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More massaging of draft paper on the FT dedoder.
git-svn-id: svn+ssh://svn.code.sf.net/p/wsjt/wsjt/branches/wsjtx@6205 ab8295b8-cf94-4d9e-aec4-7959e3be5d79
This commit is contained in:
Joe Taylor
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@ -89,23 +89,14 @@ The JT65 mode has revolutionized amateur-radio weak-signal communication
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by enabling amateur radio operators with small antennas and relatively
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low-power transmitters to communicate over propagation paths not usable
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with traditional technologies.
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One reason for the success and popularity of JT65 is its use of strong
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error-correction coding.
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The JT65 code is a short block-length, low-rate, Reed-Solomon code based
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on a 64-symbol alphabet.
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Since 2004, most JT65 decoders have used the patented
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\begin_inset Quotes eld
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\end_inset
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Koetter-Vardy
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\begin_inset Quotes erd
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\end_inset
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(KV) algebraic soft-decision decoder.
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A major reason for the success and popularity of JT65 is its use of strong
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error-correction coding: a short block-length, low-rate, Reed-Solomon code
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based on a 64-symbol alphabet.
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Since 2004, most JT65 decoders have used the patented Koetter-Vardy (KV)
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algebraic soft-decision decoder.
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The KV decoder is implemented in a closed-source program licensed to K1JT
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for use in amateur radio applications.
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We describe here a new open-source alternative called the FTRSD (or FT)
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algotithm.
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We describe here a new open-source alternative called the FT algotithm.
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It is conceptually simple, is built around the well-known Berlekamp-Massey
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errors-and-erasures algorithm, and perform at least as well as the KV decoder.
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\end_layout
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@ -116,7 +107,7 @@ Introduction
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\begin_layout Standard
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JT65 message frames consist of a short, compressed message encoded for transmiss
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ion using a Reed-Solomon code.
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ion with a Reed-Solomon code.
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Reed-Solomon codes are block codes; as such they are characterized by the
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length of their codewords,
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\begin_inset Formula $n$
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@ -134,7 +125,8 @@ ion using a Reed-Solomon code.
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.
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JT65 uses a (63,12) Reed-Solomon code with 64 possible values for each
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symbol, so each symbol represents
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symbol.
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Each symbol represents
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\begin_inset Formula $\log_{2}64=6$
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\end_inset
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@ -181,9 +173,9 @@ The minimum Hamming distance of the JT65 code is
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\end_layout
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\begin_layout Standard
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Given only a received word containing some incorrect symbols (errors), the
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received word can be decoded into the correct codeword using a deterministic,
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algebraic algorithm provided that no more than
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Given a received word containing some incorrect symbols (errors), the received
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word can be decoded into the correct codeword using a deterministic, algebraic
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algorithm provided that no more than
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\begin_inset Formula $t$
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\end_inset
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@ -199,11 +191,11 @@ For the JT65 code,
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\begin_inset Formula $t=25$
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\end_inset
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, which means that it is always possible to efficiently decode a received
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word having no more than 25 symbol errors.
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: it is always possible to efficiently decode a received word having no
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more than 25 symbol errors.
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Any one of several well-known algebraic algorithms, such as the widely
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used Berlekamp-Massey (BM) algorithm, can carry out the decoding.
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Two steps are ncessarily involved, namely
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Two steps are ncessarily involved in this process, namely
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\end_layout
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\begin_layout Enumerate
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@ -215,16 +207,16 @@ determine the correct value of the incorrect symbols
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\end_layout
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\begin_layout Standard
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If it is somehow known that certain symbols are incorrect, this information
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can be used to reduce the amount of work in step 1 and to allow step 2
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to correct more than
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If we somehow know that certain symbols are incorrect, this information
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can be used to reduce the work in step 1 and allow step 2 to correct more
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than
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\begin_inset Formula $t$
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\end_inset
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errors.
