实例介绍
【实例简介】
斯坦福大学开放课程 : 傅立叶变换及应用(Open Stanford Course : The Fourier Transform and Its Applications )课件PDF
Contents 1 Fourier series 1.1 Introduction and choices to make 1.2 Periodic phenomena 1.3 Pcriodicity: Dcfinitions, Examples, and Things to Comc 11249 1. 4 It All Adds Up 1.5 Lost at c 10 1.6 Period, Frequencies, and Spectrum 12 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 1. 11 Appendix: Morc on the Complex Inncr Product 36 1.12 Appendix: Best L Approximation by Finite Fourier Series 1.13 Fourier series in Action 1.14 Notes on Convergence of Fourier Series 50 1.15 Appendix: Pointwisc Convergence vs. Uniform Convergence 1.16 Appendix: Studying Partial Sums via the Dirichlet Kernel: The Buzz Is Back 59 1.17 Appendix: The Complex Exp Are a Basis for L2([0, 1]) 61 1.18 Appendix: More on the Gibbs Phenomenon 2 Fourier transform 65 2.1 A First look at the fourier transform 65 2.2 Appendix: Chase the Constant 75 2.3 Examples 2.4 How Docs the Graph of f(a)Compare with thc Graph of f(e? 2.5 Getting to Know Your Fourier Transform 79 involution 91 CONTENTS 3.1A米 is born. 3.2 What is Convolution, Really? 95 3.3 Properties of Convolution: Its a lot like Multiplication 97 3. For whom the bell curve tolls 99 3.5 Appendix: Evaluation of the Gaussian Integral 101 3.6 Convolution in Action I: A Little Bil on Filtering 102 3.7 Convolution in Action II: Differential equations 3.8 Appendix: Didnt We Already Solve the Heat Equation? 113 3.9 Convolution in Action III: The Central Limit Theorem 116 3.10 The Central Limit Theoren: The bell Curve Tolls for Thee 128 3.11 Appendix: The Mean and Standard Deviation for the Sum of random variables 130 3.12 More Details on the Central limit Theorem 131 3. 13 Appendix: Heisenbergs Inequality 132 4 Distributions and Their fourier transforms 135 4.1 The Day of reckoning 135 4.2 The Right Functions for Fourier Transforms: Rapidly Decreasing Functions 140 4.3 Appendix: A Very Little on Integrals 146 4.4 Appendix: The Riemann-Lebesgue lemma 150 4.5 Appendix: Smooth Windows 150 4.6 Distributions 153 4.7 Appendix: A Physical Analogy for Distributions 165 4.8 Appendix: Limits of DistributiOns 166 4.9 Appendix: Other Approximating Sequences for d .166 4.10 The Fourier Transform of a Tempered Distribution 169 4.11 Fluxions Finis: The End of Differential calculus 175 4.12 Approximations of DistributiOns .178 4.13 Appendix: The Generalized Fourier Transform Includes the Classical Fourier Transform . 179 4.14 Appendix: 1/ as a Principal Value Distribution 180 4.15 Operations on Distributions and Fourier Transforms 181 4.16 Duality, Changing Signs, Evenness and Oddness 182 4.17 A Function Times a Distribution Makes sense 185 1.18 The derivative Theorem 187 4.19 Shifts and the shift The eorem 4.20 Scaling and the Stretch Theorell 191 CONTENTS 4.21 Convolutions and the convolution Theorem 192 4.22 S Hard at Work 197 5 Sampling 209 5. 1 X Ray Diffraction: Through a Glass Darkly 5.2 The ll distribution .210 5.3 The fourier transform of ill 214 5.4 Appendix: Periodic Distributions and Fourier series 217 5.5 Appendix: How Special is Il? .221 5.6 Sampling Signals 222 5.7 Sampling and Interpolation for Bandlimited Signals 225 5.8 Interpolation a Little More generally 228 5.9 Finite Sampling for a Bandlimited Periodic Signal .230 5.10 Appendix: Timelimited vS. Bandlimited Signals 233 5.11 Appendix: Periodizing sinc Functions 235 5.12 Troubles with Sampling 237 6 Discrete fourier Transform 249 6.1 From Continuous to discrete )0 6.2 The Discrete Fourier Transform (DFT) .252 6.3 Two Grids, Reciprocally related 257 6.4 Appendix: Gauss's Problcm .258 6.5 Getting to Know Your Discrete Fourier Transform .259 6.6 Periodicity, Indexing, and Reindexing 260 6.7 Inverting the DFt and many Other Things Along the Way 6.8 Propcrtics of the DFt .271 6.9 Appendix: Different Definitions for the DFT .,,275 6.10 The FFT Algorithm 277 6.11 Zero padding 7 Linear Time-Invariant Systems 293 7.1 Linear Systems 7.2 Examples 294 7.3 Cascading Linear Systems 299 7.4 The Impulse Response 7.5 Linear Time-Invariant (LTI) Systems 302 CONTENTS 7.6 Appendix: The Linear Millennium 305 7.7 Appendix: Translating in Time and Plugging into L 306 7. 8 The Fourier Transform and LtI Systems 307 7.9Ⅵ atched filters .309 7.10 Causality 311 7. 11 The hilbert TransforIll 312 7. 12 Appendix: The Hilbert Transform of sinc 318 7.13 Filters finis .319 7. 