scholarly journals There Is Only One Fourier Transform

2018 ◽  
Author(s):  
Jens V. Fischer
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Discrete Time ◽  
Fourier Transforms ◽  
Periodic Functions ◽  
Distributional Sense ◽  
Integral Fourier Transform ◽  
The Fourier Transform

Four Fourier transforms are usually defined, the Integral Fourier transform, the Discrete-Time Fourier transform (DTFT), the Discrete Fourier transform (DFT) and the Integral Fourier transform for periodic functions. However, starting from their definitions, we show that all four Fourier transforms can be reduced to actually only one Fourier transform, the Fourier transform in the distributional sense.

scholarly journals There Is Only One Fourier Transform

2017 ◽  
Author(s):  
Jens V. Fischer
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Discrete Time ◽  
Fourier Transforms ◽  
Periodic Functions ◽  
Distributional Sense ◽  
Integral Fourier Transform ◽  
The Fourier Transform

Four Fourier transforms are usually defined, the Integral Fourier transform, the Discrete-Time Fourier transform (DTFT), the Discrete Fourier transform (DFT) and the Integral Fourier transform for periodic functions. However, starting from their definitions, we show that all four Fourier transforms can be reduced to actually only one Fourier transform, the Fourier transform in the distributional sense.

scholarly journals Four Particular Cases of the Fourier Transform

2018 ◽  
Author(s):  
Jens V. Fischer
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Discrete Fourier Transform ◽  
Discrete Time ◽  
Fourier Transforms ◽  
Summation Formula ◽  
Periodic Functions ◽  
Tempered Distributions ◽  
Integral Fourier Transform ◽  
The Fourier Transform

In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the Integral Fourier transform for periodic functions—used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense.

scholarly journals Four Particular Cases of the Fourier Transform

2018 ◽  
Author(s):  
Jens V. Fischer
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Discrete Fourier Transform ◽  
Discrete Time ◽  
Fourier Transforms ◽  
Summation Formula ◽  
Periodic Functions ◽  
Tempered Distributions ◽  
Integral Fourier Transform ◽  
The Fourier Transform

In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the Integral Fourier transform for periodic functions—used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense.

scholarly journals Four Particular Cases of the Fourier Transform

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 335 ◽  
Author(s):  
Jens Fischer
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Discrete Time ◽  
Fourier Transforms ◽  
Summation Formula ◽  
Periodic Functions ◽  
Tempered Distributions ◽  
Integral Fourier Transform ◽  
The Fourier Transform ◽  
Poisson’S Summation Formula

In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the integral Fourier transform for periodic functions—used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense.

scholarly journals More on algebraic properties of the discrete Fourier transform raising and lowering operators

4open ◽  
2019 ◽  
Vol 2 ◽  
pp. 2 ◽  
Author(s):  
Mesuma K. Atakishiyeva ◽  
Natig M. Atakishiyev ◽  
Juan Loreto-Hernández
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Fourier Transforms ◽  
Difference Operators ◽  
Discrete Fourier Transforms ◽  
Raising And Lowering Operators ◽  
Integral Fourier Transform ◽  
Symmetrical Form ◽  
Algebraic Properties ◽  
Lowering Operators

In the present work, we discuss some additional findings concerning algebraic properties of the N-dimensional discrete Fourier transform (DFT) raising and lowering difference operators, recently introduced in [Atakishiyeva MK, Atakishiyev NM (2015), J Phys: Conf Ser 597, 012012; Atakishiyeva MK, Atakishiyev NM (2016), Adv Dyn Syst Appl 11, 81–92]. In particular, we argue that the most authentic symmetrical form of discretization of the integral Fourier transform may be constructed as the discrete Fourier transforms based on the odd points N only, while in the discrete Fourier transforms on the even points N this symmetry is spontaneously broken. This heretofore undetected distinction between odd and even dimensions is shown to be intimately related with the newly revealed algebraic properties of the above-mentioned DFT raising and lowering difference operators and, of course, is very consistent with the well-known formula for the multiplicities of the eigenvalues, associated with the N-dimensional DFT. In addition, we propose a general approach to deriving the eigenvectors of the discrete number operators N(N), that avoids the above-mentioned pitfalls in the structure of each even-dimensional case N = 2L.

