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Description of Fourier Transformation for Pedestrians |
Meant to serve an "entertaining textbook," this book belongs to a rare genre. It is written for all students and practitioners who deal with Fourier transformation. Fourier series as well as continuous and discrete Fourier transformation are covered, and particular emphasis is placed on window functions. Many illustrations and easy-to-solve exercises make the book especially accessible, and its humorous style will add to the pleasure of learning from it.
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Contents of Fourier Transformation for Pedestrians |
Introduction
1 Fourier Series
1.1 Fourier Series
1.1.1 Even and Odd Functions
1.1.2 Definition of the Fourier Series
1.1.3 Calculation of the Fourier Coefficients
1.1.4 Fourier Series in Complex Notation
1.2 Theorems and Rules
1.2.1 Linearity Theorem
1.2.2 The First Shifting Rule
1.2.3 The Second Shifting Rule
1.2.4 Scaling Theorem
1.3 Partial Sums, Bessel’s Inequality, Parseval’s Equation
1.4 Gibbs’ Phenomenon
1.4.1 Dirichlet’s Integral Kernel
1.4.2 Integral Notation of Partial Sums
1.4.3 Gibbs’ Overshoot
Playground
2 Continuous Fourier Transformation
2.1 Continuous Fourier Transformation
2.1.1 Even and Odd Functions
2.1.2 The δ-Function
2.1.3 Forward and Inverse Transformation
2.1.4 Polar Representation of the Fourier Transform
2.2 Theorems and Rules
2.2.1 Linearity Theorem
2.2.2 The First Shifting Rule
2.2.3 The Second Shifting Rule
2.2.4 Scaling Theorem
2.3 Convolution, Cross Correlation, Autocorrelation, Parseval’s Theorem
2.3.1 Convolution
2.3.2 Cross Correlation
XII Contents
2.3.3 Autocorrelation
2.3.4 Parseval’s Theorem
2.4 Fourier Transformation of Derivatives
2.5 Pitfalls
2.5.1 “Turn 1 into 3”
2.5.2 Truncation Error
Playground
3 Window Functions
3.1 The Rectangular Window
3.1.1 Zeros
3.1.2 Intensity at the Central Peak
3.1.3 Sidelobe Suppression
3.1.4 3 dB-Bandwidth
3.1.5 Asymptotic Behaviour of Sidelobes
3.2 The Triangular Window (Fejer Window)
3.3 The Cosine Window
3.4 The cos2-Window (Hanning)
3.5 The Hamming Window
3.6 The Triplet Window
3.7 The Gauss Window
3.8 The Kaiser–Bessel Window
3.9 The Blackman–Harris Window
3.10 Overview over Window Functions
3.11 Windowing or Convolution?
Playground
4 Discrete Fourier Transformation
4.1 Discrete Fourier Transformation
4.1.1 Even and Odd Series and Wrap-around
4.1.2 The Kronecker Symbol or the “Discrete δ-Function”
4.1.3 Definition of the Discrete Fourier Transformation
4.2 Theorems and Rules
4.2.1 Linearity Theorem
4.2.2 The First Shifting Rule
4.2.3 The Second Shifting Rule
4.2.4 Scaling Rule/Nyquist Frequency
4.3 Convolution, Cross Correlation, Autocorrelation, Parseval’s Theorem
4.3.1 Convolution
4.3.2 Cross Correlation
4.3.3 Autocorrelation
4.3.4 Parseval’s Theorem
4.4 The Sampling Theorem
4.5 Data Mirroring
Contents XIII
4.6 Zero-padding
4.7 Fast Fourier Transformation (FFT)
Playground
5 Filter Effect in Digital Data Processing
5.1 Transfer Function
5.2 Low-pass, High-pass, Band-pass, Notch Filter
5.3 Shifting Data
5.4 Data Compression
5.5 Differentiation of Discrete Data
5.6 Integration of Discrete Data
Playground
Appendix: Solutions
References
Index
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