1 / 57

Chapter 4 The Fourier Transform

EE 207 Dr. Adil Balghonaim. Chapter 4 The Fourier Transform. Let x p (t) be a periodical wave, then expanding the periodical function. Rewriting x p (t) and X n. Fourier Transform Pairs. Fourier Transform Pairs. Fourier Transform Pairs. Finding the Fourier Transform.

britanni-hoover
Download Presentation

Chapter 4 The Fourier Transform

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. EE 207 Dr. Adil Balghonaim Chapter 4 The Fourier Transform

  2. Let xp(t) bea periodical wave, then expanding the periodical function Rewriting xp(t) and Xn

  3. Fourier Transform Pairs

  4. Fourier Transform Pairs

  5. Fourier Transform Pairs

  6. Finding the Fourier Transform

  7. Example 4-1 Find the Fourier Transform for the following function

  8. Example

  9. It was shown previously

  10. The Fourier Transform for the following function

  11. Properties of the Fourier Transform 1-Linearity Proof

  12. 2-Time-Scaling (compressing or expanding) Let Then Proof Change of variable

  13. Let

  14. Now Let Change of variable Since

  15. 3-Time-Shifting Proof

  16. Example Find the Fourier Transform of the pulse function Solution From previous Example

  17. 4-Time Transformation Proof

  18. 5-Duality ازدواجية

  19. Step 1from Known transform from the F.T Table Step 2

  20. 6- The convolution Theorem Convolution in Time Multiplication in Frequency Proof

  21. Now substitute x2(t-l) ( as the inverse Fourier Transform) in the convolution integral

  22. Exchanging the order of integration , we have

  23. The multiplication Theorem Proof Similar to the convolution theorem , left as an exercise Applying the multiplication Theorem

  24. Find the Fourier Transform of following Solution Since

  25. System Analysis with Fourier Transform

  26. 6-Frequency Shifting Proof

  27. Find the Fourier Transform of the function

  28. Since and Therefore

  29. 7-Differentiation

  30. Using integration by parts

  31. Since x(t) is absolutely integrable

  32. 7-Integration Example Find the Fourier Transform of the unit step function u(t)

More Related