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

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

3. Leo Lam © 2010-2012 Today’s menu Fourier Series

4. Trigonometric Fourier Series Leo Lam © 2010-2012 4 Set of sinusoids: fundamental frequency  0 Note a change in index

5. Trigonometric Fourier Series Leo Lam © 2010-2012 5 Orthogonality check: for m,n>0

6. Trigonometric Fourier Series Leo Lam © 2010-2012 6 Similarly: Also true: prove it to yourself at home:

7. Trigonometric Fourier Series Leo Lam © 2010-2012 7 Find coefficients: The average value of f(t) over one period (DC offset!)

8. Trigonometric Fourier Series Leo Lam © 2010-2012 8 Similarly for:

9. Compact Trigonometric Fourier Series Leo Lam © 2010-2012 9 Compact Trigonometric: Instead of having both cos and sin: Recall: Expand and equate to the LHS

10. Compact Trigonometric to e st Leo Lam © 2010-2012 10 In compact trig. form: Remember goal: Approx. f(t)  Sum of e st Re-writing: And finally:

11. Compact Trigonometric to e st Leo Lam © 2010-2012 11 Most common form Fourier Series Orthonormal:, Coefficient relationship: d n is complex: Angle of d n : Angle of d -n :

12. So for d n Leo Lam © 2010-2012 12 We want to write periodic signals as a series: And d n : Need T and  0, the rest is mechanical

13. Harmonic Series Leo Lam © 2010-2012 13 Building periodic signals with complex exp. 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 c n or d n

14. Harmonic Series Leo Lam © 2010-2012 14 Example: Expand: Fundamental freq.

15. Harmonic Series Leo Lam © 2010-2012 15 Example: Fundamental frequency: –   =GCF(1,2,5)=1 or Re-writing: d n = 0 for all other n

16. Harmonic Series Leo Lam © 2010-2012 16 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

17. Harmonic Series Leo Lam © 2010-2012 17 Graphically: (zoomed out in time) One period: t 1 to t 2 All time