The Fourier Transform and its Applications

Incarcat la data: 19 Aprilie 2011

Autor: Admin

By Brad G. Osgood - Stanford University
Licence: Creative Commons Attribution-NonCommercial-ShareAlike 3.0

Course Description:
Note: This course is being offered this summer by Stanford as an online course for credit. It can be taken individually, or as part of a master's degree or graduate certificate earned online through the Stanford Center for Professional Development.The goals for the course are to gain a facility with using the Fourier transform, both specific techniques and general principles, and learning to recognize when, why, and how it is used. Together with a great variety, the subject also has a great coherence, and the hope is students come to appreciate both.Topics include: The Fourier transform as a tool for solving physical problems. Fourier series, the Fourier transform of continuous and discrete signals and its properties. The Dirac delta, distributions, and generalized transforms. Convolutions and correlations and applications; probability distributions, sampling theory, filters, and analysis of linear systems. The discrete Fourier transform and the FFT algorithm. Multidimensional Fourier transform and use in imaging. Further applications to optics, crystallography. Emphasis is on relating the theoretical principles to solving practical engineering and science problems.
Lectures:
Lecture 1 - The Fourier Series

Previous Knowledge Recommended (Matlab), The Fourier Series, Analysis V. Synthesis, Periodic Phenomena And The Fourier Series -Periodicity In Time And Space -Reciprocal Relationship Between Domains, The Reciprocal Relationship Between Frequency And Wavelength

 
Lecture 2 - Periodicity; How Sine And Cosine Can Be Used To Model More Complex Functions

Periodicity; How Sine And Cosine Can Be Used To Model More Complex Functions, Example Of Periodizing A Signal, Discussion Of How To Model Signals With Sinusoids, "One Period, Many Frequencies" Idea In Modeling Signals, Modeling A Signal As The Sum Of Modified Sinusoids (Formula), Complex Exponential Notation, Symmetry Property Of The Complex Coefficients In The Fourier Series, Discussion Of The Generality Of The Fourier Series Representation For Modeling A Periodic Function

 
Lecture 3 - Analyzing General Periodic Phenomena As A Sum Of Simple Periodic Phenomena

Summary Of Previous Lecture (Analyzing General Periodic Phenomena As A Sum Of Simple Periodic Phenomena), Fourier Coefficients; Discussion Of How General The Fourier Series Can Be (Examples Of Discontinuous Signals), Discontinuity And Its Impact On The Generality Of The Fourier Series, Infinite Sums To Represent More General Periodic Signals, Summary Of Convergence Issues, Convergence: Continuous Case, Smooth Case (Fourier Series Converges To The Signal), Convergence: Jump Discontinuity, Convergence: General Case (Convergence On Average/ In Mean/ In Energy)

 
Lecture 4 - Wrapping Up Fourier Series; Making Sense Of Infinite Sums And Convergence

Wrapping Up Fourier Series; Making Sense Of Infinite Sums And Convergence, Integrability Of A Function (Implies Existence Of Fourier Coefficients, Convergence), Orthogonality Of Complex Exponentials (Review), The Inner Product, Norm Of F Related To The Inner Product (+ Pythagorean Theorem), Complex Exponentials As Orthonormal Functions, Fourier Coefficients As Projections Onto Complex Exponentials, Rayleigh's Identity, Application Of Fourier Series To Heat Flow

 
Lecture 5 - Continued Discussion Of Fourier Series And The Heat Equation

Continued Discussion Of Fourier Series And The Heat Equation, Transition From Fourier Series To Fourier Transforms (Periodic To Nonperiodic Phenomena), Fourier Series Analysis And Synthesis; Relation To Fourier Transform And Inverse Fourier Transform, Fourier Series/ Coefficients With Period T, Spectrum Picture For Fourier Series With Period T, Effects Of A Change In T, The Complications Of Finding The Fourier Transform By Letting T Go To Infinity (Fourier Coefficients Go To 0)

 
Lecture 6 - Correction To Heat Equation Discussion

Correction To Heat Equation Discussion, Setup For Fourier Transform Derivation From Fourier Series, Results Of The Derivation: Fourier Transform And Inverse Fourier Transform, Definition Of The Fourier Transform (Analysis), Definition Of Fourier Inversion (Synthesis), Major Secret Of The Universe: Every Signal Has A Spectrum, Which Determines The Signal, Fourier Notation, Example: Rect Function, Example: Triangle Function

 
Lecture 7 - Review Of Fourier Transform (And Inverse) Definitions

Review Of Fourier Transform (And Inverse) Definitions, Notation, Review Of Rect And Triangle Transforms, Example: Fourier Transform Of A Gaussian, The Duality Property Of The Fourier Transform, Example Of An Application Of The Duality Property

