The Fourier Transform and Its Applications Ronald N. Bracewell CONTENTS

The Fourier Transform and Its Applications Third Edition

Ronald N. Bracewell Lewis M. Terman Professor of Electrical Engineering Emeritus Stanford University

Boston Burr Ridge, IL Dubuque, IA Madison, WI New York San Francisco St. Louis Bangkok Bogota Caracas Lisbon London Madrid Mexico City Milan New Delhi Seoul Singapore Sydney Taipei Toronto

C O N T E N T S

Preface

xvii

1

Introduction

1

2

Groundwork

5

3

The Fourier Transform and Fourier's Integral Theorem Conditions for the Existence of Fourier Transforms Transforms in the Limit Oddness and Evenness Significance of Oddness and Evenness Complex Conjugates Cosine and Sine Transforms Interpretation of the Formulas

5 8 10 11 13 14 16 18

Convolution

24

Examples of Convolution Serial Products

27 30

Inversion of serial multiplication / The serial product in matrix notation I Sequences as vectors

4

Convolution by Computer The Autocorrelation Function and Pentagram Notation The Triple Correlation The Cross Correlation The Energy Spectrum

39 40 45 46 47

Notation for Some Useful Functions

55

Rectangle Function of Unit Height and Base, Il(x) Triangle Function of Unit Height and Area, A(x) Various Exponentials and Gaussian and Rayleigh Curves Heaviside's Unit Step Function, H(x) The Sign Function, sgn x The Filtering or Interpolating Function, sine x Pictorial Representation Summary of Special Symbols

55 57 57 61 65 65 68 71 IX

x

5

6

7

8

Contents

The I m p u l s e Symbol

74

The Sifting Property The Sampling or Replicating Symbol III(x) The Even and Odd Impulse Pairs n{x) and h(x) Derivatives of the Impulse Symbol Null Functions Some Functions in Two or More Dimensions The Concept of Generalized Function Particularly well-behaved functions / Regular sequences / Generalized functions / Algebra of generalized functions I Differentiation of ordinary functions

78 81 84 85 87 89 92

The Basic T h e o r e m s

105

A Few Transforms for Illustration Similarity Theorem Addition Theorem Shift Theorem Modulation Theorem Convolution Theorem Rayleigh's Theorem Power Theorem Autocorrelation Theorem Derivative Theorem Derivative of a Convolution Integral The Transform of a Generalized Function Proofs of Theorems

105 108 110 111 113 115 119 120 122 124 126 127 128

Similarity and shift theorems / Derivative theorem / Power theorem Summary of Theorems

129

O b t a i n i n g Transforms

136

Integration in Closed Form Numerical Fourier Transformation The Slow Fourier Transform Program Generation of Transforms by Theorems Application of the Derivative Theorem to Segmented Functions Measurement of Spectra Radiofrequency spectral analysis / Optical Fourier transform spectroscopy

137 140 142 145 145 147

The Two Domains

151

Definite Integral The First Moment Centroid Moment of Inertia (Second Moment) Moments Mean-Square Abscissa Radius of Gyration

152 153 155 156 157 158 159

Contents Variance Smoothness and Compactness Smoothness under Convolution Asymptotic Behavior Equivalent Width Autocorrelation Width Mean Square Widths Sampling and Replication Commute Some Inequalities Upper limits to ordinate and slope / Schwarz's inequality The Uncertainty Relation Proof of uncertainty relation /Example of uncertainty relation The Finite Difference Running Means Central Limit Theorem Summary of Correspondences in the Two Domains

9

xi 159 160 162 163 164 170 171 172 174 177 180 184 186 191

Waveforms, Spectra, Filters, a n d Linearity

198

Electrical Waveforms and Spectra Filters Generality of Linear Filter Theory Digital Filtering Interpretation of Theorems Similarity theorem / Addition theorem / Shift theorem / Modulation theorem / Converse of modulation theorem Linearity and Time Invariance Periodicity

198 200 203 204 205

10 Sampling and Series Sampling Theorem Interpolation Rectangular Filtering in Frequency Domain Smoothing by Running Means Undersampling Ordinate and Slope Sampling Interlaced Sampling Sampling in the Presence of Noise Fourier Series Gibbs phenomenon /Finite Fourier transforms /Fourier coefficients Impulse Trains That Are Periodic The Shah Symbol Is Its Own Fourier Transform

