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傅里叶变换

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Contents

1 Fourier Series 1

1.1 Introduction and Choices to Make . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Periodic Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Periodicity: Deﬁnitions, Examp les, and Things to Come . . . . . . . . . . . . . . . . . . . . 4

1.4 It All Adds Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Lost at c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6 Period, Frequencies, and Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.7 Two Examples and a Warning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.8 The Math, the Majesty, the End . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.9 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.10 Appendix: The Cauchy-Schwarz Inequality and its Consequences . . . . . . . . . . . . . . . 33

1.11 Appendix: More on the Complex Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.12 Appendix: Best L

Approximation by Finite Fourier Series . . . . . . . . . . . . . . . . . . 38

1.13 Fou rier Series in Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1.14 Notes on Convergence of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

1.15 Appendix: Pointwise Convergence vs. Uniform C onvergence . . . . . . . . . . . . . . . . . . 59

1.16 Appendix: Studying Partial Sums via the Dirichlet Kernel: The Buzz Is Back . . . . . . . . 60

1.17 Appendix: The Complex Exponentials Are a Basis for L

([0, 1]) . . . . . . . . . . . . . . . . 62

1.18 Appendix: More on the Gibbs Phen omenon . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2 Fourier Transform 65

2.1 A First Look at the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.2 Getting to Know Your Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3 Convolution 95

3.1 A ∗ is Born . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.2 What is Convolution, Really? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.3 Properties of Convolution: It’s a Lot like Multiplication . . . . . . . . . . . . . . . . . . . . 101

ii CONTENTS

3.4 Convolution in Action I: A Little Bit on Filtering . . . . . . . . . . . . . . . . . . . . . . . . 102

3.5 Convolution in Action II : Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.6 Convolution in Action III: The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . 116

3.7 The Central Limit Theorem: The Bell Curve Tolls for Thee . . . . . . . . . . . . . . . . . . 128

3.8 Fourier transform formulas under diﬀerent normalizations . . . . . . . . . . . . . . . . . . . 130

3.9 Appendix: The Mean and Standard Deviation for the Sum of Random Variables . . . . . . 131

3.10 More Details on the Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

3.11 Appendix: Heisenberg’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4 Distributions and Their Fourier Transforms 137

