The Fourier Transform and Its Applications

CONTENTS

Preface xvii

1 Introduction 1

2 Groundwork 5
2.1 The Fourier Transform and Fourier\'s Integral Theorem 5
2.2 Conditions for the Existence of Fourier Transforms 8
2.3 Transforms in the Limit 10
2.4 Oddness and Evenness 11
2.5 Significance of Oddness and Evenness 13
2.6 Complex Conjugates 14
2.7 Cosine and Sine Transforms 16
2.8 Interpretation of the Formulas 18

3 Convolution 24
3.1 Examples of Convolution 27
3.2 Serial Products 30
 Inversion of serial multiplication / The serial product in matrix notation / Sequences as vectors
3.3 Convolution by Computer 39
3.4 The Autocorrelation Function and Pentagram Notation 40
3.5 The Triple Correlation 45
3.6 The Cross Correlation 46
3.7 The Energy Spectrum 47

4 Notation for Some Useful Functions 55
4.1 Rectangle Function of Unit Height and Base, $\\Pi(x)$ 55
4.2 Triangle Function of Unit Height and Area, $\\Lambda(x)$ 57
4.3 Various Exponentials and Gaussian and Rayleigh Curves 57
4.4 Heaviside\'s Unit Step Function, $H(x)$ 61
4.5 The Sign Function, $\\sgn x$ 65
4.6 The Filtering or Interpolating Function, $\\sinc x$ 65
4.7 Pictorial Representation 68
4.8 Summary of Special Symbols 71

5 The Impulse Symbol 74
5.1 The Sifting Property 78
5.2 The Sampling or Replicating Symbol ${\\i}{\\i}{\\i}(x)$ 81
5.3 The Even and Odd Impulse Pairs ${\\i}{\\i}(x)$ and ${\\i}_{\\i}(x)$ 84
5.4 Derivatives of the Impulse Symbol 85
5.5 Null Functions 87
5.6 Some Functions in Two or More Dimensions 89
5.7 The Concept of Generalized Function 92
 Particularly well-behaved functions / Regular sequences / Generalized functions / Algebra of generalized functions / Differentiation of ordinary functions

6 The Basic Theorems 105
6.1 A Few Transforms for Illustration 105
6.2 Similarity Theorem 108
6.3 Addition Theorem 110
6.4 Shift Theorem 111
6.5 Modulation Theorem 113
6.6 Convolution Theorem 115
6.7 Rayleigh\'s Theorem 119
6.8 Power Theorem 120
6.9 Autocorrelation Theorem 122
6.10 Derivative Theorem 124
6.11 Derivative of a Convolution Integral 126
6.12 The Transform of a Generalized Function 127
6.13 Proofs of Theorems 128
 Similarity and shift theorems / Derivative theorem / Power theorem
6.14 Summary of Theorems 129

7 Obtaining Transforms 136
7.1 Integration in Closed Form 137
7.2 Numerical Fourier Transformation 140
7.3 The Slow Fourier Transform Program 142
7.4 Generation of Transforms by Theorems 145
7.5 Application of the Derivative Theorem to Segmented Functions 145
7.6 Measurement of Spectra 147
 Radio frequency spectral analysis / Optical Fourier transform spectroscopy

8 The Two Domains 151
8.1 Definite Integral 152
8.2 The First Moment 153
8.3 Centroid 155
8.4 Moment of Inertia (Second Moment) 156
8.5 Moments 157
8.6 Mean-Square Abscissa 158
8.7 Radius of Gyration 159
8.8 Variance 159
8.9 Smoothness and Compactness 160
8.10 Smoothness under Convolution 162
8.11 Asymptotic Behavior 163
8.12 Equivalent Width 164
8.14 Autocorrelation Width 170
8.15 Mean Square Widths 171
8.16 Sampling and Replication Commute 172
8.17 Some Inequalities 174
 Upper limits to ordinate and slope / Schwarz\'s inequality
8.18 The Uncertainty Relation 177
 Proof of uncertainty relation / Example of uncertainty relation
8.19 The Finite Difference 180
8.20 Running Means 184
8.21 Central Limit Theorem 186
8.22 Summary of Correspondences in the Two Domains 191

