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#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\lyxformat 474
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\begin_document
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\begin_header
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\textclass IEEEtran
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\use_default_options true
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\maintain_unincluded_children false
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\language english
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\use_package mathdots 1
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\use_package mathtools 1
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\suppress_date false
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\justification true
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\use_refstyle 1
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||||
\index Index
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||||
\shortcut idx
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||||
\color #008000
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||||
\end_index
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||||
\secnumdepth 3
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||||
\tocdepth 3
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\paragraph_separation indent
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\quotes_language english
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\html_be_strict false
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\end_header
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||||
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||||
\begin_body
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||||
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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 that could
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not be utilized using traditional technologies.
|
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One reason for the success and popularity of the JT65 mode is its use of
|
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strong error-correction coding.
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The JT65 code is a short block-length, low-rate, Reed-Solomon code based
|
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on a 64-symbol alphabet.
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Since 200?, decoders for the JT65 code have used the
|
||||
\begin_inset Quotes eld
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||||
\end_inset
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||||
|
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Koetter-Vardy
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||||
\begin_inset Quotes erd
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\end_inset
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|
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(KV) algebraic soft-decision decoder.
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The KV decoder is implemented in a closed-source program that is licensed
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to K1JT for use in amateur applications.
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This note describes a new open-source alternative to the KV decoder called
|
||||
the SFRSD decoder.
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The SFRSD decoding algorithm is shown to perform at least as well as the
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KV decoder.
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The SFRSD algorithm is conceptually simple and is built around the well-known
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Berlekamp-Massey errors-and-erasures decoder.
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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 that is encoded
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for transmission using a Reed-Solomon code.
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Reed-Solomon codes are block codes and, like all block codes, are characterized
|
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by the length of their codewords,
|
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\begin_inset Formula $n$
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\end_inset
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, the number of message symbols conveyed by the codeword,
|
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\begin_inset Formula $k$
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\end_inset
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|
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, and the number of possible values for each symbol in the codewords.
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The codeword length and the number of message symbols are specified as
|
||||
a tuple in the form
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\begin_inset Formula $(n,k)$
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\end_inset
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|
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.
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JT65 uses a (63,12) Reed-Solomon code with 64 possible values for each
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symbol, so each symbol represents
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\begin_inset Formula $\log_{2}64=6$
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\end_inset
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||||
|
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message bits.
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The source-encoded messages conveyed by a 63-symbol JT65 frame 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 the codeword consisting
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of 63 total symbols.
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\end_layout
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|
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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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|
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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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that are 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 positions.
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\end_layout
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\begin_layout Standard
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||||
Given only a received word containing some incorrect symbols (errors), the
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||||
received word can be decoded into the correct codeword using a deterministic,
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||||
algebraic, algorithm provided that no more than
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\begin_inset Formula $t$
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||||
\end_inset
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|
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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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, which means that it is always possible to efficiently decode a received
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word that contains no more than 25 symbol errors.
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\end_layout
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|
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\begin_layout Standard
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||||
There are a number of well-known algebraic algorithms that can carry out
|
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the process of decoding a received codeword that contains no more than
|
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|
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\begin_inset Formula $t$
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\end_inset
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errors.
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One such algorithm is the Berlekamp-Massey (BM) decoding algorithm.
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\end_layout
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\begin_layout Standard
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A decoder, such as BM, must carry out two tasks:
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\end_layout
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\begin_layout Enumerate
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figure out which symbols were received incorrectly
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\end_layout
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\begin_layout Enumerate
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figure out the correct value of the incorrect symbols
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\end_layout
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\begin_layout Standard
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If it is somehow known that certain symbols are incorrect, such information
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can be used in the decoding algorithm to reduce the amount of work required
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in step 1 and to allow step 2 to correct 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 fact, in the unlikely event that the location of each and every error
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is known and is provided to the BM decoder, and if no correct symbols are
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accidentally labeled as errors, then the BM decoder 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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In the decoding algorithm described herein, a list of symbols that are known
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or suspected to be incorrect is sent to the BM decoder.
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Symbols in the received word that are flagged as being incorrect 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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.
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Symbols that are not erased and that are incorrect 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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.
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The BM decoder accepts erasure information in the form of a list of indices
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corresponding to the incorrect, or suspected incorrect, symbols in the
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received word.
