FOURIER TRANSFORM – MATERIALS ANALYSIS Edited by Salih Mohammed Salih Fourier Transform – Materials Analysis Edited by Salih Mohammed Salih Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2012 InTech All chapters are Open Access distributed under the Creative Commons Attribution 3.0 license, which allows users to download, copy and build upon published articles even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. As for readers, this license allows users to download, copy and build upon published chapters even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. Notice Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published chapters. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Vana Persen Technical Editor Teodora Smiljanic Cover Designer InTech Design Team First published May, 2012 Printed in Croatia A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from orders@intechopen.com Fourier Transform – Materials Analysis, Edited by Salih Mohammed Salih p. cm. ISBN 978-953-51-0594-7 Contents Preface IX Chapter 1 Fourier Series and Fourier Transform with Applications in Nanomaterials Structure 1 Florica Matei and Nicolae Aldea Chapter 2 High Resolution Mass Spectrometry Using FTICR and Orbitrap Instruments 25 Paulo J. Amorim Madeira, Pedro A. Alves and Carlos M. Borges Chapter 3 Fourier Transform Infrared Spectroscopy for Natural Fibres 45 Mizi Fan, Dasong Dai and Biao Huang Chapter 4 Fourier Transform Infrared Spectroscopy for the Measurement of Spectral Line Profiles 69 Hassen Aroui, Johannes Orphal and Fridolin Kwabia Tchana Chapter 5 Fourier Transform Spectroscopy of Cotton and Cotton Trash 103 Chanel Fortier Chapter 6 Fourier Transformation Method for Computing NMR Integrals over Exponential Type Functions 121 Hassan Safouhi Chapter 7 Molecular Simulation with Discrete Fast Fourier Transform 137 Xiongwu Wu and Bernard R. Brooks Chapter 8 Charaterization of Pore Structure and Surface Chemistry of Activated Carbons – A Review 165 Bingzheng Li Chapter 9 Bioleaching of Galena (PbS) 191 E. R. Mejía, J. D. Ospina, M. A. Márquez and A. L. Morales VI Contents Chapter 10 Application of Hankel Transform for Solving a Fracture Problem of a Cracked Piezoelectric Strip Under Thermal Loading 207 Sei Ueda Chapter 11 Eliminating the Undamaging Fatigue Cycles Using the Frequency Spectrum Filtering Techniques 223 S. Abdullah, T. E. Putra and M. Z. Nuawi Chapter 12 Fourier Transform Sound Radiation 239 F. X. Xin and T. J. Lu Preface This book focuses on the Fourier transform applications in the analysis of some types of materials. The field of Fourier transform has seen explosive growth during the past decades, as phenomenal advances both in research and application have been made. During the preparation of this book, we found that almost all the textbooks on materials analysis have a section devoted to the Fourier transform theory. Most of those describe some formulas and algorithms, but one can easily be lost in seemingly incomprehensible mathematics. 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 19 th century, and it was proven to be useful in various applications. 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 (DSP). One of the main focuses of this book is on getting material characterization of nanomaterials through Fourier transform infrared spectroscopy (FTIR), and this fact can be taken from FTIR which gives reflection coefficient versus wave number. The Fourier transform spectroscopy is a measurement technique whereby spectra are collected based on measurements of the coherence of a radiative source, using time-domain or space-domain measurements of the electromagnetic radiation or other type of radiation. It can be applied to a variety of types of spectroscopy including optical spectroscopy, infrared spectroscopy (FTIR, FT-NIRS), nuclear magnetic resonance (NMR) and magnetic resonance spectroscopic imaging (MRSI), mass spectrometry and electron spin resonance spectroscopy. There are several methods for measuring the temporal coherence of the light, including the continuous wave Michelson or Fourier transform spectrometer and the pulsed Fourier transform spectrograph (which is more sensitive and has a much shorter sampling time than conventional spectroscopic techniques, but is only applicable in a laboratory environment). The term Fourier transform spectroscopy reflects the fact that in all these techniques, a Fourier transform is required to turn the raw data into the actual spectrum, and in many of the cases in optics involving interferometers, is based on the X Preface Wiener–Khinchin theorem. In this book, New theoretical results are appearing 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 real-life materials analysis problems, which has been made possible by the use of computers in solving practical design problems. The whole book contains twelve chapters. The chapters deal with nanomaterials structure, mass spectrometry, infrared spectroscopy for natural fibers, infrared spectroscopy to the measurements of spectral line profile, spectroscopy of cotton and cotton trash, computing NMR integrals over exponential type functions, molecular simulation, charaterization of pore structure and surface chemistry of activated carbons, bioleaching of galena, the cracked piezoelectric strip under thermal loading, eliminating the undamaging fatigue cycles, and the Fourier transform sound radiation. 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 Physics, Chemists, Agriculturalists, Electrical Engineers, Mechanical Engineers and the Signal Processing Engineers, in addition to the Academic Researchers working in these fields. Salih Mohammed Salih College of Engineering Univercity of Anbar Iraq [...]... the Fourier transform i ii Linearity If the signals x and y have the Fourier transform X and Y then the Fourier transform of α x + β y is α X + β Y Symmetry If the Fourier transform of the function h is H, then h( − f ) = ∫ ∞ −∞ H (t )exp ( −2π i f t ) dt (67) iii Scaling If h has the Fourier transform H then ∞ ∫ −∞ h( kt )exp( −2π ift )dt = 1 H ( f / k ), k ∈ R * k (68) iv Shifting If the Fourier transform. .. background of Fourier series and Fourier transform used in nanomaterials structure field 22 Fourier Transform – Materials Analysis The conclusions that can be drawn from this contribution are: i The physical periodical signals are successfully modeled using the trigonometric polynomial such us global approximation of the XRLP and the spectral distribution determination based on the Fourier analysis; ii... frequency are given by the spectral distribution named Fourier analysis The difference between Fourier series and Fourier transform is that the latter has the frequencies as argument which continuously varies Whereas Fourier transform of the signal h allows spectral decomposition of it with frequencies defined on the whole real axis 3.2 The Fourier transform for discreet signals In practice the function... coordination shell (Aldea et al., 2007) The analysis of EXAFS data for obtaining structural information [Nj, rj, σj, λj(k)] Fourier Series and Fourier Transform with Applications in Nanomaterials Structure 17 generally proceeds by the use of the Fourier transform From χ(k), the radial structure function (RSF) can be derived The single shell may be isolated by Fourier transform, Φ (r ) = ∫ ∞ −∞ k n χ ( k )WF(... k^n*Chi*WF]| 30 25 20 15 10 5 0 0 X 1 2 X 3 Distance [A] 4 5 Fig 10 The Fourier transform of the EXAFS spectrum for the nickel foil 6 20 Fourier Transform – Materials Analysis Each peak from |Φ(r)|is shifted from the true distance due to the phase shift function that is included in the EXAFS signal We proceed by taking the inverse Fourier transform given by relation (59) of the first neighboring peak, and... Because the aim of this chapter are the application of the Fourier series it will be only mentioned the basic principles of the Fourier analysis If the interval [ −T / 2, T / 2] can be decomposed in a finite number of intervals on that the function f is 4 Fourier Transform – Materials Analysis continuous and monotonic, then the function f has a Fourier series representation The next consideration is connected... in the scientific literature named as the Fourier integral (Brigham, 1988) and the expression H( f ) = ∫ ∞ −∞ h(s )exp ( −2π i f s ) ds (40) represents the Fourier transform of the function h From the relation (40) is it possible to obtain the function h by inverse Fourier transform given by 11 Fourier Series and Fourier Transform with Applications in Nanomaterials Structure h(t ) = ∫ ∞ −∞ H ( f )exp... possibility of using the analytical form of the Fourier transform instead of the numerical FFT The validity of the microstructural parameters are closely related to accuracy of the Fourier transform magnitude of the true XRLP The experimental relative intensities with respect to θ values 21 Fourier Series and Fourier Transform with Applications in Nanomaterials Structure and the nickel foil as instrumental... that it is defined by the square magnitude of the Fourier coefficients 12 Fourier Transform – Materials Analysis 12 Spectral distribution 10 8 6 4 2 0 0 0.5 1 1.5 2 2.5 3 3.5 harmonic index 4 4.5 5 Fig 5 The spectral distribution of the signal g The real component of the function (43) is represented in Fig 6 and the square magnitude of the Fourier transform is given by relation (45) and it is represented... Fourier transform infrared spectroscopy (FTIR) as well as X-ray absorption spectroscopy (XAS) dedicated to K or L near and extended edges are based on non periodical signals analysis 3.1 Mathematical background of the discreet and inverse Fourier transform Let consider the complex form of the Fourier series for a signal h defined on the interval [ −T / 2, T / 2 ] it will be introduced the Fourier transform . FOURIER TRANSFORM – MATERIALS ANALYSIS Edited by Salih Mohammed Salih Fourier Transform – Materials Analysis Edited by Salih Mohammed. Fourier Transform – Materials Analysis, Edited by Salih Mohammed Salih p. cm. ISBN 978-953-51-0594-7 Contents Preface IX Chapter 1 Fourier Series and Fourier Transform. Chapter 12 Fourier Transform Sound Radiation 239 F. X. Xin and T. J. Lu Preface This book focuses on the Fourier transform applications in the analysis of some types of materials.