9 Waveforms, Spectra, Filters, and Linearity 198
9.1 Electrical Waveforms and Spectra 198
9.2 Filters 200
9.3 Generality of Linear Filter Theory 203
9.4 Digital Filtering 204
9.5 Interpretation of Theorems 205
 Similarity theorem / Addition theorem / Shift theorem / Modulation theorem / Converse of modulation theorem
9.6 Linearity and Time Invariance 209
9.7 Periodicity 211

10 Sampling and Series 219
10.1 Sampling Theorem 219
10.2 Interpolation 224
10.3 Rectangular Filtering in Frequency Domain 224
10.4 Smoothing by Running Means 226
10.5 Undersampling 229
10.6 Ordinate and Slope Sampling 230
10.7 Interlaced Sampling 232
10.8 Sampling in the Presence of Noise 234
10.9 Fourier Series 235
 Gibbs phenomenon / Finite Fourier transforms / Fourier coefficients
10.10 Impulse Trains That Are Periodic 245
10.11 The Shah Symbol Is Its Own Fourier Transform 246

11 The Discrete Fourier Transform and the FFT 258
11.1 The Discrete Transform Formula 258
11.2 Cyclic Convolution 264
11.3 Examples of Discrete Fourier Transforms 265
11.4 Reciprocal Property 266
11.5 Oddness and Evenness 266
11.6 Examples with Special Symmetry 267
11.7 Complex Conjugates 268
11.8 Reversal Property 268
11.9 Addition Theorem 268
11.10 Shift Theorem 268
11.11 Convolution Theorem 269
11.12 Product Theorem 269
11.13 Cross-Correlation 270
11.14 Autocorrelation 270
11.15 Sum of Sequence 270
11.16 First Value 270
11.17 Generalized Parseval-Rayleigh Theorem 271
11.18 Packing Theorem 271
11.19 Similarity Theorem 272
11.20 Examples Using MATLAB 272
11.21 The Fast Fourier Transform 275
11.22 Practical Considerations 278
11.23 Is the Discrete Fourier Transform Correct? 280
11.24 Applications of the FFT 281
11.25 Timing Diagrams 282
11.26 When $N$ Is Not a Power of $2$ 283
11.27 Two-Dimensional Data 284
11.28 Power Spectra 285

12 The Discrete Hartley Transform 293
12.1 A Strictly Reciprocal Real Transform 293
12.2 Notation and Example 294
12.3 The Discrete Hartley Transform 295
12.4 Examples of DHT 297
12.5 Discussion 298
12.6 A Convolution of Algorithm in One and Two Dimensions 298
12.7 Two Dimensions 299
12.8 The Cas-Cas Transform 300
12.9 Theorems 300
12.10 The Discrete Sine and Cosine transforms 301
 Boundary value problems / Data compression application
12.11 Computing 305
12.12 Getting a Feel for Numerical Transforms 305
12.13 The Complex Hartley Transform 306
12.14 Physical Aspect of the Hartley Transformation 307
12.15 The Fast Hartley Transform 308
12.16 The Fast Algorithm 309
12.17 Running Time 314
12.18 Timing via the Stripe Diagram 315
12.19 Matrix Formulation 317
12.20 Convolution 320
12.21 Permutation 321
12.22 A Fast Hartley Subroutine 322

13 Relatives of the Fourier Transform 329
13.1 The Two-Dimensional Fourier Transform 329
13.2 Two-Dimensional Convolution 331
13.3 The Hankel Transform 335
13.4 Fourier Kernels 339
13.5 The Three-Dimensional Fourier Transform 340
13.6 The Hankel Transform in $n$ Dimensions 343
13.7 The Mellin Transform 343
13.8 The $z$ Transform 347
13.9 The Abel Transform 351
13.10 The Radon Transform and Tomography 356
 The Abel-Fourier-Hankel ring of transforms / Projection-slice theorem / Reconstruction by modified back projection
13.11 The Hilbert Transform 359
 The analytic signal/Instantaneous frequency and envelope / Causality
13.12 Computing the Hilbert Transform 364
13.13 The Fractional Fourier Transform 367
 Shift theorem / Derivative theorems / Fractional convolution theorem / Examples of transforms

