Cours Fourier Nn
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18-Dec-2015 -
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Transcript of Cours Fourier Nn
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lectronique analogique*lectronique analogiquetransformation de Fourier signal priodiquesignal non priodique
systmes linaires
amplificationamplificateuramplificateur oprationnel
filtrage
oscillateurs
lectronique analogique
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lectronique analogique*transformation de Fourier :
x(t) somme de signaux sinusodaux si x(t) est priodique, sa TF est discrte :
si x(t) est non priodique, sa TF est continue :
lectronique analogique
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lectronique analogique*transformation de Fourier d'un signal priodique :namplitude
lectronique analogique
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lectronique analogique*transformation de Fourier d'un signal priodique :tTT/2reconstruction de x(t) : la courbe rougeest la somme des 4 premires harmoniques
lectronique analogique
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lectronique analogique*transformation de Fourier d'un signal non priodique :trac de X(f) pour M=1 etT=1, T=4 et T=0,4
lectronique analogique
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lectronique analogique*transformation de Fourier la TF est linaire
dualit temps/frquence
temps "brefs" frquences levestemps "longs" frquences faibles
enjeu : augmentation des dbits de traitementde l'information frquences leves
lectronique analogique
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lectronique analogique*lectronique analogiquetransformation de Fourier signal priodiquesignal non priodique
systmes linaires
amplificationamplificateuramplificateur oprationnel
filtrage
oscillateurs
lectronique analogique
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lectronique analogique*systmes linairesS.L.x(t)y(t)la relation reliant y(t) x(t) estune quation diffrentiellelinaire coefficients constants : exemple : RCx(t)y(t)i(t)
lectronique analogique
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lectronique analogique*systmes linairesexemple : RCx(t)y(t)i(t)si x(t) est sinusodal : x(t)=Xsin(wt),alors y(t) est aussi sinusodal : y(t)=AXsin(wt+j)
lectronique analogique
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lectronique analogique*systmes linairesexemple : RCx(t)y(t)i(t)Xwt2px(t)y(t) pour RCw=0,1y(t) pour RCw=1y(t) pour RCw=10
lectronique analogique
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lectronique analogique*systmes linairesS.L.Aejwtexemple : R1/jCwX(w)IdoncA G(w) ejwtY(w)G(w) =
lectronique analogique
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lectronique analogique*systmes linaires lien avec la transformation de FourierX(w)X(f)x(t)y(t)TF-1Y(w) = G(w) X(w)Y(f) = G(f) Y(f)G(w)les signaux harmoniques sont les fonctions propresdes systmes linaires
lectronique analogique
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lectronique analogique*systmes linairesexemple : RCx(t)y(t) ?101001031041w(rd/s)|G(w)|dB
lectronique analogique
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lectronique analogique*systmes linairesexemple : 101001031041w(rd/s)ttt
lectronique analogique
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lectronique analogique*lectronique analogiquetransformation de Fourier signal priodiquesignal non priodique
systmes linaires
amplificationamplificateuramplificateur oprationnel
filtrage
oscillateurs
lectronique analogique
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lectronique analogique*amplificationsystme linaire caractris par G(f)>1 apport d'nergie amplificateur idal:Ve(w)Vs(w)A(w)i=0le courant d'entre est nulla sortie est une source de tension parfaite
lectronique analogique
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lectronique analogique*amplification amplificateur non idal (modle linaire):Ve(w)Vs(w)A(w).Ve(w)ieisReRsVsA.VeieisReRsVeRgEgRc
lectronique analogique
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lectronique analogique*amplification cascade d'amplificateurs:VeA.VeieReRsV'eVsA'.V'eR'eR'samplificateur d'entre : Re leveamplificateur de sortie : Rs faible
lectronique analogique
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lectronique analogique*amplificateur oprationnel amplificateur oprationnel idal:A.(v+-v-)ieReRs+-v+v-vsv+-v-A Re Rs 0v+-v- 0is 0
lectronique analogique
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lectronique analogique*amplificateur oprationnel exemples de montages linaires :vsveR1R2ve0vsveR1R200
lectronique analogique
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lectronique analogique*amplificateur oprationnel exemples de montages linaires :vsveR'R00vsieR00C
lectronique analogique
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lectronique analogique*lectronique analogiquetransformation de Fourier signal priodiquesignal non priodique
systmes linaires
amplificationamplificateuramplificateur oprationnel
filtrage
oscillateurs
lectronique analogique
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lectronique analogique*filtrage rduction du bruit:
V(f)sf antirepliement:
V(f)ffe2fe
lectronique analogique
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lectronique analogique*filtrage slection (ou limination) d'une bande frquentielledans le spectre d'un signal :fV(f)fp1fp2fp3s1s2s3slection d'un signal modul en amplitude
lectronique analogique
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lectronique analogique*filtrage slection (ou limination) d'une bande frquentielledans le spectre d'un signal :tfV(f)rjection de parasitesv(t)fV(f)
lectronique analogique
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lectronique analogique*filtrageSystme linaire: Les signaux harmoniques sont fonctions propres de loprateurlinaires.Fonction de transfert:Stabilit:p k et ples parties relles ngatives
lectronique analogique
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lectronique analogique*filtrageLes ples sont rels ou complexes conjugus est dcomposable en 2me ordre1er ordreUn filtre dordre quelconque peut tre raliser par la cascade de filtres du premier et du deuxime ordre.
