22 research outputs found
Density Evolution for Asymmetric Memoryless Channels
Density evolution is one of the most powerful analytical tools for
low-density parity-check (LDPC) codes and graph codes with message passing
decoding algorithms. With channel symmetry as one of its fundamental
assumptions, density evolution (DE) has been widely and successfully applied to
different channels, including binary erasure channels, binary symmetric
channels, binary additive white Gaussian noise channels, etc. This paper
generalizes density evolution for non-symmetric memoryless channels, which in
turn broadens the applications to general memoryless channels, e.g. z-channels,
composite white Gaussian noise channels, etc. The central theorem underpinning
this generalization is the convergence to perfect projection for any fixed size
supporting tree. A new iterative formula of the same complexity is then
presented and the necessary theorems for the performance concentration theorems
are developed. Several properties of the new density evolution method are
explored, including stability results for general asymmetric memoryless
channels. Simulations, code optimizations, and possible new applications
suggested by this new density evolution method are also provided. This result
is also used to prove the typicality of linear LDPC codes among the coset code
ensemble when the minimum check node degree is sufficiently large. It is shown
that the convergence to perfect projection is essential to the belief
propagation algorithm even when only symmetric channels are considered. Hence
the proof of the convergence to perfect projection serves also as a completion
of the theory of classical density evolution for symmetric memoryless channels.Comment: To appear in the IEEE Transactions on Information Theor
A Packing Lemma for Polar Codes
A packing lemma is proved using a setting where the channel is a binary-input
discrete memoryless channel , the code is
selected at random subject to parity-check constraints, and the decoder is a
joint typicality decoder. The ensemble is characterized by (i) a pair of fixed
parameters where is a parity-check matrix and is a channel
input distribution and (ii) a random parameter representing the desired
parity values. For a code of length , the constraint is sampled from where is the
indicator function of event and . Given , the codewords are chosen conditionally
independently from . It is shown
that the probability of error for this ensemble decreases exponentially in
provided the rate is kept bounded away from
with and . In the special case where is the parity-check
matrix of a standard polar code, it is shown that the rate penalty
vanishes as increases. The paper also discusses the
relation between ordinary polar codes and random codes based on polar
parity-check matrices.Comment: 5 pages. To be presented at 2015 IEEE International Symposium on
Information Theory, June 14-19, 2015, Hong Kong. Minor corrections to v
Coding Schemes for Physical Layer Network Coding Over a Two-Way Relay Channel
We consider a two-way relay channel in which two transmitters want to exchange information through a central relay. The relay observes a superposition of the trans- mitted signals from which a function of the transmitted messages is computed for broadcast. We consider the design of codebooks which permit the recovery of a function at the relay and derive information-theoretic bounds on the rates for reliable decoding at the relay.
In the spirit of compute-and-forward, we present a multilevel coding scheme that permits reliable computation (or, decoding) of a class of functions at the relay. The function to be decoded is chosen at the relay depending on the channel realization. We define such a class of reliably computable functions for the proposed coding scheme and derive rates that are universally achievable over a set of channel gains when this class of functions is used at the relay. We develop our framework with general modulation formats in mind, but numerical results are presented for the case where each node transmits using 4-ary and 8-ary modulation schemes. Numerical results demonstrate that the flexibility afforded by our proposed scheme permits substantially higher rates than those achievable by always using a fixed function or considering only linear functions over higher order fields.
Our numerical results indicate that it is favorable to allow the relay to attempt both compute-and-forward and decode-and-forward decoding. Indeed, either method considered separately is suboptimal for computation over general channels. However, we obtain a converse result when the transmitters are restricted to using identical binary linear codebooks generated uniformly at random. We show that it is impossible for this code ensemble to achieve any rate higher than the maximum of the rates achieved using compute-and-forward and decode-and-forward decoding.
Finally, we turn our attention to the design of low density parity check (LDPC) ensembles which can practically achieve these information rates with joint-compute- and-forward message passing decoding. To this end, we construct a class of two-way erasure multiple access channels for which we can exactly characterize the performance of joint-compute-and-forward message passing decoding. We derive the processing rules and a density evolution like analysis for several classes of LDPC ensembles. Utilizing the universally optimal performance of spatially coupled LDPC ensembles with message passing decoding, we show that a single encoder and de- coder with puncturing can achieve the optimal rate region for a range of channel parameters
Information reconciliation methods in secret key distribution
We consider in this thesis the problem of information reconciliation in
the context of secret key distillation between two legitimate parties.
In some scenarios of interest this problem can be advantageously
solved with low density parity check (LDPC) codes optimized for
the binary symmetric channel. In particular, we demonstrate that our
method leads to a significant efficiency improvement, with respect to
earlier interactive reconciliation methods. We propose a protocol based
on LDPC codes that can be adapted to changes in the communication
channel extending the original source. The efficiency of our protocol is
only limited by the quality of the code and, while transmitting more
information than needed to reconcile Alice’s and Bob’s sequences, it
does not reveal any more information on the original source than an
ad-hoc code would have revealed.---ABSTRACT---En esta tesis estudiamos el problema de la reconciliación de información
en el contexto de la destilación de secreto entre dos partes.
En algunos escenarios de interés, códigos de baja densidad de
ecuaciones de paridad (LDPC) adaptados al canal binario simétrico
ofrecen una buena solución al problema estudiado. Demostramos que
nuestro método mejora significativamente la eficiencia de la reconciliación.
Proponemos un protocolo basado en códigos LDPC que puede
ser adaptado a cambios en el canal de comunicaciones mediante una
extensión de la fuente original. La eficiencia de nuestro protocolo está
limitada exclusivamente por el código utilizado y no revela información
adicional sobre la fuente original que la que un código con la tasa
de información adaptada habría revelado
Coding and Signal Processing for Secure Wireless Communication
Wireless communication networks are widely deployed today and the networks are used in many applications which require that the data transmitted be secure. Due to the open nature of wireless systems, it is important to have a fundamental understanding of coding schemes that allow for simultaneously secure and reliable transmission. The information theoretic approach is able to give us this fundamental insight into the nature of the coding schemes required for security. The security issue is approached by focusing on the confidentiality of message transmission and reception at the physical layer. The goal is to design coding and signal processing schemes that provide security, in the information theoretic sense. In so doing, we are able to prove the simultaneously secure and reliable transmission rates for different network building blocks. The multi-receiver broadcast channel is an important network building block, where the rate region for the channel without security constraints is still unknown. In the thesis this channel is investigated with security constraints, and the secure and reliable rates are derived for the proposed coding scheme using a random coding argument. Cooperative relaying is next applied to the wiretap channel, the fundamental physical layer model for the communication security problem, and signal processing techniques are used to show that the secure rate can be improved in situations where the secure rate was small due to the eavesdropper enjoying a more favorable channel condition compared to the legitimate receiver. Finally, structured lattice codes are used in the wiretap channel instead of unstructured random codes, used in the vast majority of the work so far. We show that lattice coding and decoding can achieve the secrecy rate of the Gaussian wiretap channel; this is an important step towards realizing practical, explicit codes for the wiretap channel