14 The Laplace Transform 380
14.1 Convergence of the Laplace Integral 382
14.2 Theorems for the Laplace Transform 383
14.3 Transient-Response Problems 385
14.4 Laplace Transform Pairs 386
14.5 Natural Behavior 389
14.6 Impulse Response and Transfer Function 390
14.7 Initial-Value Problems 392
14.8 Setting Out Initial-Value Problems 396
14.9 Switching Problems 396

15 Antennas and Optics 406
15.1 One-Dimensional Apertures 407
15.2 Analogy with Waveforms and Spectra 410
15.3 Beam Width and Aperture Width 411
15.4 Beam Swinging 412
15.5 Arrays of Arrays 413
15.6 Interferometers 414
15.7 Spectral Sensitivity Function 415
15.8 Modulation Transfer Function 416
15.9 Physical Aspects of the Angular Spectrum 417
15.10 Two-Dimensional Theory 417
15.11 Optical Diffraction 419
15.12 Fresnel Diffraction 420
15.13 Other Applications of Fourier Analysis 422

16 Applications in Statistics 428
16.1 Distribution of a Sum 429
16.2 Consequences of the Convolution Relation 434
16.3 The Characteristic Function 435
16.4 The Truncated Exponential Distribution 436
16.5 The Poisson Distribution 438

17 Random Waveforms and Noise 446
17.1 Discrete Representation by Random Digits 447
17.2 Filtering a Random Input: Effect on Amplitude Distribution 450
 Digression on independence / The convolution relation
17.3 Effect on Autocorrelation 455
17.3 Effect on Spectrum 458
 Spectrum of random input / The output spectrum
17.4 Some Noise Records 462
17.5 Envelope of Bandpass Noise 465
17.6 Detection of a Noise Waveform 466
17.7 Measurement of Noise Power 466

18 Heat Conduction and Diffusion 475
18.1 One-Dimensional Diffusion 475
18.2 Gaussian Diffusion from a Point 480
18.3 Diffusion of a Spatial Sinusoid 481
18.4 Sinusoidal Time Variation 485

19 Dynamic Power Spectra 489
19.1 The Concept of Dynamic Spectrum 489
19.2 The Dynamic Spectrograph 491
19.3 Computing the Dynamic Power Spectrum 494
 Frequency division / Time division / Presentation
19.4 Equivalence Theorem 497
19.5 Envelope and Phase 498
19.6 Using $\\log f$ instead of $f$ 499
19.7 The Wavelet Transform 500
19.8 Adaptive Cell Placement 502
19.9 Elementary Chirp Signals (Chirplets) 502
19.10 The Wigner Distribution 504

20 Tables of $\\sinc x$, $\\sinc^2 x$, and $\\exp(-\\pi x^2)$ 508

21 Solutions to Selected Problems 513
Chapter 2 Groundwork 513
Chapter 3 Convolution 514
Chapter 4 Notation for Some Useful Functions 516
Chapter 5 The Impulse Symbol 517
Chapter 6 The Basic Theorems 522
Chapter 7 Obtaining Transforms 524
Chapter 8 The Two Domains 526
Chapter 9 Waveforms, Spectra, Filters, and Linearity 530
Chapter 10 Sampling and Series 532
Chapter 11 The Discrete Fourier Transform and the FFf 534
Chapter 12 The Hartley Transform 537
Chapter 13 Relatives of the Fourier Transform 538
Chapter 14 The Laplace Transform 539
Chapter 15 Antennas and Optics 545
Chapter 16 Applications in Statistics 555
Chapter 17 Random Waveforms and Noise 557
Chapter 18 Heat Conduction and Diffusion 565
Chapter 19 Dynamic Spectra and Wavelets 571

22 Pictorial Dictionary of Fourier Transforms 573
 Hartley Transforms of Some Functions without Symmetry 592

23 The Life of Joseph Fourier 594

Index 597