BAB I SINYAL DAN SISTEM. Input Output Sistem Environment

Transkripsi

1 BAB I SIYAL DA SISTEM.. Dfiii Sim d didfiii bgi umul obj yg diuu mmbu ro dg uju ru. Sbgi modl mmi yg mghubug r iu d ouu, umumy dibu IO Sim, ri m dlm gmbr dibwh ii : Siyl Iu Iu Ouu Sim Evirom Siyl Ouu gmbr Modl mmi IO Sim Mu dri ligug virom im d lur dri im virom dibu iyl. Sim yg mghubug iyl iu oiu dg iyl ouu oiyu dibu Sim Wu Koiyu d yg mghubug dr iyl iu diri dg dr iyl ouu dibu Sim Wu Diri SWD, ri m d gmbr d gmbr 3 y=t y SWK gmbr Sim Wu Koiyu y SWD gmbr 3 Sim Wu Diri Alii im, ri h im rbu :

2 Sbil/id bil Kul/id ul T ubh wu/brubh wu Liir/o liir Si/dimi Drmiii/oi Tool Alii = modl mmi Prm liir Trformi = jmb=mmidh u w w yg li, dg :. rformi fourir. rformi llc 3. rformi 4. rformi Fourir Diri Bd r wu oiyu dg wu diri d w frui oiyu dg w frui diri.. Sif d Klifii im. Si d dimi

3 Sim i ji lur im hy rgug d mu d iu mmoryl, dg im dimi ji lur im mgig m llu wih mmory b. Liir Sim liir ji mmuhi rii uroii, ri m d gmbr 4 dibwh ii : SWK SWK y y SWK y + y gmbr 4 Prii uroii D homogi : + = y + y c. Prgr wu Sim ubh wu ivri ji ouu im id brubh bu wluu ouuy brgr juh rgr iu. - SWK SWK y y - 3 gmbr 5. Sim ubh wu Ji y dlh ouu dri d y dlh ouu dri d = - m y = y - Sim dibu LTW u LTI Liir Tim Ivri ji liir ubh wu 3

4 d. Kuli Sim LTW dibu ul/o iciory bil lur d wu = uu SWD hy brgug d hrg-hrg dri mu blumy d rg, jdi h = uu < Ro imul SWD h h Kul o Kul gmbr 6 Ro Imul SWD ul d o Kul Ro imul SWK h h Kul o Kul gmbr 7. Ro Imul SWK Kul d o Kul C : im yg d dirlii hru ul. Sbili Sim LTW dibu bil bl i mu rb mghil lur rb BIBO = Boudd Iu Boudd Ouu 4

5 Kodii yg dirlu d cuu KPC dlh : h uu SWD h d < uu SWK bil Tid bil gmbr 8 Ro Imul LTW uu SWD, Sbil d id bil Sbili im d jug dilih dri l ol dri fugi rfr im Uu SWK l ol di blh iri umbu imgir Uu SWD l ol di dlm ligr u Jdi uu ulih ASL ii im yg rlu dlh LTW/LTI Liir ubh Wu/Liir Tim Ivri, dg mu h im rbu ul d bil. Cooh : Dihui uu SWD yg mru rformi dr mu dg lur y dg hubug : 5

6 . y = - b. y = - + b+ Pri im di. Jwb.. y = - Liiri Ji iu m ouu y = - Ji iu m ouu y = - Ambil X = 3 = + M : Y = y 3 = [-] b. y = - + b+.. Oror d q Suu oror L ouu iu = Fugi Trfr im Uu SWK L = D Prm Fugi Trfr = umror D = domior Dim = oror mju = oror igrl c d Uu SWD Lq = q D q Dim q = oror mju q = oror ud q = - 6

7 q = + Uu mglii uu im m bu dulu modl mmi iu-ouuy Cooh v L i R C Ouu im = i Iu im = v Modl mmi im : L di d + i d C = v L d d R d i v C L = L R i v C ouu iu i v L R / C C C = C LC RC Prm Modl Mmi Sim.3. Difrc Equio Modl. SWK rm Difril d y d m d y d d... y y bm... b m d d d 7

8 8 Prm 3 Modl Prm difril = ord rm difril = ofii ybu = ofii D m b = ofii mbilg = ofii Umumy > m. Ambil = d/d m : m m m m m m b b b b D y L b b y Dim : or imor L b b b D m m m m Jdi : D y= Solui d du, yiu omlmr d riulr. Solui omlmr y c ji iu =, Jdi :... y D c Did : i i i c y A y A A y A y y... Dim : y = Y i = r-r olyomil D A i = o yg dihiug dri odii wl Kmugi r D dlh riil u oml d iml u jm. M : i. y = ri uu mu r riil yg brbd ii. y = r, r, r,..., m- r uu m buh r riil mbr

9 iii. y = α co β ; α i β uu r oml yg brbd ii dri α = α + jβ iv. y = α co β ; α i β = α co β ; α i β. Solui huu riculr ji iu im d, D y y L D Ku huu ji iu oil m ouu jug oil. Ambil = A M y = L, = ji iu iudoid m bu mjdi oil. Ig : j co ji Ji = A co = R A j M y = L, = j Ji = A i = Im A j M y = L, = j Did : Solui rm : olui omlmr + olui riculr y = y c + y 9

10 .4. Rlii Sim Wu Koiyu d Sim Wu Dir Syr im d dirlii ji ul. D dirlii dlm bu :. Sruur lgug y I b. Sruur lgug y II. Rlii uu Sim Wu Diri. Sruur I uu SWD-LTW ul y i b i i i i Ambil o = y y i b i i i i i i b... b y y b... Prm 4 Sruur I SWD-LTW q -... q - b B b o q - B o... q - gmbr 9 Sruur I SWD-LTW Kul b. Sruur II uu SWD io y i i b i i i

11 y b y b y b y y y o o q b b q b q q y o i o i o o q b q q q q b b q b q X q y D q q q L Prm 5 Sruur II SWD o i o i q q q L gmbr Sruur II SWD o i o i o i i i b b q y b q y q L gmbr Sruur II SWD Rgi ol digbug L q = L q.l q y y + q - q - q - y y + q - q - q -

12 -b o q - q - -b -b y - q - -b gmbr Rgi ol Cooh.3. SWD dg y+y-+5y-+7y-3=6+4- Bu rlii y I d y II im di Jwb :

13 .5. Sbilii Sim Liir SWK bil ji bgi riil r rm D dlh gif. Pd im di ol = -6 d = -4 dlh gif, m im bil im bil. Ji Y = + j bil ji < Bidg : Drh bil Im riil gmbr 3 Bgi gif D.5.. Prm Difrc uu SWD Bu umum im LTW y M i b i i i i Prm 6 Bu umum LTW i Ambil o = y y... y b... b M M Dg oror q : Dim : q - = - d q = + m : y + q q - = b + b q b M q -M Lq = y D q q 3

