2. Leo Lam © 2010-2012 Signals and Systems EE235 Lecture 25

3. Leo Lam © 2010-2011 x squared equals 9 x squared plus 1 equals y Find value of y

4. Leo Lam © 2010-2012 Today’s menu Fourier Transform

5. Fourier Transform: Leo Lam © 2010-2012 4 Fourier Formulas: For any arbitrary practical signal And its “coefficients” (Fourier Transform):

6. Fourier Transform Properties Leo Lam © 2010-2012 5 A Fourier Transform “Pair”: f(t)  F() Re-usable! Scaling Additivity Convolution Time shift time domain Fourier transform

7. How to do Fourier Transform Leo Lam © 2010-2012 6 Three ways (or use a combination) to do it: –Solve integral –Use FT Properties (“Spiky signals”) –Use Fourier Transform table (for known signals)

8. FT Properties Example: Leo Lam © 2010-2012 7 Find FT for: We know the pair: So: -8 0 8  G()

9. More Transform Pairs: Leo Lam © 2010-2012 8 More pairs: time domain Fourier transform

10. Periodic signals: Transform from Series Leo Lam © 2010-2012 9 Integral does not converge for periodic f n s: We can get it from Fourier Series: How? Find x(t) if Using Inverse Fourier: So

11. Periodic signals: Transform from Series Leo Lam © 2010-2012 10 We see this pair: More generally, if X(w) has equally spaced impulses: Then: Fourier Series!!!

12. Periodic signals: Transform from Series Leo Lam © 2010-2012 11 If we know Series, we know Transform Then: Example: We know: We can write:

13. Leo Lam © 2010-2012 Summary Fourier Transform Pairs FT Properties

14. Duality of Fourier Transform Leo Lam © 2010-2011 13 Duality (very neat): Duality of the Fourier transform: If time domain signal f(t) has Fourier transform F(), then F(t) has Fourier transform 2 f(-) i.e. if: Then: Changed sign

15. Duality of Fourier Transform (Example) Leo Lam © 2010-2011 14 Using this pair: Find the FT of –Where T=5

16. Duality of Fourier Transform (Example) Leo Lam © 2010-2011 15 Using this pair: Find the FT of

17. Convolution/Multiplication Example Leo Lam © 2010-2011 16 Given f(t)=cos(t)e –t u(t) what is F()

18. More Fourier Transform Properties Leo Lam © 2010-2011 17 Duality Time-scaling Multiplication Differentiation Integration Conjugation time domain Fourier transform Dual of convolution 17

19. Fourier Transform Pairs (Recap) Leo Lam © 2010-2011 18 1 Review: