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D'oh! Fourier: Theory, Applications, And Derivatives [Pehme köide]

(Univ Of Southampton, Uk)
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Iconoclastic physics professor and artist Andrzej Dragan presents a unique feast of knowledge on special relativity in a straightforward, progressive manner that even a savvy high school student could follow. Encompassing the derivation of Lorentz transformations to Wigner rotations and Thomas precession; from non-inertial accelerated reference frames to event horizons, curved spacetime, and static black holes; and from the Doppler effect to relativistic structure of electromagnetism, Dragan peels back the enigmatic layers of modern physics to enable a deeper understanding of Einstein's groundbreaking theory. Comprehensive and elegantly written, full of insightful apparent paradoxes and riddles, but without any complicated math, Dragan's unique overview takes the reader well beyond the orthodox verses of standard Special Relativity to the bleeding edge of ""new-fangled"" superluminal apocrypha and their relation to Quantum Theory. The book is based on a course on Special Relativity and acclaimed by students taught by Dragan who is a leader of a research group on Relativistic Quantum Information theory at the University of Warsaw and the National University of Singapore.

Preface xiii
Style xiii
Target audience xiii
Overview of structure xiv
Ingratitude xiv
Key points (tldr) xvii
1 Basic Notions and the Nature of the Fourier Transform
1(24)
1.1 Why read this book?
1(2)
1.2 Software and reproducibility
3(2)
1.3 Notation
5(1)
1.4 Basic functions
6(3)
1.5 Analysing signals by their components: approximating functions by mathematical series
9(6)
1.5.1 Taylor series
9(3)
1.5.2 Fourier series
12(3)
1.6 What is the Fourier transform, and what can it do?
15(2)
1.7 Everyday use of the Fourier transform
17(4)
1.7.1 Transforms and speech recognition
18(1)
1.7.2 Transforms and image compression
18(1)
1.7.3 Human hearing and a transform
19(2)
1.7.4 Light and frequency
21(1)
1.8 Summary and further reading
21(4)
2 The Continuous Fourier Transform
25(50)
2.1 Continuous Fourier transform basis
25(13)
2.1.1 Continuous signals and their Fourier transform
25(4)
2.1.2 Magnitude and phase
29(1)
2.1.3 Inverse Fourier transform
30(3)
2.1.4 Fourier transform in Matlab
33(1)
2.1.5 Fourier transform pairs
34(1)
2.1.5.1 Delta function
34(1)
2.1.5.2 Sine wave
35(1)
2.1.5.3 Gaussian function
35(3)
2.2 Properties of the continuous FT
38(9)
2.2.1 Superposition
38(1)
2.2.2 Time shift
39(1)
2.2.3 Scaling in time
40(1)
2.2.4 Parseval's theorem (Rayleigh's theorem)
41(1)
2.2.5 Symmetry
42(1)
2.2.6 Differentiation
43(1)
2.2.7 Uncertainty principle
43(1)
2.2.8 Modulation
44(3)
2.3 Processing signals using the FT
47(7)
2.3.1 Convolution
47(4)
2.3.2 Correlation
51(3)
2.4 What is the importance of phase?
54(2)
2.4.1 Phase in signal reconstruction
54(1)
2.4.2 Phase in shift invariance
55(1)
2.5 Windowing the FT data
56(9)
2.5.1 Basic windowing
56(3)
2.5.2 Hanning and Hamming window operators
59(3)
2.5.3 Window duration
62(2)
2.5.4 Other windowing functions
64(1)
2.6 Filtering the FT data
65(8)
2.6.1 Basic filters and signal processing
65(1)
2.6.1.1 Low-pass, high-pass and band-pass filters
65(2)
2.6.1.2 RC networks and transfer functions: low-pass filters
67(3)
2.6.1.3 CR networks and theory: high-pass filters
70(2)
2.6.2 Bessel filters
72(1)
2.7 Summary
73(2)
3 The Discrete Fourier Transform
75(64)
3.1 The sampling theorem
75(7)
3.1.1 Sampling signals
75(3)
3.1.2 Sampling process in the frequency domain
78(4)
3.2 The discrete Fourier transform
82(13)
3.2.1 Basic DFT
82(3)
3.2.2 Inverse DFT
85(2)
3.2.3 Visualising the DFT data
87(4)
3.2.4 DFT in Matlab
91(2)
3.2.5 DFT pairs
93(1)
3.2.5.1 Pulse
93(2)
3.2.5.2 Gaussian
95(1)
3.3 Properties of the DFT
95(8)
3.3.1 Basic considerations
96(1)
3.3.2 Linearity/Superposition
96(1)
3.3.3 Time shift
96(1)
3.3.4 Time scaling
97(1)
3.3.5 Parseval's theorem (Rayleigh's theorem)
97(1)
3.3.6 Symmetry
98(1)
3.3.7 Differentiation
98(1)
3.3.8 Importance of phase - DFT
99(2)
3.3.9 Discrete data windowing functions
101(2)
3.4 Discrete convolution and correlation
103(8)
3.4.1 Discrete convolution
103(6)
3.4.2 Discrete correlation
109(2)
3.5 Digital filters; averaging and differencing samples
111(4)
3.6 The fast Fourier transform
115(21)
3.6.1 The butterfly operation and basic components of theFFT
115(1)
3.6.1.1 FFT basis
115(4)
