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

2. Leo Lam © 2010-2011 Speed of light

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

4. Fourier Series: Circuit Application Leo Lam © 2010-2011 4 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)!

5. Fourier Series: Circuit Application Leo Lam © 2010-2011 5 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

6. Fourier Series: Circuit Application Leo Lam © 2010-2011 6 Rectified sinusoids Now we know: LTI system: Transfer function: To frequency: +-+- sin(t) full wave rectifier y(t) f(t)

7. Fourier Series: Circuit Application Leo Lam © 2010-2011 7 Rectified sinusoids Now we know: LTI system: Transfer function: System response: +-+- sin(t) full wave rectifier y(t) f(t)

8. Fourier Series: Dirichlet Conditon Leo Lam © 2010-2011 8 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

9. End of Fourier Series Leo Lam © 2010-2011 9 We have accomplished: –Introduced signal orthogonality –Fourier Series derivation –Approx. periodic signals: –Fourier Series Properties Next: Fourier Transform

10. Fourier Transform: Introduction Leo Lam © 2010-2011 10 Fourier Series: Periodic Signal Fourier Transform: extends to all signals Recall time-scaling:

11. Fourier Transform: Leo Lam © 2010-2011 11 Recall time-scaling: 0 Fourier Spectra for T, Fourier Spectra for 2T,

12. Fourier Transform: Leo Lam © 2010-2011 12 Non-periodic signal: infinite period T 0 Fourier Spectra for T, Fourier Spectra for 2T,

13. Fourier Transform: Leo Lam © 2010-2011 13 Fourier Formulas: 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

14. Fourier Transform: Leo Lam © 2010-2011 14 Fourier Formulas compared: Fourier transform coefficients: Fourier transform (arbitrary signals) Fourier series (Periodic signals): Fourier series coefficients: and

15. Fourier Transform (example): Leo Lam © 2010-2011 15 Find the Fourier Transform of What does it look like? If a <0, blows up magnitude varies with  phase varies with 

16. Fourier Transform (example): Leo Lam © 2010-2011 16 Fourier Transform of Real-time signals magnitude: even phase: odd magnitudephase

17. Fourier Transform (Symmetry): Leo Lam © 2010-2011 17 Real-time signals magnitude: even – why? magnitude Even magnitude Odd phase Useful for checking answers

18. Fourier Transform/Series (Symmetry): Leo Lam © 2010-2011 18 Works for Fourier Series, too! Fourier transform (arbitrary practical signal) Fourier series (periodic functions) Fourier coefficients Fourier transform coefficients magnitude: even & phase: odd

19. Fourier Transform (example): Leo Lam © 2010-2011 19 Fourier Transform of F(  ) is purely real F() for a=1

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