LINEAR SYSTEMS THEORY

Transcription

1 Fall Introduton to Mdal Engnrng INEAR SYSTEMS THEORY Ho Kung Km Ph.D. Shool of Mhanal Engnrng Puan Natonal Unvrt

2 Evn / odd / prod funton Thn about on & n funton! Evn f - = ; Odd f - = -; Can wrt an gnal a th um of an vn and an odd part: Prod f + X = d d d o o

3 In Cartan rprntaton In polar rprntaton v u Compl funton 3 v u artan u v Ral a Imagnar a u v whr modulu or ampltud argumnt or pha

4 Important gnal funton a Eponntal p a Compl ponntal or nuod A A o n A = ampltud = patal frqun = pha 4

5 5 Rtangular funton Stp funton for for for for for for u /

6 6 Trangular funton Normalzd Gauan for for n G

7 Sn funton n n Dra mpul for d d hftng A d A alng 7

8 nar tm 8 Modlng: th pro of fndng a mathmatal rlatonhp btwn nput & output gnal o nput gnal taton output gnal rpon.g. an amplfr wth gan A; o t A t t o nar tm f th uprpoton prnpl hold; o + o + A A A.g. amplfr wth gan A; Nonlnar tm;

9 Shft-nvarant tm Shft-nvarant tm f t proprt do not hang wth patal poton; X X o hft hft hft nvarant no hang SI tm = lnar & hft-nvarant tm Impul rpon h to a Dra mpul hft varant hang wth poton PSF Pont prad funton For an mpul gnal; Thn th rpon of an SI tm; d d onvoluton; o h o d h 9

10 Convoluton d d Produr mrrorng about = b hangng to tranlatng th mrrord b = multplng to th hftd & mrrord ntgratng th rultng gnal rprntd b ara rpatng th prvou tp for ah valu of

11 Convoluton for dgtal gnal 4 3 =

12 Convoluton for multdmnonal gnal; th onvoluton valu ar rprntd b volum. Proprt ommutatvt: aoatvt: dtrbutvt: d d

13 Rpon of an SI tm 3 A H H A d h A d h A d h o d h H d S d H S o For an nput gnal nuod; Th rpon of an SI tm; whr = Fourr tranform of th PSF h = tranfr funton or fltr Invr Fourr tranform; Thn w hav; Du; h o H S S o n doman v. n doman An nput gnal an b wrttn a an ntgral of wghtd nuod wth dffrnt patal frqun

14 Frqun Rall o n = Normalzd Ampltud = = a nra o do th frqun of th ollaton th hghr th hghr th gnal roluton that on an rprnt mallr gnal dtal gnal that var mor qul 4

15 Sgnal nth An prod gnal an b ratd b a ombnaton of wghtd and hftd nuod at dffrnt frqun. bo n /3 n3 um o A A A o S d d d n d bo n /3 n3 /5 n5 /7 n7 /9 n9 / n /3 n3 /5 n5 umf. bo n /3 n3 /5 n5 um

16 Fourr tranform 6 Forward tranform dr r r S r d S S r r r d r r S r d S S r r Invr tranform Conjugat varabl f r tm dmnon "ond" tmporal frqun wth dmnon "hrtz" f r patal poton wth dmnon "mm" patal frqun wth dmnon "mm - "

17 FT{rt} A A A d A An d A n A fnt gnal n th -doman rat an nfnt gnal n th -doman. th am tru v vra 7

18 FT{tp p} 8 ral part: magnar part: modulu: pha: 4 4 a a a a d d u u a a a 4 a a 4 a 4 a a artan

19 FT{Dra mpul} d In th ampltud ptrum all patal frqun ar prnt wth ampltud. 9

20 FT{on} Th ptrum ont of two mpul at patal frqun and. A prod funton ha a drt ptrum.. not all patal frqun ar prnt. An aprod funton ha a ontnuou ptrum. o o d d d d

21 Fourr tranform par Imag pa Fourr pa o n n n G n

22 Proprt nart: S S Salng: a a S a Tranlaton: S Convoluton: S S S S Parval' thorm: d S d Sparablt: n n n n

23 h H Tranfr funton and mpul rpon or PSF ar a FT par In magng th FT of th PSF nown a th optal tranfr funton OTF. th modulu of th OTF th modulaton tranfr funton MTF Th PSF mm and OTF mm - or lp/mm haratrz th roluton of th tm. 3

24 Not If th gnal drt but nfnt thn th frqun ptrum ontnuou but prod ha ala. If th gnal drt and fnt N ampl thn th frqun ptrum drt and prod n N. 4

25 FT n polar oordnat Forward tranform S r o r n r r dd r o r n rdrd r drd Not: J r o r n ro n r n r o r Invr tranform o n dd 5

26 Samplng Sampld gnal: n III omb funton or mpul tran III n = amplng dtan n nformaton ma b lot b amplng an w rovr a ontnuou gnal ompltl from t ampl? Samplng thorm Nqut rtron f th FT of a gvn gnal band-lmtd and f th amplng frqun largr than tw th ma. patal frqun prnt n th gnal thn th ampl unqul dfn th gnal S ma If ma thn n unqul dfn S III & III K lk wth l Hn KS S K S K S K S K S S Not that KS S K bau S K K 6

27 Infnt patal tnt t a band-lmtd FT 7

28 Fnt patal tnt t a not-band-lmtd FT If th gnal not band lmtd or f t band lmtd but / ma th hftd rpla of S wll ovrlap. Thrfor th ptrum of annot b rovrd b multplaton wth a rtangular pul. Known a alang and unavodabl f th orgnal gnal not band lmtd. Patnt alwa hav a lmtd patal tnt! 8

29 Alang: A ommonl obrvd phnomnon Tan from Dr. K. Mullr Sld 9

30 Ant-alang Tan from Dr. K. Mullr Sld 3

31 Drt FT 3 Forward tranform Invr tranform Fat Fourr tranform FFT for th numbr of ampl n a powr of two nlogn flop n D n logn flop n D N q M p N nq M mp q p n m S N n M m N nq M mp n m S q p