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

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: Fundamental frequency: –   =GCF(1,2,5)=1 or Re-writing: d n = 0 for all other n

6. Harmonic Series Leo Lam © 2010-2012 6 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

7. Harmonic Series Leo Lam © 2010-2012 7 Graphically: (zoomed out in time) One period: t 1 to t 2 All time

8. Harmonic Series (example) Leo Lam © 2010-2012 8 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

9. Harmonic Series (example) Leo Lam © 2010-2012 9 One period: Turn it to: Fundamental frequency: Coefficients: t T * All basis function equally weighted and real! No phase shift! Complex conj.

10. Harmonic Series (example) Leo Lam © 2010-2012 10 From: To: Width between “spikes” is: t T Fourier spectra 0 1/T  Time domain Frequency domain

11. Exponential Fourier Series: formulas Leo Lam © 2010-2012 11 Analysis: Breaking signal down to building blocks: Synthesis: Creating signals from building blocks

12. Example: Shifted delta-train Leo Lam © 2010-2011 12 A shifted “delta-train” In this form: For one period: Find d n : time T0 T/2 *