a short message encourages experimentation with decoders of high
 computational complexity.
 As a result, on a typical fading channel the FT algorithm can extend the
 decoding threshold by many dB over the hard-decision Berlekamp-Massey decoder,
 and by a meaningful amount over the KV decoder.
 In addition to its excellent performance, the new algorithm has other desirable
 properties, not least of which is its conceptual simplicity.
 Decoding performance and complexity scale in a convenient way, providing
 steadily increasing soft-decision decoding gain as a tunable computational
 complexity parameter is increased over more than 5 orders of magnitude.
 Appreciable gain is available from our decoder even on very simple (and
 relatively slow) computers.
 On the other hand, because the algorithm benefits from a large number of
 independent decoding trials, further performance gains should be achievable
 through parallelization on high-performance computers.
\end_layout

\begin_layout Section
\begin_inset CommandInset label
LatexCommand label
name "sec:JT65-messages-and"

\end_inset

JT65 messages and Reed Solomon Codes
\end_layout

\begin_layout Standard
JT65 message frames consist of a short compressed message encoded for transmissi
on with a Reed-Solomon code.
 Reed-Solomon codes are block codes characterized by 
\begin_inset Formula $n$
\end_inset

, the length of their codewords, 
\begin_inset Formula $k$
\end_inset

, the number of message symbols conveyed by the codeword, and the number
 of possible values for each symbol in the codewords.
 The codeword length and the number of message symbols are specified with
 the notation 
\begin_inset Formula $(n,k)$
\end_inset

.
 JT65 uses a (63,12) Reed-Solomon code with 64 possible values for each
 symbol.
 Each of the 12 message symbols represents 
\begin_inset Formula $\log_{2}64=6$
\end_inset

 message bits.
 The source-encoded messages conveyed by a 63-symbol JT65 frame thus consist
 of 72 information bits.
 The JT65 code is systematic, which means that the 12 message symbols are
 embedded in the codeword without modification and another 51 parity symbols
 derived from the message symbols are added to form a codeword of 63 symbols.
 
\end_layout

\begin_layout Standard
In coding theory 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 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 decode a received word having 25 or fewer
 symbol errors.
 Any one of several well-known algebraic algorithms, such as the widely
 used Berlekamp-Massey (BM) algorithm, can carry out the decoding.
 Two steps are necessarily involved in this process.
 We must (1) determine which symbols were received incorrectly, and (2)
 find the correct value of the incorrect symbols.
 If we somehow know that certain symbols are incorrect, that 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 $d-1=n-k$
\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

 With perfect erasure information up to 51 incorrect symbols can be corrected
 for the JT65 code.
 Imperfect erasure information means that some erased symbols may be correct,
 and some other symbols in error.
 If 
\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

, 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 the reliability of individual symbols, thereby producing
 a soft-decision decoder.
\end_layout

\begin_layout Section
\begin_inset CommandInset label
LatexCommand label
name "sec:Statistical Framework"

\end_inset

Statistical Framework
\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 decoding
 of received words with more than 25 errors using the errors-and-erasures
 capability of the BM decoder.
 Algorithms of this type are generally called 
\begin_inset Quotes eld
\end_inset

reliability based
\begin_inset Quotes erd
\end_inset

 or 
\begin_inset Quotes eld
\end_inset

probabilistic
\begin_inset Quotes erd
\end_inset

 decoding methods 
\begin_inset CommandInset citation
LatexCommand cite
after "Chapter 10"
key "key-1"

\end_inset

.
 Such algorithms involve some amount of educating guessing about which received
 symbols are in error or, alternatively, about which received symbols are
 correct.
 The guesses are informed by 
\begin_inset Quotes eld
\end_inset

soft-symbol
\begin_inset Quotes erd
\end_inset

 quality metrics associated with the received symbols.
 To illustrate why it is absolutely essential to use such soft-symbol informatio
n in these algorithms it helps to consider what would happen if we tried
 to use completely random guesses, ignoring any available soft-symbol informatio
n.
\end_layout

\begin_layout Standard
As a specific example, consider a received JT65 word with 23 correct symbols
 and 40 errors.
 We do not know which symbols are in error.
 Suppose that the decoder randomly selects 
\begin_inset Formula $s=40$
\end_inset

 symbols for erasure, leaving 23 unerased symbols.
 According to Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:erasures_and_errors"

\end_inset

), the BM decoder can successfully decode this word as long as 
\begin_inset Formula $e$
\end_inset

, the number of errors present in the 23 unerased symbols, is 5 or less.
 The number of errors captured in the set of 40 erased symbols must therefore
 be at least 35.
 