14 Appendix: Geometric Series of the Vector Complex Exponentials 328 7. 15 Appendix: The Discrete Rect and ilS DFT .,,,330 8 n-dimensional fourier transform 333 8.1 Space, the Final Frontier 333 8.2 Getting to Know Your Higher Dimensional Fourier Transform 345 8.3 Appendix: The Stretch Theorem and Friends 357 8.4 Iligher dimensional fourier series 359 8.5 II, Lattices, Crystals, and Sampling .369 8.6 Crystals 8.7 Bandlimited Functions on R2 and Sampling on a Lattice 380 8.8 Appendix: The Poisson Summation Formula, again .383 8.9 Naked to the bone 384 8.10 The Radon Transform 387 8. 11 Gelling lo Know Your Radon TransforM 390 8.12 Appendix: Clarity of glass .394 8. 13 Mledical maging: Inverting the Radon transform 395 A Mathematical Background 403 A 1 Complex numbers 403 A.2 The Complex Exponential and Euler's Formula 406 A 3 Algebra and gcomctry 409 A 4 Further Applications of Euler's Formula 409 B Some references 413 Chapter 1 Fourier series 1.1 Introduction and choices to make Methods based on the Fourier transform are used in virtually all areas of science and engineering. By whom? For starters Circuit designers Spectroscopists Crystallographers Anyone working in signal processing and communications Anyone working in imaging Im 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 relevance to all people. Well 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 the Fourier transform among many possible courses. The richness of the subject both mathematically and in the range of applications, 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. Ill 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 those insights to nonperiodic phenomena, via the Fourier transform. In fact, one way of getting from Fourier series to the Fourier transform is to consider nonperiodic phenomena(and thus just about any general function) as a limiting case of periodic phenomena as the period tends to infinity. A discrete set of frequencies in the periodic 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 signa l either in the time (or spatial) domain or in the frequency domain i Bracewell, for example, starts right off with the Fourier transform and picks up a little on Fourier series later Chapter 1 Fourier Series I think this qualifies as a Major Secret of the Universe All of this was thoroughly grounded in physical applications. Most often the phenomena to be studied were modeled by the fundamental differential equations of physics(heat equation, wave equation, Laplace's equation), and the solutions were usually constrained by boundary conditions. At first the idea was to use Fourier series to lind explicit solutions This work raised hard and far reaching questions that led in different directions. It was gradually realized that setting up Fourier series (in sines and cosines) could be recast in the more general framework of orthog onality, linear operators, and eigenfunctions. That led to the general idea of working with eigenfunction expansions of solutions of differential equations. a ubiquitous line of attack in many areas and applications In the modern formulation of partial differential equations, the Fourier transform has become the basis for defining the objects of study, while still remaining a tool for solving specific equations. Much of this development depends on the remarkable relation between Fourier transforms and convolution, somethin which was also seen earlier in the Fourier series days. In an effort to apply the methods with increasing generality, mathematicians were pushed(by engineers and physicists) to reconsider how general the notion of " function"can be, and what kinds of functions can be- and should be- admitted into the operating theater of calculus. Differentiation and integration were both generalized in the service of Fourier analysis Other directions combine tools from Fourier analysis with symmetries of the objects being analyzed. This might make you think of crystals and crystallography, and you'd be right, while mathematicians think of number theory and Fourier analysis on groups. Finally, I have to mention that in the purely mathematical rcalm the qucstion of convergence of Fouricr scrics, bclicvc it or not, led G. Cantor ncar the turn of thc 20th century to investigate and invent the theory of infinite sets, and to distinguish different sizes of infinite sets, all of which led to Cantor going insane 2 Periodic phenomena To begin the course with Fourier series is to begin with periodic functions, those functions which exhibit a regularly repeating pattern. It shouldnt be necessary to try to sell periodicity as an important physical and mathematical) phenomenon- you've seen examples and applications of periodic behavior in probably almost)every class you've taken. I would only remind you that periodicity often shows up in two varieties sometimes related, sometimes not. Generally speaking we think about periodic phenomena according to whether they are periodic in time or periodic in space 1.2.1 Time and space In the case of time the phenomenon comes to you. For example, you stand at a fixed point in the ocean(or on an clectrical circuit) and the waves (or the clectrical current) wash ovcr you with a regular, rccurrin pattern of crests and troughs. The height of the wave is a periodic function of time. Sound is another example: "sound"reaches your ear as a longitudinal pressure wave, a periodic compression and rarefaction of the air. In the case of space, you come to the phenomenon. You take a picture and you observe repeating patterns Temporal and spalial periodicily come together nost llalurally in wave moLion. Take the case of one spatial dimension, and consider a singlc sinusoidal wave traveling along a string (for cxamplc). For such a wave the periodicity in time is measured by the frequency v, with dimension 1/ sec and units Hz(Hertz cycles per second), and the periodicity in space is measured by the wavelength A, with dimension length and units whatever is convenient for the particular setting. If we fix a point in space and let the time vary(lake a video of the wave motion al that point) then successive crests of the wave come past that 1.2 Periodic phenomena point v times per second, and so do successive troughs. If we fix the time and examine how the wave is spread out in space(take a snapshot instead of a video) we see that the distance between successive crests is a constant A, as is the distance between successive troughs. The frequency and wavelength are related through the equation v=Av, where v is the speed of propagation this is nothing but the wave version of speed distance/time. Thus the higher the frequency the shorter the wavelength, and the lower the frequency the longer the wavelength. If the speed is fixed, like the speed of electromagnetic waves in a vacuum, then the frequency determines the wavelength and vice versa: if you can measure one you can find the other. For sound we identify the physical property of frequency with the perceptual property of pitch for light frequency is perceived as color. Simple sinusoids are the building blocks of the most complicated wave forms that's what Fourier analysis is about 1.2.2 More on spatial periodicit Another way spatial periodicity occurs is when there is a repeating pattern or some kind of symmetry in a spatial region and physically observable quantities associated with that region have a repeating pattern that reflects this. For example, a crystal has a regular, repealing pattern of atolls in space; he arrangement of atoms is callcd a lattice. The clectron density distribution is then a periodic function of thc spatial variable (in R)that describes the crystal. I mention this example because, in contrast to the usual one-dimensional examples you might think of, here the function, in this case the electron density distribution, has three independent periods corresponding to the three directions that describe the crystal lattice Here's another example- this time in two dimensions- that is very much a natural subject for Fourier analysis. Consider these stripes of dark anld light no doubt there's some kind of spatially periodic behavior going on in the respective images. Furthermore even without stating a precise definition, it's reasonable to say that one of the patterns is"low frequency and that the others are "high frequency", meaning roughly that there are fewer stripes per unit length in the one than in the others. In two dimensions there s an extra subtlety that we see in these pictures spatial frequency,, however we ultimately define it, must be a vector quantity, not a number. We have to say that the stripes occur with a certain spacing in a certain direction Such periodic stripes are the building blocks of general two-dimensional images. When there's no color an image is a two-dimensional array of varying shades of gray, and this can be realized as a synthesis-a 4 Chapter 1 Fourier Series Fourier synthesis- of just such alternating stripes There are interesting perceptual questions in constructing images this way, and color is more complicated still. llere's a picture i got from Foundations of vision by Brian Wandell, who is in the Psychology Department here at stanford The shades of blue and yellow are the same in the two pictures- the only a change is in the frequency The closer spacing "mixes the blue and yellow to give a greenish cast. Here's a question that i know has been investigated but i dont know the answer. Show someone blue and yellow stripes of a low frequency and increase the frequency till they just start to see green. You get a number for that. Next, start with blue and yellow stripes at a high frequency so a person sees a lot of green and then lower the frequency till they see only blue and yellow. You get a number for that. Are the two numbers the same? does the orientation of the stripes make a difference? 1.3 Periodicity: Definitions, Examples, and Things to Come To be certain we all know what we're talking about, a function f(t) is periodic of period T if there is a number T>0 such that f(tI T)=f(t) for all t. If there is such a T' then the smallest one for which the equation holds is called the fundamental period of the function f 2 Every integer multiple of the fundamental period is also a pe f(t+nT)=f(t),n=0,±1,士2,,3 I'm calling the variable t here because I have to call it something, but the definition is general and is not meant to imply periodic functions of time Sometimes when people say simply " period"they mean the smallest or fundamental period. (I usually do, for example.) Sometimes they dont. Ask them what they mean 3 It's clear from the geometric picture of a repeating graph that this is true. To show it algebraically. if n>1 then we see inductively that f(t+nT)=f(t+(n-1T+r)=f(t+(n-1)T)=f(t). Then to see algebraically why negative multiples of T are also periods we have, for n21,f(t-nr)=f(t-nT+nr)=f(t) 【实例截图】
【核心代码】
斯坦福大学开放课程 : 傅立叶变换及应用(Open Stanford Course : The Fourier Transform and Its Applications )课件PDF
Contents 1 Fourier series 1.1 Introduction and choices to make 1.2 Periodic phenomena 1.3 Pcriodicity: Dcfinitions, Examples, and Things to Comc 11249 1. 4 It All Adds Up 1.5 Lost at c 10 1.6 Period, Frequencies, and Spectrum 12 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 1. 11 Appendix: Morc on the Complex Inncr Product 36 1.12 Appendix: Best L Approximation by Finite Fourier Series 1.13 Fourier series in Action 1.14 Notes on Convergence of Fourier Series 50 1.15 Appendix: Pointwisc Convergence vs. Uniform Convergence 1.16 Appendix: Studying Partial Sums via the Dirichlet Kernel: The Buzz Is Back 59 1.17 Appendix: The Complex Exp Are a Basis for L2([0, 1]) 61 1.18 Appendix: More on the Gibbs Phenomenon 2 Fourier transform 65 2.1 A First look at the fourier transform 65 2.2 Appendix: Chase the Constant 75 2.3 Examples 2.4 How Docs the Graph of f(a)Compare with thc Graph of f(e? 2.5 Getting to Know Your Fourier Transform 79 involution 91 CONTENTS 3.1A米 is born. 3.2 What is Convolution, Really? 95 3.3 Properties of Convolution: Its a lot like Multiplication 97 3. For whom the bell curve tolls 99 3.5 Appendix: Evaluation of the Gaussian Integral 101 3.6 Convolution in Action I: A Little Bil on Filtering 102 3.7 Convolution in Action II: Differential equations 3.8 Appendix: Didnt We Already Solve the Heat Equation? 113 3.9 Convolution in Action III: The Central Limit Theorem 116 3.10 The Central Limit Theoren: The bell Curve Tolls for Thee 128 3.11 Appendix: The Mean and Standard Deviation for the Sum of random variables 130 3.12 More Details on the Central limit Theorem 131 3. 13 Appendix: Heisenbergs Inequality 132 4 Distributions and Their fourier transforms 135 4.1 The Day of reckoning 135 4.2 The Right Functions for Fourier Transforms: Rapidly Decreasing Functions 140 4.3 Appendix: A Very Little on Integrals 146 4.4 Appendix: The Riemann-Lebesgue lemma 150 4.5 Appendix: Smooth Windows 150 4.6 Distributions 153 4.7 Appendix: A Physical Analogy for Distributions 165 4.8 Appendix: Limits of DistributiOns 166 4.9 Appendix: Other Approximating Sequences for d .166 4.10 The Fourier Transform of a Tempered Distribution 169 4.11 Fluxions Finis: The End of Differential calculus 175 4.12 Approximations of DistributiOns .178 4.13 Appendix: The Generalized Fourier Transform Includes the Classical Fourier Transform . 179 4.14 Appendix: 1/ as a Principal Value Distribution 180 4.15 Operations on Distributions and Fourier Transforms 181 4.16 Duality, Changing Signs, Evenness and Oddness 182 4.17 A Function Times a Distribution Makes sense 185 1.18 The derivative Theorem 187 4.19 Shifts and the shift The eorem 4.20 Scaling and the Stretch Theorell 191 CONTENTS 4.21 Convolutions and the convolution Theorem 192 4.22 S Hard at Work 197 5 Sampling 209 5. 1 X Ray Diffraction: Through a Glass Darkly 5.2 The ll distribution .210 5.3 The fourier transform of ill 214 5.4 Appendix: Periodic Distributions and Fourier series 217 5.5 Appendix: How Special is Il? .221 5.6 Sampling Signals 222 5.7 Sampling and Interpolation for Bandlimited Signals 225 5.8 Interpolation a Little More generally 228 5.9 Finite Sampling for a Bandlimited Periodic Signal .230 5.10 Appendix: Timelimited vS. Bandlimited Signals 233 5.11 Appendix: Periodizing sinc Functions 235 5.12 Troubles with Sampling 237 6 Discrete fourier Transform 249 6.1 From Continuous to discrete )0 6.2 The Discrete Fourier Transform (DFT) .252 6.3 Two Grids, Reciprocally related 257 6.4 Appendix: Gauss's Problcm .258 6.5 Getting to Know Your Discrete Fourier Transform .259 6.6 Periodicity, Indexing, and Reindexing 260 6.7 Inverting the DFt and many Other Things Along the Way 6.8 Propcrtics of the DFt .271 6.9 Appendix: Different Definitions for the DFT .,,275 6.10 The FFT Algorithm 277 6.11 Zero padding 7 Linear Time-Invariant Systems 293 7.1 Linear Systems 7.2 Examples 294 7.3 Cascading Linear Systems 299 7.4 The Impulse Response 7.5 Linear Time-Invariant (LTI) Systems 302 CONTENTS 7.6 Appendix: The Linear Millennium 305 7.7 Appendix: Translating in Time and Plugging into L 306 7. 8 The Fourier Transform and LtI Systems 307 7.9Ⅵ atched filters .309 7.10 Causality 311 7. 11 The hilbert TransforIll 312 7. 12 Appendix: The Hilbert Transform of sinc 318 7.13 Filters finis .319 7. 14 Appendix: Geometric Series of the Vector Complex Exponentials 328 7. 15 Appendix: The Discrete Rect and ilS DFT .,,,330 8 n-dimensional fourier transform 333 8.1 Space, the Final Frontier 333 8.2 Getting to Know Your Higher Dimensional Fourier Transform 345 8.3 Appendix: The Stretch Theorem and Friends 357 8.4 Iligher dimensional fourier series 359 8.5 II, Lattices, Crystals, and Sampling .369 8.6 Crystals 8.7 Bandlimited Functions on R2 and Sampling on a Lattice 380 8.8 Appendix: The Poisson Summation Formula, again .383 8.9 Naked to the bone 384 8.10 The Radon Transform 387 8. 11 Gelling lo Know Your Radon TransforM 390 8.12 Appendix: Clarity of glass .394 8. 13 Mledical maging: Inverting the Radon transform 395 A Mathematical Background 403 A 1 Complex numbers 403 A.2 The Complex Exponential and Euler's Formula 406 A 3 Algebra and gcomctry 409 A 4 Further Applications of Euler's Formula 409 B Some references 413 Chapter 1 Fourier series 1.1 Introduction and choices to make Methods based on the Fourier transform are used in virtually all areas of science and engineering. By whom? For starters Circuit designers Spectroscopists Crystallographers Anyone working in signal processing and communications Anyone working in imaging Im 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 relevance to all people. Well 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 the Fourier transform among many possible courses. The richness of the subject both mathematically and in the range of applications, 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. Ill 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 those insights to nonperiodic phenomena, via the Fourier transform. In fact, one way of getting from Fourier series to the Fourier transform is to consider nonperiodic phenomena(and thus just about any general function) as a limiting case of periodic phenomena as the period tends to infinity. A discrete set of frequencies in the periodic 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 signa l either in the time (or spatial) domain or in the frequency domain i Bracewell, for example, starts right off with the Fourier transform and picks up a little on Fourier series later Chapter 1 Fourier Series I think this qualifies as a Major Secret of the Universe All of this was thoroughly grounded in physical applications. Most often the phenomena to be studied were modeled by the fundamental differential equations of physics(heat equation, wave equation, Laplace's equation), and the solutions were usually constrained by boundary conditions. At first the idea was to use Fourier series to lind explicit solutions This work raised hard and far reaching questions that led in different directions. It was gradually realized that setting up Fourier series (in sines and cosines) could be recast in the more general framework of orthog onality, linear operators, and eigenfunctions. That led to the general idea of working with eigenfunction expansions of solutions of differential equations. a ubiquitous line of attack in many areas and applications In the modern formulation of partial differential equations, the Fourier transform has become the basis for defining the objects of study, while still remaining a tool for solving specific equations. Much of this development depends on the remarkable relation between Fourier transforms and convolution, somethin which was also seen earlier in the Fourier series days. In an effort to apply the methods with increasing generality, mathematicians were pushed(by engineers and physicists) to reconsider how general the notion of " function"can be, and what kinds of functions can be- and should be- admitted into the operating theater of calculus. Differentiation and integration were both generalized in the service of Fourier analysis Other directions combine tools from Fourier analysis with symmetries of the objects being analyzed. This might make you think of crystals and crystallography, and you'd be right, while mathematicians think of number theory and Fourier analysis on groups. Finally, I have to mention that in the purely mathematical rcalm the qucstion of convergence of Fouricr scrics, bclicvc it or not, led G. Cantor ncar the turn of thc 20th century to investigate and invent the theory of infinite sets, and to distinguish different sizes of infinite sets, all of which led to Cantor going insane 2 Periodic phenomena To begin the course with Fourier series is to begin with periodic functions, those functions which exhibit a regularly repeating pattern. It shouldnt be necessary to try to sell periodicity as an important physical and mathematical) phenomenon- you've seen examples and applications of periodic behavior in probably almost)every class you've taken. I would only remind you that periodicity often shows up in two varieties sometimes related, sometimes not. Generally speaking we think about periodic phenomena according to whether they are periodic in time or periodic in space 1.2.1 Time and space In the case of time the phenomenon comes to you. For example, you stand at a fixed point in the ocean(or on an clectrical circuit) and the waves (or the clectrical current) wash ovcr you with a regular, rccurrin pattern of crests and troughs. The height of the wave is a periodic function of time. Sound is another example: "sound"reaches your ear as a longitudinal pressure wave, a periodic compression and rarefaction of the air. In the case of space, you come to the phenomenon. You take a picture and you observe repeating patterns Temporal and spalial periodicily come together nost llalurally in wave moLion. Take the case of one spatial dimension, and consider a singlc sinusoidal wave traveling along a string (for cxamplc). For such a wave the periodicity in time is measured by the frequency v, with dimension 1/ sec and units Hz(Hertz cycles per second), and the periodicity in space is measured by the wavelength A, with dimension length and units whatever is convenient for the particular setting. If we fix a point in space and let the time vary(lake a video of the wave motion al that point) then successive crests of the wave come past that 1.2 Periodic phenomena point v times per second, and so do successive troughs. If we fix the time and examine how the wave is spread out in space(take a snapshot instead of a video) we see that the distance between successive crests is a constant A, as is the distance between successive troughs. The frequency and wavelength are related through the equation v=Av, where v is the speed of propagation this is nothing but the wave version of speed distance/time. Thus the higher the frequency the shorter the wavelength, and the lower the frequency the longer the wavelength. If the speed is fixed, like the speed of electromagnetic waves in a vacuum, then the frequency determines the wavelength and vice versa: if you can measure one you can find the other. For sound we identify the physical property of frequency with the perceptual property of pitch for light frequency is perceived as color. Simple sinusoids are the building blocks of the most complicated wave forms that's what Fourier analysis is about 1.2.2 More on spatial periodicit Another way spatial periodicity occurs is when there is a repeating pattern or some kind of symmetry in a spatial region and physically observable quantities associated with that region have a repeating pattern that reflects this. For example, a crystal has a regular, repealing pattern of atolls in space; he arrangement of atoms is callcd a lattice. The clectron density distribution is then a periodic function of thc spatial variable (in R)that describes the crystal. I mention this example because, in contrast to the usual one-dimensional examples you might think of, here the function, in this case the electron density distribution, has three independent periods corresponding to the three directions that describe the crystal lattice Here's another example- this time in two dimensions- that is very much a natural subject for Fourier analysis. Consider these stripes of dark anld light no doubt there's some kind of spatially periodic behavior going on in the respective images. Furthermore even without stating a precise definition, it's reasonable to say that one of the patterns is"low frequency and that the others are "high frequency", meaning roughly that there are fewer stripes per unit length in the one than in the others. In two dimensions there s an extra subtlety that we see in these pictures spatial frequency,, however we ultimately define it, must be a vector quantity, not a number. We have to say that the stripes occur with a certain spacing in a certain direction Such periodic stripes are the building blocks of general two-dimensional images. When there's no color an image is a two-dimensional array of varying shades of gray, and this can be realized as a synthesis-a 4 Chapter 1 Fourier Series Fourier synthesis- of just such alternating stripes There are interesting perceptual questions in constructing images this way, and color is more complicated still. llere's a picture i got from Foundations of vision by Brian Wandell, who is in the Psychology Department here at stanford The shades of blue and yellow are the same in the two pictures- the only a change is in the frequency The closer spacing "mixes the blue and yellow to give a greenish cast. Here's a question that i know has been investigated but i dont know the answer. Show someone blue and yellow stripes of a low frequency and increase the frequency till they just start to see green. You get a number for that. Next, start with blue and yellow stripes at a high frequency so a person sees a lot of green and then lower the frequency till they see only blue and yellow. You get a number for that. Are the two numbers the same? does the orientation of the stripes make a difference? 1.3 Periodicity: Definitions, Examples, and Things to Come To be certain we all know what we're talking about, a function f(t) is periodic of period T if there is a number T>0 such that f(tI T)=f(t) for all t. If there is such a T' then the smallest one for which the equation holds is called the fundamental period of the function f 2 Every integer multiple of the fundamental period is also a pe f(t+nT)=f(t),n=0,±1,士2,,3 I'm calling the variable t here because I have to call it something, but the definition is general and is not meant to imply periodic functions of time Sometimes when people say simply " period"they mean the smallest or fundamental period. (I usually do, for example.) Sometimes they dont. Ask them what they mean 3 It's clear from the geometric picture of a repeating graph that this is true. To show it algebraically. if n>1 then we see inductively that f(t+nT)=f(t+(n-1T+r)=f(t+(n-1)T)=f(t). Then to see algebraically why negative multiples of T are also periods we have, for n21,f(t-nr)=f(t-nT+nr)=f(t) 【实例截图】
【核心代码】
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