mirror of https://github.com/saitohirga/WSJT-X.git
Another set of additions to the paper, both text and figures.
git-svn-id: svn+ssh://svn.code.sf.net/p/wsjt/wsjt/branches/wsjtx@6352 ab8295b8-cf94-4d9e-aec4-7959e3be5d79
This commit is contained in:
Joe Taylor
parent
3576e0b868
commit
0f62680559
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@ -126,21 +126,18 @@ A major reason for the success and popularity of JT65 is its use of a strong
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error-correction code: a short block-length, low-rate Reed-Solomon code
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based on a 64-symbol alphabet.
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Until now, nearly all programs implementing JT65 have used the patented
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Koetter-Vardy (KV) algebraic soft-decision decoder
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Koetter-Vardy (KV) algebraic soft-decision decoder
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\begin_inset CommandInset citation
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LatexCommand cite
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key "kv2001"
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\end_inset
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, as licensed to K1JT and implemented in a closed-source program for use
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only in amateur radio applications.
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, licensed to and implemented by K1JT in a closed-source executable for
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use only in amateur radio applications.
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Since 2001 the KV decoder has been considered the best available soft-decision
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decoder for Reed Solomon codes.
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\end_layout
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\begin_layout Standard
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We describe here a new open-source alternative called the Franke-Taylor
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We describe here a new open-source alternative called the Franke-Taylor
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(FT, or K9AN-K1JT) algorithm.
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It is conceptually simple, built around the well-known Berlekamp-Massey
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errors-and-erasures algorithm, and in this application it performs even
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@ -149,8 +146,8 @@ We describe here a new open-source alternative called the Franke-Taylor
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\emph on
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WSJT-X
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\emph default
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, widely used for amateur weak-signal communication with JT65 and several
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other specialized digital modes.
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, widely used for amateur weak-signal communication with JT65 and other
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specialized digital modes.
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The program is freely available and licensed under the GNU General Public
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License.
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\end_layout
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@ -160,22 +157,22 @@ The JT65 protocol specifies transmissions that normally start one second
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into a UTC minute and last for 46.8 seconds.
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Receiving software therefore has up to several seconds to decode a message,
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before the operator sends a reply at the start of the next minute.
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With today's personal computers, this relatively long time for decoding
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a short message encourages experimentation with decoders of high computational
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complexity.
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As a result, on a typical fading channel the FT algorithm extends the decoding
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threshold by many dB over the hard-decision Berlekamp-Massey decoder, and
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by a meaningful amount over the KV decoder.
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With today's personal computers, this relatively long time available for
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decoding a short message encourages experimentation with decoders of high
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computational complexity.
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As a result, on a typical fading channel the FT algorithm can extend the
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decoding threshold by many dB over the hard-decision Berlekamp-Massey decoder,
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and by a meaningful amount over the KV decoder.
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In addition to its excellent performance, the new algorithm has other desirable
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properties---not the least of which is its conceptual simplicity.
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properties, not least of which is its conceptual simplicity.
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Decoding performance and complexity scale in a convenient way, providing
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steadily increasing soft-decision decoding gain as a tunable computational
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complexity parameter is increased over more than 5 orders of magnitude.
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This means that appreciable gain is available from our decoder even on
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very simple (and relatively slow) computers.
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Appreciable gain is available from our decoder even on very simple (and
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relatively slow) computers.
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On the other hand, because the algorithm benefits from a large number of
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independent decoding trials, it should be possible to obtain further performanc
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e gains through parallelization on high-performance computers.
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independent decoding trials, further performance gains should be achievable
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through parallelization on high-performance computers.
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\end_layout
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\begin_layout Section
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@ -943,7 +940,7 @@ Here
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\end_inset
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if the received symbol and codeword symbol are different, and
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\begin_inset Formula $p_{1\,j}$
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\begin_inset Formula $p_{1,\,j}$
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\end_inset
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is the fractional power associated with received symbol
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@ -965,12 +962,7 @@ In practice we find that
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\end_inset
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can reliably indentify the correct codeword if the signal-to-noise ratio
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for individual symbols is greater than about 4 in linear power units, or
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\begin_inset Formula $E_{s}/N_{0}\apprge6$
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\end_inset
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dB (*** check these numbers ***).
