# Course - The Fourier Transform and its Applications

Course Level: Master

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. Together with a great variety, the subject also has a great coherence, and the hope is students come to appreciate both.Topics include: The Fourier transform as a tool for solving physical problems. Fourier series, the Fourier transform of continuous and discrete signals and its properties. The Dirac delta, distributions, and generalized transforms. Convolutions and correlations and applications; probability distributions, sampling theory, filters, and analysis of linear systems. The discrete Fourier transform and the FFT algorithm. Multidimensional Fourier transform and use in imaging. Further applications to optics, crystallography. Emphasis is on relating the theoretical principles to solving practical engineering and science problems.

• Lecture 1 - The Fourier Series
• Lecture 2 - Periodicity; How Sine And Cosine Can Be Used To Model More Complex Functions
• Lecture 3 - Analyzing General Periodic Phenomena As A Sum Of Simple Periodic Phenomena
• Lecture 4 - Wrapping Up Fourier Series; Making Sense Of Infinite Sums And Convergence
• Lecture 5 - Continued Discussion Of Fourier Series And The Heat Equation
• Lecture 6 - Correction To Heat Equation Discussion
• Lecture 7 - Review Of Fourier Transform (And Inverse) Definitions
• Lecture 8 - Effect On Fourier Transform Of Shifting A Signal
• Lecture 9 - Continuing Convolution; Review Of The Formula
• Lecture 10 - Central Limit Theorem And Convolution; Main Idea
• Lecture 11 - Correction To The End Of The CLT Proof
• Lecture 12 - Cop Story
• Lecture 13 - Setting Up The Fourier Transform Of A Distribution
• Lecture 14 - Derivative Of A Distribution
• Lecture 15 - Application Of The Fourier Transform; Diffraction; Setup
• Lecture 16 - More On Results From Last Lecture (Diffraction Patterns And The Fourier Transforms)
• Lecture 17 - Review Of Main Properties Of The Shah Function
• Lecture 18 - Review Of Sampling And Interpolation Results
• Lecture 19 - Aliasing Demonstration With Music
• Lecture 20 - Review; Definition Of The DFT
• Lecture 21 - Review Of Basic DFT Definitions
• Lecture 22 - FFT Algorithm; Setup; DFT Matrix Notation
• Lecture 23 - Linear Systems; Basic Definitions
• Lecture 24 - Review Of Last Lecture; Discrete V. Continuous Linear Systems
• Lecture 25 - Review Of Last Lecture; LTI Systems And Convolution
• Lecture 26 - Approaching The Higher Dimensional Fourier Transform
• Lecture 27 - Higher Dimensional Fourier Transforms- Review
• Lecture 28 - Shift Theorem In Higher Dimensions
• Lecture 29 - Shahs
• Lecture 30 - Tomography And Inverting The Radon Transform