ISYARAT DAN SISTEM Bab 4 Deret Fourier Untuk Isyarat Periodik

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1 KE 5 ISYARA DA SISEM Bb Dr Fourir Uu Isyr Priodi Idh Susilwi, S.., M.Eg. Progrm Sudi i Elro Fuls i d Ilmu Kompur Uivrsis Mrcu Bu Yogyr 9

2 79 B A B I V DERE FOURIER UUK ISYARA PERIODIK uu Isrusiol. Umum Slh mylsi m ulih ii, mhsisw dp mlu lisis d sisis sism yg sg brmf uu mug rivis prys rum dlm dsi sism.. Khusus Slh mylsi bb ii, dihrp: - Mhsisw dp my dr Fourir isyr priodi wu oiyu. - Mhsisw dp mmhmi sif-sif dr Fourir wu oiyu. - Mhsisw dp my dr Fourir isyr priodi wu disri. - Mhsisw dp mmhmi sif-sif dr Fourir wu disri. - Mhsisw dp mu ggp frusi sism LI. - Mhsisw dp mmhmi g filr-filr pmilih frusi... ggp Sism LI rhdp Esposil Kompls ggp sism LI rhdp msu sposil mrup isyr sposil yg sm pi dg prubh mpliudo. Hl ii dp diy sbgi: s H(s s (sism wu oiyu z H[z] z (sism wu disri Suu isyr yg mghsil lur yg mrup hsil li suu os dg msuy rsbu, disbu igfucio dri sism rsbu (dlm hl ii dlh s u z. Sdg for mpliudoy disbu dg igvlu sism rsbu (dlm hl ii dlh H(s u H[z].

3 8 Briu dlh ilusrsi pd sism LI wu oiyu. rdp sism LI wu oiyu dg ggp impuls h(. Uu msu x( s m lury dp diu dg igrl ovolusi, yiu: y( dg s s h ( τx ( τdτ h ( τ h ( τ H (s s ( τ s h ( τ sτ sτ dτ dτ dτ s mrup igfucio sism H(s sτ h( τ d τ mrup igvlu sism Sdg ilusrsi pd sism LI wu disri dp dils sbgi briu. Sism LI wu disri dg ggp impuls h[] mmpuyi msu x[] z m lur sism dp diu dg umlh ovolusi sbgi y[] z z h[] h[] h[] H[z] z z h[] x[ ] z z dg

4 8 z mrup igfucio sism H [z] h[] z mrup igvlu sism.. Sif Suprposisi Sism LI mmilii sif suprposisi. Ji x( mrup ombisi liir yg diy dg prsm x ( s s s m ggp sism LI wu oiyu yg dihsil mrup umlh (suprposisi dri ggp msig-msig ompo yg mmbu msuy. Prhi: x ( x ( x ( s s s y ( y ( y ( H(s s H(s H(s s s x( y( s H(s s Dg dmii, lur y( dp diy sbgi y ( H(s s Sif suprposisi ug rdp pd sism LI wu disri. Ji msu x[] mrup ombisi liir yg diy sbgi x [] z m ggp sism LI wu disri yg dihsil ug mrup umlh (suprposisi dri ggp msig-msig ompo yg mmbu msuy. Dg cr yg sm, m lur y[] dp diy dg y [] H[z ] z

5 8... Cooh Sol d Pylsi. Sbuh sism LI wu oiyu mmpuyi hubug msu d lur sbgi briu: y( x( Ji msuy dlh x(, m ulh lur sism rsbu. Pylsi Cr Dg msu x(, m y( x( ( 6 Dlm hl ii s diulis sbgi H(s H( 6 mrup igfucio d igvlu uu Cr H(s dp dimu dg rumus s H(s h( τ dτ, dg h( δ(, shigg H(s h ( τ s s sτ δ ( τ s dτ sτ dτ δ ( τ dτ δ ( τ dτ Ji x(, m s d dg dmii H( s 6