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In the unlikely event that the location of every error can be provided
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to the BM decoder, and if no correct symbols are accidentally labeled as
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errors, the BM algorithm can correct up to
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In the unlikely event that the location of every error is known, and if
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no correct symbols are accidentally labeled as errors, the BM algorithm
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can correct up to
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\begin_inset Formula $d$
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\end_inset
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@ -233,34 +225,33 @@ If it is somehow known that certain symbols are incorrect, this information
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\end_layout
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\begin_layout Standard
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The FT algorithm creates a list of symbols suspected of being incorrect
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and sends it to the BM decoder.
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The FT algorithm creates lists of symbols suspected of being incorrect and
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sends them to the BM decoder.
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Symbols flagged in this way are called
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\begin_inset Quotes eld
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\end_inset
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erasures
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erasures,
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\begin_inset Quotes erd
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\end_inset
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, while other incorrect symbols will be called
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while other incorrect symbols will be called
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\begin_inset Quotes eld
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\end_inset
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errors
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errors.
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\begin_inset Quotes erd
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\end_inset
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.
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As already noted, with perfect erasure information up to 51 errors can
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be corrected.
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When the erasure information is imperfect, some of the erased symbols may
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be correct and some other symbols in error.
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Imperfect erasure information means that some erased symbols may be correct,
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and some other symbols in error.
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If
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\begin_inset Formula $s$
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\end_inset
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symbols are erased and the remaining unerased symbols contain
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symbols are erased and the remaining (unerased) symbols contain
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\begin_inset Formula $e$
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\end_inset
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@ -301,7 +292,7 @@ errors-only
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\begin_inset Formula $d-1=51$
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\end_inset
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for JT65), the decoder is said to be an
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for JT65), the decoder is called an
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\begin_inset Quotes eld
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\end_inset
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@ -310,8 +301,8 @@ errors-and-erasures
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\end_inset
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decoder.
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The errors-and-erasures capability of Reed-Solomon codes lies at the core
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of the FTRSD algorithm.
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The possibility of doing errors-and-erasures decoding lies at the heart
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of the FT algorithm.
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\end_layout
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@ -326,17 +317,17 @@ Do I feel lucky?
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\end_layout
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\begin_layout Standard
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The FTRSD algorithm uses a statistical argument based on the quality of
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received symbols to generate lists of symbols likely to be in error, thereby
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enabling reliable decoding of received codewords with more than 25 errors.
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As a specific example, consider a received JT65 codeword with 23 correct
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The FT algorithm uses the estimated quality of received symbols to generate
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lists of symbols considered likely to be in error, thereby enabling reliable
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decoding of received words with more than 25 errors.
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As a specific example, consider a received JT65 signal producing 23 correct
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symbols and 40 errors.
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We do not know which symbols are in error.
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Suppose that the decoder randomly chooses
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Suppose that the decoder randomly selects
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\begin_inset Formula $s=40$
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\end_inset
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symbols to erase, leaving 23 unerased symbols.
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symbols for erasure, leaving 23 unerased symbols.
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According to Eq.
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(
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\begin_inset CommandInset ref
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@ -381,15 +372,11 @@ tric probability distribution.
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\end_inset
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will be random variables.
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Let
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\begin_inset Formula $P(x|N,X,s)$
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\end_inset
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denote the conditional probability mass function for
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The conditional probability mass function for
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\begin_inset Formula $x$
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\end_inset
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, the number of erased incorrect symbols, given the stated values of
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given stated values of
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\begin_inset Formula $N$
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\end_inset
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@ -401,8 +388,7 @@ tric probability distribution.
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\begin_inset Formula $s$
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\end_inset
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.
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Then
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may be written as
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\end_layout
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\begin_layout Standard
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@ -418,7 +404,7 @@ where
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\end_inset
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is the binomial coefficient.
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[The binomial coefficient can be calculated using the
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The binomial coefficient can be calculated using the function
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\begin_inset Quotes eld
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\end_inset
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@ -430,7 +416,7 @@ nchoosek(
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\begin_inset Quotes erd
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\end_inset
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function in Gnu Octave.
