mirror of https://github.com/saitohirga/WSJT-X.git
1624 lines
36 KiB
Plaintext
1624 lines
36 KiB
Plaintext
#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\index Index
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\shortcut idx
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\end_header
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\begin_body
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\begin_layout Title
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A stochastic successive erasures soft-decision decoder for the JT65 (63,12)
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Reed-Solomon code
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\end_layout
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\begin_layout Author
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Steven J.
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Franke, K9AN and Joseph H.
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Taylor, K1JT
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\end_layout
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\begin_layout Abstract
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The JT65 protocol 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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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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Since 2004, most programs implementing JT65 have used the patented Koetter-Vard
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y (KV) algebraic soft-decision decoder, licensed to K1JT and implemented
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in a closed-source program for use in amateur radio applications.
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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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better than the KV decoder.
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\end_layout
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\begin_layout Section
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Introduction
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\end_layout
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\begin_layout Standard
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JT65 message frames consist of a short compressed message encoded for transmissi
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on with a Reed-Solomon code.
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Reed-Solomon codes are block codes characterized by
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\begin_inset Formula $n$
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\end_inset
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, the length of their codewords,
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\begin_inset Formula $k$
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\end_inset
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, the number of message symbols conveyed by the codeword, and the number
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of possible values for each symbol in the codewords.
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The codeword length and the number of message symbols are specified with
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the notation
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\begin_inset Formula $(n,k)$
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\end_inset
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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.
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Each of the 12 message symbols represents
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\begin_inset Formula $\log_{2}64=6$
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\end_inset
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message bits.
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The source-encoded messages conveyed by a 63-symbol JT65 frame thus consist
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of 72 information bits.
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The JT65 code is systematic, which means that the 12 message symbols are
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embedded in the codeword without modification and another 51 parity symbols
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derived from the message symbols are added to form a codeword of 63 symbols.
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\end_layout
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\begin_layout Standard
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The concept of Hamming distance is used as a measure of
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\begin_inset Quotes eld
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\end_inset
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distance
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\begin_inset Quotes erd
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\end_inset
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between different codewords, or between a received word and a codeword.
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Hamming distance is the number of code symbols that differ in two words
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being compared.
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Reed-Solomon codes have minimum Hamming distance
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\begin_inset Formula $d$
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\end_inset
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, where
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\begin_inset Formula
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\begin{equation}
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d=n-k+1.\label{eq:minimum_distance}
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\end{equation}
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\end_inset
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The minimum Hamming distance of the JT65 code is
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\begin_inset Formula $d=52$
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\end_inset
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, which means that any particular codeword differs from all other codewords
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in at least 52 symbol positions.
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\end_layout
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\begin_layout Standard
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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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symbols were received incorrectly, where
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\begin_inset Formula
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\begin{equation}
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t=\left\lfloor \frac{n-k}{2}\right\rfloor .\label{eq:t}
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\end{equation}
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\end_inset
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For the JT65 code
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\begin_inset Formula $t=25$
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\end_inset
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, so it is always possible to decode a received word having 25 or fewer
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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 necessarily involved in this process.
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We must (1) determine which symbols were received incorrectly, and (2)
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find the correct value of the incorrect symbols.
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If we somehow know that certain symbols are incorrect, that information
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can be used to reduce the work involved in step 1 and allow step 2 to correct
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more 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 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 $d-1=n-k$
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\end_inset
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errors.
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\end_layout
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\begin_layout Standard
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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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\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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\begin_inset Quotes eld
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\end_inset
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errors.
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\begin_inset Quotes erd
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\end_inset
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With perfect erasure information up to 51 incorrect symbols can be corrected
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for the JT65 code.
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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
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\begin_inset Formula $n-s$
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\end_inset
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symbols contain
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\begin_inset Formula $e$
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\end_inset
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errors, the BM algorithm can find the correct codeword as long as
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\begin_inset Formula
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\begin{equation}
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s+2e\le d-1.\label{eq:erasures_and_errors}
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\end{equation}
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\end_inset
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If
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\begin_inset Formula $s=0$
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\end_inset
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, the decoder is said to be an
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\begin_inset Quotes eld
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\end_inset
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errors-only
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\begin_inset Quotes erd
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\end_inset
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decoder.
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If
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\begin_inset Formula $0<s\le d-1$
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\end_inset
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, the decoder is called an
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\begin_inset Quotes eld
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\end_inset
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errors-and-erasures
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\begin_inset Quotes erd
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\end_inset
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decoder.
