6.003: Signals and Systems

Transcription

1 6.003: Sigals ad Sysems Lecure 8 March 2, : Sigals ad Sysems Mid-erm Examiaio #1 Tomorrow, Wedesday, March 3, 7:30-9:30pm. No reciaios omorrow. Coverage: Represeaios of CT ad DT Sysems Lecures 1 7 Reciaios 1 8 Homewors 1 4 Homewor 4 will o colleced or graded. Soluios are posed. Closed boo: 1 page of oes ( iches; fro ad bac). Desiged as 1-hour exam; wo hours o complee. March 2, 2010 Muliple Represeaios of CT ad DT Sysems Verbal descripios: preserve he raioale. Represeig a by a sigle sigal. Differece/differeial equaios: mahemaically compac. y[] z 0 y[ 1] ẏ() x()s 0 y() Bloc diagrams: illusrae sigal flow pahs. Y A Y z 0 R s 0 Operaor represeaios: aalyze s as polyomials. Y 1 Y A 1 z 0 R 1 s 0 A Trasforms: represeig diff. equaios wih algebraic equaios. z 1 H(z) H(s) z z 0 s s 0 Resposes o arbirary sigals Chec Yourself Alhough we have focused o resposes o simple sigals (δ[],δ()) we are geerally ieresed i resposes o more complicaed sigals. Example: Fid y[3] How do we compue resposes o a more complicaed ipu sigals? Y No problem for differece equaios / bloc diagrams. use sep-by-sep aalysis. whe he ipu is R R oe of he above 1

2 6.003: Sigals ad Sysems Lecure 8 March 2, 2010 Superposiio Brea ipu io addiive pars ad sum he resposes o he pars. Lieariy A is liear if is respose o a weighed sum of ipus is equal o he weighed sum of is resposes o each of he ipus. y[] Give x 1 [] y 1 [] ad x 2 [] he is liear if y 2 [] Superposiio wors if he is liear. αx 1 []βx 2 [] is rue for all α ad β. αy 1 []βy 2 [] Superposiio Time-Ivariace Brea ipu io addiive pars ad sum he resposes o he pars. A is ime-ivaria if delayig he ipu o he simply delays he oupu by he same amou of ime. y[] Give y[] he is ime ivaria if is rue for all 0. x[ 0 ] y[ 0 ] Reposes o pars are easy o compue if is ime-ivaria. Superposiio Brea ipu io addiive pars ad sum he resposes o he pars. Srucure of Superposiio If a is liear ad ime-ivaria (LTI) he is oupu is he sum of weighed ad shifed ui-sample resposes. y[] δ[] h[] δ[ ] h[ ] x[]δ[ ] x[]h[ ] Superposiio is easy if he is liear ad ime-ivaria. x[]δ[ ] y[] x[]h[ ] 2

3 6.003: Sigals ad Sysems Lecure 8 March 2, 2010 Respose of a LTI o a arbirary ipu. y[] LTI y[] x[]h[ ] (x h)[] Noaio is represeed wih a aseris. x[]h[ ] (x h)[] I is cusomary (bu cofusig) o abbreviae his oaio: (x h)[] h[] This operaio is called covoluio. Noaio Do o be fooled by he cofusig oaio. Cofusig (bu coveioal) oaio: x[]h[ ] h[] h[] loos lie a operaio of samples; bu i is o! x[1] h[1] (x h)[1] Srucure of x[] h[2 ] y[2] x[]h[2 ] h[] h[2 ] operaes o sigals o samples. Uambiguous oaio: x[]h[ ] (x h)[] The symbols x ad h represe DT sigals. Covolvig x wih h geeraes a ew DT sigal x h. x[]h[2 ] Srucure of x[] h[3 ] x[]h[3 ] y[3] x[]h[3 ] h[] h[3 ] Chec Yourself Which plo shows he resul of he covoluio above? oe of he above 3

4 6.003: Sigals ad Sysems Lecure 8 March 2, 2010 CT Represeig a LTI by a sigle sigal. h[] y[] The same sor of reasoig applies o CT sigals. x() Ui-sample respose h[] is a complee descripio of a LTI. Give h[] oe ca compue he respose y[] o ay arbirary ipu sigal : y[] (x h)[] x[]h[ ] x() lim x(δ)p( Δ)Δ Δ 0 where p() 1 Δ As Δ 0, Δ τ, Δ dτ, ad p() δ(): x() x(τ)δ( τ)dτ Δ Srucure of Superposiio CT If a is liear ad ime-ivaria (LTI) he is oupu is he iegral of weighed ad shifed ui-impulse resposes. δ() h() δ( τ) h( τ) of CT sigals is aalogous o covoluio of DT sigals. DT: y[] (x h)[] x[]h[ ] CT: y() (x h)() x(τ)h( τ)dτ x(τ)δ( τ) x(τ )h( τ ) x() x(τ)δ( τ)dτ y() x(τ)h( τ)dτ Chec Yourself e u() e u() Which plo shows he resul of he covoluio above? is a impora compuaioal ool. Example: characerizig LTI s Deermie he ui-sample respose h[]. Calculae he oupu for a arbirary ipu usig covoluio: y[] (x h)[] x[]h[ ] oe of he above 4

5 6.003: Sigals ad Sysems Lecure 8 March 2, 2010 Applicaios of is a impora cocepual ool: i provides a impora ew way o hi abou he behaviors of s. Images from eve he bes microscopes are blurred. Example s: microscopes ad elescopes. Image plae CCD camera z Targe plae y x Ligh source Opical axis Figure by MIT OpeCourseWare. Blurrig ca be represeed by covolvig he image wih he opical poi-spread-fucio (3D impulse respose). Measurig he impulse respose of a microscope. Image diameer 6 imes arge diameer: arge impulse. arge image Blurrig is iversely relaed o he diameer of he les. Couresy of Ahoy Paire. Used wih permissio. Images a differe focal plaes ca be assembled o form a hreedimesioal impulse respose (poi-spread fucio). Blurrig alog he opical axis is beer visualized by resamplig he hree-dimesioal impulse respose. Couresy of Ahoy Paire. Used wih permissio. Couresy of Ahoy Paire. Used wih permissio. 5

6 6.003: Sigals ad Sysems Lecure 8 March 2, 2010 Blurrig is much greaer alog he opical axis ha i is across he opical axis. The poi-spread fucio (3D impulse respose) is a useful way o characerize a microscope. I provides a direc measure of blurrig, which is a impora figure of meri for opics. Couresy of Ahoy Paire. Used wih permissio. (1990-) Why build a space elescope? Telescope images are blurred by he elescope leses AND by amospheric urbulece. h a (x, y) h d (x, y) Y amospheric blurrig blur due o mirror size h (x, y) (h a h d )(x, y) groud-based elescope Y hp://hubblesie.org Telescope blur ca be respreseed by he covoluio of blur due o amospheric urbulece ad blur due o mirror size. The mai opical compoes of he are wo mirrors. h a (θ) h d (θ) h (θ) d 12cm θ θ θ h a (θ) h d (θ) h (θ) d 1m θ θ θ [arc-secods] hp://hubblesie.org 6

7 6.003: Sigals ad Sysems Lecure 8 March 2, 2010 The diameer of he primary mirror is 2.4 meers. Hubble s firs picures of disa sars (May 20, 1990) were more blurred ha expeced. expeced poi-spread fucio early Hubble image of disa sar hp://hubblesie.org hp://hubblesie.org The parabolic mirror was groud 4 μm oo fla! Correcive Opics Space Telescope Axial Replaceme (COSTAR): eyeglasses for Hubble! Hubble COSTAR hp://hubblesie.org Hubble images before ad afer COSTAR. Hubble images before ad afer COSTAR. before afer before afer hp://hubblesie.org hp://hubblesie.org 7

8 6.003: Sigals ad Sysems Lecure 8 March 2, 2010 Images from groud-based elescope ad Hubble. Impulse Respose: Summary The impulse respose is a complee descripio of a liear, imeivaria. Oe ca fid he oupu of such a by covolvig he ipu sigal wih he impulse respose. The impulse respose is a especially useful descripio of some ypes of s, e.g., opical s, where blurrig is a impora figure of meri. hp://hubblesie.org 8

9 MIT OpeCourseWare hp://ocw.mi.edu Sigals ad Sysems Sprig 2010 For iformaio abou ciig hese maerials or our Terms of Use, visi: hp://ocw.mi.edu/erms.