lectronique analogique
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lectronique analogique*filtrageFiltre du 2me ordre normalis:Q=0,707 ButterworthQ=0,577 BesselQ=1,128 Chebyshev
lectronique analogique
Graph3
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Butterworth
Bessel
Chebychev
Feuil1
ButterworthBesselChebychevButterworthBesselChebychev
0111ButterworthBesselChebychev
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Feuil1
Butterworth
Bessel
Chebychev
Feuil2
Feuil3
Graph4
111
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0.11043152610.10480960190.1186169215
0.1034994290.09853430860.1106443491
0.09719389310.0927937250.1034587721
0.09144263960.0875303190.0969591304
0.086183330.08269386090.0910601399
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0.07693181760.07413128220.0807852477
0.07285191880.07033260530.0762946695
0.06908661120.06681438420.0721720963
0.06560458240.0635501140.0683780803
0.06237828620.06051629250.0648783499
0.05938341750.05769202910.0616430127
0.05659847060.05505871130.0586458985
0.05400436450.05259971790.0558640147
0.05158412460.05030017310.0532770924
0.04932261230.04814673440.0508672066
0.04720629310.04612740960.0486184572
0.04522303830.04423139820.0465166994
0.04336195430.04244895370.0445493153
0.04161323590.04077126430.0427050192
0.03996803830.03919034850.040973692
0.03841836760.03769896360.0393462388
0.03695698440.03629052630.0378144662
0.03557732030.03495904250.036370977
0.03427340520.03369904570.0350090785
0.03303980290.03250554280.0337227034
0.03187155530.03137396590.0325063408
0.03076413270.03030013020.0313549754
0.02971339060.0292801960.0302640351
0.02871553070.02831063590.0292293447
0.02776706720.02738820440.0282470848
0.02686479650.02650991180.0273137561
0.02600576980.02567300050.0264261477
0.02518727010.02487492380.0255813088
0.02440678980.02411332690.0247765237
0.02366201220.02338603010.0240092898
0.02295079420.0226910130.0232772975
0.0222711510.02202640150.0225784128
0.02162124210.02139045470.0219106614
0.02099935920.02078155420.0212722144
0.02040391470.02019819350.0206613758
0.01983343180.01963896940.0200765709
0.01928653540.01910257320.0195163361
0.01876194370.01858778360.0189793094
0.01825846060.01809345940.0184642222
0.01777496910.01761853350.0179698919
0.01731042530.01716200730.0174952145
0.01686385220.01672294550.0170391587
0.01643433510.01630047120.0166007602
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0.0156230930.01550204430.0157733822
0.0152398090.01512459340.0153827646
0.01487045550.01476072630.01500652
Butterworth
Bessel
Chebychev
Feuil1
ButterworthBesselChebychevButterworthBesselChebychev
0111ButterworthBesselChebychev
0.10.99995000370.99496999051.00607528730.1-0.0004342728-0.04380035780.0526096269
0.20.99920095870.97975871241.02434230110.2-0.0069431587-0.17761731830.2089021532
0.30.99597443880.95414465281.05477640880.3-0.0350361474-0.40771558490.4632081551
0.40.98744063190.91817224791.09675221230.4-0.1097801217-0.74151676210.8021703825
0.50.97014250010.87256883881.14784801150.5-0.2632893872-1.18400601251.1976877243
0.60.94088741190.81898141891.20165451120.6-0.5292468371-1.73451902811.5955924239
0.70.89799030160.7598719971.244948880.7-0.934567075-2.38519119871.9030303767
0.80.84227140070.69810145021.25729698820.8-1.4909589107-3.1216291961.9887575047
0.90.77706387850.636407011.21923998940.9-2.1908655717-3.92530090551.7217839696
10.70710678120.5771.1281-3.0102999566-4.77648373691.0461819929
1.10.63704606240.52139170291.0024737611.1-3.9165832872-5.65671768350.0214602821
1.20.57039580950.47042046220.86863507191.2-4.8764734752-6.5502759141-1.2232527924
