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T/2):" "6#/&%\"cG6#% \"kG*&.*&\"\"\"F+%\"TG!\"\"F+-%$IntG6$*&-%\"fG6#%\"tGF+-%$expG6#,$**% \"jGF+F'F+&%&omegaG6#\"\"!F+F5F+F-F+/F5;,$*&F,F+\"\"#F-F-*&F,F+FDF-F+ " }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "F(j*omega)=Int(f(t)*exp(- j*omega*t),t=-infinity..infinity):" "6#/-%\"FG6#*&%\"jG\"\"\"%&omegaGF )-%$IntG6$*&-%\"fG6#%\"tGF)-%$expG6#,$*(F(F)F*F)F2F)!\"\"F)/F2;,$%)inf inityGF8F<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Zjist\355me, \236e, ...." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Expon enci\341ln\355 tvar" }}{PARA 0 "> " 0 "" {XPPEDIT 19 1 "f(t) = Sum(c[k ]*exp(j*k*omega[0]*t),k = -infinity .. infinity):" "6#/-%\"fG6#%\"tG-% $SumG6$*&&%\"cG6#%\"kG\"\"\"-%$expG6#**%\"jGF0F/F0&%&omegaG6#\"\"!F0F' F0F0/F/;,$%)infinityG!\"\"F=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Z p\354tn\341 Fourierova transformace" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f(t)='1/(2*Pi)'*Int(F(j*omega)*exp(j* omega*t),omega=-infinity..infinity):" "6#/-%\"fG6#%\"tG*&.*&\"\"\"F+*& \"\"#F+%#PiGF+!\"\"F+-%$IntG6$*&-%\"FG6#*&%\"jGF+%&omegaGF+F+-%$expG6# *(F8F+F9F+F'F+F+/F9;,$%)infinityGF/FAF+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Z p\370edchoz\355ho je tak\351 evidentn\355, \236e Fourie r\371v obraz Diracova pulzu je roven jeho plo\232e." }{MPLTEXT 1 0 0 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "int(Dirac(t)*exp(-j*omega*t),t \+ =-infinity..infinity);" }}{PARA 0 "" 0 "" {TEXT -1 22 "Plocha Diracova pulzu." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "int(D irac(t),t=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Fouri erova transformace pulzu " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "rect:=A*(Heaviside(t+tau/2)-Heaviside(t-tau/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rectG*&%\"AG\"\"\",&-%*HeavisideG6#,&%\"tGF'*&\"\"#! \"\"%$tauGF'F'F'-F*6#,&F-F'*&F/F0F1F'F0F0F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "Jednotkov \341 funkce" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(Heavisid e(t),t=-1..1,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 516 302 302 {PLOTDATA 2 "6&-%'CURVESG6$7hn7$$!\"\"\"\"!$F*F*7$$!3ommm;p0k&*!#=F+7$ $!3wKL$3s%HaF/F+7$$!3Q+++]$ *4)*\\F/F+7$$!39+++]_&\\c%F/F+7$$!31+++]1aZTF/F+7$$!3umm;/#)[oPF/F+7$$ !3hLLL$=exJ$F/F+7$$!3*RLLLtIf$HF/F+7$$!3]++]PYx\"\\#F/F+7$$!3EMLLL7i)4 #F/F+7$$!3c****\\P'psm\"F/F+7$$!3')****\\74_c7F/F+7$$!3)3LLL3x%z#)!#>F +7$$!3KMLL3s$QM%FioF+7$$!3T,+]ivF@AFioF+7$$!3]^omm;zr)*!#@F+7$$\"32.\" 3Fp^y!RFcp$\"\"\"F*7$$\"3w0$3_]\\(o " 0 "" {MPLTEXT 1 0 49 "plot(subs(A=1,tau=1/4,rect),t=-1..1,thickness=2);" }}{PARA 13 "" 1 " " {GLPLOT2D 574 231 231 {PLOTDATA 2 "6&-%'CURVESG6$7co7$$!\"\"\"\"!$F* F*7$$!3ommm;p0k&*!#=F+7$$!3wKL$3s%HaF/F+7$$!3Q+++]$*4)*\\F/F+7$$!39+++]_&\\c%F/F+7$$!31+++]1aZTF/ F+7$$!3umm;/#)[oPF/F+7$$!3hLLL$=exJ$F/F+7$$!3*RLLLtIf$HF/F+7$$!3]++]PY x\"\\#F/F+7$$!3EMLLL7i)4#F/F+7$$!3c****\\P'psm\"F/F+7$$!3s*****\\F&*=Y \"F/F+7$$!3')****\\74_c7F/F+7$$!3`\"zp)**z7V7F/$\"\"\"F*7$$!3>$eRs3N(H 7F/F\\p7$$!3'[P4YE\\H?\"F/F\\p7$$!3%)\\(=nVjh< \"F/F\\p7$$!3=L$e9hx$\\6F/F\\p7$$!3#)*\\P4'f!e4\"F/F\\p7$$!3ZmmT5VBU5F /F\\p7$$!3#y**\\P454N*!#>F\\p7$$!3)3LLL3x%z#)FfqF\\p7$$!3gKL$e9d;J'Ffq F\\p7$$!3KMLL3s$QM%FfqF\\p7$$!3]^omm;zr)*!