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in the interpreted language GNU Octave.
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The hypergeometric probability mass function defined in Eq.
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(
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\begin_inset CommandInset ref
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@ -439,7 +425,7 @@ reference "eq:hypergeometric_pdf"
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\end_inset
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) is available in Gnu Octave as function
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) is available in GNU Octave as function
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\begin_inset Quotes eld
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\end_inset
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@ -451,15 +437,15 @@ hygepdf(
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\begin_inset Quotes erd
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\end_inset
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.]
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.
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\end_layout
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\begin_layout Paragraph
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Example 1:
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\end_layout
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\begin_layout Case
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A codeword contains
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\begin_layout Standard
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Suppose a codeword contains
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\begin_inset Formula $X=40$
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\end_inset
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@ -477,7 +463,7 @@ A codeword contains
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\begin_inset Formula $x=35$
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\end_inset
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of the erased symbols are incorrect is then
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of the erased symbols are actually incorrect is then
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\begin_inset Formula
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\[
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P(x=35)=\frac{\binom{40}{35}\binom{63-40}{40-35}}{\binom{63}{40}}=2.356\times10^{-7}.
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@ -513,10 +499,10 @@ ty of erasing 35 errors, we may safely conclude that the probability of
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Example 2:
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\end_layout
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\begin_layout Case
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How might we best choose the number of symbols to be chosen for erasure,
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so as to maximize the probability of successful decoding? By exhaustive
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search over all possible values up to
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\begin_layout Standard
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How might we best choose the number of symbols to erase, in order to maximize
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the probability of successful decoding? By exhaustive search over all possible
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values up to
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\begin_inset Formula $s=51$
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\end_inset
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@ -551,8 +537,9 @@ P(x\ge37)\simeq2\times10^{-6}.
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\end_inset
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This probability is about 8 times higher than the probability of success
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when only 40 symbols were erased, but the odds of successfully decoding
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on the first try are still only about 1 in 500,000.
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when only 40 symbols were erased.
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Nevertheless, the odds of successfully decoding on the first try are still
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only about 1 in 500,000.
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\end_layout
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@ -560,19 +547,19 @@ This probability is about 8 times higher than the probability of success
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Example 3:
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\end_layout
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\begin_layout Case
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Examples 1 and 2 show that a strategy of randomly selecting symbols to erase
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\begin_layout Standard
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Examples 1 and 2 show that a random strategy for selecting symbols to erase
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is unlikely to be successful unless we are prepared to wait a long time
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for an answer.
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So let's modify the strategy to tip the odds in our favor.
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Let the received symbol set contain
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Let the received word contain
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\begin_inset Formula $X=40$
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\end_inset
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incorrect symbols, as before, but suppose it is known that 10 symbols are
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much more reliable than the other 53.
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The 10 most reliable symbols are therefore protected from erasure, and
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erasures chosen from the smaller set of
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incorrect symbols, as before, but suppose we know that 10 symbols are significa
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ntly more reliable than the other 53.
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We might therefore protect the 10 most reliable symbols from erasure, and
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choose erasures from the smaller set of
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\begin_inset Formula $N=53$
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\end_inset
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@ -581,11 +568,9 @@ Examples 1 and 2 show that a strategy of randomly selecting symbols to erase
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\begin_inset Formula $s=45$
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\end_inset
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symbols are now chosen randomly from the set of 53 least reliable symbols,
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it is still necessary for the erased symbols to include at least 37 errors,
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as in Example 2.
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However, the probabilities are now much more favorable.
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With
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symbols are chosen randomly in this way, it is still necessary for the
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erased symbols to include at least 37 errors, as in Example 2.
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However, the probabilities are now much more favorable: with
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\begin_inset Formula $N=53$
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\end_inset
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@ -593,7 +578,7 @@ Examples 1 and 2 show that a strategy of randomly selecting symbols to erase
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\begin_inset Formula $X=40$
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\end_inset
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,
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, and
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\begin_inset Formula $s=45$
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\end_inset
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@ -610,7 +595,7 @@ reference "eq:hypergeometric_pdf"
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\end_inset
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.