Wavelet packets associated with linear canonical transform on spectrum

2021 ◽  
pp. 2150030
Author(s):  
M. Younus Bhat ◽  
Aamir H. Dar
Keyword(s):  
Fourier Transform ◽  
Multiresolution Analysis ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Integral Transforms ◽  
Wavelet Packets ◽  
Linear Canonical Transform ◽  
Fresnel Transform ◽  
The Fourier Transform ◽  
Vector Valued

The linear canonical transform (LCT) provides a unified treatment of the generalized Fourier transforms in the sense that it is an embodiment of several well-known integral transforms including the Fourier transform, fractional Fourier transform, Fresnel transform. Using this fascinating property of LCT, we, in this paper, constructed associated wavelet packets. First, we construct wavelet packets corresponding to nonuniform Multiresolution analysis (MRA) associated with LCT and then those corresponding to vector-valued nonuniform MRA associated with LCT. We investigate their various properties by means of LCT.

Fast Fourier Transform – Calculation and Interpretation

2008 ◽  
Vol 3 (4) ◽  
pp. 74-86
Author(s):  
Boris A. Knyazev ◽  
Valeriy S. Cherkasskij
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Discrete Fourier Transform ◽  
Signal Theory ◽  
Correct Interpretation ◽  
The Fourier Transform ◽  
Application Programs ◽  
Physics Problems

The article is intended to the students, who make their first steps in the application of the Fourier transform to physics problems. We examine several elementary examples from the signal theory and classic optics to show relation between continuous and discrete Fourier transform. Recipes for correct interpretation of the results of FDFT (Fast Discrete Fourier Transform) obtained with the commonly used application programs (Matlab, Mathcad, Mathematica) are given.

An analysis of the spectrum of a gravity anomaly over an inclined dike

Geophysics ◽  
1985 ◽  
Vol 50 (9) ◽  
pp. 1500-1501
Author(s):  
B. N. P. Agarwal ◽  
D. Sita Ramaiah
Keyword(s):  
Fourier Transform ◽  
Gravity Anomaly ◽  
Fourier Spectrum ◽  
Fourier Transforms ◽  
Weighted Average ◽  
Window Function ◽  
Average Spectrum ◽  
Amplitude Spectra ◽  
Distance Data ◽  
The Fourier Transform

Bhimasankaram et al. (1977) used Fourier spectrum analysis for a direct approach to the interpretation of gravity anomaly over a finite inclined dike. They derived several equations from the real and imaginary components and from the amplitude and phase spectra to relate various parameters of the dike. Because the width 2b of the dike (Figure 1) appears only in sin (ωb) term—ω being the angular frequency—they determined its value from the minima/zeroes of the amplitude spectra. The theoretical Fourier spectrum uses gravity field data over an infinite distance (length), whereas field observations are available only for a limited distance. Thus, a set of observational data is viewed as a product of infinite‐distance data with an appropriate window function. Usually, a rectangular window of appropriate distance (width) and of unit magnitude is chosen for this purpose. The Fourier transform of the finite‐distance and discrete data is thus represented by convolution operations between Fourier transforms of the infinite‐distance data, the window function, and the comb function. The combined effect gives a smooth, weighted average spectrum. Thus, the Fourier transform of actual observed data may differ substantially from theoretic data. The differences are apparent for low‐ and high‐frequency ranges. As a result, the minima of the amplitude spectra may change considerably, thereby rendering the estimate of the width of the dike unreliable from the roots of the equation sin (ωb) = 0.

The Spectra of Filters

2021 ◽  
pp. 204-268
Author(s):  
Victor Lazzarini
Keyword(s):  
Fourier Transform ◽  
Impulse Response ◽  
Discrete Time ◽  
Filter Design ◽  
Finite Impulse Response ◽  
Infinite Impulse Response ◽  
Main Body ◽  
Analysis Tool ◽  
The Fourier Transform ◽  
Definition Of

This chapter now turns to the discussion of filters, which extend the notion of spectrum beyond signals into the processes themselves. A gentle introduction to the concept of delaying signals, aided by yet another variant of the Fourier transform, the discrete-time Fourier transform, allows the operation of filters to be dissected. Another analysis tool, in the form of the z-transform, is brought to the fore as a complex-valued version of the discrete-time Fourier transform. A study of the characteristics of filters, introducing the notion of zeros and poles, as well as finite impulse response (FIR) and infinite impulse response (IIR) forms, composes the main body of the text. This is complemented by a discussion of filter design and applications, including ideas related to time-varying filters. The chapter conclusion expands once more the definition of spectrum.

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