Traitement du Signal Février 05

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Signaux Aléatoires Février 05 E. Sicard Page 4

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Filtrage Numérique Février 05!2' J. M. Dilhac Page 15

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Filtrage Numérique Février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ilhac Page 17

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Filtrage Numérique Février 05 ))(&> ),'@&&( )'$9( (9( CD &,'!' @2) 4!')( 38'3 F'> 3&) &('V 3(4 && 38, ))'$> ),'&)8!)(3 3 =% =CD & 3',(*',' &,)")& 3&,'(& 3(4 @)(&&(,,&&@&3=CD,'&!'3E(@()))'3(&!,3(&$))'$9( 3'CD&)!&'(9(%,'&('!)(&*3,&$))'$9(!( L)&!)8M'(38(*))'$9(>3=!!&&2&>3I*)3)&&$J% A@'!9(3A(&3',,',)(&')&(@% *+ µ, () & % &% & % &% & % ' % -"% "!! - - - - -.. 49(",'@&'(9()'$9(O3$)')'$,'@&'>> 23!'3&*'O'&&$*,2C:Db)&&$*,2C:D% $&>, ))'(#2)'9((>*)!&&2&O$&>&!)# ')3>)'c!&&*) ) / '& '829 # + &&$(4&&(&3,!(&&')!)(&&'(@)'$9(&% >)&,),()(&@))&(3& $3(& (9(& (,'@&' )'$9( (9( & 3',,&&% A2@' 2 &()& *\&> C)'$ ' $) '@&'D $)&% &&=2')& &(@&&'!)'=&3&)&&, &2)',&% J. M. Dilhac Page 19

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Filtrage Numérique Février 05 0 : &$)E!&3&,!(L&&(3A(>&!(L$)3,&= &! (&=&(9(%9(&(&A!!)9(E)A()A(,&% + &@)(&&(,,&&@&3A(&$)3&,&'!&&$! 9(!(2&(' 3A(2'()C!'(!&,'*'3@,)A!()&'3,9(>))>&*(4&$(4E!&,'(D% A4 'F') 3)!&' $! 9( & $3( &,'3&> *&!!Y 3& @))& 3)$( $) E)!'3 3A, ))'$>(&$) 3&, A 3* 9(A(4 &&3A, ))'$%&$)&)'&'4U T>@,)*`)&,', &UT!!)))(3&,3(&$)% & ' & %'! ' A4 'F')!( $)*(9(!!Y C&& 3&'D% &$) & )'& '4UT%3,&&*9(,'3&(&(''V)A=!&3A2$(f" (&()!'33A, ))'$4&C&=& D% ()9(&4!)&3&$(4&'3'&,3&&(&%&&$(4!(@(&&L3,&!3& 4!&&'&)=9(&% 4 UT O/!'(d/e 4. UT O/!'(d/ O!'(/gg OC/>?D!'(h/ &&$(4!,()&!(@L3*&>)&)A!()&'(3&,] δut O!'(O/ δut O!'(O O/!'(b/ O/!'(b ])A, )'( (UT O/!'(d/ O!'(/g J. M. Dilhac Page 23

Filtrage Numérique Février 05 4 UT δut. ^ /^ ^ ^ ^. 4. UT ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ / δut ^ ^ ^ ^ ^ ^ / 4 UT ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ (UT ^ ^ ^ ^ ^ ^ ^ / ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ J. M. Dilhac Page 24 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ )3*'3)A!()&'(3&,3,)δUT>'!(33()A9(',3&&'(&9( _3(_)!&3A(, ))'3A(*','3&,% UT = U T%δU T = )(&$)>'')&&)(3&@,),&3&&$(4,'(&% 6' + +!!)'&'(3A2'3)A4!&&'3)&*'35'(3&&$(4,'(&C5D + /CωD = CD% >ω A'(2)!&9()!&&$3)3,'!'&'*','&' '$')&C&35'(D3A( &$)!'39(>E)5&A&*&(!!'&(!'3*!'()&$),'(4CD% GD&*'35'(3&&$(43&,&C5D" )A','!3(*9()&&$(43&,&&'3*&9(A(4&&3A, ))'$> '3&)A@))>)5!()(L3!&'(&)*' " / CωD = + = U / T > ω( (*,(()!),*!&>)5&)A!!'4'!) '33&,$)&3)5%

