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Chapitre 5

CODAGE STATITIQUE

Présentation

Clasifct

- transc ac i poderfmnuord ºfclfgCcdunuord "ragc ac i poderfmnuord -fd"es uc"tip"es Rc loRage - lfglnfnuord ac in ufndCerfmnuord mcCCnsc PA Cosdni ufndCmoC - nanlunuord agSou Cr=f"c { "nln"oug a= "ndni 1"rmlfcCCord , - "ransc ac lfruc"uord "rdufc icC cffc=fC ac ufndCmoCCord 1 aguc"uord 2 "rffc"uord apcffc=fC, - "...oeefnsc 1 Cc"fcu acC "rmm=do"nuordC , - unur=nsc 1 mnfq=nsc a= uoufc ac lfrlfogug , - ufndC"ransc 1 "...ndscmcdu a pnil...nScu} "rdufnoducC ac ufndCmoCCord , oin dtenteq uclgis" acil oisstrfiln lmteclOtmacbt -cCCnsc MSdL àd CèCuhmc ac "rmm=do"nuord n lr=f S=u apn"...cmodcf acC mcCCnscC ap=d cblgaouc=f 1Cr=f"c apoderfmnuord, écfC =d acCuodnunofc 1=uoioCnuc=f ac ipoderfmnuord,. jc Cosdni C=llrfundu ipoderfmnuord : ufndCmcuufc arou zufc "rmlnuoSic néc" icC "nfn"ugfoCuoq=cC a= "ndni ac ufndCmoCCord. jc loRage Re "2inqoruation arou gunSiof =dc "rffcClrdand"c odvc"uoéc cdufc ic

mcCCnsc ufndCmoC lnf in Cr=f"c cu in C=ouc acC CèmSricC apoderfmnuordC MSdL ufndCmoC :

ipgmcuuc=f. tc "ransc n acC f;icC m=iuolicC. yi lfglnfc in ufndCerfmnuord a= mcCCnsc cd =d Cosdni

1ceec"u=g andC in lnfuoc ac ipgmcuuc=f } "ransc oderfmnuord : Cosdni,. yi nanluc in Cr=f"c

apoderfmnuord : in "nln"oug ac ufndCmoCCord a= "ndni. yi lcfmcu ac lfrugscf ipoderfmnuord "rdufc icC cffc=fC ac ufndCmoCCord 1andC =dc "cfunodc iomouc, cd é=c ac ic=f aguc"uord r=2cu ac ic=f "rffc"uord 1a=cC n=b lcfu=fSnuordC andC ic "ndni,. yi lcfmcu gsnicmcdu andC "cfunodC "nC apnCC=fcf ic Cc"fcu acC "rmm=do"nuordC 1ic "...oeefnsc,( cu ac unur=cf lr=f mnfq=cf =dc lfrlfogug. àd "ndni ac ufndCmoCCord arddg arou lr=érof ufndClrfucf acC mcCCnscC ac dnu=fc énfogc apr) in dg"cCCoug ap=d ufndC"ransc lcfmcuundu ac ufndCerfmcf in fclfgCcdunuord acC mcCCnscC : lnfuof ap=d ao"uorddnofc - ô =uoioCndu ipnil...nScu çô cd =dc fclfgCcdunuord acC mzmcC mcCCnscC : lnfuof ap=d ao"uorddnofc - x =uoioCndu ipnil...nScu çx. tcu nil...nScu lnfuo"=iocf cCu Cr=écdu ipcdCcmSic Sodnofc Mx( 0L( mnoC "c dpcCu lnC ic Cc=i. jégclcacilr - 4r=f"cC 4 ac mcCCnscC} lfra="uord ap=dc C=ouc ac mcCCnscC( "...n"=d apc=b gundu Cgic"uorddg andC =d cdCcmSic MacC mcCCnscC

1 M} ao"uorddnofc ac mcCCnscC lrCCoSicC - P M m0( m7(

N.L( icC m

oCrdu n=CCo nllcigC O mruC ù , - -cCCnsc } C=ouc eodoc ac CèmSricC

1"nfn"uhfcC lfoC andC A} nil...nScu ,

- çil...nScu } cdCcmSic eodo ac CèmSricC AP M n0( n7( NN( nôL oin dtenteq uclgis" acil