1.30.50924392410.42439006160.74446412491.3-5.8614828791-7.4446958995-2.5631245143
1.40.45447034170.38323625680.63731569091.4-6.8498890634-8.3306682073-3.9129077784
1.50.40613846610.34667308740.5479280071.5-7.8265175161-9.2015974358-5.2255300059
1.60.36385052710.3143018550.47428153141.6-8.7815398309-10.0530611161-6.4792757309
1.70.32699825040.28568362610.41368158851.7-9.70909142-10.8822930071-7.6666761511
1.80.29491468760.26038307660.36360027981.8-10.6060719552-11.6877449151-8.7875158255
1.90.26695537190.23799236810.32192316951.9-11.4712267104-12.468739391-9.8449553007
20.2425356250.21814220970.28696200292-12.3044892138-13.2252058391-10.8435121022
2.10.22114316440.20050523810.25739335242.1-13.1065296037-13.9574855414-11.7880534729
2.20.20233796590.18479511220.2321849652.2-13.878452407-14.6661904015-12.6833181217
2.30.18574626530.17076343960.21053028192.3-14.6215981942-15.3521021193-13.5337085621
2.40.17105241840.1581957980.19179577832.4-15.3374156268-16.0161011265-14.3432191289
2.50.15799050110.14690755970.1754803542.5-16.0273804686-16.659117109-15.1154299644
2.60.14633651890.1367398970.1611845612.6-16.6929456079-17.2820950342-15.853531186
2.70.13590155640.12755614580.14858742192.7-17.3355113913-17.8859722338-16.56035905
2.80.12652593110.11923859060.13742895592.8-17.9564091585-18.4716633227-17.2384350594
2.90.11807428830.11168567360.12749695552.9-18.55689327-19.0400506418-17.8900037124
30.11043152610.10480960190.11861692153-19.1381385238-19.5919785702-18.5170670274
3.10.1034994290.09853430860.11064434913.1-19.7012409206-20.1282505274-19.1214152316
3.20.09719389310.0927937250.10345877213.2-20.2472204346-20.6496278253-19.7046536069
3.30.09144263960.0875303190.09695913043.3-20.7770249203-21.1568297736-20.2682257653
3.40.086183330.08269386090.09106013993.4-21.2915345865-21.6505346157-20.8134337386
3.50.08136200950.07824037980.08568942643.5-21.7915666775-22.1313809977-21.3414552825
3.60.07693181760.07413128220.08078524773.6-22.2778801347-22.5999697579-21.8533587788
3.70.07285191880.07033260530.07629466953.7-22.7511801015-23.0568658937-22.3501160811
3.80.06908661120.06681438420.07217209633.8-23.2121221925-23.502600601-22.8326136048
3.90.06560458240.0635501140.06837808033.9-23.6613164875-23.9376733197-23.3016619235
40.06237828620.06051629250.06487834994-24.0993312333-24.3625537367-23.7580040922
4.10.05938341750.05769202910.06164301274.1-24.526696254-24.7776837186-24.2023228859
4.20.05659847060.05505871130.05864589854.2-24.943906079-25.1834791543-24.6352471122
4.30.05400436450.05259971790.05586401474.3-25.3514228036-25.5803316979-25.0573571319
4.40.05158412460.05030017310.05327709244.4-25.7496786989-25.9686104072-25.4691897007
4.50.04932261230.04814673440.05086720664.5-26.139078592-26.3486632773-25.8712422281
4.60.04720629310.04612740960.04861845724.6-26.520002033-26.7208186706-26.2639765319
4.70.04522303830.04423139820.04651669944.7-26.8928052692-27.0853866478-26.6478221603
4.80.04336195430.04244895370.04454931534.8-27.2578230431-27.4426602033-27.0231793364
4.90.04161323590.04077126430.04270501924.9-27.6153702301-27.7929164122-27.3904215759
50.03996803830.03919034850.0409736925-27.9657433321-28.1364174918-27.7498980203
5.10.03841836760.03769896360.03934623885.1-28.3092218396-28.4734117855-28.1019355214
5.20.03695698440.03629052630.03781446625.2-28.6460694751-28.8041346741-28.4468405077
5.30.03557732030.03495904250.0363709775.3-28.9765353297-29.1288094182-28.7849006596
5.40.03427340520.03369904570.03500907855.4-29.3008549025-29.4476479399-29.1163864166