Filtrage Numérique Février 05 )A$)&!),!(&'3&,> /ω&(*',','!)4>,'(>!'39(>3!'3.π# > A&!'(,&&'&9(/ω&&'(@'/ >ω( '(/C Mθ >@,θ Oω(!'(!!),!'3,% )&(**3',3!&/ω&((@))3)$(.π# >$)P# π# >!!) @))*'3)%*9(,>,@))&$)EP#5 #.%/ω&)&!,34ut> *',','!)4!'(@L3@&!))$% '((&$)4UT)>U/ωT&!U/ωT&!% KC MωD KC MθD 4 UT ^ ^ ^ ^ ^ ^ ^ ^ i# / i# / i / i U ( T /C >θ D 5 #. U T ω θ 5 #. * >θ /C >θ D δu T U T. U T / C >θ D. /. C >θ D U T+#. U T / U >θ T +#/. U >θ T U T.. U T / C >θ D /.C >θ D &) 2)(>' '(@)&!'!& &&))& 3) 5>!,())!'! 3 )%!'3( 3,'@')(' 3&$(4 3&,&!!Y 3&) 2)(% ) & 3*! )A9('" 5@&9(E))&A," U T. U T = + π ( + = UT%. U T U T = ( /CωD > ω( ω.π π (.GD&*'35'(3&,C5D" 5!(3&,!'4,C,A&E3!)A33A(*',','(D3& )A&!, 3&*9(,& 3(&!, 3A(&$) 3&, 9(),'9(>,A& E 3 E 3( * '!'39(% >&)A'&(!!'& 9(,,),() 3@ L D!& ) D!(,),()( (9(>)&,&&" 3,'&3 9(A( '2* 3A, ))'& 3 4UT> 3A(& & 3 '9(!'))4UT()&(*L3A'2&@'33(*> 3,),() 9(A( '2* 3@)(& 3/ω> 3A(& & 3A, ))') &!,% '(&))'&4)&,'&9(,&3,&3(4,'&% D '3('23A, ))'&34UT" '&3'&(*L3A'2&@'!'))cUT>3)$( / O " cuto!'( / gg / PCD>cUT&())!'())(& *L&3_,$()_% 3! )'&&& 3**,() 9()A4!&&' 3) 5>'2)A4!&&' 3) 5_E!'&_>!'&!'())$)A,( / O/" /CωD = B =/ U T > ω( J. M. Dilhac Page 25

Filtrage Numérique Février 05 )'3('23A, ))'&,'&3&!'(),),())'(*'&)&!,,),()" )!'3( 4UT%cUT & 3(& 3& )A&!, 3& *9(,&! )!'3( 3,'@')(' /ωscω% )A) ( 3 ),'@')(' & )A!()&' 3,>, δ(ω) ` ) ((&>)*L,$()>(,((*LA'!'(&*'35'( δ(ω). ))(&>)&$*,'! =&9( 3(&!, 3A(&$) '!'39( E 3( *>,),()&((@))!')*>A&!&'!)(&&&&'()@3&9(&'&% 9(!(L'(*'&(,>,A&9(!)(&&$3>'&&@&)A)'% 2D '3('23@)(&,),()&",),()((9(C,A&E3(, E&*&D!(!3*'*'(9(A( '2*3@)(&!'(/ω)>,9(,'&!'3E(, ))'$3(&!,%' ω)!& 3A, ))'$>)5E!'&3@CW()*D" B /U@ ωt = U T > C@ ωd( =/ 9('&9()*',''2(>KUW ωt>&(*','3&,%c,)&),&!9(d ω &(*,'3)A@))*'3)>UC.π# D# ω] 1!'&3@' L,),()&%&3(&!,&33(&!&)'% 9(&'3("9())@)(3'E ωn@)(*2)3'($3*&&3!&' &,&& 3 '2(4,),()&>!(L 3'3&% @)( )@> (,'> 3' ( &()!3 &!(L @,! 3A*'' (> ) &*'' @&!!)9( ( &!,, ))'!!& 3A'2 ) &$) '$)% ((&>) ' 3 '> 3ME,'!'()A, ))'$!')>!'&&3( 9(@)!'()A, ))'$*9()% D E! (, F E4!!!5 ' G$'!E!4!! 3 H(, IJ- '(&33(&'&9()&(')!)(&,''9(,),(),'&!'3E" ω =.π =.π ( / B( 9( 3*@> &!' 3&)A9(' 3) 5 E!'&> 3')A4!&&' 3) &*'35'(3&,C5D" B /U@ ωt = U T > =/ *','3&,>!'39(> 3!'3 C' 3 9(, ))'&!& 3', ))'& *9(,D% 5@&9(E))!'(4!&&'" U T = B /U@ ωt > @.π B B @=/ &3(4*'()&&'&!', &>()$',),())A(!'(>&&$'&&&3**,()&>,),())A(% 9((&&>,)'(&&()(!(!)(&)'>9()5@&& ))(&&!'39(>3!'3% 2)(,,'&()&3**&!&3(,),()>3&)A&!,3(!&3)*9(,>,, &(!!'&! 3A(&$) )'$9(*,@&())!& *9(,% J. M. Dilhac Page 26 @. π B