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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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On that foundation we have built a capability for using
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\begin_inset Quotes eld
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\end_inset
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soft
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\begin_inset Quotes erd
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\end_inset
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information on the reliability of individual symbols, thereby producing
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a soft-decision decoder.
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\end_layout
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\begin_layout Section
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\begin_inset CommandInset label
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LatexCommand label
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name "sec:You've-got-to"
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\end_inset
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Do I feel lucky?
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\end_layout
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\begin_layout Standard
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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, thus enabling decoding
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of received words with more than 25 errors.
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As a specific example, consider a received JT65 word with 23 correct symbols
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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 selects
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\begin_inset Formula $s=40$
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\end_inset
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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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LatexCommand ref
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reference "eq:erasures_and_errors"
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\end_inset
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), the BM decoder can successfully decode this word as long as
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\begin_inset Formula $e$
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\end_inset
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, the number of errors present in the 23 unerased symbols, is 5 or less.
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The number of errors captured in the set of 40 erased symbols must therefore
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be at least 35.
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\end_layout
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\begin_layout Standard
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The probability of selecting some particular number of incorrect symbols
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in a randomly selected subset of received symbols is governed by the hypergeome
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tric probability distribution.
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Let us define
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\begin_inset Formula $N$
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\end_inset
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as the number of symbols from which erasures will be selected,
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\begin_inset Formula $X$
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\end_inset
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as the number of incorrect symbols in the set of
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\begin_inset Formula $N$
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\end_inset
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symbols, and
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\begin_inset Formula $x$
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\end_inset
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as the number of errors in the symbols actually erased.
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In an ensemble of many received words,
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\begin_inset Formula $X$
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\end_inset
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and
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\begin_inset Formula $x$
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\end_inset
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will be random variables.
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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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with stated values of
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\begin_inset Formula $N$
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\end_inset
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,
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\begin_inset Formula $X$
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\end_inset
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, and
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\begin_inset Formula $s$
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\end_inset
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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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\begin_inset Formula
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\begin{equation}
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P(x=\epsilon|N,X,s)=\frac{\binom{X}{x}\binom{N-X}{s-\epsilon}}{\binom{N}{s}}\label{eq:hypergeometric_pdf}
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\end{equation}
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\end_inset
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where
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\begin_inset Formula $\binom{n}{k}=\frac{n!}{k!(n-k)!}$
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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 function
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\begin_inset Quotes eld
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\end_inset
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nchoosek(
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\begin_inset Formula $n,k$
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\end_inset
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)
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\begin_inset Quotes erd
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\end_inset
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in the interpreted language GNU Octave, or with one of many free online
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calculators.
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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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LatexCommand ref
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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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\begin_inset Quotes eld
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\end_inset
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hygepdf(
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\begin_inset Formula $x,N,X,s$
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\end_inset
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)
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\begin_inset Quotes erd
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\end_inset
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.
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The cumulative probability that at least
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\begin_inset Formula $\epsilon$
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\end_inset
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errors are captured in a subset of
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\begin_inset Formula $s$
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\end_inset
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erased symbols selected from a group of
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\begin_inset Formula $N$
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\end_inset
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symbols containing
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\begin_inset Formula $X$
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\end_inset
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errors is
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\begin_inset Formula
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\begin{equation}
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P(x\ge\epsilon|N,X,s)=\sum_{j=\epsilon}^{N}P(x=j|N,X,s).\label{eq:cumulative_prob}
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\end{equation}
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\end_inset
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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 Standard
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Suppose a received word contains
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\begin_inset Formula $X=40$
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\end_inset
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incorrect symbols.
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In an attempt to decode using an errors-and-erasures decoder,
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\begin_inset Formula $s=40$
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\end_inset
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symbols are randomly selected for erasure from the full set of
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\begin_inset Formula $N=n=63$
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\end_inset
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symbols.
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The probability that
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\begin_inset Formula $x=35$
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\end_inset
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of the erased symbols are actually incorrect is then
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\begin_inset Formula
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\[
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P(x=35)=\frac{\binom{40}{35}\binom{63-40}{40-35}}{\binom{63}{40}}\simeq2.4\times10^{-7}.
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\]
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\end_inset
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Similarly, the probability that
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\begin_inset Formula $x=36$
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\end_inset
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of the erased symbols are incorrect is
|
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\begin_inset Formula
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\[
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P(x=36)\simeq8.6\times10^{-9}.
