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Asach i\", Iasi, Facultatea de Electronica 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://scs 4.etc.tuiasi.ro/scslabs/Simbolice/simbolice.htm" "" }{TEXT -1 0 "" }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 16 "Breviar teoretic" }}{PARA 0 "" 0 "" {TEXT 267 16 "Scopul lucrarii:" }}{PARA 0 "" 0 "" {TEXT -1 117 "Stu diul spectrului unui semnal periodic, a spectrului unui semnal esantio nat si a unui semnal periodic si esantionat." }}{PARA 0 "" 0 "" {TEXT 268 17 "Rezumat teoretic:" }}{PARA 0 "" 0 "" {TEXT 269 17 "Semnalul pe riodic" }{TEXT -1 72 " se defineste prin repetarea unui puls generator la intervale de timp T:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "s[p](t) = sum(sg(t-n*T),n = -infinity .. infinity);" "6#/-&%\"sG6#%\"pG6#%\"tG- %$sumG6$-%#sgG6#,&F*\"\"\"*&%\"nGF2%\"TGF2!\"\"/F4;,$%)infinityGF6F:" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 135 "Pentru semnalul period ic astfel definit se deduce o formula a spectrului semnalului periodic in functie de spectrul pulsului generator." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 16 "Semnalul discret" }{TEXT -1 50 " se obtine prin esantinarea unul semnal continuu: " }}{PARA 256 " " 0 "" {XPPEDIT 18 0 "s[es](t) = s(t)*delta[Tes](t);" "6#/-&%\"sG6#%#e sG6#%\"tG*&-F&6#F*\"\"\"-&%&deltaG6#%$TesG6#F*F." }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 104 "Se deduce o formula care exprima spectru l semnalului discret in functie de spectrul semnalului continuu." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "restart:with(inttrans):" }}}{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]]):" }} }{PARA 0 "" 0 "" {TEXT -1 47 "Pentru esantionare lucram cu semnalul pe riodic:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "delta[T]:=(t,T)-> sum(Dirac(t-n*T),`n`=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&deltaG6#%\"TGR6$%\"tGF'6\"6$%)operatorG%&arrowGF+-% $sumG6$-%&DiracG6#,&9$\"\"\"*&%\"nGF79%F7!\"\"/F9;,$%)infinityGF;F?F+F +F+" }}}{PARA 0 "" 0 "" {TEXT -1 25 "Spectrul semnalului este:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "delta[Omega]:=subs(T=2*Pi/Om ega,F[dir](delta[T](t,T),t,omega));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>&%&deltaG6#%&OmegaG-%$sumG6$*&F'\"\"\"-%&DiracG6#,&%&omegaGF,*&%\"nG F,F'F,!\"\"F,/F3;,$%)infinityGF4F8" }}}{PARA 0 "" 0 "" {TEXT -1 22 "Re prezentarea grafica:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 347 "T1: =1: PLOT(rpa(eval(ts(delta[T](t,T),n=-10..10),T=T1),t,interval=-5.5..5 .5,tipgrafic=modul,culoare=[1,0,0]),AXESSTYLE(NORMAL),AXESLABELS(\"t\" ,\"\"),TITLE(\"DeltaT(t)\"));PLOT(rpa(eval(ts(delta[Omega],n=-10..10), Omega=2*Pi/T1),omega,interval=-15..15,tipgrafic=modul,culoare=[1,0,0]) ,AXESSTYLE(NORMAL),AXESLABELS(\"omega\",\"\"),TITLE(\"DeltaOmega(omega )\"));" }}{PARA 13 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{MPLTEXT 1 0 37 "Ses:=(omega)->F[dir](ses(t),t,omega):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Spes:=(omega)->F[dir](spes(t),t,ome ga):" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 "Semnale Periodice" }} {PARA 0 "" 0 "" {TEXT -1 68 "Definim pe baza pulsului generator, semna lul periodic de perioada T:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "sp:=(t)->sum(sg(t-n*T),`n`=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#spGR6#%\"tG6\"6$%)operatorG%&arrowGF(-%$sumG6$-%# sgG6#,&9$\"\"\"*&%\"nGF4%\"TGF4!\"\"/F6;,$%)infinityGF8F " 0 "" {MPLTEXT 1 0 55 "sg:=(t)->1/tau*(Heaviside(t+tau/2)-Heaviside(t-tau/2) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#sgGR6#%\"tG6\"6$%)operatorG%& arrowGF(*&,&-%*HeavisideG6#,&9$\"\"\"*&#F3\"\"#F3%$tauGF3F3F3-F/6#,&F2 F3*&#F3F6F3F7F3!