1. Leo Lam © 2010-2012 Signals and Systems EE235 Lecture 21

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

3. So for d n Leo Lam © 2010-2012 3 We want to write periodic signals as a series: And d n : Need T and  0, the rest is mechanical

4. Harmonic Series Leo Lam © 2010-2012 4 Building periodic signals with complex exp. Obvious case: sums of sines and cosines 1.Find fundamental frequency 2.Expand sinusoids into complex exponentials (“CE’s”) 3.Write CEs in terms of n times the fundamental frequency 4.Read off c n or d n

5. Harmonic Series (example) Leo Lam © 2010-2012 5 Example with (t) (a “delta train”): Write it in an exponential series: Signal is periodic: only need to do one period The rest just repeats in time t T

6. Harmonic Series (example) Leo Lam © 2010-2012 6 One period: Turn it to: Fundamental frequency: Coefficients: t T * All basis function equally weighted and real! No phase shift! Complex conj.

7. Harmonic Series (example) Leo Lam © 2010-2012 7 From: To: Width between “spikes” is: t T Fourier spectra 0 1/T  Time domain Frequency domain

8. Exponential Fourier Series: formulas Leo Lam © 2010-2012 8 Analysis: Breaking signal down to building blocks: Synthesis: Creating signals from building blocks

9. Example: Shifted delta-train Leo Lam © 2010-2011 9 A shifted “delta-train” In this form: For one period: Find d n : time T0 T/2 *

10. Example: Shifted delta-train Leo Lam © 2010-2011 10 A shifted “delta-train” Find d n : time T0 T/2 Complex coefficient!

11. Example: Shifted delta-train Leo Lam © 2010-2011 11 A shifted “delta-train” Now as a series in exponentials: time T0 T/2 0 Same magnitude; add phase! Phase of Fourier spectra 

12. Example: Shifted delta-train Leo Lam © 2010-2011 12 A shifted “delta-train” Now as a series in exponentials: 0 Phase 0 1/T Magnitude (same as non-shifted)

13. Example: Sped up delta-train Leo Lam © 2010-2012 13 Sped-up by 2, what does it do? Fundamental frequency doubled d n remains the same So: where time T/2 0 m=1 2 3 T’ is the new period

14. Lazy ways: re-using Fourier Series Leo Lam © 2010-2012 14 Example: Time scaling (last example we did): Given that: and New signal: What are the new coefficients in terms of d k ? Use the Synthesis equation: k is the integer multiple of the fundamental frequency corresponding to coefficient d k.

15. Graphical: Time scaling: Fourier Series Leo Lam © 2010-2012 15 Example: Time scaling up (graphical) New signal based on f(t): Using the Synthesis equation: Fourier spectra 0 Twice as far apart as f(t)’s

16. Graphical: Time scaling: Fourier Series Leo Lam © 2010-2012 16 Spectra change (time-scaling up): f(t) g(t)=f(2t) Does it make intuitive sense? 0 1 0 1

17. Graphical: Time scaling: Fourier Series Leo Lam © 2010-2012 17 Spectra change (time scaling down): f(t) g(t)=f(t/2) 0 1 0 1

18. Fourier Series Table Leo Lam © 2010-2012 18 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:

19. Fourier Series: Quick exercise Leo Lam © 2010-2012 19 Given: Find its exponential Fourier Series: (Find the coefficients d n and  0 )

20. Fourier Series: Fun examples Leo Lam © 2010-2012 20 Rectified sinusoids Find its exponential Fourier Series: t 0 f(t) =|sin(t)| Expand as exp., combine, integrate

21. Fourier Series: Circuit Application Leo Lam © 2010-2012 21 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)!

22. Fourier Series: Circuit Application Leo Lam © 2010-2012 22 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

23. Fourier Series: Circuit Application Leo Lam © 2010-2012 23 Rectified sinusoids Now we know: LTI system: Transfer function: To frequency: +-+- sin(t) full wave rectifier y(t) f(t)

24. Fourier Series: Circuit Application Leo Lam © 2010-2012 24 Rectified sinusoids Now we know: LTI system: Transfer function: System response: +-+- sin(t) full wave rectifier y(t) f(t)

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