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1295 lines
27 KiB
Plaintext
1295 lines
27 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 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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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 JT65 decoders have used the patented Koetter-Vardy (KV)
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algebraic soft-decision decoder, licensed to K1JT and implemented in a
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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 performs at least as well as the KV
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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 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 the 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 efficiently decode a received word having
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no 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 necessarily involved in this process, namely
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\end_layout
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\begin_layout Enumerate
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Determine which symbols were received incorrectly.
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\end_layout
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\begin_layout Enumerate
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Find 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 we somehow know that certain symbols are incorrect, this 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 $d$
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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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As already noted, with perfect erasure information up to 51 incorrect symbols
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can be corrected.
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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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(
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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 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 symbol reliability, thereby producing 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 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 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 erased symbols.
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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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given 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.
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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
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\emph on
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at least
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\emph default
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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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|
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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 codeword 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 the probabili
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ty of erasing 35 errors, we may safely conclude that the probability of
|
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randomly choosing an erasure vector that can decode the received word is
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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 successfully decoding the word on the first try are very poor,
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about 1 in 4 million.
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\end_layout
|
|
|
|
\begin_layout Paragraph
|
|
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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|
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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, and with
|
|
\begin_inset Formula $N=63$
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|
\end_inset
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|
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|
,
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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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|
, 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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|
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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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|
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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
|
|
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
|
|
are significantly more reliable than the other 53.
|
|
We might therefore protect the 10 most reliable symbols from erasure, selecting
|
|
erasures from the smaller set of
|
|
\begin_inset Formula $N=53$
|
|
\end_inset
|
|
|
|
less reliable symbols.
|
|
If
|
|
\begin_inset Formula $s=45$
|
|
\end_inset
|
|
|
|
symbols are chosen randomly for erasure in this way, it is still necessary
|
|
for the erased symbols to include at least 37 errors, as in Example 2.
|
|
However, the probabilities are now much more favorable: with
|
|
\begin_inset Formula $N=53$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $X=40$
|
|
\end_inset
|
|
|
|
, and
|
|
\begin_inset Formula $s=45$
|
|
\end_inset
|
|
|
|
, Eq.
|
|
(
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:hypergeometric_pdf"
|
|
|
|
\end_inset
|
|
|
|
) yields
|
|
\begin_inset Formula $P(x\ge37)=0.016$
|
|
\end_inset
|
|
|
|
.
|
|
Even better odds are obtained by choosing
|
|
\begin_inset Formula $s=47$
|
|
\end_inset
|
|
|
|
, which requires
|
|
\begin_inset Formula $x\ge38$
|
|
\end_inset
|
|
|
|
.
|
|
With
|
|
\begin_inset Formula $N=53$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $X=40$
|
|
\end_inset
|
|
|
|
, and
|
|
\begin_inset Formula $s=47$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $P(x\ge38)=0.027$
|
|
\end_inset
|
|
|
|
.
|
|
The odds for successful decoding on the first try are now about 1 in 38.
|
|
A few hundred independently randomized tries would be enough to all-but-guarant
|
|
ee production of a valid codeword by the BM decoder.
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "sec:The-decoding-algorithm"
|
|
|
|
\end_inset
|
|
|
|
The Franke-Taylor decoding algorithm
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Example 3 shows how reliable information about symbol quality should make
|
|
it possible to decode received frames having a large number of errors.
|
|
In practice the number of errors in the received word is unknown, so we
|
|
use a stochastic algorithm to assign high erasure probability to low-quality
|
|
symbols and relatively low probability to high-quality symbols.
|
|
As illustrated by Example 3, a good choice of erasure probabilities can
|
|
increase the chance of a successful decode by many orders of magnitude.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The FT algorithm uses two quality indices made available by a noncoherent
|
|
64-FSK demodulator.
|
|
The demodulator identifies the most likely value for each symbol based
|
|
on the largest signal-plus-noise power in 64 frequency bins.
|
|
The fractions of total power in the two bins containing the largest and
|
|
second-largest powers (denoted by
|
|
\begin_inset Formula $p_{1}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $p_{2}$
|
|
\end_inset
|
|
|
|
, respectively) are passed to the decoder from the demodulator as
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
soft-symbol
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
information.
|
|
The decoder derives two metrics from
|
|
\begin_inset Formula $\{p_{1},p_{2}\}:$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
\begin_inset Formula $p_{1}$
|
|
\end_inset
|
|
|
|
-rank: the rank
|
|
\begin_inset Formula $\{1,2,\ldots,63\}$
|
|
\end_inset
|
|
|
|
of the symbol's fractional power,
|
|
\begin_inset Formula $p_{1}$
|
|
\end_inset
|
|
|
|
in the sorted list of
|
|
\begin_inset Formula $p_{1}$
|
|
\end_inset
|
|
|
|
values.
|
|
High ranking symbols have larger signal-to-noise ratio than those with
|
|
lower rank.