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Even better odds are obtained with
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Even better odds are obtained by choosing
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\begin_inset Formula $s=47$
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\end_inset
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@ -627,7 +612,7 @@ reference "eq:hypergeometric_pdf"
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\begin_inset Formula $X=40$
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\end_inset
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, and
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, and
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\begin_inset Formula $s=47$
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\end_inset
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@ -636,8 +621,9 @@ reference "eq:hypergeometric_pdf"
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\end_inset
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.
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These odds are the best so far, about 1 in 38.
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The odds for successful decoding on the first try are now about 1 in 38.
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A few hundred independently randomized tries would be enough to all-but-guarant
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ee production of a valid codeword from the BM decoder.
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\end_layout
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\begin_layout Section
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@ -647,23 +633,21 @@ name "sec:The-decoding-algorithm"
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\end_inset
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The FTRSD decoding algorithm
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The FT decoding algorithm
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\end_layout
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\begin_layout Standard
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Example 3 shows how reliable information about symbol quality might lead
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to an algorithm capable of decoding received frames with a large number
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of errors.
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In practice the number of errors in the received word is unknown, so it
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is better use a stochastic algorithm to assign a high probability of erasure
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to low-quality symbols and a relatively low probability to high-quality
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symbols.
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As illustrated by Example 3, a good choice of erasure probabilities can
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increase the chance of a successful decode by many orders of magnitude.
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Example 3 shows how reliable information about symbol quality should make
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it possible to decode received frames having a large number of errors.
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In practice the number of errors in the received word is unknown, so we
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use a stochastic algorithm to assign a high erasure probability to low-quality
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symbols and a relatively low probability to high-quality symbols.
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As illustrated by Example 3, a good choice of these probabilities can increase
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the chance of a successful decode by many orders of magnitude.
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\end_layout
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\begin_layout Standard
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The FTRSD algorithm uses two quality indices made available by a noncoherent
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The FT algorithm uses two quality indices made available by a noncoherent
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64-FSK demodulator.
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The demodulator identifies the most likely value for each symbol based
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on which of 64 frequency bins contains the the largest signal-plus-noise
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@ -710,9 +694,8 @@ soft-symbol
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\end_inset
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values.
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High ranking symbols have larger signal-to-noise ratio than lower ranked
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symbols.
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High ranking symbols have larger signal-to-noise ratio than those with
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lower rank.
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\end_layout
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\begin_layout Itemize
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@ -728,8 +711,8 @@ soft-symbol
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\end_layout
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\begin_layout Standard
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The FTRSD decoder uses a table of symbol error probabilities derived from
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a large dataset of received words that have been successfully decoded.
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The FT decoder uses a table of symbol error probabilities derived from a
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large dataset of received words that have been successfully decoded.
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The table provides an estimate of the
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\emph on
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a-priori
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@ -743,50 +726,40 @@ a-priori
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\end_inset
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metrics.
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These probabilities will be close to 1 for low-quality symbols and close
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to 0 for high-quality symbols.
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Recall from Examples 2 and 3 that the best performance was obtained when
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These probabilities are close to 1 for low-quality symbols and close to
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0 for high-quality symbols.
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Recall from Examples 2 and 3 that best performance was obtained with
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\begin_inset Formula $s>X$
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\end_inset
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.
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Correspondingly, the FTRSD algorithm works best when the probability of
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erasing a symbol is somewhat larger than the probability that the symbol
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is incorrect.
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Correspondingly, the FT algorithm works best when the probability of erasing
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a symbol is somewhat larger than the probability that the symbol is incorrect.
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Empirically, we found good decoding performance when the symbol erasure
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probability is about 1.3 times the symbol error probability.
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\end_layout
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\begin_layout Standard
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The FTRSD algorithm tries successively to decode the received word using
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educated guesses to select symbols for erasure.
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For each iteration an independent stochastic erasure vector is generated
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based on the symbol erasure probabilities.