#@F\\p7$$\"3%pJL$ezw5VFfqF\\ p7$$\"3s*)***\\PQ#\\\")FfqF\\p7$$\"3ilm\"z\\1A-\"F/F\\p7$$\"3GKLLe\"*[ H7F/F\\p7$$\"3i:/w7J(GC\"F/F\\p7$$\"3'*)\\(=nqDc7F/F+7$$\"3K#e9;-T'p7F /F+7$$\"3kl;/w\\-$G\"F/F+7$$\"3MKe*[)Gz48F/F+7$$\"3.***\\PzglL\"F/F+7$ $\"3UK$e9h'4!R\"F/F+7$$\"3ylm;HCjV9F/F+7$$\"3aKLekSq]:F/F+7$$\"3I***** **pvxl\"F/F+7$$\"3g)***\\7JFn=F/F+7$$\"3#z****\\_qn2#F/F+7$$\"3U)***\\ i&p@[#F/F+7$$\"3B)****\\2'HKHF/F+7$$\"3ElmmmZvOLF/F+7$$\"3i******\\2go PF/F+7$$\"3UKL$eR<*fTF/F+7$$\"3m******\\)Hxe%F/F+7$$\"3ckm;H!o-*\\F/F+ 7$$\"3y)***\\7k.6aF/F+7$$\"3#emmmT9C#eF/F+7$$\"33****\\i!*3`iF/F+7$$\" 3%QLLL$*zym'F/F+7$$\"3wKLL3N1#4(F/F+7$$\"3Nmm;HYt7vF/F+7$$\"3Y*******p (G**yF/F+7$$\"3]mmmT6KU$)F/F+7$$\"3fKLLLbdQ()F/F+7$$\"3[++]i`1h\"*F/F+ 7$$\"3W++]P?Wl&*F/F+7$F\\pF+-%'COLOURG6&%$RGBG$\"#5F)F+F+-%*THICKNESSG 6#\"\"#-%+AXESLABELSG6$Q\"t6\"Q!Fiy-%%VIEWG6$;F(F\\p%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "int(A*exp(-I*omega*t),t=-tau /2..tau/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**^#!\"\"\"\"\"%\"AGF&, &-%$expG6#*(^##F&\"\"#F&%$tauGF&%&omegaGF&F&-F*6#*(^##F%F/F&F0F&F1F&F% F&F1F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "F_rec:=simplify(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&F_recG,$**\"\"#\"\"\"%\"AGF(% &omegaG!\"\"-%$sinG6#,$*(F'F+%$tauGF(F*F(F(F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "with(inttrans);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7/%)addtableG%(fourierG%+fouriercosG%+fouriersinG%'hankelG%(hilb ertG%+invfourierG%+invhilbertG%+invlaplaceG%*invmellinG%(laplaceG%'mel linG%*savetableG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "fourier (rect,t,omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"#\"\"\"%\"A GF&%&omegaG!\"\"-%$sinG6#,$*(F%F)%$tauGF&F(F&F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "Jak bylo uvedeno, Fourier\371v obraz Diracova \+ pulzu je roven jeho plo\232e. Nyn\355 pou\236ijme pro v\375po\350et p \370\355m\375 p\370\355kaz." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fourier(Dirac(t),t,omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Nepou\236\355vat funkci piecewi se" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "rc:=piecewise(t>-tau/2 and t%#rcG-%*PIECEWISE G6$7$%\"AG32,$*&\"\"#!\"\"%$tauG\"\"\"F/%\"tG2F2,$*&F.F/F0F1F17$\"\"!% *otherwiseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot(subs(A= 1,tau=1/4,rc),t=-1..1,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 526 241 241 {PLOTDATA 2 "6&-%'CURVESG6$7co7$$!\"\"\"\"!$F*F*7$$!3ommm;p0k& *!#=F+7$$!3wKL$3s%HaF/F+7$$ !3Q+++]$*4)*\\F/F+7$$!39+++]_&\\c%F/F+7$$!31+++]1aZTF/F+7$$!3umm;/#)[o PF/F+7$$!3hLLL$=exJ$F/F+7$$!3*RLLLtIf$HF/F+7$$!3]++]PYx\"\\#F/F+7$$!3E MLLL7i)4#F/F+7$$!3c****\\P'psm\"F/F+7$$!3s*****\\F&*=Y\"F/F+7$$!3')*** *\\74_c7F/F+7$$!3`\"zp)**z7V7F/$\"\"\"F*7$$!3>$eRs3N(H7F/F\\p7$$!3'[P4 YE\\H?\"F/F\\p7$$!3%)\\(=nVjh<\"F/F\\p7$$!3=L$e 9hx$\\6F/F\\p7$$!3#)*\\P4'f!e4\"F/F\\p7$$!3ZmmT5VBU5F/F\\p7$$!3#y**\\P 454N*!#>F\\p7$$!3)3LLL3x%z#)FfqF\\p7$$!3gKL$e9d;J'FfqF\\p7$$!3KMLL3s$Q M%FfqF\\p7$$!3]^omm;zr)*!#@F\\p7$$\"3%pJL$ezw5VFfqF\\p7$$\"3s*)***\\PQ #\\\")FfqF\\p7$$\"3ilm\"z\\1A-\"F/F\\p7$$\"3GKLLe\"*[H7F/F\\p7$$\"3i:/ w7J(GC\"F/F\\p7$$\"3'*)\\(=nqDc7F/F+7$$\"3K#e9;-T'p7F/F+7$$\"3kl;/w\\- $G\"F/F+7$$\"3MKe*[)Gz48F/F+7$$\"3.