1. Leo Lam © 2010-2012 Signals and Systems EE235 Lecture 19

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

3. Visualize dot product Leo Lam © 2010-2012 3 In general, for d-dimensional a and b For signals f(t) and x(t) For signals f(t) and x(t) to be orthogonal from t 1 to t 2 For complex signals Fancy word: What does it mean physically?

4. Orthogonal signal (example) Leo Lam © 2010-2012 4 Are x(t) and y(t) orthogonal? Yes. Orthogonal over any timespan!

5. Orthogonal signal (example 2) Leo Lam © 2010-2012 5 Are a(t) and b(t) orthogonal in [0,2  ]? a(t)=cos(2t) and b(t)=cos(3t) Do it…(2 minutes)

6. Orthogonal signal (example 3) Leo Lam © 2010-2012 6 x(t) is some even function y(t) is some odd function Show a(t) and b(t) are orthogonal in [-1,1]? Need to show: Equivalently: We know the property of odd function: And then?

7. Orthogonal signal (example 3) Leo Lam © 2010-2012 7 x(t) is some even function y(t) is some odd function Show x(t) and y(t) are orthogonal in [-1,1]? Change in variable v=-t Then flip and negate: Same, QED 1

8. x 1 (t) t x 2 (t) t x 3 (t) t T T T T/2 x 1 (t)x 2 (t) t T x 2 (t)x 3 (t) t T Orthogonal signals Any special observation here? Leo Lam © 2010-2012

9. Summary Intro to Fourier Series/Transform Orthogonality Periodic signals are orthogonal=building blocks

10. Fourier Series Leo Lam © 2010-2012 10 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

11. Fourier Series: Orthogonality Leo Lam © 2010-2012 11 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

12. Fourier Series: Orthogonality in signals Leo Lam © 2010-2012 12 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)?

13. Fourier Series: Signal representation Leo Lam © 2010-2012 13 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

14. Fourier Series: Signal representation Leo Lam © 2010-2012 14 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!

15. Fourier Series: Parseval’s Theorem Leo Lam © 2010-2012 15 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

16. Fourier Series: Orthonormal basis Leo Lam © 2010-2012 16 x n (t) – orthonormal basis: –Trigonometric functions (sinusoids) –Exponentials –Wavelets, Walsh, Bessel, Legendre etc... Fourier Series functions