\"\"F=F3F7F=F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 56 "Sp ectrul semnalului generator si al semnalului periodic:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Sg(omega);Sp(omega);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*&-%$sinG6#,$*&%$tauG\"\"\"%&omegaGF+#F+\"\"#F +*&F*F+F,F+!\"\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$sumG6$,$*&* &-%$sinG6#*&*(%$tauG\"\"\"%\"nGF0%#PiGF0F0%\"TG!\"\"F0-%&DiracG6#,&%&o megaGF0*&*(\"\"#F0F1F0F2F0F0F3F4F4F0F0F1F4F " 0 "" {MPLTEXT 1 0 195 "pl ot(eval([sg(t),ts(sp(t),n=-10..10)],[tau=1,T=3]),t=-10..10, view=[-6.2 ..6.2,-0.5..1.5],title=\"Semnalul periodic (rosu) si pulsul generator \+ (negru)\", color=[black,red],style=LINE, thickness=2 );" }{TEXT -1 0 " " }}{PARA 13 "" 1 "" {GLPLOT2D 398 217 217 {PLOTDATA 2 "6)-%'CURVESG6$ 7co7$$!#5\"\"!$F*F*7$$!3!pmmm\"p0k&*!#s%HaF/F+7$$!3]******\\$*4)*\\F/F+7$$!3o******\\_& \\c%F/F+7$$!3%)******\\1aZTF/F+7$$!3Imm;/#)[oPF/F+7$$!3%HLLL=exJ$F/F+7 $$!3lKLLL2$f$HF/F+7$$!3%)****\\PYx\"\\#F/F+7$$!3gLLLL7i)4#F/F+7$$!3o)* **\\P'psm\"F/F+7$$!3?****\\74_c7F/F+7$$!3PlmT5VBU5F/F+7$$!3M:LL$3x%z#) !#=F+7$$!3'*=Le9rc&H(F\\pF+7$$!3gAL$e9d;J'F\\pF+7$$!3UC$e9;-(>eF\\pF+7 $$!3BEL3xruF`F\\pF+7$$!3pw&*)4VeZ?&F\\pF+7$$!39Fe*[op<3&F\\pF+7$$!3gx? 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43 "Spectrul unui semnal periodic este discret." }{TEXT -1 127 " Spectrul semnalului periodic este o versiune discretizata a spec trului semnalului generator cu o rata de esantionare egala cu " } {XPPEDIT 18 0 "Omega;" "6#%&OmegaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 260 4 "Obs." }{TEXT -1 130 ":S-a preferat o forma a semnalului g enerator simpla, cu suport finit. Din acest motiv spectrul semnalului \+ generator este infinit. " }}{PARA 0 "" 0 "" {TEXT -1 59 " Semnalul periodic se obtine prin produs de convolutie:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "sp(t) = int(sg(tau)*delta[T](t-tau),tau = -infinity .. \+ infinity);" "6#/-%#spG6#%\"tG-%$intG6$*&-%#sgG6#%$tauG\"\"\"-&%&deltaG 6#%\"TG6#,&F'F0F/!\"\"F0/F/;,$%)infinityGF8F<" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 " Spectrul semnalului periodic se obtine pri n produs algebric:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "Sp(omega) = Sg( omega)*delta[Omega](omega);" "6#/-%#SpG6#%&omegaG*&-%#SgG6#F'\"\"\"-&% &deltaG6#%&OmegaG6#F'F," }{TEXT -1 1 "." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 19 "Semnale esantionate" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ses:=(t)->sg(t)*delta[T](t,Tes);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sesGR6#%\"tG6\"6$%)operatorG%&arrowGF(*&-%#sgG6#9$\"\"\"-&%&d eltaG6#%\"TG6$F0%$TesGF1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "sg:=(t)->(sin(omega0*t)/(omega0*t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#sgGR6#%\"tG6\"6$%)operatorG%&arrowGF(*&-%$sinG6#*&%' omega0G\"\"\"9$F2F2*&F1F2F3F2!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 58 "Spectrul semnalului generator si al semnalului esantionat:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Sg(omega);sum(Omega*S[g](ome ga-n*Omega),`n`=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&%#PiG\"\"\",(!\"\"F&-%*HeavisideG6#,&%&omegaGF(%'omega0GF&F&-F* 6#,&F-F&F.F&F&F&F&F.F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sumG6$*&% &OmegaG\"\"\"-&%\"SG6#%\"gG6#,&%&omegaGF(*&%\"nGF(F'F(!\"\"F(/F2;,$%)i nfinityGF3F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 269 "PLOT(rpa(c ollect(expand(ts(eval(ses(t),[Tes=1,omega0=1]),n=-20..20)),Dirac),t,in terval=-10..10,tipgrafic=real,culoare=[1,0,0]),rpa(eval(sg(t),omega0=1 ),t,interval=-10..10,tipgrafic=real,culoare=[0,0,0]),TITLE(\"Semnalul \+ original (negru) si semnalul esantionat (rosu)\"));" }}{PARA 13 "" 1 " " {GLPLOT2D 590 200 200 {PLOTDATA 2 "6O-%'CURVESG6&7dw7$$!#5\"\"!$F*F* 7$$!3m'[(\\**y%[*)*!#cN.)*F/F+7$$!3(>_/4)HY+(*F/F+7$$!3! 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" }}{PARA 0 "" 0 "" {TEXT -1 56 " Semnalul esantionat se obtine prin produs algeb ric:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "ses(t) = sg(t)*delta[T](t);" "6#/-%$sesG6#%\"tG*&-%#sgG6#F'\"\"\"-&%&deltaG6#%\"TG6#F'F," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 " Spectrul semnalului esanti onat se obtine prin produs de convolutie:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "Ses(omega) = int(Sg(w)*delta[Omega](w-omega),w = -infin ity .. infinity)/(2*Pi);" "6#/-%$SesG6#%&omegaG*&-%$intG6$*&-%#SgG6#% \"wG\"\"\"-&%&deltaG6#%&OmegaG6#,&F0F1F'!\"\"F1/F0;,$%)infinityGF9F=F1 *&\"\"#F1%#PiGF1F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 110 "S- a ales o frecventa de esantionare care sa respecte teorema esantionari i astfel incit sa nu apara fenomen de " }{TEXT 262 8 "aliasing" } {TEXT -1 1 "." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 32 "Semnale periodi ce si esantionate" }}{PARA 0 "" 0 "" {TEXT -1 46 "Definim semnalul per iodic armonic de pulsatie " }{XPPEDIT 18 0 "omega0;" "6#%'omega0G" } {TEXT -1 34 " provenit dintr-un puls generator:" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 91 "sp:=(t)->cos(omega0*t):sg:=cos(omega0*t)*(Heav iside(t+Pi/omega0)-Heaviside(t-3*Pi/omega0)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "spes:=(t)->sp(t)*delta[T](t,Tes);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%spesGR6#%\"tG6\"6$%)operatorG%&arrowGF(*&-%#s pG6#9$\"\"\"-&%&deltaG6#%\"TG6$F0%$TesGF1F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 36 "Spectrul semnalului esantionat este:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Spes(omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$sumG6$,$*&*&%#PiG\"\"\"-%&DiracG6#,(%&omegaGF+%'omega0G!\"\"*& *(\"\"#F+%\"nGF+F*F+F+%$TesGF2F2F+F+F7F2F5/F6;,$%)infinityGF2F;#F+F5*& F " 0 "" {MPLTEXT 1 0 279 "PLOT(rpa(collect(e xpand(ts(eval(spes(t),[Tes=1,omega0=1]),n=-20..20)),Dirac),t,interval= -10..10,tipgrafic=real,culoare=[1,0,0]),rpa(eval(sp(t),omega0=1),t,int erval=-10..10,tipgrafic=real,culoare=[0,0,0]),TITLE(\"Semnalul periodi c (negru) si semnalul periodic esantionat (rosu)\"));" }}{PARA 13 "" 1 "" {GLPLOT2D 398 184 184 {PLOTDATA 2 "6O-%'CURVESG6&7dw7$$!#5\"\"!$F *F*7$$!3m'[(\\**y%[*)*!#cN.)*F/F+7$$!3(>_/4)HY+(*F/F+7$$ !3!ye*F/F+7$$!3MkGd96#\\4*F/F+7$$!3'e3, -\"*yF/F+7$$!3)*e>Ryck(y(F/F+7$$!3C$yc8n:Fp(F/F+7$$!3Ma3QwK%4$y'F/F+7$$!3e3=Os/$*ymF/F+7$$!35U%)oPDa%e'F/F+7$$!3S\\)pRf^8['F /F+7$$!3uW*)ydpD%Q'F/F+7$$!3[g@V'3lFG'F/F+7$$!3Qzd:J#QN='F/F+7$$!3SQxa 4plzgF/F+7$$!3gb7D]ogzfF/F+7$$!3E^.297HxeF/F+7$$!3ou],.G#ex&F/F+7$$!3K $yc8(Qe#o&F/F+7$$!3AT$oO`@dd&F/F+7$$!3EBY#\\eU,[&F/F+7$$!37hAX!\\N#y`F /F+7$$!3U\\**)zR(p!G&F/F+7$$!3#\\.29)QRs^F/F+7$$!3%*[)pR*4hy]F/F+7$$!3 1Z%*)ydm@(\\F/F+7$$!3!f?T#[S:v[F/F+7$$!3GtY$peP!pZF/F+7$$!3Cyc8FqZxYF/ F+7$$!3]Hf=P-?tXF/F+7$$!3m],.1SZsWF/F+7$$!3_@W)od8=P%F/F+7$$!3#>['HfM_ rUF/F+7$$!3MJjE`g-rSF/F+7$$!3'zrV([bvrRF/F+7$$!3eS #['HBGnQF/F+7$$!3o8Fa3*3Fx$F/F+7$$!3MY$pQd(=oOF/F+7$$!3a=Pu[V2oNF/F+7$ $!3W]+,-[?oMF/F+7$$!3+LoOt1*QO$F/F+7$$!3->Qw_N\"yE$F/F+7$$!3Gzf>RuTpJF /F+7$$!3Oa4>QkxgIF/F+7$$!31kFb54TiHF/F+7$$!3!4MoOt8='GF/F+7$$!3'))*)zf **3&fFF/F+7$$!3%Q&3$f?F/ F+7$$!3Y],.1!zD'>F/F+7$$!3Ax_069be=F/F+7$$!31v],.I5a^-05**GN\"F/F+7$$! 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