Delta Functions

Delta Functions has now been updated, restructured and modernised into a second edition, to answer specific difficulties typically found by students encountering delta functions for the first time. In particular, the treatment of the Laplace transform has been revised with this in mind. The chapter on Schwartz distributions has been considerably extended and the book is supplemented by a fuller review of Nonstandard Analysis and a survey of alternative infinitesimal treatments of generalised functions. Dealing with a difficult subject in a simple and straightforward way, the text is readily accessible to a broad audience of scientists, mathematicians and engineers. It can be used as a working manual in its own right, and serves as a preparation for the study of more advanced treatises. Little more than a standard background in calculus is assumed, and attention is focused on techniques, with a liberal selection of worked examples and exercises. Second edition has been updated, restructured and modernised to answer specific difficulties typically found by students encountering delta functions for the first time Attention is focused on techniques, with a liberal selection of worked examples and exercises Readily accessible to a broad audience of scientists, mathematicians and engineers and can be used as a working manual in its own right

Delta Functions has now been updated, restructured and modernised into a second edition, to answer specific difficulties typically found by students encountering delta functions for the first time. In particular, the treatment of the Laplace transform has been revised with this in mind. The chapter on Schwartz distributions has been considerably extended and the book is supplemented by a fuller review of Nonstandard Analysis and a survey of alternative infinitesimal treatments of generalised functions. Dealing with a difficult subject in a simple and straightforward way, the text is readily accessible to a broad audience of scientists, mathematicians and engineers. It can be used as a working manual in its own right, and serves as a preparation for the study of more advanced treatises. Little more than a standard background in calculus is assumed, and attention is focused on techniques, with a liberal selection of worked examples and exercises. Second edition has been updated, restructured and modernised to answer specific difficulties typically found by students encountering delta functions for the first time Attention is focused on techniques, with a liberal selection of worked examples and exercises Readily accessible to a broad audience of scientists, mathematicians and engineers and can be used as a working manual in its own right

Designed to follow an introductory text on psychoacoustics, this book takes readers through the mathematics of signal processing from its beginnings in the Fourier transform to advanced topics in modulation, dispersion relations, minimum phase systems, sampled data, and nonlinear distortion. While organised like an introductory engineering text on signals, the examples and exercises come from research on the perception of sound. A unique feature of this book is its consistent application of the Fourier transform, which unifies topics as diverse as cochlear filtering and digital recording. More than 250 exercises are included, many of them devoted to practical research in perception, while others explore surprising auditory illusions generated by special signals. Periodic signals, aperiodic signals, and noise -- along with their linear and nonlinear transformations -- are covered in detail. More advanced mathematical topics are treated in the appendices. A working knowledge of elementary calculus is the only prerequisite. Indispensable for researchers and advanced students in the psychology of auditory perception.

Signal analysis and signal treatment are integral parts of all types of Nuclear Magnetic Resonance. In the last ten years, much has been achieved in the development of dimensional spectra. At the same time new NMR techniques such as NMR Imaging and multidimensional spectroscopy have appeared, requiring entirely new methods of signal analysis. Up until now, most NMR texts and reference books limited their presentation of signal processing to a short introduction to the principles of the Fourier Transform, signal convolution, apodisation and noise reduction. To understand the mathematics of the newer signal processing techniques, it was necessary to go back to the primary references in NMR, chemometrics and mathematics journals. The objective of this book is to fill this void by presenting, in a single volume, both the theory and applications of most of these new techniques to Time-Domain, Frequency-Domain and Space-Domain NMR signals. Details are provided on many of the algorithms used and a companion CD-ROM is also included which contains some of the computer programs, either as source code or in executable form. Although it is aimed primarily at NMR users in the medical, industrial and academic fields, it should also interest chemometricians and programmers working with other techniques.

Gathering an extensive range of mathematical topics into a plenary reference/text for solving science and engineering problems, Advanced Mathematical Models in Science and Engineering elucidates integral methods, field equation derivations, and operations applicable to modern science systems. Applying academic skills to practical problems in science and engineering, the author reviews basic methods of integration and series solutions for ordinary differential equations; introduces derivations and solution methods for linear boundary value problems in one dimension, covering eigenfunctions and eigenfunction expansions, orthogonality, and adjoint and self-adjoint systems; discusses complex variables, calculus, and integrals as well as application of residues and the integration of multivalued functions; considers linear partial differential equations in classical physics and engineering with derivations for the topics of wave equations, heat flow, vibration, and strength of materials; clarifies the calculus for integral transforms; explains Green's functions for ordinary and partial differential equations for unbounded and bounded media; examines asymptotic methods; presents methods for asymptotic solutions of ordinary differential equations; and more.

Signals and Systems Primer with MATLAB® equally emphasizes the fundamentals of both analog and digital signals and systems. To ensure insight into the basic concepts and methods, the text presents a variety of examples that illustrate a wide range of applications, from microelectromechanical to worldwide communication systems. It also provides MATLAB functions and procedures for practice and verification of these concepts. Taking a pedagogical approach, the author builds a solid foundation in signal processing as well as analog and digital systems. The book first introduces orthogonal signals, linear and time-invariant continuous-time systems, discrete-type systems, periodic signals represented by Fourier series, Gibbs's phenomenon, and the sampling theorem. After chapters on various transforms, the book discusses analog filter design, both finite and infinite impulse response digital filters, and the fundamentals of random digital signal processing, including the nonparametric spectral estimation. The final chapter presents different types of filtering and their uses for random digital signal processing, specifically, the use of Wiener filtering and least mean squares filtering. Balancing the study of signals with system modeling and interactions, this text will help readers accurately develop mathematical representations of systems.

This second edition of Generalized Functions has been strengthened in many ways. The already extensive set of examples has been expanded. Since the publication of the first edition, there has been tremendous growth in the subject and I have attempted to incorporate some of these new concepts. Accordingly, almost all the chapters have been revised. The bibliography has been enlarged considerably. Some of the material has been reorganized. For example, Chapters 12 and 13 of the first edition have been consolidated into Chapter 12 of this edition by a judicious process of elimination and addition of the subject matter. The new Chapter 13 explains the interplay between the theories of moments, asymptotics, and singular perturbations. Similarly, some sections of Chapter 15 have been revised and included in earlier chapters to improve the logical flow of ideas. However, two sections are retained. The section dealing with the application of the probability theory has been revised, and I am thankful to Professor Z.L. Crvenkovic for her help. The new material included in this chapter pertains to the modern topics of periodic distributions and microlocal theory. I have demonstrated through various examples that familiarity with the generalized functions is very helpful for students in physical sciences and technology. For instance, the reader will realize from Chapter 6 how the generalized functions have revolutionized the Fourier analysis which is being used extensively in many fields of scientific activity.

Explaining and comparing the various standard types of generalised functions which have been developed during the 20th Century, this text also contains accounts of recent non-standard theories of distributions, ultradistributions and Stato-hyperfunctions. The book could readily be used as a main text on generalised functions for mathematical undergraduates in final year analysis courses, as it presupposes little more than a general mathematical background. It also makes a valuable reference text for non-specific applied mathematics students, such as physicists or electrical engineers, needing to gain expertise in the application of generalised functions to physical problems, without any prior acquaintance of the specialised subject matter. An ideal companion book to Delta Functions, also by Professor Hoskins. Explains and compares the various standard types of generalised functions that have been developed during the 20th Century Contains accounts of recent non-standard theories of distributions, ultradistributions and Stato-hyperfunctions