D'oh! Fourier: Theory, Applications, and Derivatives

Preface
Style
Target Audience
Overview of Structure
In Gratitude
Key points (tldr)
Basic Notions and the Nature of the Fourier Transform:
Why Read this Book?
Software and Reproducibility
Notation
Basic Functions
Analysing Signals by Their Components: Approximating Functions by Mathematical Series:
Taylor Series
Fourier Series
What is the Fourier Transform, and What Can It Do?
Everyday Use of the Fourier Transform:
Transforms and Speech Recognition
Transforms and Image Compression
Human Hearing and a Transform
Light and Frequency
Summary and Further Reading
The Continuous Fourier Transform:
Continuous Fourier Transform Basis:
Continuous Signals and Their Fourier Transform
Magnitude and Phase
Inverse Fourier Transform
Fourier Transform in Matlab
Fourier Transform Pairs
Properties of the Continuous FT:
Superposition
Time Shift
Scaling in Time
Parseval's Theorem (Rayleigh's Theorem)
Symmetry
Differentiation
Uncertainty Principle
Modulation
Processing Signals Using the FT:
Convolution
Correlation
What is the Importance of Phase?:
Phase in Signal Reconstruction
Phase in Shift Invariance
Windowing the FT Data:
Basic Windowing
Hanning and Hamming Window Operators
Window Duration
Other Windowing Functions
Filtering The FT Data:
Basic Filters and Signal Processing
Bessel Filters
Summary
The Discrete Fourier Transform:
The Sampling Theorem:
Sampling Signals
Sampling Process in the Frequency Domain
The Discrete Fourier Transform:
Basic DFT
Inverse DFT
Visualising the DFT Data
DFT in Matlab
DFT Pairs
Properties of The DFT:
Basic Considerations
Linearity/Superposition
Time Shift
Time Scaling
Parseval's Theorem (Rayleigh's Theorem)
Symmetry
Differentiation
Importance of Phase — DFT
Discrete Data Windowing Functions
Discrete Convolution and Correlation:
Discrete Convolution
Discrete Correlation
Digital Filters; Averaging and Differencing Samples
The Fast Fourier Transform:
The Butterfly Operation and Basic Components of the FFT
Decimation in Time
Radix 2 FFT
Computational Time for FFT Compared with DFT
Optimising the FFT
Even Faster FFT Algorithms
Summary
The Two-Dimensional Fourier Transform:
2-D Functions and Images:
Image Formation
Human Vision
Sampling Images
Discrete Images
Discrete Image Frequency Components
2-D Fourier Transform and Its Inverse:
2-D Continuous Fourier Transform and Separability
2-D Discrete Fourier Transform
Properties of The 2-D Discrete Fourier Transform:
Displaying Images
Rotation
Scaling
Shift Invariance
The Importance of Phase
Computational Cost of 2-D DFT and FFT
Image Processing via the Fourier Transform:
Convolution
Computational Considerations of Image Convolution and Template Convolution
Correlation
Filtering
Summary
Variants of the Fourier Transform:
Cosine and Sine Transforms, Including the Discrete Cosine Transform:
1-D Continuous Transforms
1-D Discrete Cosine and Sine Transforms
2-D Discrete Cosine Transform
Walsh–Hadamard Transform:
Walsh Transform
Walsh–Hadamard Transform
Hartley Transform
Image Compression Properties of Fourier, DCT, Walsh and Hartley Transforms
Laplace, Mellin and Fourier Mellin:
Laplace and Mellin Transforms
Fourier–Mellin Transform
Z-Transform
Wavelets:
Filter Banks and Signal Analysis
Gabor Wavelets
Summary
Applications of the Fourier Transform:
Overview
Fourier Transforms:
The Continuous Fourier Transform and Fourier Optics
Magnitude and Phase, and Beamforming
Properties of the Fourier Transform:
Superposition and Fingerprint Analysis
Invariance and Image Texture Analysis
Invariance and Image Registration
Differentiation and Image Feature Extraction
Processing Signals Using the Fourier Transform:
Convolution Theorem and Ear Biometrics
Deconvolution and Image Enhancement
Speech Recognition and Correlation
The Importance of Phase and Phase Congruency
Filtering and Denoising, and Image Enhancement
Variants of the Fourier Transform, and Coding
Summary
Who and What was Fourier?:
Nature and Origins of the Fourier Transform:
The Basic Nature and Definitions of the Fourier Transform
On the Development of the Fourier Transform
Baron Jean Baptiste Joseph Fourier
Final Summary
Ready Reference Time:
Summary of Fourier Transforms and Their Variants
Summary of Properties of the Continuous Fourier Transform
Continuous Fourier Transform Pairs
Summary of Properties of the Discrete Fourier Transform
Discrete Fourier Transform Pairs
References
Index

D'oh! Fourier introduces the Fourier transform and is aimed at undergraduates in Computer Science, Mathematics, and Applied Sciences, as well as for those wishing to extend their education. Formulated around ten key points, this accessible book is light-hearted and illustrative, with many applications. The basis and deployment of the Fourier transform are covered applying real-world examples throughout inductively rather than the theoretical approach deductively.The key components of the textbook are continuous signals analysis, discrete signals analysis, image processing, applications of Fourier analysis, together with the origin and nature of the transform itself. D'oh! Fourier is reproducible via MATLAB/Octave and is supported by a comprehensive website which provides the code contained within the book.