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

2. Leo Lam © 2010-2011 Exceptional

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

4. Example from yesterday (clarification) Leo Lam © 2010-2011 4 One period: Turn it to: Fundamental frequency: Coefficients: t T *

5. Exponential Fourier Series: formulas Leo Lam © 2010-2011 5 Analysis: Breaking signal down to building blocks: Synthesis: Creating signals from building blocks

6. Example: Shifted delta-train Leo Lam © 2010-2011 6 A shifted “delta-train” In this form: For one period: Find d n : time T0 T/2 *

7. Example: Shifted delta-train Leo Lam © 2010-2011 7 A shifted “delta-train” Find d n : time T0 T/2 Complex coefficient!

8. Example: Shifted delta-train Leo Lam © 2010-2011 8 A shifted “delta-train” Now as a series in exponentials: time T0 T/2 0 Same magnitude; add phase! Phase of Fourier spectra 

9. Example: Shifted delta-train Leo Lam © 2010-2011 9 A shifted “delta-train” Now as a series in exponentials: 0 Phase 0 1/T Magnitude (same as non-shifted)

10. Example: Sped up delta-train Leo Lam © 2010-2011 10 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!

11. Lazy ways: re-using Fourier Series Leo Lam © 2010-2011 11 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