1. Leo Lam © 2010-2012 Signals and Systems EE235 Lecture 22

2. Leo Lam © 2010-2012 Today’s menu Fourier Series (periodic signals)

3. Leo Lam © 2010-2012 It’s here! Solve Given Solve

4. Reminder from last week Leo Lam © 2010-2012 4 We want to write periodic signals as a series: And d n : Need T and  0, the rest is mechanical

5. Harmonic Series Leo Lam © 2010-2012 5 Example (your turn): Write it in an exponential series: d 0 =-5, d 2 =d -2 =1, d 3 =1/2j, d -3 =-1/2j, d 4 =1

6. Harmonic Series Leo Lam © 2010-2012 6 Graphically: (zoomed out in time) One period: t 1 to t 2 All time

7. Harmonic Series (example) Leo Lam © 2010-2012 7 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

8. Harmonic Series (example) Leo Lam © 2010-2012 8 One period: Turn it to: Fundamental frequency: Coefficients: t T * All basis function equally weighted and real! No phase shift! Complex conj.

9. Harmonic Series (example) Leo Lam © 2010-2012 9 From: To: Width between “spikes” is: t T Fourier spectra 0 1/T  Time domain Frequency domain

10. Exponential Fourier Series: formulas Leo Lam © 2010-2012 10 Analysis: Breaking signal down to building blocks: Synthesis: Creating signals from building blocks

11. Example: Shifted delta-train Leo Lam © 2010-2011 11 A shifted “delta-train” In this form: For one period: Find d n : time T0 T/2 *

12. Example: Shifted delta-train Leo Lam © 2010-2011 12 A shifted “delta-train” Find d n : time T0 T/2 Complex coefficient!

13. Example: Shifted delta-train Leo Lam © 2010-2011 13 A shifted “delta-train” Now as a series in exponentials: time T0 T/2 0 Same magnitude; add phase! Phase of Fourier spectra 

14. Example: Shifted delta-train Leo Lam © 2010-2011 14 A shifted “delta-train” Now as a series in exponentials: 0 Phase 0 1/T Magnitude (same as non-shifted)

15. Example: Sped up delta-train Leo Lam © 2010-2011 15 Sped-up by 2, what does it do? Fundamental frequency doubled d n remains the same (why?) For one period: time T/2 0 m=1 2 3 Great news: we can be lazy!

16. Lazy ways: re-using Fourier Series Leo Lam © 2010-2011 16 Standard notation: “ ” means “a given periodic signal has Fourier series coefficients ” Given, find where is a new signal based on Addition, time-scaling, shift, reversal etc. Direct correlation: Look up table! Textbook Ch. 3.1 & everywhere online: http://saturn.ece.ndsu.nodak.edu/ecewiki/ima ges/3/3d/Ece343_Fourier_series.pdf http://saturn.ece.ndsu.nodak.edu/ecewiki/ima ges/3/3d/Ece343_Fourier_series.pdf