6 8. Uu sism LI yg sm dg pd cooh sol o., m ulh lur sism i msuy dlh x( cos cos 7 Pylsi Msu x( mrup ombisi lir yg rdiri s du ompo, shigg sol ii dp dislsi dg sif suprposisi yg dimilii sism LI. Msu x( dp diuri mggu rumus Eulr mdi: x ( cos cos 7 { } { 7 7 } m lur y( dp dimu sbgi y( x( cos ( ( { ( } cos{ 7( } 7( 7( Eigfucio d igvlu uu msig-msig ompo diprlih pd bl.. bl.. Eigfucio d igvlu uu msig-msig ompo uu cooh sol o. Kompo Eigfucio Eigvlu ( ½ ( ½ 7( 7 ½ 7( 7 ½... Sol-sol mbh. Sbuh sism LI wu oiyu mmpuyi hubug lur d msu yg diy dg prsm

7 8 y( x( Ji msuy dlh x(, m ulh lur sism.. Sbuh sism LI wu oiyu mmpuyi hubug lur d msu yg diy dg prsm y( x( x( Ji msuy dlh x(, m ulh lur sism... Rprssi Dr Fourir Pd Isyr Priodi Wu Koiyu Uu isyr priodi wu oiyu x( yg mmpuyi priod dsr d frusi dsr /, m dr Fourir s x( didfiisi sbgi briu: dg x( x( x( d d Kofisi disbu ofisi dr Fourir u ofisi sprl dri x(. Kofisi (yiu s disbu ompo dc u os dri x(, yg diu olh: x( d Bsry muu bsry siyl x( pd sip hrmoi dri ompo dsr. Pd subb briu dibri bbrp cooh sol briu pylsiy dlm hl my sbuh isyr mdi dr Fouriry.

8 85... Cooh Sol d Pylsi. Isyr x( si mmpuyi frusi dsr. ulh dr Fourir uu my x(. Pylsi Slh su cr uu mu dr Fourir uu x( si dlh dg mguri x( mggu rumus Eulr. { } si x ( Dri pguri x( dp dihui bhw d uu ili yg li.. Isyr x( didfiisi sbgi briu: x( si cos cos ( / ylh x( dlm dr Fourir. Pylsi Isyr x( mmpuyi frusi dsr d dp diuri mdi: { } { } cos cos si x( m

9 86 x( d dp dihui bhw: uu ( ( > Kofisi dp diplo spri pd gmbr Gmbr. Plo ofisi-ofisi dr Fourir pd sol o.. Isyr x( pd gmbr. dp diy sbgi < < <,, ( x ylh isyr rsbu dlm dr Fourir.

10 87 X( - - Gmbr. Isyr x( uu sol o. Pylsi Isyr x( mrup isyr yg priodi, m uu su priod x( dp dimu ofisi. S, m [ ] (. d d x( S, m [ ] [ ] d d x(

11 88 si ( si( si( Ig bhw Sbgi gmbr, m dp dimisl suu sus i shigg u. Dg pmisl ii dp dimu ili-ili ofisi dr Fourir x( uu brbgi hrg. si si( m si( / si si( / si si(5 / d srusy 5 Kofisi dp diplo spri pd gmbr Gmbr. Plo ofisi-ofisi dr Fourir pd sol o.

12 89.. Sol-sol mbh. Uu sol yg sm dg cooh sol o., ulh ofisi dr Fourir dri x( i:. 8 b. 6 c. Ap simpul d?. ulh dr Fourir uu isyr x( yg diprlih pd gmbr. briu ii. X( Gmbr. Isyr x( uu sol o... Sif-sif Dr Fourir Wu Koiyu Uu pig mudh dlm pmbhs g sif-sif dr Fourir wu oiyu, m ofisi dr Fourir dri sbuh isyr x(, yiu, diulis dg osi: x( Ariy, isyr x( dp diy dg dr Fourir dg ofisiofisi ( Fourir Sris Dr Fourir. Briu dlh sif-sif dr Fourir wu oiyu:. Liris Ji x( d y( dlh isyr priodi dg priod d x( y( b m uu isyr z( yg didfiisi sbgi z( A x( B y( brlu sif liris sbgi briu: z ( c A B b

13 9 yiu ofisi dr Fourir dri z( dlh c A B b, dg A d B dlh os.. Prgsr wu Ji x( dlh isyr priodi dg priod dg x( d y( mrup isyr rgsr wu dri x( yg diy sbgi y( x( m brlu sif y(.. yiu ofisi dr Fourir dri y( mrup prli dg.. Wu-bli Ji x( dlh isyr priodi dg priod dg x( m uu isyr wu bli dri x( yiu x(- brlu sif sbgi briu: x(. Prli Ji x( d y( dlh isyr priodi dg priod d x( y( m brlu b x ( y( h l l b l yiu ofisi dr Fourir dri prli x( d y( mrup umlh ovolusi disri dri d b. 5. Psl wu (im scllig Ji isyr x( dp diy dg dr Fourir sbgi

14 9 x( m isyr x( yg rsl wu (sbsr α mmpuyi dr Fourir yg diy sbgi x( α ( α 6. Koug d simri oug Ji x( dlh isyr priodi dg priod dg m x( x( dim x ( dlh oug ompls dri x( d dlh oug ompls dri. Ji x( mrup bilg riil muri, m x ( x( d ofisi dr Fourir mdi simri oug, yiu:.5. Dr Fourir Isyr Priodi Wu Disri Ji isyr priodi wu oiyu x[] priodi dg priod dsr d frusi dsr /, m x[] dp diy sbgi dr Fourir wu disri sbgi briu: x [ ] dg dlh ofisi dr Fourir uu isyr x[], d didfiisi dg pry

15 9 x[] x[].5.. Cooh Sol d Pylsi. Ji x[] si, m u dr Fourir uu x[]. Pylsi Isyr x[] si priodi hy i / mrup bilg bul u prbdig bilg bul. Ji, m x[] priodi dg priod dsr d x[] dp diuri mdi dr Fourir sbgi briu: x[] si si Dg dmii, ofisi dr Fouriry dlh...? uu yg li Uu hrg yg li, dp dicri dg cr briu. Olh r x[] priodi dg priod dsr, m ofisi-ofisi dr Fourir x[] ug brulg dg priod, shigg d Misl dimbil priod dsr 5, m dp diu:

16 ( ( ( ( ( ( Kofisi dp diplo spri pd gmbr.5. / / Gmbr.5 Plo ofisi-ofisi dr Fourir pd sol o.. Uu sol yg sm dg sol o., m misl / mrup prbdig bilg bul sbgi briu: M u m ylh ofisi-ofisi dr Fourir uu x[]. Pylsi Dg msubsiusi M x[] dp diy mbli sbgi: M pd prsm isyr x[], m

17 9 M M M si si x[] Shigg M M Isyr x[] priodi dg priod dsr, i dipilih M d 5, m diprolh: M M ( ( ( ( ( ( ( ( M 7 M 7 M M 8 5 M 5 M 5 M 8 5 M Kofisi dp diplo spri pd gmbr.6.

18 95 / / Gmbr.6 Plo ofisi-ofisi dr Fourir pd sol o.. ulh dr Fourir uu isyr x[] si cos Pylsi Isyr x[] priodi dg priod dsr, d dp diuri mdi: x[] si cos Shigg diprolh:. ulh ofisi-ofisi dr Fourir uu isyr x[] i isyr x[] diprlih pd gmbr.7. X[] - - Gmbr.7 Isyr x[] uu sol o.

19 96 Pylsi Dri gmbr isyr x[] di s, dihui bhw x[] uu hrg < <, m dp diy: Dg mgggp bhw m, m...,,, ; si,5 si.,5,5 * m m (m m ± ± C * Suu m m mrup sbuh dr gomri dg suu, shigg umlh dr gomri rsbu dp dihui mggu rumus umlh dr gomri. Sdg uu, ±, ±, m ofisi dr Fouriry dlh

20 97 Sbgi cooh, ofisi-ofisi uu ( 5 dp digmbr uu brbgi ili, misly. Uu, ±, ±, m 5 Uu, ±, ±, m,5 si si si si si si Shigg diprolh: si, si si si 5 si, si si si

21 si, si si si si, 7 si si si 5 9 si, 9 si Kr sify yg priodi dg priod, m Kofisi dp diplo spri pd gmbr Gmbr.8 Plo ofisi-ofisi dr Fourir pd sol o.

22 Sol-sol mbh. Uu sol yg sm dg pd cooh sol o., ulh ofisi-ofisi dr Fourir uu x[] i. 5 b. c. 5 Gmbr plo dri ofisi-ofisi rsbu.. Uu sol yg sm dg pd cooh sol o., ulh ofisi-ofisi dr Fourir uu x[] i. b. Gmbr plo dri ofisi-ofisi rsbu..6. Sif-sif Dr Fourir Wu Disri Uu isyr wu disri x[] yg priodi dg priod d mmpuyi ofisi dr Fourir, m hubug ii diulis sbgi briu: x[] Briu ii dlh sif-sif dr Fourir wu disri.. Prli Ji x[] d y[] dlh isyr priodi dg priod d x[] y[] m brlu b x[].y[] d l l b l. Difrsisi prm Ji x[] dlh isyr priodi dg priod d x[]

23 m ofisi dr Fourir yg ssui dg difrsisi prm dri x[] diy sbgi: x [] x[ ] dim frusi dsr x[] dlh /..7. Dr Fourir d Sism LI Dlm wu oiyu, i x( s mrup ipu u msu sism LI wu oiyu, m lury dlh dg y( H(s s H(s sτ h( τ d τ d h( mrup ggp impuls sism LI. Ji s m s shigg msu LI mrup sposil ompls dg frusi. Dlm hl ii, m H(s H( yg diy: H( h( τ sτ dτ H( disbu dg isilh ggp frusi (frqucy rspos sism LI. briu Ji isyr msu x( diy dlm dr Fourir sbgi x [] m lur sism LI dp diy dg y [] H( dim dlm hl ii s, d ofisi dri y( dlh b H(

24 Dg cr yg sm, dlm wu disri i x[] z mrup msu sism LI wu disri, m lury dlh dg y[] H(z z H (z h[] z d h[] dlh ggp impuls sism LI. Ji hrg z dipilih sdmii rup shigg z, m d z z Dg dmii, m diprolh prsm yg my ggp frusiy, yiu: ( H h[] Ji isyr msu x[] mrup isyr priodi yg diy dlm dr Fourir sbgi x [] m lur sism LI (dg ggp impuls h[] dlh y [] H( dim ofisi y[] dlh b H( /.7.. Cooh Sol d Pylsi. Sbuh sism LI wu oiyu mmpuyi ggp impuls h( u( Ji msu sism sism ii dlh

25 dg x[] ¼ ½ m ulh ggp frusi d lur sism LI rsbu. Pylsi ggp frusi dp dimu sbgi briu: H( h( ( u( d d d d ( Olh r isyr x( mmpuyi frusi dsr m lur sism LI rsbu dlh dg y[] b b H( shigg dp diu bsry b uu hrg-hrg yg brbd sbgi briu.

26 b b b b b b H( H( H( H(6 6 H( H( H( 6 6 b. Sbuh sism LI wu disri mmpuyi ggp impuls h[] α u[] dim < α <. Ji msu sism ii dlh x[ ] cos m ulh ggp frusi sism rsbu. Pylsi x[ ] cos m dp diprolh ggp frusi uu / d /, sbgi briu: H / ( ( / [ α ] h[ ] α α ( / u[ ] / ( /

27 d ( [ ] / / ( / ( / ( / u[ ] h[ ] H α α α Scr umum, ggp frusiy dp diy dg ( α H Cr li uu mu ggp frusi sism dlh sbgi briu. Msu x[] dp diulis sbgi dr Fourir x[] d ggp frusiy dlh ( α α α α u[] h[] H Dg dmii, lur sism dp diy sbgi: / / / / / H( H( H( y[] α α

28 5.7.. Sol-sol mbh. Sbuh sism LI wu disri mmpuyi ggp impuls h[] m ulh rprssi dr Fourir dri y[] i msu x[] dlh isyr priodi dg priod 6 d diy sbgi briu x [],,, ± ±, ±. Sbuh sism LI wu oiyu dg ggp impuls h( m ulh rprssi dr Fourir dri y( i msu x( spri pd gmbr briu. X( Gmbr.9 Isyr x( uu sol o..8. Filr-filr Pmilih Frusi ggp frusi suu filr dlm wu oiyu, scr gris bsr dibd mdi:. Lowpss idl Filr ii mlw frusi rdh s. Prhi gmbr.. H( c c Gmbr. ggp frusi filr lowpss idl (wu oiyu

29 6. Highpss idl Filr ii mlw frusi iggi s. Prhi gmbr.. H( c c Gmbr. ggp frusi filr highpss idl (wu oiyu. Bdpss idl ggp frusi filr bdpss idl diprlih pd gmbr.. H( Gmbr. ggp frusi filr bdpss idl (wu oiyu Sdg ggp frusi suu filr dlm wu disri, scr gris bsr ug dibd mdi:. Lowpss idl ggp frusi filr bdpss idl diprlih pd gmbr.. H( Gmbr. ggp frusi filr lowpss idl (wu disri. Highpss idl ggp frusi filr bdpss idl diprlih pd gmbr..

30 7 H( Gmbr. ggp frusi filr highpss idl (wu disri. Bdpss idl ggp frusi filr bdpss idl diprlih pd gmbr.5. H( Gmbr.5 ggp frusi filr bdpss idl (wu disri.8.. Cooh Sol d Pylsi. Sbuh rgi filr lowpss RC sdrh pd gmbr.6. Ji msu sism dlh V s ( d lury dlh V c (, ulh prsm yg my hubug r lur d msu sism. R V s ( C V c ( i( Gmbr.6 Rgi filr RC lowpss sdrh Pylsi Dg V s ( sbgi msu sism d V c ( sbgi lury, m uu rgi RC di s brlu:

31 8 RC d d V ( V ( V ( R R i( V ( V ( V ( V ( V ( C C C C S S S Ji msu V s ( m lur V c ( hrus mdi V c ( H( Shigg diprolh Ji RC d d RC { H( } H( RC H( H( H( H( [ RC ] H( H( RC m H( d > m H( << ggp frusi filr ii dp digmbr scr grfis pd gmbr.7. IH(JI -/RC /RC Gmbr.7 ggp filr lowpss RC pd sol o. Filr lowpss RC sdrh pd cooh sol o. ii mrup filr yg o-idl.. Sbuh rgi filr highpss RC sdrh pd gmbr.8. Ji msu sism dlh V s ( d lury dlh V R (, ulh prsm yg my hubug r lur d msu sism.

32 9 V R ( R V s ( C i( Gmbr.8 Rgi filr RC highpss sdrh Pylsi Dg V s ( sbgi msu sism d V R ( sbgi lury, m uu rgi RC di s brlu: V ( V ( V ( R C VR ( i(d V ( C S VR ( VR ( d V ( C S R VR ( V ( d V ( RC R S d d RC VR ( VR ( RC V S( d d S Ji msu V s ( m lur V R ( hrus mdi V R ( H( Shigg diprolh d RC d RC d d [ H(. ] RC. H( (RC d VR ( VR ( RC V S( d d H(. RC d H(. RC. H( RC RC H( RC rlih dri prsm ggp frusi sism, bhw i >> /RC m rdi rdm. Dg li, i mdi ol m H( << d i /RC m H(.

33 ggp frusi filr ii dp digmbr scr grfis pd gmbr.9. IH(JI -/RC /RC Gmbr.9 ggp filr highpss RC pd sol o. Filr highpss RC sdrh pd cooh sol o. ii mrup filr yg o-idl.