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A few more editorial changes.
git-svn-id: svn+ssh://svn.code.sf.net/p/wsjt/wsjt/branches/wsjtx@6306 ab8295b8-cf94-4d9e-aec4-7959e3be5d79
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Joe Taylor
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@ -189,28 +189,18 @@ t=\left\lfloor \frac{n-k}{2}\right\rfloor .\label{eq:t}
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\end_inset
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For the JT65 code,
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For the JT65 code
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\begin_inset Formula $t=25$
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\end_inset
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, so it is always possible to efficiently decode a received word having
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no more than 25 symbol errors.
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, so it is always possible to decode a received word having 25 or fewer
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symbol errors.
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Any one of several well-known algebraic algorithms, such as the widely
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used Berlekamp-Massey (BM) algorithm, can carry out the decoding.
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Two steps are necessarily involved in this process, namely
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\end_layout
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\begin_layout Enumerate
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Determine which symbols were received incorrectly.
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\end_layout
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\begin_layout Enumerate
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Find the correct value of the incorrect symbols.
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\end_layout
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\begin_layout Standard
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If we somehow know that certain symbols are incorrect, this information
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Two steps are necessarily involved in this process.
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We must (1) determine which symbols were received incorrectly, and (2)
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find the correct value of the incorrect symbols.
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If we somehow know that certain symbols are incorrect, that information
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can be used to reduce the work involved in step 1 and allow step 2 to correct
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more than
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\begin_inset Formula $t$
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@ -220,7 +210,7 @@ If we somehow know that certain symbols are incorrect, this information
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In the unlikely event that the location of every error is known and if
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no correct symbols are accidentally labeled as errors, the BM algorithm
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can correct up to
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\begin_inset Formula $d-1$
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\begin_inset Formula $d-1=n-k$
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\end_inset
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errors.
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@ -246,8 +236,8 @@ errors.
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\begin_inset Quotes erd
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\end_inset
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As already noted, with perfect erasure information up to 51 incorrect symbols
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can be corrected.
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With perfect erasure information up to 51 incorrect symbols can be corrected
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for the JT65 code.
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Imperfect erasure information means that some erased symbols may be correct,
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and some other symbols in error.
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If
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@ -447,11 +437,7 @@ hygepdf(
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\end_inset
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.
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The cumulative probability that
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\emph on
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at least
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\emph default
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The cumulative probability that at least
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\begin_inset Formula $\epsilon$
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\end_inset
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@ -521,15 +507,15 @@ P(x=36)\simeq8.6\times10^{-9}.
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\end_inset
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Since the probability of erasing 36 errors is so much smaller than the probabili
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ty of erasing 35 errors, we may safely conclude that the probability of
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randomly choosing an erasure vector that can decode the received word is
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approximately
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Since the probability of erasing 36 errors is so much smaller than that
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for erasing 35 errors, we may safely conclude that the probability of randomly
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choosing an erasure vector that can decode the received word is approximately
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\begin_inset Formula $P(x=35)\simeq2.4\times10^{-7}$
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\end_inset
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.
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The odds of successfully decoding the word on the first try are very poor,
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The odds of producing a valid codeword on the first try are very poor,
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about 1 in 4 million.
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\end_layout
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@ -660,7 +646,7 @@ reference "eq:hypergeometric_pdf"
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\end_inset
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.
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The odds for successful decoding on the first try are now about 1 in 38.
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The odds for producing a codeword on the first try are now about 1 in 38.
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A few hundred independently randomized tries would be enough to all-but-guarant
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ee production of a valid codeword by the BM decoder.
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\end_layout
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@ -682,15 +668,15 @@ Example 3 shows how statistical information about symbol quality should
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use a stochastic algorithm to assign high erasure probability to low-quality
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symbols and relatively low probability to high-quality symbols.
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As illustrated by Example 3, a good choice of erasure probabilities can
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increase the chance of producing a codeword by many orders of magnitude.
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Note that at this stage we treat any codeword selected by errors-and-erasures
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decoding as only a
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increase by many orders of magnitude the chance of producing a codeword.
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Note that at this stage we must treat any codeword obtained by errors-and-erasu
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res decoding as no more than a
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\emph on
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candidate
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\emph default
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.
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The next task is to find a metric that can reliably select one of many
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proffered candidates as the codeword that was actually transmitted.
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Our next task is to find a metric that can reliably select one of many
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proffered candidates as the codeword actually transmitted.
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\end_layout
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\begin_layout Standard
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@ -704,11 +690,11 @@ The FT algorithm uses quality indices made available by a noncoherent 64-FSK
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\begin_inset Formula $i=1,64$
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\end_inset
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is the spectral bin number and
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is the frequency index and
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\begin_inset Formula $j=1,63$
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\end_inset
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the symbol number.
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the symbol index.
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The most likely value for symbol
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\begin_inset Formula $j$
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\end_inset
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@ -719,8 +705,8 @@ The FT algorithm uses quality indices made available by a noncoherent 64-FSK
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\end_inset
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.
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The fraction of total power in the two bins containing the largest and
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second-largest powers (denoted by
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The fractions of total power in the two bins containing the largest and
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second-largest powers, denoted respectively by
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\begin_inset Formula $p_{1}$
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\end_inset
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@ -728,16 +714,8 @@ The FT algorithm uses quality indices made available by a noncoherent 64-FSK
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\begin_inset Formula $p_{2}$
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\end_inset
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, respectively) are passed from demodulator to decoder as
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\begin_inset Quotes eld
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\end_inset
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soft-symbol
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\begin_inset Quotes erd
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\end_inset
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information.
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The decoder then derives two metrics from
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, are passed from demodulator to decoder as soft-symbol information.
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The FT decoder derives two metrics from
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\begin_inset Formula $p_{1}$
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\end_inset
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@ -745,7 +723,7 @@ and
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\begin_inset Formula $p_{2}$
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\end_inset
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:
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, namely
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\end_layout
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\begin_layout Itemize
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@ -757,7 +735,7 @@ and
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\end_inset
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of the symbol's fractional power
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\begin_inset Formula $p_{1}$
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\begin_inset Formula $p_{1,\,j}$
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\end_inset
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in a sorted list of
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@ -825,7 +803,7 @@ educated guesses
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For each iteration a stochastic erasure vector is generated based on the
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symbol erasure probabilities.
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The erasure vector is sent to the BM decoder along with the full set of
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63 received hard-decision symbols.
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63 hard-decision symbol values.
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When the BM decoder finds a candidate codeword it is assigned a quality
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metric
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\begin_inset Formula $d_{s}$
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@ -879,7 +857,7 @@ In practice we find that
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\end_inset
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dB.
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We also find that weaker signals can often be decoded by using soft-symbol
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We also find that weaker signals frequently can be decoded by using soft-symbol
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information beyond that contained in
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\begin_inset Formula $p_{1}$
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\end_inset
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@ -893,11 +871,8 @@ and
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\begin_inset Formula $u$
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\end_inset
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, the average signal-plus-noise power in all
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\begin_inset Formula $n$
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\end_inset
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symbols according to a candidate codeword's symbol values:
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, the average signal-plus-noise power in all symbols, according to a candidate
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codeword's symbol values:
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\end_layout
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\begin_layout Standard
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@ -917,7 +892,7 @@ Here the
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\end_layout
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\begin_layout Standard
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The correct codeword produces a value for
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The correct JT65 codeword produces a value for
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\begin_inset Formula $u$
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\end_inset
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@ -925,11 +900,12 @@ The correct codeword produces a value for
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\begin_inset Formula $n=63$
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\end_inset
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bins of signal-plus-noise, while incorrect codewords have at most
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bins containing both signal and noise power.
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Incorrect codewords have at most
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\begin_inset Formula $k=12$
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\end_inset
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bins with signal-plus-noise and at least
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such bins and at least
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\begin_inset Formula $n-k=51$
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\end_inset
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@ -938,8 +914,8 @@ The correct codeword produces a value for
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\begin_inset Formula $S(i,\,j)$
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\end_inset
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is normalized so that its median value (essentially the average noise level)
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is unity, the correct codeword is expected to yield
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has been normalized so that its median value (essentially the average noise
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level) is unity, the correct codeword is expected to yield the metric value
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\end_layout
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\begin_layout Standard
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@ -954,23 +930,30 @@ where
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\begin_inset Formula $y$
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\end_inset
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is the signal-to-noise ratio in power units and the quoted one standard
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deviation uncertainty range assumes Gaussian statistics.
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Incorrect codewords will yield at most
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is the signal-to-noise ratio (in linear power units) and the quoted one-standar
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d-deviation uncertainty range assumes Gaussian statistics.
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Incorrect codewords will yield metric values no larger than
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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u=\frac{n-k\pm\sqrt[]{n-k}}{n}+\frac{k\pm\sqrt[]{k}}{n}(1+y)\approx1\pm0.13+(0.19\pm0.06)\,y.
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u=\frac{n-k\pm\sqrt{n-k}}{n}+\frac{k\pm\sqrt{k}}{n}(1+y).
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\]
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\end_inset
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For JT65 this expression evaluates to
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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u\approx1\pm0.13+(0.19\pm0.06)\,y.
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\]
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\end_inset
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As a specific example, consider signal strength
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\begin_inset Formula $y=4$
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\end_inset
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@ -980,11 +963,12 @@ As a specific example, consider signal strength
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\end_inset
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dB.
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(For JT65, the corresponding SNR in 2500 Hz bandwidth is
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For JT65, the corresponding SNR in 2500 Hz bandwidth is
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\begin_inset Formula $-23.7$
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\end_inset
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dB.) The correct codeword is then expected to yield
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dB.
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The correct codeword is then expected to yield
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\begin_inset Formula $u\approx5.0\pm$
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\end_inset
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@ -993,13 +977,13 @@ As a specific example, consider signal strength
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\end_inset
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or less.
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A threshold set at
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We find that a threshold set at
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\begin_inset Formula $u_{0}=4.4$
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\end_inset
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, about 8 standard deviations above the expected maximum for incorrect codewords
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, serves reliably to distinguish the correct codeword from all other candidates,
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with a very small probability of false decodes.
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(about 8 standard deviations above the expected maximum for incorrect codewords
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) reliably serves to distinguish correct codewords from all other candidates,
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while ensuring a very small probability of false decodes.
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\end_layout
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\begin_layout Standard
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@ -1124,14 +1108,15 @@ The fraction of time that
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\begin_inset Formula $X$
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\end_inset
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, the number of symbols received incorrectly, is less than some number
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, the number of symbols received incorrectly, is expected to be less than
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some number
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\begin_inset Formula $D$
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\end_inset
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depends of course on signal-to-noise ratio.
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depends on signal-to-noise ratio.
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For the case of additive white Gaussian noise (AWGN) and noncoherent 64-FSK
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demodulation this probability is easily calculated, and representative
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examples for
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demodulation this probability is easy to calculate.
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Representative examples for
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\begin_inset Formula $D=25,$
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\end_inset
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@ -1174,15 +1159,15 @@ reference "fig:bodide"
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\end_inset
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as filled squares with connecting lines.
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The rightmost curve with solid squares shows that on the AWGN channel the
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hard-decision BM decoder should succeed about 90% of the time at
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for a range of SNRs as filled squares with connecting lines.
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The rightmost such curve shows that on the AWGN channel the hard-decision
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BM decoder should succeed about 90% of the time at
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\begin_inset Formula $E_{s}/N_{0}=7.5$
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\end_inset
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dB, 99% of the time at 8 dB, and 99.98% at 8.5 dB.
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The righmost curve with open squares shows that simulated results agree
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with theory to within 0.2 dB.
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For comparison, the righmost curve with open squares shows that simulated
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results agree with theory to within less than 0.2 dB.
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\end_layout
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@ -1273,28 +1258,20 @@ WSJT-X
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\end_layout
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\begin_layout Standard
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Received JT65 words with
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\begin_inset Formula $X>25$
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\end_inset
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incorrect symbols can be decoded if sufficient information is available
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concerning individual symbol reliabilities.
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Received JT65 words with more than 25 incorrect symbols can be decoded if
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sufficient information on individual symbol reliabilities is available.
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Using values of
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\begin_inset Formula $T$
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\end_inset
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that are practical with today's personal computers and the soft-symbol
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information described above, we find that the FT algorithm produces correct
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decodes most of the time up to
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\begin_inset Formula $X\approx40$
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information described above, we find that the FT algorithm nearly always
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produces correct decodes up to
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\begin_inset Formula $X=40$
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\end_inset
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, with some additional decodes in the range
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\begin_inset Formula $X=41$
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\end_inset
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to 43.
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As a specific example, Figure
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, and some additional decodes are found in the range 41 to 43.
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As an example, Figure
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:N_vs_X"
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@ -1302,16 +1279,16 @@ reference "fig:N_vs_X"
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\end_inset
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plots the number of stochastic erasure trials required to find the correct
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codeword versus the number of hard-decision errors.
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This result was obtained with 1000 simulated frames at
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codeword versus the number of hard-decision errors for a run with 1000
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simulated transmissions at
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\begin_inset Formula $SNR=-24$
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\end_inset
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dB, just slightly above the decoding threshold.
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Note that the mean and variance of the required number of trials both increase
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Note that both mean and variance of the required number of trials increase
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steeply with the number of errors in the received word.
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Execution time of the FT algorithm is roughly proportional to the number
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of trials.
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of required trials.
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\end_layout
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@ -1343,9 +1320,9 @@ name "fig:N_vs_X"
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\end_inset
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The number of trials needed to decode a received word vs the Hamming distance
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between the received word and the decoded codeword plotted for 1000 simulated
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frames with no fading.
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Number of trials needed to decode a received word versus Hamming distance
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between the received word and the decoded codeword, for 1000 simulated
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frames on an AWGN channel with no fading.
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The SNR in 2500 Hz bandwidth is -24 dB (
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\begin_inset Formula $E_{s}/N_{o}=5.7$
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\end_inset
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@ -1376,8 +1353,8 @@ Comparisons of decoding performance are usually presented in the professional
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, the signal-to-noise ratio per information bit.
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Results of simulations using the Berlekamp-Massey, Koetter-Vardy, and Franke-Ta
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ylor decoding algorithms on the (63,12) code are shown inthis way in Figure
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ylor decoding algorithms on the (63,12) code are presented in this way in
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Figure
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:WER"
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@ -1441,7 +1418,8 @@ Word error rate (WER) as a function of
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\end_layout
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\begin_layout Standard
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Plots like Figure
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Because of the importance of error-free transmission in commercial applications,
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plots like that in Figure
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:WER"
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@ -1452,38 +1430,39 @@ reference "fig:WER"
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\begin_inset Formula $10^{-6}$
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\end_inset
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or less, because of the importance of error-free transmission in commercial
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applications.
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or less, .
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The circumstances for minimal amateur-radio QSOs are very different, however.
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Error rates on the order of 0.1, or ever higher, may be acceptable.
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In this case the essential information is better presented in a plot like
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Figure
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:Psuccess"
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\end_inset
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showing the percentage of transmissions copied correctly as a function
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of signal-to-noise ratio.
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Error rates of order 0.1, or ever higher, may be acceptable.
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In this case the essential information is better presented in a plot showing
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the percentage of transmissions copied correctly as a function of signal-to-noi
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se ratio.
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\end_layout
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\begin_layout Standard
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In Figure
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Figure
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:Psuccess"
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\end_inset
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we plot the results of simulations for signal-to-noise ratios ranging from
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-18 to -30 dB, again using 1000 simulated signals for each point.
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For each decoding algorithm we include three curves: one for the AWGN channel
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and no fading, and two more for Doppler spreads of 0.2 and 1.0 Hz.
|
||||
(For reference, we note that the JT65 symbol rate is about 2.69 Hz.
|
||||
presents the results of simulations for signal-to-noise ratios ranging
|
||||
from
|
||||
\begin_inset Formula $-18$
|
||||
\end_inset
|
||||
|
||||
to
|
||||
\begin_inset Formula $-30$
|
||||
\end_inset
|
||||
|
||||
dB, again using 1000 simulated signals for each plotted point.
|
||||
We include three curves for each decoding algorithm: one for the AWGN channel
|
||||
and no fading, and two more for simulated Doppler spreads of 0.2 and 1.0
|
||||
Hz.
|
||||
For reference, we note that the JT65 symbol rate is about 2.69 Hz.
|
||||
The simulated Doppler spreads are comparable to those encountered on HF
|
||||
ionospheric paths and for EME at VHF and lower UHF bands.)
|
||||
ionospheric paths and for EME at VHF and lower UHF bands.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
@ -1545,11 +1524,16 @@ Hinted Decoding
|
||||
|
||||
\begin_layout Standard
|
||||
...
|
||||
TBD ...
|
||||
Still to come ...
|
||||
\end_layout
|
||||
|
||||
\begin_layout Section
|
||||
Summary
|
||||
Summary
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
...
|
||||
Still to come ...
|
||||
\end_layout
|
||||
|
||||
\begin_layout Bibliography
|
||||
|
||||
Loading…
Reference in New Issue
Block a user