Derivative Of A Distribution

Lecture Description

Derivative Of A Distribution, Example: Derivative Of A Unit Step, Example: Derivative Of Sgn(X), Applications To The Fourier Transform (Using The Derivative Theorem), Caveat To Distributions: Multiplying Distributions, Distributions*Functions, Special Case: The Delta Function And Sampling, Convolution In Distributions, Special Case: Convolution When T = Delta, The Scaling Property Of Delta

Course Description

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.

Related Resources

Transcript   |  Problem Set 4   |  Problem set 4 Solutions

Course Index

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