1. Leo Lam © 2010-2011 Signals and Systems EE235 Leo Lam

2. Leo Lam © 2010-2011 Today’s menu Fourier Series (Exponential form)

3. Fourier Series Table Leo Lam © 2010-2011 3 Added constant only affects DC term Linear ops Time scale Same d k, scale  0 reverse Shift in time –t 0 Add linear phase term –jk   t 0 Fourier Series Properties:

4. Fourier Series: Quick exercise Leo Lam © 2010-2011 4 Given: Find its exponential Fourier Series: (Find the coefficients d n and  0 )

5. Fourier Series: Fun examples Leo Lam © 2010-2011 5 Rectified sinusoids Find its exponential Fourier Series: t 0 f(t) =|sin(t)| Expand as exp., combine, integrate

6. Fourier Series: Circuit Application Leo Lam © 2010-2011 6 Rectified sinusoids Now we know: Circuit is an LTI system: Find y(t) Remember: +-+- sin(t) full wave rectifier y(t) f(t) Where did this come from? S Find H(s)!

7. Fourier Series: Circuit Application Leo Lam © 2010-2011 7 Finding H(s) for the LTI system: e st is an eigenfunction, so Therefore: So: Shows how much an exponential gets amplified at different frequency s

8. Fourier Series: Circuit Application Leo Lam © 2010-2011 8 Rectified sinusoids Now we know: LTI system: Transfer function: To frequency: +-+- sin(t) full wave rectifier y(t) f(t)

9. Fourier Series: Circuit Application Leo Lam © 2010-2011 9 Rectified sinusoids Now we know: LTI system: Transfer function: System response: +-+- sin(t) full wave rectifier y(t) f(t)

10. Fourier Series: Dirichlet Conditon Leo Lam © 2010-2011 10 Condition for periodic signal f(t) to exist has exponential series: Weak Dirichlet: Strong Dirichlet (converging series): f(t) must have finite maxima, minima, and discontinuities in a period All physical periodic signals converge

11. End of Fourier Series Leo Lam © 2010-2011 11 We have accomplished: –Introduced signal orthogonality –Fourier Series derivation –Approx. periodic signals: –Fourier Series Properties Next: Fourier Transform

12. Fourier Transform: Introduction Leo Lam © 2010-2011 12 Fourier Series: Periodic Signal Fourier Transform: extends to all signals Recall time-scaling:

13. Fourier Transform: Leo Lam © 2010-2011 13 Recall time-scaling: 0 Fourier Spectra for T, Fourier Spectra for 2T,

14. Fourier Transform: Leo Lam © 2010-2011 14 Non-periodic signal: infinite period T 0 Fourier Spectra for T, Fourier Spectra for 2T,

15. Fourier Transform: Leo Lam © 2010-2011 15 Fourier Formulas: (Rigorous derivation in Ch.4) For any arbitrary practical signal And its “coefficients” (Fourier Transform): F(  ) is complex Rigorous math derivation in Ch. 4 (not required reading, but recommended) Time domain to Frequency domain

16. Fourier Transform: Leo Lam © 2010-2011 16 Fourier Formulas compared: Fourier transform coefficients: Fourier transform (arbitrary signals) Fourier series (Periodic signals): Fourier series coefficients: and

17. Fourier Transform (example): Leo Lam © 2010-2011 17 Find the Fourier Transform of What does it look like? If a <0, blows up magnitude varies with  phase varies with 

18. Fourier Transform (example): Leo Lam © 2010-2011 18 Fourier Transform of Real-time signals magnitude: even phase: odd magnitudephase

19. Fourier Transform (Symmetry): Leo Lam © 2010-2011 19 Real-time signals magnitude: even – why? magnitude Even magnitude Odd phase Useful for checking answers

20. Fourier Transform/Series (Symmetry): Leo Lam © 2010-2011 20 Works for Fourier Series, too! Fourier transform (arbitrary practical signal) Fourier series (periodic functions) Fourier coefficients Fourier transform coefficients magnitude: even & phase: odd

21. Fourier Transform (example): Leo Lam © 2010-2011 21 Fourier Transform of F(  ) is purely real F() for a=1

22. Fourier Transform (delta function): Leo Lam © 2010-2011 22 Fourier Transform of Standard Fourier Transform pair notation

23. Leo Lam © 2010-2011 Summary Fourier Transform intro Inverse etc.