Fourier Transform
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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
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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
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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...more
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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...more
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Measure, fourier series, and fourier transforms
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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
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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
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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
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More On Results From Last Lecture (Diffraction Patterns And The Fourier Transforms)
Stanford / Mathematics
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
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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
