Sinusoids : theory and technological applications

Sinusoids: Theory and Technological Applications explains how sinusoids and Fourier transforms are used in a variety of application areas, including signal processing, GPS, optics, x-ray crystallography, radioastronomy, poetry and music as sound waves, and the medical sciences. With more than 200 illustrations, the book discusses electromagnetic force and sychrotron radiation comprising all kinds of waves, including gamma rays, x-rays, UV rays, visible light rays, infrared, microwaves, and radio waves. It also covers topics of common interest, such as quasars, pulsars, the Big Bang theory, Olbers' paradox, black holes, Mars mission, and SETI.

The book begins by describing sinusoids--which are periodic sine or cosine functions--using well-known examples from wave theory, including traveling and standing waves, continuous musical rhythms, and the human liver. It next discusses the Fourier series and transform in both continuous and discrete cases and analyzes the Dirichlet kernel and Gibbs phenomenon. The author shows how invertibility and periodicity of Fourier transforms are used in the development of signals and filters, addresses the general concept of communication systems, and explains the functioning of a GPS receiver. The author then covers the theory of Fourier optics, synchrotron light and x-ray diffraction, the mathematics of radioastronomy, and mathematical structures in poetry and music. The book concludes with a focus on tomography, exploring different types of procedures and modern advances. The appendices make the book as self-contained as possible.

Author Notes

Prem K. Kythe is a Professor Emeritus of Mathematics at the University of New Orleans. He is the author/coauthor of ten books and author of 46 research papers. His research interests encompass the fields of complex analysis, continuum mechanics, and wave theory, including boundary element methods, finite element methods, conformal mappings, PDEs and boundary value problems, linear integral equations, computation integration, fundamental solutions of differential operators, Green's functions, and coding theory.

