FOURIER ANALYSIS Signals and System Analysis

Size: px

Start display at page:

Download "FOURIER ANALYSIS Signals and System Analysis"

Sylvia Sherman
5 years ago
Views:

1 FOURIER ANALYSIS

2 Isc Nwo

3 Whi ligh cosiss of sv compos

4 J Bpis Josph Fourir Bor: Mrch 768 i Auxrr, Bourgog, Frc Did: 6 My 83 i Pris, Frc

5 Fourir Sris A priodic sigl of priod T sisfis ft f for ll f f for ll

6 Fourir Alysis

7 Th siusoid Asi φ hs h sigl frqucy. Fourir provd: y priodic sigl ohr h siusoid will coi mor h o frqucy d i ddiio showd how hs frqucy compos c b clculd.

8 Priodic fucio f hs h Fourir sris b f si cos d f d f cos d f b si

9 Ths rsuls r sily provd mkig us of orhogoliy si m cos d for ll m d if m si m si d if m if m if m cos m cos d if m if m

10 f or cos b si f si cos b / -m vlu, d h frqucy of h h hrmoic d b mpliuds of h h hrmoic cosiusoidl d siusoidl compos wh wri A cos-φ.

11 b b A whr A A A b A A b cos si si cos cos si cos si cos ϕ ϕ ϕ ϕ

12 Th Fourir sris my lso b wri f A cos ϕ wih φ d / A.

13 f A cos ϕ Th h hrmoic hs mpliud A b / d phs φ - b /

14 For priodic fucio f wih priod T, cosidr T d d,, h Fourir sris si cos T b T f / / T T d f T / / cos T T d T f T / / si T T d T f T b

15 Fourir sris complx form Sic cos j -j / d si j - -j /j. j j j j j b f

16 j j j j j j j j j j j j j j j c jb jb jb jb jb jb jb j b f

17 f c j whr c / c - jb /. c - jb / c f j d

18 I rms of T f c j c T T f j d

19 Exmpl: Squr wv wih priod of < < lswhr f / / / / d d f si si cos cos / / / / d d f

20 ,4,6,...,3,5,7,... / si si / / d d f b... cos5 5 cos3 3 cos cos,3,5... / f

21 5 3 / squr

22 squr rms rms rms rms

23 %Dmo for Fourir Sris: A Squr Wv h-*pi:.5:4*pi; mpsizh; Mmp; squr.5.5*sigcosh; subplo5,,; ploh,squr; xis[ - *pi 4*pi -.5 ]; ylbl'squr'; N4; for k:m Fk.5; d; cou; for ::N coucou; f//pi*-^-/*cos.*h; for k:m FkFkfk; d; d; cou subplo5,,; ploh,f; xis[ - *pi 4*pi -.5 ]; ylbl' rms'; for k:m Fk.5; d; cou; for ::N coucou; f//pi*-^-/*cos.*h; for k:m FkFkfk; d; d; cou subplo5,,3; ploh,f; xis[ - *pi 4*pi -.5 ]; ylbl'8 rms'; N3; for k:m Fk.5; d; cou; for ::N coucou; f//pi*-^-/*cos.*h; for k:m FkFkfk; d; d; cou subplo5,,4; ploh,f; xis[ - *pi 4*pi -.5 ]; ylbl'6 rms'; %...

24 Fourir Trsform Cosidr sigl x of priod T d lows frqucy compo /T. Th Fourir sris i complx form is j c x T T j d x T c / / T T j j d x T x / /

25 Now s ->, T T->ifiiy -> ruig vribl, i h limi h bov givs d d x d x d x j j j j Dfi d x X j d X x j

26 X j x d j x X d x d X r Fourir rsform pirs. Th Fourir rsform c b rgrdd s spcil cs, s j, of h doubl sidd Lplc rsform..

27 F ourir A lysis

28 Ev d odd sigls: If w us h fc h h xpoil c b sprd io si d cosi rms i is sy o show h. If x is rl d v h X x cos d which will b v d rl i.. If x is rl d odd h X j x si d which will b odd d imgiry i.

29 Fourir Trsform Exmpl : x δ, fid is Fourir rsform: X j δ d δ d

30 Fourir Trsform Exmpl : x, fid is Fourir rsform: x is v fucio X cosd Usig formul X cosbd b cosd >

31 Ivrs Fourir Trsform Exmpl : δ X δ δ d d d X x j j

32 Ivrs Fourir Trsform Exmpl : [ ] δ δ X [ ] [ ] d d d d d d X x j j j j j j j j cos δ δ δ δ δ δ

33 x X δ δ cos δ δ - - si j δ - -j δ -

Mrcoi, Guglilmo, Mrchs 874-937, Ili lcricl gir d Nobl lur, h ivor of h firs prcicl rdio-siglig sysm. Bor i Bolog d ducd h Uivrsiy of Bolog.

34 Mrcoi, Guglilmo, Mrchs , Ili lcricl gir d Nobl lur, h ivor of h firs prcicl rdio-siglig sysm. Bor i Bolog d ducd h Uivrsiy of Bolog. 89 h bcm irsd i wirlss lgrphy, 895 h hd dvlopd pprus wih which h succdd i sdig sigls o poi fw kilomrs wy by ms of dirciol. H formd 897 Mrcoi's Wirlss Tlgrph Compy, Ld., i Lodo. I 899 h sblishd commuicio cross h Eglish Chl, 9 h commuicd sigls cross h Alic Oc bw Corwll, Egld, d Nwfoudld, Cd.

35 Ampliud Modulio A crrir high frqucis c c b rdid hrough spc x Acos c A low frqucy mssg sigl m y K Bcos m Muliplicio of hs sigls v xy gk Bcos m Acos c g - h gi of h muliplir

36 v gakcos c gab/cos c m gab/cos c m 3 frqucy compos hv b grd cos c cos c m cos c m sid frqucy compos c m c m c

37 For muli-frqucy mssg sigls: Sid bds roud crrir frqucy will b grd cos c cos c m, cos c m, cos c m, cos c m, sid bds sid bds c

38 AM crrir Voic AM sigl

39 omg_c; omg_m4; A; B; K.4; :.:; xa*cosomg_c.*; ybk*siomg_m.*; zx.*y; subplo3,,; plo,x; xis[ - ]; ylbl'crrir'; il'am'; subplo3,,; plo,y; ylbl'voic'; xis[ - ]; subplo3,,3; plo,z; ylbl'am sigl'; xlbl''; xis[ - ];

40 AM crrir Voic AM sigl

41 Rcificio Low-pss filrig Amplificio

42 Frqucy Modulio Crrir sigl x Acos c. Mssg sigl y Bcos m Sic c, if h crrir frqucy is o cos d whr rprss vryig crrir frqucy. I frqucy modulio is giv by givig c ky c ky d

43 Th frqucy moduld sigl v is hus giv by v Acos Acos c ky d Acos c kb m d cos [ kb / si ] Acos Acos c βsi m whr β kb/ m. c m m

44 Furhr lysis ivolv cosβsi d siβsi, Bssl fucios will b iroducd. Thr r ifii umbr of sidbds frqucis c ± m, whr is,, 3,

45 FM crrir Voic FM sigl

46 omg_c5; omg_m4; A; B; K3; :.:; Nlgh; xa*cosomg_c.*; yb*siomg_m.*; bk*b/omg_m; for k:n zka*cosomg_c * k b *siomg_m *k; d; subplo3,,; plo,x; xis[ - ]; ylbl'crrir'; il'fm'; subplo3,,; plo,y; ylbl'voic'; xis[ - ]; subplo3,,3; plo,z; ylbl'fm sigl'; xlbl''; xis[ - ];

47 Spcrum of FM sigl

EEE 303: Signals and Linear Systems

EEE 303: Signals and Linear Systems 33: Sigls d Lir Sysms Orhogoliy bw wo sigls L us pproim fucio f () by fucio () ovr irvl : f ( ) = c( ); h rror i pproimio is, () = f() c () h rgy of rror sigl ovr h irvl [, ] is, { }{ } = f () c () d =