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

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

3. Compact Trigonometric Fourier Series Leo Lam © 2010-2011 3 Compact Trigonometric: Instead of having both cos and sin: Recall: Expand and equate to the LHS

4. Compact Trigonometric to e st Leo Lam © 2010-2011 4 In compact trig. form: Remember goal: Approx. f(t)  Sum of e st Re-writing: And finally: d n is complex!

5. Compact Trigonometric to e st Leo Lam © 2010-2011 5 Most common form Fourier Series Orthonormal:, Coefficient relationship: d n is complex: Angle of d n : Angle of d -n :

6. Harmonic Series Leo Lam © 2010-2011 6 Building periodic signals with complex exp. A “Harmonic Series” 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 d n

7. Harmonic Series Leo Lam © 2010-2011 7 Example: Expand: Fundamental freq.

8. Harmonic Series Leo Lam © 2010-2011 8 Example: Fundamental frequency: –   =GCD(1,2,5)=1 or Re-writing: d n = 0 for all other n

9. Harmonic Series Leo Lam © 2010-2011 9 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

10. Harmonic Series Leo Lam © 2010-2011 10 Graphically: (zoomed out in time) One period: t 1 to t 2 All time

11. Leo Lam © 2010-2011 Summary Fourier series Tomorrow: tons of examples (and some lazy ways to do things!)