Preface	p. xv
Notations, Definitions, and Acronyms	p. xix
1 Introduction	p. 1
1.1 Definitions	p. 1
1.2 Continuous Sinusoids	p. 3
1.3 Discrete Sinusoids	p. 9
1.4 Harmonic Series	p. 10
1.5 Traveling and Standing Waves	p. 10
1.5.1 Vibrating String	p. 15
1.6 Wave Propagation and Dispersion	p. 19
1.6.1 Heat Waves	p. 24
1.7 Applications of Sinusoids	p. 25
1.8 Historical Notes	p. 26
2 Fourier Series	p. 31
2.1 Orthogonality	p. 31
2.2 Completeness and Uniform Convergence	p. 35
2.3 Fourier Series	p. 36
2.3.1 Some Special Cases	p. 38
2.3.2 Amplitude and Phase Form	p. 38
2.3.3 Fourier Sine Series	p. 40
2.3.4 Fourier Cosine Series	p. 41
2.3.5 Uniform Convergence	p. 41
2.4 Dirac Delta Function	p. 42
2.5 Delta Function and Dirichlet Kernel	p. 45
2.6 Gibbs Phenomenon	p. 47
2.6.1 Sigma-Approximation	p. 48
2.7 Square Waveform	p. 50
2.8 Examples of Fourier Series	p. 52
2.9 Complex Fourier Coefficients	p. 53
3 Fourier Transforms	p. 57
3.1 Definitions	p. 57
3.1.1 Fourier Integral	p. 59
3.1.2 Properties of Fourier Transforms	p. 59
3.1.3 Fourier Transforms of the Derivatives of a Function	p. 60
3.1.4 Convolution Theorems for Fourier Transform	p. 61
3.1.5 Fourier Transforms of Some Typical Signals	p. 63
3.1.6 Poisson's Summation Formula	p. 67
3.1.7 Sine and Cosine Transforms	p. 68
3.2 Finite Fourier Transforms	p. 70
3.2.1 Properties	p. 72
3.2.2 Finite Fourier Transforms of Derivatives	p. 72
3.2.3 Periodic Extensions	p. 72
3.2.4 Convolution	p. 73
3.2.5 Differentiation with Respect to a Parameter	p. 74
3.3 Fourier Transforms of Two Variables	p. 75
3.3.1 Local Spatial Frequency	p. 75
3.3.2 Fourier Transform of Signals	p. 77
3.3.3 Circle Function	p. 78
3.3.4 Amplitude Transmittance Function	p. 79
3.3.5 Convolution	p. 80
3.3.6 Symmetry of Fourier Transform	p. 81
3.3.7 Image Rotation	p. 81
3.3.8 Other Properties of 2-D Fourier Transform	p. 81
3.3.9 Grating	p. 84
3.4 Fourier Transforms: Discrete Case	p. 85
3.4.1 Discrete Signal	p. 87
3.5 Fast Fourier Transform	p. 90
3.5.1 Radix-2 Algorithm for FFT	p. 91
3.6 Multiple Fourier Transform	p. 92
3.7 Fourier Slice Theorem	p. 93
4 Signals and Filters	p. 97
4.1 Description of Signals	p. 97
4.2 Convolution and Signals	p. 100
4.3 Theory of Signals	p. 103
4.3.1 Types of Signals	p. 103
4.3.2 Continuous Deterministic Signals	p. 104
4.3.3 Discrete Deterministic Signals	p. 104
4.3.4 Bandpass Signals	p. 108
4.3.5 Continuous-Time Domain	p. 109
4.3.6 Linear and Nonlinear Systems	p. 110
4.3.7 Time-Invariant and Time-Varying Systems	p. 110
4.3.8 Causal and Noncausal Systems	p. 110
4.3.9 Linear Time-Invariant Systems	p. 111
4.4 Random Signals	p. 112
4.4.1 Discrete Case	p. 113
4.4.2 Random Sequence of Pulses	p. 113
4.4.3 Sampling	p. 114
4.4.4 Amplitude	p. 115
4.4.5 Frequency	p. 117
4.4.6 Superposition of Signals	p. 118
4.4.7 Periodic Signals	p. 119
4.5 Discrete Fourier Analysis	p. 120
4.5.1 Fourier Series of Elementary Waveforms	p. 123
4.6 Fourier Resynthesis: Discrete Case	p. 127
4.6.1 Additive Synthesis	p. 128
4.6.2 Time and Phase Shifts	p. 129
4.6.3 Nonperiodic Signals	p. 132
4.7 Theory of Filters	p. 133
4.7.1 Delay Network	p. 134
4.7.2 Classification of Filters	p. 136
4.7.3 Properties and Uses	p. 148
5 Communication Systems	p. 151
5.1 Communication Channels	p. 151
5.2 Noise	p. 152
5.3 Quantization	p. 154
5.4 Digital Signals	p. 159
5.4.1 Amplitude and Phase Characteristic	p. 160
5.5 Interference	p. 161
5.5.1 Light Sources	p. 163
5.5.2 Laser Beam	p. 163
5.5.3 Astronomical Interferometry	p. 163
5.5.4 Acoustic Interferometry	p. 164
5.5.5 Quantum Interference	p. 164
5.6 Nonlinear Systems	p. 165
5.7 SDR System	p. 165
5.8 Data Transmission	p. 167
5.9 Space Exploration	p. 168
5.9.1 Error-Correcting Codes	p. 169
5.9.2 Combustion	p. 173
5.9.3 Combustion Instability	p. 173
5.10 Mars Project and Beyond	p. 174
5.10.1 SETI	p. 179
6 Global Positioning System	p. 181
6.1 GPS Structure	p. 181
6.1.1 Bancroft's Method	p. 182
6.1.2 Trilateration Method	p. 184
6.1.3 GPS Segments	p. 187
6.1.4 Navigation	p. 192
6.1.5 Accuracy	p. 193
6.1.6 Applications	p. 194
6.2 CDMA Principle	p. 195
6.2.1 CDMA Signals	p. 197
6.2.2 Differential GPS (DGPS) Techniques	p. 197
6.2.3 GPS Error Sources	p. 198
6.3 C/A Code Architecture	p. 200
6.3.1 Gold Codes	p. 201
6.3.2 C/A Code and Data Format	p. 204
6.3.3 GPS Signal Waveform: L1 Channel	p. 205
6.3.4 DSSS	p. 205
6.3.5 BPSK	p. 206
6.3.6 GPS Frequency	p. 206
6.3.7 Navigation	p. 209
6.3.8 GPS Signal Structure	p. 209
