w4kek/WSJT-X
1
0
Fork 0
mirror of https://github.com/saitohirga/WSJT-X.git synced 2024-11-25 21:58:38 -05:00
Code Issues Packages Projects Releases Wiki Activity

Use u_i, sigma_i for Eqs 10 and 11, and other edits.

Browse Source 
git-svn-id: svn+ssh://svn.code.sf.net/p/wsjt/wsjt/branches/wsjtx@6360 ab8295b8-cf94-4d9e-aec4-7959e3be5d79
This commit is contained in:
Joe Taylor 2016-01-06 19:04:42 +00:00
parent cf5f3916c7
commit 20381b971a
1 changed files with 80 additions and 25 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

93
lib/ftrsd/ftrsd_paper/ftrsd.lyx
View File

@ -1051,17 +1051,39 @@ where
\end_inset
In contrast, worst-case incorrect codewords will yield
\end_layout
\begin_layout Standard
In contrast, the expected value and standard deviation of the
\begin_inset Formula $u$
\end_inset
-metrics with expectation value and standard deviation given by
-metric for a randomly selected incorrect codeword (selected from a population
of all
\begin_inset Quotes eld
\end_inset
worst case
\begin_inset Quotes erd
\end_inset
codewords,
\emph on
i.e.
\emph default
, those with
\begin_inset Formula $k-1$
\end_inset
symbols identical to corresponding ones in the correct word) are given
by
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
\bar{u}_{2}=1+\left(\frac{k-1}{n}\right)y,\label{eq:u2-exp}
\bar{u}_{i}=1+\left(\frac{k-1}{n}\right)y,\label{eq:u2-exp}
\end{equation}
\end_inset
@ -1072,39 +1094,72 @@ In contrast, worst-case incorrect codewords will yield
\begin_layout Standard
\begin_inset Formula
\begin{equation}
\sigma_{2}=\frac{1}{n}\left[n+2y(k-1)\right]^{1/2}.\label{eq:sigma2}
\sigma_{i}=\frac{1}{n}\left[n+2y(k-1)\right]^{1/2}.\label{eq:sigma2}
\end{equation}
\end_inset
If
\begin_inset Formula $u$
\end_inset
\end_layout
\begin_layout Standard
If tests on a number of tested candidate codewords yield largest and second-larg
est metrics
is evaluated for a large number of candidate codewords, one of which is
correct, we should expect the largest value
\begin_inset Formula $u_{1}$
\end_inset
to be drawn from a population with statistics described by
\begin_inset Formula $\bar{u}_{1}$
\end_inset
and
\begin_inset Formula $u_{2},$
\begin_inset Formula $\sigma_{1}.$
\end_inset
respectively, we expect the ratio
\begin_inset Formula $r=u_{2}/u_{1}$
\end_inset
to be significantly smaller in cases where the candidate associated with
If no tested codeword is correct,
\begin_inset Formula $u_{1}$
\end_inset
is in fact the correct codeword.
On the other hand, if none of the tested candidates is correct,
\begin_inset Formula $r$
is likely to come from the
\begin_inset Formula $(\bar{u}_{i},\,\sigma_{i})$
\end_inset
population and to be several standard deviations above the mean.
In either case the second-largest value,
\begin_inset Formula $u_{2},$
\end_inset
will likely come from the
\begin_inset Formula $(\bar{u}_{i},\,\sigma_{i})$
\end_inset
population, again several standard deviations above the mean.
\end_layout
\begin_layout Standard
If no tested codeword is correct or the signal-to-noise ratio
\begin_inset Formula $y$
\end_inset
is too small for decoding to be possible, the ratio
\begin_inset Formula $r=u_{2}/u_{1}$
\end_inset
will likely be close to 1.
On the other hand, correctly identified codewords will produce
\begin_inset Formula $u_{1}$
\end_inset
significantly larger than
\begin_inset Formula $u_{2}$
\end_inset
and thus smaller values of
\begin_inset Formula $r$
\end_inset
.
We therefore apply a ratio threshold test, say
\begin_inset Formula $r<r_{1}$
\end_inset