From 7f9024bf62b3a4bae221e69bfbac3ad43cb66ee4 Mon Sep 17 00:00:00 2001
From: Steven Franke <s.j.franke@icloud.com>
Date: Thu, 26 Nov 2015 03:57:42 +0000
Subject: [PATCH] Remove redundant file.

git-svn-id: svn+ssh://svn.code.sf.net/p/wsjt/wsjt/branches/wsjtx@6186 ab8295b8-cf94-4d9e-aec4-7959e3be5d79
---
 lib/sfrsd2/sfrsd.lyx | 755 -------------------------------------------
 1 file changed, 755 deletions(-)
 delete mode 100644 lib/sfrsd2/sfrsd.lyx

diff --git a/lib/sfrsd2/sfrsd.lyx b/lib/sfrsd2/sfrsd.lyx
deleted file mode 100644
index 3aae759c2..000000000
--- a/lib/sfrsd2/sfrsd.lyx
+++ /dev/null
@@ -1,755 +0,0 @@
-#LyX 2.1 created this file. For more info see http://www.lyx.org/
-\lyxformat 474
-\begin_document
-\begin_header
-\textclass IEEEtran
-\use_default_options true
-\maintain_unincluded_children false
-\language english
-\language_package default
-\inputencoding auto
-\fontencoding global
-\font_roman default
-\font_sans default
-\font_typewriter default
-\font_math auto
-\font_default_family default
-\use_non_tex_fonts false
-\font_sc false
-\font_osf false
-\font_sf_scale 100
-\font_tt_scale 100
-\graphics default
-\default_output_format default
-\output_sync 0
-\bibtex_command default
-\index_command default
-\paperfontsize default
-\spacing single
-\use_hyperref false
-\papersize default
-\use_geometry false
-\use_package amsmath 1
-\use_package amssymb 1
-\use_package cancel 1
-\use_package esint 1
-\use_package mathdots 1
-\use_package mathtools 1
-\use_package mhchem 1
-\use_package stackrel 1
-\use_package stmaryrd 1
-\use_package undertilde 1
-\cite_engine basic
-\cite_engine_type default
-\biblio_style plain
-\use_bibtopic false
-\use_indices false
-\paperorientation portrait
-\suppress_date false
-\justification true
-\use_refstyle 1
-\index Index
-\shortcut idx
-\color #008000
-\end_index
-\secnumdepth 3
-\tocdepth 3
-\paragraph_separation indent
-\paragraph_indentation default
-\quotes_language english
-\papercolumns 1
-\papersides 1
-\paperpagestyle default
-\tracking_changes false
-\output_changes false
-\html_math_output 0
-\html_css_as_file 0
-\html_be_strict false
-\end_header
-
-\begin_body
-
-\begin_layout Title
-A stochastic successive erasures soft-decision decoder for the JT65 (63,12)
- Reed-Solomon code
-\end_layout
-
-\begin_layout Author
-Steven J.
- Franke, K9AN and Joseph H.
- Taylor, K1JT
-\end_layout
-
-\begin_layout Abstract
-The JT65 mode has revolutionized amateur-radio weak-signal communication
- by enabling amateur radio operators with small antennas and relatively
- low-power transmitters to communicate over propagation paths that could
- not be utilized using traditional technologies.
- One reason for the success and popularity of the JT65 mode is its use of
- strong error-correction coding.
- The JT65 code is a short block-length, low-rate, Reed-Solomon code based
- on a 64-symbol alphabet.
- Since 200?, decoders for the JT65 code have used the 
-\begin_inset Quotes eld
-\end_inset
-
-Koetter-Vardy
-\begin_inset Quotes erd
-\end_inset
-
- (KV) algebraic soft-decision decoder.
- The KV decoder is implemented in a closed-source program that is licensed
- to K1JT for use in amateur applications.
- This note describes a new open-source alternative to the KV decoder called
- the SFRSD decoder.
- The SFRSD decoding algorithm is shown to perform at least as well as the
- KV decoder.
- The SFRSD algorithm is conceptually simple and is built around the well-known
- Berlekamp-Massey errors-and-erasures decoder.
