The Fourier Transform and Its Applications

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The Fourier Transform and Its Applications, Ronald N. Bracewell, 3rd Edition, McGraw-Hill, 2000

Preface XVll

1 Introduction 1

2 Groundwork 5
The Fourier Transform and Fourier's Integral Theorem 5
Conditions for the Existence of Fourier Transforms 8
Transforms in the Limit 10
Oddness and Evenness 11
Significance of Oddness and Evenness 13
Complex Conjugates 14
Cosine and Sine Transforms 16
Interpretation of the Formulas 18

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

4 Notation for Some Useful Functions 55
Rectangle Function of Unit Height and Base, n(x) 55
Triangle Function of Unit Height and Area, A(x) 57
Various Exponentials and Gaussian and Rayleigh Curves 57
Heaviside's Unit Step Function, H(x) 61
The Sign Function, sgn x 65
The Filtering or Interpolating Function, sine x 65
Pictorial Representation 68
Summary of Special Symbols 71

5 The Impulse Symbol 74
The Sifting Property 78
The Sampling or Replicating Symbol III(x) 81
The Even and Odd Impulse Pairs n(x) and II (x) 84
Derivatives of the hnpulse Symbol 85
Null Functions 87
Some Functions in Two or More Dimensions 89
The Concept of Generalized Function 92
Particularly well-behaved functions I Regular sequences I Generalized functions I
Algebra of generaliud functions / Differentiation of ordinary functions

6 The Basic Theorems 105
A Few Transforms for Illustration 105
Similarity Theorem 108
Addition Theorem 110
Shift Theorem 111
Modulation Theorem 113
Convolution Theorem 115
Rayleigh's Theorem 119
Power Theorem 120
Autocorrelation Theorem 122
Derivative Theorem 124
Derivative of a Convolution Integral 126
The Transform of a Generalized Function 127
Proofs of Theorems 128
Similarity and shift theorems / Derivative theorem / Power theorem
Summary of Theorems 129

7 Obtaining Transforms 136
Integration in Gosed Form 137
Numerical Fourier Transformation 140
The Slow Fourier Transform Program 142
Generation of Transforms by Theorems 145
Application of the Derivative Theorem to Segmented Functions 145
Measurement of Spectra 147
Radiofrequency spectral analysis / Optical Fourier transform spectroscopy

8 The Two Domains 151
Definite Integral 152
The First Moment 153
Centroid 155
Moment of Inertia (Second Moment) 156
Moments 157
Mean-Square Abscissa 158
Radius of Gyration 159
Variance  159
Smoothness and Compactness  160
Smoothness under Convolution  162
Asymptotic Behavior  163
Equivalent Width  164
Autocorrelation Width  170
Mean Square Widths 171
Sampling and Replication Commute  172
Some Inequalities  174
         Upper limits to ordinate and slope / Schwarz's inequality
The Uncertainty Relation  177
         Proof of uncertainty relation / Example of uncertainty relation
The Finite Difference  180
Running Means  184
Central Limit Theorem  186
Summary of Correspondences in the Two Domains  191

9 Waveforms, Spectra, Filters, and Linearity 198
Electrical Waveforms and Spectra 198
Filters 200
Generality of Linear Filter Theory 203
Digital Filtering 204
Interpretation of Theorems 205
      Similarity theorem / Addition theorem / Shift theorem / Modulation theorem /
      Converse of modulation theorem
Linearity and TIme Invariance 209
Periodicity 211

10 Sampling and Series 219
Sampling Theorem 219
Interpolation 224
Rectangular Filtering in Frequency Domain 224
Smoothing by Running Means 226
Undersampling 229
Ordinate and !5lope Sampling 230
Interlaced Sampling 232
Sampling in the Presence of Noise 234
Fourier Series 235
Gibbs phenomenon / Finite Fourier transforms / Fourier coefficients
Impulse Trains That Pile Periodic 245
The Shah Symbol Is Its Own Fourier Transform 246