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for individual symbols is greater than about 4 in linear power units.
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We also find that significantly weaker signals can be decoded by using
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soft-symbol information beyond that contained in
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\begin_inset Formula $p_{1}$
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@ -1117,7 +1109,7 @@ est metrics
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will likely be close to 1.
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We therefore apply a ratio threshold test, say
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\begin_inset Formula $r<r_{0}$
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\begin_inset Formula $r<r_{1}$
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\end_inset
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, to identify codewords with high probability of being correct.
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@ -1128,7 +1120,7 @@ reference "sec:Theory,-Simulation,-and"
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\end_inset
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, we have used simulations to set an empirical acceptance threshold
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, we use simulations to set an empirical acceptance threshold
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\begin_inset Formula $r_{0}$
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\end_inset
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@ -1145,21 +1137,32 @@ Technically the FT algorithm is a list decoder.
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is retained.
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As with all such algorithms, a stopping criterion is necessary.
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FT accepts a codeword unconditionally if the Hamming distance and soft
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distance
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FT accepts a codeword unconditionally if the Hamming distance
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\begin_inset Formula $X$
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\end_inset
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and soft distance
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\begin_inset Formula $d_{s}$
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\end_inset
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are less than some conservatively specified limits.
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Secondary acceptance criteria
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\begin_inset Formula $d_{s}<d_{0}$
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are less than conservatively specified limits
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\begin_inset Formula $X_{0}$
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\end_inset
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and
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\begin_inset Formula $r<r_{0}$
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\begin_inset Formula $d_{0}$
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\end_inset
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are used to validate additional decodes.
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.
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Secondary acceptance criteria
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\begin_inset Formula $d_{s}<d_{1}$
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\end_inset
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and
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\begin_inset Formula $r<r_{1}$
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\end_inset
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are used to validate additional decodes that did not pass the first test.
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A timeout is used to limit the algorithm's execution time if no acceptable
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codeword is found in a reasonable number of trials,
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\begin_inset Formula $T$
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@ -1227,7 +1230,7 @@ If BM decoding was not successful, go to step 2.
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\begin_layout Enumerate
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Calculate the hard-decision Hamming distance
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\begin_inset Formula $h$
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\begin_inset Formula $X$
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\end_inset
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between the candidate codeword and the received symbols, the corresponding
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@ -1244,7 +1247,7 @@ Calculate the hard-decision Hamming distance
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\begin_inset Formula $u$
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\end_inset
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is the largest one encountered so far, preserve the previous value of
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is the largest one encountered so far, preserve any previous value of
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\begin_inset Formula $u_{1}$
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\end_inset
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@ -1261,7 +1264,7 @@ Calculate the hard-decision Hamming distance
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\begin_layout Enumerate
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If
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\begin_inset Formula $h<h_{0}$
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\begin_inset Formula $X<X_{0}$
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\end_inset
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and
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@ -1290,7 +1293,7 @@ If
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\end_inset
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and
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\begin_inset Formula $r<r_{1}$
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\begin_inset Formula $r<r_{1},$
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\end_inset
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go to step 10.
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@ -1301,11 +1304,7 @@ Otherwise, declare decoding failure and exit.
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\end_layout
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\begin_layout Enumerate
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An acceptable codeword with
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\begin_inset Formula $u_{max}>u_{0}$
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\end_inset
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has been found.
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An acceptable codeword has been found.
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Declare a successful decode and return this codeword.
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\end_layout
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@ -1316,7 +1315,7 @@ An acceptable codeword with
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\begin_layout Standard
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Inspiration for the FT decoding algorithm came from a number of sources,
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particularly references
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particularly references
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\begin_inset CommandInset citation
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LatexCommand cite
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key "lhmg2010"
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@ -1330,7 +1329,7 @@ key "lk2008"
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\end_inset
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and the textbook by Lin and Costello
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and the textbook by Lin and Costello
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\begin_inset CommandInset citation
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LatexCommand cite
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key "lc2004"
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@ -1365,8 +1364,8 @@ key "ls2009"
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is applied to higher-rate Reed-Solomon codes on a binary-input channel
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with BPSK-modulated symbols.
