Lectures (Video)

- 1. The Fourier Series
- 2. Periodicity, Modeling A Signal
- 3. Convergence
- 4. Inner Product, Complex Exponentials
- 5. Fourier Transforms
- 6. Fourier Inversion
- 7. Duality Property
- 8. Stretch Theorem Formula, Convolution Formula
- 9. Continuing Convolution
- 10. Central Limit Theorem And Convolution
- 11. Best Class Of Signals For Fourier Transforms
- 12. Generalized Functions
- 13. Fourier Transform Of A Distribution
- 14. The Delta Function And Sampling
- 15. Application Of The Fourier Transform
- 16. Shah Function, Poisson Summation Formula
- 17. Interpolation Problem
- 18. Sampling Rate, Nyquist Rate, Aliasing
- 19. Discrete Version Of The Fourier Transform
- 20. Discrete Complex Exponential Vector
- 21. Review Of Basic DFT Definitions
- 22. FFT Algorithm
- 23. Linear Systems
- 24. Impulse Response, Schwarz Kernel Theorem
- 25. Fourier Transform For LTI Systems
- 26. Higher Dimensional Fourier Transform
- 27. Fourier Transforms Of Seperable Functions
- 28. Shift Theorem
- 29. Shahs, Lattices, And Crystallography
- 30. Tomography And Inverting The Radon Transform

## Fourier Transform and its Applications

### Course Summary

This course is based on

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. As a tool for applications it is used in virtually all areas of science and engineering. In electrical engineering Fourier methods are found in all varieties of signal processing, from communications and circuit design to imaging and optics. (Includes a full set of video lectures.)
*The Fourier Transform and its Applications EE261*made available by*Stanford University: Stanford Engineering Everywhere*under the*Creative Commons Attribution 3.0*license.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.

### Reading Material

1.**The Fourier Transform and its Applications**

R. N. Bracewell,

*The Fourier Transform and its Applications*, McGraw Hill, 1986

(Click the image below for the link to the 1999 edition)

### Course Material

1.**Course Information (pdf file)**

2.

**Course Reader (30 MB pdf file)**

3.

**References**

4.

**Formula Sheet**

5.

**Course Reader (Chapter 8), with all exponents (2 MB pdf file)**