11 The Discrete Fourier Transform and the FFT The Discrete Transform Formula Cyclic Convolution Examples of Discrete Fourier Transforms

209 211

219 219 224 224 226 229 230 232 234 235 245 246

258 258 264 265

Contents

Xll

Reciprocal Property Oddness and Evenness Examples with Special Symmetry Complex Conjugates Reversal Property Addition Theorem Shift Theorem Convolution Theorem Product Theorem Cross-Correlation Autocorrelation Sum of Sequence First Value Generalized Parseval-Rayleigh Theorem Packing Theorem Similarity Theorem Examples Using MATLAB The Fast Fourier Transform Practical Considerations Is the Discrete Fourier Transform Correct? Applications of the FFT Timing Diagrams When N Is Not a Power of 2 Two-Dimensional Data Power Spectra

12 The Discrete Hartley Transform A Strictly Reciprocal Real Transform Notation and Example The Discrete Hartley Transform Examples of DHT Discussion A Convolution of Algorithm in One and Two Dimensions Two Dimensions The Cas-Cas Transform Theorems The Discrete Sine and Cosine transforms

266 266 267 268 268 268 268 269 269 270 270 270 270 271 271 272 272 275 278 280 281 282 283 284 285 293 293 294 295 297 298 298 299 300 300 301

Boundary value problems / Data compression application

Computing Getting a Feel for Numerical Transforms The Complex Hartley Transform Physical Aspect of the Hartley Transformation The Fast Hartley Transform The Fast Algorithm Running Time

305 305 306 307 308 309 314

Contents

xia

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 2 Transform The Abel Transform The Radon Transform and Tomography

315 317 320 321 322 329 329 331 335 339 340 343 343 347 351 356

The Abel-Fourier-Hankel ring of transforms /Projection-slice theorem I Reconstruction by modified back projection

The Hilbert Transform

359

The analytic signal / Instantaneous frequency and envelope I Causality

Computing the Hilbert Transform The Fractional Fourier Transform

364 367

Shift theorem / Derivative theorems / Fractional convolution theorem / Examples of transforms

,14 The Laplace Transform ^Convergence of the Laplace Integral place Transform

380 382 383 385 386 389

390

xiv

Contents Modulation Transfer Function Physical Aspects of the Angular Spectrum Two-Dimensional Theory Optical Diffraction Fresnel Diffraction Other Applications of Fourier Analysis

16 Applications in Statistics Distribution of a Sum Consequences of the Convolution Relation The Characteristic Function The Truncated Exponential Distribution The Poisson Distribution

17 Random Waveforms and Noise Discrete Representation by Random Digits Filtering a Random Input: Effect on Amplitude Distribution

416 417 417 419 420 422

428 429 434 435 436 438

446 447 450

Digression on independence / The convolution relation

Effect on Autocorrelation Effect on Spectrum

455 458

Spectrum of random input / The output spectrum

Some Noise Records Envelope of Bandpass Noise Detection of a Noise Waveform Measurement of Noise Power

18 Heat Conduction and Diffusion One-Dimensional Diffusion Gaussian Diffusion from a Point Diffusion of a Spatial Sinusoid Sinusoidal Time Variation

19 Dynamic Power Spectra The Concept of Dynamic Spectrum The Dynamic Spectrograph Computing the Dynamic Power Spectrum

462 465 466 466

475 475 480 481 485

489 489 491 494

Frequency division I Time division / Presentation

Equivalence Theorem Envelope and Phase Using log / instead of / The Wavelet Transform Adaptive Cell Placement Elementary Chirp Signals (Chirplets) The Wigner Distribution

497 498 499 500 502 502 504

Contents

xv

20 Tables of sine x, sine2 x, and exp (—TTX2) 21 Solutions to Selected Problems Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19

Groundwork Convolution Notation for Some Useful Functions The Impulse Symbol The Basic Theorems Obtaining Transforms The Two Domains Waveforms, Spectra, Filters, and Linearity Sampling and Series The Discrete Fourier Transform and the FFT The Hartley Transform Relatives of the Fourier Transform The Laplace Transform Antennas and Optics Applications in Statistics Random Waveforms and Noise Heat Conduction and Diffusion Dynamic Spectra and Wavelets

22 Pictorial Dictionary of Fourier Transforms

508 513 513 514 516 517 522 524 526 530 532 534 537 538 539 545 555 557 565 571 573 592

Hartley Transforms of Some Functions without Symmetry 594

23 The Life of Joseph Fourier 597

Index