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As already noted, if the erasure information is perfect, then up to 51
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errors will be corrected.
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When the erasure information is imperfect, then some of the erased symbols
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will actually be correct, and some of the unerased symbols will be in error.
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If a total of
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\begin_inset Formula $n_{era}$
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\end_inset
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symbols are erased and the remaining unerased symbols contain
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\begin_inset Formula $n_{err}$
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\end_inset
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errors, then 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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n_{era}+2n_{err}\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 $n_{era}=0$
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\end_inset
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, then 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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||||
|
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decoder and it can correct up to
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\begin_inset Formula $t$
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\end_inset
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errors (
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\begin_inset Formula $t$
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\end_inset
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=25 for JT65).
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If
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\begin_inset Formula $0<n_{era}\le d-1$
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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), then 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-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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|
||||
\end_layout
|
||||
|
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\begin_layout Standard
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For the JT65 code, (
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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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) says that if
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\begin_inset Formula $n_{era}$
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\end_inset
|
||||
|
||||
symbols are declared to be erased, then the BM decoder will find the correct
|
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codeword as long as the remaining un-erased symbols contain no more than
|
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|
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\begin_inset Formula $\left\lfloor \frac{51-n_{era}}{2}\right\rfloor $
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\end_inset
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errors.
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The errors-and-erasures capability of the BM decoder is a very powerful
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feature that serves as the core of the new soft-decision decoder described
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herein.
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||||
\end_layout
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||||
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\begin_layout Standard
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It will be helpful to have some understanding of the errors and erasures
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tradeoff described by (
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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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) to appreciate how the new decoder algorithm works.
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Section
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "sec:Errors-and-erasures-decoding-exa"
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\end_inset
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describes some examples that should illustrate how the errors-and-erasures
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capability can be combined with some information about the quality of the
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received symbols to enable development of a decoding algorithm that can
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reliably decode received words that contain many more than 25 errors.
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Section describes the SFRSD decoding algorithm.
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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:Errors-and-erasures-decoding-exa"
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\end_inset
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You've got to ask yourself.
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Do I feel lucky?
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\end_layout
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||||
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\begin_layout Standard
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||||
Consider a particular received codeword that contains 40 incorrect symbols
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and 23 correct symbols.
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It is not known which 40 symbols are in error.
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\begin_inset Foot
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status open
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\begin_layout Plain Layout
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In practice the number of errors will not be known either, but this is not
|
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a serious problem.
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||||
\end_layout
|
||||
|
||||
\end_inset
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||||
|
||||
Suppose that the decoder randomly chooses 40 symbols to erase (
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\begin_inset Formula $n_{era}=40$
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\end_inset
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||||
|
||||
), leaving 23 unerased symbols.
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||||
According to (
|
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\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "eq:erasures_and_errors"
|
||||
|
||||
\end_inset
|
||||
|
||||
), the BM decoder can successfully decode this word as long as the number
|
||||
of errors present in the 23 unerased symbols is 5 or less.
|
||||
This means that the number of errors captured in the set of 40 erased symbols
|
||||
must be at least 35.
|
||||
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||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
The probability of selecting some particular number of bad symbols in a
|
||||
randomly selected subset of the codeword symbols is governed by the hypergeomet
|
||||
ric probability distribution.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Define:
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||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
\begin_inset Formula $N$
|
||||
\end_inset
|
||||
|
||||
= number of symbols in a codeword (63 for JT65),
|
||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
\begin_inset Formula $K$
|
||||
\end_inset
|
||||
|
||||
= number of incorrect symbols in a codeword,
|
||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
\begin_inset Formula $n$
|
||||
\end_inset
|
||||
|
||||
= number of symbols erased for errors-and-erasures decoding,
|
||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
\begin_inset Formula $k$
|
||||
\end_inset
|
||||
|
||||
= number of incorrect symbols in the set of erased symbols.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Let
|
||||
\begin_inset Formula $X$
|
||||
\end_inset
|
||||
|
||||
be the number of incorrect symbols in a set of
|
||||
\begin_inset Formula $n$
|
||||
\end_inset
|
||||
|
||||
symbols chosen for erasure.
