{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Helvetica" 1 10 255 0 0 1 2 1 2 0 1 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Helvetica" 1 10 128 0 0 1 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "Helvetica" 0 0 128 0 0 1 1 2 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Hel vetica" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Helvetica" 1 16 128 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Maple Out put" -1 11 1 {CSTYLE "" -1 -1 "Helvetica" 1 10 128 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 3 0 3 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Helvetica" 1 10 128 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 3 0 3 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Helvetica" 1 10 128 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 3 0 3 0 2 2 0 1 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Helvetica" 1 16 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 3 0 3 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE " " -1 -1 "Helvetica" 1 10 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 3 0 3 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 3 0 3 0 2 2 0 1 }} {SECT 0 {PARA 18 "" 0 "" {TEXT -1 13 "Seria Fourier" }}{PARA 18 "" 0 " " {TEXT -1 23 "Partea a III-a: Exemple" }}{PARA 19 "" 0 "" {TEXT -1 86 "Universitatea Tehnica \"Gh. Asachi\", Iasi, Facultatea de Electron ica si Telecomunicatii" }}{PARA 19 "" 0 "" {TEXT -1 43 "Laboratorul de Semnale, Circuite si Sisteme" }}{PARA 19 "" 0 "" {URLLINK 17 "http:// scs4.etc.tuiasi.ro" 4 "http://scs4.etc.tuiasi.ro/scslabs/Simbolice/sim bolice.htm" "" }{TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 16 "B reviar teoretic" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "restart:w ith(inttrans):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "libname:= \"../SCSlib\",libname:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "a ddtable(fourier,sg(t),Sg(omega),t,omega);" }{TEXT -1 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 33 "Pornind de la semnalul generator " }{XPPEDIT 18 0 "sg(t);" "6#-%#sgG6#%\"tG" }{TEXT -1 43 " se obtine semnalul periodic \+ de perioada T:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "sp(t):=sum (sg(t-n*T),`n`=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>-%#spG6#%\"tG-%$sumG6$-%#sgG6#,&F'\"\"\"*&%\"nGF/%\"TGF/!\"\"/F1;,$ %)infinityGF3F7" }}}{PARA 0 "" 0 "" {TEXT -1 72 "Semnalul periodic se \+ poate descompune sub forma seriei Fourier complexe:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "spsfc(t):=subs(T=2*Pi/Omega,SFC(sp(t),t)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%&spsfcG6#%\"tG-%$sumG6$,$*&*(%& OmegaG\"\"\"-%#SgG6#*&%\"nGF/F.F/F/-%$expG6#**^#F/F/F4F/F.F/F'F/F/F/%# PiG!\"\"#F/\"\"#/F4;,$%)infinityGF;FA" }}}{PARA 0 "" 0 "" {TEXT -1 5 " unde " }{XPPEDIT 18 0 "Omega = 2*Pi/T;" "6#/%&OmegaG*(\"\"#\"\"\"%#PiG F'%\"TG!\"\"" }{TEXT -1 47 " iar coeficientii seriei Fourier complexe \+ sunt " }{XPPEDIT 18 0 "c[n] = Sg(n*Omega)/T;" "6#/&%\"cG6#%\"nG*&-%#Sg G6#*&F'\"\"\"%&OmegaGF-F-%\"TG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 63 "Semnalul periodic se descompune sub forma seriei Fourie r reale:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "spsfr(t):=subs(T =2*Pi/Omega,SFR(sp(t),t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%&spsf rG6#%\"tG,&*&*&%&OmegaG\"\"\"-%#SgG6#\"\"!F,F,%#PiG!