"The Fourier analysis is mainly presented from a practical point of view, where the mathematical theory is very simplified. This book is mainly written for broad readership of graduate students and researchers in physics, computer science, and engineering with special interest in signal processing. ... Doubtless, this textbook will stimulate the practical education in the Fourier analysis and its applications in signal processing." (Manfred Tasche, zbMATH 1407.94002, 2019)

Contents

1 Signals 11

1.1 Basic Signals

1.1.1 Unit-impulse Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1.2 Unit-step Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.1.3 Unit-ramp Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.1.4 Sinusoids and Complex Exponentials . . . . . . . . . . . . . . . . . . . . . . . 14

1.2 Classification of Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.1 Continuous, Discrete, and Digital Signals . . . . . . . . . . . . . . . . . . . . 19

1.2.2 Periodic and Aperiodic Signals . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.3 Even- and Odd-symmetric Signals . . . . . . . . . . . . . . . . . . . . . . . . 20

1.2.4 Energy and Power Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.2.5 Deterministic and Random Signals . . . . . . . . . . . . . . . . . . . . . . . . 24

1.2.6 Causal and Noncausal Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.3 Signal Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.3.1 Time Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.3.2 Time Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.4 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2 The Discrete Fourier Transform 33

2.1 The Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2 The Complex Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2.1 Euler’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2.2 Real Sinusoid in terms of Complex Exponentials . . . . . . . . . . . . . . . . 35

2.3 The DFT and the IDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3.1 The DFT and the IDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3.2 The Criterion of Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3.3 The Matrix form of the DFT and IDFT . . . . . . . . . . . . . . . . . . . . . 41

2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.4.1 Fourier Boundary Descriptor . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Properties of the DFT 53

3.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3 Circular Time Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4 Circular Frequency Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5 Circular Time-reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.6 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.7 Transform of Complex Conjuagtes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.8 Circular Convolution and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.8.1 Circular convolution of Time-domain Sequences . . . . . . . . . . . . . . . . . 56

3.8.2 Circular Convolution of Frequency-domain Sequences . . . . . . . . . . . . . 58

3.8.3 Circular Correlation of Time-domain Sequences . . . . . . . . . . . . . . . . . 59

3.9 Sum and Difference of Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.10 Upsampling of a Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.11 Zero Padding the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.12 Symmetry Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.13 Parseval’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.14 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4 Two-Dimensional DFT 67

4.1 Two-Dimensional DFT as two 1-D DFTs . . . . . . . . . . . . . . . . . . . . . . . . 68

4.1.1 Computation of the 2-D DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2 The 2-D DFT and IDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3 DFT Representation of Real-valued Signals . . . . . . . . . . . . . . . . . . . . . . . 78

4.4 Properties of the 2-D DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5 Convolution and Correlation 89

5.1 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.1.1 Linear Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.1.2 Circular Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.1.3 2-D Linear Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.2.1 The Linear Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.3.1 Lowpass Filtering of Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.3.2 Highpass Filtering of Images . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.3.3 Object Detection in Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.3.4 Orthogonal Frequency Division Modulation . . . . . . . . . . . . . . . . . . . 107

5.3.5 Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6 Aliasing and Leakage 119

6.1 Aliasing Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.2 Leakage Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.2.1 Modeling Data Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.2.2 Tapered Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.2.3 Hann and Hamming windows . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.2.4 Reducing the Spectral Leakage . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.3 Picket-Fence Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7 Fourier Series 131

7.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.1.1 FS as the Limiting Case of the DFT . . . . . . . . . . . . . . . . . . . . . . . 132

7.1.2 Gibbs Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.2 Properties of the Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.2.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.2.2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.2.3 Time-shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

7.2.4 Frequency-shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7.2.5 Convolution in the Time-domain . . . . . . . . . . . . . . . . . . . . . . . . . 146

7.2.6 Convolution in the Frequency-domain . . . . . . . . . . . . . . . . . . . . . . 147

7.2.7 Time-scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

7.2.8 Time-differentiation and Integration . . . . . . . . . . . . . . . . . . . . . . . 148

7.2.9 Parseval’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.3 Applications of the Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.3.1 Analysis of Rectified Power Supply . . . . . . . . . . . . . . . . . . . . . . . . 150

7.3.2 Steady-state Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.4 Numerical Evaluation of the Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . 153