14 b bq -.. bm q - q - q - Jdi : D qy = q Solui : y = y c + y.5.. Solui omlmr ji dr mu = Dq y c = q. =, m D q = dg olui, y c = r riil d uggl, dim r = r olyomil D q dg olui :. r riil d uggl y = r. r riil d jm jumlh m buh 3. r oml i uggl 4. r oml d jm jumlh m buh.5.3. Solui riulr ji dr mu d Dq y = q y = q = Lq D q u huu ji iu oil, mbil = A Did : y = [Lq ] q Sbili im SWD, bil ji mliudo r oliomil Dq < u di dlm ligr u. Cooh :. Suu rm Y+3 8 y y+ 5 y = 8.5 Dg Y =, y = 3, y = 5.6. Klifii Siyl Siyl dlh buh fugi yg d dirri cr fii muu vribl yg mgdug iformi. Siyl d dilifii dlm : cum ri 4

15 . Coiou Tim d Diri Tim Sigl b. Alog d Digil Sigl c. Rl d Koml Sigl d. Drmiiic d Rdom Sigl. Ev d Odd Sigl f. Priodi d o riodi Sigl g. Ergy d Powr Sigl Adu Mcm-mcm iyl-iyl Dr dlh. Ui S Koiyu u u u,, gmbr 4 Ui oiu u. Ui diri u u 3 u,, gmbr 5 Ui diri u 3. Fugi gim Π,.5.5, liy

16 gmbr 6 Ui fugi gi m 4. Fugi giig Ʌ - gmbr 7 Ui fugi gi ig 5. Fugi ic Sic i c gmbr 8 Ui fugi ic i = uu = ±, ±,... = uu = 6. Fugi imul dl oiyu δ,, liy gmbr 9 imul dl oiyu 7. Fugi imul diri δ δ,, liy 6

17 Ori Pcrmi, Pl wu d rgr. Siyl oiyu f f b b f. iyl diri f b b f + gr iri, - gr = omri, / = i crmi Cooh :. Dihui iyl oiyu f f -.5 f,.5,, liy - 7

18 . f omri f -,.5 f,, liy.5 f,,-.5.5, -, =, liy. f f i 3 f , 6, -3 8

19 f 6 3, liy 3. f-3 gr 3 f f 3 \, - -3, , , -3 5, liy 4. f- f f, -,.5-, - +.5, -, liy 5. f-.5 f-.5 9

20 -.5 - f. 5, - -.5,.5-.5, , -4, liy 6. f3- = f--3 = f-+3_ bli, gr 3 f3-=f f. 5, - 3-, , , 3, liy.7. Kovri Siyl Wu Koiyu Siyl Wu Diri Siyl briu : Kovri iyl oiyu diri d digmbr ri gmbr bgi Wu oiyu mlig rorui Wu diri Pg oiyu Pg diri - - Frui oiyu Ω mlig rorui Frui diri gmbr Hubug Pg Koiyu d Pg Diri.8. Ro Imul Ro imul dlh ro u ouu im ji muy dibri fugi imul/ dl. δ SWD h δ SWK h Ro imul

21 gmbr Ro Imul Sim rig digmbr dg ro imuly r dg ro imuly d dilih h im rbu ul d bil SWD h y SWK h y gmbr Ro Imul. Ro Imul SWD Cooh : Dihui SWD-LTW y + y- + y y - = Ro imul im ji = M y = h Jdi h + h- + h h- = Kr ul m h = uu < M = h = = ; h- =, h - = d = h = h + h = = = h + h + h +...= = D Ig olui rbd y = y c + y Dq y = q

22 y c Dq y c = m D q = m y c = A r y y = L q dim q = uu = i iu di ii bu oil m y = cooh : y-,8y- +,5y- = ro imul h,8h- +,5h- = δ m = h,8h- +,5h- = δ = h = = h,8h +,5h- = δ = h =,8 = h,8h +,5h = δ = h =,8.,8,5 =,49 Solui y c Dim y m A r A r A r,8q,5q Jdi L q c,8q q,5q A,5 B, y 3 q q = y c = A + B = = y c =,5A +,3B =,8 q q,8q,5 q q,5 q,3 M y A +,6B =,6 A + B = -,9B =,6 B = -,5 d A =,5 h,5,5,5,3 u

23 Bgim ji iuy uroii? Kr im liir m ouuy jug uroii. Ro Imul SWK Dri r Difril b d y d d y dy... d y m d d d m d d b... b b m d d m Prm 7 Ro imul SWK Dlm oror L D y D y Ro imul y = h ji iu = δ Dim y = h = y + y y = h r y A c i i r = r dri oliomil D ji iu uroii m ouu uroii rii liiri dg mcri rlbih dhulu h uu u hrg δ jdi d y d d y dy... d d y dg odii wl =, li du ii dg d mudi diigri dri - higg + did 3

24 Cooh : d y d d d y dy d 4 3y 3 d d d 4 3 y 3 L y c A 3 B = y c = A + B = y c dy c d 3A B 3 -A = A = - ½ d B = ½ 3 h Iu Iu d d dy h 3 d 3 Jdi h h h 4

25 h,5 3,5 u.9. Kovolui Kovolui dlh uu ori rli ligu jumlh. Dlm w wu d digu uu md ro im rhd mu bb. Jdi mru rformi dri mu lur..9.. Pjumlh Kovolui Torm Ji dlh iu uu SWD-LTW d y ouu im b dim y = T [] m : y h h Prm 8 Kovolui Dim mu ori di didfiii bgi oror ovolui * higg h h h y * h = ro imul h y Sif : y = * h. Commuiv * h = h * 5

26 . Aoiiv * [y * ] = [ * y] * 3. Diribuiv uu ori jumlh * [y * ] = * y * 4. Mmilii lm idiy * = * = 5. Kovolui dri uu dr ul mlig rud dg * - = - Cooh : SWD- LTW h h,5 3 D ouu im y = * h gmbr 3 Kovolui d h.9.. Igrl Kovolui Prii d ify m dg SWD dim digi dg h y y * h h d h - Prm 9 Kovolui Igrl d 6

27 cr grfi : Gmbr d h - dim h - = h- - yiu h yg dicrmi mudi digr juh mudi di igrl Diii mru vribl igri, jdi u hrg mudi lu ori di dim hil igri mru lu bwh urv Cooh : L=H R=Ω y Hiug y Jwb h y h = - u h- h h y * d S < < τ τ 7