¨ Définitions

•d nllciic uessage ur=uc C=ouc eodoc ac "nfn"uhfcC lfoC andC =d a"phamet ç } cdCcmSic eodo ac sMumo"es 1lnf cbcmlic } icuufcC( "...oeefcC( N,. •d nllciic socrle Re uessages 4( ic CèCuhmc q=o lfra=ou =dc C=ouc ucmlrfciic ac mcCCnscC m o( "...n"=d apc=b gundu lfoC andC ipcdCcmSic - acC mcCCnscC lrCCoSicC. - cCu nllcig Riltionnaire Re uessages 1r= mruC,. jc mcCCnsc ufndCmoC cCu cd enou =d ucbuc erfmg lnf acC fhsicC Cèdunboq=cC ac mcCCnscC gigmcdunofcC nllcigC mruC } - P Mm0( m7( N L( "...nq=c mru Cpg"fou lnf =dc C=ouc eodoc eobgc ac CèmSricC lfoC andC =d nil...nScu ç. Ed erd"uord acC nllio"nuordC cdéoCnsgcC( icC Cr=f"cC ac mcCCnsc 4 lc=écdu =uoioCcf acC ao"uorddnofcC ac dnu=fc ufhC aoeegfcducC } acl=oC icC mcCCnscC g"fouC =uoioCndu icC "nfn"uhfcC ac ipnil...nScu( icC "...oeefcC( icC "nfn"uhfcC ac lrd"u=nuord cu ac unS=inuord v=Cq=pn=b mcCCnscC éoC=ciC r) icC mcCCnscC Crdu acC omnscC d=mgfoCgcC r)( lnf cbcmlic( "...nq=c mru cCu =d lobci Cpg"foéndu "rmmc in C=ouc ac H CèmSricC 1SodnofcC, lfoC C=f ipnil...nScu Mx( 0L 1r"ucu,.

Clasifcteen u-lteri-smteE4Qç//•/ 0>DHR

- Rgeodouord ac i pclmtsaca-ntcu ac i ptlasifct &InlertitcRey a p=d gégdcmcdu E } y1E, P {irs

7ëfMEL Unités} Sou 1 êydnfè =doî Co irs7,

dnu 1 /çu=fni =doî Co jrsc,} 0 dnuP0(DDV Sou

Co Cr=f"c Comlic

CdP A y1Cd, P ∑oP03dy1m ao,

&EntropieQ a p=dc énfonSic nignurofc aoC"fhuc 5 }

Q15, P E

58 y15, T P ∑oP03dloy15o, P {∑oP03dloirs71lo,

& ëfrlfogugC ac i pcdufrloc - Q ³x 3 Q cCu "rduod=c ( Cèmgufoq=c 3 Q1l

0( N ( l/, £irs7d

- Co 1l

0( N( l

d, cu 1q0(N qd, Crdu 7 fglnfuouordC ac lfrSnSoiougC

PPA ∑

oP03dloirs71 qo2 lo, £x "nf jrs b U b { 0 jpclmtsaca-nt y ap=d gégdcmcdu E ac lfrSnSoioug ëf1 E , cCu ageodoc lnf } y 1 E , P irs 7 ,E1ëf0 P { irs7 ëf1 E ,

· "cmnfq=cC

{ Co ëf1 E , P 027 nirfC y 1 E , P 0 1od"cfuou=ac =doug, { Co ëf1 E , P 0 nirfC y 1 E , P x } ipod"cfuou=ac cCu d=iic lr=f =d gégdcmcdu "cfunod. { jp=doug cCu ic hca 1Binary Unit, : dc lnC "rderdafc néc" ic Sou } Binary digit.

