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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.

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Chapter 4 The Fourier Transform

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  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. ExampleFind the Fourier Transform for the following function

  8. Example

  9. It was shown previously

  10. The Fourier Transform for the following function

  11. ExampleFind the Fourier Transform for the delta function x(t) = d(t)

  12. Properties of the Fourier Transform 1-Linearity Proof

  13. 2-Time-Scaling (compressing or expanding) Let Then Proof Change of variable

  14. Let

  15. Now Let Change of variable Since

  16. 3-Time-Shifting Proof

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

  18. 4-Time Transformation Proof

  19. 5-Duality ازدواجية

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

  21. 6- The convolution Theorem Convolution in Time Multiplication in Frequency Proof

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

  23. Exchanging the order of integration , we have

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

  25. Find the Fourier Transform of following Solution Since

  26. System Analysis with Fourier Transform

  27. 6-Frequency Shifting Proof

  28. ExampleFind the Fourier Transform for

  29. Find the Fourier Transform of the function

  30. Method 1 Since and Therefore

  31. Method 2

  32. 7-Differentiation

  33. Using integration by parts

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