Numerical algorithms for the computation of the Smith normal form of integral matrices,

Numerical algorithms for the computation of the Smith normal form of integral matrices are described. More specifically, the compound matrix method, methods based on elementary row or column operations and methods using modular or p-adic arithmetic are presented. A variety of examples and numerical results are given illustrating the execution of the algorithms

Lossless and low-cost integer-based lifting wavelet transform

Discrete wavelet transform (DWT) is a powerful tool for analyzing real-time signals, including aperiodic, irregular, noisy, and transient data, because of its capability to explore signals in both the frequency- and time-domain in different resolutions. For this reason, they are used extensively in a wide number of applications in image and signal processing. Despite the wide usage, the implementation of the wavelet transform is usually lossy or computationally complex, and it requires expensive hardware. However, in many applications, such as medical diagnosis, reversible data-hiding, and critical satellite data, lossless implementation of the wavelet transform is desirable. It is also important to have more hardware-friendly implementations due to its recent inclusion in signal processing modules in system-on-chips (SoCs). To address the need, this research work provides a generalized implementation of a wavelet transform using an integer-based lifting method to produce lossless and low-cost architecture while maintaining the performance close to the original wavelets. In order to achieve a general implementation method for all orthogonal and biorthogonal wavelets, the Daubechies wavelet family has been utilized at first since it is one of the most widely used wavelets and based on a systematic method of construction of compact support orthogonal wavelets. Though the first two phases of this work are for Daubechies wavelets, they can be generalized in order to apply to other wavelets as well. Subsequently, some techniques used in the primary works have been adopted and the critical issues for achieving general lossless implementation have solved to propose a general lossless method. The research work presented here can be divided into several phases. In the first phase, low-cost architectures of the Daubechies-4 (D4) and Daubechies-6 (D6) wavelets have been derived by applying the integer-polynomial mapping. A lifting architecture has been used which reduces the cost by a half compared to the conventional convolution-based approach. The application of integer-polynomial mapping (IPM) of the polynomial filter coefficient with a floating-point value further decreases the complexity and reduces the loss in signal reconstruction. Also, the “resource sharing” between lifting steps results in a further reduction in implementation costs and near-lossless data reconstruction. In the second phase, a completely lossless or error-free architecture has been proposed for the Daubechies-8 (D8) wavelet. Several lifting variants have been derived for the same wavelet, the integer mapping has been applied, and the best variant is determined in terms of performance, using entropy and transform coding gain. Then a theory has been derived regarding the impact of scaling steps on the transform coding gain (GT). The approach results in the lowest cost lossless architecture of the D8 in the literature, to the best of our knowledge. The proposed approach may be applied to other orthogonal wavelets, including biorthogonal ones to achieve higher performance. In the final phase, a general algorithm has been proposed to implement the original filter coefficients expressed by a polyphase matrix into a more efficient lifting structure. This is done by using modified factorization, so that the factorized polyphase matrix does not include the lossy scaling step like the conventional lifting method. This general technique has been applied on some widely used orthogonal and biorthogonal wavelets and its advantages have been discussed. Since the discrete wavelet transform is used in a vast number of applications, the proposed algorithms can be utilized in those cases to achieve lossless, low-cost, and hardware-friendly architectures

Cocyclic Hadamard Matrices: An Efficient Search Based Algorithm

This dissertation serves as the culmination of three papers. “Counting the decimation classes of binary vectors with relatively prime fixed-density presents the first non-exhaustive decimation class counting algorithm. “A Novel Approach to Relatively Prime Fixed Density Bracelet Generation in Constant Amortized Time presents a novel lexicon for binary vectors based upon the Discrete Fourier Transform, and develops a bracelet generation method based upon the same. “A Novel Legendre Pair Generation Algorithm expands upon the bracelet generation algorithm and includes additional constraints imposed by Legendre Pairs. It further presents an efficient sorting and comparison algorithm based upon symmetric functions, as well as multiple unique Legendre Pairs

Fundamentals of computer systems architecture

In the study guide "Fundamentals of computer systems architecture" the questions of presentation of information in different systems of calculation, execution of logical and arithmetic operations are considered. Each chapter provides the necessary theoretical information, examples of presentation of information and examples of execution of arithmetic and logical operations, given tasks for self-execution and control questions. For the students of specialties 121 – “Software Engineering” and 123 – “Computer Engineering”

Numerical issues and computational problems in algebraic control theory

The work of this thesis concerns computational issues arising from various fields of Algebraic Control Theory. Efficient algorithms covering the following classes of problems are developed. (i) Exterior Algebra Computations: For given matrices [Please see formulas inside thesis] algorithms achieving the computation of [Please see formulas inside thesis] are formulated. An algorithm for the evaluation of Plucker matrices is also proposed. Most of these algorithms are used in the development of a unifying numerical algorithm for the solution of the Determinantal Assignment Problem. (ii) Numerical Techniques for handling nonqeneric computations: Several numerical tools for the diagnosis of certain properties in an "almost sense", and the definition of procedures attaining the termination of algorithms are developed. (iii) Evaluation of the Greatest Common Divisor of polynomials: A new numerical algorithm for the evaluation of the greatest common divisor of any set of polynomials is formulated. (iv) Almost Zero Computations: Algorithms achieving the evaluation of the Prime almost zero of a polynomial set and the computation of the zero radius are given. Useful comments about the achievement of improved bounds for the zero-trapping region are also presented

A fast procedure for generating random numbers by a modification of the Marsaglia-Maclaren method

Marsaglia and Maclaren combined two linear congruential generators in order to produce a pseudo random number sequence uniformly distributed in the range [0,2³⁵]. Their method is a considerable improvement compared with the primitive linear congruential method at the cost of greater generation time. In this thesis, a simple modification of the Marsaglia-Maclaren method is presented in which there is an alleviation of the increased generation time, and a slight further increase in randomness. The modified generator is tested extensively in a variety of statistical tests and simulation problems

Number theoretic techniques applied to algorithms and architectures for digital signal processing

Many of the techniques for the computation of a two-dimensional convolution of a small fixed window with a picture are reviewed. It is demonstrated that Winograd's cyclic convolution and Fourier Transform Algorithms, together with Nussbaumer's two-dimensional cyclic convolution algorithms, have a common general form. Many of these algorithms use the theoretical minimum number of general multiplications. A novel implementation of these algorithms is proposed which is based upon one-bit systolic arrays. These systolic arrays are networks of identical cells with each cell sharing a common control and timing function. Each cell is only connected to its nearest neighbours. These are all attractive features for implementation using Very Large Scale Integration (VLSI). The throughput rate is only limited by the time to perform a one-bit full addition. In order to assess the usefulness to these systolic arrays a 'cost function' is developed to compare them with more conventional techniques, such as the Cooley-Tukey radix-2 Fast Fourier Transform (FFT). The cost function shows that these systolic arrays offer a good way of implementing the Discrete Fourier Transform for transforms up to about 30 points in length. The cost function is a general tool and allows comparisons to be made between different implementations of the same algorithm and between dissimilar algorithms. Finally a technique is developed for the derivation of Discrete Cosine Transform (DCT) algorithms from the Winograd Fourier Transform Algorithm. These DCT algorithms may be implemented by modified versions of the systolic arrays proposed earlier, but requiring half the number of cells

M-Sequences Related to the Multifocal Electroretinogram: Identification of Appropriate Primitive Polynomials to Avoid Cross-Contamination in Multifocal Electroretinogram Responses

The basis of multifocal ERG is the use of a decimated m-sequence for simultaneous and independent stimulation of many areas of the visual pathway. The purpose of this thesis is to investigate the effects of cross contamination from higher orders of the response. To examine the effects of cross contamination a series of primitive polynomials were found by constructing finite fields. The first order ERG response is formed by cross correlating the m-sequence with the physiological response. A second order response is formed by investigating particular flash sequences of the stimulation sequence and is formed by cross correlation of a second order m-sequence with the physiological response. Zech Logarithms were used to identify cross contamination between the various first and second order sequences. Tables of good and bad primitive polynomials were constructed for degree 12 to degree 16 and the effects of window length and decimation length examined. If we decimate the sequence into 128 areas, and look at a window of length 16, cross-contamination occurs in all sequences generated from primitive polynomials of degree less than or equal 12, but only 26% in the case of degree 14, and 5.6% for degree 16. Finally, selected good and bad primitive polynomials were used to generate decimated m-sequences for a multifocal electrophysiological experiment to demonstrate the practical effects of cross-contamination. Trace arrays showing uncontaminated discreet physiological responses from 61 individual elements were recorded using the example good primitive polynomial whereas additional waveforms were present on the trace array when the same experiment was repeated with a bad primitive polynomial. The use of finite field theory to generate primitive polynomials and zech logorithm analysis enables us to predict which primitive polynomials are suitable for m- sequence generation for multifocal electroretinography. Practical investigations support the theoretical analysis. This has important implications for developers of multifocal electrophysiology systems