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Fourier Transforms Using Mathematica

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Fourier Transforms Using Mathematica





Fourier Transforms Using Mathematica
Goodman, J.W.
発行元 SPIE
発行年/月 2020年11月
装丁 ソフトカバー
ページ数 110 ページ
ISBN 978-1-5106-3855-6


The Fourier transform is a ubiquitous tool used in most areas of engineering and physical sciences. The purpose of this book is two-fold: (1) to introduce the reader to the properties of Fourier transforms and their uses, and (2) to introduce the reader to the program MathematicaR and demonstrate its use in Fourier analysis. Unlike many other introductory treatments of the Fourier transform, this treatment will focus from the start on both one-dimensional and two-dimensional transforms, the latter of which play an important role in optics and digital image processing, as well as in many other applications. It is hoped that by the time readers have completed this book, they will have a basic understanding of Fourier analysis and Mathematica.

The PDF is a Read Me First file with links to the Mathematica interactive book and to the free Wolfram Player for readers who do not have the full Mathematica program.




1 Introduction
1.1 Why Mathematica?
1.2 What the Reader Should Know at the Start
1.3 Why Study the Fourier Transform?
2 Some Useful 1D and 2D Functions
2.1 User-Defined Names for Useful Functions
2.2 Dirac Delta Functions
2.3 The Comb Function
3 Definition of the Continuous Fourier Transform
3.1 The 1D Fourier Transform and Inverse Fourier Transform
3.2 The 2D Fourier Transform and Inverse Fourier Transform
3.3 Fourier Transform Operators in Mathematica
3.4 Transforms in-the-Limit
3.5 A Table of Some Frequently Encountered Fourier Transforms
4 Convolutions and Correlations
4.1 Convolution Integrals
4.2 The Central Limit Theorem
4.3 Correlation Integrals
5 Some Useful Properties of Fourier Transforms
5.1 Symmetry Properties of Fourier Transforms
5.2 Area and Moment Properties of 1D Fourier Transforms
5.3 Area and Moment Properties of 2D Fourier Transforms
5.4 Fourier Transform Theorems
5.5 The Projection-Slice Theorem
5.6 Widths in the x Domain and the u Domain
6 Fourier Transforms in Polar Coordinates
6.1 Using the 2D Fourier Transform for Circularly Symmetric Functions
6.2 The Zero-Order Hankel Transform
6.3 The Projection Transform Method
6.4 Polar-Coordinate Functions with a Simple Harmonic Phase
7 Linear Systems and Fourier Transforms
7.1 The Superposition Integral for Linear Systems
7.2 Invariant Linear Systems and the Convolution Integral
7.3 Transfer Functions of Linear Invariant Systems
7.4 Eigenfunctions of Linear Invariant Systems
8 Sampling and Interpolation
8.1 The Sampling Theory in One Dimension
8.2 The Sampling Theory in Two Dimensions
9 From Fourier Transforms to Fourier Series
9.1 Periodic Functions and Their Fourier Transforms
9.2 Example of a Complex Fourier Series
9.3 Mathematica Commands for Fourier Series
9.4 Other Types of Fouier Series
9.5 Circular Harmonic Expansions
10 The Discrete Fourier Transform
10.1 Sampling in Both Domains
10.2 Vectors and Matrices in Mathematica
10.3 The Discrete Fourier Transform (DFT)
10.4 The DFT and Mathematica
10.5 DFT Properties and Theorems
10.6 Discrete Convolutions and Correlations
10.7 The Fast Fourier Transform
11 The Fresnel Transform
11.1 Definition of the 1D Fresnel Transform
11.2 Approximations to the Bandwidth of the Interval-Limited Quadratic-Phase Exponential
11.3 Equivalent Bandwidth of the Interval-Limited Quadratic-Phase Exponentiald
11.4 The 2D Fresnel Transform
11.5 The Fresnel Transform of Circularly Symmetric Functions
11.6 Examples of Fresnel Transforms
11.7 The Frensel-Diffraction Transfer Function
11.8 The Discrete Frensel-Diffraction Integral
12 Fractional Fourier Transforms
12.1 Definition of the Fractional Fourier Transform
12.2 Mathematica Calculation of the Fractional Fourier Transform
12.3 Relationship between the Fractional Fourier Transform and the Fresnel-Diffraction Integral
13 Other Transforms Related to the Fourier Transform
13.1 The Abel Transform
13.2 The Radon Transform
13.3 The Hilbert Transform
13.4 The Analytic Signal
13.5 The Laplace Transform
13.6 The Mellin Transform
14 Fourier Transforms and Digital Image Processing with Mathematica
14.1 Inputting an Image into Mathematica
14.2 Some Elemental Properties of Images
14.3 Some Image Point Manipulations
14.4 Blurring and Sharpening in Mathematica
14.5 Fourier Domain Processing of Images