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Radar High-Speed Target Detection via Coherent Integration Transform offers a systematic presentation of high-speed radar target detection methods using coherent integration transforms including the signal model, derivations of coherent integration transforms, and definitions of related key concepts.The authors present mathematical models and design principles necessary to analyze the behavior of each type of coherent integration transform, and based on this, they introduce and convey new approaches and techniques for designing such transforms, which will help to achieve efficient signal integrators and detectors, especially in the challenging low signal-to-noise ratio (SNR) environments.The book will be of interest to graduate students and engineering professionals in statistical signal processing, signal detection and estimation, and radar signal processing.

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A guide to wave-field computational methods based on contrast source type of integral equations Forward and Inverse Scattering Algorithms Based on Contrast Source Integral Equations presents a text that examines wave-field computational methods based on contrast source type of integral equations and the computational implementation in wave-field based imaging methods. Written by a noted expert on the topic, the book provides a guide to efficient methods for calculating wave fields in a known inhomogeneous medium. The author provides a link between the fundamental scattering theory and its discrete counterpart and discusses the forward scattering problem based on the contrast-source integral equations. The book fully describes the calculation of wave fields inside and outside a scattering object with general shape and material property and reviews the inverse scattering problem, in which material properties are resolved from wave-field measurements outside the scattering object. The theoretical approach is the inverse of the forward scattering problem that determines how radiation is scattered, based on the scattering object. This important book: Provides a guide to the effects of scalar waves, acoustic waves and electromagnetic waves Describes computer modeling in 1D, 2D and 3D models Includes an online site for computer codes with adjustable configurations Written for students, researchers, and professionals, Forward and Inverse Scattering Algorithms Based on Contrast Source Integral Equations offers a guide to wave-field computational methods based on contrast source type of integral equations and the computational implementation in wave-field based imaging methods.

The author shows that the finite time type II blow up solutions for the energy critical nonlinear wave equation $ \Box u = -u^5 $ on $\mathbb R^3+1$ constructed in Krieger, Schlag, and Tataru (2009) and Krieger and Schlag (2014) are stable along a co-dimension three manifold of radial data perturbations in a suitable topology, provided the scaling parameter $\lambda (t) = t^-1-\nu $ is sufficiently close to the self-similar rate, i. e. $\nu >0$ is sufficiently small. Our method is based on Fourier techniques adapted to time dependent wave operators of the form $ -\partial _t^2 + \partial _r^2 + \frac 2r\partial _r +V(\lambda (t)r) $ for suitable monotone scaling parameters $\lambda (t)$ and potentials $V(r)$ with a resonance at zero.

Fourier Series, Fourier Transforms, and Function Spaces is designed as a textbook for a second course or capstone course in analysis for advanced undergraduate or beginning graduate students. By assuming the existence and properties of the Lebesgue integral, this book makes it possible for students who have previously taken only one course in real analysis to learn Fourier analysis in terms of Hilbert spaces, allowing for both a deeper and more elegant approach. This approach also allows junior and senior undergraduates to study topics like PDEs, quantum mechanics, and signal processing in a rigorous manner. Students interested in statistics (time series), machine learning (kernel methods), mathematical physics (quantum mechanics), or electrical engineering (signal processing) will find this book useful. With 400 problems, many of which guide readers in developing key theoretical concepts themselves, this text can also be adapted to self-study or an inquiry-based approach. Finally, of course, this text can also serve as motivation and preparation for students going on to further study in analysis.

This third edition builds on the introduction of spectral analysis as a means of investigating wave propagation and transient oscillations in structures. Each chapter of the textbook has been revised, updated and augmented with new material, such as a modified treatment of the curved plate and cylinder problem that yields a relatively simple but accurate spectral analysis. Finite element methods are now integrated into the spectral analyses to gain further insights into the high-frequency problems. In addition, a completely new chapter has been added that deals with waves in periodic and discretized structures. Examples for phononic materials meta-materials as well as genuine atomic systems are given.