|
||||
Then
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
P(X=k)=\frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}\label{eq:hypergeometric_pdf-1}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
where
|
||||
\begin_inset Formula $\binom{n}{m}=\frac{n!}{m!(n-m)!}$
|
||||
\end_inset
|
||||
|
||||
is the binomial coefficient.
|
||||
The binomial coefficient can be calculated using the
|
||||
\begin_inset Quotes eld
|
||||
\end_inset
|
||||
|
||||
nchoosek(n,k)
|
||||
\begin_inset Quotes erd
|
||||
\end_inset
|
||||
|
||||
function in Gnu Octave.
|
||||
The hypergeometric probability mass function is available in Gnu Octave
|
||||
as function
|
||||
\begin_inset Quotes eld
|
||||
\end_inset
|
||||
|
||||
hygepdf(k,N,K,n)
|
||||
\begin_inset Quotes erd
|
||||
\end_inset
|
||||
|
||||
.
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Case
|
||||
A codeword contains
|
||||
\begin_inset Formula $K=40$
|
||||
\end_inset
|
||||
|
||||
incorrect symbols.
|
||||
In an attempt to decode using an errors-and-erasures decoder,
|
||||
\begin_inset Formula $n=40$
|
||||
\end_inset
|
||||
|
||||
symbols are randomly selected for erasure.
|
||||
The probability that
|
||||
\begin_inset Formula $35$
|
||||
\end_inset
|
||||
|
||||
of the erased symbols are incorrect is:
|
||||
\begin_inset Formula
|
||||
\[
|
||||
P(X=35)=\frac{\binom{40}{35}\binom{63-40}{40-35}}{\binom{63}{40}}=2.356\times10^{-7}.
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
Similarly:
|
||||
\begin_inset Formula
|
||||
\[
|
||||
P(X=36)=8.610\times10^{-9}.
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
Since the probability of catching 36 errors is so much smaller than the
|
||||
probability of catching 35 errors, it is safe to say that the probability
|
||||
of randomly selecting an erasure vector that can decode the received word
|
||||
is essentially equal to
|
||||
\begin_inset Formula $P(X=35)\simeq2.4\times10^{-7}$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
The odds of successfully decoding the word on the first try are about 1
|
||||
in 4 million.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Case
|
||||
A codeword contains
|
||||
\begin_inset Formula $K=40$
|
||||
\end_inset
|
||||
|
||||
incorrect symbols.
|
||||
It is interesting to work out the best choice for the number of symbols
|
||||
that should be selected at random for erasure if the goal is to maximize
|
||||
the probability of successfully decoding the word.
|
||||
By exhaustive search, it turns out that the best case is to erase
|
||||
\begin_inset Formula $n=45$
|
||||
\end_inset
|
||||
|
||||
symbols, in which case the word will be decoded if the set of erased symbols
|
||||
contains at least 37 errors.
|
||||
With
|
||||
\begin_inset Formula $N=63$
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\begin_inset Formula $K=40$
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\begin_inset Formula $n=45$
|
||||
\end_inset
|
||||
|
||||
, then
|
||||
\begin_inset Formula
|
||||
\[
|
||||
P(X\ge37)\simeq2\times10^{-6}.
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
This probability is about 8 times higher than the probability of success
|
||||
when only
|
||||
\begin_inset Formula $40$
|
||||
\end_inset
|
||||
|
||||
symbols were erased, and the odds of successfully decoding on the first
|
||||
try are roughly 1 in 500,000.
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Case
|
||||
Cases 1 and 2 illustrate the fact that a strategy that tries to guess which
|
||||
symbols to erase is not going to be very successful unless we are prepared
|
||||
to wait all day for an answer.
|
||||
Consider a slight modification to the strategy that can tip the odds in
|
||||
our favor.
|
||||
Suppose that the codeword contains
|
||||
\begin_inset Formula $K=40$
|
||||
\end_inset
|
||||
|
||||
incorrect symbols, as before.
|
||||
In this case it is known that 10 of the symbols are much more reliable
|
||||
than the other 53 symbols.
|
||||
The 10 most reliable symbols are all correct and these 10 symbols are protected
|
||||
from erasure, i.e.
|
||||
the set of erasures is chosen from the smaller set of 53 less reliable
|
||||
symbols.
