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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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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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, Stat...
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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 Univers...
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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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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 Exponen...
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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...
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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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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, H...
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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 Hi...
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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 ...
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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, ...
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