Preface

During the preparation of this book, we found that almost all the textbooks on signal analysis have a section devoted to the Fourier transform theory. The Fourier transform is a mathematical operation with many applications in physics, and engineering that express a mathematical function of time as a function of frequency, known as its frequency spectrum; Fourier's theorem guarantees that this can always be done. The basic idea behind all those horrible looking formulas is rather simple, even fascinating: *it is possible to form any function as a summation of a series of sine and cosine terms of increasing frequency*. In other words, any space or time varying data can be transformed into a different domain called the *frequency space*. A fellow called *Joseph Fourier* first came up with the idea in the 19th century, and it was proven to be useful in various applications, mainly in signal processing. As far as we can tell, Gauss was the first to propose the techniques that we now call the Fast Fourier Transform (FFT) for calculating the coefficients in a trigonometric expansion of an asteroid's orbit in 1805. However, it was the seminal paper by Cooley and Tukey in 1965 that caught the attention of the science and engineering community and, in a way, founded the discipline of digital signal processing. While the Discrete Fourier Transform (DFT) can be applied to any complex valued series, in practice for large series it can take considerable time to compute, the time taken being proportional to the square of the number on points in the series. It is hard to overemphasis the importance of the DFT, convolution, and fast algorithms. The FFT may be the most important numerical algorithm in science, engineering, and applied mathematics. New theoretical results are still appearing, advances in computers and hardware continually restate the basic questions, and new applications open new areas for research. It is hoped that this book will provide the background, references and incentive to encourage further research and results in this area as well as provide tools for practical applications. One of the attractive features of this book is the inclusion of extensive simple, but practical, examples that expose the reader to reallife signal analysis problems, which has been made possible by the use of computers in solving practical design problems. The aim of this book is to expand the applications of Fourier transform into main three sections:

The first section deals with *Electromagnetic Field and Microwave Applications*. It consists of five chapters. The chapters are related to: the computation of transient near-field radiated by electronic devices, analysis of novel optically-inspired phenomena at

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microwaves, the computation of lightning electromagnetic field, beamforming and DOA estimation, and the solar microwave radiation.

The chapters of the second section discuss some advanced methods used in Fourier transform analysis which are related to the *Medical Applications*. This section consists of three chapters. The chapters are related to: spectral analysis of global behaviour of C. Elegans chromosomes, spectral analysis of heart rate variability in women, and the cortical specification of a fast Fourier transform supports a convolution model of visual perception.

The third section includes the *Fourier and Helbert Transform Applications*. This section consists of four chapters. The chapters are concerns to: the Fourier convolution theorem over finite fields (error control coding), application of the weighted energy method in the partial Fourier space to linearized viscous conservation laws with nonconvex condition, Fourier transform methods for option pricing, and the Hilbert transform applications.

Finally, we would like to thank all the authors who have participated in this book for their valuable contribution. Also we would like to thank all the reviewers for their valuable notes. While there is no doubt that this book may have omitted some significant findings in the Fourier transform field, we hope the information included will be useful for electrical engineers, control engineers, communication engineers, signal processing engineers, medical researchers, and the mathematicians, in addition to the academic researchers working in the above fields.

> **Salih Mohammed Salih**  College of Engineering University of Anbar Iraq
