1,804 research outputs found
The Saddlepoint Approximation: Unified Random Coding Asymptotics for Fixed and Varying Rates
This paper presents a saddlepoint approximation of the random-coding union
bound of Polyanskiy et al. for i.i.d. random coding over discrete memoryless
channels. The approximation is single-letter, and can thus be computed
efficiently. Moreover, it is shown to be asymptotically tight for both fixed
and varying rates, unifying existing achievability results in the regimes of
error exponents, second-order coding rates, and moderate deviations. For fixed
rates, novel exact-asymptotics expressions are specified to within a
multiplicative 1+o(1) term. A numerical example is provided for which the
approximation is remarkably accurate even at short block lengths.Comment: Accepted to ISIT 2014, presented without publication at ITA 201
Asymptotic Estimates in Information Theory with Non-Vanishing Error Probabilities
This monograph presents a unified treatment of single- and multi-user
problems in Shannon's information theory where we depart from the requirement
that the error probability decays asymptotically in the blocklength. Instead,
the error probabilities for various problems are bounded above by a
non-vanishing constant and the spotlight is shone on achievable coding rates as
functions of the growing blocklengths. This represents the study of asymptotic
estimates with non-vanishing error probabilities.
In Part I, after reviewing the fundamentals of information theory, we discuss
Strassen's seminal result for binary hypothesis testing where the type-I error
probability is non-vanishing and the rate of decay of the type-II error
probability with growing number of independent observations is characterized.
In Part II, we use this basic hypothesis testing result to develop second- and
sometimes, even third-order asymptotic expansions for point-to-point
communication. Finally in Part III, we consider network information theory
problems for which the second-order asymptotics are known. These problems
include some classes of channels with random state, the multiple-encoder
distributed lossless source coding (Slepian-Wolf) problem and special cases of
the Gaussian interference and multiple-access channels. Finally, we discuss
avenues for further research.Comment: Further comments welcom
An asymptotically optimal push-pull method for multicasting over a random network
We consider allcast and multicast flow problems where either all of the nodes
or only a subset of the nodes may be in session. Traffic from each node in the
session has to be sent to every other node in the session. If the session does
not consist of all the nodes, the remaining nodes act as relays. The nodes are
connected by undirected links whose capacities are independent and identically
distributed random variables. We study the asymptotics of the capacity region
(with network coding) in the limit of a large number of nodes, and show that
the normalized sum rate converges to a constant almost surely. We then provide
a decentralized push-pull algorithm that asymptotically achieves this
normalized sum rate without network coding.Comment: 13 pages, extended version of paper presented at the IEEE
International Symposium on Information Theory (ISIT) 2012, minor revision to
text to address review comments, to appear in IEEE Transactions in
information theor
Exact Asymptotics for the Random Coding Error Probability
Error probabilities of random codes for memoryless channels are considered in
this paper. In the area of communication systems, admissible error probability
is very small and it is sometimes more important to discuss the relative gap
between the achievable error probability and its bound than to discuss the
absolute gap. Scarlett et al. derived a good upper bound of a random coding
union bound based on the technique of saddlepoint approximation but it is not
proved that the relative gap of their bound converges to zero. This paper
derives a new bound on the achievable error probability in this viewpoint for a
class of memoryless channels. The derived bound is strictly smaller than that
by Scarlett et al. and its relative gap with the random coding error
probability (not a union bound) vanishes as the block length increases for a
fixed coding rate.Comment: Full version of the paper in ISIT2015 with some corrections and
refinement
The Dispersion of Nearest-Neighbor Decoding for Additive Non-Gaussian Channels
We study the second-order asymptotics of information transmission using
random Gaussian codebooks and nearest neighbor (NN) decoding over a
power-limited stationary memoryless additive non-Gaussian noise channel. We
show that the dispersion term depends on the non-Gaussian noise only through
its second and fourth moments, thus complementing the capacity result
(Lapidoth, 1996), which depends only on the second moment. Furthermore, we
characterize the second-order asymptotics of point-to-point codes over
-sender interference networks with non-Gaussian additive noise.
Specifically, we assume that each user's codebook is Gaussian and that NN
decoding is employed, i.e., that interference from the unintended users
(Gaussian interfering signals) is treated as noise at each decoder. We show
that while the first-order term in the asymptotic expansion of the maximum
number of messages depends on the power of the interferring codewords only
through their sum, this does not hold for the second-order term.Comment: 12 pages, 3 figures, IEEE Transactions on Information Theor
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