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Some more progress on the sfrsd document. IEEE Layout won't work with enumerate style, so switched to a different style for now.
git-svn-id: svn+ssh://svn.code.sf.net/p/wsjt/wsjt/branches/wsjtx@6197 ab8295b8-cf94-4d9e-aec4-7959e3be5d79
This commit is contained in:
Steven Franke
parent
864a1f24a6
commit
a5e6dd2063
@ -2,7 +2,7 @@
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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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\textclass paper
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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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@ -28,7 +28,7 @@
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\spacing single
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\use_hyperref false
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\papersize default
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\use_geometry false
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\use_geometry true
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\use_package amsmath 1
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\use_package amssymb 1
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\use_package cancel 1
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@ -52,6 +52,10 @@
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\shortcut idx
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\color #008000
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\end_index
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\leftmargin 1in
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\topmargin 1in
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\rightmargin 1in
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\bottommargin 1in
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\secnumdepth 3
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\tocdepth 3
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\paragraph_separation indent
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@ -109,6 +113,10 @@ Koetter-Vardy
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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 that is encoded
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for transmission using a Reed-Solomon code.
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@ -216,11 +224,11 @@ 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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determine 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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determine the correct value of the incorrect symbols
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\end_layout
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\begin_layout Standard
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@ -270,24 +278,24 @@ errors
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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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\begin_inset Formula $n_{e}$
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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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\begin_inset Formula $x$
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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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n_{e}+2x\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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\begin_inset Formula $n_{e}=0$
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\end_inset
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, then the decoder is said to be an
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@ -308,7 +316,7 @@ errors-only
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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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\begin_inset Formula $0<n_{e}\le d-1$
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\end_inset
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(
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@ -336,13 +344,13 @@ 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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\begin_inset Formula $n_{e}$
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\end_inset
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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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\begin_inset Formula $\left\lfloor \frac{51-n_{era}}{2}\right\rfloor $
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\begin_inset Formula $\left\lfloor \frac{51-n_{e}}{2}\right\rfloor $
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\end_inset
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errors.
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@ -362,24 +370,17 @@ 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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Section NN describes some examples that illustrate ho w 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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received symbols to enable a decoding algorithm to reliably decode received
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words that contain many more than 25 errors.
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Section NN 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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name "sec:You've-got-to"
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\end_inset
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@ -390,8 +391,7 @@ You've got to ask yourself.
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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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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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@ -402,8 +402,9 @@ In practice the number of errors will not be known either, but this is not
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\end_inset
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.
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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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\begin_inset Formula $n_{e}=40$
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\end_inset
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), leaving 23 unerased symbols.
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@ -415,7 +416,11 @@ 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 the number
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of errors present in the 23 unerased symbols is 5 or less.
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of errors,
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\begin_inset Formula $x$
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\end_inset
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, present in the 23 unerased symbols is 5 or less.
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This means that the number of errors captured in the set of 40 erased symbols
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must be at least 35.
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@ -432,53 +437,71 @@ Define:
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\end_layout
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\begin_layout Itemize
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\begin_inset Formula $N$
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\begin_inset Formula $n$
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\end_inset
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= number of symbols in a codeword (63 for JT65),
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\end_layout
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\begin_layout Itemize
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\begin_inset Formula $K$
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\begin_inset Formula $X$
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\end_inset
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= number of incorrect symbols in a codeword,
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\end_layout
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\begin_layout Itemize
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\begin_inset Formula $n$
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\begin_inset Formula $n_{e}$
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\end_inset
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= number of symbols erased for errors-and-erasures decoding,
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\end_layout
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\begin_layout Itemize
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\begin_inset Formula $k$
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\begin_inset Formula $x$
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\end_inset
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= number of incorrect symbols in the set of erased symbols.
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\end_layout
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\begin_layout Standard
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Let
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In an ensemble of received words,
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\begin_inset Formula $X$
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\end_inset
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be the number of incorrect symbols in a set of
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\begin_inset Formula $n$
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and
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\begin_inset Formula $x$
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\end_inset
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symbols chosen for erasure.
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will be random variables.
