Number of $S$-random interleaving sequences
$begingroup$
I do not know if this problem has already been addressed in number theory known with another name, but I have been some time unsuccessfully trying to find an answer.
In turbo coding a $S$-random interleaver is a scrambling of the bits with the constraint that the distance of the previous $S$ indices of the scrambled sequence have distance $S$ with the next index, that is, $|pi(i)-pi(i-j)|> S,j=1,..., S$, where $pi(i)$ refers to the index of the bit after the permutation. This problem is then to rearrange a sequence of natural numbers $1,...,N$ so that the distance between one of the numbers with its previous $S$ numbers is greater or equal to $S$.
Computational algorithms to obtain such sequences exist, with no guarantee of convergence, and with a reasonable computational time if $S$ is chosen such that $Sleqsqrt{frac{N}{2}}$.
I am wondering if it is possible to determine analytically how many $S$-random sequences can be constructed for a given value of $N$.
number-theory coding-theory interleaving
asked May 15 '18 at 10:23
Josu Etxezarreta MartinezJosu Etxezarreta Martinez
9741517
$endgroup$
add a comment |
$begingroup$
I do not know if this problem has already been addressed in number theory known with another name, but I have been some time unsuccessfully trying to find an answer.
In turbo coding a $S$-random interleaver is a scrambling of the bits with the constraint that the distance of the previous $S$ indices of the scrambled sequence have distance $S$ with the next index, that is, $|pi(i)-pi(i-j)|> S,j=1,..., S$, where $pi(i)$ refers to the index of the bit after the permutation. This problem is then to rearrange a sequence of natural numbers $1,...,N$ so that the distance between one of the numbers with its previous $S$ numbers is greater or equal to $S$.
Computational algorithms to obtain such sequences exist, with no guarantee of convergence, and with a reasonable computational time if $S$ is chosen such that $Sleqsqrt{frac{N}{2}}$.
I am wondering if it is possible to determine analytically how many $S$-random sequences can be constructed for a given value of $N$.
number-theory coding-theory interleaving
asked May 15 '18 at 10:23
Josu Etxezarreta MartinezJosu Etxezarreta Martinez
9741517
$endgroup$
$begingroup$
Sounds difficult to me, but I haven't played too much with turbo interleavers. +1 for asking this well. Anyway, a remark I want to make is that it may not be necessary to maximize $S$ to get optimal performance from the turbo code. Avoiding "short cycles" (in terms of LLRs of some bits feeding back to themselves too quickly) is also important. But, my experience is very limited.
$endgroup$
– Jyrki Lahtonen
May 15 '18 at 12:46
$begingroup$
Yeah, I am aware of the fact that the maximization of $S$ is not necessary for the optimal performance of Turbo codes, as more elavorated interleaving schemes might imply avoiding "short cycles". However, $S$-random interleavers do at least decorrelate the input bits feeding the Turbo code, and so they might imply an increase in the performance of Turbo codes, at least at short block length. That's the reason for me being interested in such problem. Thanks for the comment!
$endgroup$
– Josu Etxezarreta Martinez
May 15 '18 at 13:55
add a comment |
$begingroup$
I do not know if this problem has already been addressed in number theory known with another name, but I have been some time unsuccessfully trying to find an answer.
In turbo coding a $S$-random interleaver is a scrambling of the bits with the constraint that the distance of the previous $S$ indices of the scrambled sequence have distance $S$ with the next index, that is, $|pi(i)-pi(i-j)|> S,j=1,..., S$, where $pi(i)$ refers to the index of the bit after the permutation. This problem is then to rearrange a sequence of natural numbers $1,...,N$ so that the distance between one of the numbers with its previous $S$ numbers is greater or equal to $S$.
Computational algorithms to obtain such sequences exist, with no guarantee of convergence, and with a reasonable computational time if $S$ is chosen such that $Sleqsqrt{frac{N}{2}}$.
I am wondering if it is possible to determine analytically how many $S$-random sequences can be constructed for a given value of $N$.
number-theory coding-theory interleaving
asked May 15 '18 at 10:23
Josu Etxezarreta MartinezJosu Etxezarreta Martinez
9741517
$endgroup$
I do not know if this problem has already been addressed in number theory known with another name, but I have been some time unsuccessfully trying to find an answer.
In turbo coding a $S$-random interleaver is a scrambling of the bits with the constraint that the distance of the previous $S$ indices of the scrambled sequence have distance $S$ with the next index, that is, $|pi(i)-pi(i-j)|> S,j=1,..., S$, where $pi(i)$ refers to the index of the bit after the permutation. This problem is then to rearrange a sequence of natural numbers $1,...,N$ so that the distance between one of the numbers with its previous $S$ numbers is greater or equal to $S$.
Computational algorithms to obtain such sequences exist, with no guarantee of convergence, and with a reasonable computational time if $S$ is chosen such that $Sleqsqrt{frac{N}{2}}$.
