1. Leo Lam © 2010-2013 Signals and Systems EE235

2. Courtesy of Phillip Leo Lam © 2010-2013

3. Today’s menu Fourier Series

4. Leo Lam © 2010-2013 4 Fourier Series/Transform: Build signals out of complex exponentials Established “orthogonality” x(t) to X(j  ) Oppenheim Ch. 3.1-3.5 Schaum’s Ch. 5

5. Fourier Series: Orthogonality Leo Lam © 2010-2013 5 Vectors as a sum of orthogonal unit vectors Signals as a sum of orthogonal unit signals How much of x and of y to add? x and y are orthonormal (orthogonal and normalized with unit of 1) x y a = 2x + y of x of y a

6. Fourier Series: Orthogonality in signals Leo Lam © 2010-2013 6 Signals as a sum of orthogonal unit signals For a signal f(t) from t 1 to t 2 Orthonormal set of signals x 1 (t), x 2 (t), x 3 (t) … x N (t) of Does it equal f(t)?

7. Fourier Series: Signal representation Leo Lam © 2010-2013 7 For a signal f(t) from t 1 to t 2 Orthonormal set of signals x 1 (t), x 2 (t), x 3 (t) … x N (t) Let Error: of

8. Fourier Series: Signal representation Leo Lam © 2010-2013 8 For a signal f(t) from t 1 to t 2 Error: Let {x n } be a complete orthonormal basis Then: Summation series is an approximation Depends on the completeness of basis Does it equal f(t)? of Kind of!

9. Fourier Series: Parseval’s Theorem Leo Lam © 2010-2013 9 Compare to Pythagoras Theorem Parseval’s Theorem Generally: c a b Energy of vector Energy of each of orthogonal basis vectors All x n are orthonormal vectors with energy = 1

10. Fourier Series: Orthonormal basis Leo Lam © 2010-2013 10 x n (t) – orthonormal basis: –Trigonometric functions (sinusoids) –Exponentials –Wavelets, Walsh, Bessel, Legendre etc... Fourier Series functions

11. Trigonometric Fourier Series Leo Lam © 2010-2013 11 Set of sinusoids: fundamental frequency  0 Note a change in index

12. Trigonometric Fourier Series Leo Lam © 2010-2013 12 Orthogonality check: for m,n>0

13. Trigonometric Fourier Series Leo Lam © 2010-2013 13 Similarly: Also true: prove it to yourself at home:

14. Trigonometric Fourier Series Leo Lam © 2010-2013 14 Find coefficients: The average value of f(t) over one period (DC offset!)

15. Trigonometric Fourier Series Leo Lam © 2010-2013 15 Similarly for:

16. Compact Trigonometric Fourier Series Leo Lam © 2010-2013 16 Compact Trigonometric: Instead of having both cos and sin: Recall: Expand and equate to the LHS

17. Compact Trigonometric to e st Leo Lam © 2010-2013 17 In compact trig. form: Remember goal: Approx. f(t)  Sum of e st Re-writing: And finally:

18. Compact Trigonometric to e st Leo Lam © 2010-2013 18 Most common form Fourier Series Orthonormal:, Coefficient relationship: d n is complex: Angle of d n : Angle of d -n :

19. So for d n Leo Lam © 2010-2013 19 We want to write periodic signals as a series: And d n : Need T and  0, the rest is mechanical

20. Harmonic Series Leo Lam © 2010-2013 20 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

21. Harmonic Series Leo Lam © 2010-2013 21 Example: Expand: Fundamental freq.