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stream interlace and deinterlace (Theorem)

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:

\begin{displaymath} c_{n}=\left\{ \begin{array}{cc} b_{n-\left\lfloor n z \right... ...ta _{c}= \frac{\Delta _{a}\Delta _{b}}{\Delta _{a}+\Delta _{b}}\end{displaymath}

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 }\... ...=\frac{\Delta _{c}\Delta _{b}}{\left\vert \Delta _{c}-\Delta _{b}\right\vert } $ and $ b_{n} = c_{n+\left\lfloor \frac{n\Delta _{b}}{\Delta _{a}}\right\rfloor} ,\ \D... ...b}=\frac{\Delta _{c}\Delta _{a}}{\left\vert \Delta _{c}-\Delta_{a}\right\vert }$

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. MR 38:3214
[2]
Michal Widera, Deterministic method of data sequence processing, Vol. IV, ISSN 1732-1360, Annales UMCS (2006), 314-331. UMCS Annales AI



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See Also: Fraenkel's partition theorem, Beatty sequence, data stream

Other names:  interlace deinterlace
Also defines:  stream junction method
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Cross-references: integers, complementary, bracket function, references, Fraenkel partition theorem, primary, constant terms, sequence, interval, data stream

This is version 13 of stream interlace and deinterlace, born on 2005-12-27, modified 2007-06-23.
Object id is 7543, canonical name is WideraInterlaceAndDeinterlace.
Accessed 9420 times total.

Classification:
AMS MSC11B83 (Number theory :: Sequences and sets :: Special sequences and polynomials)

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