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\]
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\end_inset
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Since the probability of erasing 36 errors is so much smaller than that
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for erasing 35 errors, we may safely conclude that the probability of randomly
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choosing an erasure vector that can decode the received word is approximately
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\begin_inset Formula $P(x=35)\simeq2.4\times10^{-7}$
|
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\end_inset
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|
.
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The odds of producing a valid codeword on the first try are very poor,
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about 1 in 4 million.
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\end_layout
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|
|
\begin_layout Paragraph
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Example 2:
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\end_layout
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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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|
, it turns out that for
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\begin_inset Formula $X=40$
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\end_inset
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the best strategy is to erase
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\begin_inset Formula $s=45$
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\end_inset
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symbols.
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Decoding will then be assured if the set of erased symbols contains at
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least 37 errors.
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With
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\begin_inset Formula $N=63$
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\end_inset
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|
,
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\begin_inset Formula $X=40$
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\end_inset
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, and
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\begin_inset Formula $s=45$
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\end_inset
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|
, the probability of successful decode in a single try is
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|
\begin_inset Formula
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|
\[
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P(x\ge37)\simeq1.9\times10^{-6}.
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\]
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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.
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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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|
|
\begin_layout Paragraph
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|
Example 3:
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\end_layout
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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 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 we know that 10 received symbols
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are significantly more reliable than the other 53.
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We might therefore protect the 10 most reliable symbols from erasure, selecting
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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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less reliable symbols.
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|
If
|
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\begin_inset Formula $s=45$
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\end_inset
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|
symbols are chosen randomly for erasure in this way, it is still necessary
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for the 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
|
|
|
|
,
|
|
\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
|
|
\begin_inset Formula $i=1,64$
|
|
\end_inset
|
|
|
|
is the frequency index and
|
|
\begin_inset Formula $j=1,63$
|
|
\end_inset
|
|
|
|
the symbol index.
|
|
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 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.
|
|
We found empirically that good decoding performance is obtained when the
|
|
symbol erasure probability is about 1.3 times the symbol error probability.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The FT algorithm tries successively to decode the received word using independen
|
|
t
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
educated guesses
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
to select symbols for erasure.
|
|
For each iteration a stochastic erasure vector is generated based on the
|
|
symbol erasure probabilities.
|
|
The erasure vector is sent to the BM decoder along with the full set of
|
|
63 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 power units, or
|
|
\begin_inset Formula $E_{s}/N_{0}\apprge6$
|
|
\end_inset
|
|
|
|
dB.
|
|
We also find that weaker signals frequently 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
|
|
\[
|
|
u=\frac{1}{n}\sum_{j=1}^{n}S(c_{j},\,j).
|
|
\]
|
|
|
|
\end_inset
|
|
|
|
Here the
|
|
\begin_inset Formula $c_{j}$
|
|
\end_inset
|
|
|
|
's are the symbol values for the candidate codeword being tested.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The correct JT65 codeword produces a value for
|
|
\begin_inset Formula $u$
|
|
\end_inset
|
|
|
|
equal to average of
|
|
\begin_inset Formula $n=63$
|
|
\end_inset
|
|
|
|
bins containing both signal and noise power.
|
|
Incorrect codewords have at most
|
|
\begin_inset Formula $k=12$
|
|
\end_inset
|
|
|
|
such bins and at least
|
|
\begin_inset Formula $n-k=51$
|
|
\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, the correct codeword is expected to yield the metric value
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula
|
|
\[
|
|
u=(1\pm n^{-\frac{1}{2}})(1+y)\approx(1.0\pm0.13)(1+y),
|
|
\]
|
|
|
|
\end_inset
|
|
|
|
where
|
|
\begin_inset Formula $y$
|
|
\end_inset
|
|
|
|
is the signal-to-noise ratio (in linear power units) and the quoted one-standar
|
|
d-deviation uncertainty range assumes Gaussian statistics.
|
|
Incorrect codewords will yield metric values no larger than
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula
|
|
\[
|
|
u=\frac{n-k\pm\sqrt{n-k}}{n}+\frac{k\pm\sqrt{k}}{n}(1+y).
|
|
\]
|
|
|
|
\end_inset
|
|
|
|
For JT65 this expression evaluates to
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula
|
|
\[
|
|
u\approx1\pm0.13+(0.19\pm0.06)\,y.