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Our 64-ary input channel with 64-FSK modulation required us to develop
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unique methods for assigning erasure probabilities and for defining an
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acceptance criteria to select the best codeword from the list of candidates.
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unique methods for assigning erasure probabilities and for defining acceptance
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criteria to select the best codeword from the list of candidates.
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\end_layout
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@ -1381,21 +1380,24 @@ Hinted Decoding
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\end_layout
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\begin_layout Standard
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The FT algorithm is completely general: it recovers with equal sensitivity
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The FT algorithm is completely general: with equal sensitivity it recovers
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any one of the
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\begin_inset Formula $2^{72}\approx4.7\times10^{21}$
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\end_inset
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different messages that can be transmitted using the JT65 protocol.
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In many circumstances it's easy to imagine a much smaller list of messages
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(say, a few thousand or less) that may be among the most likely ones to
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be received.
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For example, one such situation exists when making short ham-radio contacts
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exchanging minimal amounts of information such as callsigns, signal reports,
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perhaps a Maidenhead locator, and acknowledgments.
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Similarly, on the EME path or on a VHF or UHF band with limited geographical
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coverage, the most likely received messages will often originate from callsigns
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that have been decoded before.
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different messages that can be transmitted with the JT65 protocol.
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In some circumstances it's easy to imagine a
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\emph on
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much
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\emph default
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smaller list of messages (say, a few thousand messages or less) that may
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be among the most likely ones to be received.
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One such situation exists when making short ham-radio contacts that exchange
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minimal information including callsigns, signal reports, perhaps Maidenhead
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locators, and acknowledgments.
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On the EME path or on a VHF or UHF band with limited geographical coverage,
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the most likely received messages often originate from callsigns that have
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been decoded before.
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Saving a list of previously decoded callsigns makes it easy to generate
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lists of hypothetical messages and their corresponding codewords, at very
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little computational expense.
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@ -1420,13 +1422,14 @@ hinted decoding;
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\begin_inset Quotes eld
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\end_inset
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Deep Search
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deep search
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\begin_inset Quotes erd
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\end_inset
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algorithm.
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In certain limited situations it can provide enhanced sensitivity for the
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principal task of any decoder, namely to determine what message was sent.
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principal task of any decoder, namely to determine precisely what message
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was sent.
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\end_layout
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\begin_layout Standard
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@ -1459,7 +1462,8 @@ small enough
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\begin_inset Quotes erd
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\end_inset
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for adequate confidence, while still ensuring that false decodes are rare.
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to establish adequate confidence, while still ensuring that false decodes
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are rare.
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Because tested candidate codewords are drawn from a list typically no longer
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than a few thousand, rather than
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\begin_inset Formula $2^{72},$
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@ -1469,22 +1473,26 @@ small enough
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\begin_inset Formula $r_{2}$
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\end_inset
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can be a more relaxed limit than the ones
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\begin_inset Formula $r_{0}$
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can set a more relaxed limit than
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\begin_inset Formula $r_{1},$
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\end_inset
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and
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\begin_inset Formula $r_{1}$
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as used in the FT algorithm.
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For the limited subset of messages established by operator experience as
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\begin_inset Quotes eld
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\end_inset
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used in the FT algorithm.
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For the limited subset of messages considered as likely, hinted decodes
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can be obtained at lower signal levels than would be required for decodes
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selected from the full universe of
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likely,
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\begin_inset Quotes erd
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\end_inset
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hinted decodes can be obtained at lower signal levels than required for
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decodes obtained from the full universe of
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\begin_inset Formula $2^{72}$
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\end_inset
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distinct messages.
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possible messages.