Following an introduction to the basis of the fast Fourier transform (FFT), this book focuses on the implementation details on FFT for parallel computers. FFT is an efficient implementation of the discrete Fourier transform (DFT), and is widely used for many applications in engineering, science, and mathematics. Presenting many algorithms in pseudo-code and a complexity analysis, this book offers a valuable reference guide for graduate students, engineers, and scientists in the field who wish to apply FFT to large-scale problems.Parallel computation is becoming indispensable in solving the large-scale problems increasingly arising in a wide range of applications. The performance of parallel supercomputers is steadily improving, and it is expected that a massively parallel system with hundreds of thousands of compute nodes equipped with multi-core processors and accelerators will be available in the near future. Accordingly, the book also provides up-to-date computational techniquesrelevant to the FFT in state-of-the-art parallel computers. Following the introductory chapter, Chapter 2 introduces readers to the DFT and the basic idea of the FFT. Chapter 3 explains mixed-radix FFT algorithms, while Chapter 4 describes split-radix FFT algorithms. Chapter 5 explains multi-dimensional FFT algorithms, Chapter 6 presents high-performance FFT algorithms, and Chapter 7 addresses parallel FFT algorithms for shared-memory parallel computers. In closing, Chapter 8 describes parallel FFT algorithms for distributed-memory parallel computers.

The present book on ‘‘Laplace and Fourier Transforms''has been written as a textbook according to the latest guidelines and syllabus in Mathematics issued by the U.G.C. for various universities. The text of the book has been prepared with the following salient features: (i) The language of the book is simple and easy to understand. (ii) Each topic has been presented in a systematic, simple, lucid and exhaustive manner. (iii) A large number of important solved examples properly selected from the previous university question papers have been provided to enable the students to have a clear grasp of the subject and to equip them for attempting problems in the university examination without any difficulty. (iv) Apart from providing a large number of examples, different type of questions in ample quantity have been provided for a thorough practice to the students. (v) A large number of ‘notes'and ‘remarks'have been added for better understanding of the subject.

This book helps in giving a qualitative feel for the properties of Fourier series and Fourier transforms by using the illustrative powers of computer graphics. It is useful for wide variety of students as it focuses on qualitative aspects and the flexibility with regard to program modification.

Die Autoren klassifizieren kurz die in Natur und Technik auftretenden Signale und stellen die Ecksteine der „analogen“ Fourier-Transformation als Basis für das essential dar. Sie zeigen anschaulich den Zusammenhang von Periodifizierung und Abtastung von Signalen im Zeit- und Frequenzbereich und leiten daraus das für die Digitalisierung fundamentale Abtasttheorem ab. Darauf aufbauend wird weitgehend visuell die diskrete Fourier-Transformation erklärt und deren Handhabung an einem Rechenbeispiel gezeigt. Es folgt eine Kurzdarstellung ausgewählter Grundbausteine der Übertragungstechnik und eine robuste Erläuterung des Mehrträger-Verfahrens und des orthogonalen Frequenzmultiplexverfahrens OFDM als Kernstück der Signalübertragung im Internet.Die Autoren:Prof. Dr.-Ing. habil. Jörg Lange war in leitender Position im Entwicklungsbereich Mobilfunk in einem Technologiekonzern tätig, bevor er in Ruhestand ging. Prof. Dr.-Ing. Tatjana Lange lehrte vor ihrem Ruhestand Automatisierungstechnik an der Hochschule Merseburg und ist weiterhin in der Forschung aktiv.

This book is derived from lecture notes for a course on Fourier analysis for engineering and science students at the advanced undergraduate or beginning graduate level. Beyond teaching specific topics and techniques—all of which are important in many areas of engineering and science—the author's goal is to help engineering and science students cultivate more advanced mathematical know-how and increase confidence in learning and using mathematics, as well as appreciate the coherence of the subject. He promises the readers a little magic on every page. The section headings are all recognizable to mathematicians, but the arrangement and emphasis are directed toward students from other disciplines. The material also serves as a foundation for advanced courses in signal processing and imaging. There are over 200 problems, many of which are oriented to applications, and a number use standard software. An unusual feature for courses meant for engineers is a more detailed and accessible treatment of distributions and the generalized Fourier transform. There is also more coverage of higher-dimensional phenomena than is found in most books at this level. A thoroughly enjoyable yet careful mathematical perspective of the underlying concepts and many applications of modern signal analysis. —Les Atlas, University of Washington Osgood leads his readers from the basics to the more sophisticated parts of applicable Fourier analysis with a lively style, a light touch on the technicalities, and an eye toward communications engineering. This book should be a great resource for students of mathematics, physics, and engineering alike. —Gerald B. Folland, University of Washington Fourier analysis with a swing in its step. —Tom Körner, University of Cambridge