28 h h y * d r d. d BAB II AALISIS FOURIER PADA SIYAL WAKTU KOTIYU.. Pdhulu Alii Fourir, dimbil lh ilmuw fii Prci J Bi Fourir , mmch bhw hmir luruh fugi riodi d dirri bgi iuoidl d luruh frui yg ri. Krj Fourir lh mmbw dm yg y dlm ori im liir huuy dlm ori omuii. Bu hy mmbu i uu d mgri iyl riodi mllui 8

29 rum fruiy i jug d mgmbg ghu dr d jumlh iyl riodi mllui rformi Fourir. Koribui Fourir d diimul olh ry bhw iyl yg digu dlm girig d dirri dlm domi wu d domi frui, lbih dri iu rriy mmuyi hubug cr ui higg gubh rri iyl rbu dlm u domi mgubh rriy dlm domi yg li... Dr Fourir Uu Siyl Prodi Siyl di riodi ji mmuhi rm dibwh ii uu jumlh ili T, + T= Prm Dr fourir uu iyl riodi ili rcil T yg mmuhi Prm di dibu bgi riod iyl. Br f o =/T di frui dr iyl Hr d Ω o =f o dlh frui dr dlm rdi rdi. Siuoidl dg frui f o, dim dlh igr, di bgi hrmoi.... Bu Trigoomri d Dr Fourir Ji dlh fugi riodi dg riod T, hmir dlm i u d diri dg orm Fourir ri Prm dibwh ii : co b i o Prm Torm fourir o o Prm rbu dibu bgi dr fourir uu o i d b dibu ofii fourir. Prm. rbu muju bhw iyl riodi 9

30 X X X gmbr 4 Bbr iyl riodi yg rig digu d didi dg jumlh jumlh iyl iuoidl dg riod T=Ωo. Alii Fourir d iyl wu oiyu bruju uu mmidh iyl dri w ru w frui, yiu dg mggu Dr Fourir Wu Koiyu DFWK uu iyl riodi, d dg Trformi Fourir Wu Koiyu TFWK uu iyl id riodi riodi. Dr bgim Prm d jug diy dlm bu bgi briu : d co o Prm Dr o Hubug ofii d b rhd d d φ d diuju bgi briu : = d co υ b = d i υ Prm 3 ofii d b o Uu rlu rhiug ofii fourir digu if dr orogoli iuoidl. Bbr igrl ig yg d cr mudh dihil uu o d bilg bul d m m : o T i d.5 o Prm 4 ofii fourir o T co d.6 o Prm 5 ofii fourir I 3

31 o T i co m d.7 o Prm 6 ofii fourir II o o T ji m i o i mo d.8 T / ji m Prm 7 ofii fourir V o T ji m i o i mo d.9 T / ji m Prm 8 ofii V Ki d mggu idi di u mgvlui ofii fourir. Prm, i li du ii dri Prm dg d d mgigrl jg u riod, mil mi +. Dg mggu Prm 4 d Prm 5 m ii hilg, culi. Shigg i rolh : T d. T Prm 9 ofii V Bgiu ul gli du ii Prm dg co Ω d mgigrl jg u riod, i rolh hrmoi - : b T T T T co d Prm Hrmoi d b i d Prm rmoi d b..... Dr Fourir Wu Koiyu Bu Eoil 3

32 Bu oil dri dr Fourir didr d idi Eulr,yg d diuli bgi briu : j j co.3 Prm Idi ulr j j i.4 j Prm 3 Idi ulr Prm di i ubiui dlm Prm, i rolh : co b i j j b j j j jb j j.5 jb Prm 4 Idi ulr 3 Sljuy d lbih didrh dg ubiui dibwh ii : Shigg : c c c jb jb j c.6 Prm 5 Idi ulr 3 3

33 Dmii jug, dg mubiui igrl d b dri Prm d Prm, i rolh : c T T j d Prm 6 Trformi fourir.7 Prm 5 d Prm 6 di mdifiii dr fourir bu oil yg m Prm 6 mru rm lii uu mgli rm bu riodi dlm omo fourir. Sdg Prm 5 di bgi rm ii yg d digu mi bu glombg dri omo Fourir. Hubug r ofii fourir bu rigoomri dg bu oil diuju bgi briu : j jb d Prm 7 Kofii fourir c.8 Cooh. Ki rhi glombg bu rgi ri d cooh 3-, mily lbr ul iggi A d riod T, m : c T T j d T T T A j d T T. j d T T. j d A A T A j d A j T A j j j A A T j j 33

34 A i A i c T T Uu = dg L Hoil dirolh A T A T A c, higg : T j i c.9 Prm 8 Fugi L Hoil..3. Pgruh Simri Sbuh iyl riodi di fugi g ji mmuhi rm : = -. Prm 9 Fugi g d di fugi gjil ji : = --. Prm 3 Fugi gjil Prm 9 d Prm 3 di muju bhw fugi g dlh imri d umbu vril d =, d fugi gjil dlh iimri rhd umbu vril. Bbr cooh fugi g d gjil diuju d gmbr 5 Dg dmii bhw i Ω dlh gjil d co Ω dlh g uu luruh ili. Hil dri jumlh dri bbr fugi gjil dlh fugi gjil, d hil dri jumlh jumlh fugi g dlh fugi g. Shigg Dr Fourir uu fugi riodi gjil d brii hy iu d o. Sm hly, dr fourir uu fugi riodi g d brii hy coiu d o. Ko brili ol ji drh dibwh gh ilu oiif m dg drh dibwhgh ilu gif. Ji i mggu bu oil, ofii yg mucul dlh mjir uu fugi gjil d rl uu fugi g. b c 34

35 gmbr 5 Cooh fugi g d gjil fugi g b fugi gjil c fugi g Dri rg di d i hmi bhw, ji buh iyl riodi mmuyi imri gjil u g, m gigrl d dilu hy gh riod, olh r gh riod yg liy idi. Dg mgli hil igrl gh riod rbu dg, dirolh hil hir igrl lg. Sbuh fugi riodi di mmuyi imri gh glombg ji : T Prm 3 Simri gh glombg Ji imri ii d diviulii dg c bhw, ji bgi gh glombg gif dri glombg digr jg gh riod m mru crmi byg dri bgi gh-glombg oiif rhd umbu wu. Bbr bu glombg yg mmuyi imri gh glombg diuju dg gmbr 6 gmbr 6 Cooh bu glombg dg gh-glombg..4. Sif-if Dr Fourir Wu Koiyu 35