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\end_layout
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\begin_layout Section
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@ -1497,10 +1505,6 @@ name "sec:Theory,-Simulation,-and"
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Decoder Performance Evaluation
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\end_layout
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\begin_layout Subsection
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Simulated results on the AWGN channel
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\end_layout
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\begin_layout Standard
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Comparisons of decoding performance are usually presented in the professional
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literature as plots of word error rate versus
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@ -1514,8 +1518,8 @@ Comparisons of decoding performance are usually presented in the professional
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.
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For weak-signal amateur radio work, performance is more conveniently presented
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as the probability of successfully decoding a received word versus signal-to-no
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ise ratio in a 2500 Hz reference bandwidth,
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as the probability of successfully decoding a received word plotted against
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signal-to-noise ratio in a 2500 Hz reference bandwidth,
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\begin_inset Formula $\mathrm{SNR}{}_{2500}$
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\end_inset
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@ -1536,12 +1540,36 @@ reference "sec:Appendix:SNR"
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\end_inset
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.
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Examples of both types of plot are included in the following discussion,
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where we describe a number of simulations carried out to compare performance
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of the FT algorithm with others, and with theoretical expectations.
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We have also used simulations to establish suitable default values for
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the acceptance parameters
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\begin_inset Formula $h_{0},$
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\end_inset
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\begin_inset Formula $d_{0},$
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\end_inset
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\begin_inset Formula $d_{1},$
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\end_inset
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and
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\begin_inset Formula $r_{1}.$
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\end_inset
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\end_layout
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\begin_layout Subsection
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Simulated results on the AWGN channel
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\end_layout
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\begin_layout Standard
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Results of simulations using the BM, FT, and KV decoding algorithms on the
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JT65 (63,12) code are presented in terms of word error-rate vs
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JT65 code are presented in terms of word error rate versus
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\begin_inset Formula $E_{b}/N_{o}$
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\end_inset
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@ -1556,9 +1584,9 @@ reference "fig:bodide"
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For these tests we generated at least 1000 signals at each signal-to-noise
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ratio, assuming the additive white gaussian noise (AWGN) channel, and processed
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the data using each algorithm.
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For word error-rates less than 0.1 it was necessary to process 10,000 or
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For word error rates less than 0.1 it was necessary to process 10,000 or
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even 100,000 simulated signals in order to capture enough errors to make
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the estimates of word-error-rate statistically meaningful.
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the measurements statistically meaningful.
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As a test of the fidelity of our numerical simulations, Figure
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\begin_inset CommandInset ref
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LatexCommand ref
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|
|
@ -1566,8 +1594,7 @@ reference "fig:bodide"
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\end_inset
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also shows theoretical results (filled squares) for comparison with the
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BM results.
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also shows theoretical results for comparison with the BM results.
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The simulated BM results agree with theory to within about 0.1 dB.
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This difference between simulated BM results and theory is caused by small
|
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errors in the estimates of time- and frequency-offset of the received signal
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|
|
@ -1628,29 +1655,23 @@ Word error rates as a function of
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\begin_inset Formula $E_{b}/N_{0},$
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\end_inset
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|
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the signal-to-noise ratio per bit.
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The single curve marked with filled squares shows a theoretical prediction
|
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for the BM decoder.
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Open squares illustrate simulation results for an AWGN channel with the
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BM, FT (
|
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\begin_inset Formula $T=10^{5}$
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\end_inset
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||||
|
||||
) and KV (
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||||
the signal-to-noise ratio per information bit.
|
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Theory: theoretical prediction for the hard-decision BM decoder.
|
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The remaining curves represent simulation results on an AWGN channel for
|
||||
the BM, KV, and FT decoders.
|
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The KV algorithm was executed with complexity coefficient
|
||||
\begin_inset Formula $\lambda=15$
|
||||
\end_inset
|
||||
|
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) decoders used in program
|
||||
, the most aggressive setting historically used in the
|
||||
\emph on
|
||||
WSJT-X
|
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WSJT
|
||||
\emph default
|
||||
.
|
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The KV results are for decoding complexity coefficient
|
||||
\begin_inset Formula $\lambda=15$
|
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programs.
