stream interlace and deinterlace

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:

$c_{n}=\left\{\begin{array}[]{cc}b_{n-\left\lfloor nz\right\rfloor}&\left% \lfloor nz\right\rfloor=\left\lfloor\left(n+1\right)z\right\rfloor\\ a_{\left\lfloor nz\right\rfloor}&\left\lfloor nz\right\rfloor\neq\left\lfloor% \left(n+1\right)z\right\rfloor\end{array}\right.,z=\frac{\Delta_{b}}{\Delta_{a% }+\Delta_{b}},\Delta_{c}=\frac{\Delta_{a}\Delta_{b}}{\Delta_{a}+\Delta_{b}}$

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.

$a_{n}=c_{n+\left\lceil\frac{(n+1)\Delta_{a}}{\Delta_{b}}\right\rceil}\ ,\ % \Delta_{a}=\frac{\Delta_{c}\Delta_{b}}{\left|\Delta_{c}-\Delta_{b}\right|}$ and $b_{n}=c_{n+\left\lfloor\frac{n\Delta_{b}}{\Delta_{a}}\right\rfloor},\ \Delta_{% b}=\frac{\Delta_{c}\Delta_{a}}{\left|\Delta_{c}-\Delta_{a}\right|}$

This sequences are the Fraenkel partition theorem instance.

References

[1: ] Aviezri S. Fraenkel, The bracket function and complementary sets of integers, Canad. J. Math. 21 (1969), 6–27. http://www.ams.org/mathscinet-getitem?mr=38:3214MR 38:3214
[2: ] Michal Widera, Deterministic method of data sequence processing, Vol. IV, ISSN 1732-1360, Annales UMCS (2006), 314–331. http://www.annales.umcs.lublin.pl/AI/index.htmlUMCS Annales AI

Title	stream interlace and deinterlace
Canonical name	StreamInterlaceAndDeinterlace
Date of creation	2013-03-22 15:37:08
Last modified on	2013-03-22 15:37:08
Owner	michal (7107)
Last modified by	michal (7107)
Numerical id	16
Author	michal (7107)
Entry type	Theorem
Classification	msc 11B83
Synonym	interlace deinterlace
Related topic	FraenkelsPartitionTheorem
Related topic	BeattySequence
Related topic	DataStream
Defines	stream junction method