\end_layout

\begin_layout Standard
The probability of selecting some particular number of incorrect symbols
 in a randomly selected subset of received symbols is governed by the hypergeome
tric probability distribution.
 Let us define 
\begin_inset Formula $N$
\end_inset

 as the number of symbols from which erasures will be selected, 
\begin_inset Formula $X$
\end_inset

 as the number of incorrect symbols in the set of 
\begin_inset Formula $N$
\end_inset

 symbols, and 
\begin_inset Formula $x$
\end_inset

 as the number of errors in the symbols actually erased.
 In an ensemble of many received words 
\begin_inset Formula $X$
\end_inset

 and 
\begin_inset Formula $x$
\end_inset

 will be random variables but for this example we will assume that 
\begin_inset Formula $X$
\end_inset

 is known and that only 
\begin_inset Formula $x$
\end_inset

 is random.
 The conditional probability mass function for 
\begin_inset Formula $x$
\end_inset

 with stated values of 
\begin_inset Formula $N$
\end_inset

, 
\begin_inset Formula $X$
\end_inset

, and 
\begin_inset Formula $s$
\end_inset

 may be written as
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
P(x=\epsilon|N,X,s)=\frac{\binom{X}{\epsilon}\binom{N-X}{s-\epsilon}}{\binom{N}{s}}\label{eq:hypergeometric_pdf}
\end{equation}

\end_inset

where 
\begin_inset Formula $\binom{n}{k}=\frac{n!}{k!(n-k)!}$
\end_inset

 is the binomial coefficient.
 The binomial coefficient can be calculated using the function 
\family typewriter
nchoosek(n,k)
\family default
 in the interpreted language GNU Octave, or with one of many free online
 calculators.
 The hypergeometric probability mass function defined in Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:hypergeometric_pdf"

\end_inset

) is available in GNU Octave as function 
\family typewriter
hygepdf
\family default
(x,N,X,s).
 The cumulative probability that at least 
\begin_inset Formula $\epsilon$
\end_inset

 errors are captured in a subset of 
\begin_inset Formula $s$
\end_inset

 erased symbols selected from a group of 
\begin_inset Formula $N$
\end_inset

 symbols containing 
\begin_inset Formula $X$
\end_inset

 errors is
\begin_inset Formula 
\begin{equation}
P(x\ge\epsilon|N,X,s)=\sum_{j=\epsilon}^{s}P(x=j|N,X,s).\label{eq:cumulative_prob}
\end{equation}

\end_inset


\end_layout

\begin_layout Paragraph
Example 1:
\end_layout

\begin_layout Standard
Suppose a received word contains 
\begin_inset Formula $X=40$
\end_inset

 incorrect symbols.
 In an attempt to decode using an errors-and-erasures decoder, 
\begin_inset Formula $s=40$
\end_inset

 symbols are randomly selected for erasure from the full set of 
\begin_inset Formula $N=n=63$
\end_inset

 symbols.
 The probability that 
\begin_inset Formula $x=35$
\end_inset

 of the erased symbols are actually incorrect is then
\begin_inset Formula 
\[
P(x=35)=\frac{\binom{40}{35}\binom{63-40}{40-35}}{\binom{63}{40}}\simeq2.4\times10^{-7}.
\]

\end_inset

Similarly, the probability that 
\begin_inset Formula $x=36$
\end_inset

 of the erased symbols are incorrect is
\begin_inset Formula 
\[
P(x=36)\simeq8.6\times10^{-9}.
\]

\end_inset

Since the probability of erasing 36 errors is so much smaller than that
 for 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 producing a valid codeword on the first try are very poor,
 about 1 in 4 million.
\end_layout

\begin_layout Paragraph
Example 2:
\end_layout

\begin_layout Standard
How might we best choose the number of symbols to erase, in order to maximize
 the probability of successful decoding? By exhaustive search over all possible
 values up to 
\begin_inset Formula $s=51$
\end_inset

, it turns out that for 
\begin_inset Formula $X=40$
\end_inset

 the best strategy is to erase 
\begin_inset Formula $s=45$
\end_inset

 symbols.
 Decoding will then be assured if the set of erased symbols contains at
 least 37 errors.
 With 
\begin_inset Formula $N=63$
\end_inset

, 
\begin_inset Formula $X=40$
\end_inset

, and 
\begin_inset Formula $s=45$
\end_inset

, the probability of successful decode in a single try is 
\begin_inset Formula 
\[
P(x\ge37)\simeq1.9\times10^{-6}.
\]

\end_inset

This probability is about 8 times higher than the probability of success
 when only 40 symbols were erased.
 Nevertheless, the odds of successfully decoding on the first try are still
 only about 1 in 500,000.
 
