Fourier Transformation for Pedestrians (Fourier Series)

Preface......Page 7
Preface of the Translator......Page 9
Contents......Page 11
Introduction......Page 15
1.1 Fourier Series......Page 17
1.2 Theorems and Rules......Page 27
1.3 Partial Sums, Bessel’s Inequality, Parseval’s Equation......Page 35
1.4 Gibbs’ Phenomenon......Page 38
Playground......Page 44
2.1 Continuous Fourier Transformation......Page 47
2.2 Theorems and Rules......Page 56
2.3 Convolution, Cross Correlation, Autocorrelation, Parseval’s Theorem......Page 60
2.4 Fourier Transformation of Derivatives......Page 72
2.5 Pitfalls......Page 74
Playground......Page 80
3.1 The Rectangular Window......Page 83
3.2 The Triangular Window (Fejer Window)......Page 87
3.3 The Cosine Window......Page 88
3.4 The cos2-Window (Hanning)......Page 89
3.5 The Hamming Window......Page 91
3.6 The Triplet Window......Page 92
3.7 The Gauss Window......Page 93
3.8 The Kaiser–Bessel Window......Page 94
3.9 The Blackman–Harris Window......Page 95
3.10 Overview over Window Functions......Page 98
3.11 Windowing or Convolution?......Page 101
Playground......Page 102
4.1 Discrete Fourier Transformation......Page 103
4.2 Theorems and Rules......Page 110
4.3 Convolution, Cross Correlation, Autocorrelation, Parseval’s Theorem......Page 113
4.4 The Sampling Theorem......Page 119
4.5 Data Mirroring......Page 123
4.6 How to Get Rid of the “Straight-jacket” Periodic Continuation? By Using Zero-padding!......Page 126
4.7 Fast Fourier Transformation (FFT)......Page 132
Playground......Page 140
5.1 Transfer Function......Page 145
5.2 Low-pass, High-pass, Band-pass, Notch Filter......Page 146
5.3 Shifting Data......Page 153
5.5 Differentiation of Discrete Data......Page 155
5.6 Integration of Discrete Data......Page 157
Playground......Page 161
Playground of Chapter 1......Page 163
Playground of Chapter 2......Page 173
Playground of Chapter 3......Page 183
Playground of Chapter 4......Page 187
Playground of Chapter 5......Page 203
References......Page 211
Index......Page 213