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Let
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\begin_inset Formula $P(x|(X,n_{e}))$
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\end_inset
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denote the conditional probability mass function for the number of incorrect
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symbols,
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\begin_inset Formula $x$
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\end_inset
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, given that the number of incorrect symbols in the codeword is X and the
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number of erased symbols is
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\begin_inset Formula $n_{e}$
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\end_inset
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.
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Then
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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=k)=\frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}\label{eq:hypergeometric_pdf-1}
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P(x|(X,n_{e}))=\frac{\binom{X}{x}\binom{n-X}{n_{e}-x}}{\binom{n}{n_{e}}}\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 $\binom{n}{m}=\frac{n!}{m!(n-m)!}$
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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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@ -486,17 +509,31 @@ where
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\begin_inset Quotes eld
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\end_inset
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nchoosek(n,k)
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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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function in Gnu Octave.
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The hypergeometric probability mass function is available in Gnu Octave
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as function
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The hypergeometric probability mass function defined in (
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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(k,N,K,n)
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hygepdf(
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\begin_inset Formula $x,n,X,n_{e}$
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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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@ -504,14 +541,18 @@ hygepdf(k,N,K,n)
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\end_layout
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\begin_layout Paragraph
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Case 1
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\end_layout
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\begin_layout Case
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A codeword contains
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\begin_inset Formula $K=40$
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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 $n=40$
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\begin_inset Formula $n_{e}=40$
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\end_inset
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symbols are randomly selected for erasure.
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@ -522,7 +563,7 @@ A codeword contains
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of the erased symbols are incorrect is:
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\begin_inset Formula
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\[
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P(X=35)=\frac{\binom{40}{35}\binom{63-40}{40-35}}{\binom{63}{40}}=2.356\times10^{-7}.
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P(x=35)=\frac{\binom{40}{35}\binom{63-40}{40-35}}{\binom{63}{40}}=2.356\times10^{-7}.
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\]
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\end_inset
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@ -530,7 +571,7 @@ P(X=35)=\frac{\binom{40}{35}\binom{63-40}{40-35}}{\binom{63}{40}}=2.356\times10^
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Similarly:
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\begin_inset Formula
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\[
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P(X=36)=8.610\times10^{-9}.
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P(x=36)=8.610\times10^{-9}.
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\]
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\end_inset
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@ -547,37 +588,40 @@ Since the probability of catching 36 errors is so much smaller than the
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in 4 million.
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\end_layout
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\begin_layout Case
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A codeword contains
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\begin_inset Formula $K=40$
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\end_inset
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\begin_layout Paragraph
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Case 2
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\end_layout
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incorrect symbols.
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\begin_layout Case
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It is interesting to work out the best choice for the number of symbols
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that should be selected at random for erasure if the goal is to maximize
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the probability of successfully decoding the word.
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By exhaustive search, it turns out that the best case is to erase
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By exhaustive search, it turns out that if
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\begin_inset Formula $X=40$
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\end_inset
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, then the best strategy is to erase
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\begin_inset Formula $n=45$
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\end_inset
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symbols, in which case the word will be decoded if the set of erased symbols
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contains at least 37 errors.
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With
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\begin_inset Formula $N=63$
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\begin_inset Formula $n=63$
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\end_inset
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,
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\begin_inset Formula $K=40$
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\begin_inset Formula $X=40$
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\end_inset
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,
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\begin_inset Formula $n=45$
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\begin_inset Formula $n_{e}=45$
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\end_inset
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, then
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\begin_inset Formula
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\[
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P(X\ge37)\simeq2\times10^{-6}.
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P(x\ge37)\simeq2\times10^{-6}.
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\]
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\end_inset
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@ -592,6 +636,10 @@ This probability is about 8 times higher than the probability of success
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\end_layout
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\begin_layout Paragraph
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Case 3
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\end_layout
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\begin_layout Case
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Cases 1 and 2 illustrate the fact that a strategy that tries to guess which
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symbols to erase is not going to be very successful unless we are prepared
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@ -599,7 +647,7 @@ Cases 1 and 2 illustrate the fact that a strategy that tries to guess which
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Consider a slight modification to the strategy that can tip the odds in
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our favor.