\end_layout

\begin_layout Paragraph
Example 3:
\end_layout

\begin_layout Standard
Examples 1 and 2 show that a random strategy for selecting symbols to erase
 is unlikely to be successful unless we are prepared to wait a long time
 for an answer.
 So let's modify the strategy to tip the odds in our favor.
 Let the received word contain 
\begin_inset Formula $X=40$
\end_inset

 incorrect symbols, as before, but suppose we know that 10 received symbols
 are significantly more reliable than the other 53.
 We might therefore protect the 10 most reliable symbols from erasure, selecting
 erasures from the smaller set of 
\begin_inset Formula $N=53$
\end_inset

 less reliable symbols.
 If 
\begin_inset Formula $s=45$
\end_inset

 symbols are chosen randomly for erasure in this way, it is still necessary
 for the erased symbols to include at least 37 errors, as in Example 2.
 However, the probabilities are now much more favorable: with 
\begin_inset Formula $N=53$
\end_inset

, 
\begin_inset Formula $X=40$
\end_inset

, and 
\begin_inset Formula $s=45$
\end_inset

, Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:hypergeometric_pdf"

\end_inset

) yields 
\begin_inset Formula $P(x\ge37)=0.016$
\end_inset

.
 Even better odds are obtained by choosing 
\begin_inset Formula $s=47$
\end_inset

, which requires 
\begin_inset Formula $x\ge38$
\end_inset

.
 With 
\begin_inset Formula $N=53$
\end_inset

, 
\begin_inset Formula $X=40$
\end_inset

, and 
\begin_inset Formula $s=47$
\end_inset

, 
\begin_inset Formula $P(x\ge38)=0.027$
\end_inset

.
 The odds for producing a codeword on the first try are now about 1 in 38.
 A few hundred independently randomized tries would be enough to all-but-guarant
ee production of a valid codeword by the BM decoder.
\end_layout

\begin_layout Section
\begin_inset CommandInset label
LatexCommand label
name "sec:The-decoding-algorithm"

\end_inset

The Franke-Taylor decoding algorithm
\end_layout

\begin_layout Standard
Example 3 shows how statistical information about symbol quality should
 make it possible to decode received frames having a large number of errors.
 In practice the number of errors in the received word is unknown, so we
 use a stochastic algorithm to assign high erasure probability to low-quality
 symbols and relatively low probability to high-quality symbols.
 As illustrated by Example 3, a good choice of erasure probabilities can
 increase by many orders of magnitude the chance of producing a codeword.
 Note that at this stage we must treat any codeword obtained by errors-and-erasu
res decoding as no more than a 
\emph on
candidate
\emph default
.
 Our next task is to find a metric that can reliably select one of many
 proffered candidates as the codeword actually transmitted.
\end_layout

\begin_layout Standard
The FT algorithm uses quality indices made available by a noncoherent 64-FSK
 demodulator.
 The demodulator computes the power spectrum 
\begin_inset Formula $S(i,j)$
\end_inset

 for each signalling interval; for the JT65 protocol the frequency index
 and symbol index have values 
\begin_inset Formula $i=$
\end_inset

1 to 64 and 
\begin_inset Formula $j=$
\end_inset

1 to 63.
 The most likely value for symbol 
\begin_inset Formula $j$
\end_inset

 is taken as the frequency bin with largest signal-plus-noise power over
 all values of 
\begin_inset Formula $i$
\end_inset

.
 The fractions of total power in the two bins containing the largest and
 second-largest powers, denoted respectively by 
\begin_inset Formula $p_{1}$
\end_inset

 and 
\begin_inset Formula $p_{2}$
\end_inset

, are computed and passed from demodulator to decoder as soft-symbol information.
 The FT decoder derives two metrics from 
\begin_inset Formula $p_{1}$
\end_inset

 and 
\begin_inset Formula $p_{2}$
\end_inset

, namely 
\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 $p_{1,\,j}$
\end_inset

 in a 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
We use an empirical table of symbol error probabilities derived from a large
 dataset of received words that were successfully decoded.
 The table provides an estimate of the 
\emph on
a priori
\emph default
 probability of symbol error based on the 
\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 candidate codewords are produced with
 higher probability when 
\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.
 For the JT65 code 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 educated guesses 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 hard-decision symbol values.
 When the BM decoder finds a candidate codeword it is assigned a quality
 metric 
\begin_inset Formula $d_{s}$
\end_inset

, the soft distance between the received word and the codeword: 
\begin_inset Formula 
\begin{equation}
d_{s}=\sum_{j=1}^{n}\alpha_{j}\,(1+p_{1,\,j}).\label{eq:soft_distance}
\end{equation}

\end_inset

Here 
\begin_inset Formula $\alpha_{j}=0$
\end_inset

 if received symbol 
\begin_inset Formula $j$
\end_inset

 is the same as the corresponding symbol in the codeword, 
\begin_inset Formula $\alpha_{j}=1$
\end_inset

 if the received symbol and codeword symbol are different, and 
\begin_inset Formula $p_{1,\,j}$
\end_inset

 is the fractional power associated with received symbol 
\begin_inset Formula $j$
\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
In practice we find that 
\begin_inset Formula $d_{s}$
\end_inset

 can reliably indentify the correct codeword if the signal-to-noise ratio
 for individual symbols is greater than about 4 in linear power units.
 We also find that significantly weaker signals can be decoded by using
 soft-symbol information beyond that contained in 
\begin_inset Formula $p_{1}$
\end_inset

 and 
\begin_inset Formula $p_{2}$
\end_inset

.
 To this end we define an additional metric 
\begin_inset Formula $u$
\end_inset