***\\PzglL\"F/F+7$$\"3UK$e9h'4!R\"F /F+7$$\"3ylm;HCjV9F/F+7$$\"3aKLekSq]:F/F+7$$\"3I*******pvxl\"F/F+7$$\" 3g)***\\7JFn=F/F+7$$\"3#z****\\_qn2#F/F+7$$\"3U)***\\i&p@[#F/F+7$$\"3B )****\\2'HKHF/F+7$$\"3ElmmmZvOLF/F+7$$\"3i******\\2goPF/F+7$$\"3UKL$eR <*fTF/F+7$$\"3m******\\)Hxe%F/F+7$$\"3ckm;H!o-*\\F/F+7$$\"3y)***\\7k.6 aF/F+7$$\"3#emmmT9C#eF/F+7$$\"33****\\i!*3`iF/F+7$$\"3%QLLL$*zym'F/F+7 $$\"3wKLL3N1#4(F/F+7$$\"3Nmm;HYt7vF/F+7$$\"3Y*******p(G**yF/F+7$$\"3]m mmT6KU$)F/F+7$$\"3fKLLLbdQ()F/F+7$$\"3[++]i`1h\"*F/F+7$$\"3W++]P?Wl&*F /F+7$F\\pF+-%'COLOURG6&%$RGBG$\"#5F)F+F+-%+AXESLABELSG6$Q\"t6\"Q!Fey-% *THICKNESSG6#\"\"#-%%VIEWG6$;F(F\\p%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "fourier(rc,t,omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%(fourierG6%-%*PIECEWISEG6$7$%\"AG32,$*&\"\"#!\"\"%$tauG\"\"\"F 0%\"tG2F3,$*&F/F0F1F2F27$\"\"!%*otherwiseGF3%&omegaG" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 89 "Fourierova transformace periodick\375ch funkc\355 a vztah mezi \+ Fourierovou tranzformac\355 a \370adou" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 480 "V\375sledkem Fourierova obrazu je tedy spojit\341 funkce , narozd\355l od Fourierovy \370ady. Dal\232\355 odli\232nost\355 je f yzik\341ln\355 rozm\354r t\351to funkce (obrazu) oproti fyzik\341ln \355mu rozm\354ru koeficient\371 \370ady (viz koment\341\370 d\341le). Pro n\341\232 p\370\355pad obd\351ln\355hov\351ho puzlu dostaneme ted y velmi podobn\375 v\375raz jeho obrazu v porovn\341n\355 se vztahem p ro fourierovy koeficienty periodicky se opakuj\355c\355ho pr\371b\354h u. Stejn\375 vztah dostaneme, pokud ve Fourierov\354 obrazu p\370ejdem e od spojit\351ho kmito\350tov\351ho spektra k diskr\351tn\355mu (" } {XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 4 " -> " }{XPPEDIT 18 0 "k*omega0;" "6#*&%\"kG\"\"\"%'omega0GF%" }{TEXT -1 45 ") a ten pak d \354l\355me z\341kladn\355 periodou sign\341lu." }{MPLTEXT 1 0 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "F_rec;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "subs(omega=k*omega0,F_rec)/T;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "ck;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"#\"\"\" %\"AGF&%&omegaG!\"\"-%$sinG6#,$*(F%F)%$tauGF&F(F&F&F&F&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*.\"\"#\"\"\"%\"TG!\"\"%\"AGF&-%$sinG6#,$**F%F (%\"kGF&%'omega0GF&%$tauGF&F&F&F/F(F0F(F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.\"\"#\"\"\"%\"TG!\"\"%\"AGF&-%$sinG6#,$**F%F(%\"kGF &%'omega0GF&%$tauGF&F&F&F/F(F0F(F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Obecn\354 tedy plat\355 n\341sleduj\355c\355 pravidlo: Pokud z n\341me Fourier\371v obraz \350asov\354 omezen\351 funkce s dobou trv \341n\355 " }{XPPEDIT 18 0 "tau;" "6#%$tauG" }{TEXT -1 143 ", m\371 \236eme jednodu\232e ur\350it koeficienty Fourierovy \370ady periodick \351 funkce, kter\341 vznikla z p\371vodn\355 funkce jej\355m periodic k\375m opakov\341n\355m s periodou " }{XPPEDIT 18 0 "tau < T;" "6#2%$t auG%\"TG" }{TEXT -1 19 ". Tyto koeficienty " }{XPPEDIT 18 0 "c[k];" "6 #&%\"cG6#%\"kG" }{TEXT -1 46 " ur\350\355me tak, \236e do Fourierova o brazu dosad\355me " }{XPPEDIT 18 0 "omega = k*omega[0];" "6#/%&omegaG* &%\"kG\"\"\"&F$6#\"\"!F'" }{TEXT -1 64 " (p\370ejdeme od spojit\351ho \+ kmito\350tov\351ho spektra k diskr\351tn\355mu, kde " }{XPPEDIT 18 0 " omega[0] = 2*Pi/T;" "6#/&%&omegaG6#\"\"!