36 Dg mmhmi if-if dr fourir, m mmrmudh dlm mu dr fourir dri bbr fugi riodi. Uu iu i mmuli dg ori uru dfril d gigrl Difril d Igri Siyl Priodi Ji dr fourir uu iyl riodi dihui, m dfril u igrl iyl rbu d dihui ji uru u mgigrl i-i bgi dri dr rbu. Lbih jly, ji uru dr du ii dr fourir oml yg dibri rm.6 dirolh : d d j j c j. c. d d.3 Prm 3 Diffril d igri iyl riodi Dri Prm 3 d i lih bhw gruh difril dlh gli ofii fourir dg jω. Dg li, ji ofii dr fourir oml dibri olh jω c. Ii muju bhw gruh uru mi mgiud hrmoi yg lbih iggi. Dmii hly igri dlh bli uru. Pgruh igri mmbgi ofii bu oml d dr fourir olh jω. Shigg, igri mmuyi gruh gurg mgiud d hrmoi yg lbih iggi dri iyl rbu. Dlm igri ii hru dihiug jug o igri. Hl ii d dirolh cr drh, omo dc dri igri dirolh dri drh dibwh u ur uh fugi iu Pgruh Prgr wu d bu glombg Dibri buh iyl dg dr Fouriry dlh : j c Prm 33 Dr fourir..4 36

37 Mil i dfiii iyl liy y, yg mmuyi orli dg bgi briu : y = -τ.5 Prm 34 Dr fourir Siyl d y diuju dlm gmbr -6. Muju bhw rlmb rbu d dirolh dri rformi drh dg mgubh vribl mjdi -τ. Kouiy, dr fourir uu y dirolh dg ggi mjdi -τ dlm rm.4, higg dirolh : j c Prm 35 Dr fourir 3..6 Kofii Fourir uu fugi yg lh digr, Ĉ dirolh : Ĉ = c. -jωτ.7 Prm 36 Kofii fourir Ii muju bhw du ofii Fourir rbu dlh bilg oml dg mliud m, i rgum brbd : y τ gmbr 7. Prgr jg umbu wu Did lh i iggug bhw hmir i u buh fugi riodi d diri dlm bu dr fourir. Hl ii br uu hmir luruh 37

38 iyl yg digu dlm mlh girig, i rlu li mghui odii yg dirlu gr dr ovrg uu fugi yg dibri. Kodii ii di Dirichl Codiio. Ji buh fugi mmuhi odii ii m dr fourir dg ili rhigg ifii, dijmi ovrg uu luruh ili, dg gculi d ii dioiyu iyl. Kodii rbu dlh ri dibwh ii :. Fugi rbu d diigrl cr mul boluly igrbl jg bbr riod, higg : T d.8 Prm 37 boluly igrbl. Fugi hru mmuyi ili mim dm miim rb fii jg riod uh. 3. Fugi hru mmuyi jumlh dioiui rb jg riod uh. Dlm yy, d dibui bhw ji uu fugi mmuhi ig odii rbu, m dr Fourir Trucud mru d rbi uu iyl liy, dibdig dg bbr fugi hrmoi liy..3. Sim Dg Iu Priodi.3.. Ro Sdy-S Trhd Iu Siuoidl Dlm Bb I i lh mmbh olui riculr rm difril liir uu fugi iuoidl yg m lh dirolh g ui mggu ljbr oml. Sbgim hl ii, dr d dlh idi d didr d ggi oror dg jω, dim Ω dlh frui iuoidl dlm rdi rdi. Dibri buh im liir yg dijl olh rm difril dibwh ii, diuli dlm oi oror : y u D.9 Prm 38 Prm diffril 38

39 Dim D d dlh oliomil dlh oror difril. Ki miju u huu dim fugi iu u dlh iuoidl, dibri olh : A u co.3 Prm 39 Fugi iu iuiodl Mil dg idi Eulr i uli bgi briu : u j j A U U Prm 4 Idi ulr co.3 Dim : D U. 5A j.3 Prm 4 Idi ulr U. 5A.33 j Prm 4 Idi ulr C bhw cr umum U d U dlh bilg oml d ojugy U di bgi hor yg mrri iuoidl, yg m brii iformi mgi mgiud d h iuoidl rbu. Dlm bb rdhulu, omo dy- dirolh dg mggu uroii : y D D j j ju U.34 Prm 43 Komo dy j Ji rg i mdfiii : 39

40 y D Prm 44 Komo dy j j M.35 Dim M dlh modulu d bilg oml d dlh rgum, i d muli rm di : y Y j Y j Prm 45 Komo dy.36 Dim bilg oml j j M U MA Y.37 Prm 46 Komo dy D Y ojug oml Dg mubiui Y d Y dlm rm.36 di, i rolh olui dy- : y MAco.38 Prm 47 Komo dy 3 Cooh -3 : Rgi liri ri gmbr dibwh dg mu im bgi briu : co 3 8co Prm 48 Komo dy 4 Ki mghiug lur im dy- y Mohm Mohm µf µf gmbr 8 Rgi RLC Loo Rgi rbu ji y dlm oror : 4

41 3 P 3P y, L m y 3 P 3P Dlm u ii, iu rdiri ig omo, dg frui, 3, d 6 rdi. Olh r iu i mgvlui im dlm frui rbu d mudi mgli iliy cr bruru dg hor iu uu mmrolh hor ouu yg ui. P 3P 3 P j j974 P j P 3P j66658 P j P 3P Ahiry gg lur dy- dibri olh : v 6.69co co Prm 49 Tgg lur dy.4. Srum Siyl Priodi Pd umumy, fugi riodi rdiri fugi iuoidl dr d jumlh hrmoiy cr ori rhigg. Ji fugi brgug d rubh ili d i-i hrmoi. Ji i mggu bu oil uu dr Fourir, m c yg cr umum dlh bilg oml,mgdug iformi g mgiud d h i-i omo glombg. Ji c mgiud c dilo rhd Ω i mmrolh rum frui mgiud fugi riodi rbu. Dg jl yg m i d mmlo rum frui dri h c. Dlm ggu yg umum, rum dri mgiud di bgi rum frui, mugi r id brubh rhd rgr wu, dim rum dri h brubh rhd rgr jg umbu wu..5. Dy R-R Siyl Priodi Brdr Prvl d orm, dy rr iyl riodi dlh : 4

42 P T T d Prm 5 Dy r-r iyl riodi.4 Dim T dlh riod iyl. D diuju bhw ui P mrri dy r-r bry yg diirim dlm rior ohm ji dlh ru yg mgliriy u ggy. Sif orogoli r hrmoi dlm dr Fourir, bgim diuju dlm bu Prm d Prm, bhw : mmugi uu mghiug dy r-r dlm omo ofii Fourir hru mgrj igri ri yg dibri dlm rm.4. Hl ii d diuju dg mudh, mllui bu Prm d Prm, bhw : P T T d c Prm 5 Dy r-r iyl riodi.4 Dim c dlh ofii dr Fourir oml. Prm di jug mugi diri dlm bu ofii dr Fourir rigoomri, higg : P b Prm 5 Kofii dr fourir rigoomri Trformi Fourir Pd bh.4 lh dijl bhw crum frui d iyl riodi dlh dir d d dirolh dg dr Fourir. Dg d bu oil, dim ili rly dlh dir rhd frui, c = f dg ili igr. c j 4

43 c T j Srg i mmbh o rum frui dri iyl riodi. Hl ii g ig r hmr i i dihd d iyl yg id riodi. Uu rlu ii digu Trformi Fourir. Uu mmljri mlh ii i gu d gruh i riod d iyl. Dlm b ru, i riod mdi rhigg, i mmrol iyl riodi. Dg li buh iyl rodi d digg bgi iyl riodi dg rod rhigg. = uu T T Shigg d uu iyl riodi g cil. Kouiy, crum dir uu fugi rod digi dg crum oiyu uu fugi riodi. j d Prm 53 Igrl crum dicr.47 j d.48 Prm 54 Igrl crum oiou Dim crum frui dlh rformi Fourir dri d dlh ivr rformi Fourir d. Sbgim d dr Fourir, fugi hru mmuhi Diricl Codiio gr rformi Fourir d. Kodii ii mjmi bhw igrl yg didfiii dlm rm.38 ovrg. Kodii rbu dlh. hru boluly igrbl, yiu: d 43

44 .49 Prm 55 boluly igrbl. X hru mmuyi jumlh mim d miim rhigg dlm irvl wu rb 3. hru mmuyi jumlh dioiyu rhig dlm jumlh irvl d i-i dioiyu rhigg.6.. Bbr Pg Trformi Fourir Dlm bh ii i mmbri cooh mcrimcri rformi Fourir uu bbr fugi riodi, mudi diiui cri ivr rformi Fouriry..6.. Sif-Sif Trformi Fourir Dg mgl if-if rformi Fourir mbu i mgri hubug rri iyl d w wu d w frui. Sli iu jug mmbu i mmrolh rformi Fourir dg lbih mudh. Dlm hl ii i i bhw : X = F [] muju bhw X dlh rformi Fourir dri X = F - [X] dlh rformi Fourir dri X.6... Liiri Sif liiri diuju : X = F [] d X = F [] M F[ + ] = [X + X] 44

45 Prm 56 Sif liiri.6... Duli Dlm bh rdhulu, bhw rformi Fourir dri ul gim dlh fugi ic, d rformi fugi ic dlh ul gim. Dmii jug rformi uu o dlh imul d imul dlh uu o. Hubug duli rformi Fourir diuju bb : Ji X = F [] m F [X] = - Prm 57 Prii duli Turu D Igrl Dlm Kw Wu Srigli i rlu mu rformi Fourir uu fugi wu dg mmrolhy dri fugi wu yg li yg lh dihui rformi Fouriry. Prm briu d mmbu i uu rlu ii. Ji X = F [] m F [] = j D F X j X j Prm 58 Turu d igrl dlm w wu Pl Wu D Frui JiX F mf X Dim dlh o rl Prm 59 Pl wu d frui Prgr Wu Thorm : JiX F j mf X Prm 6 Prgr wu 45

46 Prgr Frui Thorm : jo mx o F JiX F Prm 6 Prgr frui Turu D Igrl Dlm Kw Frui Thorm : dx JiX F d.6.3. Kovolui m : F j Prm 6 Turu d igrl dlm w frui Kovolui Dlm Kw Wu Torm : Mil: X = F [], H= F[h, d ji Y = F [y] dim, h, d y dihubug olh igrl ovolui bb : y h d Y H X Prm 63 Kovolui dlm w wu Kovolui Dlm Kw Frui Torm: Mil : X = F [] d Y = F [y] m : F y X u y u du Prm 64 Kovolui dlm w frui.6.4. Trformi Fourir Pd Fugi Priodi 46

47 Trformi Fourir Pd Suu Ko Pd cooh.8 lh diuju bhw ivr rformi Fourir uu fugi imul ui dlh u o / lh r iu, rformi Fourir uu o dlh A Trformi Fourir Pd Siuoidl Trformi Fourir uu iuoidl d diu dri ry bu oil oml bgi briu: co o i o j o j o j jo jo Prm 65 Eoil oml Kouiy, uu mgvlui rformi fugi rbu, rm li i hru mmrolhrformi fourir d fugi oil. Ki d mgrjy dg mcri ivr rformi fourir uu fugi imul -. j o d jo jo o F F co F i o o] j o o 47

48 Hil ii mmbr i uu d li uu mjl moduli. Prli iyl moduli dg mbw iuoidl m dg ovolui dlm w frui. Tbl Pg Trformi Fourir X X Krg j o. A-o A Imul d =o. A A Fugi 3. A A Hy br uu > j 4. A A j Hy br uu > A 5. A j A j 6. A co o j Ao 7. A i o j A 8. A j Hy br uu > Hy br uu > Hy br uu > Hy br uu > 9. A j A o. A co o A o o. A i o ja o o 48

49 T. A P T AT i c Pul gi m dg iggi A, -T/ <<T/ AW W 3. i c AP W Fugi ic Tbl Pg Trformi Fourir Tbl. T. Liri F[+ = X +. Duli F[X]=- 3. Difril w wu F 4. Igri w wu F[/ ] = /j Xj+X 5. Pl wu u frui F = X X 6. ovolui w wu F h d X H F y X u y u du 7. ovolui w frui jo 8. rgr wu F o X jo 9. rgr frui X o F. Difril frui F dx j d. Igri frui F j X u du 49

50 .7. Ergi Siyl Kdug rgi iyl didfiii bgi : E d Prm 66 Ergi igl Igrl di mrri rgi ol yg didiii dlm hmb u ohm dg ru yg mlluiy u gg. Hy dy dri iyl rodi yg d dirolh dri ofii dr fourir, rg i muju bhw rgi iyl riodi d dirolh dri rformi fourir. Dg mggu ivr rformi fourir, i d muli rm bgi briu : E j dd j X dd X X d Prm 67 Ivr rformi fourir Dim dlh fugi wu y, X -=X*. Shigg i rolh E d = [ X ] d Prm ii mrri bu yg dihil dri orm Prvl uu fugi wu y riodi dg rgi rb Fugi [X] dihui bgi rgy crl diy fucio d iyl yg muju diribui rgi ol d iyl rbu. 5

51 BAB III AALISIS FOURIER UTUK SIYAL WAKTU DISKRIT Dlm bb i lh mmljri bgim iyl oiyu wu brulg d digmbr bgi buh uroii iuoid dlm bu buh fourir dr. Dlm bb ii i mgulgi mbli wu diri dlm ori yg diljri dlm bb blumy. Dlm bgi briuy i mmuli dg gmbr dr fourir dri iyl wu diri oiyu. Ii diiui dg buh gji dri Trformi fourir wu diri TFWD, d rulg rum frui d iyl-iyl wu diri riodi. 3.. Dr Fourir Uu Siyl-Siyl Wu Diri Brulg Dlm bb, buh iyl SWD didfiii riodi ji bbr igr oif, iyl rbu mmuhi rm : X= + Prm 68 Dr fourir iyl wu diri ili rcil higg r. 3. b dim riod iyl. Jug lh diuju dlm bb hy uu u iyl wu rm brulg yg lig drh brbu : X = co / 5

52 X = b i / Ki mdfiii frui udu iyl-iyl brulg wu diri bg : = / Dg u rdi. Uu iyl-iyl oiyu frui udu dilmbg mmuyi u rdi r di. Frui udu id d diuhi olh mbrg ili. Ti uu fugi brulg oiyu wu id d b ili uu. Jdi d diimul : Co + = co Dim dlh mbrg igr. Ii muju bhw rm iuoid yg idi dihil uu frui yg diih d jr. Sbgi oui dri hil ii dlh bhw hy ili-ili rb dri hrmoihrmoi bb iyl rbu. Dlm u iyl wu diri i bolh mggu bu rigoomri : o= / Brdr co o mmuyi hrmoi-hrmoi bb: co o = co = co o co [+]o = co o co [+]o = co o co [+]o = co o Prm 69 ili hrmoi Hubug yg ru dg mudh dimbg uu hrmoi-hrmoi dri bu i [+]o 5

53 3.. Pry Dr Fourir Didr d hubug : j jo j Dim dlh mbrg igr uu yrgm oi, m ui u r dioi bb : W j jo Su iyl oml wu diri yg brulg dg riod dibri olh rm : w j mu dri iyl iyl ii mmuyi frui yg mru li dri frui dr W d dim hrmoi. Lbih lju fruifrui ii mmuhi hubug orhogoli : i w w w w i Dim i mbrg igr. Ki jbr mbrg iyl brulg wu diri dg riod dlm bu dr Fourir, bb : w,,, 53

54 Dim, w,,,,...,. Yg lig ig uu diig bhw dr Fourir iyl wu diri dlh buh jumlh ili-ili yg jumlhy rb, ri yg diuju d r. 3.3, dim ii brbd li dr Fourir uu iyl-iyl oiyu wu yg rdiri dri ili-ili yg jumlhy rb. Trd omo dc d omo oo frui yg bry /. Hrmoi-hrmoi frui yg bry /, dim =,3,4,-. Pig jug dic bhw hy omoomo dri dr yg hru divlui Pmbui Prm Dr Fourir Prm 3.3 d 3.4 dlh mru ry dr Fourir uu wu diri. Cooh 3. Cri ofii Fourir wu diri glombg o ri dirlih dlm gb 3.4 uu = Ro Tu Trhd Mu Priodi Ro u dri im wu diri bil rhd mu riodi d diu cr yg m ri cr-cr yg digu uu im wu oiyu. Cooh 3. Tu ouu u dy dri im, bil im mmuyi rm im d rm mu riodi dg riod. Prm im diri : Y+-.7y++.y=u++u Prm 7 Sim diri j Prm mu riodi dibri olh : U= 54

55 3.5. Trformi Fourir Wu Diri Di bb ii i mmbh r dr Fourir uu iyl-iyl wu diri id riodi. Ii dili dg cr yg digu uu myli u wu oiyu. Tiju yg mru dr Fouriryg mru iyl wu diri yg brili ol uu rg <=<=. Sljuy dlh iyl rb wu. Ki d myuu buh iyl riodi dg riod >. Siyliyl ii diuju dlm gmbr 3.6. Pggu cr yg m ri bb 3., dr Fourir uu d diuli bgi briu : Dim, jo o jo Kr dlh ol uu >, i d my rm 3.9 dlm bu ri di bwh ii : o jo Prm 7 Dr fourir iyl wu diri Ahiry r dlh id ol uu <=<= m : jo Yg rlu diig bhw o 55

56 Mrilh rg i mguji gruh yg diimbul dg mbh. Dri dfiii, mdi uu ili yg mig ru. Ki mguji gruh yg diimbul dg mbh lim ~ Bil muju ili higg, m o. uu mu crum oiu i mu dulu rm limu dri crum diri rm b dlh : X j Prm 7 Prm limu ru diri dim mggi o d dibri olh rm : X Srg i d muli rm dlm bu X uu mghil : ~ j o X o. Prm 73 Dr fourir iyl wu diri Bil dri rm 3.3 diubiui i d muli Prm 73 mjdi ~ j o o X o Prm 74 Dr fourir iyl wu diri Kr, o uu ~ ili rb dri. Kouiy dlh mih dir omo-omo frui mgubh o mjdi d d r 3.37 mjdi jo X d Prm 3.34 d 3.38 mdfiii Trformi Fourir Wu Diri TFWD d Trformi Fourir Wu Diri IvrTFWDI 56

57 Cooh 3.4 Cri rm rformi Fourir wu diri X dri iyl ul o briu ii:, uu, uuygl i 3. Fourir Wu Diri Siyl-Siyl Priodi Sri hly u uu wu oiyu, mu oil mglmi rgr frui, yg m uu mu iyl yg brbu iuoid diubh dulu dlm bu oil. Eoil oml d diy dlm bu : A jo Prm 75 Eoil oml Trformi fouriry dibri olh : X o Prm 76 Trformi fourir Srg i mcri TFWD uu fugi riodi coiu d iu. Uu coiu i mmuyi rm : Aco o A jo jo Prm 77 Fugi riodi iu Dg mmf hil yg did d Prm 76 d Prm 77 i md : X o o Prm 78 Fugi riodi iu dlh Dg cr yg m i d muju bhw TFWD dri fugi iu A Ai o j jo jo

58 58 Adlh o o j X Sif-Sif TFWD Bbr if TFWD dlh. Priodi u brulg X X b. Liri X X c. Prgr wu d frui o X m X o m X o j o j d. Pl wu d frui / X m X. Difrii d jumlh j m X X m f. Difrii dlm frui d dx j g. Torm rvl d X h. Kovolui Ji X H my h y i. Kovolui riodi d ovolui irulr

59 y m m ~ m ~ m R m 3.4 Trformi Fourir Diri Miu ori dri rformi Fourir wu diri g brgu uu mmhmi ro ymlig d u by r li, i id d mm X dlm mmori omur r di dlh buh fugi yg oiyu. U imlmi dri omur digil i hru mdiri frui. Hl ii mmbuuh buh o dri rformi Fourir Diri TFD. TFD mgmbil r yg cuu lu dlm dui golh iyl digil. Trformi Fourir Diri mmilii rm bb : X K X X j /,,,,3.... X. Prm 79 Trformi fourir iyl wu diri Ivr dri TFD dlh X o X w,,,,...,. Prm 8 Ivr rformi fourir iyl wu diri Prbd r TFWD d TFD dlh bgi briu : Ki mmuyi iyl rb wu i bu mru iyl riodi dg duri. Ki md buh crum frui oiyu uu dr iyl ii,, dg mmbu mdi ili brhigg d mggg bhw riod dr iyl ii dlh rb. Dg y ii, TFWD mmroy mjdi buh iyl fugi oiyu dri frui, bliy dg TFD i mggg iyl ii, dlh riodi dg riod, dim dlh rb. Hil dri TFD dlh fugi diri dri frui r mu omo frui dlh igrl rli dri frui dr o = /. Sljuy lih dlh g miri dg dr Fourir diri. Kouiy 59

60 crum diri dg ili rl d rli igrl dri frui dr o. Prm briu mdfiii rformi Fourir diri d rformi Fourir diri ivr X o X w,,,,..., prm-rm ii i dirogrm d comur. Lbih lju bil i mmbdig du rm di, i mmuyi dr lgorim yg bi u mghiug rformi briuy. Ag gli riil yg dibuuh uu omui TFD dlh Bbr Sif Dri TFD Ji F[]=X, =,,,3,, M :. Lir. rgr wu 3. rgr frui 4. duli X X m m X w m w X m 5. ovolui 6. rli i i mod y i X Y 6

61 7. orm rvl y i mod Y i i X 3.6 Pr TFD Slh u ry dlh uu ym idifii. Ki lh mmbh g rmlh idifii ym di bb ri uu r dovolui d diuju dlm ub bb. bhw org d mu dr ro imul dri ym liir ji mu d lur d y. Sorg jug bi mggu lgorim FFT uu md TFD X d Y. TFD H dri dr ro imul mudi cr lgug d dihui mllui rm 3.74 H =Y/X Prm 8 Dr ro imul Ji mgigi, org d mu dr ro imul h dg mjumlh TFD ivr dri H. Dlm rjl ljuy lgorim FF rig digu uu mu modl ym dri d iu ouuy. Suu hru bgimu rlu dic bhwim yg did ljuy d cr igifi digruhi olh hdir oi dlm d iu-ouu. Hil ii biy d diig dg mfilr d blum mggu lgorim FFT. 6

62 BAB IV TRASFORMASI LAPLACE Alii uu SWK: Kr dy dl lii dg TFWK r li : Kuli igri d TFWK Pd lii ri rgi llu dihd dg bilg oml +j, dg TFWK hy brj dlm drh Jdi rformi Llc ri hly TFWK mrubh iyl w wu mjdi w frui frui oml 4.. Trformi Llc Bilrl TLB diuru dri TFWK : X. j X d j d Prm 8 Trformi llc bilrl Bil fugi X dim fugi oil mru for ovrgi, m m TFWK dri y dlh: Y Y Y X.. j j Jdi dri r di X j d d j. j d Dfiii vribl frui oml Bil = +j d=j.d d = d/j 6

63 M X. d X d dibu g TLB Bidg dlh : oi : X = [] = - [X] c: rformi Llc Kovrgi TLB : Trigri cr mul ji iliy urg dri higg 63

64 4.. Trformi Llc Su Sii Dfiii : Dibri uu iyl ul, m S j. d TLSS = j. d Kovrgi TLSS ji Lim Sif-Sif Trformi Llc Su Sii..... Liiri. b.. u c. d. Prgr Wu. X. Prgr Frui. X Frui Pl wu d. X d. X X d f. X d X d Diffrii Wu d X g. d h. X d Igri Wu d X. Prli dg d. d X d u du Pmbgi rhd 64

65 Kovolui Wu. i.. d X X c j j. c j j. X u X u. du Kovolui frui 4.4. Pg Tl D Gmbr Di Bidg S. Fugi Eoil Kul A.. u A jω Bidg S α X Pol σ X b. Fugi Ui S A A A. d A u jω A σ U d. c. Fugi Coiu Kul X=A.Coβ..u jω A jβ ro -jβ σ X j.j,5. A j,5. A j A. A d. Fugi Siu Kul X=A.Siβ..u jω A jβ -jβ σ 65

66 X A. j A.. j -j,5. A/ j j,5. A/ j j A.. Fugi Coiu Trdm Kul X=A. -α..coβ..u jω jβ -α -jβ σ X= [A. -α..coβ..u] {Gu : if rgr frui} f. Fugi Siu Trdm Kul X=A. -α..siβ..u jω jβ -α -jβ σ X= [A. -α..siβ..u] Gu if c. M gi dg + α dim Jdi X A. A. A. Si.. u g. Fugi Ui Rm Kul X=.u jω ol σ 66

67 X.!. d d lih. bl.i gri.. Ambil =, m X =/ Au dri if : [.] = -dx/d Dim X = [ u ] = / Jdi h. Fugi Imul d d. u δ δ-τ τ.. d d. Sif Prgr Wu Fugi = A.. X X Sif Prgr Frui. A.. A/ A. M. A... A d d A Sif Prgr Wu A 4.5. Ivr Trformi Llc Uu mgmbli rl iyl fugi wu X X 67

68 X j j X j d D dili lw dfiii di u mlih g TLSSy A.δ-τ A.u A. -.u A.Coβ.u A.Siβ.u A. -α.coβ..u A. -α.siβ..u A. A.u. -.u A A. A. A.. A.! Lbih mudh dili dg cr yg rhir dg mlih Bu olyomil X = /D lih r D Pg rformiy Bu X = /D dlm i ril 4.6. Solui Dg Pyui Kofii Fugi riol X

69 Ei Pril Uu Ar DS Siml Pol A A A A D X A A A A X M X A 4.8. Ar DS, Mulil Pol-Siml r r A A A A A X dim. r r r X A r r r X d d A. r r r X d d A.!... r r r i X d d A! Ei Pril: DS Koml Kojug Siml Pol X

70 4.. Mod Grfi Uu mgvlui ofii ril dri X dg cr mggmbr vor digrm mu ol-or im. Dihui : X D K... m... ili dri X di = X K Prli Prli Jr Jr Lgug Lgug Si Si Zro Pol Evlui ol dri X A. X X KA K Prli Prli Jr Jr Lgug Lgug Si Si Zro Pol 4.. Torm ili Awl D Ahir Digu uu mmudh mcri uu d wl = d d hir = dri uu fugi wu mllui uu frui. Torm ili wl, ji 7

71 M X X Lim. X Torm ili hir, ji X X M lim lim. X 4.. Alii TLSS Alii yg ig dri TLSS. Solui rm diffril b. Mcri ro imul im c. Mcri ro lg rgi li rgi d. Alii SWK 4.3. Solui Prm Difril Sif diffrii d. X X d Bu umum d. X. X d d d.... d d 4.4. Ro Imul Sim dy d d 3. y d. dg y d 7

72 4.5. Solui Lg Rgi RLC i. Rior R V = R.i V=R.I I=G.V i=g.v ii. Iduor L di V L V L.. l L. i L d l V i L. L L V d i i L L iii. Kior C di Vc V L V d C. C. V. C. V. l c i d V V c C 4.6. Alii SWK Dibri SWK LTW X h Y. y y = b b m. m 7

73 Ro im y = *h Trformi Llc [ ]. Y = [b + b b m. m ]X Fugi Trfr Sim H Y X h m b b... bm... - H Ro dy Y y H.X - H. X Sbili SWK SWK bil ji d hy ji : i. Sbil dlm ri BIBO ii. Ro imul rigri cr mul H D iii. Lim h = iv. Ar rl D < v. L ol diblh iri umbu imjir 4.7. Trformi Llc Bilrl TLB diuru dri TFWK : X. j X d j d 73

74 bil fugi X dim fugi oil mru forr ovrgi, m m TFWK dri y dlh: Y Y Y X.. j j Jdi dri r di j d d X j. j d Dfiii vribl frui oml Bil = +j d=j.d d = d/j X. d X Bidg dlh : d dibu g TLB oi : X = [] = - [X] c: rformi Llc Kovrgi TLB : 74

75 Trigri cr mul ji iliy urg dri higg 75

76 BAB V TRASFORMASI Z Pd bb ii dibh rformi- dg wu diri d ym LTI. Pd wu diri im LTI dg ro imul h[], lur y[] d mu oil dim, dlh y[] = T{ } = H H = h [ ] 5.. Dfiii Fugi H mru rformi- dri h[]. Uu wu diri iyl [], mjdi X X = [ ] Vribl mru uu ili oml yg dirri dg, dim, = r j r dlh mgiud d Ω dlh br udu. 5.. Rgio of Covrgc ROC ROC mru rg ili d vribl oml dim drh ii dil Pryr d ROC. ROC id brii bbr ol.. Ji [] mru uu fii-quc. Dim ili [] =, culi d, dim ili d dlh fii. 3. Ji [] mru ii dim [] = uu < < - d X ovrg uu bbr ili, m ROC : rm r m u > rm mru mgiud rbr d bbr ol X. 4. Ji [] mru ii iri dim [] = uu > > X ovrg uu bbr ili, m ROC 76

77 r mi u < rmi r mi mru mgiud rcilr d bbr ol X. 5. Ji [] mru du ii dim [] id dibi olh ii uu iri d X ovrg uu bbr ili m ROC : Dim, r r r d r mru mgiud d du ol d X Trformi- uu bbr commo quc. Ui Imul [] X = [ ] M []. Ui S u[] Sig =, m u[] uu mu Tbl 3 Tbl Trformi- X[] X ROC [] Smu u[], -u[--], [-m] u[] - u[--] u[] - u[--] m,,,, Smu cul ji m> u ji m < 77

78 + u[],[ ] co u[] co co i u[] r co u[] r i u[] i co co r co r i co r liy r r r 5.5. Sif dri Trformi-Z. Liir Ji [] X ROC = R [] X ROC = R Kmudi ] [ ] X X. Prgr [ Ji [] X ROC = R X[- ] X R = R { } 3. Mulilii olh Ji [] X ROC = R Kmudi, [] X R = 4. Tim Rvrl R Ji [] X ROC = R Kmudi, [-] X R = R 78

79 5. Difrii Ji [] X ROC = R Kmudi, dx [] - R = R d 6. Kombii Ji [] X ROC = R Kmudi, 7. Kovolui [ ] X X R = R { } Ji [] X ROC = R [] X ROC = R Kmudi, ' [] * [] X. X R R R 5.6. Ivr Trformi-Z Ivr dri rformi- dri X dlh []. Yg dirri dg :. Ivr Dim, [] = X [] = j c X C mru courclocwi. d. Mggu bl rformi- Dim, X = X X X... X mru fugi dri ivr rformi []... []. Dg if liir mjdi: [] = []+...+ [] 79

80 8 3. Powr Sri X = ] [ =...+ [-] + [-] + []+ [] + [] Mggu Pch Pril X = m D Aumi m d mu ol dlh drh M : c c c c c c X... Dim, X c X c D diuli mjdi: X = c c c c c... Cooh Sol :. Tu Trformi- dri. [] = ] [ u b. [] = ] [ u. X = u ] [ Ji u Shigg mjdi, X = b. X = u ] [

81 8 = Ji u Shigg mjdi, X =. Ji dihui, liy Dim, d buh b. Tuju ROC d X d -l culi mugi = u =. X = ] [ Uu yg id brili u rb m X ovrg. Ji < d > m duy oiif d d muy gif d. 3. Tu X d gmbr ol-ro dg ROC :. [] = ] [ 3 ] [ u u b. [] = ] [ ] [ 3 u u c. [] = ] [ 3 ] [ u u. Brdr bl rformi-: ] [ u ] [ 3 u 3 3 D dilih ROC mglmi ovrl :

82 8 X = Im Im R R b b. Brdr bl rformi-: ] [ 3 u 3 3 ] [ u - D dilih ROC mglmi ovrl : X = c. Brdr bl rformi-: ] [ u ] [ 3 u D dilih bhw ROC id lig ovrl m id d commo ROC d [] id mmuyi X. 4. Tu ivr rformi-z dri : X = Mggu i ch ril :

83 X Z c Dim, c Dg ubiui m did : Sig ili =, did 4 4 M dirolh ili X = Kr ROC dlh, m [] mru uu fugi ii. [] = u[ ] 5. Tu fugi H dri gmbr briu : [] / q[-] q[] /3 y[] Brdr gmbr di dirolh : q[] = [] + q[ ] y[] = q[] + q[ ] 3 Mggu rformi- m dirolh : Q = X + Q Y = Q + Q 3 83

84 84 Abil diuli ulg : X Q 3 X Q M H d dihui: H = 3 3 X Y