Exponentials
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Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODEs) deal with functions of one variable, which can often be thought of as time. Topics include: Solution of first-order ODE's by analytical, graphical and numerical methods; Linear ODE's,...more
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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)
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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
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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
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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...more
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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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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
