Preface

Chapter 8 **Mineralogical Characterization of Chalcopyrite**

E.R. Mejía, J.D. Ospina, L. Osorno, M.A. Márquez and A.L. Morales

**Bioleaching 197**

**VI** Contents

The Fourier transform is important in engineering, mathematics, and physical sciences. Its discrete counterpart, the discrete Fourier transform (DFT), which is typically computed us‐ ing the fast Fourier transform (FFT), has revolutionized modern society, as it is ubiquitous in digital electronics and signal processing. This book focuses on Fourier transform applica‐ tions in signal processing techniques and physical sciences. The field of signal processing has seen an explosive growth during the past decades, as phenomenal advances both in re‐ search and applications have been made. During the preparation of this book, we found that almost all the textbooks on signal processing and physics have a section devoted to the Fourier transform theory. The basic idea is that *it is possible to form any function as a summa‐ tion of a series of sine and cosine terms of increasing frequency.* In other words, any space or timevarying data can be transformed into a different domain called the *frequency space. Joseph Fourier* first proposed the idea of Fourier transform in the 19th century, and it had proven to be useful for various applications, mainly in signal processing for many applications. It can be said that Gauss was the first scientist who proposed the techniques that we now call the 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 atten‐ tion of the science and engineering community and, in a way, proposed the discipline of digital signal processing. The FFT may be the most important numerical algorithm in sci‐ ence, engineering, and applied mathematics. New theoretical results are still appearing, ad‐ vances 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 back‐ ground, references, and incentive to encourage further research and results in this field as well as provide tools for practical applications. One of the features of this book is that the inclusion is simple, and practical examples that expose the reader to real-life signal process‐ ing have been given. The whole book contains eight chapters, and it divided into two sec‐ tions. The first section consists of five chapters that deal with signal processing while the last three chapters deal with physical sciences.

In the first chapter, a unified fast hybrid recursive Fourier transform based on Jacket matrix has been derived. The proposed analysis proves that discrete cosine transform-II (DCT-II), discrete sine transform-II (DST-II), and DFT can be unified by using the diagonal sparse ma‐ trix based on the Jacket matrix and recursive structure based on some modifications. In chapter two, a new acquisition algorithm for global navigation satellite system has been in‐ troduced. Two multipath interference mitigation algorithms based on the decoupled param‐ eter estimation algorithms are presented in this chapter. The FFT based approach is used to determine the sensitivity of output parameters in chapter three. The signal influenced by sensitive parameter variation is compared with a reference signal. In chapter 4, theorems about convergence of integrals of products, based on a version of Riemann-Lebesgue Lem‐ ma function are presented. Fourier transform using Henstock-Kurzweil Integral is used in this chapter. The real and the complex number fields are extended for treating the Fourier transform for functional in chapter five. Two kinds of Fourier transform theories to the propagator for fields harmonic oscillator are applied.

In chapter six, section two, it is shown that XRD analysis provides more information for un‐ derstanding the physical properties of nanomaterial structure. In chapter seven, a new ap‐ proach is introduced for studying the irregular Gabor transform, and proved some non harmonic sets of Fourier expansions for Gabor systems instead of sets of complex exponen‐ tials. In chapter eight, the mineral phases formed during chalcopyrite bioleaching are stud‐ ied using *Acidithiobacillusferrooxidans* bacteria in the absence of ferrous sulfate and elemental sulfur based on Fourier transform infrared spectroscopy, scanning electron microscopy with energy dispersive X-ray spectroscopy, and X-ray diffraction.

Finally, I would like to thank all the authors who have participated in this book for their valuable contributions. Also, I 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 engi‐ neers, communication engineers, signal processing engineers, physicians and mathemati‐ cians, in addition to the academic researchers working in this field.

**Asst. Prof. Salih Mohammed Salih, SMIEEE**

Renewable Energy Research Center University of Anbar, Iraq

**Section 1**

**Signal Processing**

about convergence of integrals of products, based on a version of Riemann-Lebesgue Lem‐ ma function are presented. Fourier transform using Henstock-Kurzweil Integral is used in this chapter. The real and the complex number fields are extended for treating the Fourier transform for functional in chapter five. Two kinds of Fourier transform theories to the

In chapter six, section two, it is shown that XRD analysis provides more information for un‐ derstanding the physical properties of nanomaterial structure. In chapter seven, a new ap‐ proach is introduced for studying the irregular Gabor transform, and proved some non harmonic sets of Fourier expansions for Gabor systems instead of sets of complex exponen‐ tials. In chapter eight, the mineral phases formed during chalcopyrite bioleaching are stud‐ ied using *Acidithiobacillusferrooxidans* bacteria in the absence of ferrous sulfate and elemental sulfur based on Fourier transform infrared spectroscopy, scanning electron microscopy with

Finally, I would like to thank all the authors who have participated in this book for their valuable contributions. Also, I 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 engi‐ neers, communication engineers, signal processing engineers, physicians and mathemati‐

**Asst. Prof. Salih Mohammed Salih, SMIEEE**

Renewable Energy Research Center

University of Anbar, Iraq

propagator for fields harmonic oscillator are applied.

VIII Preface

energy dispersive X-ray spectroscopy, and X-ray diffraction.

cians, in addition to the academic researchers working in this field.

**Chapter 1**
