Mathematical Preliminaries for Transforms, Subbands, and Wavelets

Transcription

1 Mahmaical Prlimiaris for rasforms, Subbads, ad Wavls C.M. Liu Prcpual Sigal Procssig Lab Collg of Compur Scic Naioal Chiao-ug Uivrsiy hp:// Offic: EC538 (03)

2 Radig Assigm 2 Rviw /pp / You should kows his alrady opics: Do/ir produc Vcor spac Subspac Basis Orhogoal & orhoormal ss

3 3 Vcor Spac

4 4 Vcor Spac-- Exampl

5 5 Subspac ad Basis

6 6 Ir Produc

7 7 Ir Produc Exampls

8 8 Orhogoal ad Orhoormal Ss

9 9 Orhoormal Ss

10 10 Orhoormal Ss

11 11 5. Orhoormal Ss

12 Fourir Sris 12 Origially dvlopd i 1812 o sudy ha diffusio quaios Giv a priodic fucio f() w/ priod : f ( ) f ( + ), ± 1, ± 2,K rigoomric form: 2π f ( ) a + a cosω + b si ω, ω0 Eulr s idiy: i φ cosφ + i siφ, i 1. Expoial form: i 0 f ( ) c ω

13 Fourir Sris (2) 13 Basis: { i ω } 0 Ir produc: 1 ( ), ( ) 0 + f g f ( ) g( )* d, ( 0 0 0) + + iw imw 0 i im ω0 ω0 i ( m) ω0, 1 0 d 1 0 d

14 Fourir Sris (3) 14 L m: iω 1 0 iω0 ( ) 0, i ω d 1 0 L k m 0 iω 1 0 imω0, 0 1 ikω 0 ikω 0 d 1 ikω ( ik 2π 1) 0 0 ( ikω 1) 0 ik 2π cos( 2kπ ) + i si(2kπ ) 1, ω0 2π

15 Fourir Sris Advaags 15 f() (or f(x)) rprss h sigal as a fucio of im/spac A Fourir sris givs us a frqucy-basd rprsaio: { 2iω } flucuas wic as fas as { iω } { 4iω } flucuas wic as fas as { 2iω } {c } giv us a masur how much of ach of h diffr basis flucuaios ar prs i h sigal Rcall ha 1 cycl pr scod 1 Hz

16 Fourir rasform 16 Cosidr h priodic xsio of a o-priodic fucio f(): f ( ) f ( ), P > Fourir sris xpasio: 1 iω0 2 fp( ) c, ( ) c f P 2 1 iω 0 d

17 Fourir rasform (2) 17 L C(, ) c, Δω ω 0 Δ 2 2 ) ( ), ( i P d f C ω i P C f Δω ), ( ) ( o rcovr f() from f P () w l Δ Δ d f d f i P i P ω ω ω ) ( ) ( lim 2 2 0

18 Fourir rasform (3) 18 Fourir rasform Ivrs Fourir rasform

19 F Propris: Prsval s horm 19 Ergy prsrvaio: No: 1/2π facor is a rsul of Hz-o-radias covrsio

20 F Propris: Modulaio 20 Jusificaio:

21 F Propris: Covoluio horm 21 Covoluio horm

22 Liar Sysms 22 L L b a liar sysm wih ipu f() ad oupu g(): g() L[f()] Propris Homogiy L[f 1 () + f 2 ()] L[f 1 ()] + L[f 2 ()] Scalig L[α f()] α L[f()] oghr kows as suprposiio

23 Liar Sysms (2) 23 W ar irsd i im-ivaria sysms: L[f( - 0 )] g( - 0 ) Scalig & dlay L[cos(ω 0 )] α cos(ω 0 ( - d )) or [ ω ] 0 i ω0 ( d ) α L i Phas: φ ω 0 d

24 Liar Sysms (3) 24 Gai fucio: α(ω) Phas fucio: φ(ω) rasfr fucio Covoluio H ( ω) α( ω) iφ ( ω) G ( ω) H ( ω) F( ω)

25 25 Impuls Rspos

26 Impuls Rspos (2) 26 by suprposiio:

27 Impuls Rspos (3) 27 L 0 Dirac dla fucio

28 Impuls Rspos (4) 28 Impuls rspos fucio Assumig im-ivariac: [ h( )] H ( ω) F Dla fucio sifig propry:

29 Filrs 29 Filr: A liar sysm ha prmis crai frqucy compos o pass hrough whil auaig h rs Low-pass filrs Allows oly compos blow frqucy W Hz Idal low-pass filr: H ( ω) Similarly High pass filr Bad pass filr: iαω ω < 2πW 0 ohrwis Badwidh W

30 Filr Exampls 30 Idal low-pass Idal high-pass Ral low-pass Idal bad-pass

31 31 Idal Samplig: Frqucy Viw

32 Fourir sris xpasio Cofficis hrough Ir Producs Hc Idal Samplig: Frqucy Viw (2) 32 W i P c F ω ω 2 1 ) ( W i W W P F W c ω π π ω π ) ( 4 1 ω ω π ω d F W c W i 2 ) ( W f W c W i W W F W c ω π π ω π ) ( 4 1

33 Idal Samplig: Frqucy Viw (3) 33 Rcosruc f() from c.

34 34 Idal Samplig im Domai Viw

35 35 Idal Samplig im Domai Viw

36 36 Idal Samplig im Domai Viw

37 Discr Fourir rasform (DF) 37 So far, w cosidrd coiuous fucios Wha abou discr sigal sampls? Rcall h Fourir sris of priodic fucio f() wih priod d f c ik k 0 0 ) ( 1 ω Assum w sampl h fucio N im durig ach priod. N k i N k i N k N f d N f F π ω δ ) ( 1 0

38 DF (2) 38 akig 1 ad Dfiig h h priodic cofficis f f N F k N 1 0 f 2π k i N Sic k c i ω0 f ( ) k for N : f f ( N ) k c k 2π k i N

39 39 DF (3)

40 Rfrcs ad Homwork 40 Rfrcs Chaprs 2, 4, A.V. Opphim ad R.W. Schafr, Discr- im Sigal Procssig, Pric Hall, Ic., Eglwood Cliffs, Nw Jrsy, 2d Ediio, hp:// Chapr 5, Sv J. Lo, Liar Algbra wih Applicaios, Svh Ediio Homworks 2, 3, 4