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dc.contributor.advisorGoertz, Norbert
dc.contributor.advisorO'Carroll, Liam
dc.contributor.authorMitchell, David G. M.
dc.date.accessioned2010-11-17T10:14:55Z
dc.date.available2010-11-17T10:14:55Z
dc.date.issued2009
dc.identifier.urihttp://hdl.handle.net/1842/4330
dc.description.abstractDigital communications have now become a fundamental part of modern society. In communications, channel coding is an effective way to reduce the information rate down to channel capacity so that the information can be transmitted reliably through the channel. This thesis is devoted to studying the mathematical theory and analysis of channel codes that possess a useful diagonal structure in the parity-check and generator matrices. The first aspect of these codes that is studied is the ability to describe the parity-check matrix of a code with sliding diagonal structure using polynomials. Using this framework, an efficient new method is proposed to obtain a generator matrix G from certain types of parity-check matrices with a so-called defective cyclic block structure. By the nature of this method, G can also be completely described by a polynomial, which leads to efficient encoder design using shift registers. In addition, there is no need for the matrices to be in systematic form, thus avoiding the need for Gaussian elimination. Following this work, we proceed to explore some of the properties of diagonally structured lowdensity parity-check (LDPC) convolutional codes. LDPC convolutional codes have been shown to be capable of achieving the same capacity-approaching performance as LDPC block codes with iterative message-passing decoding. The first crucial property studied is the minimum free distance of LDPC convolutional code ensembles, an important parameter contributing to the error-correcting capability of the code. Here, asymptotic methods are used to form lower bounds on the ratio of the free distance to constraint length for several ensembles of asymptotically good, protograph-based LDPC convolutional codes. Further, it is shown that this ratio of free distance to constraint length for such LDPC convolutional codes exceeds the ratio of minimum distance to block length for corresponding LDPC block codes. Another interesting property of these codes is the way in which the structure affects the performance in the infamous error floor (which occurs at high signal to noise ratio) of the bit error rate curve. It has been suggested that “near-codewords” may be a significant factor affecting decoding failures of LDPC codes over an additive white Gaussian noise (AWGN) channel. A near-codeword is a sequence that satisfies almost all of the check equations. These nearcodewords can be associated with so-called ‘trapping sets’ that exist in the Tanner graph of a code. In the final major contribution of the thesis, trapping sets of protograph-based LDPC convolutional codes are analysed. Here, asymptotic methods are used to calculate a lower bound for the trapping set growth rates for several ensembles of asymptotically good protograph-based LDPC convolutional codes. This value can be used to predict where the error floor will occur for these codes under iterative message-passing decoding.en
dc.contributor.sponsorScottish Funding Councilen
dc.contributor.sponsorRoyal Academy of Engineeringen
dc.language.isoenen
dc.publisherThe University of Edinburghen
dc.relation.hasversionD. G. M. Mitchell, L. O’Carroll, and N. Goertz, A new method of encoding block codes with polynomials, Proc. International Symposium on Communications Theory and Applications (Ambleside, England), July 2007.en
dc.relation.hasversionD. G. M. Mitchell, L. O’Carroll, and N. Goertz,"Towards efficient encoding using polynomials", Proc. International ITG Conference on Source and Channel Coding (Ulm, Germany), Jan. 2008.en
dc.relation.hasversionD. G. M. Mitchell, A. E. Pusane, and D. J. Costello, Jr., Asymptotic trapping set analysis of regular protograph-based LDPC convolutional codes, Proc. Information Theory and Applications Workshop (San Diego, CA), Feb. 2009.en
dc.relation.hasversionD. G. M. Mitchell, A. E. Pusane, and D. J. Costello, Jr., Trapping set analysis of protograph-based LDPC convolutional codes, Proc. IEEE International Symposium on Information Theory (Seoul, Korea), July, 2009.en
dc.relation.hasversionD. G. M. Mitchell, A. E. Pusane, N. Goertz, and D. J. Costello, Jr., Free distance bounds for protograph-based regular LDPC convolutional codes, Proc. International Symposium on Turbo Codes and Related Topics (Lausanne, Switzerland), Aug. 2008.en
dc.relation.hasversionD. G. M. Mitchell, A. E. Pusane, K. Sh. Zigangirov, and D. J. Costello, Jr., Asymptotically good LDPC convolutional codes based on protographs, Proc. IEEE International Symposium on Information Theory (Toronto, Canada), July 2008.en
dc.subjectchannel codingen
dc.subjectmathematical theoryen
dc.subjectpolynomialsen
dc.subjectdefective cyclic block structureen
dc.subjectGaussian eliminationen
dc.subjectTanner graphen
dc.titleMathematical approach to channel codes with a diagonal matrix structureen
dc.typeThesis or Dissertationen
dc.type.qualificationlevelDoctoralen
dc.type.qualificationnamePhD Doctor of Philosophyen


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