3.6.1.2 FFT computation and speed
119(2)
3.6.1.3 Extending the FFT
121(3)
3.6.2 Decimation in time
124(4)
3.6.3 Radix 2 FFT
128(2)
3.6.4 Computational time for FFT compared with DFT
130(1)
3.6.4.1 Improvement in speed vs DFT
130(2)
3.6.4.2 Speeding convolution via the convolution theorem
132(2)
3.6.5 Optimising the FFT
134(1)
3.6.6 Even faster FFT algorithms
135(1)
3.7 Summary
136(3)
4 The Two-Dimensional Fourier Transform
139(40)
4.1 2-D functions and images
139(10)
4.1.1 Image formation
139(1)
4.1.2 Human vision
140(2)
4.1.3 Sampling images
142(3)
4.1.4 Discrete images
145(4)
4.1.5 Discrete image frequency components
149(1)
4.2 2-D Fourier transform and its inverse
149(6)
4.2.1 2-D continuous Fourier transform and separability
149(2)
4.2.2 2-D discrete Fourier transform
151(4)
4.3 Properties of the 2-D discrete Fourier transform
155(8)
4.3.1 Displaying images
155(1)
4.3.1.1 Transforms and their repetition
155(1)
4.3.1.2 Intensity normalisation
156(1)
4.3.2 Rotation
157(2)
4.3.3 Scaling
159(1)
4.3.4 Shift invariance
160(1)
4.3.5 The importance of phase
161(1)
4.3.6 Computational cost of 2-D DFT and FFT
162(1)
4.4 Image processing via the Fourier transform
163(15)
4.4.1 Convolution
163(1)
4.4.1.1 Image convolution
163(1)
4.4.1.2 Template convolution
164(2)
4.4.1.3 Filtering an image via convolution
166(2)
4.4.2 Computational considerations of image convolution and template convolution
168(2)
4.4.3 Correlation
170(1)
4.4.3.1 Image correlation
170(1)
4.4.3.2 Template correlation/template matching
171(1)
4.4.3.3 Finding objects by template correlation/matching
172(3)
4.4.4 Filtering
175(1)
4.4.4.1 Low-and high-pass filtering
175(2)
4.4.4.2 Unsharp masking
177(1)
4.5 Summary
178(1)
5 Variants of the Fourier Transform
179(38)
5.1 Cosine and sine transforms, including the discrete cosine transform
180(12)
5.1.1 1-D continuous transforms
180(1)
5.1.2 1-D discrete cosine and sine transforms
180(1)
5.1.2.1 Discrete cosine transform and compression
180(3)
5.1.2.2 Basic coding
183(3)
5.1.2.3 Relationship between the DCT and the DFT
186(3)
5.1.2.4 Other properties of the DCT
189(1)
5.1.2.5 Discrete sine transform
189(1)
5.1.3 2-D discrete cosine transform
190(2)
5.2 Walsh--Hadamard transform
192(5)
5.2.1 Walsh transform
192(1)
5.2.1.1 The 1-D transform
192(3)
5.2.1.2 The 2-D Walsh transform
195(1)
5.2.2 Walsh--Hadamard transform
196(1)
5.3 Hartley transform
197(3)
5.4 Image compression properties of Fourier, DCT, Walsh and Hartley transforms
200(3)
5.5 Laplace, Mellin and Fourier Mellin
203(5)
5.5.1 Laplace and Mellin transforms
203(1)
5.5.1.1 Laplace transform and basic systems analysis
203(2)
5.5.1.2 Mellin transform for scale invariance
205(2)
5.5.2 Fourier-Mellin transform
207(1)
5.6 z-transform
208(1)
5.7 Wavelets
209(7)
5.7.1 Filter banks and signal analysis
209(1)
5.7.2 Gabor wavelets
210(6)
5.8 Summary
216(1)
6 Applications of the Fourier Transform
217(36)
6.1 Overview
217(1)
6.2 Fourier transforms
218(4)
6.2.1 The continuous Fourier transform and Fourier optics
218(1)
6.2.2 Magnitude and phase, and beamforming
219(3)
6.3 Properties of the Fourier transform
222(10)
6.3.1 Superposition and fingerprint analysis
222(1)
6.3.2 Invariance and image texture analysis
222(5)
6.3.3 Invariance and image registration
227(1)
6.3.4 Differentiation and image feature extraction
228(1)
6.3.4.1 Template convolution and edge detection
228(3)
6.3.4.2 z-transform and Fourier analysis, and edge detection operators
231(1)
6.4 Processing signals using the Fourier transform
232(7)
6.4.1 Convolution theorem and ear biometrics
232(2)
6.4.2 Deconvolution and image enhancement
234(1)
6.4.3 Speech recognition and correlation
235(4)
6.5 The importance of phase and phase congruency
239(7)
6.6 Filtering and denoising, and image enhancement
246(2)
6.7 Variants of the Fourier transform, and coding
248(3)
6.8 Summary
251(2)
7 Who and What was Fourier?
253(10)
7.1 Nature and origins of the Fourier transform
253(5)
7.1.1 The basic nature and definitions of the Fourier transform
253(3)
7.1.2 On the development of the Fourier transform
256(2)
7.2 Baron Jean Baptiste Joseph Fourier
258(3)
7.3 Final summary
261(2)
8 Ready Reference Time
263(8)
8.1 Summary of Fourier transforms and thier variants
263(1)
8.2 Summary of properties of the continuous Fourier transform
264(1)
8.3 Continuous Fourier transform pairs
265(2)
8.4 Summary of properties of the discrete Fourier transform
267(1)
8.5 Discrete Fourier transform pairs
267(4)
References 271(8)
Index 279