1. Leo Lam © 2010-2012 Signals and Systems EE235 Lecture 23

2. Leo Lam © 2010-2012 Today’s menu Fourier Series Example Fourier Transform

3. Leo Lam © 2010-2012 Motivation

4. Fourier Series: Quick exercise Leo Lam © 2010-2012 4 Given: Find its exponential Fourier Series: (Find the coefficients d n and  0 )

5. Fourier Series: Fun examples Leo Lam © 2010-2012 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 © 2010-2012 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 © 2010-2012 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 © 2010-2012 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 © 2010-2012 9 Rectified sinusoids Now we know: LTI system: Transfer function: System response: +-+- sin(t) full wave rectifier y(t) f(t)

10. Leo Lam © 2010-2012 Summary Fourier Series circuit example

11. Fourier Series: Dirichlet Conditon Leo Lam © 2010-2012 11 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

12. End of Fourier Series Leo Lam © 2010-2012 12 We have accomplished: –Introduced signal orthogonality –Fourier Series derivation –Approx. periodic signals: –Fourier Series Properties Next: Fourier Transform

13. Fourier Transform: Introduction Leo Lam © 2010-2012 13 Fourier Series: Periodic Signal Fourier Transform: extends to all signals Recall time-scaling:

14. Fourier Transform: Leo Lam © 2010-2012 14 Recall time-scaling: 0 Fourier Spectra for T, Fourier Spectra for 2T,

15. Fourier Transform: Leo Lam © 2010-2012 15 Non-periodic signal: infinite period T 0 Fourier Spectra for T, Fourier Spectra for 2T,

16. Fourier Transform: Leo Lam © 2010-2012 16 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

17. Fourier Transform: Leo Lam © 2010-2012 17 Fourier Formulas compared: Fourier transform coefficients: Fourier transform (arbitrary signals) Fourier series (Periodic signals): Fourier series coefficients: and

18. Fourier Transform (example): Leo Lam © 2010-2012 18 Find the Fourier Transform of What does it look like? If a <0, blows up magnitude varies with  phase varies with 