FOURIER TRANSFORMS  APPROACH TO SCIENTIFIC PRINCIPLES Edited by Goran S. Nikolić Fourier Transforms - Approach to Scientific Principles Edited by Goran S. Nikolić Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2011 InTech All chapters are Open Access articles distributed under the Creative Commons Non Commercial Share Alike Attribution 3.0 license, which permits to copy, distribute, transmit, and adapt the work in any medium, so long as the original work is properly cited. 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. Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. 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ISBN 978-953-307-231-9 free online editions of InTech Books and Journals can be found at www.intechopen.com Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Preface IX Theoretical Description of the Fourier Transform of the Absolute Amplitude Spectra and Its Applications 1 Levente Csoka and Vladimir Djokovic Gaussian and Fourier Transform (GFT) Method and Screened Hartree-Fock Exchange Potential for First-principles Band Structure Calculations 15 Tomomi Shimazaki and Yoshihiro Asai Low Complexity Fourier Transforms using Multiple Square Waves 37 Khoirul Anwar and Minoru Okada Orbital Stability of Periodic Traveling Wave Solutions 45 Jaime Angulo Pava and Fábio Natali Approach to Fundamental Properties of the Henstock-Fourier Transform 71 Fco. Javier Mendoza Torres, J. Alberto Escamilla Reyna and Ma. Guadalupe Raggi Cárdenas Three Dimensional Reconstruction Strategies Using a Profilometrical Approach based on Fourier Transform 87 Pedraza-Ortega Jesus Carlos, Gorrostieta-Hurtado Efren, Aceves-Fernandez Marco Antonio, Sotomayor-Olmedo Artemio, Ramos-Arreguin Juan Manuel, Tovar-Arriaga Saul and Vargas-Soto Jose Emilio Quadratic Discrete Fourier Transform and Mutually Unbiased Bases 103 Maurice R. Kibler Orthogonal Discrete Fourier and Cosine Matrices for Signal Processing 139 Daechul Park and Moon Ho Lee Contents Contents VI Optimized FFT Algorithm and its Application to Fast GPS Signal Acquisition 157 Lin Zhao, Shuaihe Gao, Jicheng Ding and Lishu Guo Homogenization of Nonlocal Electrostatic Problems by Means of the Two-Scale Fourier Transform 175 Niklas Wellander Time-resolved Fourier Transform Infrared Emission Spectroscopy: Application to Pulsed Discharges and Laser Ablation 189 Svatopluk Civiš and Vladislav Chernov Weighting Iterative Fourier Transform Algorithm for Kinoform Implemented with Liquid-Crystal SLM 225 Alexander Kuzmenko, Pavlo Iezhov and Jin-Tae Kim Two-Dimensional Quaternionic Windowed Fourier Transform 247 Mawardi Bahri and Ryuichi Ashino High Frame Rate Ultrasonic Imaging through Fourier Transform using an Arbitrary Known Transmission Field 261 Hu Peng High-Accuracy and High-Security Individual Authentication by the Fingerprint Template Generated Using the Fractional Fourier Transform 281 Reiko Iwai and Hiroyuki Yoshimura Fourier Transform Mass Spectrometry for the Molecular Level Characterization of Natural Organic Matter: Instrument Capabilities, Applications, and Limitations 295 Rachel L. Sleighter and Patrick G. Hatcher Enhanced Fourier Transforms for X-Ray Scattering Applications 321 Benjamin Poust and Mark Goorsky Fourier Transform on Group-Like Structures and Applications 341 Massoud Amini, Mehrdad Kalantar, Hassan Myrnouri and Mahmood M. Roozbahani Reduced Logic and Low-Power FFT Architectures for Embedded Systems 381 Erdal Oruklu, Jafar Saniie and Xin Xiao Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Contents VII The Effect of Local Field Dispersion on the Spectral Characteristics of Nanosized Particles and their Composites 405 T.S. Perova, I.I. Shaganov and K. Berwick Fourier Transform Based Hyperspectral Imaging 427 Marco Q. Pisani and Massimo E. Zucco Application of Fast Fourier Transform for Accuracy Evaluation of Thermal-Hydraulic Code Calculations 447 Andrej Prošek and Matjaž Leskovar Chapter 20 Chapter 21 Chapter 22 Pref ac e APPROACH TO SCIENTIFIC PRINCIPLES: THEORY-METHODOLOGY-APPLICATIONS “Our real problem is not our strength today; it is rather the vital necessity of action today to ensure our strength tomorrow.” – Calvin Coolidge – This book provides a broad treatment of the principles and theory of Fourier Trans- form Infrared Spectroscopy (FTIR) as it is used in the physical, chemical, mathemati- cal, biological sciences, as well as in medicine and technology. It is wri en at a scientif- ic-technical level, with mathematics used to augment, rather than replace, clear verbal descriptions of the phenomena. The book is intended to allow the reader to understand FTIR at a fundamental and scientifi c level, and to see illustrations of the applications of FTIR in diff erent area. Emphasis is on the study of new Fourier transform methods and diff erent strategies using the Fourier transform, but the book also includes the main principles of FTIR spectrophotometer, i.e. Michelson’s interferometer, and the principles of FTIR imaging and localized spectroscopy. Last couple of years have seen a steady progress and a number of advances in the FTIR area. New methods have been developed and deeper results have been obtained, but new problems have also emerged. This volume gives an overview of recent methods developed by authors for the study of these basic issues, and presents old and new applications for FTIR. These methods are based in the theory of totally positive opera- tors, the equations, the theory of analytic perturbations for linear operators, Fourier analysis, the Poisson summation theorem and the theory of elliptic functions. Some of the authors of the volume are the pioneers in the study of the existence and nonlinear stability of periodic traveling wave solutions for nonlinear dispersive equations, new methods and applications. Thus, in this volume we have: - proposed a novel screened Harteee-Fock (HF) exchange potential; - proposed multiple square wave for Fourier transform, which is suitable for digital communication systems where the power consumption constraint is considered; - developed the Gaussian Fourier transform (GFT) method which is suitable to employ well-established quantum chemical theories and methodologies; - defi ned a norm with which the Lebesgue-integrable functions space becomes a Banach space with good properties; X Preface - represented the inverse of the DFT matrix following the factorization process of the jacket transform, as well as DCT/DFT matrices via one hybrid architecture; - optimized FFT algorithm and applied it to a fast GPS signal acquisition; - presented a new iterative Fourier transform method to synthesize kinoforms; - presented diff erent strategies using the Fourier transform for three dimensional reconstruction purposes; - developed an extended, more general HFR method for 2D imaging to widen the imaged area; - presented the Fourier transform mass spectrometry for the molecular level characterization of natural organic ma er; - introduced a new method for enhancing Fourier transforms of x-ray sca ering data; - applied Fourier transform spectroscopy to Fabry-Perot hyperspectral imaging. In this volume of the book we have described the main principles of Fourier trans- form and IR spectrophotometer, i.e. Michelson’s interferometer. The interferogram have been defi ned and the main formulae that lead to Fourier transform calculation of the measured spectrum from the interferogram have been described. The questions of frequency modulation, apodization and phase correction have been addressed based on those formulae. The principal diff erences between the Fourier and dispersive spec- trophotometers and the real eff ects of the multiplexing advantage have been discussed next. Naturally, some of the aspects have disadvantages which are discussed here as well. We have also touched on some theorems and their consequences in the Fourier transform spectrophotometer. These aspects have been considered mainly from the viewpoint of photocurrent spectroscopy of non-crystalline semiconductors. Many oth- er general aspects are covered by other chapters in the second volume of the book. The Fourier transforms play an important role used in physical optics, optical infor- mation processing, linear systems theory and the other areas. In this volume, authors present a new aspect of Fourier transform, and methodologies for fi rst-principle band structure calculations using Fourier transform technique. For instance, Fourier transform was designed to solve diff erent problems in diff erent areas of mathematics. Thus, some of the integral (for example Henstock-Kurzweil) can be applied to the diff erential equations theory, integral equations theory, Fourier anal- ysis, probability, statistics, etc. Today, Lebesgue integral is the main integral used in various areas of mathematics, for example Fourier analysis. However, many functions (e.g. functions that have a “bad” oscillatory behavior) which are not Lebesgue-inte- grable are Henstock-Kurzweil-integrable. Therefore, it seems a natural way to study Fourier analysis by using this integral. In one of the chapters the basic theorem is investigated how the Fourier transform of absolute amplitude spectra can be defi ned in a closed form including a description of the theory of repeated FT for one and two dimensional signals, delta functions and how the theory can be carried over to arbitrary functions. It also includes a direct ap- plication to wood anatomy. On the other hand, the study of the existence and nonlinear stability of traveling wave solutions for nonlinear dispersive evolution equations has grown into a large fi eld in [...]... chapter, it will be shown how the presented theory can be applied to the analysis of the wood anatomy, specifically to determination of the transition point between juvenile and mature wood 2 Fourier Transforms - Approach to Scientific Principles 2 Problem statement We will start these theoretical considerations with familiar one-dimensional Fourier transform (FT) of a given function f ( x ) , F( k ) =... matrix element of the Hartree term is 16 Fourier Transforms - Approach to Scientific Principles determined in the real-space integration including Gaussian-based atomic orbtials and plane waves We can employ a recursive relation to achieve the integration, as discussed later Conversely, we can employ the effective core potential (ECP) instead of explicitly taking into account core electrons in the GFT... original spectrum Reciprocate of Eq (30) was further used to determine the FT spectrum of the absolute amplitude spectrum from a density function of a tree Similarly to Eq (30) we can generate formula for two dimensional signals (pictures) as, ∑∑ cos(2π f0mx)cos(2π f0nx) ⇒|∑∑ e−i 2 π f m n m n 0m e−i 2 π f0 n | (31) 6 Fourier Transforms - Approach to Scientific Principles It should also be emphasized that... 6 can be noticed, with could also justify our approach of using forwarded Fourier transformation of the absolute spectrum for determination of the demarcation zone between juvenile and mature wood The texture of the 3D picture obtained from the forwarded FT of the absolute spectrum exhibit obvious annual ring pattern 10 Fourier Transforms - Approach to Scientific Principles Fig 5 The power spectrum... and makes the determination of the changes related to the distance from the pith impossible 12 Fourier Transforms - Approach to Scientific Principles In this chapter we presented the FT of an amplitude spectrum theorem that can find direct application in studying of a wood anatomy In spite of its simplicity, to our best knowledge there is no reference in the literature regarding the use of forwarded... conjugate of f (by definition Eq (17) is a relationship between FT and its autocorrelation function) Using Eq (17), F( k ) can be expressed as, |F( k )|= ∞ ⎡ ∫−∞ ⎣F[|F( k )| ]( 2 ⎤ )⎦ e i 2 π k d = (18) 4 Fourier Transforms - Approach to Scientific Principles = ∞ ⎡ ∞ ∫−∞ ⎢ ∫−∞ f (x ) f ( x + ⎣ )dx ⎤ e i 2 π k d ⎥ ⎦ (19) Therefore, the Fourier transform of the absolute amplitude spectrum is ∞ F[|F( k )|](... dα ,*j dβ , j exp ( ik ⋅ Q ) K k j (5) 18 Fourier Transforms - Approach to Scientific Principles ( Where f FD EF − λ jk respectively ) and E F are the Fermi–Dirac distribution function and the Fermi energy, 2.2 Gaussian and Fourier Transform (GFT) method In the crystal orbital method, the calculation of the Hartree term is the most time-consuming part due to the long-range behavior of the Coulomb potential... complex effect of the interaction among these waves 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 60 120 180 240 300 Distance from the pith to the bark [mm] Fig 2 The density function of a sugi tree (obtained by laser scanning of the X-ray image) 8 Fourier Transforms - Approach to Scientific Principles Density value [g/cm3] 0.04 0.03 0.02 0.01 0 0 1 2 3 4 5 6 Frequency [1/mm] 7 8 Fig 3 The amplitude spectrum... authentication method Discrete Fourier transform (DFT) is useful for constructing mutually unbiased bases One of chapters studies a quadratic transformation generalization, called quadratic discrete Fourier transform, which makes it possible to derive mutually unbiased bases Although the main goal of the chapter is to introduce the notion of quadratic discrete Fourier transform and to apply it to mutually unbiased... sum of the slices from the bottom to the top of the image The highest peak in the spectrum refers to the transition point of juvenile and mature wood 90 The highest peak indicate the boundary line between 75 juvenil and mature wood 60 amplitude X=169 dpi 45 30 15 0 -15 100 120 140 160 180 200 220 240 260 Pixel [dpi] Fig 7 The sum of pixel slices from the bottom to the top of the forwarded FT of the . FOURIER TRANSFORMS  APPROACH TO SCIENTIFIC PRINCIPLES Edited by Goran S. Nikolić Fourier Transforms - Approach to Scientific Principles Edited by Goran. can be applied to the analysis of the wood anatomy, specifically to determination of the transition point between juvenile and mature wood. Fourier Transforms - Approach to Scientific Principles. e APPROACH TO SCIENTIFIC PRINCIPLES: THEORY-METHODOLOGY-APPLICATIONS “Our real problem is not our strength today; it is rather the vital necessity of action today to ensure our strength tomorrow.”