31,250 research outputs found
Finite Length Analysis of Rateless Codes and Their Application in Wireless Networks
Mobile communication systems are undergoing revolutionary developments as a result of the rapidly growing demands for high data rates and reliable communication connections. The key features of the next-generation mobile communication systems are provision of high-speed and robust communication links. However, wireless communications still need to address the same challenge–unreliable communication connections, arising from a number of causes including noise, interference, and distortion because of hardware imperfections or physical limitations. Forwarding error correction (FEC) codes are used to protect source information by adding redundancy. With FEC codes, errors among the transmitted message can be corrected by the receiver. Recent work has shown that, by applying rateless codes (a class of FEC codes), wireless transmission efficiency and reliability can be dramatically improved. Unlike traditional codes, rateless codes can adapt to different channel conditions. Rateless codes have been widely used in many multimedia broadcast/multicast applications. Among the known rate- less codes, two types of codes stand out: Luby transform (LT) codes and Raptor codes. However, our understanding of LT codes and Raptor codes is still in- complete due to the lack of complete theoretical analysis on the decoding error performance of these codes. Particularly, this thesis focuses on the decoding error performance of these codes under maximum-likelihood (ML) decoding, which provides a benchmark on the optimum system performance for gauging other decoding schemes. In this thesis, we discuss the effectiveness of rateless codes in terms of the success probability of decoding. It is defined as the probability that all source symbols can be successfully decoded with a given number of success- fully received coded symbols under ML decoding. This thesis provides a detailed mathematical analysis on the rank profile of general LT codes to evaluate the decoding success probability of LT codes under ML decoding. Furthermore, by analyzing the rank of the product of two random coefficient matrices, this thesis derived bounds on the decoding success probability of Raptor codes with a systematic low-density generator matrix (LDGM) code as the pre-code under ML decoding. Additionally, by resorting to stochastic geometry analysis, we develop a rateless codes based broadcast scheme. This scheme allows a base station (BS) to broadcast a given number of symbols to a large number of users, without user acknowledgment, while being able to pro- vide a performance guarantee on the probability of successful delivery. Further, the BS has limited statistical information about the environment including the spatial distribution of users (instead of their exact locations and number) and the wireless propagation model. Based on the analysis of finite length LT codes and Raptor codes, an upper and a lower bound on the number of transmissions required to meet the performance requirement are obtained. The technique and analysis developed in this thesis are useful for designing efficient and reliable wireless broadcast strategies. It is of interest to implement rateless codes into modern communication systems
Cyclone Codes
We introduce Cyclone codes which are rateless erasure resilient codes. They
combine Pair codes with Luby Transform (LT) codes by computing a code symbol
from a random set of data symbols using bitwise XOR and cyclic shift
operations. The number of data symbols is chosen according to the Robust
Soliton distribution. XOR and cyclic shift operations establish a unitary
commutative ring if data symbols have a length of bits, for some prime
number . We consider the graph given by code symbols combining two data
symbols. If such random pairs are given for data symbols, then a
giant component appears, which can be resolved in linear time. We can extend
Cyclone codes to data symbols of arbitrary even length, provided the Goldbach
conjecture holds.
Applying results for this giant component, it follows that Cyclone codes have
the same encoding and decoding time complexity as LT codes, while the overhead
is upper-bounded by those of LT codes. Simulations indicate that Cyclone codes
significantly decreases the overhead of extra coding symbols
Inactivation Decoding of LT and Raptor Codes: Analysis and Code Design
In this paper we analyze LT and Raptor codes under inactivation decoding. A
first order analysis is introduced, which provides the expected number of
inactivations for an LT code, as a function of the output distribution, the
number of input symbols and the decoding overhead. The analysis is then
extended to the calculation of the distribution of the number of inactivations.
In both cases, random inactivation is assumed. The developed analytical tools
are then exploited to design LT and Raptor codes, enabling a tight control on
the decoding complexity vs. failure probability trade-off. The accuracy of the
approach is confirmed by numerical simulations.Comment: Accepted for publication in IEEE Transactions on Communication
Analysis of the Second Moment of the LT Decoder
We analyze the second moment of the ripple size during the LT decoding
process and prove that the standard deviation of the ripple size for an LT-code
with length is of the order of Together with a result by Karp
et. al stating that the expectation of the ripple size is of the order of
[3], this gives bounds on the error probability of the LT decoder. We also give
an analytic expression for the variance of the ripple size up to terms of
constant order, and refine the expression in [3] for the expectation of the
ripple size up to terms of the order of , thus providing a first step
towards an analytic finite-length analysis of LT decoding.Comment: 5 pages, 1 figure; submitted to ISIT 200
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