Fourier Transforms and Wavelet

Discrete wavelet transform (DWT) algorithms have become standard tools for discrete-time signal and image processing in several areas in research and industry. As DWT provides both frequency and location information of the analyzed signal, it is constantly used to solve and treat more and more advanced problems.

This book focuses on the Discrete Fourier Transform (DFT), discrete convolution, and, particularly, the fast algorithms to calculate them. These topics have been at the center of digital signal processing since its beginning, and new results in hardware, theory and applications continue to keep them important and exciting.

This book aims to provide information about Fourier transform to those needing to use infrared spectroscopy, by explaining the fundamental aspects of the Fourier transform, and techniques for analyzing infrared data obtained for a wide number of materials.

The book focuses on Fourier transform applications in electromagnetic field and microwave, medical applications, error control coding, methods for option pricing, and Helbert transform application.

The use of the wavelet transform to analyze the behaviour of the complex systems from various fields started to be widely recognized and applied successfully during the last few decades.

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.

This text introduces the basic concepts of function spaces and operators, both from the continuous and discrete viewpoints. Emphasis is given on ideas and intuition, avoiding the heavy computations which are usually involved in the study of wavelets.

This monograph is concerned with wavelet harmonic analysis on the sphere. By starting with orthogonal polynomials and functional Hilbert spaces on the sphere, the foundations are laid for the study of spherical harmonics such as zonal functions.

The essential mathematics required to understand and apply Fourier analysis is explained. The tutorial style of writing, combined with diagrams, offers a visually intuitive and rigorous account of Fourier methods.

The main purpose of this book is to provide a modern review about recent advances in Fourier transforms as the most powerful analytical tool for high-tech application in electrical, electronic, and computer engineering, as well as Fourier transform spectral techniques with a wide range of biological, biomedical, biotechnological, pharmaceutical, and nanotechnological applications. The confluence of Fourier transform methods with high tech opens new opportunities for detection and handling of atoms and molecules using nanodevices, with potential for a large variety of scientific and technological applications.

The Discrete wavelet transform (DWT) algorithms have a firm position in processing of signals in several areas of research and industry. As DWT provides both octave-scale frequency and spatial timing of the analyzed signal, it is constantly used to solve and treat more and more advanced problems.

The field of signal processing has seen explosive growth during the past decades; almost all textbooks on signal processing have a section devoted to the Fourier transform theory. For this reason, this book focuses on the Fourier transform applications in signal processing techniques. The book chapters are related to DFT, FFT, OFDM, estimation techniques and the image processing techqniques.

This book is intended to give the necessary mathematical background on wavelets, and explore the main applications, both current and potential, to computer graphics. The emphasis is put on the connection between wavelets and the tools and concepts which should be familiar to any skilled computer graphics person: Fourier techniques, pyramidal schemes, spline representations.