, the average signal-plus-noise power in all symbols according to a candidate
 codeword's symbol values:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
u=\frac{1}{n}\sum_{j=1}^{n}S(c_{j},\,j).\label{eq:u-metric}
\end{equation}

\end_inset

Here the 
\begin_inset Formula $c_{j}$
\end_inset

's are the symbol values for the candidate codeword being tested.
 The correct JT65 codeword produces a value for 
\begin_inset Formula $u$
\end_inset

 equal to the average of 
\begin_inset Formula $n=63$
\end_inset

 bins containing both signal and noise power.
 Incorrect codewords have at most 
\begin_inset Formula $k-1=11$
\end_inset

 such bins and at least 
\begin_inset Formula $n-k+1=52$
\end_inset

 bins containing noise only.
 Thus, if the spectral array 
\begin_inset Formula $S(i,\,j)$
\end_inset

 has been normalized so that its median value (essentially the average noise
 level) is unity, 
\begin_inset Formula $u$
\end_inset

 for the correct codeword has expectation value (average over many random
 realizations)
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\bar{u}_{1}=1+y,\label{eq:u1-exp}
\end{equation}

\end_inset

where 
\begin_inset Formula $y$
\end_inset

 is the signal-to-noise ratio in linear power units.
 If we assume Gaussian statistics and a large number of trials, the standard
 deviation of measured values of 
\begin_inset Formula $u_{1}$
\end_inset

 is
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\sigma_{1}=\left(\frac{1+2y}{n}\right)^{1/2}.\label{eq:sigma1}
\end{equation}

\end_inset

In contrast, worst-case incorrect codewords will yield 
\begin_inset Formula $u$
\end_inset

-metrics with expectation value and standard deviation given by
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\bar{u}_{2}=1+\left(\frac{k-1}{n}\right)y,\label{eq:u2-exp}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\sigma_{2}=\frac{1}{n}\left[n+2y(k-1)\right]^{1/2}.\label{eq:sigma2}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
If tests on a number of tested candidate codewords yield largest and second-larg
est metrics 
\begin_inset Formula $u_{1}$
\end_inset

 and 
\begin_inset Formula $u_{2},$
\end_inset

 respectively, we expect the ratio 
\begin_inset Formula $r=u_{2}/u_{1}$
\end_inset

 to be significantly smaller in cases where the candidate associated with
 
\begin_inset Formula $u_{1}$
\end_inset

 is in fact the correct codeword.
 On the other hand, if none of the tested candidates is correct, 
\begin_inset Formula $r$
\end_inset

 will likely be close to 1.
 We therefore apply a ratio threshold test, say 
\begin_inset Formula $r<r_{1}$
\end_inset

, to identify codewords with high probability of being correct.
 As described in Section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Theory,-Simulation,-and"

\end_inset

, we use simulations to set an empirical acceptance threshold 
\begin_inset Formula $r_{1}$
\end_inset

 that maximizes the probability of correct decodes while ensuring a low
 rate of false decodes.
\end_layout

\begin_layout Standard
Technically the FT algorithm is a list decoder.
 Among the list of candidate codewords found by the stochastic search algorithm,
 only the one with the largest 
\begin_inset Formula $u$
\end_inset

 is retained.
 As with all such algorithms, a stopping criterion is necessary.
 FT accepts a codeword unconditionally if the Hamming distance 
\begin_inset Formula $X$
\end_inset

 and soft distance 
\begin_inset Formula $d_{s}$
\end_inset

 are less than specified limits 
\begin_inset Formula $X_{0}$
\end_inset

 and 
\begin_inset Formula $d_{0}$
\end_inset

.
 Secondary acceptance criteria 
\begin_inset Formula $d_{s}<d_{1}$
\end_inset

 and 
\begin_inset Formula $r<r_{1}$
\end_inset

 are used to validate additional codewords that did not pass the first test.
 A timeout is used to limit the algorithm's execution time if no acceptable
 codeword is found in a reasonable number of trials, 
\begin_inset Formula $T$
\end_inset

.
 Today's personal computers are fast enough that 
\begin_inset Formula $T$
\end_inset

 can be set as large as 
\begin_inset Formula $10^{5},$
\end_inset

 or even higher.
 Pseudo-code for the FT algorithm is presented in an accompanying text box.
\end_layout

\begin_layout Standard
\begin_inset Float algorithm
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
Pseudo-code for the FT algorithm.
\end_layout

\end_inset


\end_layout

\begin_layout Enumerate
For each received symbol, define the erasure probability as 1.3 times the
 
\emph on
a priori
\emph default
 symbol-error probability determined from soft-symbol information 
\begin_inset Formula $\{p_{1}\textrm{-rank},\,p_{2}/p_{1}\}$
\end_inset

.
 
