scholarly journals Approach to Fundamental Properties of the Henstock-Fourier Transform

10.5772/15629 ◽  
2011 ◽  
Author(s):  
Fco. Javier Mendoza Torres ◽  
J. Alberto Escamilla Reyna ◽  
Ma. Guadalupe Raggi Crdenas
Keyword(s):  
Fourier Transform ◽  
Fundamental Properties

Continuous quaternion fourier and wavelet transforms

2014 ◽  
Vol 12 (04) ◽  
pp. 1460003 ◽  
Author(s):  
Mawardi Bahri ◽  
Ryuichi Ashino ◽  
Rémi Vaillancourt
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Heat Equation ◽  
Wavelet Transforms ◽  
Quaternion Algebra ◽  
Two Dimensional ◽  
Quaternion Fourier Transform ◽  
Fundamental Properties

A two-dimensional (2D) quaternion Fourier transform (QFT) defined with the kernel [Formula: see text] is proposed. Some fundamental properties, such as convolution, Plancherel and vector differential theorems, are established. The heat equation in quaternion algebra is presented as an example of the application of the QFT to partial differential equations. The wavelet transform is extended to quaternion algebra using the kernel of the QFT.

scholarly journals The Formalization of Discrete Fourier Transform in HOL

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Zhiping Shi ◽  
Yupeng Zhang ◽  
Yong Guan ◽  
Liming Li ◽  
Jie Zhang
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Order Logic ◽  
Theorem Prover ◽  
Formal Definition ◽  
Accurate Analysis ◽  
Higher Order Logic ◽  
Safety Critical ◽  
Fundamental Properties ◽  
Definition Of

Traditionally, Discrete Fourier Transform (DFT) is performed with numerical or symbolic computation, which cannot guarantee 100% accurate analysis which may be necessary for safety-critical applications. Machine theorem proving is one of the formal methods that perform accurate analysis with completeness to some extent. This paper proposes the formalization of DFT in a higher-order logic theorem prover named HOL. We propose the formal definition of DFT and verify the fundamental properties of DFT. Two case studies are presented to illustrate usefulness and correctness of the formalized DFT, including formal verifications of Fast Fourier Transform (FFT) and cosine frequency shift.

scholarly journals Chirp Signal Transform and Its Properties

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Mio Horai ◽  
Hideo Kobayashi ◽  
Takashi G. Nitta
Keyword(s):  
Fourier Transform ◽  
Differential Equations ◽  
Discrete Fourier Transform ◽  
Chirp Signal ◽  
Constant Amplitude ◽  
Continuous Version ◽  
Fundamental Properties ◽  
Chirp Signals ◽  
Zero Autocorrelation

The chirp signalexp(iπ(x-y)2)is a typical example of CAZAC (constant amplitude zero autocorrelation) sequence. Using the chirp signals, the chirp z transform and the chirp-Fourier transform were defined in order to calculate the discrete Fourier transform. We define a transform directly from the chirp signals for an even or odd numberNand the continuous version. We study the fundamental properties of the transform and how it can be applied to recursion problems and differential equations. Furthermore, whenNis not prime and  N=ML, we define a transform skippedLand develop the theory for it.

Special affine wavelet transform and the corresponding Poisson summation formula

2021 ◽  
pp. 2050086
Author(s):  
Firdous A. Shah ◽  
Aajaz A. Teali ◽  
Azhar Y. Tantary
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
Wavelet Transforms ◽  
Inversion Formula ◽  
Summation Formula ◽  
Poisson Summation Formula ◽  
Time Frequency ◽  
Fundamental Properties ◽  
Stationary Signals ◽  
Poisson Summation

In the article, “Windowed special affine Fourier transform” in J. Pseudo-Differ. Oper. Appl. (2020), we introduced the notion of windowed special affine Fourier transform (WSAFT) as a ramification of the special affine Fourier transform. Keeping in view the fact that the WSAFT is not befitting for in the context of non-stationary signals, we continue our endeavor and introduce the notion of the special affine wavelet transform (SAWT) by combining the merits of the special affine Fourier and wavelet transforms. Besides studying the fundamental properties of the SAWT including orthogonality relation, inversion formula and range theorem, we also demonstrate that the SAWT admits the constant [Formula: see text]-property in the time–frequency domain. Moreover, we formulate an analog of the well-known Poisson summation formula for the proposed SAWT.

Comparison of Calculated and Recorded Electron Energy-Loss Spectra of Aluminium and Carbon

1990 ◽  
Vol 48 (2) ◽  
pp. 64-65
Author(s):  
L. Reimer ◽  
R. Oelgeklaus
Keyword(s):  
Fourier Transform ◽  
Energy Loss ◽  
Electron Energy ◽  
Rotational Symmetry ◽  
Mean Free Path ◽  
Single Scattering ◽  
Specimen Thickness ◽  
Two Dimensional ◽  
Image Position ◽  
Electron Energy Loss

Quantitative electron energy-loss spectroscopy (EELS) needs a correction for the limited collection aperture α and a deconvolution of recorded spectra for eliminating the influence of multiple inelastic scattering. Reversely, it is of interest to calculate the influence of multiple scattering on EELS. The distribution f(w,θ,z) of scattered electrons as a function of energy loss w, scattering angle θ and reduced specimen thickness z=t/Λ (Λ=total mean-free-path) can either be recorded by angular-resolved EELS or calculated by a convolution of a normalized single-scattering function ϕ(w,θ). For rotational symmetry in angle (amorphous or polycrystalline specimens) this can be realised by the following sequence of operations :(1)where the two-dimensional distribution in angle is reduced to a one-dimensional function by a projection P, T is a two-dimensional Fourier transform in angle θ and energy loss w and the exponent -1 indicates a deprojection and inverse Fourier transform, respectively.

FR-IR Molecular Microanalysis System

1990 ◽  
Vol 48 (2) ◽  
pp. 270-271
Author(s):  
John A. Reffner ◽  
William T. Wihlborg
Keyword(s):  
Fourier Transform ◽  
Fourier Transform Infrared ◽  
Integrated System ◽  
Morphological Examination ◽  
Fully Integrated ◽  
Ir Microscopy ◽  
Normal Functions ◽  
Infrared Microspectroscopy ◽  
Infrared Spectral ◽  
Ft Ir

The IRμs™ is the first fully integrated system for Fourier transform infrared (FT-IR) microscopy. FT-IR microscopy combines light microscopy for morphological examination with infrared spectroscopy for chemical identification of microscopic samples or domains. Because the IRμs system is a new tool for molecular microanalysis, its optical, mechanical and system design are described to illustrate the state of development of molecular microanalysis. Applications of infrared microspectroscopy are reviewed by Messerschmidt and Harthcock.Infrared spectral analysis of microscopic samples is not a new idea, it dates back to 1949, with the first commercial instrument being offered by Perkin-Elmer Co. Inc. in 1953. These early efforts showed promise but failed the test of practically. It was not until the advances in computer science were applied did infrared microspectroscopy emerge as a useful technique. Microscopes designed as accessories for Fourier transform infrared spectrometers have been commercially available since 1983. These accessory microscopes provide the best means for analytical spectroscopists to analyze microscopic samples, while not interfering with the FT-IR spectrometer’s normal functions.

The extended Fourier algorithm: Application in discrete optics and electron holography

1994 ◽  
Vol 52 ◽  
pp. 918-919
Author(s):  
E. Voelkl ◽  
L. F. Allard
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Sampling Rate ◽  
Electron Holography ◽  
Optical Bench ◽  
Fourier Space ◽  
Ccd Cameras ◽  
Simulated Image ◽  
Initial Sampling ◽  
Fourier Algorithm

The conventional discrete Fourier transform can be extended to a discrete Extended Fourier transform (EFT). The EFT allows to work with discrete data in close analogy to the optical bench, where continuous data are processed. The EFT includes a capability to increase or decrease the resolution in Fourier space (thus the argument that CCD cameras with a higher number of pixels to increase the resolution in Fourier space is no longer valid). Fourier transforms may also be shifted with arbitrary increments, which is important in electron holography. Still, the analogy between the optical bench and discrete optics on a computer is limited by the Nyquist limit. In this abstract we discuss the capability with the EFT to change the initial sampling rate si of a recorded or simulated image to any other(final) sampling rate sf.

Sign in / Sign up

Export Citation Format

Share Document