This book provides an introduction to image processing, an overview of the transforms which are most widely used in the field of image processing, and an introduction to the application of multiscale transforms in image processing. The book is divided into three parts, with the first part offering the reader a basic introduction to image processing. The second part of the book starts with a chapter on Fourier analysis and Fourier transforms, wavelet analysis, and ends with a chapter on new multiscale transforms. The final part of the book deals with all of the most important applications of multiscale transforms in image processing. The chapters consist of both tutorial and highly advanced material, and as such the book is intended to be a reference text for graduate students and researchers to obtain state-of-the-art knowledge on specific applications. The technique of solving problems in the transform domain is common in applied mathematics and widely used in research and industry, but is a somewhat neglected subject within the undergraduate curriculum. It is hoped that faculty can use this book to create a course that can be offered early in the curriculum and fill this void. Also, the book is intended to be used as a reference manual for scientists who are engaged in image processing research, developers of image processing hardware and software systems, and practising engineers and scientists who use image processing as a tool in their applications.

This book investigates the convergence and summability of both one-dimensional and multi-dimensional Fourier transforms, as well as the theory of Hardy spaces. To do so, it studies a general summability method known as theta-summation, which encompasses all the well-known summability methods, such as the Fejér, Riesz, Weierstrass, Abel, Picard, Bessel and Rogosinski summations. Following on the classic books by Bary (1964) and Zygmund (1968), this is the first book that considers strong summability introduced by current methodology. A further unique aspect is that the Lebesgue points are also studied in the theory of multi-dimensional summability. In addition to classical results, results from the past 20-30 years – normally only found in scattered research papers – are also gathered and discussed, offering readers a convenient “one-stop” source to support their work. As such, the book will be useful for researchers, graduate and postgraduate students alike.

This monograph addresses the fundamental concepts of frequency spectrum analysis using discrete Fourier transforms. Topics range from the discretization of measured physical phenomenon to the interpretation of periodogram spectral content in terms of the Fourier basis sinusoids. The text also discusses peculiarities that arise due to discretization, such as leakage and aliasing, and why de-trending and windows mitigate such effects. Included examples explain the details of implementation and interpretation, with MATLAB source code provided for a number of frequency-spectrum-analysis tools.

A classic treatment of Fourier transforms from the acclaimed Annals of Mathematics Studies seriesPrinceton University Press is proud to have published the Annals of Mathematics Studies since 1940. One of the oldest and most respected series in science publishing, it has included many of the most important and influential mathematical works of the twentieth century. The series continues this tradition as Princeton University Press publishes the major works of the twenty-first century.To mark the continued success of the series, all books are available in paperback and as ebooks.

The purpose of this book is to present results on the subject of weak convergence in function spaces to study invariance principles in statistical applications to dependent random variables, U-statistics, censor data analysis. Different techniques, formerly available only in a broad range of literature, are for the first time presented here in a self-contained fashion. Contents:Weak convergence of stochastic processesWeak convergence in metric spacesWeak convergence on C[0, 1] and D[0,∞)Central limit theorem for semi-martingales and applicationsCentral limit theorems for dependent random variablesEmpirical processBibliography

工学系では必修と言って良いラプラス・フーリエ変換を、多くの演習問題を解くことで理解でき、その利用を学べる。ラプラス変換編では多くの自然現象・工学上の課題が微分方程式で表わされることを示す。フーリエ変換編では周波数というイメージしにくい分野を扱うため、表計算ソフトを利用して視覚的に確認しながら進められるよう工夫してある。各章末には演習問題と解を掲載しており確実な理解を助ける。

The discrete Fourier transform (DFT) is an extremely useful tool that finds application in many different disciplines. However, its use requires caution. The aim of this book is to explain the DFT and its various artifacts and pitfalls and to show how to avoid these (whenever possible), or at least how to recognize them in order to avoid misinterpretations. This concentrated treatment of the DFT artifacts and pitfalls in a single volume is, indeed, new, and it makes this book a valuable source of information for the widest possible range of DFT users. Special attention is given to the one and two dimensional cases due to their particular importance, but the discussion covers the general multidimensional case, too. The book favours a pictorial, intuitive approach which is supported by mathematics, and the discussion is accompanied by a large number of figures and illustrative examples, some of which are visually attractive and even spectacular. Mastering the Discrete Fourier Transform in One, Two or Several Dimensions is intended for scientists, engineers, students and any readers who wish to widen their knowledge of the DFT and its practical use. This book will also be very useful for ‘naive'users from various scientific or technical disciplines who have to use the DFT for their respective applications. The prerequisite mathematical background is limited to an elementary familiarity with calculus and with the continuous and discrete Fourier theory.

At the international conference on'Harmonic Analysis and Integral Transforms', conducted by one of the authors at the Mathematical Research Institute in Oberwolfach (Black Forest) in August 1965, it was felt that there was a real need for a book on Fourier analysis stressing (i) parallel treatment of Fourier series and Fourier trans­ forms from a transform point of view, (ii) treatment of Fourier transforms in LP(lRn)_ space not only for p = 1 and p = 2, (iii) classical solution of partial differential equations with completely rigorous proofs, (iv) theory of singular integrals of convolu­ tion type, (v) applications to approximation theory including saturation theory, (vi) multiplier theory, (vii) Hilbert transforms, Riesz fractional integrals, Bessel potentials, (viii) Fourier transform methods on locally compact groups. This study aims to consider these aspects, presenting a systematic treatment of Fourier analysis on the circle as well as on the infinite line, and of those areas of approximation theory which are in some way or other related thereto. A second volume is in preparation which goes beyond the one-dimensional theory presented here to cover the subject for functions of several variables. Approximately a half of this first volume deals with the theories of Fourier series and of Fourier integrals from a transform point of view.

The Fourier transform of large multidimensional data sets is an essen­ tial computation in many scientific and engineering fields, including seismology, X-ray crystallography, radar, sonar and medical imaging. Such fields require multidimensional arrays for complete and faithful modelling. Classically, a set of data is processed one dimension at a time, permitting control over the size of the computation and calling on well-established I-dimensional programs. The rapidly increasing availability of powerful computing chips, vector processors, multinode boards and parallel machines has provided new tools for carrying out multidimensional computations. Multidimensional processing offers a wider range of possible implementations as compared to I-dimensional the greater flexibility of movement in the data in­ processing, due to dexing set. This increased freedom, along with the massive size data sets typically found in multidimensional applications, places intensive demands on the communication aspects of the computation. The writ­ ing of code that takes into account all the algorithmic possibilities and matches these possibilities to the communication capabilities of the tar­ get architecture is an extremely time-consuming task. A major goal of this text is to provide a sufficiently abstra

This book is intended to be a comprehensive reference to multiplicative com­ plexity theory as applied to digital signal processing computations. Although a few algorithms are included to illustrate the theory, I concentrated more on the develop­ ment of the theory itself. Howie Johnson's infectious enthusiasm for designing efficient DfT algorithms got me interested in this subject. I am grateful to Prof. Sid Burrus for encouraging and supporting me in this effort. I would also like to thank Henrik Sorensen and Doug Jones for many stimulating discussions. lowe a great debt to Shmuel Winograd, who, almost singlehandedly, provided most of the key theoretical results that led to this present work. His monograph, Arithmetic Complexity o/Computations, introduced me to the mechanism behind the proofs of theorems in multiplicative complexity. enabling me to return to his earlier papers and appreciate the elegance of his methods for deriving the theory. The second key work that influenced me was the paper by Louis Auslander and Winograd on multiplicative complexity of semilinear systems defined by polynomials. After reading this paper, it was clear to me that this theory could be applied to many impor­ tant computational problems. These influences can be easily discerned in the present work.