|
|
\]
|
|
|
|
\end_inset
|
|
|
|
As a specific example, consider signal strength
|
|
\begin_inset Formula $y=4$
|
|
\end_inset
|
|
|
|
, corresponding to
|
|
\begin_inset Formula $E_{s}/N_{0}=6$
|
|
\end_inset
|
|
|
|
dB.
|
|
For JT65, the corresponding SNR in 2500 Hz bandwidth is
|
|
\begin_inset Formula $-23.7$
|
|
\end_inset
|
|
|
|
dB.
|
|
The correct codeword is then expected to yield
|
|
\begin_inset Formula $u\approx5.0\pm$
|
|
\end_inset
|
|
|
|
0.6, while incorrect codewords will give
|
|
\begin_inset Formula $u\approx2.0\pm0.3$
|
|
\end_inset
|
|
|
|
or less.
|
|
We find that a threshold set at
|
|
\begin_inset Formula $u_{0}=4.4$
|
|
\end_inset
|
|
|
|
(about 8 standard deviations above the expected maximum for incorrect codewords
|
|
) reliably serves to distinguish correct codewords from all other candidates,
|
|
while ensuring a very small probability 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
|
|
\begin_inset Formula $u>u_{0}$
|
|
\end_inset
|
|
|
|
.
|
|
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.
|
|
\end_layout
|
|
|
|
\begin_layout Paragraph
|
|
Algorithm pseudo-code:
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
For each received symbol, define the erasure probability as 1.3 times the
|
|
|
|
\emph on
|
|
a priori
|
|
\emph default
|
|
symbol-error probability determined from soft-symbol information
|
|
\begin_inset Formula $\{p_{1}\textrm{-rank},\,p_{2}/p_{1}\}$
|
|
\end_inset
|
|
|
|
.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Make independent stochastic decisions about whether to erase each symbol
|
|
by using the symbol's erasure probability, allowing a maximum of 51 erasures.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Attempt errors-and-erasures decoding by using the BM algorithm and the set
|
|
of 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 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, set
|
|
\begin_inset Formula $u_{max}=u$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
If
|
|
\begin_inset Formula $u_{max}>u_{0}$
|
|
\end_inset
|
|
|
|
, go to step 8.
|
|
|
|
\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.
|
|
Otherwise, declare decoding failure and exit.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
An acceptable codeword with
|
|
\begin_inset Formula $u_{max}>u_{0}$
|
|
\end_inset
|
|
|
|
has been found.
|
|
Declare a successful decode and return this codeword .
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
Theory and Simulations
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The fraction of time that
|
|
\begin_inset Formula $X$
|
|
\end_inset
|
|
|
|
, the number of symbols received incorrectly, is expected to be less than
|
|
some number
|
|
\begin_inset Formula $D$
|
|
\end_inset
|
|
|
|
depends on signal-to-noise ratio.
|
|
For the case of additive white Gaussian noise (AWGN) and noncoherent 64-FSK
|
|
demodulation this probability is easy to calculate.
|
|
Representative examples for
|
|
\begin_inset Formula $D=25,$
|
|
\end_inset
|
|
|
|
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
\bar no
|
|
\strikeout off
|
|
\uuline off
|
|
\uwave off
|
|
\noun off
|
|
\color none
|
|
|
|
\begin_inset Formula $D=40$
|
|
\end_inset
|
|
|
|
|
|
\family default
|
|
\series default
|
|
\shape default
|
|
\size default
|
|
\emph default
|
|
\bar default
|
|
\strikeout default
|
|
\uuline default
|
|
\uwave default
|
|
\noun default
|
|
\color inherit
|
|
, and
|
|
\begin_inset Formula $D=43$
|
|
\end_inset
|
|
|
|
are plotted in Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:bodide"
|
|
|
|
\end_inset
|
|
|
|
for a range of SNRs as filled squares with connecting lines.
|
|
The rightmost such curve shows that on the AWGN channel the hard-decision
|
|
BM decoder should succeed about 90% of the time at
|
|
\begin_inset Formula $E_{s}/N_{0}=7.5$
|
|
\end_inset
|
|
|
|
dB, 99% of the time at 8 dB, and 99.98% at 8.5 dB.
|
|
For comparison, the righmost curve with open squares shows that simulated
|
|
results agree with theory to within less than 0.2 dB.
|
|
|
|
\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 $E_{s}/N_{0},$
|
|
\end_inset
|
|
|
|
the signal-to-noise ratio in bandwidth equal to the symbol rate.
