Generalized Transforms
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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....more
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The global AIDS pandemic furnishes a case study for many of the themes addressed throughout the course. While in the developed West the disease largely afflicts concentrated high-risk groups such as intravenous drug users and the sexually promiscuous, in Southern Africa it is much more a generalized disease of poverty. In countries such as Botswana and Swaziland, the economic and social consequences of the disease have created a vicious...more
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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...more
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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)
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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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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
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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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Guest Lecturer: Jacob Mattingley, Logistics, Agenda, Convex Set, Convex Cone, Polyhedra, Positive Semidefinite Cone, Operations That Preserve Convexity, Intersection, Affine Function, Generalized Inequalities, Minimum And Minimal Elements, Supporting Hyperlane Theorem, Minimum And Minimal Elements Via Dual Inequalities
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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
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Interior-Point Methods (Cont.), Example, Barrier Method (Review), Complexity Analysis Via Self-Concordance, Total Number Of Newton Iterations, Generalized Inequalities, Logarithmic Barrier And Central Path, Barrier Method, Course Conclusion, Further Topics
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Jordan Canonical Form, Generalized Modes, Cayley-Hamilton Theorem, Proof Of C-H Theorem, Linear Dynamical Systems With Inputs & Outputs, Block Diagram, Transfer Matrix, Impulse Matrix, Step Matrix
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Spatial Descriptions, Generalized Coordinates, Operational Coordinates, Rotation Matrix, Example - Rotation Matrix, Translations, Example - Homogeneous Transform, Operators, General Operators