I am wondering if it is possible to determine analytically how many $S$-random sequences can be constructed for a given value of $N$.
number-theory coding-theory interleaving
number-theory coding-theory interleaving
asked May 15 '18 at 10:23
Josu Etxezarreta MartinezJosu Etxezarreta Martinez
9741517
asked May 15 '18 at 10:23
Josu Etxezarreta MartinezJosu Etxezarreta Martinez
9741517
edited Dec 13 '18 at 14:47
Josu Etxezarreta Martinez
asked May 15 '18 at 10:23
Josu Etxezarreta MartinezJosu Etxezarreta Martinez
9741517
asked May 15 '18 at 10:23
Josu Etxezarreta MartinezJosu Etxezarreta Martinez
9741517
asked May 15 '18 at 10:23
Josu Etxezarreta MartinezJosu Etxezarreta Martinez
9741517
9741517
$begingroup$
Sounds difficult to me, but I haven't played too much with turbo interleavers. +1 for asking this well. Anyway, a remark I want to make is that it may not be necessary to maximize $S$ to get optimal performance from the turbo code. Avoiding "short cycles" (in terms of LLRs of some bits feeding back to themselves too quickly) is also important. But, my experience is very limited.
$endgroup$
– Jyrki Lahtonen
May 15 '18 at 12:46
$begingroup$
Yeah, I am aware of the fact that the maximization of $S$ is not necessary for the optimal performance of Turbo codes, as more elavorated interleaving schemes might imply avoiding "short cycles". However, $S$-random interleavers do at least decorrelate the input bits feeding the Turbo code, and so they might imply an increase in the performance of Turbo codes, at least at short block length. That's the reason for me being interested in such problem. Thanks for the comment!
$endgroup$
– Josu Etxezarreta Martinez
May 15 '18 at 13:55
add a comment |
$begingroup$
Sounds difficult to me, but I haven't played too much with turbo interleavers. +1 for asking this well. Anyway, a remark I want to make is that it may not be necessary to maximize $S$ to get optimal performance from the turbo code. Avoiding "short cycles" (in terms of LLRs of some bits feeding back to themselves too quickly) is also important. But, my experience is very limited.
$endgroup$
– Jyrki Lahtonen
May 15 '18 at 12:46
$begingroup$
Yeah, I am aware of the fact that the maximization of $S$ is not necessary for the optimal performance of Turbo codes, as more elavorated interleaving schemes might imply avoiding "short cycles". However, $S$-random interleavers do at least decorrelate the input bits feeding the Turbo code, and so they might imply an increase in the performance of Turbo codes, at least at short block length. That's the reason for me being interested in such problem. Thanks for the comment!
$endgroup$
– Josu Etxezarreta Martinez
May 15 '18 at 13:55
$begingroup$
Sounds difficult to me, but I haven't played too much with turbo interleavers. +1 for asking this well. Anyway, a remark I want to make is that it may not be necessary to maximize $S$ to get optimal performance from the turbo code. Avoiding "short cycles" (in terms of LLRs of some bits feeding back to themselves too quickly) is also important. But, my experience is very limited.
$endgroup$
– Jyrki Lahtonen
May 15 '18 at 12:46
$begingroup$
Sounds difficult to me, but I haven't played too much with turbo interleavers. +1 for asking this well. Anyway, a remark I want to make is that it may not be necessary to maximize $S$ to get optimal performance from the turbo code. Avoiding "short cycles" (in terms of LLRs of some bits feeding back to themselves too quickly) is also important. But, my experience is very limited.
$endgroup$
– Jyrki Lahtonen
May 15 '18 at 12:46
$begingroup$
Yeah, I am aware of the fact that the maximization of $S$ is not necessary for the optimal performance of Turbo codes, as more elavorated interleaving schemes might imply avoiding "short cycles". However, $S$-random interleavers do at least decorrelate the input bits feeding the Turbo code, and so they might imply an increase in the performance of Turbo codes, at least at short block length. That's the reason for me being interested in such problem. Thanks for the comment!
$endgroup$
– Josu Etxezarreta Martinez
May 15 '18 at 13:55
$begingroup$
Yeah, I am aware of the fact that the maximization of $S$ is not necessary for the optimal performance of Turbo codes, as more elavorated interleaving schemes might imply avoiding "short cycles". However, $S$-random interleavers do at least decorrelate the input bits feeding the Turbo code, and so they might imply an increase in the performance of Turbo codes, at least at short block length. That's the reason for me being interested in such problem. Thanks for the comment!
$endgroup$
– Josu Etxezarreta Martinez
May 15 '18 at 13:55
add a comment |
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$begingroup$
Sounds difficult to me, but I haven't played too much with turbo interleavers. +1 for asking this well. Anyway, a remark I want to make is that it may not be necessary to maximize $S$ to get optimal performance from the turbo code. Avoiding "short cycles" (in terms of LLRs of some bits feeding back to themselves too quickly) is also important. But, my experience is very limited.
$endgroup$
– Jyrki Lahtonen
May 15 '18 at 12:46
$begingroup$
Yeah, I am aware of the fact that the maximization of $S$ is not necessary for the optimal performance of Turbo codes, as more elavorated interleaving schemes might imply avoiding "short cycles". However, $S$-random interleavers do at least decorrelate the input bits feeding the Turbo code, and so they might imply an increase in the performance of Turbo codes, at least at short block length. That's the reason for me being interested in such problem. Thanks for the comment!
$endgroup$
– Josu Etxezarreta Martinez
May 15 '18 at 13:55