11 The Discrete Fourier Transform and the FFT  258
The Discrete Transform Formula  258
Cyclic Convolution  264
Examples of Discrete Fourier Transforms  265
Reciprocal Property  266
Oddness and Evenness  266
Examples with Special Symmetry  267
Complex Conjugates  268
Reversal Property  268
Addition Theorem 268
Shift Theorem  268
Convolution Theorem  269
Product Theorem  269
Cross-Correlation  270
Autocorrelation  270
Sum of Sequence  270
First Value  270
Generalized Parseval-Rayleigh Theorem  271
Packing Theorem  271
Similarity Theorem  272
Examples Using MATLAB  272
The Fast Fourier Transform  275
Practical Considerations  278
Is the Discrete Fourier Transform Correct?  280
Applications of the FFT  281
Timing Diagrams  282
When N Is Not a Power of 2  283
Two-Dimensional Data  284
Power Spectra  285

12 The Discrete Hartley Transform  293
A Strictly Reciprocal Real Transform  293
Notation and Example  294
The Discrete Hartley Transform  295
Examples of DHT  297
Discussion  298
A Convolution of Algorithm in One and Two Dimensions  298
Two Dimensions  299
The Cas-Cas Transform  300
Theorems  300
The Discrete Sine and Cosine transforms  301
      Boundary value problems J Data compression application
Computing  305
Cetting a Feel for Numerical Transforms  305
The Complex Hartley Transform  306
Physical Aspect of the Hartley Transformation  307
The Fast Hartley Transform  308
The Fast Algorithm  309
Running lime  314
Timing via the Stripe Diagram
Matrix Formulation
Convolution
Permutation
A Fast Hartley Subroutine

13 Relatives of the Fourier Transform
The Two-Dimensional Fourier Transform
Two-Dimensional Convolution
The Hankel Transform
Fourier Kernels
The Three-Dimensional Fourier Transform
The Hankel Transform in n Dimensions
The Mellin Transform
The z Transform
The Abel Transform
The Radon Transform and Tomography
The Abel-Fourier-Hnnkel ring of tmnsforrns / Projection-slice theorem /
Reconstruction by modified back projection
The Hilbert Transform
The analytic signal / Instl11ztaneous frequency l11zd envelope / Causality
Computing the Hilbert Transform
The Fractional Fourier Transform
Shift theorem / Derivntive theorems / Fractio1U1l convolution theorem /
Examples of transforms

14 The Laplace Transform
Convergence of the Laplace Integral
Theorems for the Laplace Transform
Transient-Response Problems
Laplace Transform Pairs
Natural Behavior
Impulse Response and Transfer Function
Initial-Value Pwblems
Setting Out Initial-Value Problems
Switching Problems

15 Antennas and Optics
One-Dimensional Apertures
Analogy with Waveforms and Spectra
Beam Width and Aperture Width
Beam Swinging
Arrays of Arrays
Interferometers
Spectral Sensitivity Function
Modulation Transfer Function 416
Physical Aspects of the Angular Spectrum 417
Two-Dimensional Theory 417
Optical Diffraction 419
Fresnel Diffraction 420
Other Applications of Fourier Analysis 422

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

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

18 Heat Conduction and Diffusion 475
One-Dimensional Diffusion 475
Gaussian Diffusion from a Point 480
Diffusion of a Spatial Sinusoid 481
Sinusoidal TIme Variation 485

19 Dynamic Power Spectra 489
The Concept of Dynamic Spectrum 489
The Dynamic Spectrograph 491
Computing the Dynamic Power Spectrum 494
Frequency division I Time division J Presentation
Equivalence Theorem 497
Envelope and Phase 498
Using log f instead of f 499
The Wavelet Transform 500
Adaptive Cell Placement 502
Elementary Chirp Signals (Chirplets) 502
The Wigner Distribution 504

20 Tables of sinc x, sinr x, and exp ( -1Tr) 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 Transfonns 573
Hartley Transforms of Some Functions without Symmetry 592

23 The Life of Joseph Fourier 594

Index 597