CT Correlation (2A) Young Won Lim 9/9/14

Transcription

1 CT Correlation (2A)

2 Copyright (c) Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later ersion published by the Free Software Foundation; with no Inariant Sections, no Front-Coer Texts, and no Back-Coer Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. This document was produced by using OpenOffice and Octae.

3 Two types of improper integrals x (τ) y (τ) x(τ) y(τ+t) d τ y (τ+t ) x (τ) y (τ) x (τ t) x(τ t) y(τ) d τ t +t x (τ) y (t τ) d τ y (t τ) x (τ) x (t τ) y (τ) x (t τ) y(τ) d τ +t +t CT Correlation (2A) 3

4 Shift only s. Flip-Shift x (τ) y (τ) x (τ) y (τ) y (τ+t ) x (τ) y (τ) x (τ t) x (τ) y (τ) t +t y ( τ) x (τ) y (t τ) x (τ) x (t τ) y (τ) x ( τ) y (τ) +t +t CT Correlation (2A) 4

5 Correlation and Conolution Integrals Correlation x(τ) y (τ+t) d τ = x( τ t) y (τ) d τ Conolution x(τ) y (t τ) d τ = x(t τ) y (τ) d τ y (τ+t ) x (τ) y (τ) x (τ t) t +t y (t τ) x (τ t) x (τ) y (τ) +t +t CT Correlation (2A) 5

6 Conentional Notations Correlation x(τ) y (τ+t) d τ = x( τ t) y (τ) d τ Conolution x(τ) y (t τ) d τ = x(t τ) y (τ) d τ real x(t), y(t) R xy (τ) = x (t) y (t + τ) d t = x(t τ) y(t) d t = R yx (τ) x(t ) y (t ) = x (t τ) y (τ) d τ = x( τ) y(t τ) d τ = y(t) x(t) CT Correlation (2A) 6

7 Conentional Notations x (t) h(t ) y (t) = x(t) h(t) R xy (τ) = x(t) y(t ) real x(t), y(t) R xy (τ) = x(t) y(t ) = x(t) y (t + τ) d t = x (t τ) y (t) d t = R yx (τ) y (t) = x(t) h(t) = x (t τ)h(τ) d τ = x(τ)h(t τ) d τ = h(t) x(t) CT Correlation (2A) 7

8 Correlation and Conolution real x(t), y(t), h(t) Correlation + R xy (τ) = + x(t) y(t+ τ) d t = x(t τ) y (t) d t Conolution x (t) h(t ) = x (t τ)h(τ) d τ = x (τ)h(t τ) d τ x( t) X ( f ) R xy (τ) = x( τ) y(τ) R xy (τ) X ( f )Y ( f ) CT Correlation (2A) 8

9 Correlation of y(t) with δ(t) () x (t) = δ(t) y (t) = h(t) R xy (τ) = h(t) τ x (t) = δ(t ) y (t) = h(t ) R xy (τ) = h(t) τ x (t) y (t) R xy (τ)=? τ CT Correlation (2A) 9

10 Correlation of y(t) with x(t) (2) x (t) y (t) R xy (τ) + + τ x (t) y (t) R xy (τ) + + τ x (t) y (t) R xy (τ) τ CT Correlation (2A) 0

11 Conolution: delayed response of h(t) (3) Linear System δ(t ) h t y t δ(t ) h(t ) x () δ(t ) x () h(t ) x ()h(t ) input alue at time x() delayed impulse response x() h(t - ) x(t ) = x ()δ(t ) d y t = x ()h(t ) d CT Correlation (2A)

12 Correlation and Conolution real x(t), y(t), h(t) Correlation R xy (τ) = x (t) y (t +τ) d t = x(t τ) y (t ) d t y t = x ()h(t ) d = x(t )h() d R xy (τ) = = x(t τ) y (t ) d t x(t τ) [ x(t ξ)h(ξ) d ξ ] d t = h(ξ) [ x(t τ)x (t ξ) d t ] d ξ = h(ξ) R xx (τ ξ) d ξ = h(τ) R xx (τ) CT Correlation (2A) 2

13 Correlation and Conolution real x(t), y(t), h(t) Correlation + R xy (τ) = + x(t) y(t+ τ) d t = x(t τ) y (t) d t x(t ) = x ()δ(t ) d y t = x ()h(t ) d x (t τ) = x ()δ(t τ ) d y (t + τ) = x ()h(t +τ ) d R xy ( τ) = x(t) y(t + τ) d t + = + = [ x()δ(t ) d ] [ x(ξ) h(t +τ ξ) d ξ ] d t [ x() x(ξ)δ(t )h(t +τ ξ) d ξ d ] d t + = x() x(ξ) [ + = x( ) x(ξ) [ = x() x(ξ) h(+ τ ξ) d ξ d δ(t ) h(t + τ ξ) d t ] d ξ d δ(t ) h( +τ ξ) d t ] d ξ d CT Correlation (2A) 3

14 Correlation and Conolution R xy ( τ) = x(t) y(t + τ) d t = x()x(ξ) h(+ τ ξ) d ξ d = x()x(ξ) h( ξ+τ) d ξ d = R xx ( ξ) h( ξ+τ) d CT Correlation (2A) 4

15 Conolution: Commutatie Law x(t) h t y(t ) LTI x() Flip Shift h() h( ) h(t ) x()h(t ) d = y(t) h() x(t ) d = y (t) Flip Shift h() x() x( ) x(t ) h(t) x(t) y(t ) LTI CT Correlation (2A) 5

16 Conolution: delayed response of x(t) () Impulse response h(t) δ(t ) h t y t δ(t ) Impulse response x(t) x(t) y t h t delayed response by y(t ) =? CT Correlation (2A) 6

17 Conolution: delayed response of x(t) (2) Time Inariant h(t) x(t) h(t) Time Inariant x(t) + h(t) Time Inariant x(t) + CT Correlation (2A) 7

18 Conolution: delayed response of x(t) (3) Linear System δ(t ) x(t) y(t ) h(t) δ(t ) h() δ(t ) x(t ) h() x(t ) input alue at time h() delayed impulse response h() x(t - ) h() x(t ) h(t) h(t) = h()δ(t ) d y t = h()x (t ) d CT Correlation (2A) 8

19 FIR and IIR δ(t ) Impulse response h(t) h t y t Finite Impulse Response Finite Duration δ(t ) Impulse response x(t) x(t) y t Infinite Impulse Response Infinite Duration CT Correlation (2A) 9

20 CT Correlation (2A) 20

21 Impulse Response t x t Impulse response h t y t t 0.5 delayed response by 0.5 t delayed response by CT Correlation (2A) 2

22 LTI System x t y t =? t t.5 t t t t h t h t t h t t t.5 t t h t h t h t t.5 h t t x t = t t t t.5 t t y t = h t t h t t.5 h t t CT Correlation (2A) 22

23 Computing y(t ) : Output at t = t δ(t t + ) t t.5 t h() h(t t + ) t t x(t ) = δ(t t + ) + δ(t t +.5) + δ(t t ) δ(t t +.5) h (.5) h (t t +.5) t.5 t.5 y(t )=? δ(t t ) h(0) h (t t ) t t t CT Correlation (2A) 23

24 Computing y(t ) : delayed impulse response h(t) delayed impulse response h(t) x(t ) y(t) h () h(0) h(.5) t t t.5 t t x t = t t t t.5 t t y t = h h.5 h 0 x(t ) y(t) t t t.5 t t x t = t t t t.5 t t y t = h t t h t t.5 h t t CT Correlation (2A) 24

25 Computing y(t ) : flip and shift x(t) x t = t t t t.5 t t Change of ariables t x() = δ( t + ) + δ( t +.5) + δ( t ) Flip around y axis and then shift to the right by t t x(t ) = δ(t t + ) + δ(t t +.5) + δ(t t ) y t = x t h d = t t h d h t t impulse response delayed by t t t.5 h d h t t.5 impulse response delayed by t.5 t t h d h t t impulse response delayed by t y t = h t t h t t.5 h t t y t = h h.5 h 0 CT Correlation (2A) 25

26 Computing y(t ) : flip and shift h(t) h(t) h() Change of ariables t Flip around y axis and then shift to the right by t t h(t ) y t = x h t d = t h t d h t t impulse response delayed by t t.5 h t d h t t.5 impulse response delayed by t.5 t h t d h t t impulse response delayed by t y t = h t t h t t.5 h t t y t = h h.5 h 0 CT Correlation (2A) 26

27 Computing y(t ) : commutatiity () y t = x h t d = t h t d t.5 h t d t h t d h(t t + ) h(t t +.5) h(t t ) y t = x t h d = t t h d t t.5 h d t t h d Flip and Shift h(t) x() δ(t t +.5) h(t ) δ(t t + ) δ(t t ) h () h(.5) h(0) t t t.5 t t CT Correlation (2A) 27

28 Computing y(t ) : commutatiity (2) y t = x h t d = t h t d t.5 h t d t h t d h(t t + ) h(t t +.5) h(t t ) y t = x t h d = t t h d t t.5 h d t t h d Flip and shift input x(t) δ(t t + ) δ(t t +.5) δ(t t ) x(t ) h() h(0) h(.5) h ().5 CT Correlation (2A) 28

29 Computing y(t ) : commutatiity (3) δ( t ) δ( t +.5) δ( t + ) δ( t + ) δ( t +.5) δ( t ) Flip around y-axis t t +.5 t + t t.5 t Shift to the right by t δ( t ) δ( t +.5) δ( t + ) δ( +.5) δ( ) δ( + ) t t +.5 t +.5 δ(t t + ) δ(t t +.5) δ(t t ) x(t ) h() h(0) h (.5) h ().5 CT Correlation (2A) 29

30 CT Correlation (2A) 30

31 Computing y(), y(t) x 0 x(t) delayed impulse response at t= x 0 h x(t) delayed impulse response at t x(0)h(t ) h(t) t x.5 delayed impulse response at t= x.5 h.5 delayed impulse response at t x.5 h t.5 h(t.5) x t delayed impulse response at t= x h 0 delayed impulse response at t= x h t h(t ) t CT Correlation (2A) 3

32 Computing y() : shift & flip x(t) x 0 x(t) delayed impulse response at t= x 0 h x( ) h() y (t) x 0 h h t x.5 delayed impulse response at t= x.5 h.5 x.5 h.5 h t.5 x delayed impulse response at t= h t x h 0 x h 0 CT Correlation (2A) 32

33 Computing y(t) : shift & flip x(t) x( ) h() y () x(t ) h() y (t) x h x 0 h h x t x t t h t = x(0) h(t) t x h x.5 h.5 h x t x t t.5 h t.5 = x.5 h t.5 t x h x h 0 h x t x t t h t = x ()h(t ) t CT Correlation (2A) 33

34 Computing y(t) : the earliest and latest inputs x (t - ) x t the earliest input alue x (0) y (t) x 0 h t t x.5 h t.5 x t x t t the latest input alue x () x h t t CT Correlation (2A) 34

35 Computing y(t), y(s) x (t - ) y (t) x (s - ) y (s) h x t x t t h t = x 0 h t h x s x s s h s = x 0 h s t s h x t x t t.5 h t.5 = x.5 h t.5 h x s x s s.5 h s.5 = x.5 h s.5 t s h x t x t t h t x h t h x s x s s h s = x h s t s CT Correlation (2A) 35

36 Computing y(t) : earliest and latest inputs the latest input alue that affects the output alue y(t) x (t - ) y (t) the earliest input alue that affects the output alue y(t) x t h t y t = x t h d h this alue of h(t) is used first CT Correlation (2A) 36

37 CT Correlation (2A) 37

38 Computing y(t) : delayed impulse response x t y t =? t these inputs affects the output alue at t t delayed impulse response the earliest input alue the latest input alue that affects the output alue y(t) CT Correlation (2A) 38

39 Computing y(t) : multiplication sequence x x x t t t t h(t ) h x()h(t ) d x(t )h() d CT Correlation (2A) 39

40 References [] [2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003