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

2. Leo Lam © 2010-2012 Today’s menu Today: Fourier Series –“Orthogonality” –Fourier Series etc.

3. Visualize dot product Leo Lam © 2010-2012 3 Let a x be the x component of a Let a y be the y component of a Take dot product of a and b In general, for d-dimensional a and b x-axis a y-axis b

4. Visualize dot product Leo Lam © 2010-2012 4 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?

5. Orthogonal signal (example) Leo Lam © 2010-2012 5 Are x(t) and y(t) orthogonal? Yes. Orthogonal over any timespan!

6. Orthogonal signal (example 2) Leo Lam © 2010-2012 6 Are a(t) and b(t) orthogonal in [0,2  ]? a(t)=cos(2t) and b(t)=cos(3t) Do it…(2 minutes)

7. Orthogonal signal (example 3) Leo Lam © 2010-2012 7 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?

8. Orthogonal signal (example 3) Leo Lam © 2010-2012 8 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

9. 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

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