LINEAR SYSTEMS THEORY

Transcription

1 Introducton to Mdcl Engnrng INEAR SYSTEMS THEORY Ho Kyung Km, Ph.D. School of Mchncl Engnrng Pun Ntonl Unvrty Evn / odd / prodc functon Thn bout con & n functon! Evn f - ; Odd f - -; d d o d Cn wrt ny gnl th um of n vn nd n odd prt: Prodc f X o

2 Compl functon In Crtn rprntton, y u, y v, y Imgnry In polr rprntton, y, y φ, y v, y, y whr, y u, y v, y modulu or mpltud φ v, y φ, y rctn u, y rgumnt or ph u Rl 3 Importnt gnl functon Eponntl p Compl ponntl or nuod A φ A [ co φ n φ ] A mpltud ptl frquncy φ ph 4

3 5 Rctngulr functon wdth Stp functon or Hvd functon > < for for for for for for u > < / 6 Trngulr functon b Normlzd Gun < Λ for for σ µ σ n G

4 Snc functon n nc Drc mpul δ for δ d δ d hftng A δ d A clng 7 nr ytm nput gnl ctton o { } o output gnl rpon.g., n mplfr wth gn A; { t } A t t o Modlng: th proc of fndng mthmtcl rltonhp btwn nput & output gnl nr ytm f th uprpoton prncpl hold; { c c } c { } c { } o.g., mplfr wth gn A; { c c } o A c { } c { } c A c c c A Nonlnr ytm; { c c } c c c c 8

5 Shft-nvrnt ytm Shft-nvrnt ytm f t proprt do not chng wth ptl poton; { } X X o hft hft hft nvrnt no chng SI ytm lnr & hft-nvrnt ytm Impul rpon, h, to Drc mpul hft vrnt chng wth poton PSF Pont prd functon For n rbtrry gnl; Thn, th rpon of n SI ytm; ξ δ ξ dξ { } ξ { δ ξ } dξ convoluton; o ξ h ξ dξ o h 9 Convoluton ξ ξ dξ ξ ξ dξ Procdur mrrorng bout ξ by chngng ξ to ξ trnltng th mrrord by ξ multplyng to th hftd & mrrord ntgrtng th rultng gnl rprntd by r rptng th prvou tp for ch vlu of

6 Convoluton for dgtl gnl Convoluton for multdmnonl gnl;, y, y ξ, y ζ ξ, ζ dξdζ th convoluton vlu r rprntd by volum. Proprt commuttvty: octvty: 3 3 dtrbutvty: 3 3

7 Rpon of n SI ytm For n nput gnl nuod; A Th rpon of n SI ytm; o A A { } A ξ ξ h ξ dξ h ξ dξ ξ h ξ dξ H H whr H ξ h ξ d ξ Fourr trnform of th PSF h trnfr functon or fltr Invr Fourr trnform; Thn, w hv; o S d S H d FT { S H } Dcu; h n domn v. S S H n domn o FT { } FT FT { S H } o [ ] Any nput gnl cn b wrttn n ntgrl of wghtd nuod wth dffrnt ptl frqunc o 3 Frquncy Rcll co φ n φ Normlzd Ampltud ncr, o do th frquncy of th ocllton th hghr, th hghr th gnl roluton, tht, on cn rprnt mllr gnl dtl gnl tht vr mor qucly 4

8 Sgnl ynth Any prodc gnl cn b crtd by combnton of wghtd nd hftd nuod t dffrnt frqunc. bo n /3 n3 um o A A A co φ n φ φ φ S d d d Invr FT!!! d bo n /3 n3 /5 n5 /7 n7 /9 n9 / n /3 n3 /5 n5 umf. bo n /3 n3 /5 n5 um Fourr trnform Forwrd trnform S r { r } r I r dr Invr trnform { S } r I S d r S r r I r r r dr { r } r { S } r r r I S d r r r Conjugt vrbl f r tm dmnon "cond", tmporl frquncy wth dmnon "hrtz" f r ptl poton wth dmnon "mm", ptl frquncy wth dmnon "mm - " 6

9 FT{rct} 7 A fnt gnl n th -domn crt n nfnt gnl n th -domn. th m tru vc vr nc n A A A d A d A A I A A A / / FT{tp p} 8 rl prt: mgnry prt: modulu: ph: { } 4 4 d d u u I rctn

10 FT{Drc mpul} 9 { } d δ δ I FT{con} Th pctrum cont of two mpul t ptl frqunc nd. A prodc functon h dcrt pctrum.., not ll ptl frqunc r prnt. An prodc functon h contnuou pctrum. { } co co d d d d δ δ I

11 Fourr trnform pr Img pc Fourr pc δ δ co n δ δ δ δ Λ nc nc G σ n Proprt nrty: c c cs cs Sclng: S Trnlton: S modfyng only t ph pctrum Convoluton: S S S S Prvl' thorm: Sprblty: I d S d { nc nc y } I{ nc } I{ nc y }

12 h H Trnfr functon nd mpul rpon or PSF r n FT pr In mgng, th FT of th PSF nown th optcl trnfr functon OTF. th modulu of th OTF th modulton trnfr functon MTF Th PSF mm nd OTF mm - or lp/mm chrctrz th roluton of th ytm. 3 Not If th gnl dcrt but nfnt, thn th frquncy pctrum contnuou but prodc h l. If th gnl dcrt nd fnt N mpl, thn th frquncy pctrum dcrt nd prodc n N. 4

13 FT n polr coordnt Forwrd trnform S, y r coθ yr n θ, y r, θ r, θ y y ddy r coθ yr n θ rdrdθ r drdθ Not: r J r θ coθ n θ rco θ n θ r y y n θ r coθ r θ Invr trnform co φ y n φ, φ, y ddφ 5 Smplng Smpld gnl: n III comb functon or mpul trn III δ n n mplng dtnc nformton my b lot by mplng cn w rcovr contnuou gnl compltly from t mpl? Smplng thorm Nyqut crtron f th FT of gvn gnl bnd-lmtd nd f th mplng frquncy lrgr thn twc th m. ptl frquncy prnt n th gnl, thn th mpl unquly dfn th gnl S > m If > m thn n unquly dfn S I{ III } { } & I III K δ lk wth l Hnc, K S S K S K S K S K S S Not tht KS S K bcu S K K 6

14 Infnt ptl tnt t bnd-lmtd FT 7 Fnt ptl tnt t not-bnd-lmtd FT If th gnl S not bnd lmtd, or f t bnd lmtd but / m, th hftd rplc of S wll ovrlp. Thrfor, th pctrum of S cnnot b rcovrd by multplcton wth rctngulr pul. Known lng nd unvodbl f th orgnl gnl not bnd lmtd. Ptnt lwy hv lmtd ptl tnt! 8

15 Alng: A commonly obrvd phnomnon Tn from Dr. K. Mullr Sld 9 Ant-lng Tn from Dr. K. Mullr Sld 3

16 Dcrt FT 3 Forwrd trnform Invr trnform Ft Fourr trnform FFT for th numbr of mpl n powr of two nlogn flop n D n logn flop n D,, N q M p N nq M mp y y q p n m S,, N n M m N nq M mp y n m S y q p