The Fourier Transform and its Applications

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Contents:
Fourier series – Periodicity; How Sine And Cosine Can Be Used To Model More Complex Functions – Analyzing General Periodic Phenomena As A Sum Of Simple Periodic Phenomena – Wrapping Up Fourier Series; Making Sense Of Infinite Sums And Convergence – Continued Discussion Of Fourier Series And The Heat Equation – Correction To Heat Equation Discussion – Review Of Fourier Transform (And Inverse) Definitions – Effect On Fourier Transform Of Shifting A Signal-Continuing Convolution: Review Of The Formula – Central Limit Theorem And Convolution; Main Idea – Correction To The End Of The CLT Proof-Cop Story – Setting Up The Fourier Transform Of A Distribution – Derivative Of A Distribution-Application Of The Fourier Transform: Diffraction: Setup

More On Results From Last Lecture – Review Of Main Properties Of The Shah Function – Review Of Sampling And Interpolation Results – Aliasing Demonstration With Music – Review: Definition Of The DFT – Review Of Basic DFT Definitions – FFT Algorithm: Setup: DFT Matrix Notation – Linear Systems: Basic Definitions-Discrete Vs Continuous Linear Systems – LTI Systems And Convolution – Approaching The Higher Dimensional Fourier Transform-Higher Dimensional Fourier Transforms – Review – Shift Theorem In Higher Dimensions – Shahs – Tomography And Inverting The Radon Transform

Course Curriculum

Fourier series Details 52:7
Periodicity; How Sine And Cosine Can Be Used To Model More Complex Functions Details 52:57
Analyzing General Periodic Phenomena As A Sum Of Simple Periodic Phenomena Details 50:46
Wrapping Up Fourier Series; Making Sense Of Infinite Sums And Convergence Details 52:7
Continued Discussion Of Fourier Series And The Heat Equation Details 52:1
Correction To Heat Equation Discussion Details 47:52
Review Of Fourier Transform (And Inverse) Definitions Details 47:49
Effect On Fourier Transform Of Shifting A Signal Details 50:37
Continuing Convolution: Review Of The Formula Details 54:23
Central Limit Theorem And Convolution; Main Idea Details 54:58
Correction To The End Of The CLT Proof Details 50:56
Cop Story Details 52:56
Setting Up The Fourier Transform Of A Distribution Details 49:25
Derivative Of A Distribution Details 0:54
Application Of The Fourier Transform: Diffraction: Setup Details 52:9
More On Results From Last Lecture Details 49:43
Review Of Main Properties Of The Shah Function Details 41:13
Review Of Sampling And Interpolation Results Details 51:10
Aliasing Demonstration With Music Details 52:6
Review: Definition Of The DFT Details 52:53
Review Of Basic DFT Definitions Details 52:59
FFT Algorithm: Setup: DFT Matrix Notation Details 51:16
Linear Systems: Basic Definitions Details 51:4
Discrete Vs Continuous Linear Systems Details 57:5
LTI Systems And Convolution Details 53:17
Approaching The Higher Dimensional Fourier Transform Details 53:56
Higher Dimensional Fourier Transforms- Review Details 49:57
Shift Theorem In Higher Dimensions Details 49:7
Shahs Details 50:6
Tomography And Inverting The Radon Transform Details 47:9

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