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Suppose that the codeword contains
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\begin_inset Formula $K=40$
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\begin_inset Formula $X=40$
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\end_inset
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incorrect symbols, as before.
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@ -610,45 +658,81 @@ Cases 1 and 2 illustrate the fact that a strategy that tries to guess which
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the set of erasures is chosen from the smaller set of 53 less reliable
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symbols.
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If
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\begin_inset Formula $n=40$
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\begin_inset Formula $n_{e}=45$
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\end_inset
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symbols are chosen randomly from the set of
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\begin_inset Formula $N=53$
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\begin_inset Formula $n=53$
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\end_inset
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least reliable symbols, it is still necessary for the erased symbols to
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include at least 35 errors (as in Case 1).
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include at least 37 errors (as in Case 2).
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In this case, with
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\begin_inset Formula $N=53$
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\begin_inset Formula $n=53$
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\end_inset
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,
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\begin_inset Formula $K=40$
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\begin_inset Formula $X=40$
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\end_inset
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,
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\begin_inset Formula $n=35$
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\begin_inset Formula $n_{e}=45$
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\end_inset
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,
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\begin_inset Formula $P(X=35)=0.001$
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\begin_inset Formula $P(x\ge37)=0.016$
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\end_inset
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! Now, the situation is much better.
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The odds of decoding the word on the first try are approximately 1 in 1000.
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The odds are even better if 41 symbols are erased, in which case
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\begin_inset Formula $P(X=35)=0.0042$
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The odds of decoding the word on the first try are approximately 1 in 62.5!
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\end_layout
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||||
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\begin_layout Standard
|
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Even better odds are obtained with
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\begin_inset Formula $n_{e}=47$
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\end_inset
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||||
|
||||
, giving odds of about 1 in 200!
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which requires
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\begin_inset Formula $x\ge38$
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\end_inset
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||||
|
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.
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||||
With
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\begin_inset Formula $n=53$
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\end_inset
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||||
|
||||
,
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||||
\begin_inset Formula $X=40$
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||||
\end_inset
|
||||
|
||||
,
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||||
\begin_inset Formula $n_{e}=47$
|
||||
\end_inset
|
||||
|
||||
,
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||||
\begin_inset Formula $P(x\ge38)=0.0266$
|
||||
\end_inset
|
||||
|
||||
, which makes the odds the best so far; about 1 in 38.
|
||||
|
||||
\end_layout
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||||
|
||||
\begin_layout Section
|
||||
\begin_inset CommandInset label
|
||||
LatexCommand label
|
||||
name "sec:The-decoding-algorithm"
|
||||
|
||||
\end_inset
|
||||
|
||||
The SFRSD decoding algorithm
|
||||
\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 quality of just 10 of the 63 symbols, it is possible to devise an algorithm
|
||||
that can 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
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||||
\end_inset
|
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@ -659,86 +743,220 @@ guessing
|
||||
|
||||
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.
|
||||
likely to be in error.
|
||||
In practice, because the number of errors in the received word is unknown,
|
||||
rather than erase a fixed number of symbols, it is better use a stochastic
|
||||
algorithm which assigns a relatively high probability of erasure to the
|
||||
lowest quality symbols and a relatively low probability of erasure to the
|
||||
highest quality symbols.
|
||||
As illustrated by case 3, a good choice of the erasure probabilities can
|
||||
increase the probability of a successful decode by many 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 SFRSD algorithm uses information available from the demodulator to assign
|
||||
a variable probability of erasure to each received symbol.
|
||||
The erasure probability for a symbol is determined using two quality indices
|
||||
that are derived from information provided by the demodulator.
|
||||
that are derived from the the JT65 64-FSK demodulator.
|
||||
The noncoherent 64-FSK demodulator identifies the most likely received
|
||||
symbol based on which of 64 frequency bins contains the the largest signal
|
||||
plus noise power.
|
||||
The percentage of the total signal plus noise 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 Section
|
||||
The decoding algorithm
|
||||
\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 power percentage,
|
||||
\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 lower ranked
|
||||
symbols.
|
||||
|
||||
\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 is not much better than
|
||||
the second most likely symbol
|
||||
\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.
|
||||
The decoder has a built-in 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 that is expected based on the
|
||||
\begin_inset Formula $p_{1}$
|
||||
\end_inset
|
||||
|
||||
-rank and
|
||||
\begin_inset Formula $p_{2}/p_{1}$
|
||||
\end_inset
|
||||
|
||||
metrics.
|
||||
These
|
||||
\emph on
|
||||
a-priori
|
||||
\emph default
|
||||
symbol error probabilities will be close to 1 for lower-quality symbols
|
||||
and closer to 0 for high-quality symbols.
|
||||
Recall, from Case 2, that the best performance was obtained when
|
||||
\begin_inset Formula $n_{e}>X$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
Correspondingly, the SFRSD algorithm works best when the probability of
|
||||
erasing a symbol is somewhat larger than the probability that the symbol
|
||||
is incorrect.
|
||||
Empirically, it was determined that good performance of the SFRSD algorithm
|
||||
is obtained when the symbol erasure probability is somewhat larger than
|
||||
the prior estimate of symbol error probability.
|
||||
It has been empirically determined that choosing the erasure probabilities
|
||||
to be a factor of
|
||||
\begin_inset Formula $1.3$
|
||||
\end_inset
|
||||
|
||||
larger than the symbol error probabilities gives the best results.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
For each received word:
|
||||
The SFRSD algorithm successively tries to decode the received word.
|
||||
In each iteration, an independent stochastic erasure vector is generated
|
||||
based on a-priori symbol erasure probabilities.
|
||||
Technically, the algorithm is a list-decoder, potentially generating a
|
||||
list of candidate codewords.
|
||||
Each codeword on the list is assigned a quality metric, defined to be the
|
||||
soft distance between the received word and the codeword.
|
||||
Among the list of candidate codewords found by this stochastic search algorithm
|
||||
, only the one with the smallest soft-distance from the received word is
|
||||
kept.
|
||||
As with all such algorithms, a stopping criterion is necessary.
|
||||
SFRSD accepts a codeword unconditionally if its soft distance is smaller
|
||||
than an acceptance threshold,
|
||||
\begin_inset Formula $d_{a}$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
A timeout is employed to limit the execution time of the algorithm.
|
||||
|
||||
\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.
|
||||
\begin_layout Paragraph
|
||||
Algorithm
|
||||
\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.
|
||||
\begin_layout Enumerate
|
||||
For each symbol in the received word, find the erasure probability from
|
||||
the erasure-probability matrix and the
|
||||
\begin_inset Formula $\{p_{1}\textrm{-rank},p_{2}/p_{1}\}$
|
||||
\end_inset
|
||||
|
||||
soft-symbol information.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Enumerate
|
||||
Make independent decisions about whether or not to erase each symbol in
|
||||
the word using the symbol's erasure probability.
|
||||
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.
|
||||
\begin_layout Enumerate
|
||||
Attempt BM errors-and-erasures decoding with the set of erased symbols that
|
||||
was determined in step 2.
|
||||
If the BM decoder is successful go to step 5.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
4.
|
||||
\begin_layout Enumerate
|
||||
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.
|
||||
\begin_layout Enumerate
|
||||
Calculate the soft distance,
|
||||
\begin_inset Formula $d_{s}$
|
||||
\end_inset
|
||||
|
||||
, between the candidate codeword and the received word.
|
||||
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 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
|
||||
\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 codeword with
|
||||
\begin_inset Formula $d_{s}\le d_{a}$
|
||||
\end_inset
|
||||
|
||||
has been found.
|
||||
Declare that is successful.
|
||||
Return the best codeword found so far.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Section
|
||||
Results
|
||||
\end_layout
|
||||
|
||||
\begin_layout Section
|
||||
Summary
|
||||
\end_layout
|
||||
|
||||
\begin_layout Bibliography
|
||||
|
||||
Loading…
Reference in New Issue
Block a user