{ •d lc=u =uoioCcf in erd"uord irsnfou...mc dglgfocd : in lin"c a= irsnfou...mc cd SnCc 7( andC

"c "nC ip=doug cCu ic l a 1/nu=fni àdou P 0(DDV Sou,. •d "rdCoahfc mnoducdndu q=c icC gégdcmcduC E Crdu cd enou acC fgnioCnuordC ap=dc énfonSic nignurofc aoC"fhuc 5. •d ageodou nirfC iptlasifct Q "rmmc gundu ipod"cfuou=ac mrècddc ac in énfonSic nignurofc 5. 4o rd "rdCoahfc cd enou "...nq=c gégdcmcdu xi( o Î 80( dT( "rmmc =dc

fgnioCnuord ap=dc énfonSic nignurofc 5 1o.c. 5 cCu =dc énfonSic nignurofc : énic=fC andC M x1( x2(

N( xn L, }

Q15, P E

5 M y15, L P ∑oP0..d ëfM5 P boL . y1bo, P ∑oP0..d lo.y1bo,( néc" lo P ëfM5 P boL

jpcdufrloc aglcda ac in iro ac lfrSnSoioug ac 5 mnoC dpcCu lnC erd"uord acC énic=fC lfoCcC lnf

5. Eiic Cpcblfomc cd SouC 1r= dnuC, cu fclfgCcduc ic drmSfc mrècd ac SouC dg"cCCnofcC : "racf

cd Sodnofc icC aoeegfcducC fgnioCnuordC ac 5. •d "rdCoahfc mnoducdndu =dc Cr=f"c apoderfmnuordC 4 ageodoc lnf =d cdCcmSic ac mcCCnsc mo lrCCoSicC 1ao"uorddnofc, } 4Mm

0( m7( N( m/L( cu lnf =d mg"ndoCmc apgmoCCord ac mcCCnscC uci

q=c } Cd P Mma0( ma7( N( madL néc" ma0 } 0cf mcCCnsc gmoC( N( mad } dohmc mcCCnsc gmoC.

Attention } ipodao"c O o ù andC ao ageodou ipodao"c ucmlrfci andC in C=ouc ac mcCCnscC gmoC lnf in

Cr=f"c. ao ageodou ipodao"c a= ohmc mcCCnsc gmoC andC ic ao"uorddnofc - acC mcCCnscC lrCCoSicC( cd sgdgfni } / ¹ d. jc "...rob acC m ao Cpceec"u=c Ccird =dc iro ac lfrSnSoioug arddgc. jpgmoCCord ap=dc Cr=f"c

aoC"fhuc apoderfmnuordC "rffcClrda ard" : =dc C=ouc ac énfonSicC nignurofcC 5o( oÎ 80( dT } =dc

énfonSic nignurofc nCCr"ogc : in fgnioCnuord ac "...nq=c mcCCnsc gmoC m ao ac in Cr=f"c. jn lfrSnSoioug ac C d Cpcblfomc nirfC "rmmc =d lfra=ou ac lfrSnSoiougC "rdaouorddciicC }

ëf1C

d, P ëfM50 P ma0L ëfM57 P ma7 2 50 P ma0L N ëfM 5d P mad 2 50 P ma0( N( 5d{0 P mad{0L RndC ic "nC ac Cr=f"cC ComlicC( icC d énfonSicC nignurofcC 5o Crdu odaglcdanducC cu ac mzmc iro ard" } "1o( v, Î 80( dT b 80( /T( ëfM 5o P mv L P lv( cu ëfMCdL P la0.la7Nlad

? y 1 Cd , P { irs7 ëfMCdL P { irs7 1 la0.la7Nlad , P ∑oP0..d { irs7 lao P ∑oP0..d y 1 ma0 ,

y 1 C d , P ∑oP0..d y 1 ma0 , RndC ic "nC ap=dc Cr=f"c aoC"fhuc ac O d ù mcCCnscC mo( r= "...nq=c mcCCnsc mo cCu nCCr"og : in lfrSnSoioug l o( ipcdufrloc Q ac in Cr=f"c 4 cCu arddgc lnf } d 0oo

7olirs.l,41Q

¨ Propriétés de l"entropie :

0oo=∑

=( nirfC Q15, A x } ipcdufrloc cCu lrCouoéc. { 4rocdu 1l

0( l7( N( ld, cu 1q0( q7( N( qd, ac=b iroC ac lfrSnSoioug( nirfC

0£∑

=d 0ooo 7 olqirsl .