4.1 The Day of Reckoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

4.2 The Right Fu nctions for Fourier Transforms: Rapidly Decreasing Functions . . . . . . . . . 142

4.3 A Very Little on Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

4.4 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

4.5 A Physical Analogy for Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

4.6 Limits of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

4.7 The Fourier Transform of a Tempered Distribution . . . . . . . . . . . . . . . . . . . . . . . 168

4.8 Fluxions Finis: The End of Diﬀerential Calculus . . . . . . . . . . . . . . . . . . . . . . . . 174

4.9 Approximations of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

4.10 The Generalized Fourier Transform Includes the Classical Fourier Transform . . . . . . . . 178

4.11 Operations on Distributions and Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . 179

4.12 Duality, Changing Signs, Evenness and Oddness . . . . . . . . . . . . . . . . . . . . . . . . 179

4.13 A Function Times a Distribution Makes Sense . . . . . . . . . . . . . . . . . . . . . . . . . . 182

4.14 The Derivative Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

4.15 S hifts and the Shift Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

4.16 S caling and the Stretch Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

4.17 Convolutions an d the Convolution Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

4.18 δ Hard at Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

4.19 Appendix: The Riemann-Lebesgue lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

4.20 Appendix: Smooth Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

4.21 Appendix: 1/x as a Principal Value Distribution . . . . . . . . . . . . . . . . . . . . . . . . 209

4.22 Appendix: A formula for δ on a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

5 III, Sampling, and Interpolation 213

5.1 X-Ray Diﬀraction: Th rough a Glass Darkly

. . . . . . . . . . . . . . . . . . . . . . . . . . 213

5.2 The III Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

CONTENTS iii

5.3 The Fourier Transform of III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

5.4 Periodic Distributions and Four ier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

5.5 Sampling Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

5.6 Sampling and Interpolation for Bandlimited Signals . . . . . . . . . . . . . . . . . . . . . . 227

5.7 Interpolation a Little More Generally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

5.8 Finite Sampling for a Bandlimited Period ic Signal . . . . . . . . . . . . . . . . . . . . . . . 233

5.9 Troubles with Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

5.10 Appendix: How Special is III? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

5.11 Appendix: Timelimited vs. Bandlimited Signals . . . . . . . . . . . . . . . . . . . . . . . . . 250

6 Discrete Fourier Transform 253

6.1 From Continuous to Discrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

6.2 The Discrete Fourier Transform (DFT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

6.3 Two Grids, Reciprocally Related . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

6.4 Appendix: Gauss’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

6.5 Getting to Know Your Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . 263

6.6 Periodicity, Indexing, and Reindexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

6.7 Inverting the DFT and Many Other Things Along the Way . . . . . . . . . . . . . . . . . . 266

6.8 Properties of the DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

6.9 Diﬀerent Deﬁnitions for the DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

6.10 The FFT Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

6.11 Zero Padding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

7 Linear Time-Invariant Systems 297

7.1 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

7.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

7.3 Cascading Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

7.4 The Impulse Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

7.5 Linear Time-Invariant (LTI) Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

7.6 Appendix: The Linear Millennium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

7.7 Appendix: Translating in Time and Plugging into L . . . . . . . . . . . . . . . . . . . . . . 310

7.8 The Fourier Transform and LTI Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

7.9 Matched Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

7.10 Caus ality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

7.11 The Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

7.12 Appendix: The Hilbert Transform of sinc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

iv CONTENTS

7.13 Filters Finis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

7.14 Appendix: Geometric Series of the Vector Complex Exponentials . . . . . . . . . . . . . . . 332

7.15 Appendix: The Discrete Rect and its DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

8 n-dimensional Fourier Transform 337

8.1 Space, the Final Fr ontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

8.2 Getting to Know Your Higher Dimensional Fourier Transform . . . . . . . . . . . . . . . . . 349

8.3 Higher Dimensional Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

8.4 III, Lattices, Crystals, and Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

8.5 Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

8.6 Bandlimited Functions on R

and Sampling on a Lattice . . . . . . . . . . . . . . . . . . . . 385

8.7 Naked to the Bone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

8.8 The Radon Tran s form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

8.9 Getting to Know Your Radon Tran s form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

8.10 Appendix: Clarity of Glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

8.11 Medical Imaging: Inverting the Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . 398

8.12 Appendix: Line impulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

A Complex Numbers and Complex Exponentials 415

A.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

A.2 The Complex Exponential and Euler’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . 418

A.3 Algebra and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

A.4 Further Applications of Euler’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

Chapter 1

Fourier Series

1.1 Introduction and Choices to Make

Methods based on the Fourier transform are used in virtually all areas of engineering and science and by

virtually all engineers and scientists. For starters:

• Circuit designers

• Spectroscopists

• Crystallographers

• Anyone working in signal processing and communications

• Anyone working in imaging

I’m expecting that many ﬁelds and many interests will be represented in the class, and this brings up an

important issue for all of us to be aware of. With the diversity of interests and backgrounds present not

all examples and applications will be familiar and of r elevance to all people. We’ll all have to cut each

other some slack, and it’s a chance for all of us to branch out. Along the same lines, it’s also important for

you to realize that this is one course on th e Fou rier transf orm among many possible courses. The richness

of the subject, both mathematically and in the range of app lications, means that we’ll be making choices

almost constantly. Books on the subject do not look alike, nor do they look like these notes — even the

notation used for basic objects and operations can vary from book to book. I’ll try to point out when a

certain choice takes us along a certain path, and I’ll try to say something of what the alternate paths may

be.

The very ﬁrst choice is where to start, and my choice is a brief treatment of Fourier series.

Fourier analysis

was originally concerned with representing and analyzing periodic phenomena, via Fourier series, and later

with extending th ose ins ights to nonperiodic phenomena, via the Fourier transform. In fact, one way of

getting from Fourier series to the Fourier transform is to consider nonperiodic ph en omena (and thus just

about any general function) as a limiting case of p eriodic phenomena as th e period tends to inﬁnity. A

discrete set of frequencies in the perio dic case becomes a continuum of frequencies in the nonperiodic case,

the spectrum is born, and with it comes the most important principle of the subject:

Every signal has a spectrum and is determined by its spectrum. You can analyze the signal

either in the time (or spatial) domain or in the frequen cy domain.

Bracewell, for example, starts right oﬀ with the Fourier transform and picks up a little on Fourier series later.

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