Digital 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.

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A theoretical study reducing the general diffusion solution in a multicomponent system to an eigenvalue problem is carried out. Certain properties of the diffusion coefficient matrix are investigated. Special solutions are obtained tor a ternary finite diffusion couple and for a moving phase boundary in an infinite medium involving diffusion on both sides or the interface. The latter solution is used to study the kinetics of the growth of proeutectoid ferrite in ternary Fe-Mn-C austenite. Information on the iron-rich corner of the ternary Fe-Mn-C phase diagram in the temperature range from 725°C to 790°C has been obtained experimentally, and the diffusion coefficient of manganese in a-iron has been determined. / Thesis / Master of Science (MS)

This thesis is devoted to the study and designs of two important proportional weirs having the discharge-head characteristics of Q α H 2/3 In the first design a geometrically simple weir in the form of a rectangular weir over a inverted V-notch (Chimney weir) is presented. This weir gives for all flows above a threshold depth a discharge proportional to H 2/3 within a maximum percentage error of ±1.5, (measured above a reference plane) within certain limits of head. Second design is concerned with the self-basing weir in which a portion of the weir above the crest acts as a base. This design is achieved by using the complementary weir profile of a Quadratic weir above the parabolic base which has the significant property of fast convergence. This weir gives discharge for all flows above the threshold depth, proportional to (head)2/3 measured above a reference plane, with increasing accuracy as head increases. Experiments with these two weirs confirm the theory by giving a constant average Coefficient of Discharge (Cd) of 0.62. The importance of these weirs as a sensitive discharge measuring device in field and laboratory is highlighted.