 
Lecture 8 - Effect On Fourier Transform Of Shifting A Signal

Effect On Fourier Transform Of Shifting A Signal, Resulting Delay Formula (Shift Theorem), Effect Of Scaling The Time Signal, Stretch Theorem Formula/ Interpretation, Convolution In Context Of Fourier Transforms; Multiplying Two Signals In Frequency, Resulting Convolution Formula

 
Lecture 9 - Continuing Convolution: Review Of The Formula

Continuing Convolution: Review Of The Formula, Situiation In Which It Arose, Example Of Convolution: Filtering, The Ideas Behind Filtering, Terminology, Interpreting Convolution In The Time Domain, General Properties Of Convolution In The Time Domain, Derivative Theorem For Fourier Transforms, Heat Equation On An Infinite Rod

 
Lecture 10 - Central Limit Theorem And Convolution; Main Idea

Central Limit Theorem And Convolution; Main Idea, Introduction, Normalization Of The Gaussian, The Gaussian In Probability; Pictorial Demonstration With Convolution, The Setup For The CLT, Key Result: Distribution Of Sums And Convolution (With Proof), Other Assumptions Needed To Set Up CLT, Statement Of The Central Limit Theorem, Using The Fourier Transform To Prove The CLT

 
Lecture 11 - Correction To The End Of The CLT Proof

Correction To The End Of The CLT Proof, Discussion Of The Convergence Of Integrals; Approaches To Making A More Robust Definition Of The Fourier Transform, Examples Of Problematic Signals, How To Approach Solving The Problem; Choosing Basic Phenomena To Use To Explain Others, Identifying The Best Class Of Signals For Fourier Transforms; + Their Properties, The Definition Of The Class Of Rapidly Decreasing Functions, Rationale For Why These Properties Are Used (Derivative Theorem, Parseval's)

 
Lecture 12 - Cop Story

Cop Story, Review Of Rapidly Decreasing Functions, Generalized Functions (Distributions) (Delta Function, Etc.), Viewing Delta As A Limit V. Operationally, Definition Of A Distribution, Delta As A Distribution, Discussion Of How To Consider Ordinary Functions In This Space; Pairing Through Integration

 
Lecture 13 - Setting Up The Fourier Transform Of A Distribution

Setting Up The Fourier Transform Of A Distribution, Example Of Delta As A Distribution, Distributions Induced By Functions (Includes Many Functions), The Fourier Transform Of A Distribution, The Class Of Tempered Distributions, FT Of A Tempered Distribution, Definition Of The Fourier Transform (By How It Operates On A Test Function), The Inverse Fourier Transform (Proof), Calculations Of Fourier Transforms Using This Definition (Distributions)

 
Lecture 14 - Derivative Of A Distribution

Derivative Of A Distribution, Example: Derivative Of A Unit Step, Example: Derivative Of Sgn(X), Applications To The Fourier Transform (Using The Derivative Theorem), Caveat To Distributions: Multiplying Distributions, Distributions*Functions, Special Case: The Delta Function And Sampling, Convolution In Distributions, Special Case: Convolution When T = Delta, The Scaling Property Of Delta

 
Lecture 15 - Application Of The Fourier Transform: Diffraction: Setup

Application Of The Fourier Transform: Diffraction: Setup, Representation Of Electric Field, Approach Using Huyghens' Principle, Discussion Of The Phase Change Associated With Different Paths, Use Of The Fraunhofer Approximation, Aperture Function, Result; In General And For Single/ Double Slits

 
Lecture 16 - More On Results From Last Lecture (Diffraction Patterns And The Fourier Transforms)

More On Results From Last Lecture (Diffraction Patterns And The Fourier Transforms), Setup For Crystallography Discussion (History, Concepts), 1-Dimensional Version, The Fourier Transform Of The Shah Function, Trick: Poisson Summation Formula, Proof Of The Poisson Summation Formula, Fourier Transform Of The Shah Function: Result, Fourier Transform Of The Shah Function With Spacing P, Application To Crystals

 
Lecture 17 - Review Of Main Properties Of The Shah Function

Review Of Main Properties Of The Shah Function, Setup For The Interpolation Problem, Bandwidth Assumption, Solving For Exact Interpolation For Bandlimited Signals, Periodizing The Signal By Convolution With The Shah Function, Solution Of The Interpolation Problem

 
Lecture 18 - Review Of Sampling And Interpolation Results

Review Of Sampling And Interpolation Results, Terminology: Sampling Rate, Nyquist Rate, Issues With The Interpolation Formula In Practical Applications, Aliasing And Interpolation, Main Argument In Aliasing, Example Of Aliasing: Cosine

 
Lecture 19 - Aliasing Demonstration With Music

Aliasing Demonstration With Music, Transition To Discrete! The DFT, The Plan For Transitioning To Discrete Time, Creating A Discrete Signal From F(T) Creating A Discrete Version Of The Fourier Transform Of The Sampled Version Of F(T), Summary Of What We Just Did, Summary Of Results (Formulas), Moving From Continuous To Discrete Variables, Final Result: The DFT

