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

2. Leo Lam © 2010-2011 Stanford The Stanford Linear Accelerator Center was known as SLAC, until the big earthquake, when it became known as SPLAC. SPLAC? Stanford Piecewise Linear Accelerator.

3. Leo Lam © 2010-2011 Today’s menu Today: Fourier Series –1 st topic “Orthogonality”

4. Fourier Series: Introduction Leo Lam © 2010-2011 4 Fourier Series/Transform: Build signals out of complex exponentials –Periodic signals –Extend to more general signals Why? –Convolution: hard –Multiplication: easy (frequency domain) Some signals are more easily handled in frequency domain

5. Fourier Series: Why Complex Exp? Leo Lam © 2010-2011 5 Complex exponentials are nice signals –Eigenfunctions to LTI –Convolution (in t)  Multiplication (in ) Frequency: directly related to sensory Harmonics: Orthogonality (later today) –Orthogonality simplifies math

6. The beauty of Fourier Series Leo Lam © 2010-2011 6 Recall: Write x(t) in terms of e st (Fourier/Laplace Transform) The input is a sum of weighted shifted impulses The output is a sum of weighted shifted impulses S Special input:

7. The beauty of Fourier Series Leo Lam © 2010-2011 7 Write x(t) in terms of e st (Fourier/Laplace Transform) Make life easier by approximation: Output: LTI Sum of weighted eigenfunctions Sum of scaled weighted eigenfunctions

8. Definition: Approximation error Leo Lam © 2010-2011 8 Approximating f(t) by cx(t): Choose c so f(t) is as close to cx(t) as possible Minimizing the error energy: Which gives: error Dot-product

9. Dot product: review Leo Lam © 2010-2011 9 Dot product between two vectors Vectors (and signals) are orthogonal if their dot product is zero. f x Angle between the two vectors

10. Vector orthogonality Leo Lam © 2010-2011 10 Vectors (and signals) are orthogonal if their dot product is zero. Dot product: length of x projected onto a unit vector f Orthogonal: cos()=0 Perpendicular vectors=no projection f x f x Key idea

11. Visualize dot product Leo Lam © 2010-2011 11 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

12. Visualize dot product Leo Lam © 2010-2011 12 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?

13. Orthogonal signal (example) Leo Lam © 2010-2011 13 Are x(t) and y(t) orthogonal? Yes. Orthogonal over any timespan!

14. Orthogonal signal (example 2) Leo Lam © 2010-2011 14 Are a(t) and b(t) orthogonal in [0,2  ]? a(t)=cos(2t) and b(t)=cos(3t) Do it…(2 minutes)

15. Orthogonal signal (example 3) Leo Lam © 2010-2011 15 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?

16. Orthogonal signal (example 3) Leo Lam © 2010-2011 16 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

17. 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 17 Orthogonal signals Any special observation here?

18. Leo Lam © 2010-2011 Summary Intro to Fourier Series/Transform Orthogonality Periodic signals are orthogonal=building blocks