\"\"#F,\"\"#-%$sum G6$,&*&*(F+F,,&-F.6#*&%\"nGF,F+F,F,-F.6#,$F>F2F,F,-%$cosG6#*(F?F,F+F,F 'F,F,F,F1F2F3*&**^#F3F,F+F,,&F " 0 "" {MPLTEXT 1 0 53 "Sp(omega)=sub s(T=2*Pi/Omega,FOURIER(sp(t),t,omega0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#SpG6#%&omegaG-%$sumG6$*(%&OmegaG\"\"\"-%#SgG6#*&%\"nGF-F,F-F --%&DiracG6#,&%'omega0GF-F1!\"\"F-/F2;,$%)infinityGF8F<" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Mod de lucru" }}{PARA 0 "" 0 "" {TEXT -1 133 "In afara de functiile incluse in libraria standard Maple, in acea sta lucrare vor fi utilizate cateva functii din libraria aditionala " }{TEXT 256 7 "SCSlib." }}{PARA 0 "" 0 "" {TEXT -1 55 "Pentru calculare a transformatei Fourier a unor semnale:" }}{PARA 15 "" 0 "" {TEXT -1 8 "functia " }{TEXT 257 7 "FOURIER" }{TEXT -1 65 " \+ " }{HYPERLNK 17 "help" 1 "..\\Help\\Helpfile.mws" "Functia FOURIER" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 30 "Pentru trunchierea unor serii:" }}{PARA 15 "" 0 " " {TEXT -1 8 "functia " }{TEXT 258 2 "ts" }{TEXT -1 77 " \+ " } {HYPERLNK 17 "help" 1 "..\\Help\\Helpfile.mws" "Functia ts" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 45 "Pentru reprezentarea spectrelor u nor semnale:" }}{PARA 15 "" 0 "" {TEXT -1 8 "functia " }{TEXT 259 3 "r pa" }{TEXT -1 74 " \+ " }{HYPERLNK 17 "help" 1 "..\\Help\\Helpfile.mws " "Functia rpa" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 29 "libname:=\"../SCSlib\",libname:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "F:=table([dir=FOURIER,inv=inttrans[ invfourier]]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Sg:=(omeg a)->F[dir](sg(t),t,omega):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Sp:=(omega)->F[dir](sp(t),t,omega):" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 7 "Exemplu" }}{PARA 0 "" 0 "" {TEXT -1 77 "Pentru semnalul pe riodic obtinut prin repetarea pulsului generator de forma: " } {XPPEDIT 18 0 "PIECEWISE([cos(t), abs(t) < 1/2*Pi],[0, 1/2*Pi <= abs(t )])" "6#-%*PIECEWISEG6$7$-%$cosG6#%\"tG2-%$absG6#F**(\"\"\"F0\"\"#!\" \"%#PiGF07$\"\"!1*(F0F0F1F2F3F0-F-6#F*" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "sg:=(t)->cos(Pi/tau*t)*(Heaviside(t+tau/2)-Heaviside (t-tau/2));plot(eval(sg(t),tau=Pi/2),t=-1.1*Pi..1.1*Pi,numpoints=1000, color=[black,red],title=\"sg(t)\",view=[DEFAULT,-0.5..1.5], style=LINE , thickness = 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#sgGR6#%\"tG6\" 6$%)operatorG%&arrowGF(*&-%$cosG6#*&*&%#PiG\"\"\"9$F3F3%$tauG!\"\"F3,& -%*HeavisideG6#,&F4F3*&#F3\"\"#F3F5F3F3F3-F96#,&F4F3*&#F3F>F3F5F3F6F6F 3F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 399 229 229 {PLOTDATA 2 "6(-%'CU RVESG6$7hin7$$!3?++/7>vbM!#<$\"\"!F,7$$!3GBz0?M^[MF*F+7$$!3ZC$))oC:AW$ F*F+7$$!3_TPX#HK^V$F*F+7$$!3I_%*f(Q-!GMF*F+7$$!3S_$Q*pj!4U$F*F+7$$!3+D I@`uK9MF*F+7$$!3Ix'4gO:vS$F*F+7$$!3C$=zcHq/S$F*F+7$$!3q(ey*=yW$R$F*F+7 $$!3)o\"*>&QVA'Q$F*F+7$$!3a'4+I*=')zLF*F+7$$!3U-fG'G*psLF*F+7$$!3Yq.Ap s]lLF*F+7$$!3%[PH\"GkdeLF*F+7$$!3K3BVgDG_LF*F+7$$!3YzX)Ha)zWLF*F+7$$!3 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=huceF$F*F+7$$\"3%=`sgfJFG$F*F+7$$\"3sM_\\.C\"**G$F*F+7$$\"3kNiF*=ElH$ F*F+7$$\"3u\"y$f5'*H.LF*F+7$$\"3@L+cs#y2J$F*F+7$$\"3sNTR)e\\vJ$F*F+7$$ \"3+h:_;UT$F*F+7$ $\"3^&*))G)H'*4U$F*F+7$$\"3)RG-ukX$GMF*F+7$$\"3yKPsx[qMMF*F+7$$\"39+!R \\o3?W$F*F+7$$\"3Qq^pZY()[MF*F+7$$\"3?++/7>vbMF*F+-%'COLOURG6&%$RGBGF, F,F,-%&TITLEG6#Q&sg(t)6\"-%&STYLEG6#%%LINEG-%+AXESLABELSG6$Q\"tF`exQ!6 \"-%*THICKNESSG6#\"\"#-%%VIEWG6$%(DEFAULTG;$!\"&!