| Version 3 |
Version 2 |
| By the operation of interlace of two data streams $(a_n,\Delta _{a})$ and $(b_n,\Delta _{b}) $ |
By the operation of interlace of two data streams $(a_n,\Delta _{a})$ and $(b_n,\Delta _{b}) $ |
| we will understand the operation creating a new data stream |
we will understand the operation creating a new data stream |
| $(c_n,\Delta _{c})$. |
$(c_n,\Delta _{c})$. |
| If $\Delta _{a}$ and $\Delta _{b}$ are constants then |
If $\Delta _{a}$ and $\Delta _{b}$ are constants then |
| the resulting sequence $c_n$ and the interval $\Delta _{c}$ can be calculated with the help of the following formula: |
the resulting sequence $c_n$ and the interval $\Delta _{c}$ can be calculated with the help of the following formula: |
|
|
| $ |
$ |
| c_{n}=\left\{ |
c_{n}=\left\{ |
| \begin{array}{cc} |
\begin{array}{cc} |
| b_{n-\left\lfloor n z \right\rfloor } & \left\lfloor n z |
b_{n-\left\lfloor n z \right\rfloor } & \left\lfloor n z |
| \right\rfloor =\left\lfloor \left( n+1\right) z \right\rfloor \\ |
\right\rfloor =\left\lfloor \left( n+1\right) z \right\rfloor \\ |
| a_{\left\lfloor n z \right\rfloor } & \left\lfloor n 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 |
\neq \left\lfloor \left( n+1\right) z \right\rfloor |
| \end{array} |
\end{array} |
| \right. , z =\frac{\Delta _{b}}{\Delta _{a}+\Delta _{b}},\Delta _{c}= |
\right. , z =\frac{\Delta _{b}}{\Delta _{a}+\Delta _{b}},\Delta _{c}= |
| \frac{\Delta _{a}\Delta _{b}}{\Delta _{a}+\Delta _{b}} \label{interlace} |
\frac{\Delta _{a}\Delta _{b}}{\Delta _{a}+\Delta _{b}} \label{interlace} |
| $ |
$ |
|
|
| Where: |
Where: |
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| $\Delta _{a,b,c}$ are the values that determine constant |
$\Delta _{a,b,c}$ are the values that determine constant |
| time interval between tuples in streams $A$,$B$ and $C$. |
time interval between tuples in streams $A$,$B$ and $C$. |
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| $n$ denote position of tuple |
$n$ denote position of tuple |
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| $a_{n},b_{n},c_{n}$ denotes tuples of stream $A$, $B$ and $C$. |
$a_{n},b_{n},c_{n}$ denotes tuples of stream $A$, $B$ and $C$. |
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Deinterlace is defined by:
|
Deinterlace is definied by:
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|
| $ |
$ |
| a_{n} = c_{n+ \left\lceil \frac{(n+1)\Delta _{a}}{\Delta _{b}} \right\rceil }\ |
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 } |
,\ \Delta _{a}=\frac{\Delta _{c}\Delta _{b}}{\left\vert \Delta _{c}-\Delta _{b}\right\vert } |
| \label{deinterlace_a} |
\label{deinterlace_a} |
| $ |
$ |
| and |
and |
| $ |
$ |
| b_{n} = c_{n+\left\lfloor \frac{n\Delta _{b}}{\Delta _{a}}\right\rfloor} |
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} |
,\ \Delta _{b}=\frac{\Delta _{c}\Delta _{a}}{\left\vert \Delta _{c}-\Delta_{a}\right\vert } \label{deintrlace_b} |
| $ |
$ |
|
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| This sequences are the Fraenkel Partition Theorem instance. |
This sequences are the Fraenkel Partition Theorem instance. |
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| {\Huge \bf References} |
{\Huge \bf References} |
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| \begin{description} |
\begin{description} |
| \item[ [1] ] Aviezri S. Fraenkel, {\em The bracket function and complementary sets of integers}, Canad. J. |
\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 |
Math. {\bf 21} (1969), 6--27. {\bf \PMlinkexternal{MR |
| 38:3214}{http://www.ams.org/mathscinet-getitem?mr=38:3214}} |
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}} |
\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}} |
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|
| \end{description} |
\end{description} |