*(\"\"#\"\"\"%#PiGF*%\"TG!\"\" " }{TEXT -1 33 ") a v\375sledek pak d\354l\355me periodou " }{XPPEDIT 18 0 "T;" "6#%\"TG" }{TEXT -1 86 ". To tak\351 vypl\375v\341 z v\375 \232e uveden\375ch vztah\371 - ze vztahu pro koeficienty Fourierovy \+ \370ady " }{XPPEDIT 18 0 "c[k];" "6#&%\"cG6#%\"kG" }{TEXT -1 23 " a pr o Fourier\371v obraz " }{XPPEDIT 18 0 "F(j*omega);" "6#-%\"FG6#*&%\"jG \"\"\"%&omegaGF(" }{TEXT -1 2 ". " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Pou\236ijme toto pravidlo na n\341\232 p\370 \355pad. 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"" {TEXT -1 165 "Jak\375 je tedy vztah mezi spektrem peri odick\351ho sign\341lu vypo\350ten\375m pomoc\355 Fourierovy transform ace a koeficienty jeho Fourierova rozvoje. Nap\370\355klad pro spektru m funkce " }{XPPEDIT 18 0 "A*cos(omega0*t);" "6#*&%\"AG\"\"\"-%$cosG6# *&%'omega0GF%%\"tGF%F%" }{TEXT -1 47 " plat\355 v\375\232e uveden\375 \+ vztah. Vzhledem k~tomu, \236e " }{XPPEDIT 18 0 "A*cos(omega0*t) = A/2* (exp(I*omega0*t)+exp(-I*omega0*t));" "6#/*&%\"AG\"\"\"-%$cosG6#*&%'ome ga0GF&%\"tGF&F&*(F%F&\"\"#!\"\",&-%$expG6#*(%\"IGF&F+F&F,F&F&-F26#,$*( F5F&F+F&F,F&F/F&F&" }{TEXT -1 45 ", jsou koeficienty komplexn\355 Four ierovy \370ady " }{XPPEDIT 18 0 "c[-1] = c[1];" "6#/&%\"cG6#,$\"\"\"! \"\"&F%6#F(" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "A/2;" "6#*&%\"AG\"\"\" \"\"#!\"\"" }{TEXT -1 107 ". Jejich grafick\351 zn\341zorn\354n\355 je uvedeno na n\341sledij\355c\355m obr\341zku (a). 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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::<:ry:::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::Z:5:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Plocha Diracov\375ch pulz\371 je p \370itom \372m\354rn\341 velikosti koeficientu " }{XPPEDIT 18 0 "ck;" "6#%#ckG" }{TEXT -1 54 " Fourierova rozvoje. Je-li fyzik\341ln\355 roz m\354r amplitudy " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 141 " nap \370. [V], pak plocha tohoto pulzu m\341 fyzik\341ln\355 rozm\354r [V \+ * s * rad / s]=[V * rad]. Pokud tedy chceme zn\341t velikost p\370\355 slu\232n\351ho koeficientu " }{XPPEDIT 18 0 "ck;" "6#%#ckG" }{TEXT -1 59 " sta\350\355 vyd\354lit plochu odpov\355daj\355c\355ho Diracova pu lzu \350lenem " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" } {TEXT -1 162 ". To tak\351 evidentn\355 ze vztahu pro zp\354tnou Fouri erovu transformaci. 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[o\"F`w$\"3+=x3>XSi6FZ7$$\"33++Dr\"[8v\"F`w$\"3h&)f^**z+B&*Fhn7$$\"3++ ++Ijy5=F`w$\"3aaF_YLqnzFhn7$$\"31+]P/)fT(=F`w$\"3!\\%*oHkr\")e'Fhn7$$ \"31+]i0j\"[$>F`w$\"3IXFk(>w?\\&Fhn7$$\"\"#F*$\"3&fm7E%4e;XFhn-%'COLOU RG6&%$RGBG$\"#5F)F+F+-%+AXESLABELSG6$Q\"t6\"Q!F]]l-%*THICKNESSG6#F`\\l -%%VIEWG6$;F(F_\\l%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "fourier(ex2*Heaviside(t-t0),t,omega) assuming a>0, t0 >0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%\"UG\"\"\",&%\"aGF%*&%&omega GF%^#F%F%F%!