6.3.9 C/A Code Acquisition	p. 211
6.3.10 Serial Search Algorithm	p. 212
6.3.11 Time Domain Correlation	p. 213
6.3.12 Domain Correlation	p. 216
6.3.13 Delayed Signal	p. 219
6.4 P Code Architecture	p. 221
6.4.1 P Code Acquisition	p. 226
6.4.2 P Code Spectral Density	p. 228
6.5 Computational Aspects	p. 229
7 Fourier Optics	p. 231
7.1 Physical Optics	p. 231
7.1.1 Abbe Sine Condition	p. 233
7.2 Scalar Diffraction Theory	p. 234
7.2.1 Helmholtz Equation	p. 236
7.2.2 Helmholtz-Kirchhoff Integral Theorem	p. 236
7.2.3 Fresnel-Kirchhoff Diffraction	p. 239
7.2.4 Rayleigh-Sommerfeld Diffraction	p. 240
7.2.5 Fresnel Diffraction	p. 244
7.2.6 Huygens-Fresnel Principle	p. 244
7.2.7 Fresnel Approximation	p. 245
7.2.8 Fraunhofer Diffraction	p. 251
7.3 Quasi-Optics	p. 254
7.4 Electromagnetic Spectrum	p. 259
7.5 Electromagnetic Radiation	p. 264
7.6 Adaptive Additive Algorithm	p. 265
8 X-Ray Crystallography	p. 267
8.1 Historical Notes	p. 267
8.2 Bragg's Law	p. 267
8.3 X-Ray Diffraction	p. 270
8.3.1 Electron Distribution	p. 271
8.4 Generation of X-Rays	p. 273
8.5 DNA	p. 274
8.6 Hydrogen Atom	p. 278
8.6.1 Hydrogen Spectrum	p. 283
8.7 Units of Measurement	p. 286
8.7.1 Ultrashort Pulse	p. 288
8.8 Laser	p. 289
8.8.1 Luminescence	p. 293
9 Radioastronomy	p. 297
9.1 Historical Notes	p. 297
9.2 Synchrotron Radiation	p. 298
9.3 Radioastronomy	p. 301
9.4 Radio Interferometry	p. 302
9.5 Aperture Synthesis	p. 306
9.6 Big Bang, Quasars, and Pulsars	p. 308
9.6.1 Quasars	p. 309
9.6.2 Pulsars	p. 312
9.6.3 Olbers Paradox	p. 313
9.7 Black Holes	p. 315
9.7.1 A Mathematical Theory	p. 316
9.7.2 Black Hole Solutions	p. 316
10 Acoustics, Poetry, and Music	p. 321
10.1 Introduction	p. 321
10.2 Harmonic Analysis	p. 323
10.3 Huygens Principle Revisited	p. 325
10.4 Modulation	p. 326
10.4.1 Audio Signals	p. 327
10.5 Rhythm and Music	p. 329
10.5.1 Pitch, Loudness, and Timbre	p. 329
10.5.2 Music Bars	p. 332
10.5.3 Timbre	p. 335
10.5.4 Harmonics	p. 336
10.5.5 Musical Scales	p. 340
10.5.6 Tempered Scale	p. 341
10.5.7 Resonance	p. 343
10.6 Drums	p. 344
10.6.1 Vibrating Rectangular Membrane	p. 346
10.6.2 Frequency Modulation	p. 347
10.7 Music Synthesizers	p. 349
11 Computerized Axial Tomography	p. 351
11.1 Introduction	p. 351
11.2 Types of Tomography	p. 354
11.2.1 Straight-Ray Imaging	p. 354
11.2.2 Series Expansion Method	p. 354
11.2.3 Sinogram	p. 355
11.2.4 B-Scan Imaging	p. 356
11.2.5 Reflection Tomography	p. 359
11.3 Magnetic Resonance Imaging	p. 363
11.3.1 Nonlocality of Radon Inversion	p. 368
11.4 Modern Tomography	p. 369
11.4.1 Nuclear Medicine	p. 371
A Tables of Fourier Transforms	p. 373
A.1 Complex Fourier Transform Pairs	p. 373
A.2 Fourier Cosine Transform Pairs	p. 375
A.3 Fourier Sine Transform Pairs	p. 376
B Hilbert Transforms	p. 377
B.1 Hilbert Transforms	p. 377
B.2 Inverse Hilbert Transforms	p. 379
B.3 Discrete Hilbert Transforms	p. 380
C Radon Transforms	p. 383
C.1 Radon Transform	p. 383
C.2 Inverse Radon Transform	p. 385
C.3 2-D Radon Transform	p. 387
C.4 Discrete Radon Transform	p. 388
C.4.1 DR.T and Sinogram	p. 390
C.4.2 Inverse DRT: Continuous Case	p. 391
C.4.3 IDRT as Approximate IFT	p. 392
C.4.4 Relationship with Fourier Transform	p. 393
C.4.5 Tomographic Reconstruction Methods	p. 394
C.4.6 Practical Issues	p. 395
D Maxwell's Equations and Solitons	p. 397
D.1 Maxwell's Equations	p. 397
D.2 sine-Gordon Equation	p. 399
D.3 K-dV Equation	p. 405
D.4 Nonlinear Schrödinger Equation	p. 407
E Modulation	p. 409
E.1 Definition	p. 409
E.2 Phase Modulation and FM	p. 411
E.3 Waveshaping	p. 413
E.4 BPSK Modulation	p. 418
F Boolean and Bitwise Operations	p. 419
G Galois Field	p. 421
G.1 Galois Field Arithmetic	p. 421
G.2 LFSR Encoder	p. 424
H GPS Geometry	p. 429
H.1 Earth	p. 429
H.2 User Location	p. 430
H.3 Altitude	p. 431
H.4 Sidereal Day	p. 433
H.5 Kepler's Laws	p. 433
I Gold Codes	p. 435
I.1 Definition	p. 435
I.2 m-Sequences	p. 436
I.3 Gold Code Generator	p. 438
J Doppler Effect	p. 439
K Bessel Functions	p. 443
K.1 Bessel functions of the First Kind	p. 443
K.2 Modified Bessel Functions	p. 444
K.3 Airy Functions	p. 445
K.4 Hankel Transforms	p. 445
L Heisenberg's Uncertainty Principle	p. 449
M Classical Latin Prosody	p. 453
M.1 Rhythm and Meter	p. 453
M.2 Feet	p. 455
M.3 List of Feet	p. 456
M.4 Division of Rhythms	p. 458
M.5 Roman Concept of Music and Drama	p. 460
Bibliography	p. 463
Index	p. 479

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