Download presentation

Presentation is loading. Please wait.

Published byElena Moubray Modified over 2 years ago

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

2
Leo Lam © 2010-2011 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 © 2010-2011 4 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

5
Fourier Series: Orthogonality Leo Lam © 2010-2011 5 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 © 2010-2011 6 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 © 2010-2011 7 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 © 2010-2011 8 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 © 2010-2011 9 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 © 2010-2011 10 x n (t) – orthonormal basis: –Trigonometric functions (sinusoids) –Exponentials –Wavelets, Walsh, Bessel, Legendre etc... Fourier Series functions

11
Trigonometric Fourier Series Leo Lam © 2010-2011 11 Set of sinusoids: fundamental frequency 0 Note a change in index

12
Trigonometric Fourier Series Leo Lam © 2010-2011 12 Orthogonality check: for m,n>0

13
Trigonometric Fourier Series Leo Lam © 2010-2011 13 Similarly: Also true: prove it to yourself at home:

14
Trigonometric Fourier Series Leo Lam © 2010-2011 14 Find coefficients:

15
Trigonometric Fourier Series Leo Lam © 2010-2011 15 Similarly for:

16
Compact Trigonometric Fourier Series Leo Lam © 2010-2011 16 Compact Trigonometric: Instead of having both cos and sin: Recall: Expand and equate to the LHS

17
Compact Trigonometric to e st Leo Lam © 2010-2011 17 In compact trig. form: Remember goal: Approx. f(t) Sum of e st Re-writing: And finally:

18
Leo Lam © 2010-2011 Summary Fourier series Periodic signals into sum of exp.

Similar presentations

OK

Leo Lam © 2010-2012 Signals and Systems EE235 Lecture 21.

Leo Lam © 2010-2012 Signals and Systems EE235 Lecture 21.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on bond length definition Ppt on brain machine interface download Ppt on conservation of natural vegetation and wildlife Ppt on solar system in hindi Economics for kids ppt on batteries Ppt on indian equity market Ppt on food of different states of india Ppt on non biodegradable waste examples Ppt on 108 ambulance service Ppt on going places