\"\"-%$expG6#*(^#F+F%F)F%%#t0GF%F%" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "\332l oha 2.1.3" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "M\341me ur\350it frek ven\350n\355 p\370enos obvodu (do kmito\350tu " }{XPPEDIT 18 0 "fm;" " 6#%#fmG" }{TEXT -1 156 ", kde u\236 je p\370enos obvod u t\351m\354 \370 nulov\375. Obvod m\341me vybudit pulzem a kmito\350tov\375 p\370e nos pak ur\350it z jeho odezvy. Jak \372zk\375 pulz je t\370eba pro vy buzen\355 obvodu? " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Spektrum pulzu." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "F_rec;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"#\" \"\"%\"AGF&-%$sinG6#,$*(F%!\"\"%&omegaGF&%$tauGF&F&F&F.F-F&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "L\351pe je to vid\354t z tohoto vz tahu " }{BITMAP 77 43 43 1 "?TMWsCB:n@>?fCV?F:k:;::::yyyyyy::::::::::: ::::::::::::::::::::::::::Z:vyyuy::::::::::::::::::::::::::::::::::::: Z:Zy=:::::::::::::::::::::XweYWF\\KG\\KFd;jyyyyyI::::::::::::B:ry::::: :::::::::::::::::fpohpGhkGPv\\QvUXmUPv\\QvyyyyyY::::::::::::<:xI:::::: ::::::::::::::::Rg>obHopqxpqXxyyyyy;:::::::::::B:ry::::::::::::::::::: ::::fkGhkUXmUxyyyyy=::::::::::::<:xI:::::::::::::::::::::bwgiwixoixyyy ySIuSI@kb@SIuSyyyyyyA:::::::::::Z:Zy=:::::::::::::::::::::@vZAvYxMZxyy oX\\KFdKkEXkky;:::::::::::B:ry::::::::::::::::::::::VmUXmwgiwwdQ?eohpo xyyyyy=::::::::::::<:xI:::::::::::::::::::::::::::::::::::::B:ry:::::: :::::::::::::ZF`kG`kFhK:::::ZyyyyyY::::::::::<:xI::::::::::::::::::::: ::::::::::::::::B:ry:::::::::::::::::::::::::::::::::::::Z:Zy=:::::::: :::::::::::@FitOkDdKWdSEUSuOiDP=Xj=xyyyyyaQMplPpxyywY::::::::::::<:xI: ::::::::::::::::::rIyCJ?ir?y`^gf_gFasFagf_gQytQyYlHkZpqxpyYxI::::::::: ::B:ry:::::::::::::::::::ZQwdRwyyyy\\Qv\\ydQ?E^Wpr@qryyiyMplPp:xI::::: ::::::B:ry:::::::::::::::::::ZolZAvZkZ;gaRj@qr@EHeWgeWgvYxI::::::::::: B:ryfP\\KFdKFr`qv`yaymZsI>]RN]yva:gpohpymyCU^SSxyyqy;::::::::::bwfkGhk ymy;SHmSHUlPuUYumPplPF\\[R:Xoe\\?xI::::::::::B:ry:fiwgifawfyYxIroiPFBm SHmyay=pBq>vY:::::::::::::::::::::::::::::<:xI:SawgiwwYxIRj>]RN]QytQyY xIBm^GSuyvY:::::::::::::::::::::::::::::<:xI:BdKWditOktOiDPrIyCJkZfmys yZpMFryvY:J@C]:::::::::::::::::::::::::vYxIB:ry::^GjyCd>]RN=vYXGry^GlX pmxysy::::::::::::::::::::::::::::Z:Zy=:J`CdKWdyyyyY]vay>bymySjVmUXm\\ mGSgeWg?ir?irvYxI:::::::::::::::::::::::::::B:ry::ZwKpSYvDxIyAVM\\KFdK F::eWgewyvY::::::::::::TIuUYuewgiwyay=:XGkZR:jysy::nXpBgiwxIyA::Z:Zy=: :lHrOfiwgiymy;::::::::::::::::RYfpohPvY::jTS]NVhC?aymy;:Rgfiwgiymy;::B :ry:::>O\\;U^ymy;::::::::::::::::e?Zy=:::PEK_pMusyvY::jxRgvYxI::B:ry:: :ZQKN]dymy;::::::::::::::::y[;Ir;yYxI::::U^Ulysy:::FRX?yay=::<:xI:::ro Nv\\QVvY::::::::::::::::lFkB@TIry::Zpmesy^W\\;Saymy;::B@kb@S`ymy;:B:ry ::::::::::Zfk@qBfkGhkQytQqxpqxfKFrO\\OkEXkmPp:^sFasfCUk:vtQytymy;r`^Gx IyA::Z;mYrynX\\;?aymy;::RYVmUtyvY::<:xI::::::::::B@^WJusOiDPyayMFroZyM F:ry>=:xI:RY:vY:::jGSYpqeRgZy=:::hCryvY::<:xI::::::::::r`vYxIS`yva:[yM F:ry>=:xIk:Zy=:\\;e?vY::::::::::PEkZymy;:B:ry::::::::::ZfmxB@kb@kZ;Ir; yYxIkZQ=xIk:ZyMF:ry:ZRjTsyvY:\\;k::::::Zymy;Rg>=jy;:B:ry::::::::::ZQkx RgNVDHyayMFroZyMF:ry>=:xI:B@fPxIyA:::::::::ZoLFryvY::<:xI::::::::::rIn xpqxyMps`qv`ry>=XGry>=hCbw>=lHyay=:x=e_ymy;:::::::::Rgvay>bymy;:B:ry:: ::::::::ZQyTqv`qv`fK\\TNaTuCUvayroixogAZFasFYmUTY>obtyvY::pAUn@yIyAZ;m