 
Lecture 20 - Review: Definition Of The DFT

Review: Definition Of The DFT, Sample Points, Relationship Between N And Spacing In Time/Frequency, Complex Exponentials In The Discrete DFT, DFT Written With Discrete Complex Exponential Vector, Periodicity Of Inputs And Outputs In The DFT (More On This In Next Lecture), Orthogonality Of The Vector Of Discrete Complex Exponentials, Note On Orthonormality Of Discrete Complex Exponential Vector (Or Lack Thereof), Consequence Of Orthogonality: Inverse DFT

 
Lecture 21 - Review Of Basic DFT Definitions

Review Of Basic DFT Definitions, Special Case: Value Of The DFT At 0, Two Special Signals: One Vector, Delta Vector, DFT Of Deltas, Complex Exponentials, DFT As Nxn Matrix Multiplication, Periodicity Of Input/Output Signals In The DFT, Result Of Periodicity: Indexing, Result Of Periodicity: Duality

 
Lecture 22 - FFT Algorithm: Setup: DFT Matrix Notation

FFT Algorithm: Setup: DFT Matrix Notation, One Intuition Behind FFT: Factoring Matrix, Our Approach: Split Order N Into Two Order N/2, Iterate, Notation (To Keep Track Of Powers Of Complex Exponentials), Plugging New Notation Into DFT; Split Into Even And Odd Indices, Result For Indices 0 To N/2-1, Result For Indices N/2 To N-1, Summary Of Results (DFT As Combination Of 2 Half Order Dfts)

 
Lecture 23 - Linear Systems: Basic Definitions

Linear Systems: Basic Definitions, Direct Proportionality As Example, Special Cases Of Linear Systems, Eigenvectors And Eigenvalues, The Spectral Theorem And Finding A Basis Of Eigenvectors, Matrix Multiplication = Only Example Of Finite Dimensional Linear Systems, Integration Against A Kernel Generalizing Matrix Multiplication, Example: The Fourier Transform

 
Lecture 24 - Review Of Last Lecture: Discrete V. Continuous Linear Systems

Review Of Last Lecture: Discrete V. Continuous Linear Systems, Cascading Linear Systems, Derivation Of The Impulse Response, Schwarz Kernel Theorem, Example: Impulse Response For Fourier Transform, Example: Switch, Special Case: Convolution, Time Invariance, Result: If A System Is Given By Convolution, It Is Time Invariant; Converse True As Well, Two Main Ideas Sumarized (Linear->Integration Against Kernel, Time Invariant If Given By Convolution)

 
Lecture 25 - Review Of Last Lecture: LTI Systems And Convolution

Review Of Last Lecture: LTI Systems And Convolution, Comment On Time Invariant Discrete Systems, The Fourier Transform For LTI Systems; Complex Exponentials As Eigenfunctions, Discussion Of Sine And Cosine V. Complex Exponentials As Eigenfunctions (Generally They Are Not), Discrete Version (Discrete Complex Exponentials Are Eigenvectors), Discrete Results From A Matrix Perspective

 
Lecture 26 - Approaching The Higher Dimensional Fourier Transform

Approaching The Higher Dimensional Fourier Transform, Notation: Thinking In Terms Of Vectors, Definition Of The Higher Dimensional Fourier Transform, Inverse Fourier Transform, Reciprocal Relationship Between Spatial And Frequency Domain, One Dimensional Case: Reciprocal Relationship, 2-D Case: Visualizing Higher Dimensional Complex Exponentials, Results: Visualizing 2-D Complex Exponentials

 
Lecture 27 - Higher Dimensional Fourier Transforms- Review

Higher Dimensional Fourier Transforms- Review, Fourier Transforms Of Seperable Functions (Ex: 2-D Rect), Result: Formula For Fourier Transform Of A Seperable Function, Example: 2-D Gaussian, Radial Functions, Proof That The Fourier Transform Of A Radial Function Is Also Radial, Convolution In Higher Dimensions

 
Lecture 28 - Shift Theorem In Higher Dimensions

Shift Theorem In Higher Dimensions, Shift Theorem: Result, Stretch Theorem Derivation, Stretch Theorem Result, Special Case: Scaling, Special Case: Rotation, What Reciprocal Means In Higher Dimensions (Inverse Transpose), Deltas In Higher Dimensions (Basic Properties, Scaling)

 
Lecture 29 - Shahs

Shahs, Lattices, And Crystallography, 2-D Shah, Crystals As Lattices, The Fourier Transform Of The Shah Function Of An Oblique Lattice, Relation To Crystals; Notation, Concepts, And Results, Application To Medical Imaging: Tomography

 
Lecture 30 - Tomography And Inverting The Radon Transform

Tips For Filling Out Evals, Tomography And Inverting The Radon Transform; Setup, Introducing Coordinates, Delta Along A Line, The Integral Of U Along A Line, Inverting The Radon Transform

 

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