450 research outputs found
An Upper Bound on the Capacity of non-Binary Deletion Channels
We derive an upper bound on the capacity of non-binary deletion channels.
Although binary deletion channels have received significant attention over the
years, and many upper and lower bounds on their capacity have been derived,
such studies for the non-binary case are largely missing. The state of the art
is the following: as a trivial upper bound, capacity of an erasure channel with
the same input alphabet as the deletion channel can be used, and as a lower
bound the results by Diggavi and Grossglauser are available. In this paper, we
derive the first non-trivial non-binary deletion channel capacity upper bound
and reduce the gap with the existing achievable rates. To derive the results we
first prove an inequality between the capacity of a 2K-ary deletion channel
with deletion probability , denoted by , and the capacity of the
binary deletion channel with the same deletion probability, , that is,
. Then by employing some existing upper
bounds on the capacity of the binary deletion channel, we obtain upper bounds
on the capacity of the 2K-ary deletion channel. We illustrate via examples the
use of the new bounds and discuss their asymptotic behavior as .Comment: accepted for presentation in ISIT 201
A Note on the Deletion Channel Capacity
Memoryless channels with deletion errors as defined by a stochastic channel
matrix allowing for bit drop outs are considered in which transmitted bits are
either independently deleted with probability or unchanged with probability
. Such channels are information stable, hence their Shannon capacity
exists. However, computation of the channel capacity is formidable, and only
some upper and lower bounds on the capacity exist. In this paper, we first show
a simple result that the parallel concatenation of two different independent
deletion channels with deletion probabilities and , in which every
input bit is either transmitted over the first channel with probability of
or over the second one with probability of , is nothing
but another deletion channel with deletion probability of . We then provide an upper bound on the concatenated
deletion channel capacity in terms of the weighted average of ,
and the parameters of the three channels. An interesting consequence
of this bound is that which
enables us to provide an improved upper bound on the capacity of the i.i.d.
deletion channels, i.e., for . This
generalizes the asymptotic result by Dalai as it remains valid for all . Using the same approach we are also able to improve upon existing upper
bounds on the capacity of the deletion/substitution channel.Comment: Submitted to the IEEE Transactions on Information Theor
On Coding over Sliced Information
The interest in channel models in which the data is sent as an unordered set
of binary strings has increased lately, due to emerging applications in DNA
storage, among others. In this paper we analyze the minimal redundancy of
binary codes for this channel under substitution errors, and provide several
constructions, some of which are shown to be asymptotically optimal up to
constants. The surprising result in this paper is that while the information
vector is sliced into a set of unordered strings, the amount of redundant bits
that are required to correct errors is order-wise equivalent to the amount
required in the classical error correcting paradigm
Achievable Rates for Noisy Channels with Synchronization Errors
Cataloged from PDF version of article.We develop several lower bounds on the capacity of binary input symmetric output channels with synchronization errors, which also suffer from other types of impairments such as substitutions, erasures, additive white Gaussian noise (AWGN), etc. More precisely, we show that if a channel suffering from synchronization errors as well as other type of impairments can be decomposed into a cascade of two component channels where the first one is another channel with synchronization errors and the second one is a memoryless channel (with no synchronization errors), a lower bound on the capacity of the original channel in terms of the capacity of the component synchronization error channel can be derived. A primary application of our results is that we can employ any lower bound derived on the capacity of the component synchronization error channel to find lower bounds on the capacity of the (original) noisy channel with synchronization errors. We apply the general ideas to several specific classes of channels such as synchronization error channels with erasures and substitutions, with symmetric q-ary outputs and with AWGN explicitly, and obtain easy-to-compute bounds. We illustrate that, with our approach, it is possible to derive tighter capacity lower bounds compared to the currently available bounds in the literature for certain classes of channels, e.g., deletion/substitution channels and deletion/AWGN channels (for certain signal-to-noise ratio (SNR) ranges). © 2014 IEEE
Upper bounds on the capacity of deletion channels using channel fragmentation
Cataloged from PDF version of article.We study memoryless channels with synchronization
errors as defined by a stochastic channel matrix allowing for
symbol drop-outs or symbol insertions with particular emphasis
on the binary and non-binary deletion channels. We offer
a different look at these channels by considering equivalent
models by fragmenting the input sequence where different
subsequences travel through different channels. The resulting
output symbols are combined appropriately to come up with an
equivalent input–output representation of the original channel
which allows for derivation of new upper bounds on the channel
capacity. We consider both random and deterministic types
of fragmentation processes applied to binary and nonbinary
deletion channels. With two specific applications of this idea,
a random fragmentation applied to a binary deletion channel
and a deterministic fragmentation process applied to a nonbinary
deletion channel, we prove certain inequality relations among the
capacities of the original channels and those of the introduced
subchannels. The resulting inequalities prove useful in deriving
tighter capacity upper bounds for: 1) independent identically
distributed (i.i.d.) deletion channels when the deletion probability
exceeds 0.65 and 2) nonbinary deletion channels. Some extensions
of these results, for instance, to the case of deletion/substitution
channels are also explored
Fundamental Bounds and Approaches to Sequence Reconstruction from Nanopore Sequencers
Nanopore sequencers are emerging as promising new platforms for
high-throughput sequencing. As with other technologies, sequencer errors pose a
major challenge for their effective use. In this paper, we present a novel
information theoretic analysis of the impact of insertion-deletion (indel)
errors in nanopore sequencers. In particular, we consider the following
problems: (i) for given indel error characteristics and rate, what is the
probability of accurate reconstruction as a function of sequence length; (ii)
what is the number of `typical' sequences within the distortion bound induced
by indel errors; (iii) using replicated extrusion (the process of passing a DNA
strand through the nanopore), what is the number of replicas needed to reduce
the distortion bound so that only one typical sequence exists within the
distortion bound.
Our results provide a number of important insights: (i) the maximum length of
a sequence that can be accurately reconstructed in the presence of indel and
substitution errors is relatively small; (ii) the number of typical sequences
within the distortion bound is large; and (iii) replicated extrusion is an
effective technique for unique reconstruction. In particular, we show that the
number of replicas is a slow function (logarithmic) of sequence length --
implying that through replicated extrusion, we can sequence large reads using
nanopore sequencers. Our model considers indel and substitution errors
separately. In this sense, it can be viewed as providing (tight) bounds on
reconstruction lengths and repetitions for accurate reconstruction when the two
error modes are considered in a single model.Comment: 12 pages, 5 figure
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