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Leo Lam © 2010-2013 Signals and Systems EE235. Courtesy of Phillip Leo Lam © 2010-2013.

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Presentation on theme: "Leo Lam © 2010-2013 Signals and Systems EE235. Courtesy of Phillip Leo Lam © 2010-2013."— Presentation transcript:

1 Leo Lam © Signals and Systems EE235

2 Courtesy of Phillip Leo Lam ©

3 Today’s menu Fourier Series

4 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 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 © 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: The average value of f(t) over one period (DC offset!)

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 Compact Trigonometric to e st Leo Lam © 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 © 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 © 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 © Example: Expand: Fundamental freq.


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