\end_layout

\begin_layout Enumerate
Make independent stochastic decisions about whether to erase each symbol
 by using the symbol's erasure probability, allowing a maximum of 51 erasures.
\end_layout

\begin_layout Enumerate
Attempt errors-and-erasures decoding by using the BM algorithm and the set
 of erasures determined in step 2.
 If the BM decoder produces a candidate codeword, go to step 5.
\end_layout

\begin_layout Enumerate
If BM decoding was not successful, go to step 2.
\end_layout

\begin_layout Enumerate
Calculate the hard-decision Hamming distance 
\begin_inset Formula $X$
\end_inset

 between the candidate codeword and the received symbols, the corresponding
 soft distance 
\begin_inset Formula $d_{s}$
\end_inset

, and the quality metric 
\begin_inset Formula $u$
\end_inset

.
 If 
\begin_inset Formula $u$
\end_inset

 is the largest one encountered so far, preserve any previous value of 
\begin_inset Formula $u_{1}$
\end_inset

 as 
\begin_inset Formula $u_{2}$
\end_inset

 and then set 
\begin_inset Formula $u_{1}=u$
\end_inset

.
\end_layout

\begin_layout Enumerate
If 
\begin_inset Formula $X<X_{0}$
\end_inset

 and 
\begin_inset Formula $d_{s}<d_{0}$
\end_inset

, go to step 10.
\end_layout

\begin_layout Enumerate
If the number of trials is less than the timeout limit
\begin_inset Formula $T,$
\end_inset

 go to 2.
 
\begin_inset Formula $ $
\end_inset


\end_layout

\begin_layout Enumerate
If 
\begin_inset Formula $d_{s}<d_{1}$
\end_inset

 and 
\begin_inset Formula $r<r_{1},$
\end_inset

 go to step 10.
\end_layout

\begin_layout Enumerate
Otherwise, declare decoding failure and exit.
\end_layout

\begin_layout Enumerate
An acceptable codeword has been found.
 Declare a successful decode and return this codeword.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Inspiration for the FT decoding algorithm came from a number of sources,
 particularly references 
\begin_inset CommandInset citation
LatexCommand cite
key "lhmg2010"

\end_inset

 and 
\begin_inset CommandInset citation
LatexCommand cite
key "lk2008"

\end_inset

 and the textbook by Lin and Costello 
\begin_inset CommandInset citation
LatexCommand cite
key "lc2004"

\end_inset

.
 After developing this algorithm, we became aware that our approach is conceptua
lly similar to a 
\begin_inset Quotes eld
\end_inset

stochastic erasures-only list decoding algorithm
\begin_inset Quotes erd
\end_inset

, described in reference 
\begin_inset CommandInset citation
LatexCommand cite
key "ls2009"

\end_inset

.
 The algorithm in 
\begin_inset CommandInset citation
LatexCommand cite
key "ls2009"

\end_inset

 is applied to higher-rate Reed-Solomon codes on a binary-input channel
 with BPSK-modulated symbols.
 Our 64-ary input channel with 64-FSK modulation required us to develop
 unique methods for assigning erasure probabilities and for defining acceptance
 criteria to select the best codeword from the list of candidates.
 
\end_layout

\begin_layout Section
\begin_inset CommandInset label
LatexCommand label
name "sec:Hinted-Decoding"

\end_inset

Hinted Decoding
\end_layout

\begin_layout Standard
The FT algorithm is completely general: with equal sensitivity it recovers
 any one of the 
\begin_inset Formula $2^{72}\approx4.7\times10^{21}$
\end_inset

 different messages that can be transmitted with the JT65 protocol.
 In some circumstances it's easy to imagine a 
\emph on
much
\emph default
 smaller list of messages (say, a few thousand messages or less) that we
 can guess may be among the most likely ones to be received.
 One such situation exists when making short ham-radio contacts that exchange
 minimal information including callsigns, signal reports, perhaps Maidenhead
 locators, and acknowledgments.
 On the EME path or on a VHF or UHF band with limited geographical coverage,
 the most likely received messages often originate from callsigns that have
 been decoded before.
 Saving a list of previously decoded callsigns and associated locators makes
 it easy to generate lists of hypothetical messages and their corresponding
 codewords at very little computational expense.
 The resulting candidate codewords can be tested in the same way as those
 generated by the probabilistic method described in Setcion 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:The-decoding-algorithm"

\end_inset

.
 We call this approach 
\begin_inset Quotes eld
\end_inset

hinted decoding;
\begin_inset Quotes erd
\end_inset

 it is sometimes referred to as the 
\begin_inset Quotes eld
\end_inset

deep search
\begin_inset Quotes erd
\end_inset

 algorithm.
 In certain limited situations it can provide enhanced sensitivity for the
 principal task of any decoder, namely to determine precisely what message
 was sent.
\end_layout

\begin_layout Standard
For hinted decoding we again invoke a ratio threshold test, but in this
 case we use it to answer a more limited question.
 Over the full list of messages considered likely, we want to know whether
 
\begin_inset Formula $r=u_{2}/u_{1}$
\end_inset

, the ratio of second-largest to largest
\begin_inset Formula $u$
\end_inset

-metric, is small enough for us to be confident the codeword associated
 with 
\begin_inset Formula $u_{1}$
\end_inset

 is the one that was transmitted.
 Once again we will set an empirical limit, say 
\begin_inset Formula $r_{2},$
\end_inset

 that is small enough to establish adequate confidence, while still ensuring
 that false decodes are rare.
 Because tested candidate codewords are drawn from a list typically no longer
 than a few thousand, rather than 
\begin_inset Formula $2^{72},$
\end_inset

 
\begin_inset Formula $r_{2}$
\end_inset

 can be a more relaxed limit than the 
\begin_inset Formula $r_{1}$
\end_inset

 used in the FT algorithm.
 For the limited subset of messages that operator experience suggests to
 be likely, hinted decodes can be obtained at lower signal levels than required
 for those obtained from the full universe of 
\begin_inset Formula $2^{72}$
\end_inset

 possible messages.
\end_layout

\begin_layout Section
\begin_inset CommandInset label
LatexCommand label
name "sec:Theory,-Simulation,-and"

\end_inset

Decoder Performance Evaluation
\end_layout

\begin_layout Standard
Comparisons of decoding performance are usually presented in the professional
 literature as plots of word error rate versus 
\begin_inset Formula $E_{b}/N_{0}$
\end_inset

, the ratio of the energy collected per information bit to the one-sided
 noise power spectral density.
 For weak-signal amateur radio work, performance is more conveniently presented
 as the probability of successfully decoding a received word plotted against
 signal-to-noise ratio in a 2500 Hz reference bandwidth, 
\begin_inset Formula $\mathrm{SNR}{}_{2500}$
\end_inset

.
 The relationship between 
\begin_inset Formula $E_{b}/N_{o}$
\end_inset

 and 
\begin_inset Formula $\mathrm{SNR}{}_{2500}$
\end_inset

 is described in Appendix 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Appendix:SNR"

\end_inset

.
 Examples of both presentations are included in the following discussion,
 where we describe simulations carried out to compare performance of FT
 with other algorithms, and with theoretical expectations.
 We have also used simulations to establish suitable default values for
 the acceptance parameters 
\begin_inset Formula $X_{0},$
\end_inset

 
\begin_inset Formula $d_{0},$
\end_inset

 
\begin_inset Formula $d_{1},$
\end_inset

 
\begin_inset Formula $r_{1},$
\end_inset

 and 
\begin_inset Formula $r_{2}.$
\end_inset


\end_layout

\begin_layout Subsection
Simulated results on the AWGN channel
\end_layout

\begin_layout Standard
Results of simulations using the BM, FT, and KV decoding algorithms on the
 JT65 code are presented in terms of word error rate versus 
\begin_inset Formula $E_{b}/N_{o}$
\end_inset

 in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:bodide"

\end_inset

.
 For these tests we generated at least 1000 signals at each signal-to-noise
 ratio, assuming the additive white gaussian noise (AWGN) channel, and we
 processed the data using each algorithm.
 For word error rates less than 0.1 it was necessary to process 10,000 or
 even 100,000 simulated signals in order to capture enough errors to make
 the measurements statistically meaningful.
 As a test of the fidelity of our numerical simulations, Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:bodide"

\end_inset

 also shows results calculated from theory for comparison with the BM results.
 The simulated BM results agree with theory to within about 0.1 dB.
 This difference between simulated BM results and theory is caused by small
 errors in the estimates of time- and frequency-offset of the received signal
 in the simulated data.
 Such 
\begin_inset Quotes eld
\end_inset

sync losses
\begin_inset Quotes erd
\end_inset

 are not accounted for in the idealized theoretical results.
 
\end_layout

\begin_layout Standard
As expected, the soft-decision algorithms, FT and KV, are about 2 dB better
 than the hard-decision BM algorithm.
 In addition, FT has a slight edge (about 0.2 dB) over KV.
 On the other hand, the execution time for FT with timeout parameter 
\begin_inset Formula $T=10^{5}$
\end_inset

 is longer than the execution time for the KV algorithm.
 Nevertheless, the execution time required for the FT algorithm with 
\begin_inset Formula $T=10^{5}$
\end_inset

 is small enough to be practical on most of today's home computers.
 
\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_bodide.pdf

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:bodide"

\end_inset

Word error rates as a function of 
\begin_inset Formula $E_{b}/N_{0},$
\end_inset

 the signal-to-noise ratio per information bit.
 Theory: theoretical prediction for the hard-decision BM decoder.
 The remaining curves represent simulation results on an AWGN channel for
 the BM, KV, and FT decoders.
 The KV algorithm was executed with complexity coefficient 
\begin_inset Formula $\lambda=15$
\end_inset

, the most aggressive setting historically used in the 
\emph on
WSJT
\emph default
 programs.
 The FT alrithm was run with timeout setting 
\begin_inset Formula $T=10^{5}.$
\end_inset

 
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Because of the importance of error-free transmission in commercial applications,
 plots like that in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:bodide"