\"\"$\"#:Fffx" 1 6 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 33 "Spectrul pulsului generator este:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Sg(omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*(,,*(^#!\"\"\"\"\")%#PiG\"\"#F)-%&DiracG6#*&,&*&%$ta uGF)%&omegaGF)F(F+F)F)F3F(F)F)**^#F)F)F-F))F3F,F))F4F,F)F)**F'F)-F.6#* &,&F2F)F+F)F)F3F(F)F7F)F8F)F)*(F6F)F*F)F:F)F)*&F,F)F3F)F)F)F+F)-%$cosG 6#,$F2#F)F,F)F)*&F1F)F=F)F(" }}}{PARA 0 "" 0 "" {TEXT -1 41 "Semnalul \+ periodic este definit sub forma:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "sp:=(t)->sum(sg(t-n*T),`n`=-infinity..infinity);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#spGR6#%\"tG6\"6$%)operatorG%&arrowGF(-%$sumG6 $-%#sgG6#,&9$\"\"\"*&%\"nGF4%\"TGF4!\"\"/F6;,$%)infinityGF8F " 0 "" {MPLTEXT 1 0 13 "SFC(sp(t),t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sumG6$,$*&**%#PiG\"\"\"-%$cosG6 #*&*(%$tauGF*%\"nGF*F)F*F*%\"TG!\"\"F*F0F*-%$expG6#*&**^#\"\"#F*F1F*F) F*%\"tGF*F*F2F3F*F**(F2F*,&F.F:F)F3F*,&F.F:F)F*F*F3!\"#/F1;,$%)infinit yGF3FC" }}}{PARA 0 "" 0 "" {TEXT -1 47 "Seria Fourier reala a semnalul ui periodic este:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "SFR(sp( t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%$tauG\"\"\"*&%\"TGF&%#P iGF&!\"\"\"\"#-%$sumG6$,$*&**F)F&-%$cosG6#*&*(F%F&%\"nGF&F)F&F&F(F*F&F %F&-F36#,$*&*(F7F&F)F&%\"tGF&F&F(F*F+F&F&*(F(F&,&F5F+F)F*F&,&F5F+F)F&F &F*!\"%/F7;F&%)infinityGF&" }}}{PARA 0 "" 0 "" {TEXT -1 37 "Spectrul a cestui semnal este discret:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Sp(omega);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%$sumG6$,,*&*.^#\" \"#\"\"\"-%$cosG6#*&*(%$tauGF+%\"nGF+%#PiGF+F+%\"TG!\"\"F+F4F+-%&Dirac G6#*&,&*&F1F+F2F+F*F4F+F+*&F4F+F1F+F5F+F3F+-F76#,&%&omegaGF+*&*(F*F+F2 F+F3F+F+F4F5F5F+F+*&,&F;!\"#F4F+F+F:F+F5F+*&*0^#!\")F+F,F+F6F+)F1F*F+) F2F*F+F3F+F=F+F+*(F4F+FDF+F:F+F5F+*&*0^#\"\")F+F,F+-F76#*&FDF+*&F4F+F1 F+F5F+FJF+FKF+F3F+F=F+F+*(F4F+FDF+F:F+F5F+*&*.^#FEF+F,F+F4F+FQF+F3F+F= F+F+*&FDF+F:F+F5F+*&*,\"\"%F+F,F+F4F+F1F+F=F+F+*&FDF+F:F+F5F+/F2;,$%)i nfinityGF5F[o" }}}{PARA 0 "" 0 "" {TEXT -1 91 "Reprezentarea formelor \+ de unda ale pulsului generator si semnalului periodic corespunzator:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "plot(eval([sg(t),ts(sp(t) ,n=-5..5),cos(2*t)],[tau=Pi/2,T=2*Pi]),t=-5*Pi..5*Pi,numpoints=1000,co lor=[black,red,gray],title=\"sg(t) (negru), sp(t) (rosu), cos(w0*t) (g ri)\",view=[DEFAULT,-1.5..1.5], style=LINE, thickness = 2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6*-%'CURVESG6$7\\jn7$$!3 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"Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1 8" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" " Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curv e 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 19 "Probleme. Intrebari" }}{PARA 0 "" 0 "" {TEXT -1 56 "1. Refaceti calculele pentru un puls generator deplasat." }} {PARA 0 "" 0 "" {TEXT -1 153 "2. Cum este spectrul pulsului generator? Comparati cu spectrul pulsului dreptunghiular si a pulsului triunghiu lar. Cum este spectrul semnalului periodic?" }}{PARA 0 "" 0 "" {TEXT -1 197 "3. Comparati aproximarea cu numar finit de componente din SFR \+ pentru puls generator dreptunghiular si pentru puls generator de tip c osinus. In ce caz aproximarea este mai buna? Explicati concluzia!" }} {PARA 0 "" 0 "" {TEXT -1 57 "4. Refaceti calculele pentru puls generat or de tip Gauss." }}}}{MARK "6 8 0" 0 }{VIEWOPTS 1 1 1 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }