Fourier transforms - approach to scientific principles

482 Pages·2011·21.635 MB·English

by  Goran S Nikolić

FOURIER TRANSFORMS (cid:883)
APPROACH TO
SCIENTIFIC PRINCIPLES
Edited by Goran S. Nikolić
Fourier Transforms - Approach to Scientific Principles
Edited by Goran S. Nikolić
Published by InTech
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Contents
Preface IX
Chapter 1 Theoretical Description of the Fourier Transform
of the Absolute Amplitude Spectra and Its Applications 1
Levente Csoka and Vladimir Djokovic
Chapter 2 Gaussian and Fourier Transform (GFT) Method
and Screened Hartree-Fock Exchange Potential
for First-principles Band Structure Calculations 15
Tomomi Shimazaki and Yoshihiro Asai
Chapter 3 Low Complexity Fourier Transforms
using Multiple Square Waves 37
Khoirul Anwar and Minoru Okada
Chapter 4 Orbital Stability of Periodic Traveling Wave Solutions 45
Jaime Angulo Pava and Fábio Natali
Chapter 5 Approach to Fundamental Properties
of the Henstock-Fourier Transform 71
Fco. Javier Mendoza Torres, J. Alberto Escamilla Reyna
and Ma. Guadalupe Raggi Cárdenas
Chapter 6 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
Chapter 7 Quadratic Discrete Fourier Transform
and Mutually Unbiased Bases 103
Maurice R. Kibler
Chapter 8 Orthogonal Discrete Fourier and
Cosine Matrices for Signal Processing 139
Daechul Park and Moon Ho Lee
VI Contents
Chapter 9 Optimized FFT Algorithm
and its Application to Fast GPS Signal Acquisition 157
Lin Zhao, Shuaihe Gao, Jicheng Ding and Lishu Guo
Chapter 10 Homogenization of Nonlocal Electrostatic Problems
by Means of the Two-Scale Fourier Transform 175
Niklas Wellander
Chapter 11 Time-resolved Fourier Transform
Infrared Emission Spectroscopy:
Application to Pulsed Discharges and Laser Ablation 189
Svatopluk Civiš and Vladislav Chernov
Chapter 12 Weighting Iterative Fourier Transform Algorithm
for Kinoform Implemented with Liquid-Crystal SLM 225
Alexander Kuzmenko, Pavlo Iezhov and Jin-Tae Kim
Chapter 13 Two-Dimensional Quaternionic
Windowed Fourier Transform 247
Mawardi Bahri and Ryuichi Ashino
Chapter 14 High Frame Rate Ultrasonic Imaging through Fourier
Transform using an Arbitrary Known Transmission Field 261
Hu Peng
Chapter 15 High-Accuracy and High-Security Individual
Authentication by the Fingerprint Template Generated
Using the Fractional Fourier Transform 281
Reiko Iwai and Hiroyuki Yoshimura
Chapter 16 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
Chapter 17 Enhanced Fourier Transforms
for X-Ray Scattering Applications 321
Benjamin Poust and Mark Goorsky
Chapter 18 Fourier Transform on Group-Like
Structures and Applications 341
Massoud Amini, Mehrdad Kalantar,
Hassan Myrnouri and Mahmood M. Roozbahani
Chapter 19 Reduced Logic and Low-Power FFT Architectures
for Embedded Systems 381
Erdal Oruklu, Jafar Saniie and Xin Xiao
Contents VII
Chapter 20 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
Chapter 21 Fourier Transform Based Hyperspectral Imaging 427
Marco Q. Pisani and Massimo E. Zucco
Chapter 22 Application of Fast Fourier Transform for Accuracy
Evaluation of Thermal-Hydraulic Code Calculations 447
Andrej Prošek and Matjaž Leskovar

Preface
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 writt 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 matt er;
- introduced a new method for enhancing Fourier transforms of x-ray
scatt 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