Fourier Transform
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DC Or Static Gain Matrix, Discretization With Piecewise Constant Inputs, Causality, Idea Of State, Change Of Coordinates, Z-Transform, Symmetric Matrices, Quadratic Forms, Matrix Nom, And SVD, Eigenvalues Of Symmetric Matrices, Interpretations Of Eigenvalues Of Symmetric Matrices, Example: RC Circuit
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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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Laplace transform of e^at.
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Solving a non-homogeneous differential equation using the Laplace Transform.
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Solution Via Laplace Transform And Matrix Exponential, Laplace Transform Solution Of X_^ = Ax, Harmonic Oscillator Example, Double Integrator Example, Characteristic Polynomial, Eigenvalues Of A And Poles Of Resolvent, Matrix Exponential, Time Transfer Property
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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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Measure, fourier series, and fourier transforms
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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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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
