1. Fourier Transform Comp344 Tutorial Kai Zhang

2. Outline Fourier Transform (FT) Properties Fourier Transform of regular signals Exercises

3. Popular forms of FT

4. Period and Frequency Imagine a rod spinning around the center, taking 2 seconds for one round Period T = 2 (second) Ordinary frequency f = 0.5 (Hz or times/second) Angular frequency ω = 1 (degree/second) Remember

5. FT Properties Property 1: time domain shifting (or delay) Proof

6. FT Properties Property 2: frequency domain shifting Proof

7. FT Properties Property 3: scaling Proof

8. FT Properties Property 4: time domain differentiation Proof Question: what about

9. FT Properties Property 5: Symmetry Proof:

10. Some common FT-pairs Impulse function

11. Some common FT-pairs Complex exponential function Using the frequency domain shifting property, and symmetry property

12. Some common FT-pairs Sine function By using the Euler formula and delay property

13. Some common FT-pairs Rectangular function

14. Some common FT-pairs Gaussian function

15. More Example Let f(t)-F(w) be a FT-pair. Now compute the FT of g(t) = f(t)cos(t). using Euler’s Formula and FT frequency shifting property

16. Exercise Compute the FT of the following signals u(t)cos( ω 0 t) u(t)sin( ω 0 t)e -at e -|a|t u(t)e -at u(t)te -at Here u(t) is the heavyside step function