5.50.03303980290.03250554280.03372270345.5-29.619251052-29.7608515449-29.4415523353
5.60.03187155530.03137396590.03250634085.6-29.9319348685-30.0686115928-29.760638317
5.70.03076413270.03030013020.03135497545.7-30.2391064745-30.3711101175-30.0738707178
5.80.02971339060.0292801960.03026403515.8-30.5409557595-30.668520403-30.381463356
5.90.02871553070.02831063590.02922934475.9-30.8376630558-30.9610075167-30.6836184273
60.02776706720.02738820440.02824708486-31.1293997608-31.2487288055-30.9805273371
6.10.02686479650.02650991180.02731375616.1-31.4163289104-31.5318343559-31.2723714603
6.20.02600576980.02567300050.02642614776.2-31.6986057074-31.8104674216-31.5593228345
6.30.02518727010.02487492380.02558130886.3-31.9763780105-32.0847648216-31.8415447945
6.40.02440678980.02411332690.02477652376.4-32.249786785-32.3548573105-32.1191925533
6.50.02366201220.02338603010.02400928986.5-32.5189665216-32.6208699232-32.3924137353
6.60.02295079420.0226910130.02327729756.6-32.7840456226-32.8829222963-32.6613488668
6.70.0222711510.02202640150.02257841286.7-33.0451467612-33.1411289684-32.9261318264
6.80.02162124210.02139045470.02191066146.8-33.3023872146-33.3955996593-33.1868902608
6.90.02099935920.02078155420.02127221446.9-33.5558791736-33.6464395316-33.4437459681
70.02040391470.02019819350.02066137587-33.8057300307-33.8937494353-33.6968152525
7.10.01983343180.01963896940.02007657097.1-34.0520426474-34.1376261357-33.9462092512
7.20.01928653540.01910257320.01951633617.2-34.2949156049-34.378162528-34.1920342388
7.30.01876194370.01858778360.01897930947.3-34.5344434363-34.6154478371-34.4343919083
7.40.01825846060.01809345940.01846422227.4-34.7707168441-34.8495678059-34.6733796323
7.50.01777496910.01761853350.01796989197.5-35.0038229038-35.080604871-34.9090907061
7.60.01731042530.01716200730.01749521457.6-35.2338452532-35.3086383286-35.1416145737
7.70.01686385220.01672294550.01703915877.7-35.4608642705-35.5337444895-35.3710370379
7.80.01643433510.01630047120.01660076027.8-35.6849572406-35.7559968252-35.5974404569
7.90.01602101670.01589376160.01617911667.9-35.9061985113-35.975466105-35.8209039269
80.0156230930.01550204430.01577338228-36.1246596395-36.192220526-36.0415034536
8.10.0152398090.01512459340.01538276468.1-36.340409529-36.4063258338-36.2593121119
8.20.01487045550.01476072630.015006528.2-36.5535145589-36.6178454381-36.4744001954
Feuil1
Butterworth
Bessel
Chebychev
Feuil2
Butterworth
Bessel
Chebychev
Feuil3
-
lectronique analogique*filtrageGabarit dun filtre: H(w)critre de " gain plat "dans la bande passanteslectivitphase linaireTransposition de frquence:Exemple: s=w0/sFiltre PB normalis Filtre PHs=w0/sFiltre Passe-BasFiltre Passe-Hauts=s+w02/sFiltre Passe-BasFiltre Passe-Bande
lectronique analogique
-
lectronique analogique*filtrageFiltres de Butterworth:Filtre maximally flat:si N est pair, les ples sont les racines de s2N=ejp, donc sk=ekjp/2N.Ex: N=4xxxxsi N est impair, les ples sont les racines de s2N=ej2p, donc sk=ekjp/N.Ex: N=3xxxxxx
lectronique analogique
-
lectronique analogique*filtrageFiltres de Chebychev:Plus slectif que B.:Les polynmes de C. sont dfinis par: TN+1(x)=2xTN(x)-TN-1(x) avec, T0(x)=1 et T1(x)=x.
lectronique analogique
-
lectronique analogique*filtrageFiltres de Bessel:Pour quun signal ne soit pas dform par un systme linaire,il faut quil subisse un retard pur: s(t)=A.e(t-t).S(f)=A.E(f).exp(-j2pft)Le gain du systme est donc G(f)=A.exp(-j2pft).La phase du filtre varie linairement avec la frquence.Un tel filtre est non causal donc non physique, le filtre de Besselest celui qui approche le mieux un filtre phase linaire.
BN est un polynme de Bessel dfini par:BN(s)=(2N-1)BN-1(s)+s2BN-2(s)avec B0=1 et B1(s)=s+1
lectronique analogique
-
lectronique analogique*filtrageComparaison des fonctions de transfert (filtres dordre 3)Phase comparedes filtres de Butterworthet de Bessel
lectronique analogique
Graph5
10.99955042340.99999
10.99820675460.9999600003
0.99999999960.99598403630.9999100013
0.9999999980.9929068760.9998400043
0.99999999220.9890087780.9997500104
0.99999997670.98433125820.9996400216
0.99999994120.97892278550.9995100399
0.99999986890.97283760380.9993600681
0.99999973430.96613448450.999190109
0.99999950.95887546430.9990001659
0.99999430470.91649356040.9977508356
0.99996800150.86952445430.9960026214
0.9998779520.82404192420.9937563397
0.99963569920.78391864080.9910129975
0.99908213250.75127212970.9877737647
0.99795827010.72717446320.9840399456
0.99587379670.71225175110.9798129512
0.99227787670.70710678120.9750942764
0.98644054010.71260792530.9698854841
0.97745779940.73008339550.9641882
0.96429778580.76137545670.9580041206
0.94590448150.80845631070.9513350367
0.9213641610.87157553710.9441828764
0.89011382510.94326863020.936549767
0.85213705940.99565732790.9284381188
0.80807163990.97745779940.9198507308
0.75916951440.8652188820.9107909168
0.70710678120.70710678120.9012626522
0.65370709140.55972757790.8912707369
0.60067254110.4429561620.8808209729
0.54939575480.35499130520.86992035
0.50087830890.2890445770.8585772359
0.45573827840.23901904710.8468015624
0.41427107260.20043121460.8346050024
0.37652979990.17014906930.8220011292
0.34240279150.1459983520.8090055503
0.31167753830.12645451050.7956360103
0.28408832970.11043152610.7819124532
0.25934910320.09714220990.7678570417
0.23717457190.08600609550.753494127
0.21729281060.07658814870.7388501675
0.19945198470.06855739240.7239535954
0.18342325960.06165852070.7088346333
0.16900134110.05569210120.6935250629
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0.05680390050.01599692890.4497591648
0.05365822040.01503849410.4360716295
0.05073982150.014156470.42268867
0.04802865610.01334326570.4096203975
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0.04315792770.01189733330.3844581294
0.04096766870.01125345140.3723744829
0.03892285850.01065588820.3606264835
0.03701166050.01010049480.3492151479
0.03522336990.00958356610.3381400833
0.03354828970.00910178560.327399629
0.03197762290.00865217860.3169909947
0.03050337720.00823207090.3069103945
0.02911828050.00783905360.2971531747
0.02781570720.00747095280.2877139338
0.0265896120.00712580290.2785866355
0.02543447210.00680182340.2697647128
0.02434523520.00649739870.2612411639
0.02331727380.00621106030.2530086395
0.0223463440.00594147160.2450595219
0.0214285490.00568741370.2373859956
0.02056030690.00544777370.229980111
0.01973832110.00522153370.2228338401
0.01895955440.00500776190.2159391256
0.01822120540.00480560370.2092879241
0.01752068750.00461427420.2028722433
0.01685561010.00443305220.196684173
0.01622376140.00426127350.1907159133
0.0156230930.00409832620.1849597962
0.01505170610.00394364570.1794083048
0.01450783880.00379671050.1740540889
0.01398985480.00365703820.1688899764
0.01349623320.00352418230.1639089837
0.01302555880.00339772940.159104322
0.01257651420.00327729550.1544694027
0.01214787130.00316252450.1499978406
0.01173848490.00305308530.1456834552
0.01134728610.00294866980.1415202713
0.01097327620.00284899130.1375025178
0.01061552160.00275378260.133624626
0.0102731490.00266279440.129881227
0.00994534060.00257579410.1262671482
0.00963133040.00249256440.1227774092
0.00933040010.0024129020.1194072181
0.00904187570.00233661690.1161519659
0.00876512470.0022635310.1130072224
0.00849955270.00219347740.1099687305
0.0082446010.00212629980.1070324008
0.0079997440.00206185130.1041943067
0.00776448710.0019999940.1014506787
0.00753836450.00194059840.0987978987
0.00732093710.00188354270.0962324949
0.0071117910.00182871220.0937511366
0.00691053580.0017759990.0913506285
0.00671680270.00172530140.0890279058
0.00653024380.00167652370.086780029
0.00635052990.00162957530.084604179
0.006177350.0015843710.0824976523
0.00601040980.00154083020.0804578564
0.0058494310.00149887680.078482305
0.00569414970.00145843870.0765686139
0.00554431610.0014194480.0747144967
0.00539969340.00138184020.0729177604
0.00526005710.00134555420.0711763019
0.00512519410.00131053240.0694881041
0.00499490210.00127671990.0678512318
0.00486898930.00124406480.0662638289
0.0047472730.00121251790.0647241144
0.004629580.00118203230.0632303797
0.00451574540.00115256360.061780985
0.00440561230.00112406970.0603743568
0.00429903160.00109651040.0590089844
Butterworth-mod
Chebychev-mod
Bessel-mod
Feuil1
ButterworthBesselChebychevButterworthBesselChebychev
0111ButterworthBesselChebychev
0.10.99995000370.99496999051.00607528730.1-0.0004342728-0.04380035780.0526096269
0.20.99920095870.97975871241.02434230110.2-0.0069431587-0.17761731830.2089021532
0.30.99597443880.95414465281.05477640880.3-0.0350361474-0.40771558490.4632081551
0.40.98744063190.91817224791.09675221230.4-0.1097801217-0.74151676210.8021703825
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0.70.89799030160.7598719971.244948880.7-0.934567075-2.38519119871.9030303767
0.80.84227140070.69810145021.25729698820.8-1.4909589107-3.1216291961.9887575047
0.90.77706387850.636407011.21923998940.9-2.1908655717-3.92530090551.7217839696
10.70710678120.5771.1281-3.0102999566-4.77648373691.0461819929
1.10.63704606240.52139170291.0024737611.1-3.9165832872-5.65671768350.0214602821
1.20.57039580950.47042046220.86863507191.2-4.8764734752-6.5502759141-1.2232527924
1.30.50924392410.42439006160.74446412491.3-5.8614828791-7.4446958995-2.5631245143
1.40.45447034170.38323625680.63731569091.4-6.8498890634-8.3306682073-3.9129077784
1.50.40613846610.34667308740.5479280071.5-7.8265175161-9.2015974358-5.2255300059
1.60.36385052710.3143018550.47428153141.6-8.7815398309-10.0530611161-6.4792757309
1.70.32699825040.28568362610.41368158851.7-9.70909142-10.8822930071-7.6666761511
1.80.29491468760.26038307660.36360027981.8-10.6060719552-11.6877449151-8.7875158255
1.90.26695537190.23799236810.32192316951.9-11.4712267104-12.468739391-9.8449553007
20.2425356250.21814220970.28696200292-12.3044892138-13.2252058391-10.8435121022
2.10.22114316440.20050523810.25739335242.1-13.1065296037-13.9574855414-11.7880534729
2.20.20233796590.18479511220.2321849652.2-13.878452407-14.6661904015-12.6833181217
2.30.18574626530.17076343960.21053028192.3-14.6215981942-15.3521021193-13.5337085621
2.40.17105241840.1581957980.19179577832.4-15.3374156268-16.0161011265-14.3432191289
2.50.15799050110.14690755970.1754803542.5-16.0273804686-16.659117109-15.1154299644
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2.70.13590155640.12755614580.14858742192.7-17.3355113913-17.8859722338-16.56035905
2.80.12652593110.11923859060.13742895592.8-17.9564091585-18.4716633227-17.2384350594
2.90.11807428830.11168567360.12749695552.9-18.55689327-19.0400506418-17.8900037124
30.11043152610.10480960190.11861692153-19.1381385238-19.5919785702-18.5170670274
3.10.1034994290.09853430860.11064434913.1-19.7012409206-20.1282505274-19.1214152316
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3.30.09144263960.0875303190.09695913043.3-20.7770249203-21.1568297736-20.2682257653
3.40.086183330.08269386090.09106013993.4-21.2915345865-21.6505346157-20.8134337386
3.50.08136200950.07824037980.08568942643.5-21.7915666775-22.1313809977-21.3414552825
3.60.07693181760.07413128220.08078524773.6-22.2778801347-22.5999697579-21.8533587788
3.70.07285191880.07033260530.07629466953.7-22.7511801015-23.0568658937-22.3501160811
3.80.06908661120.06681438420.07217209633.8-23.2121221925-23.502600601-22.8326136048
3.90.06560458240.0635501140.06837808033.9-23.6613164875-23.9376733197-23.3016619235
40.06237828620.06051629250.06487834994-24.0993312333-24.3625537367-23.7580040922
4.10.05938341750.05769202910.06164301274.1-24.526696254-24.7776837186-24.2023228859
4.20.05659847060.05505871130.05864589854.2-24.943906079-25.1834791543-24.6352471122
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Feuil1
Butterworth
Bessel
Chebychev
Feuil2
Butterworth
Bessel
Chebychev
Feuil3
Butterworth-modButterworthChebychev-modBessel-modBessel
0.011-0.02000033340.99955042340.99999-0.01
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5.950.004747273-4.37461954080.00121251790.0647241144-3.6929121996
60.00462958-4.3774620290.00118203230.0632303797-3.7015324507
6.050.0045157454-4.38025682840.00115256360.061780985-3.7100091714
6.10.0044056123-4.38300513540.00112406970.0603743568-3.7183458405
6.150.0042990316-4.38570810660.00109651040.0590089844-3.72654583
Feuil3
Butterworth-mod
Chebychev-mod
Bessel-mod
Graph6
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Butterworth
Chebychev
Bessel
Feuil1
ButterworthBesselChebychevButterworthBesselChebychev
0111ButterworthBesselChebychev
0.10.99995000370.99496999051.00607528730.1-0.0004342728-0.04380035780.0526096269
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0.70.89799030160.7598719971.244948880.7-0.934567075-2.38519119871.9030303767
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0.90.77706387850.636407011.21923998940.9-2.1908655717-3.92530090551.7217839696
10.70710678120.5771.1281-3.0102999566-4.77648373691.0461819929
1.10.63704606240.52139170291.0024737611.1-3.9165832872-5.65671768350.0214602821
1.20.57039580950.47042046220.86863507191.2-4.8764734752-6.5502759141-1.2232527924
1.30.50924392410.42439006160.74446412491.3-5.8614828791-7.4446958995-2.5631245143
1.40.45447034170.38323625680.63731569091.4-6.8498890634-8.3306682073-3.9129077784
1.50.40613846610.34667308740.5479280071.5-7.8265175161-9.2015974358-5.2255300059
1.60.36385052710.3143018550.47428153141.6-8.7815398309-10.0530611161-6.4792757309
1.70.32699825040.28568362610.41368158851.7-9.70909142-10.8822930071-7.6666761511
1.80.29491468760.26038307660.36360027981.8-10.6060719552-11.6877449151-8.7875158255
1.90.26695537190.23799236810.32192316951.9-11.4712267104-12.468739391-9.8449553007
20.2425356250.21814220970.28696200292-12.3044892138-13.2252058391-10.8435121022
2.10.22114316440.20050523810.25739335242.1-13.1065296037-13.9574855414-11.7880534729
2.20.20233796590.18479511220.2321849652.2-13.878452407-14.6661904015-12.6833181217
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30.11043152610.10480960190.11861692153-19.1381385238-19.5919785702-18.5170670274
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3.70.07285191880.07033260530.07629466953.7-22.7511801015-23.0568658937-22.3501160811
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3.90.06560458240.0635501140.06837808033.9-23.6613164875-23.9376733197-23.3016619235
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4.10.05938341750.05769202910.06164301274.1-24.526696254-24.7776837186-24.2023228859
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70.02040391470.02019819350.02066137587-33.8057300307-33.8937494353-33.6968152525
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80.0156230930.01550204430.01577338228-36.1246596395-36.192220526-36.0415034536
8.10.0152398090.01512459340.01538276468.1-36.340409529-36.4063258338-36.2593121119
8.20.01487045550.01476072630.015006528.2-36.5535145589-36.6178454381-36.4744001954
Feuil1
Butterworth
Bessel
Chebychev
Feuil2
Butterworth
Bessel
Chebychev
Feuil3
Butterworth-modButterworthButterworthChebychev-modChebychevBessel-modBesselBessel
0.011-0-0.02000033340.9995504234-0.00390585120.99999-0.0000868592-0.01
0.021-0.0000000003-0.04000266790.9982067546-0.01558991370.9999600003-0.0003474402-0.02
0.030.9999999996-0.0000000032-0.06000900970.9959840363-0.03495244840.9999100013-0.0007817535-0.03
0.040.999999998-0.0000000178-0.08002137420.992906876-0.06182963530.9998400043-0.0013898165-0.04
0.050.9999999922-0.0000000679-0.10004179140.989008778-0.09599707520.9997500104-0.0021716535-0.05
0.060.9999999767-0.0000002026-0.12007231020.9843312582-0.13717446650.9996400216-0.0031272959-0.06
0.070.9999999412-0.0000005109-0.14011500330.9789227855-0.18503125380.9995100399-0.004256782-0.07
0.080.9999998689-0.0000011385-0.16017197140.9728376038-0.23919301320.9993600681-0.0055601573-0.08
0.090.9999997343-0.000002308-0.18024534830.9661344845-0.29924832450.999190109-0.0070374743-0.09
0.10.9999995-0.0000043429-0.20033730460.9588754643-0.3647558790.9990001659-0.0086887926-0.0999999999
0.150.9999943047-0.0000494686-0.30115488240.9164935604-0.75741164380.9977508356-0.0195579964-0.1499999989
0.20.9999680015-0.0002779396-0.4027909490.8695244543-1.21436399040.9960026214-0.034790371-0.1999999919
0.250.999877952-0.0010601599-0.50558105490.8240419242-1.68101384890.9937563397-0.0544017546-0.2499999616
0.30.9996356992-0.0031648533-0.60990703080.7839186408-2.11458016640.9910129975-0.0784129909-0.2999998631
0.350.9990821325-0.0079761573-0.71619839580.7512721297-2.48405444160.9877737647-0.1068502575-0.3499995992
0.40.9979582701-0.0177523699-0.8249255870.7271744632-2.76722762020.9840399456-0.1397454336-0.3999989854
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Feuil3
Butterworth-mod
Chebychev-mod
Bessel-mod
Butterworth
Chebychev
Bessel
Graph7
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Butterworth
Bessel
Feuil1
ButterworthBesselChebychevButterworthBesselChebychev
0111ButterworthBesselChebychev
0.10.99995000370.99496999051.00607528730.1-0.0004342728-0.04380035780.0526096269
0.20.99920095870.97975871241.02434230110.2-0.0069431587-0.17761731830.2089021532
0.30.99597443880.95414465281.05477640880.3-0.0350361474-0.40771558490.4632081551
0.40.98744063190.91817224791.09675221230.4-0.1097801217-0.74151676210.8021703825
0.50.97014250010.87256883881.14784801150.5-0.2632893872-1.18400601251.1976877243
0.60.94088741190.81898141891.20165451120.6-0.5292468371-1.73451902811.5955924239
0.70.89799030160.7598719971.244948880.7-0.934567075-2.38519119871.9030303767
0.80.84227140070.69810145021.25729698820.8-1.4909589107-3.1216291961.9887575047
0.90.77706387850.636407011.21923998940.9-2.1908655717-3.92530090551.7217839696
10.70710678120.5771.1281-3.0102999566-4.77648373691.0461819929
1.10.63704606240.52139170291.0024737611.1-3.9165832872-5.65671768350.0214602821
1.20.57039580950.47042046220.86863507191.2-4.8764734752-6.5502759141-1.2232527924
1.30.50924392410.42439006160.74446412491.3-5.8614828791-7.4446958995-2.5631245143
1.40.45447034170.38323625680.63731569091.4-6.8498890634-8.3306682073-3.9129077784
1.50.40613846610.34667308740.5479280071.5-7.8265175161-9.2015974358-5.2255300059
1.60.36385052710.3143018550.47428153141.6-8.7815398309-10.0530611161-6.4792757309
1.70.32699825040.28568362610.41368158851.7-9.70909142-10.8822930071-7.6666761511
1.80.29491468760.26038307660.36360027981.8-10.6060719552-11.6877449151-8.7875158255
1.90.26695537190.23799236810.32192316951.9-11.4712267104-12.468739391-9.8449553007
20.2425356250.21814220970.28696200292-12.3044892138-13.2252058391-10.8435121022
2.10.22114316440.20050523810.25739335242.1-13.1065296037-13.9574855414-11.7880534729
2.20.20233796590.18479511220.2321849652.2-13.878452407-14.6661904015-12.6833181217
2.30.18574626530.17076343960.21053028192.3-14.6215981942-15.3521021193-13.5337085621
2.40.17105241840.1581957980.19179577832.4-15.3374156268-16.0161011265-14.3432191289
2.50.15799050110.14690755970.1754803542.5-16.0273804686-16.659117109-15.1154299644
2.60.14633651890.1367398970.1611845612.6-16.6929456079-17.2820950342-15.853531186
2.70.13590155640.12755614580.14858742192.7-17.3355113913-17.8859722338-16.56035905
2.80.12652593110.11923859060.13742895592.8-17.9564091585-18.4716633227-17.2384350594
2.90.11807428830.11168567360.12749695552.9-18.55689327-19.0400506418-17.8900037124
30.11043152610.10480960190.11861692153-19.1381385238-19.5919785702-18.5170670274
3.10.1034994290.09853430860.11064434913.1-19.7012409206-20.1282505274-19.1214152316
3.20.09719389310.0927937250.10345877213.2-20.2472204346-20.6496278253-19.7046536069
3.30.09144263960.0875303190.09695913043.3-20.7770249203-21.1568297736-20.2682257653
3.40.086183330.08269386090.09106013993.4-21.2915345865-21.6505346157-20.8134337386
3.50.08136200950.07824037980.08568942643.5-21.7915666775-22.1313809977-21.3414552825
3.60.07693181760.07413128220.08078524773.6-22.2778801347-22.5999697579-21.8533587788
3.70.07285191880.07033260530.07629466953.7-22.7511801015-23.0568658937-22.3501160811
3.80.06908661120.06681438420.07217209633.8-23.2121221925-23.502600601-22.8326136048
3.90.06560458240.0635501140.06837808033.9-23.6613164875-23.9376733197-23.3016619235
40.06237828620.06051629250.06487834994-24.0993312333-24.3625537367-23.7580040922
4.10.05938341750.05769202910.06164301274.1-24.526696254-24.7776837186-24.2023228859
4.20.05659847060.05505871130.05864589854.2-24.943906079-25.1834791543-24.6352471122
4.30.05400436450.05259971790.05586401474.3-25.3514228036-25.5803316979-25.0573571319
4.40.05158412460.05030017310.05327709244.4-25.7496786989-25.9686104072-25.4691897007
4.50.04932261230.04814673440.05086720664.5-26.139078592-26.3486632773-25.8712422281
4.60.04720629310.04612740960.04861845724.6-26.520002033-26.7208186706-26.2639765319
4.70.04522303830.04423139820.04651669944.7-26.8928052692-27.0853866478-26.6478221603
4.80.04336195430.04244895370.04454931534.8-27.2578230431-27.4426602033-27.0231793364
4.90.04161323590.04077126430.04270501924.9-27.6153702301-27.7929164122-27.3904215759
50.03996803830.03919034850.0409736925-27.9657433321-28.1364174918-27.7498980203
5.10.03841836760.03769896360.03934623885.1-28.3092218396-28.4734117855-28.1019355214
5.20.03695698440.03629052630.03781446625.2-28.6460694751-28.8041346741-28.4468405077
5.30.03557732030.03495904250.0363709775.3-28.