7.4.1 Aliasing effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

8 The Discrete-Time Fourier Transform 157

8.1 The DTFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

8.1.1 DTFT as the Limiting case of the DFT . . . . . . . . . . . . . . . . . . . . . 158

8.1.2 The DTFT of a Discrete Periodic Signal . . . . . . . . . . . . . . . . . . . . . 164

8.1.3 Determination of the DFT from the DTFT . . . . . . . . . . . . . . . . . . . 165

8.2 Properties of the Discrete-Time Fourier Transform . . . . . . . . . . . . . . . . . . . 166

8.2.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

8.2.2 Time-shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

8.2.3 Frequency-shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

8.2.4 Convolution in the Time-domain . . . . . . . . . . . . . . . . . . . . . . . . . 167

8.2.5 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

8.2.6 Convolution in the Frequency-domain . . . . . . . . . . . . . . . . . . . . . . 167

8.2.7 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

8.2.8 Time-reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

8.2.9 Time-expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

8.2.10 Frequency-differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

8.2.11 Summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

8.2.12 Parseval’s theorem and the Energy Transfer Function . . . . . . . . . . . . . 172

8.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

8.3.1 Transfer Function and the System Response . . . . . . . . . . . . . . . . . . . 172

8.3.2 Digital Filter Design using the DTFT . . . . . . . . . . . . . . . . . . . . . . 174

8.3.3 Digital Differentiator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

8.3.4 Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

8.4 Approximation of the Discrete-Time Fourier Transform . . . . . . . . . . . . . . . . 177

8.4.1 Approximation of the inverse DTFT by the IDFT . . . . . . . . . . . . . . . 178

8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

9 The Fourier Transform 181

9.1 The FT as a limiting case of the FS . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

9.1.1 Existence of the FT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

9.1.2 Determination of the FS from the FT . . . . . . . . . . . . . . . . . . . . . . 184

9.2 Properties of the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

9.2.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

9.2.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

9.2.3 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

9.2.4 Time-shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

9.2.5 Frequency-shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

9.2.6 Convolution in the Time-domain . . . . . . . . . . . . . . . . . . . . . . . . . 190

9.2.7 Convolution in the Frequency-domain . . . . . . . . . . . . . . . . . . . . . . 190

9.2.8 Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

9.2.9 Cross-correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

9.2.10 Time-reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

9.2.11 Time-scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

9.2.12 Time-differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

9.2.13 Time-integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

9.2.14 Frequency-differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

9.2.15 Parseval’s Theorem and the Energy Transfer Function . . . . . . . . . . . . . 196

9.3 Fourier Transform of Mixed Class of Signals . . . . . . . . . . . . . . . . . . . . . . . 197

9.3.1 The FT of a Continuous Periodic Signal . . . . . . . . . . . . . . . . . . . . . 197

9.3.2 The FT of a Sampled Signal and the Aliasing Effect . . . . . . . . . . . . . . 198

9.3.3 The FT of a Sampled Aperiodic Signal and the DTFT of the Corresponding

Discrete Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

9.3.4 The FT of a Sampled Periodic Signal and the DFT of the Corresponding

Discrete Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

9.3.5 Reconstruction of the Continuous Signal from its Sampled Version . . . . . . 202

9.4 Applications of the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 203

9.4.1 Transfer Function and the System Response . . . . . . . . . . . . . . . . . . . 203

9.4.2 Ideal Filters and their Unrealizability . . . . . . . . . . . . . . . . . . . . . . 205

9.5 Approximation of the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . 205

9.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

10 Fast Computation of the DFT 209

10.1 Half-Wave Symmetry of Periodic Waveforms . . . . . . . . . . . . . . . . . . . . . . . 209

10.2 The PM DIF DFT Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

10.3 The PM DIT DFT Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

10.3.1 Basics of the PM DIT DFT Algorithm . . . . . . . . . . . . . . . . . . . . . . 218

10.4 Efficient Computation of the DFT of Real Data . . . . . . . . . . . . . . . . . . . . . 219

10.4.1 Two DFTs of Real Data Simultaneously . . . . . . . . . . . . . . . . . . . . . 221

10.4.2 DFT of a Single Real Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . 223

10.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

A Transform Pairs and Properties 229

B Useful Mathematical Formulas 237

C Bibliography 239

D Solutions to selected problems 247

E Index

Dr. D. Sundararajan holds a B.E. in Electrical Engineering from Madras University and an M.Tech. in Electrical Engineering from the Indian Institute of Technology Chennai (IIT Chennai). He obtained his Ph.D. in Electrical Engineering at Concordia University, Montreal, Canada in 1988. As the principal inventor of the latest family of discrete Fourier transform (DFT) algorithms, he holds three patents (granted by the US, Canada and Britain). Further, he has published several papers in IEEE Transactions and in the Proceedings of the IEEE Conference, and he is the author of five books. He has taught undergraduate and graduate classes in digital signal processing, digital image processing, engineering mathematics, programming, operating systems and digital logic design at Concordia University, Canada, Nanyang Technological University, Singapore, and Adhiyamaan College of Engineering, India. He has also conducted workshops on Digital image processing, MATLAB and LaTeX.

Over the course of his engineering career, he has held positions at the National Aerospace Laboratory, Bangalore, and the National Physical Laboratory, New Delhi, where he worked on the design of digital and analog signal processing systems.

This book sheds new light on Transform methods, which dominate the study of linear time-invariant systems in all areas of science and engineering, such as circuit theory, signal/image processing, communications, controls, vibration analysis, remote sensing, biomedical systems, optics and acoustics. It presents Fourier analysis primarily using physical explanations with waveforms and/or examples, only using mathematical formulations to the extent necessary for its practical use. Intended as a textbook for senior undergraduates and graduate level Fourier analysis courses in engineering and science departments, and as a supplementary textbook for a variety of application courses in science and engineering, the book is also a valuable reference for anyone – student or professional – specializing in practical applications of Fourier analysis. The prerequisite for reading this book is a sound understanding of calculus, linear algebra, signals and systems, and programming at the undergraduate level.

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