D'oh! Fourier

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.

Sample Chapter(s)
Preface

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Contents:

• 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

Readership: Aimed at undergraduates with a mathematical background who cover Fourier as part of their undergraduate curriculum. The target curricula include courses on signal processing, communications, speech analysis and understanding, image processing, and computer vision. The book is also aimed at people who are interested in furthering their knowledge on Fourier, for whom maths might be less practiced.