YR]UV]q^yva:WmUXmpqxpyYXGS]ymy;::R]fiwgiymy;:B:ry::::::::::::::::::::: ZbLp:yAZyvA?aoLF:q^w;lHkZp=xI::rO>OxIyA:Z:Zy=:::::::::::::lFy[fawfyYxI :::::br>]RN=Zy=X?yaymY:r`NVxIS]Qmysy::nX\\KFNVjy;:B:ry:::::::::::::ZQM pBulxysy::::::^GPuTQuysyZolZsy^STYuwY\\Dk:fPPEyay=:jTsOiDPry::Z:Zy=::: ::::::::::::::::::jesOiDPgiwgAyAfKxIyAZfK`_sfCUkZR:X_RwYxIbwvDXGyay=:: <:xI::::::::::::::::::::::Bd>obHoymy;:::::::Bd>OxIyA::Z:Zy=::::::::::: :::::::::::jTcwgiwyay=::::::JuSgeWgyay=:::<:xI:::::::::::::::::::::::: :::::::::::::B:ry:::::::::::::::::::::::::::::::::::::Z:3:" }{TEXT -1 10 ". Dosa\357me " }{XPPEDIT 18 0 "omega = 2*Pi*f;" "6#/%&omegaG*(\"\" #\"\"\"%#PiGF'%\"fGF'" }{TEXT -1 6 ", tj. " }{XPPEDIT 18 0 "omega*tau/ 2 = Pi*f*tau;" "6#/*(%&omegaG\"\"\"%$tauGF&\"\"#!\"\"*(%#PiGF&%\"fGF&F 'F&" }{TEXT -1 18 " a ozna\350me sou\350in " }{XPPEDIT 18 0 "f*t = x; " "6#/*&%\"fG\"\"\"%\"tGF&%\"xG" }{TEXT -1 93 ". Pak vyneseme, jak se \+ zmen\232uje spektrum v z\341vislosti na tomto sou\350inu (relativn\354 v\371\350i plo\232e " }{XPPEDIT 18 0 "A*tau;" "6#*&%\"AG\"\"\"%$tauGF %" }{TEXT -1 4 "). " }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(sin(Pi*x)/(Pi*x),x=0..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 639 353 353 {PLOTDATA 2 "6%-%'CURVESG6$7fn7$$\"3`*****\\n5;\"o!#@$\"2i 3IyO#******!#<7$$\"3#******\\8ABO\"!#?$\"3u;8Mr%p*****!#=7$$\"33+++-K[ V?F1$\"3UJmf58$*****F47$$\"3#)******pUkCFF1$\"3%>y+d)y()****F47$$\"3s* ****\\Smp3%F1$\"3y`\\3W_s****F47$$\"3k******R&)G\\aF1$\"3'eSr\"[:^**** F47$$\"3Y******4G$R<)F1$\"3k@k^.5!*)***F47$$\"3%******zqd)*3\"!#>$\"3u 6edyi/)***F47$$\"3*)*****>c'yM;FQ$\"3c.e')[Wg&***F47$$\"3')*****fT:(z@ FQ$\"3GICS)['=#***F47$$\"3#*******zZ*z7$FQ$\"33!4jk:8R)**F47$$\"33+++X TFwSFQ$\"3g;BSg,ps**F47$$\"3&******4z_\"4iFQ$\"3u=)3>X-n$**F47$$\"3o** ****R&phN)FQ$\"37e!ptmOb))*F47$$\"3++++*=)H\\5F4$\"3T+h]p)p)>)*F47$$\" 3%******z/3uC\"F4$\"33x3')z?+Y(*F47$$\"35+++J$RDX\"F4$\"3]xREje`c'*F47 $$\"37+++)R'ok;F4$\"33(Qs%=BN]&*F47$$\"3-+++1J:w=F4$\"3_jju$4m4V*F47$$ \"33+++3En$4#F4$\"3iSdeZmQ%H*F47$$\"3-+++/RE&G#F4$\"3?!QVG/9G;*F47$$\" 3\")*****\\K]4]#F4$\"3QcJ6#fGC+*F47$$\"3$******\\PAvr#F4$\"3-!G5Y2aF4$\"3s0\")QN5LQeF47$$\"3\\+++yXu9c F4$\"3-5KI27yjbF47$$\"3d******[y))GeF4$\"3)Qsq**f&zw_F47$$\"3.+++i_QQg F4$\"33p#R9'*HM*\\F47$$\"3A+++!y%3TiF4$\"3DoP#fA\\ur%F47$$\"35+++O![hY 'F4$\"3$f*=;n\")o4WF47$$\"3I+++#Qx$omF4$\"3$G`B4iAE8%F47$$\"3s*****RP+ V)oF4$\"3,>dtj+&p$QF47$$\"3Y*****ppe*zqF4$\"3e+U%pef(pNF47$$\"3u*****R #\\'QH(F4$\"3Uf[E6*)4zKF47$$\"3#******H,M^\\(F4$\"3%eBgW*)*e2IF47$$\"3 S+++0#=bq(F4$\"3ZSUDs)fks#F47$$\"3Y*****p?27\"zF4$\"3K9Z(H$=%[X#F47$$ \"3a+++IXaE\")F4$\"3M.3NP6au@F47$$\"37+++l*RRL)F4$\"3r>tdy)*34>F47$$\" 3i*****HvJga)F4$\"3)RGopi)yU;F47$$\"3s*****HJnjv)F4$\"3oWhO>IS%Q\"F47$ $\"3k******[Qk\\*)F4$\"3?0*4/)fW_6F47$$\"3w******o0;r\"*F4$\"334vS;SnN *)FQ7$$\"3[*****\\w(Gp$*F4$\"33Xzmeit(o'FQ7$$\"37+++!oK0e*F4$\"3QB+9\" focO%FQ7$$\"33+++<5s#y*F4$\"3YC(fe8C$>AFQ7$$\"\"\"\"\"!$\"3mt$>D$=<)*Q !#M-%'COLOURG6&%$RGBG$\"#5!\"\"$Ff]lFf]lFa^l-%+AXESLABELSG6$Q\"x6\"Q!F f^l-%%VIEWG6$;Fa^lFd]l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Jestli\236e zvol\355me " }{XPPEDIT 18 0 "fm*tau = 1/10;" "6#/*&%#f mG\"\"\"%$tauGF&*&F&F&\"#5!\"\"" }{TEXT -1 58 ", pak spektrum takov \351ho pulzu vykazuje pro tento kmito\350et " }{XPPEDIT 18 0 "f = fm; " "6#/%\"fG%#fmG" }{TEXT -1 42 " pokles cca o 2%, co\236 je mo\236n \351 tolerovat. " }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "evalf(subs(x=0.1,sin(Pi*x)/(Pi*x)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+IkJO)*!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Prakticky \+ nap\370. pro " }{XPPEDIT 18 0 "fm = 100;" "6#/%#fmG\"$+\"" }{TEXT -1 12 "kHz vych\341z\355 " }{XPPEDIT 18 0 "tau = 1*mu;" "6#/%$tauG*&\"\" \"F&%#muGF&" }{TEXT -1 2 "s." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(100e3*tau=0.1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++5!#:" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "\332loha 2.1.4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "S:=1/1000;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "u2:=1/2*exp(-1000*t)*Heaviside(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG#\"\"\"\"%+5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u2G,$*& #\"\"\"\"\"#F(*&-%$expG6#,$*&\"%+5F(%\"tGF(!\"\"F(-%*HeavisideG6#F1F(F (F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "U1:=fourier(Dirac(t) *S,t,omega);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "U2:=fourier(u2,t,om ega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#U1G#\"\"\"\"%+5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#U2G,$*&\"\"\"F'*&\"\"#F',&\"%+5F'*&%&omeg aGF'^#F'F'F'F'!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "P:= U2/U1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG,$*&\"$+&\"\"\",&\"%+5 F(*&%&omegaGF(^#F(F(F(!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "evalf(numer(P)/1000,1)/evalf((denom(P)/1000),1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&$\"\"&!\"\"\"\"\",&$F(\"\"!F(*&^#$F(!\"$F(%&om egaGF(F(F'F(" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "P\370\355klad 2.1.4" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 32 "u1:=2*exp(-2000*t)*Heaviside(t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "u2:=(8*exp(-2000*t)-6*exp(-3000*t)-2*exp(-1 000*t))*Heaviside(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u1G,$*(\" \"#\"\"\"-%$expG6#,$*&\"%+?F(%\"tGF(!\"\"F(-%*HeavisideG6#F/F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u2G*&,(*&\"\")\"\"\"-%$expG6#,$*&\" %+?F)%\"tGF)!\"\"F)F)*&\"\"'F)-F+6#,$*&\"%+IF)F0F)F1F)F1*&\"\"#F)-F+6# ,$*&\"%+5F)F0F)F1F)F1F)-%*HeavisideG6#F0F)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "U1:=fourier(u1,t,omega);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "U2:=fourier(u2,t,omega);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "P:=U2/U1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#U1G,$* &\"\"#\"\"\",&\"%+?F(*&%&omegaGF(^#F(F(F(!\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#U2G*,^#\"%+S\"\"\"%&omegaGF(,&\"%+?F(*&F)F(^#F(F(F(! \"\",&\"%+IF(F,F(F.,&\"%+5F(F,F(F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"PG**^#\"%+?\"\"\"%&omegaGF(,&\"%+IF(*&F)F(^#F(F(F(!\"\",&\"%+5F(F, F(F." }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 "V\375po\350ty pomoc\355 Laplaceovy transformace " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Laplaceova transformace Diraco va pulzu." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "lap lace(Dirac(t),t,p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Laplaceova transformace jednotkov \351ho skoku." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "laplace(Heaviside(t),t,p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\" \"F$%\"pG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "P\370\355kaz \" laplace\" evidentn\354 ignoruje funk\350n\355 hodnoty pro " }{XPPEDIT 18 0 "t < 0;" "6#2%\"tG\"\"!" }{TEXT -1 21 " (integruje se od 0)." } {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "laplace(1,t,p); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$%\"pG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "N\341sleduj\355 v\375po\350ty n\354kter \375ch v\375\232e uveden\375ch p\370\355klad\371 pomoc\355 Laplaceovy \+ transformace." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "laplace(U*exp(-a*t),t,p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"UG\"\"\",&%\"pGF%%\"aGF%!\"\"" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 38 "Je nutn\351 vyn\341sobit jednotkov\375m skokem." } {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "laplace(U*exp(- a*(t-t0)),t,p);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "laplace(U*exp(-a *(t-t0))*Heaviside(t-t0),t,p) assuming t0>0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%\"UG\"\"\"-%$expG6#*&%\"aGF%%#t0GF%F%,&%\"pGF%F*F%! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(%\"UG\"\"\",&%\"pGF%%\"aGF%! \"\"-%$expG6#,$*&F'F%%#t0GF%F)F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "invlaplace(%,p,t) assuming t0>0;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*(%\"UG\"\"\"-%$expG6#,$*&%\"aGF%,&%\"tGF%%#t0G!\"\"F %F/F%-%*HeavisideG6#F,F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "u1:=2*exp(-2000*t)*Heaviside(t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "u2:=(8*exp(-2000*t)-6*exp(-3000*t)-2*exp(-1000*t))*Heaviside(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u1G,$*(\"\"#\"\"\"-%$expG6#,$*&\" %+?F(%\"tGF(!\"\"F(-%*HeavisideG6#F/F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u2G*&,(*&\"\")\"\"\"-%$expG6#,$*&\"%+?F)%\"tGF)!\"\"F)F)*&\" \"'F)-F+6#,$*&\"%+IF)F0F)F1F)F1*&\"\"#F)-F+6#,$*&\"%+5F)F0F)F1F)F1F)-% *HeavisideG6#F0F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "U1:=la place(u1,t,p);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "U2:=laplace(u2,t, p);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "P:=U2/U1;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#U1G,$*&\"\"#\"\"\",&%\"pGF(\"%+?F(!\"\"F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#U2G,$*,\"%+S\"\"\"%\"pGF(,&F)F(\"%+ ?F(!\"\",&F)F(\"%+IF(F,,&F)F(\"%+5F(F,F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG,$**\"%+?\"\"\"%\"pGF(,&F)F(\"%+IF(!\"\",&F)F(\"%+5F(F,F( " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "P\370edpokl\341d\341 se, \236 e funkce \"za\350\355n\341\" v nule. " }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "invlaplace(U2,p,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\")\"\"\"-%$expG6#,$*&\"%+?F&%\"tGF&!\"\"F&F&*&\" \"'F&-F(6#,$*&\"%+IF&F-F&F.F&F.*&\"\"#F&-F(6#,$*&\"%+5F&F-F&F.F&F." }} }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "13 1 0" 53 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }