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

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Presentation on theme: "Leo Lam © 2010-2011 Signals and Systems EE235 Leo Lam."— Presentation transcript:

1 Leo Lam © Signals and Systems EE235 Leo Lam

2 Leo Lam © Today’s menu Orthogonality (last slide) Fourier Series

3 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 3 Orthogonal signals Any special observation here?

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

5 Fourier Series: Orthogonality Leo Lam © 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 © 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 © 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 © For a signal f(t) from t1 to t2 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 © 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 © x n (t) – orthonormal basis: –Trigonometric functions (sinusoids) –Exponentials –Wavelets, Walsh, Bessel, Legendre etc... Fourier Series functions

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

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

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

14 Trigonometric Fourier Series Leo Lam © Find coefficients:

15 Trigonometric Fourier Series Leo Lam © Similarly for:

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

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

18 Leo Lam © Summary Fourier series Periodic signals into sum of exp.


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