- 
-\end_layout
-
-\begin_layout Standard
-JT65 message frames consist of a short, compressed, message that is encoded
- for transmission using a Reed-Solomon code.
- Reed-Solomon codes are block codes and, like all block codes, are characterized
- by the length of their codewords, 
-\begin_inset Formula $n$
-\end_inset
-
-, the number of message symbols conveyed by the codeword, 
-\begin_inset Formula $k$
-\end_inset
-
-, and the number of possible values for each symbol in the codewords.
- The codeword length and the number of message symbols are specified as
- a tuple in the form 
-\begin_inset Formula $(n,k)$
-\end_inset
-
-.
- JT65 uses a (63,12) Reed-Solomon code with 64 possible values for each
- symbol, so each symbol represents 
-\begin_inset Formula $\log_{2}64=6$
-\end_inset
-
- message bits.
- The source-encoded messages conveyed by a 63-symbol JT65 frame consist
- of 72 bits.
- The JT65 code is systematic, which means that the 12 message symbols are
- embedded in the codeword without modification and another 51 parity symbols
- derived from the message symbols are added to form the codeword consisting
- of 63 total symbols.
- 
-\end_layout
-
-\begin_layout Standard
-The concept of Hamming distance is used as a measure of 
-\begin_inset Quotes eld
-\end_inset
-
-distance
-\begin_inset Quotes erd
-\end_inset
-
- between different codewords, or between a received word and a codeword.
- Hamming distance is the number of code symbols that differ in the two words
- that are being compared.
- Reed-Solomon codes have minimum Hamming distance 
-\begin_inset Formula $d$
-\end_inset
-
-, where 
-\begin_inset Formula 
-\begin{equation}
-d=n-k+1.\label{eq:minimum_distance}
-\end{equation}
-
-\end_inset
-
-The minimum Hamming distance of the JT65 code is 
-\begin_inset Formula $d=52$
-\end_inset
-
-, which means that any particular codeword differs from all other codewords
- in at least 52 positions.
- 
-\end_layout
-
-\begin_layout Standard
-Given only a received word containing some incorrect symbols (errors), the
- received word can be decoded into the correct codeword using a deterministic,
- algebraic, algorithm provided that no more than 
-\begin_inset Formula $t$
-\end_inset
-
- symbols were received incorrectly, where
-\begin_inset Formula 
-\begin{equation}
-t=\left\lfloor \frac{n-k}{2}\right\rfloor .\label{eq:t}
-\end{equation}
-
-\end_inset
-
-For the JT65 code, 
-\begin_inset Formula $t=25$
-\end_inset
-
-, which means that it is always possible to efficiently decode a received
- word that contains no more than 25 symbol errors.
- 
-\end_layout
-
-\begin_layout Standard
-There are a number of well-known algebraic algorithms that can carry out
- the process of decoding a received codeword that contains no more than
- 
-\begin_inset Formula $t$
-\end_inset
-
- errors.
- One such algorithm is the Berlekamp-Massey (BM) decoding algorithm.
-\end_layout
-
-\begin_layout Standard
-A decoder, such as BM, must carry out two tasks: 
-\end_layout
-
-\begin_layout Enumerate
-figure out which symbols were received incorrectly 
-\end_layout
-
-\begin_layout Enumerate
-figure out the correct value of the incorrect symbols 
-\end_layout
-
-\begin_layout Standard
-If it is somehow known that certain symbols are incorrect, such information
- can be used in the decoding algorithm to reduce the amount of work required
- in step 1 and to allow step 2 to correct more than 
-\begin_inset Formula $t$
-\end_inset
-
- errors.
- In fact, in the unlikely event that the location of each and every error
- is known and is provided to the BM decoder, and if no correct symbols are
- accidentally labeled as errors, then the BM decoder can correct up to 
-\begin_inset Formula $d$
-\end_inset
-
- errors! 
-\end_layout
-
-\begin_layout Standard
-In the decoding algorithm described herein, a list of symbols that are known
- or suspected to be incorrect is sent to the BM decoder.
- Symbols in the received word that are flagged as being incorrect are called
- 
-\begin_inset Quotes eld
-\end_inset
-
-erasures
-\begin_inset Quotes erd
-\end_inset
-
-.
- Symbols that are not erased and that are incorrect will be called 
-\begin_inset Quotes eld
-\end_inset
-
-errors
-\begin_inset Quotes erd
-\end_inset
-
-.
- The BM decoder accepts erasure information in the form of a list of indices
- corresponding to the incorrect, or suspected incorrect, symbols in the
- received word.
- As already noted, if the erasure information is perfect, then up to 51
- errors will be corrected.
- When the erasure information is imperfect, then some of the erased symbols
- will actually be correct, and some of the unerased symbols will be in error.
- If a total of 
-\begin_inset Formula $n_{era}$
-\end_inset
-
- symbols are erased and the remaining unerased symbols contain 
-\begin_inset Formula $n_{err}$
-\end_inset
-
- errors, then the BM algorithm can find the correct codeword as long as
- 
-\begin_inset Formula 
-\begin{equation}
-n_{era}+2n_{err}\le d-1\label{eq:erasures_and_errors}
-\end{equation}
-
-\end_inset
-
-If 
-\begin_inset Formula $n_{era}=0$
-\end_inset
-
-, then the decoder is said to be an 
-\begin_inset Quotes eld
-\end_inset
-
-errors-only
-\begin_inset Quotes erd
-\end_inset
-
- decoder and it can correct up to 
-\begin_inset Formula $t$
-\end_inset
-
- errors (
-\begin_inset Formula $t$
-\end_inset
-
-=25 for JT65).
- If 
-\begin_inset Formula $0<n_{era}\le d-1$
-\end_inset
-
- (
-\begin_inset Formula $d-1=51$
-\end_inset
-
- for JT65), then the decoder is said to be an 
-\begin_inset Quotes eld
-\end_inset
-
-errors-and-erasures
-\begin_inset Quotes erd
-\end_inset
-
- decoder.
- 
-\end_layout
-
-\begin_layout Standard
-For the JT65 code, (
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:erasures_and_errors"
-
-\end_inset
-
-) says that if 
-\begin_inset Formula $n_{era}$
-\end_inset
-
- symbols are declared to be erased, then the BM decoder will find the correct
- codeword as long as the remaining un-erased symbols contain no more than
- 
-\begin_inset Formula $\left\lfloor \frac{51-n_{era}}{2}\right\rfloor $
-\end_inset
-
- errors.
- The errors-and-erasures capability of the BM decoder is a very powerful
- feature that serves as the core of the new soft-decision decoder described
- herein.
- 
-\end_layout
-
-\begin_layout Standard
-It will be helpful to have some understanding of the errors and erasures
- tradeoff described by (
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:erasures_and_errors"
-
-\end_inset
-
-) to appreciate how the new decoder algorithm works.
- Section 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:Errors-and-erasures-decoding-exa"
-
-\end_inset
-
- describes some examples that should illustrate how the errors-and-erasures
- capability can be combined with some information about the quality of the
- received symbols to enable development of a decoding algorithm that can
- reliably decode received words that contain many more than 25 errors.
- Section describes the SFRSD decoding algorithm.
-\end_layout
-
-\begin_layout Section
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:Errors-and-erasures-decoding-exa"
-
-\end_inset
-
-You've got to ask yourself.
- Do I feel lucky?
-\end_layout
-
-\begin_layout Standard
-Consider a particular received codeword that contains 40 incorrect symbols
- and 23 correct symbols.
- It is not known which 40 symbols are in error.
- 
-\begin_inset Foot
-status open
-
-\begin_layout Plain Layout
-In practice the number of errors will not be known either, but this is not
- a serious problem.
-\end_layout
-
-\end_inset
-
- Suppose that the decoder randomly chooses 40 symbols to erase (
-\begin_inset Formula $n_{era}=40$
-\end_inset
-
-), leaving 23 unerased symbols.
- According to (
-\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.
- 
-\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:
-\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