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
\begin_inset Formula $p_{2}/p_{1}$
|
|
\end_inset
|
|
|
|
: when
|
|
\begin_inset Formula $p_{2}/p_{1}$
|
|
\end_inset
|
|
|
|
is not small compared to 1, the most likely symbol value is only slightly
|
|
more reliable than the second most likely one.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The FT decoder uses a table of symbol error probabilities derived from a
|
|
large dataset of received words that have been successfully decoded.
|
|
The table provides an estimate of the
|
|
\emph on
|
|
a-priori
|
|
\emph default
|
|
probability of symbol error based on a given symbol's
|
|
\begin_inset Formula $p_{1}$
|
|
\end_inset
|
|
|
|
-rank and
|
|
\begin_inset Formula $p_{2}/p_{1}$
|
|
\end_inset
|
|
|
|
metrics.
|
|
These probabilities are close to 1 for low-quality symbols and close to
|
|
0 for high-quality symbols.
|
|
Recall from Examples 2 and 3 that best performance was obtained with
|
|
\begin_inset Formula $s>X$
|
|
\end_inset
|
|
|
|
.
|
|
Correspondingly, the FT algorithm works best when the probability of erasing
|
|
a symbol is somewhat larger than the probability that the symbol is incorrect.
|
|
We found empirically that good decoding performance is obtained when the
|
|
symbol erasure probability is about 1.3 times the symbol error probability.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The FT algorithm tries successively to decode the received word using independen
|
|
t
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
educated guesses
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
to select symbols for erasure.
|
|
For each iteration a stochastic erasure vector is generated based on the
|
|
symbol erasure probabilities.
|
|
The erasure vector is sent to the BM decoder along with the full set of
|
|
63 received symbols.
|
|
When the BM decoder finds a candidate codeword it is assigned a quality
|
|
metric
|
|
\begin_inset Formula $d_{s}$
|
|
\end_inset
|
|
|
|
defined as the soft distance between the received word and the codeword,
|
|
where
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
d_{s}=\sum_{i=1}^{n}\alpha_{i}\,(1+p_{1,i}).\label{eq:soft_distance}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
Here
|
|
\begin_inset Formula $\alpha_{i}=0$
|
|
\end_inset
|
|
|
|
if received symbol
|
|
\begin_inset Formula $i$
|
|
\end_inset
|
|
|
|
is the same as the corresponding symbol in the codeword,
|
|
\begin_inset Formula $\alpha_{i}=1$
|
|
\end_inset
|
|
|
|
if the received symbol and codeword symbol are different, and
|
|
\begin_inset Formula $p_{1,i}$
|
|
\end_inset
|
|
|
|
is the fractional power associated with received symbol
|
|
\begin_inset Formula $i$
|
|
\end_inset
|
|
|
|
.
|
|
Think of the soft distance as made up of two terms: the first is the Hamming
|
|
distance between the received word and the codeword, and the second ensures
|
|
that if two candidate codewords have the same Hamming distance from the
|
|
received word, a smaller soft distance will be assigned to the one where
|
|
differences occur in symbols of lower estimated reliability.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Technically the FT algorithm is a list decoder, potentially generating a
|
|
list of candidate codewords.
|
|
Among the list of candidate codewords found by the stochastic search algorithm,
|
|
only the one with the smallest soft distance from the received word is
|
|
retained.
|
|
As with all such algorithms, a stopping criterion is necessary.
|
|
FT accepts a codeword unconditionally if its soft distance is smaller than
|
|
an empirically determined acceptance threshold,
|
|
\begin_inset Formula $d_{a}$
|
|
\end_inset
|
|
|
|
.
|
|
A timeout is used to limit the algorithm's execution time if no codewords
|
|
within soft distance
|
|
\begin_inset Formula $d_{a}$
|
|
\end_inset
|
|
|
|
of the received word are found in a reasonable number of trials.
|
|
\end_layout
|
|
|
|
\begin_layout Paragraph
|
|
Algorithm pseudo-code:
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
For each received symbol, define the erasure probability as 1.3 times the
|
|
|
|
\emph on
|
|
a priori
|
|
\emph default
|
|
symbol-error probability determined from soft-symbol information
|
|
\begin_inset Formula $\{p_{1}\textrm{-rank},\, p_{2}/p_{1}\}$
|
|
\end_inset
|
|
|
|
.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Make independent stochastic decisions about whether to erase each symbol
|
|
by using the symbol's erasure probability, allowing a maximum of 51 erasures.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Attempt errors-and-erasures decoding by using the BM algorithm and the set
|
|
of erasures determined in step 2.
|
|
If the BM decoder is successful go to step 5.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
If decoding is not successful, go to step 2.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Calculate the soft distance
|
|
\begin_inset Formula $d_{s}$
|
|
\end_inset
|
|
|
|
between the candidate codeword and the received symbols.
|
|
Set
|
|
\begin_inset Formula $d_{s,min}=d_{s}$
|
|
\end_inset
|
|
|
|
if the soft distance is the smallest one encountered so far.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
If
|
|
\begin_inset Formula $d_{s,min}\le d_{a}$
|
|
\end_inset
|
|
|
|
, go to 8.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
If the number of trials is less than the maximum allowed number, go to 2.
|
|
Otherwise, declare decoding failure and exit.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
A
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
best
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
codeword with
|
|
\begin_inset Formula $d_{s,min}\le d_{a}$
|
|
\end_inset
|
|
|
|
has been found.
|
|
Declare a successful decode and return this codeword .
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
Results and Comparison with KVASD
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Possible figures:
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
histogram of
|
|
\begin_inset Formula $s$
|
|
\end_inset
|
|
|
|
(number of erasures) for successful decodes with HF and EME data
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
histogram of
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
ntrials
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
(or execution time)
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
Number of decodes vs.
|
|
ntrials
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
Probability of successful decode vs.
|
|
Es/No or S/N in 2500 Hz BW
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename fig_psuccess.pdf
|
|
lyxscale 120
|
|
scale 120
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "fig:Psuccess"
|
|
|
|
\end_inset
|
|
|
|
Percentage of JT65 messages (
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
words
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
) successfully decoded as a function of SNR in 2.5 kHz bandwidth.
|
|
These results are for the idealized situation of a non-fading signal in
|
|
additive white Gaussian noise (AWGN).
|
|
Results are shown for the hard-decision Berlekamp-Massey (BM) and soft-decision
|
|
Franke-Taylor (FT) decoding algorithms.
|
|
Results for the FT algorithm are shown for two different sets of time-out
|
|
and acceptance criteria.
|
|
FT-1 was obtained with a limit of
|
|
\begin_inset Formula $10^{4}$
|
|
\end_inset
|
|
|
|
erasure vectors and with acceptance criteria
|
|
\begin_inset Formula $d_{a}<72$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $n_{hard}<42$
|
|
\end_inset
|
|
|
|
, and FT-2 corresponds to
|
|
\begin_inset Formula $10^{5}$
|
|
\end_inset
|
|
|
|
erasure vectors and
|
|
\begin_inset Formula $d_{a}<76$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $n_{hard}<44$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\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 $E_{s}/N_{o}$
|
|
\end_inset
|
|
|
|
for non-fading signals in AWGN.
|
|
This is the same data that is shown in Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:Psuccess"
|
|
|
|
\end_inset
|
|
|
|
, but plotted as the fraction of frames that were not decoded vs symbol
|
|
SNR.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
wide false
|
|
sideways false
|
|
status collapsed
|
|
|
|
\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
|
|
The number of trials needed to decode a received word vs the Hamming distance
|
|
between the received word and the decoded codeword plotted for 1000 simulated
|
|
frames with no fading.
|
|
The SNR in 2500 Hz bandwidth is -24 dB (
|
|
\begin_inset Formula $E_{s}/N_{o}=5.7$
|
|
\end_inset
|
|
|
|
dB).
|
|
Execution time will be roughly proportional to the number of trials.
|
|
The mean and variance of the number of trials (and execution time) increase
|
|
with the number of errors in the received word.
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename fig_wer2.pdf
|
|
lyxscale 120
|
|
scale 120
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
Word error rate (WER) as a function of
|
|
\begin_inset Formula $E_{s}/N_{o}$
|
|
\end_inset
|
|
|
|
for Rayleigh-fading with Doppler-spread
|
|
\begin_inset Formula $\sigma_{f}=0.2$
|
|
\end_inset
|
|
|
|
Hz.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Itemize
|
|
other...
|
|
?
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
Summary
|
|
\end_layout
|
|
|
|
\begin_layout Bibliography
|
|
\begin_inset CommandInset bibitem
|
|
LatexCommand bibitem
|
|
key "key-1"
|
|
|
|
\end_inset
|
|
|
|
"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
|
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key "key-3"
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\end_inset
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\begin_inset Quotes erd
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\end_inset
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Stochastic Erasure-Only List Decoding Algorithms for Reed-Solomon Codes,
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\begin_inset Quotes erd
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\end_inset
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Chang-Ming Lee and Yu T.
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Su, IEEE Signal Processing Letters, Vol.
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16, No.
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8, August 2009.
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\end_layout
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\begin_layout Bibliography
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\begin_inset CommandInset bibitem
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LatexCommand bibitem
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key "key-4"
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\end_inset
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“Algebraic soft-decision decoding of Reed-Solomon codes,” R.
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Köetter and A.
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Vardy, IEEE Trans.
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Inform.
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Theory, Vol.
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49, Nov.
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2003.
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\end_layout
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\begin_layout Bibliography
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\begin_inset CommandInset bibitem
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LatexCommand bibitem
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key "key-5"
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\end_inset
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Berlekamp-Massey decoder written by Phil Karn, http://www.ka9q.net/code/fec/
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\end_layout
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\end_body
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\end_document
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