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The erasure vector is provided to the BM decoder along with the full set
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of 63 received symbols.
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If the BM decoder finds a candidate codeword it is assigned a quality metric,
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defined as the soft distance,
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The FT algorithm tries successively to decode the received word using independen
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t educated guesses to select symbols for erasure.
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For each iteration an stochastic erasure vector is generated based on the
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symbol erasure probabilities.
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The erasure vector is sent to the BM decoder along with the full set of
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63 received symbols.
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When the BM decoder finds a candidate codeword it is assigned a quality
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metric
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\begin_inset Formula $d_{s}$
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\end_inset
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, between the received word and the codeword:
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defined as the soft distance between the received word and the codeword:
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\begin_inset Formula
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\begin{equation}
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d_{s}=\sum_{i=1}^{n}(1+p_{1,i})\alpha_{i}.\label{eq:soft_distance}
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d_{s}=\sum_{i=1}^{n}\alpha_{i}\,(1+p_{1,i}).\label{eq:soft_distance}
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\end{equation}
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\end_inset
|
||||
|
||||
Here
|
||||
\begin_inset Formula $p_{1,i}$
|
||||
\end_inset
|
||||
|
||||
is the fractional power associated with received symbol
|
||||
\begin_inset Formula $i$
|
||||
\end_inset
|
||||
|
||||
;
|
||||
\begin_inset Formula $\alpha_{i}=0$
|
||||
\end_inset
|
||||
|
||||
@ -794,28 +767,36 @@ Here
|
||||
\begin_inset Formula $i$
|
||||
\end_inset
|
||||
|
||||
is the same as the corresponding symbol in the codeword, and
|
||||
is the same as the corresponding symbol in the codeword,
|
||||
\begin_inset Formula $\alpha_{i}=1$
|
||||
\end_inset
|
||||
|
||||
if the received symbol and codeword symbol are different.
|
||||
This soft distance can be written as two terms, the first of which is just
|
||||
the Hamming distance between the received word and the codeword.
|
||||
The second term ensures that if two candidate codewords have the same Hamming
|
||||
distance from the received word, a smaller distance will be assigned to
|
||||
the one where the different symbols occurred in lower quality symbols.
|
||||
if the received symbol and codeword symbol are different, and
|
||||
\begin_inset Formula $p_{1,i}$
|
||||
\end_inset
|
||||
|
||||
is the fractional power associated with received symbol
|
||||
\begin_inset Formula $i$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
Think of the soft distance as two terms: the first is the Hamming distance
|
||||
between the received word and the codeword, and the second ensures that
|
||||
if two candidate codewords have the same Hamming distance from the received
|
||||
word, a smaller distance will be assigned to the one where differences
|
||||
occur in symbols of lower quality.
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Technically the FT algorithm is a list-decoder, potentially generating a
|
||||
Technically the FT algorithm is a list decoder, potentially generating a
|
||||
list of candidate codewords.
|
||||
Among the list of candidate codewords found by this stochastic search algorithm
|
||||
, only the one with the smallest soft-distance from the received word is
|
||||
retained.
|
||||
As with all such algorithms, a stopping criterion is necessary.
|
||||
FTRSD accepts a codeword unconditionally if its soft distance is smaller
|
||||
than an empirically determined acceptance threshold,
|
||||
FT accepts a codeword unconditionally if its soft distance is smaller than
|
||||
an empirically determined acceptance threshold,
|
||||
\begin_inset Formula $d_{a}$
|
||||
\end_inset
|
||||
|
||||
|
||||
@ -2348,8 +2348,8 @@ void MainWindow::guiUpdate()
|
||||
m_ntx=7;
|
||||
ui->rbGenMsg->setChecked(true);
|
||||
} else {
|
||||
m_ntx=6;
|
||||
ui->txrb6->setChecked(true);
|
||||
//JHT 11/29/2015 m_ntx=6;
|
||||
// ui->txrb6->setChecked(true);
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user