|
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The FT alrithm was run with timeout setting
|
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\begin_inset Formula $T=10^{5}.$
|
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\end_inset
|
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|
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, the most aggressive setting that has historically been used in earlier
|
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versions of the WSJT programs.
|
||||
|
||||
\end_layout
|
||||
|
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|
|
@ -1702,15 +1723,15 @@ reference "fig:bodide"
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\end_inset
|
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|
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in this format along with additional FT results for
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\begin_inset Formula $T=10^{4},10^{3},10^{2}$
|
||||
\begin_inset Formula $T=10^{4},\:10^{3},\:10^{2}$
|
||||
\end_inset
|
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|
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and
|
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\begin_inset Formula $10^{1}$
|
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\begin_inset Formula $10$
|
||||
\end_inset
|
||||
|
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.
|
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The KV results are plotted with open triangles.
|
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The KV results are plotted with open squares.
|
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It is apparent that the FT decoder produces more decodes than KV when
|
||||
\begin_inset Formula $T=10^{4}$
|
||||
\end_inset
|
||||
|
|
@ -1747,24 +1768,19 @@ name "fig:WER2"
|
|||
|
||||
\end_inset
|
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|
||||
Percent of JT65 messages copied as a function of SNR in 2.5 kHz bandwidth.
|
||||
Solid lines with filled round circles are results from the FT decoder with
|
||||
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 $T=10^{5},10^{4},10^{3},10^{2}$
|
||||
\end_inset
|
||||
|
||||
and
|
||||
\begin_inset Formula $10$
|
||||
\end_inset
|
||||
|
||||
, respectively, from left to right.
|
||||
The dashed line with open triangles is the KV decoder with complexity coefficie
|
||||
nt
|
||||
\begin_inset Formula $\lambda=15$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
Results from the BM algorithm are also shown with filled triangles.
|
||||
Results from the BM algorithm are shown with a dashed line and crosses.
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
|
@ -1809,7 +1825,7 @@ reference "fig:N_vs_X"
|
|||
\begin_inset Formula $X\le25$
|
||||
\end_inset
|
||||
|
||||
because all such words were successfully decoded by the BM algorithm.
|
||||
because all such words are successfully decoded by the BM algorithm.
|
||||
Figure
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
|
|
@ -1826,8 +1842,8 @@ reference "fig:N_vs_X"
|
|||
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 also provide insight into the mean and variance of the execution
|
||||
time for the FT algorithm, as execution time will be roughly proportional
|
||||
These results provide insight into the mean and variance of the execution
|
||||
time for the FT algorithm, since execution time will be roughly proportional
|
||||
to the number of required trials.
|
||||
\end_layout
|
||||
|
||||
|
|
@ -1859,13 +1875,21 @@ 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 -24 dB (
|
||||
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).
|
||||
dB.
|
||||
|
||||
\end_layout
|
||||
|
||||
|
|
@ -1880,7 +1904,7 @@ Number of trials needed to decode a received word versus Hamming distance
|
|||
\end_layout
|
||||
|
||||
\begin_layout Subsection
|
||||
Simulated results for hinted decoding and Rayleigh fading
|
||||
Simulated results for Rayleigh fading and hinted decoding
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
|
@ -1904,9 +1928,11 @@ reference "fig:Psuccess"
|
|||
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.
|
||||
For reference, we note that the JT65 symbol rate is about 2.69 Hz.
|
||||
The simulated Doppler spreads are comparable to those encountered on HF
|
||||
ionospheric paths and for EME at VHF and lower UHF bands.
|
||||
For reference, we note that the JT65 symbol rate is about 2.69 Hz.
|
||||
(*** A little more description of hinted decoding is needed here, and new
|
||||
data for the DS curves.***)
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
|
@ -1948,7 +1974,14 @@ Deep Search
|
|||
\begin_inset Quotes erd
|
||||
\end_inset
|
||||
|
||||
) matched-filter algorithm.
|
||||
) 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
|
||||
|
|
|
|||
Loading…
Reference in New Issue