|
||||
If
|
||||
\begin_inset Formula $n=40$
|
||||
\end_inset
|
||||
|
||||
symbols are chosen randomly from the set of
|
||||
\begin_inset Formula $N=53$
|
||||
\end_inset
|
||||
|
||||
least reliable symbols, it is still necessary for the erased symbols to
|
||||
include at least 35 errors (as in Case 1).
|
||||
In this case, with
|
||||
\begin_inset Formula $N=53$
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\begin_inset Formula $K=40$
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\begin_inset Formula $n=35$
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\begin_inset Formula $P(X=35)=0.001$
|
||||
\end_inset
|
||||
|
||||
! Now, the situation is much better.
|
||||
The odds of decoding the word on the first try are approximately 1 in 1000.
|
||||
The odds are even better if 41 symbols are erased, in which case
|
||||
\begin_inset Formula $P(X=35)=0.0042$
|
||||
\end_inset
|
||||
|
||||
, giving odds of about 1 in 200!
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Case 3 illustrates how, with the addition of some reliable information about
|
||||
the quality of just 10 of the 63 symbols, it is possible to decode received
|
||||
words containing a relatively large number of errors using only the BM
|
||||
errors-and-erasures decoder.
|
||||
The key to improving the odds enough to make the strategy of
|
||||
\begin_inset Quotes eld
|
||||
\end_inset
|
||||
|
||||
guessing
|
||||
\begin_inset Quotes erd
|
||||
\end_inset
|
||||
|
||||
at the erasure vector useful for practical implementation is to use information
|
||||
about the quality of the received symbols to decide which ones are most
|
||||
likely to be in error, and to assign a relatively high probability of erasure
|
||||
to the lowest quality symbols and a relatively low probability of erasure
|
||||
to the highest quality symbols.
|
||||
It turns out that a good choice of the erasure probabilities can increase
|
||||
the probability of a successful decode by several orders of magnitude relative
|
||||
to a bad choice.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Rather than selecting a fixed number of symbols to erase, the SFRSD algorithm
|
||||
uses information available from the demodulator to assign a variable probabilit
|
||||
y of erasure to each received symbol.
|
||||
Symbols that are determined to be of low quality and thus likely to be
|
||||
incorrect are assigned a high probability of erasure, and symbols that
|
||||
are likely to be correct are assigned low erasure probabilities.
|
||||
The erasure probability for a symbol is determined using two quality indices
|
||||
that are derived from information provided by the demodulator.
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Section
|
||||
The decoding algorithm
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Preliminary setup: Using a large dataset of received words that have been
|
||||
successfully decoded, estimate the probability of symbol error as a function
|
||||
of the symbol's metrics P1-rank and P2/P1.
|
||||
The resulting matrix is scaled by a factor (1.3) and used as the erasure-probabi
|
||||
lity matrix in step 2.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
For each received word:
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
1.
|
||||
Determine symbol metrics for each symbol in the received word.
|
||||
The metrics are the rank {1,2,...,63} of the symbol's power percentage and
|
||||
the ratio of the power percentages of the second most likely symbol and
|
||||
the most likely symbol.
|
||||
Denote these metrics by P1-rank and P2/P1.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
2.
|
||||
Use the erasure probability for each symbol, make independent decisions
|
||||
about whether or not to erase each symbol in the word.
|
||||
Allow a total of up to 51 symbols to be erased.
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
3.
|
||||
Attempt errors-and-erasures decoding with the erasure vector that was determine
|
||||
d in step 3.
|
||||
If the decoder is successful, it returns a candidate codeword.
|
||||
Go to step 5.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
4.
|
||||
If decoding is not successful, go to step 2.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
5.
|
||||
If a candidate codeword is returned by the decoder, calculate its soft
|
||||
distance from the received word and save the codeword if the soft distance
|
||||
is the smallest one encountered so far.
|
||||
If the soft distance is smaller than threshold dthresh, delare a successful
|
||||
decode and return the codeword.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
6.
|
||||
If the number of trials is equal to the maximum allowed number, exit and
|
||||
return the current best codeword.
|
||||
Otherwise, go to 2
|
||||
\end_layout
|
||||
|
||||
\begin_layout Bibliography
|
||||
\begin_inset CommandInset bibitem
|
||||
LatexCommand bibitem
|
||||
key "key-1"
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\end_body
|
||||
\end_document
|
||||
Loading…
Reference in New Issue
Block a user