Download presentation

Presentation is loading. Please wait.

Published byTristan Croll Modified over 3 years ago

1
Leo Lam © 2010-2013 Signals and Systems EE235

2
Courtesy of Phillip Leo Lam © 2010-2013

3
Today’s menu Fourier Series

4
Leo Lam © 2010-2013 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 © 2010-2013 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 © 2010-2013 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 © 2010-2013 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 © 2010-2013 8 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 © 2010-2013 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 © 2010-2013 10 x n (t) – orthonormal basis: –Trigonometric functions (sinusoids) –Exponentials –Wavelets, Walsh, Bessel, Legendre etc... Fourier Series functions

11
Trigonometric Fourier Series Leo Lam © 2010-2013 11 Set of sinusoids: fundamental frequency 0 Note a change in index

12
Trigonometric Fourier Series Leo Lam © 2010-2013 12 Orthogonality check: for m,n>0

13
Trigonometric Fourier Series Leo Lam © 2010-2013 13 Similarly: Also true: prove it to yourself at home:

14
Trigonometric Fourier Series Leo Lam © 2010-2013 14 Find coefficients: The average value of f(t) over one period (DC offset!)

15
Trigonometric Fourier Series Leo Lam © 2010-2013 15 Similarly for:

16
Compact Trigonometric Fourier Series Leo Lam © 2010-2013 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 © 2010-2013 17 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 © 2010-2013 18 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 © 2010-2013 19 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 © 2010-2013 20 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 © 2010-2013 21 Example: Expand: Fundamental freq.

Similar presentations

OK

SUBTRACTING INTEGERS 1. CHANGE THE SUBTRACTION SIGN TO ADDITION

SUBTRACTING INTEGERS 1. CHANGE THE SUBTRACTION SIGN TO ADDITION

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on mars one astronauts Ppt on switching devices in ece Ppt on sri lanka history Ppt on tourist places in india Ppt on london stock exchange Ppt on microcontroller based digital thermometer Ppt on business environment nature concept and significance of dreams Ppt on trial and error supernatural Ppt on cross site scripting example Knowledge based view ppt on android