\end_inset

 often extend downward to even smaller error rates, say 
\begin_inset Formula $10^{-6}$
\end_inset

 or less.
 The circumstances for minimal amateur-radio QSOs are very different, however.
 Error rates of order 0.1 or higher may be acceptable.
 In this case the essential information is better presented in a plot showing
 the percentage of transmissions copied correctly as a function of signal-to-noi
se ratio.
 Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:WER2"

\end_inset

 shows in this format the FT results for 
\begin_inset Formula $T=10^{5}$
\end_inset

 and the KV results from Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:bodide"

\end_inset

, along with additional FT results for 
\begin_inset Formula $T=10^{4},\:10^{3},\:10^{2}$
\end_inset

 and 
\begin_inset Formula $10$
\end_inset

.
 It is apparent that the FT decoder produces more decodes than KV when 
\begin_inset Formula $T=10^{4}$
\end_inset

 or larger.
 It also provides a very significant gain over the hard-decision BM decoder
 even when limited to 10 or fewer trials.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename fig_wer2.pdf
	lyxscale 120

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:WER2"

\end_inset

Percent of JT65 messages copied as a function of SNR in 2500 Hz bandwidth.
 Solid lines with filled circles are results from the FT decoder; numbers
 adjacent to the curves specify values of the timeout parameter 
\begin_inset Formula $T.$
\end_inset

 The dotted line with open squares is the KV decoder with complexity coefficient
 
\begin_inset Formula $\lambda=15$
\end_inset

.
 Results from the BM algorithm are shown with a dashed line and crosses.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Timeout parameter 
\begin_inset Formula $T$
\end_inset

 is the maximum number of symbol-erasure trials allowed for a particular
 attempt at decoding a received word.
 Most successful decodes take only a small fraction of the maximum allowed
 number of trials.
 Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:N_vs_X"

\end_inset

 shows the number of stochastic erasure trials required to find the correct
 codeword vs.
 the number of hard-decision errors in the received word, for a run with
 1000 simulated transmissions at 
\begin_inset Formula $\mathrm{SNR}=-24$
\end_inset

 dB, just slightly above the decoding threshold.
 The timeout parameter was 
\begin_inset Formula $T=10^{5}$
\end_inset

 for this run.
 No points are shown for 
\begin_inset Formula $X\le25$
\end_inset

 because all such words are successfully decoded by the BM algorithm.
 Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:N_vs_X"

\end_inset

 shows that the FT algorithm decoded received words with as many as 
\begin_inset Formula $X=43$
\end_inset

 symbol errors.
 The results also show that, on average, the number of trials increases
 with the number of errors in the received word.
 The variability of the decoding time also increases dramatically with the
 number of errors in the received word.
 These results provide insight into the mean and variance of the execution
 time for the FT algorithm, since execution time is roughly proportional
 to the number of required trials.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename fig_ntrials_vs_nhard.pdf
	lyxscale 120

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:N_vs_X"

\end_inset

Number of trials needed to decode a received word versus Hamming distance
 
\begin_inset Formula $X$
\end_inset

 between the received word and the decoded codeword, for 1000 simulated
 frames on an AWGN channel with no fading.
 The SNR in 2500 Hz bandwidth is 
\begin_inset Formula $-24$
\end_inset

 dB, which corresponds to 
\begin_inset Formula $E_{b}/N_{o}=5.1$
\end_inset

 dB.
 
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Simulated results for Rayleigh fading and hinted decoding
\end_layout

\begin_layout Standard
Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Psuccess"

\end_inset

 presents the results of simulations for signal-to-noise ratios ranging
 from 
\begin_inset Formula $-18$
\end_inset

 to 
\begin_inset Formula $-30$
\end_inset

 dB, again using 1000 simulated signals for each plotted point.
 We include three curves for each decoding algorithm: one for the AWGN channel
 and no fading, and two more for simulated Doppler spreads of 0.2 and 1.0
 Hz.
 These simulated Doppler spreads are comparable to those encountered on
 HF ionospheric paths and also for EME at VHF and the lower UHF bands.
 For reference, we note that the JT65 symbol rate is about 2.69 Hz.
 
\end_layout

\begin_layout Standard
(*** A little more description is needed here, along with new data for the
 DS curves.***)
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename fig_psuccess.pdf
	lyxscale 90

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Psuccess"

\end_inset

Percentage of JT65 messages successfully decoded as a function of SNR in
 2500 Hz bandwidth.
 Results are shown for the hard-decision Berlekamp-Massey (BM) and soft-decision
 Franke-Taylor (FT) decoding algorithms.
 Curves labeled DS correspond to the hinted-decode (
\begin_inset Quotes eld
\end_inset

Deep Search
\begin_inset Quotes erd
\end_inset

) algorithm.
 Numbers adjacent to the curves are the simulated Doppler spreads in Hz.
 The curve labeled Sync illustrates the dependence of proper time and frequency
 synchronization in the decoder presently implemented in 
\emph on
WSJT-X
\emph default
.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Section
Summary
\end_layout

\begin_layout Standard
...
 Still to come ...
\end_layout

\begin_layout Bibliography
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "1"
key "kv2001"

\end_inset

“Algebraic soft-decision decoding of Reed-Solomon codes,” R.
 Köetter and A.
 Vardy, IEEE Trans.
 Inform.
 Theory, Vol.
 49, Nov.
 2003.
\end_layout

\begin_layout Bibliography
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "2"
key "lhmg2010"

\end_inset

"Stochastic Chase Decoding of Reed-Solomon Codes", Camille Leroux, Saied
 Hemati, Shie Mannor, Warren J.
 Gross, IEEE Communications Letters, Vol.
 14, No.
 9, September 2010.
\end_layout

\begin_layout Bibliography
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "3"
key "lk2008"

\end_inset

"Soft-Decision Decoding of Reed-Solomon Codes Using Successive Error-and-Erasure
 Decoding," Soo-Woong Lee and B.
 V.
 K.
 Vijaya Kumar, IEEE 
\begin_inset Quotes eld
\end_inset

GLOBECOM
\begin_inset Quotes erd
\end_inset

 2008 proceedings.
\end_layout

\begin_layout Bibliography
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "4"
key "lc2004"

\end_inset

Error Control Coding, 2nd edition, Shu Lin and Daniel J.
 Costello, Pearson-Prentice Hall, 2004.
\end_layout

\begin_layout Bibliography
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "5"
key "ls2009"

\end_inset


\begin_inset Quotes erd
\end_inset

Stochastic Erasure-Only List Decoding Algorithms for Reed-Solomon Codes,
\begin_inset Quotes erd
\end_inset

 Chang-Ming Lee and Yu T.
 Su, IEEE Signal Processing Letters, Vol.
 16, No.
 8, August 2009.
\end_layout

\begin_layout Bibliography
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "6"
key "karn"

\end_inset

Berlekamp-Massey decoder written by Phil Karn, http://www.ka9q.net/code/fec/
\end_layout

\begin_layout Section
\start_of_appendix
\begin_inset CommandInset label
LatexCommand label
name "sec:Appendix:SNR"

\end_inset

Appendix: Signal to Noise Ratios
\end_layout

\begin_layout Standard
The signal to noise ratio in a bandwidth, 
\begin_inset Formula $B$
\end_inset

, that is at least as large as the bandwidth occupied by the signal is:
\begin_inset Formula 
\begin{equation}
\mathrm{SNR}_{B}=\frac{P_{s}}{N_{o}B}\label{eq:SNR}
\end{equation}

\end_inset

where 
\begin_inset Formula $P_{s}$
\end_inset

 is the signal power (W), 
\begin_inset Formula $N_{o}$
\end_inset

 is one-sided noise power spectral density (W/Hz), and 
\begin_inset Formula $B$
\end_inset

 is the bandwidth in Hz.
 In amateur radio applications, digital modes are often compared based on
 the SNR defined in a 2.5 kHz reference bandwidth, 
\begin_inset Formula $\mathrm{SNR}_{2500}$
\end_inset

.
 
\end_layout

\begin_layout Standard
In the professional literature, decoder performance is characterized in
 terms of 
\begin_inset Formula $E_{b}/N_{o}$
\end_inset

, the ratio of the energy collected per information bit, 
\begin_inset Formula $E_{b}$
\end_inset

, to the one-sided noise power spectral density, 
\begin_inset Formula $N_{o}$
\end_inset

.
 Denote the duration of a channel symbol by 
\begin_inset Formula $\tau_{s}$
\end_inset

 (for JT65, 
\begin_inset Formula $\tau_{s}=0.3715\,\mathrm{s}$
\end_inset

).
 Signal power is related to the energy per symbol by 
\begin_inset Formula 
\begin{equation}
P_{s}=E_{s}/\tau_{s}.\label{eq:signal_power}
\end{equation}

\end_inset

The total energy in a received JT65 message consisting of 
\begin_inset Formula $n=63$
\end_inset

 channel symbols is 
\begin_inset Formula $63E_{s}$
\end_inset

.
 The energy collected for each of the 72 bits of information conveyed by
 the message is then
\begin_inset Formula 
\begin{equation}
E_{b}=\frac{63E_{s}}{72}=0.875E_{s.}\label{eq:Eb_Es}
\end{equation}

\end_inset

Using equations (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:SNR"

\end_inset

)-(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Eb_Es"

\end_inset

), 
\begin_inset Formula $\mathrm{SNR}_{2500}$
\end_inset

 can be written in terms of 
\begin_inset Formula $E_{b}/N_{o}$
\end_inset

:
\begin_inset Formula 
\[
\mathrm{SNR}_{2500}=1.23\times10^{-3}\frac{E_{b}}{N_{o}}.
\]

\end_inset

If all quantities are expressed in dB, then:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\mathrm{SNR}_{2500}=(E_{b}/N_{o})_{\mathrm{dB}}-29.1\,\mathrm{dB}.
\]

\end_inset


\end_layout

\end_body
\end_document