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

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

3. Harmonic Series Leo Lam © 2010-2011 3 Graphically: (zoomed out in time) One period: t 1 to t 2 All time

4. Harmonic Series (example) Leo Lam © 2010-2011 4 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

5. Harmonic Series (example) Leo Lam © 2010-2011 5 One period: Turn it to: Fundamental frequency: Coefficients: t T * All basis function equally weighted and real! No phase shift! Complex conj.

6. Harmonic Series (example) Leo Lam © 2010-2011 6 From: To: Width between “spikes” is: t T Fourier spectra 0 1/T  Time domain Frequency domain

7. Exponential Fourier Series: formulas Leo Lam © 2010-2011 7 Analysis: Breaking signal down to building blocks: Synthesis: Creating signals from building blocks

8. Example: Shifted delta-train Leo Lam © 2010-2011 8 A shifted “delta-train” In this form: For one period: Find d n : time T0 T/2

9. Example: Shifted delta-train Leo Lam © 2010-2011 9 A shifted “delta-train” Find d n : time T0 T/2 Complex coefficient!

10. Example: Shifted delta-train Leo Lam © 2010-2011 10 A shifted “delta-train” Now as a series in exponentials: time T0 T/2 0 Same magnitude; add phase! Phase of Fourier spectra

11. Example: Shifted delta-train Leo Lam © 2010-2011 11 A shifted “delta-train” Now as a series in exponentials: 0 Phase 0 1/T Magnitude (same as non-shifted)

12. Example: Sped up delta-train Leo Lam © 2010-2011 12 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!

13. Lazy ways: re-using Fourier Series Leo Lam © 2010-2011 13 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

14. Lazy ways: re-using Fourier Series Leo Lam © 2010-2011 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 © 2010-2011 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 © 2010-2011 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 © 2010-2011 17 Spectra change (time scaling down): f(t) g(t)=f(t/2) 0 1 0 1/2

18. Leo Lam © 2010-2011 Summary Fourier series Examples Fourier series properties