w4kek/WSJT-X
1
0
Fork 0
mirror of https://github.com/saitohirga/WSJT-X.git synced 2025-07-30 20:52:28 -04:00
Code Issues Packages Projects Releases Wiki Activity

Another incremental draft of the FTRSD paper. Expect more changes fairly soon!

Browse Source 
git-svn-id: svn+ssh://svn.code.sf.net/p/wsjt/wsjt/branches/wsjtx@6350 ab8295b8-cf94-4d9e-aec4-7959e3be5d79
This commit is contained in:
Joe Taylor 2016-01-05 17:36:15 +00:00
parent 6595c329a8
commit ccef778b02
2 changed files with 2214 additions and 2027 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

BIN
lib/ftrsd/ftrsd_paper/fig_wer.pdf
View File

Binary file not shown.

605
lib/ftrsd/ftrsd_paper/ftrsd.lyx
View File

@ -93,24 +93,6 @@ LatexCommand tableofcontents
\end_layout
\begin_layout Abstract
The JT65 protocol 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 not usable
with traditional technologies.
A major reason for the success and popularity of JT65 is its use of a strong
error-correction code: a short block-length, low-rate Reed-Solomon code
based on a 64-symbol alphabet.
Since 2004, most programs implementing JT65 have used the patented Koetter-Vard
y (KV) algebraic soft-decision decoder, licensed to K1JT and implemented
in a closed-source program for use in amateur radio applications.
We describe here a new open-source alternative called the Franke-Taylor
(FT, or K9AN-K1JT) algorithm.
It is conceptually simple, built around the well-known Berlekamp-Massey
errors-and-erasures algorithm, and in this application it performs even
better than the KV decoder.
\end_layout
\begin_layout Section
\begin_inset CommandInset label
LatexCommand label
@ -122,32 +104,78 @@ Introduction and Motivation
\end_layout
\begin_layout Standard
The following paragraph may not belong here - feel free to get rid of it,
change it, whatever.
The JT65 protocol has revolutionized amateur-radio weak-signal communication
by enabling operators with small or compromise antennas and relatively
low-power transmitters to communicate over propagation paths not usable
with traditional technologies.
The protocol was developed in 2003 for Earth-Moon-Earth (EME, or
\begin_inset Quotes eld
\end_inset
moonbounce
\begin_inset Quotes erd
\end_inset
) communication, where the scattered return signals are always weak.
It was soon found that JT65 also facilitates worldwide communication on
the HF bands with low power, modest antennas, and efficient spectral usage.
\end_layout
\begin_layout Standard
The Franke-Taylor (FT) decoder is a probabilistic list-decoder that we have
developed for use in the short block-length, low-rate Reed-Solomon code
used in JT65.
JT65 provides a unique sandbox for playing with decoding algorithms.
Several seconds are available for decoding a single 63-symbol message.
This is a long time! The luxury of essentially unlimited time allows us
to experiment with decoders that have high computational complexity.
The payoff is that we can extend the decoding threshold by many dB over
the hard-decision, Berlekamp-Massey decoder on a typical fading channel,
and by a meaningful amount over the KV decoder, long considered to be the
best available soft-decision decoder.
In addition to its excellent performance, the FT algorithm has other desirable
properties, not the least of which is its conceptual simplicity.
Decoding performance and complexity scale in a useful way, providing steadily
increasing soft-decision decoding gain as a tunable computational complexity
parameter is increased over more than 5 orders of magnitude.
This means that appreciable gain should be available from our decoder even
on very simple (and slow) computers.
On the other hand, because the algorithm requires a large number of independent
decoding trials, it should be possible to obtain significant performance
gains through parallelization on high-performance computers.
A major reason for the success and popularity of JT65 is its use of a strong
error-correction code: a short block-length, low-rate Reed-Solomon code
based on a 64-symbol alphabet.
Until now, nearly all programs implementing JT65 have used the patented
Koetter-Vardy (KV) algebraic soft-decision decoder
\begin_inset CommandInset citation
LatexCommand cite
key "kv2001"
\end_inset
, as licensed to K1JT and implemented in a closed-source program for use
only in amateur radio applications.
Since 2001 the KV decoder has been considered the best available soft-decision
decoder for Reed Solomon codes.
\end_layout
\begin_layout Standard
We describe here a new open-source alternative called the Franke-Taylor
(FT, or K9AN-K1JT) algorithm.
It is conceptually simple, built around the well-known Berlekamp-Massey
errors-and-erasures algorithm, and in this application it performs even
better than the KV decoder.
The FT algorithm is implemented in the popular program
\emph on
WSJT-X
\emph default
, widely used for amateur weak-signal communication with JT65 and several
other specialized digital modes.
The program is freely available and licensed under the GNU General Public
License.
\end_layout
\begin_layout Standard
The JT65 protocol specifies transmissions that normally start one second
into a UTC minute and last for 46.8 seconds.
Receiving software therefore has up to several seconds to decode a message,
before the operator sends a reply at the start of the next minute.
With today's personal computers, this relatively long time for decoding
a short message encourages experimentation with decoders of high computational
complexity.
As a result, on a typical fading channel the FT algorithm extends the decoding
threshold by many dB over the hard-decision Berlekamp-Massey decoder, and
by a meaningful amount over the KV decoder.
In addition to its excellent performance, the new algorithm has other desirable
properties---not the least of which is its conceptual simplicity.
Decoding performance and complexity scale in a convenient way, providing
steadily increasing soft-decision decoding gain as a tunable computational
complexity parameter is increased over more than 5 orders of magnitude.
This means that appreciable gain is available from our decoder even on
very simple (and relatively slow) computers.
On the other hand, because the algorithm benefits from a large number of
independent decoding trials, it should be possible to obtain further performanc
e gains through parallelization on high-performance computers.
\end_layout
\begin_layout Section
@ -195,7 +223,8 @@ on with a Reed-Solomon code.
\end_layout
\begin_layout Standard
The concept of Hamming distance is used as a measure of
In coding theory the concept of Hamming distance is used as a measure of
\begin_inset Quotes eld
\end_inset
@ -412,8 +441,8 @@ n.
\end_layout
\begin_layout Standard
As a specific example, we will consider a received JT65 word with 23 correct
symbols and 40 errors.
As a specific example, consider a received JT65 word with 23 correct symbols
and 40 errors.
We do not know which symbols are in error.
Suppose that the decoder randomly selects
\begin_inset Formula $s=40$
@ -509,17 +538,9 @@ where
is the binomial coefficient.
The binomial coefficient can be calculated using the function
\begin_inset Quotes eld
\end_inset
nchoosek(
\begin_inset Formula $n,k$
\end_inset
)
\begin_inset Quotes erd
\end_inset
\family typewriter
nchoosek(n,k)
\family default
in the interpreted language GNU Octave, or with one of many free online
calculators.
The hypergeometric probability mass function defined in Eq.
@ -531,18 +552,10 @@ reference "eq:hypergeometric_pdf"
\end_inset
) is available in GNU Octave as function
\begin_inset Quotes eld
\end_inset
hygepdf(
\begin_inset Formula $x,N,X,s$
\end_inset
)
\begin_inset Quotes erd
\end_inset
.
\family typewriter
hygepdf
\family default
(x,N,X,s).
The cumulative probability that at least
\begin_inset Formula $\epsilon$
\end_inset
@ -792,15 +805,16 @@ The FT algorithm uses quality indices made available by a noncoherent 64-FSK
\begin_inset Formula $S(i,j)$
\end_inset
for each signalling interval; for the JT65 protocol
\begin_inset Formula $i=1,64$
for each signalling interval; for the JT65 protocol the frequency index
and symbol index have values
\begin_inset Formula $i=$
\end_inset
is the frequency index and
\begin_inset Formula $j=1,63$
1 to 64 and
\begin_inset Formula $j=$
\end_inset
the symbol index.
1 to 63.
The most likely value for symbol
\begin_inset Formula $j$
\end_inset
@ -820,7 +834,7 @@ The FT algorithm uses quality indices made available by a noncoherent 64-FSK
\begin_inset Formula $p_{2}$
\end_inset
, are passed from demodulator to decoder as soft-symbol information.
, are computed and passed from demodulator to decoder as soft-symbol information.
The FT decoder derives two metrics from
\begin_inset Formula $p_{1}$
\end_inset
@ -841,7 +855,7 @@ The FT algorithm uses quality indices made available by a noncoherent 64-FSK
\end_inset
of the symbol's fractional power
\begin_inset Formula $p_{1,\, j}$
\begin_inset Formula $p_{1,\,j}$
\end_inset
in a sorted list of
@ -891,21 +905,14 @@ a priori
.
Correspondingly, the FT algorithm works best when the probability of erasing
a symbol is somewhat larger than the probability that the symbol is incorrect.
We found empirically that good decoding performance is obtained when the
symbol erasure probability is about 1.3 times the symbol error probability.
For the JT65 code we found empirically that good decoding performance is
obtained when the symbol erasure probability is about 1.3 times the symbol
error probability.
\end_layout
\begin_layout Standard
The FT algorithm tries successively to decode the received word using independen
t
\begin_inset Quotes eld
\end_inset
educated guesses
\begin_inset Quotes erd
\end_inset
to select symbols for erasure.
t educated guesses to select symbols for erasure.
For each iteration a stochastic erasure vector is generated based on the
symbol erasure probabilities.
The erasure vector is sent to the BM decoder along with the full set of
@ -918,7 +925,7 @@ educated guesses
, the soft distance between the received word and the codeword:
\begin_inset Formula
\begin{equation}
d_{s}=\sum_{j=1}^{n}\alpha_{j}\,(1+p_{1,j}).\label{eq:soft_distance}
d_{s}=\sum_{j=1}^{n}\alpha_{j}\,(1+p_{1,\,j}).\label{eq:soft_distance}
\end{equation}
\end_inset
@ -936,7 +943,7 @@ Here
\end_inset
if the received symbol and codeword symbol are different, and
\begin_inset Formula $p_{1,j}$
\begin_inset Formula $p_{1\,j}$
\end_inset
is the fractional power associated with received symbol
@ -958,13 +965,14 @@ In practice we find that
\end_inset
can reliably indentify the correct codeword if the signal-to-noise ratio
for individual symbols is greater than about 4 in power units, or
for individual symbols is greater than about 4 in linear power units, or
\begin_inset Formula $E_{s}/N_{0}\apprge6$
\end_inset
dB.
We also find that weaker signals frequently can be decoded by using soft-symbol
information beyond that contained in
dB (*** check these numbers ***).
We also find that significantly weaker signals can be decoded by using
soft-symbol information beyond that contained in
\begin_inset Formula $p_{1}$
\end_inset
@ -977,15 +985,15 @@ In practice we find that
\begin_inset Formula $u$
\end_inset
, the average signal-plus-noise power in all symbols, according to a candidate
, the average signal-plus-noise power in all symbols according to a candidate
codeword's symbol values:
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
u=\frac{1}{n}\sum_{j=1}^{n}S(c_{j},\, j).
\]
\begin{equation}
u=\frac{1}{n}\sum_{j=1}^{n}S(c_{j},\,j).\label{eq:u-metric}
\end{equation}
\end_inset
@ -994,15 +1002,11 @@ Here the
\end_inset
's are the symbol values for the candidate codeword being tested.
\end_layout
\begin_layout Standard
The correct JT65 codeword produces a value for
The correct JT65 codeword produces a value for
\begin_inset Formula $u$
\end_inset
equal to average of
equal to the average of
\begin_inset Formula $n=63$
\end_inset
@ -1017,18 +1021,23 @@ The correct JT65 codeword produces a value for
bins containing noise only.
Thus, if the spectral array
\begin_inset Formula $S(i,\, j)$
\begin_inset Formula $S(i,\,j)$
\end_inset
has been normalized so that its median value (essentially the average noise
level) is unity, the correct codeword is expected to yield the metric value
level) is unity,
\begin_inset Formula $u$
\end_inset
for the correct codeword has expectation value (average over many random
realizations)
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
u=(1\pm n^{-\frac{1}{2}})(1+y)\approx(1.0\pm0.13)(1+y),
\]
\begin{equation}
\bar{u}_{1}=1+y,\label{eq:u1-exp}
\end{equation}
\end_inset
@ -1036,60 +1045,95 @@ where
\begin_inset Formula $y$
\end_inset
is the signal-to-noise ratio (in linear power units) and the quoted one-standar
d-deviation uncertainty range assumes Gaussian statistics.
Incorrect codewords will yield metric values no larger than
is the signal-to-noise ratio in linear power units.
If we assume Gaussian statistics and a large number of trials, the standard
deviation of measured values of
\begin_inset Formula $u_{1}$
\end_inset
is
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
u=\frac{n-k+1\pm\sqrt{n-k+1}}{n}+\frac{k-1\pm\sqrt{k-1}}{n}(1+y).
\]
\begin{equation}
\sigma_{1}=\left(\frac{1+2y}{n}\right)^{1/2}.\label{eq:sigma1}
\end{equation}
\end_inset
For JT65 this expression evaluates to
In contrast, worst-case incorrect codewords will yield
\begin_inset Formula $u$
\end_inset
-metrics with expectation value and standard deviation given by
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
u\approx1\pm0.11+(0.17\pm0.05)\, y.
\]
\begin{equation}
\bar{u}_{2}=1+\left(\frac{k-1}{n}\right)y,\label{eq:u2-exp}
\end{equation}
\end_inset
As a specific example, consider signal strength
\begin_inset Formula $y=4$
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
\sigma_{2}=\frac{1}{n}\left[n+2y(k-1)\right]^{1/2}.\label{eq:sigma2}
\end{equation}
\end_inset
, corresponding to
\begin_inset Formula $E_{s}/N_{0}=6$
\end_layout
\begin_layout Standard
If tests on a number of tested candidate codewords yield largest and second-larg
est metrics
\begin_inset Formula $u_{1}$
\end_inset
dB.
For JT65, the corresponding SNR in 2500 Hz bandwidth is
\begin_inset Formula $-23.7$
and
\begin_inset Formula $u_{2},$
\end_inset
dB.
The correct codeword is then expected to yield
\begin_inset Formula $u\approx5.0\pm$
respectively, we expect the ratio
\begin_inset Formula $r=u_{2}/u_{1}$
\end_inset
0.6, while incorrect codewords will give
\begin_inset Formula $u\approx1.7\pm0.3$
to be significantly smaller in cases where the candidate associated with
\begin_inset Formula $u_{1}$
\end_inset
or less.
We find that a threshold set at
\begin_inset Formula $u_{0}=4.4$
is in fact the correct codeword.
On the other hand, if none of the tested candidates is correct,
\begin_inset Formula $r$
\end_inset
(about 8 standard deviations above the expected maximum for incorrect codewords
) reliably serves to distinguish correct codewords from all other candidates,
while ensuring a very small probability of false decodes.
will likely be close to 1.
We therefore apply a ratio threshold test, say
\begin_inset Formula $r<r_{0}$
\end_inset
, to identify codewords with high probability of being correct.
As described in Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Theory,-Simulation,-and"
\end_inset
, we have used simulations to set an empirical acceptance threshold
\begin_inset Formula $r_{0}$
\end_inset
that maximizes the probability of correct decodes while ensuring a low
rate of false decodes.
\end_layout
\begin_layout Standard
@ -1101,11 +1145,21 @@ Technically the FT algorithm is a list decoder.
is retained.
As with all such algorithms, a stopping criterion is necessary.
FT accepts a codeword unconditionally if
\begin_inset Formula $u>u_{0}$
FT accepts a codeword unconditionally if the Hamming distance and soft
distance
\begin_inset Formula $d_{s}$
\end_inset
.
are less than some conservatively specified limits.
Secondary acceptance criteria
\begin_inset Formula $d_{s}<d_{0}$
\end_inset
and
\begin_inset Formula $r<r_{0}$
\end_inset
are used to validate additional decodes.
A timeout is used to limit the algorithm's execution time if no acceptable
codeword is found in a reasonable number of trials,
\begin_inset Formula $T$
@ -1121,10 +1175,25 @@ Technically the FT algorithm is a list decoder.
\end_inset
or even higher.
Pseudo-code for the FT algorithm is presented in an accompanying text box.
\end_layout
\begin_layout Paragraph
Algorithm pseudo-code:
\begin_layout Standard
\begin_inset Float algorithm
wide false
sideways false
status open
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Pseudo-code for the FT algorithm.
\end_layout
\end_inset
\end_layout
\begin_layout Enumerate
@ -1134,7 +1203,7 @@ For each received symbol, define the erasure probability as 1.3 times the
a priori
\emph default
symbol-error probability determined from soft-symbol information
\begin_inset Formula $\{p_{1}\textrm{-rank},\, p_{2}/p_{1}\}$
\begin_inset Formula $\{p_{1}\textrm{-rank},\,p_{2}/p_{1}\}$
\end_inset
.
@ -1150,17 +1219,6 @@ Make independent stochastic decisions about whether to erase each symbol
Attempt errors-and-erasures decoding by using the BM algorithm and the set
of erasures determined in step 2.
If the BM decoder produces a candidate codeword, go to step 5.
\begin_inset Foot
status open
\begin_layout Plain Layout
Our implementation of the FT-algorithm is based on the excellent open-source
BM decoder written by Phil Karn, KA9Q.
\end_layout
\end_inset
\end_layout
\begin_layout Enumerate
@ -1168,8 +1226,12 @@ If BM decoding was not successful, go to step 2.
\end_layout
\begin_layout Enumerate
Calculate the hard-decision Hamming distance between the candidate codeword
and the received symbols, the corresponding soft distance
Calculate the hard-decision Hamming distance
\begin_inset Formula $h$
\end_inset
between the candidate codeword and the received symbols, the corresponding
soft distance
\begin_inset Formula $d_{s}$
\end_inset
@ -1182,8 +1244,16 @@ Calculate the hard-decision Hamming distance between the candidate codeword
\begin_inset Formula $u$
\end_inset
is the largest one encountered so far, set
\begin_inset Formula $u_{max}=u$
is the largest one encountered so far, preserve the previous value of
\begin_inset Formula $u_{1}$
\end_inset
as
\begin_inset Formula $u_{2}$
\end_inset
and then set
\begin_inset Formula $u_{1}=u$
\end_inset
.
@ -1191,11 +1261,14 @@ Calculate the hard-decision Hamming distance between the candidate codeword
\begin_layout Enumerate
If
\begin_inset Formula $u_{max}>u_{0}$
\begin_inset Formula $h<h_{0}$
\end_inset
, go to step 8.
and
\begin_inset Formula $d_{s}<d_{0}$
\end_inset
, go to step 10.
\end_layout
\begin_layout Enumerate
@ -1204,7 +1277,27 @@ If the number of trials is less than the timeout limit
\end_inset
go to 2.
Otherwise, declare decoding failure and exit.
\begin_inset Formula $ $
\end_inset
\end_layout
\begin_layout Enumerate
If
\begin_inset Formula $d_{s}<d_{1}$
\end_inset
and
\begin_inset Formula $r<r_{1}$
\end_inset
go to step 10.
\end_layout
\begin_layout Enumerate
Otherwise, declare decoding failure and exit.
\end_layout
\begin_layout Enumerate
@ -1216,26 +1309,31 @@ An acceptable codeword with
Declare a successful decode and return this codeword.
\end_layout
\end_inset
\end_layout
\begin_layout Standard
The inspiration for the FT decoding algorithm came from a number of sources,
Inspiration for the FT decoding algorithm came from a number of sources,
particularly references
\begin_inset CommandInset citation
LatexCommand cite
key "key-2"
key "lhmg2010"
\end_inset
and
\begin_inset CommandInset citation
LatexCommand cite
key "key-3"
key "lk2008"
\end_inset
and the textbook by Lin and Costello
\begin_inset CommandInset citation
LatexCommand cite
key "key-1"
key "lc2004"
\end_inset
@ -1252,7 +1350,7 @@ stochastic erasures-only list decoding algorithm
, described in reference
\begin_inset CommandInset citation
LatexCommand cite
key "key-4"
key "ls2009"
\end_inset
@ -1260,15 +1358,15 @@ key "key-4"
The algorithm in
\begin_inset CommandInset citation
LatexCommand cite
key "key-4"
key "ls2009"
\end_inset
is applied to higher-rate Reed-Solomon codes on a binary-input channel
over which BPSK-modulated symbols are transmitted.
with BPSK-modulated symbols.
Our 64-ary input channel with 64-FSK modulation required us to develop
our own unique methods for assigning erasure probabilities and for defining
an acceptance criteria to select the best codeword from the list of candidates.
unique methods for assigning erasure probabilities and for defining an
acceptance criteria to select the best codeword from the list of candidates.
\end_layout
@ -1283,21 +1381,110 @@ Hinted Decoding
\end_layout
\begin_layout Standard
To be written...
\end_layout
The FT algorithm is completely general: it recovers with equal sensitivity
any one of the
\begin_inset Formula $2^{72}\approx4.7\times10^{21}$
\end_inset
\begin_layout Section
\begin_inset CommandInset label
LatexCommand label
name "sec:Implementation-in-WSJT-X"
different messages that can be transmitted using the JT65 protocol.
In many circumstances it's easy to imagine a much smaller list of messages
(say, a few thousand or less) that may be among the most likely ones to
be received.
For example, one such situation exists when making short ham-radio contacts
exchanging minimal amounts of information such as callsigns, signal reports,
perhaps a Maidenhead locator, and acknowledgments.
Similarly, on the EME path or on a VHF or UHF band with limited geographical
coverage, the most likely received messages will often originate from callsigns
that have been decoded before.
Saving a list of previously decoded callsigns makes it easy to generate
lists of hypothetical messages and their corresponding codewords, at very
little computational expense.
The resulting candidate codewords can be tested in the same way as those
generated by the probabilistic method described in Setcion
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:The-decoding-algorithm"
\end_inset
Implementation in WSJT-X
.
We call this approach
\begin_inset Quotes eld
\end_inset
hinted decoding;
\begin_inset Quotes erd
\end_inset
it is sometimes referred to as the
\begin_inset Quotes eld
\end_inset
Deep Search
\begin_inset Quotes erd
\end_inset
algorithm.
In certain limited situations it can provide enhanced sensitivity for the
principal task of any decoder, namely to determine what message was sent.
\end_layout
\begin_layout Standard
To be written...
For hinted decoding we again invoke a ratio threshold test, but in this
case we use it to answer a more limited question.
Over the full list of messages considered likely, we want to know whether
\begin_inset Formula $r=u_{2}/u_{1}$
\end_inset
, the ratio of second-largest to largest
\begin_inset Formula $u$
\end_inset
-metric, is small enough for us to be confident the codeword associated
with
\begin_inset Formula $u_{1}$
\end_inset
is the one that was transmitted.
Once again we will set an empirical limit, say
\begin_inset Formula $r_{2},$
\end_inset
for what is
\begin_inset Quotes eld
\end_inset
small enough
\begin_inset Quotes erd
\end_inset
for adequate confidence, while still ensuring that false decodes are rare.
Because tested candidate codewords are drawn from a list typically no longer
than a few thousand, rather than
\begin_inset Formula $2^{72},$
\end_inset
\begin_inset Formula $r_{2}$
\end_inset
can be a more relaxed limit than the ones
\begin_inset Formula $r_{0}$
\end_inset
and
\begin_inset Formula $r_{1}$
\end_inset
used in the FT algorithm.
For the limited subset of messages considered as likely, hinted decodes
can be obtained at lower signal levels than would be required for decodes
selected from the full universe of
\begin_inset Formula $2^{72}$
\end_inset
distinct messages.
\end_layout
\begin_layout Section
@ -1316,7 +1503,7 @@ Simulated results on the AWGN channel
\begin_layout Standard
Comparisons of decoding performance are usually presented in the professional
literature as plots of word error-rate versus
literature as plots of word error rate versus
\begin_inset Formula $E_{b}/N_{0}$
\end_inset
@ -1326,10 +1513,10 @@ Comparisons of decoding performance are usually presented in the professional
\end_inset
.
In amateur radio circles performance is usually plotted as the probability
of successfully decoding a received word vs signal-to-noise ratio in a
2.5 kHz reference bandwidth,
\begin_inset Formula $\mathrm{SNR}{}_{2.5\,\mathrm{kHz}}$
For weak-signal amateur radio work, performance is more conveniently presented
as the probability of successfully decoding a received word versus signal-to-no
ise ratio in a 2500 Hz reference bandwidth,
\begin_inset Formula $\mathrm{SNR}{}_{2500}$
\end_inset
.
@ -1338,7 +1525,7 @@ Comparisons of decoding performance are usually presented in the professional
\end_inset
and
\begin_inset Formula $\mathrm{SNR}{}_{2.5\,\mathrm{kHz}}$
\begin_inset Formula $\mathrm{SNR}{}_{2500}$
\end_inset
is described in Appendix
@ -1787,19 +1974,24 @@ Summary
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "1"
key "key-1"
key "kv2001"
\end_inset
Error Control Coding, 2nd edition, Shu Lin and Daniel J.
Costello, Pearson-Prentice Hall, 2004.
“Algebraic soft-decision decoding of Reed-Solomon codes,” R.
Köetter and A.
Vardy, IEEE Trans.
Inform.
Theory, Vol.
49, Nov.
2003.
\end_layout
\begin_layout Bibliography
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "2"
key "key-2"
key "lhmg2010"
\end_inset
@ -1814,7 +2006,7 @@ key "key-2"
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "3"
key "key-3"
key "lk2008"
\end_inset
@ -1837,7 +2029,19 @@ GLOBECOM
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "4"
key "key-4"
key "lc2004"
\end_inset
Error Control Coding, 2nd edition, Shu Lin and Daniel J.
Costello, Pearson-Prentice Hall, 2004.
\end_layout
\begin_layout Bibliography
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "5"
key "ls2009"
\end_inset
@ -1855,28 +2059,11 @@ Stochastic Erasure-Only List Decoding Algorithms for Reed-Solomon Codes,
8, August 2009.
\end_layout
\begin_layout Bibliography
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "5"
key "key-5"
\end_inset
“Algebraic soft-decision decoding of Reed-Solomon codes,” R.
Köetter and A.
Vardy, IEEE Trans.
Inform.
Theory, Vol.
49, Nov.
2003.
\end_layout
\begin_layout Bibliography
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "6"
key "key-6"
key "karn"
\end_inset
@ -1922,7 +2109,7 @@ where
is the bandwidth in Hz.
In amateur radio applications, digital modes are often compared based on
the SNR defined in a 2.5 kHz reference bandwidth,
\begin_inset Formula $\mathrm{SNR}_{2.5\,\mathrm{kHz}}$
\begin_inset Formula $\mathrm{SNR}_{2500}$
\end_inset
.
@ -1994,7 +2181,7 @@ reference "eq:Eb_Es"
\end_inset
),
\begin_inset Formula $\mathrm{SNR}_{2.5\,\mathrm{kHz}}$
\begin_inset Formula $\mathrm{SNR}_{2500}$
\end_inset
can be written in terms of
@ -2004,7 +2191,7 @@ reference "eq:Eb_Es"
:
\begin_inset Formula
\[
\mathrm{SNR}_{2.5\,\mathrm{kHz}}=1.23\times10^{-3}\frac{E_{b}}{N_{o}}.
\mathrm{SNR}_{2500}=1.23\times10^{-3}\frac{E_{b}}{N_{o}}.
\]
\end_inset
@ -2015,7 +2202,7 @@ If all quantities are expressed in dB, then:
\begin_layout Standard
\begin_inset Formula
\[
SNR_{2.5\,\mathrm{kHz}}=(E_{b}/N_{o})_{\mathrm{dB}}-29.1\,\mathrm{dB}.
\mathrm{SNR}_{2500}=(E_{b}/N_{o})_{\mathrm{dB}}-29.1\,\mathrm{dB}.
\]
\end_inset