Ed ceecu(

ard" ( )0111=-=)) ====id70lqid70lqlid70lqirsld 0ood 0ood 0ooo od 0ooo 7 o. { jpcdufrloc ap=dc énfonSic nignurofc 5 : d énic=fC lrCCoSicC cCu mnbomnic cu én=u irs7d irfCq=c in iro ac 5 cCu =doerfmc. Ed lfcdndu q

0 P q7 P N P qd P d0 1iro =doerfmc,( andC

in lfrlfogug lfg"gacduc }

0£∑

=d 0ooo 7 olqirsl ∑∑ d 0oo 7od 0oo

7oqirsllirsl

=-£Ûd 0o7 odirsl,51Q1 dirsldirs,51Q7d 0oo

7=-£Û∑

=1 { jpcdufrloc cCu "rduod=c cu Cèmgufoq=c. RndC in C=ouc ac "c "r=fC( dr=C =uoioCcfrdC CèCugmnuoq=cmcdu in SnCc 7 a= irsnfou...mc.

Exemple simple :

•d "rdCoahfc =dc Cr=f"c 4( : iro =doerfmc( q=o cdéroc acC mcCCnscC : lnfuof ac ipnil...nScu

efnd»noC : 7N "nfn"uhfcC 1n(S("( N( O,. •d nvr=uc : "cu nil...nScu ic "nfn"uhfc O cCln"c ù "rmmc

Cglnfnuc=f acC mruC.

jpnil...nScu cCu ard" "rdCuou=g ac 7P "nfn"uhfcC } Q14, P { =7P

0o77P0irs7P0P irs717P, P D(PQ SouC

apoderfmnuord lnf "nfn"uhfc. Ed fgnioug( ipcdufrloc cCu éroCodc ac D SouC apoderfmnuord lnf

"nfn"uhfc C=f =d ufhC sfnda drmSfc ac ucbucC efnd»noC. C

Clasifcton

C dequog"e"rg"rmOu

Cbjaétojto-ERRha(

C C C

Codage riaitrithme oéitqau

p iaCtre5ODCAGtO5E DCD CtGSTIDCADCSDttQUDtCSGADCIrGCADCT5rPQPGIG é

TCsCnT

ofCdddd"dfCTuc ntre5ODCRrSrUElDc p grAQUDCte5CelCQITRQPD CC2CsCqCQ ofCQmfCddfCQM- p bl 5rTGDCADCIQCtre5ODCCènicCD CIrlUeDe5CSr/DllDCADtCSr tpOrADtCCbnlc hsbno:Dqe de faA ftuuapaCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC p GICD*Gt DCQeCSrGltCelCOrADCG55éAeO GPIDCAéORG++5QPIDC,é5G+GQl Ca ?éUQIG é tGCTGADCIQC+r5SDCaCCCTGsCMplGnCtGCMCsCmCCCs0CClGsCp IrUmCTGc h sbno:Dqe de lExyy1ynoD5 Rér5ESDCte5CIDCOrAQUDCtQltCP5eG Cc 1C C

uretCQ,rltC,eCAQltCIQC5Dttre5ODCT5éOéADl DCaC2C3é+GlG GrlCAeCOrAQUDCD CT5rT5Gé ét4CMeDCIDtC

rP5DO G+tCAeCOrAQUDCtrl CT5GlOGTQIDSDl CADC 5QltO5G5DCADtCGl+r5SQ GrltCD CADC5éAeG5DCIQCMeQl G éC

ADCt/SPrIDtCléODttQG5DtC6CIQC5DT5étDl Q GrlCADCI7Gl+r5SQ Grl"CblC2CrT GSGtQl CIDCOrAQUDC4fCrlC treRQG DCArlOC5éAeG5DCQeCSQ*GSeSCOD DCMeQl G éCADCt/SPrIDtCaC C

8lCOrltGAE5DCelDCtre5ODCtGSTIDCA7Gl+r5SQ GrlCRrSrUElDCiCsCqSofCSmfCdfCSu-CSelGDCA7elDC

IrGCADCT5rPQPGIG éCTCsCqCT

ofCTmfCdfCTuC-Cr9CTGCsC:5qSCsCSG-"CC iGC; GCDt CIDCSr pOrADCOr55DtTrlAQl CQeCSDttQUDCSGfCrlCQTTDIIDClGCsCln;GcCIDClrSP5DCADC OQ5QO E5DtCQTTQ5 DlQl C6CI7QITRQPD C2CngQ5An2cCsCMcCléODttQG5DtCQeCOrAQUDCADCSGfClGCDt CArlOCIQC

IrlUeDe5CAeCSr pOrADC;

G"C I7QITRQPD C2CDt CArlOCéUQIDC6C

2CMC4COQ5QO E5Dt"C3DCOD DCGléUQIG éfCrlCAéAeG CMeDCbnlcC

³CMIrUè

m"C C :re5CrT GSGtD5CIDCOrAQUDfCrlCtreRQG DCArlOC5éAeG5DCbnlcfCIQCIrlUeDe5CSr/DllDCADtCSr tp OrADt"CgD DCIrlUeDe5CSr/DllDClDCTDe CTQtC= 5DC5DlAeDCGl+é5GDe5DC6C

MIrUè

m"CuéQlSrGltfCADe*C Rér5ESDtCAéSrl 5Dl CMe7GICDt CTrttGPIDCA7rP DlG5CI7éUQIG éCbnlcCsC

MIrUè

mCD CA7rP DlG5CelCOrADC rT GSQICaC C

¨ sbno:Dqe de faA ftuuap 3

C >lDCtre5ODCA7Gl+r5SQ GrlCiCA7Dl 5rTGDCèCOrAéDCADC+Q?rlCAéORG++5QPIDCTQ5CelCQITRQPD C6CMC

OQ5QO E5DtCDt C DIIDCMeDCaCbnlcC

³CMIrUè

mCD CGICD*Gt DCQeCSrGltCelCOrADCG55éAeO GPIDCA7elDCIrGC

ArlléDCMeGC,é5G+GDCaCbnlcC

mC/Co"C<7éUQIG éCDt CQ DGl DCtGCTGCsCMplGCnG"D"ClGCsCpCIrUMCTGcCD CrlC

QCQIr5tCelCOrAQUDCrT GSQI"C

C @DSQ5MeD CaC C

3QltCIDCOQtCTQ5 GOeIGD5CA7elCI7QITRQPD CPGlQG5DCq1fCo-fCrlCQCMCsCm"CiGCIQC5DIQ GrlCl

GCsCpCIrUmCTGCC

Dt C ,é5G+GéDfC rlC QC QIr5tC bnlcC sC èCaC I7Dl 5rTGDC Dt CIQC Pr5lDC Gl+é5GDe5DC ADC I7DltDSPIDC ADtC

IrlUeDe5tCSr/DllDtCADtCSr tpOrADtCD COD DCPr5lDCGl+é5GDe5DCDt CQ DGl D"C C ¨ sbno:Dqe de lExyy1y rm: ue Aodage rapr v:mti 3 C Are DCtre5ODCA7Gl+r5SQ GrlCRrSrUElDCDt C DIIDCMe7GICD*Gt DCelCOrAQUDCG55éAeO GPIDCTre5CIDMeDIC IQCIrlUeDe5CSr/DllDCADtCSr tpOrADtCDt CQettGC,rGtGlDCMeDCI7rlC,De CADCtQCPr5lDCGl+é5GDe5DC

MIrUè

m"CCodage riaitrithme oéitqau hCoder de 0apo 2 lbappop hCoder a:tibqnithmer-Aodage ep vuoAè de i4ée Aodage d5tpie:'auuer(

TrttGPGIG étCA7QAQT Q GrlCDlCIGUlD

h

Coder de Emccqap

CCT5GlOGTDtCADCPQtDa

p tGCTG.CT5CCs0CClG³l5C p IDtCmCOrADtCIDtCSrGltCT5rPQPIDtCtrl CADCS=SDCIrlUeDe5 p IDtCmCOrADtCIDtCSrGltCT5rPQPIDtCnADCIrlUeDe5CSQ*cCrl CIDCS=SDCC

T5é+G*DCADCIrlUeDe5Cl

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