stream interlace and deinterlaceWideraInterlaceAndDeinterlace2005-12-27 08:14:052007-06-23 13:43:48Theoremstream junction method% this is the default PlanetMath preamble. as your knowledge
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% define commands here{\it Interlace} is the method to create a new data stream from
two data streams, each of which has a constant time interval sequence.
Formally, suppose $A = (a , \Delta_{a} )$ and $B = (b, \Delta_{b})$
are two data streams, each have a constant time sequence. For convenience,
we use $\Delta_{a}$ and $\Delta_{b}$ to also denote the constant terms
of each of those sequences. We construct a new data stream $C = (c, \Delta_{c})$, also having constant time interval, as follows:
$
c_{n}=\left\{
\begin{array}{cc}
b_{n-\left\lfloor n z \right\rfloor } & \left\lfloor n z
\right\rfloor =\left\lfloor \left( n+1\right) z \right\rfloor \\
a_{\left\lfloor n z \right\rfloor } & \left\lfloor n z \right\rfloor
\neq \left\lfloor \left( n+1\right) z \right\rfloor
\end{array}
\right. , z =\frac{\Delta _{b}}{\Delta _{a}+\Delta _{b}},\Delta _{c}=
\frac{\Delta _{a}\Delta _{b}}{\Delta _{a}+\Delta _{b}} \label{interlace}
$
{\it Deinterlace } is the method of constructing two data streams , $A$ and
$B$,
each having constant time interval, from a given data stream $C$ and primary interlace value $\Delta $ of computed stream, where
$C$ has constant time interval.
$
a_{n} = c_{n+ \left\lceil \frac{(n+1)\Delta _{a}}{\Delta _{b}} \right\rceil }\
,\ \Delta _{a}=\frac{\Delta _{c}\Delta _{b}}{\left\vert \Delta _{c}-\Delta _{b}\right\vert }
\label{deinterlace_a}
$
and
$
b_{n} = c_{n+\left\lfloor \frac{n\Delta _{b}}{\Delta _{a}}\right\rfloor}
,\ \Delta _{b}=\frac{\Delta _{c}\Delta _{a}}{\left\vert \Delta _{c}-\Delta_{a}\right\vert } \label{deintrlace_b}
$
This sequences are the Fraenkel partition theorem instance.
{\Huge \bf References}
\begin{description}
\item[ [1] ] Aviezri S. Fraenkel, {\em The bracket function and complementary sets of integers}, Canad. J.
Math. {\bf 21} (1969), 6--27. {\bf \PMlinkexternal{MR
38:3214}{http://www.ams.org/mathscinet-getitem?mr=38:3214}}
\item[ [2] ] Michal Widera, {\em Deterministic method of data sequence processing}, Vol. IV, ISSN 1732-1360, Annales UMCS (2006), 314--331. {\bf \PMlinkexternal{UMCS Annales AI}{http://www.annales.umcs.lublin.pl/AI/index.html}}
\end{description}