1 / 42

Understanding Fourier Transform: From Impulses to Waves

This lecture discusses the Fourier transform, explaining how it can be used to represent signals as a sum of impulses or waves. It covers the Fourier series in both real and complex forms, unit period signals, and strategies for handling non-unit periods. The lecture also explores the properties and applications of the Fourier transform, including convolution, noise suppression, and filtering. Homework assignment on the uncertainty property is given.

bourke
Download Presentation

Understanding Fourier Transform: From Impulses to Waves

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 38655 BMED-2300-02 Lecture 6: Fourier Transform Ge Wang, PhD Biomedical Imaging Center CBIS/BME, RPI wangg6@rpi.edu February 2, 2018

  2. BB Schedule for S18 Office Hour: Ge Tue & Fri 3-4 @ CBIS 3209 | wangg6@rpi.edu Kathleen Mon 4-5 & Thurs 4-5 @ JEC 7045 | chens18@rpi.edu

  3. As a Sum of Impulses

  4. As Sum of Waves

  5. Fourier Series (Real Form)

  6. Fourier Series(Complex Form) Unit Period

  7. When Period Isn’t Unit

  8. Common Sense • Simple versus Complex Methods • Divide and Conquer Strategies

  9. Outline

  10. When Period Isn’t Unit

  11. Inserting Coefficients Right Hand Side: Inner products at infinitely many discrete frequency points u=n/T, and for a sufficiently large Tand all integer n the interval for u is dense on the whole number axis, and the distance between adjacent frequencies is infinitesimal Δu=1/T.

  12. Δu=1/T u u=n/T -T/2 T/2 Inner products at many discrete points u=n/T, and for a sufficiently large Tand all integer n the interval for u is dense on the whole axis, and the distance between adjacent frequencies is infinitesimal Δu=1/T.

  13. Forward & Inverse Transforms (since u=n/T) (since du=1/T)

  14. Rectangular/Gate Function

  15. Periodization

  16. As Period Gets Larger

  17. Fourier Transform Pair

  18. Example 1: Gate Function

  19. Sinc Function

  20. Example 2: Triangle Function

  21. Sinc2

  22. More Examples

  23. Basic Properties

  24. Linearity

  25. Shift

  26. Scaling

  27. Example

  28. Derivation

  29. Paired Combs

  30. Convolution Theorem

  31. Why?

  32. Why? • For a shift-invariant linear system, a sinusoidal input will only generate a sinusoidal output at the same frequency. Therefore, a convolution in the t-domain must be a multiplication in the Fourier domain. • The above invariability only holds for sinusoidal functions. Therefore, the convolution theorem exists only with the Fourier transform. • If you are interested, you could write a paper out of these comments.

  33. Parseval's Identity

  34. Why?

  35. 2D Fourier Transform

  36. Noise Suppression FT IFT

  37. Low-/High-pass Filtering

  38. Example: 2D Rectangle Function Rectangle of Sides X and Y, Centered at Origin

  39. Rotation Property

  40. Why?

  41. Homework for BB06 Read about the uncertainty property of Fourier transform, and write no more than three sentences to explain what it is. Analytically compute the Fourier transform of exp(bt)u(-t), where b is positive, u(t) is the step function (u(t)=1 for positive t and 0 otherwise). Due date: One week from now (by midnight next Friday). Please upload your report to MLS. https://www.youtube.com/watch?v=1hX_MUh8wfk

More Related