|
|
Filled squares illustrate theoretical values for
|
|
\begin_inset Formula $D=25,$
|
|
\end_inset
|
|
|
|
|
|
\family roman
|
|
\series medium
|
|
\shape up
|
|
\size normal
|
|
\emph off
|
|
\bar no
|
|
\strikeout off
|
|
\uuline off
|
|
\uwave off
|
|
\noun off
|
|
\color none
|
|
|
|
\begin_inset Formula $D=40$
|
|
\end_inset
|
|
|
|
|
|
\family default
|
|
\series default
|
|
\shape default
|
|
\size default
|
|
\emph default
|
|
\bar default
|
|
\strikeout default
|
|
\uuline default
|
|
\uwave default
|
|
\noun default
|
|
\color inherit
|
|
, and
|
|
\begin_inset Formula $D=43$
|
|
\end_inset
|
|
|
|
.
|
|
Open squares illustrate measured results for the BM and FT (
|
|
\begin_inset Formula $T=10^{5}$
|
|
\end_inset
|
|
|
|
) decoders in program
|
|
\emph on
|
|
WSJT-X
|
|
\emph default
|
|
.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Received JT65 words with more than 25 incorrect symbols can be decoded if
|
|
sufficient information on individual symbol reliabilities is available.
|
|
Using values of
|
|
\begin_inset Formula $T$
|
|
\end_inset
|
|
|
|
that are practical with today's personal computers and the soft-symbol
|
|
information described above, we find that the FT algorithm nearly always
|
|
produces correct decodes up to
|
|
\begin_inset Formula $X=40$
|
|
\end_inset
|
|
|
|
, and some additional decodes are found in the range 41 to 43.
|
|
As an example, Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:N_vs_X"
|
|
|
|
\end_inset
|
|
|
|
plots the number of stochastic erasure trials required to find the correct
|
|
codeword versus the number of hard-decision errors for a run with 1000
|
|
simulated transmissions at
|
|
\begin_inset Formula $SNR=-24$
|
|
\end_inset
|
|
|
|
dB, just slightly above the decoding threshold.
|
|
Note that both mean and variance of the required number of trials increase
|
|
steeply with the number of errors in the received word.
|
|
Execution time of the FT algorithm 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
|
|
scale 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
|
|
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 (
|
|
\begin_inset Formula $E_{s}/N_{o}=5.7$
|
|
\end_inset
|
|
|
|
dB).
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
Comparison with Berlekamp-Massey and Koetter-Vardy
|
|
\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 signal-to-noise ratio per information bit.
|
|
Results of simulations using the Berlekamp-Massey, Koetter-Vardy, and Franke-Ta
|
|
ylor decoding algorithms on the (63,12) code are presented in this way in
|
|
Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:WER"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
For these tests we generated 1000 signals at each signal-to-noise ratio,
|
|
assuming the additive white gaussian noise (AWGN) channel, and processed
|
|
the data using each algorithm.
|
|
As expected, the soft-decision algorithms FT and KV are about 2 dB better
|
|
than the hard-decision BM algorithm.
|
|
FT has a slight edge (about 0.2 dB) over KV with the default settings for
|
|
each algorithm, as implemented in our JT65 decoders.
|
|
\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_wer.pdf
|
|
lyxscale 120
|
|
scale 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:WER"
|
|
|
|
\end_inset
|
|
|
|
Word error rate (WER) as a function of
|
|
\begin_inset Formula $E_{b}/N_{0}$
|
|
\end_inset
|
|
|
|
for non-fading signals in AWGN.
|
|
|
|
\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:WER"
|
|
|
|
\end_inset
|
|
|
|
often extend downward to much 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 ever 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.
|
|
|
|
\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.
|
|
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.
|
|
\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
|
|
scale 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
|
|
|
|
) matched-filter algorithm.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
Hinted Decoding
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
...
|
|
Still to come ...
|
|
\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
|
|
key "key-1"
|
|
|
|
\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
|
|
key "key-2"
|
|
|
|
\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
|
|
key "key-3"
|
|
|
|
\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
|
|
key "key-4"
|
|
|
|
\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
|
|
key "key-5"
|
|
|
|
\end_inset
|
|
|
|
Berlekamp-Massey decoder written by Phil Karn, http://www.ka9q.net/code/fec/
|
|
\end_layout
|
|
|
|
\end_body
|
|
\end_document
|