Filtrage Numérique Février 05 GD&*'35'(!3" D '23A'!'&,&&&!'(),),()3)5" (!!'&'&4UT)`!'(,),())!))3A(3&@)(&3A(5 ^()!),'&3(=!4UT% M.πW# ^CD33'&3&&()&3&()!),'&!,3& &',&&&%'(')>!'(!'&3(&!,>&ee>.,=,)&C,=,)O33' ()!),'D%, **&E()!)!3(4!(&9(A)*((&&,),())!$% &,&'>'&(!!'&9()&@)(&3 M.πW#!(@L,),()&(*'&!'( '(&>$&'"))&3!3!&3&@)(&3&, ))'&%!!),'(9(" (!!'&'&),),()!&)3A(5E/.0O. /!'&>()&(,,(C@',!_= &3&*)&(9(&_D!'&&3(@&&3,),()31/&!,=,)>E! 3A(&$))'$9(4CD, ))'!)*','3&,4UT%!&,),()&)'&,O.4C. / D. O//&% &)!'&&2)3A2)(),!&,),())&,,&9(&!'))&34UT>))) )$(3)*L3',(N&,,3&&'(&&()&'&,&!'&&2)&" D ),),()3)5,&&'&3!&9()A,9(&'!)3&, ))'& 34UT%3(!&!'(*(, '&>&&),),()3)5&&&() j, >)&&(3&'>3','!, % 2D )A,9(&'3&, ))'&34UT,&&'&3!&9(),),()3A(&!,! ) 5% 3(4&& 3A,9(&'& ) 4&(!& '!3)9() ) A=!& 3A,9(&'3A, ))'&")=3',!3A*''%,D,9(&',),()')L3(>,A&),&'!)9('(&,'&3'&,% )'& / O O,% J. M. Dilhac Page 27

Filtrage Numérique Février 05 4CD KC*D * K ; / * # / /# 5O < < / / #. /* O5#. /A ; K / / 8 < < / *O# / 8 / # 5 5 5 5 5 5 5 5 5 5 5 5 33( $)E@'//&#///O//a&>&'5 O/WF% )A!!),'3( '3 '!'&9()*9(,4)C*4!9(!( *)!)D!&3&)&!,34CD&')'35 >&'* 4 O1WF[)9()> ')$' A&3',,!2)9(33&&$(4)@2)&% J. M. Dilhac Page 28

Filtrage Numérique Février 05 2D &*''35'(!3" '2(4@(4',, E3()'23A'!'&,&&&!'(),),()3) 5%&&()&)&!)(&,'(&&',(43&'')=' (W=C?71D>,'(&&'(&) '3A)$' 3A),!')%'(&))'&3',)!,!% &(!!'&9()'23A, ))'&&(!(&&,33(4>&'" O. >()% 4UT!'&&3)'&())('2!3A, ))'&>)5!(&A," B. B @π. >0 B /U@ ωt = U. T + U. +T > 0 @π B.@π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b4UT C '&),& &!,()'V4UT&M(&!'39(3!'3 / D% (>, 9( C,A&E3, ))'D34UW ωt,'&!'3*e(_ '9(_ 34AUT% ()!),'! M.πW# &)(3! &$4U.T4C.PT J. M. Dilhac Page 29

Filtrage Numérique Février 05 '(@,!'2)>3A(&*L&9()*L,$()&'()&&`)('2M,*& 3A'3( (,'(!')) (4!'& 3,,'3 3& 3**&!'3&> ()A!)(3 3&!& 3&, ))'& 3 4UT% &)A&!, 3&*9(,&>,' 3ME 3>)*L 3@&A!!', )!)(&!'&&2) 3)A!()&'( 3&, δ[ω]> )(3),'@')('% '2(&& *L& 4& C$() '( 3 :)> :),W> $> $> X&] D '(&'**(,'!'& 3** ) ')$&& 3& & '9(!&&)'!!'3&&(!!)&%,!'3@(>)*L,$()&)))(!'(3&$(3(4&3M,&>& )!!'(3,3&&3*2)!)(3% 1GD$(4E!)(&(&3&'&" A)=& 3 5'(!( &A!!)9( E 3& &$(4 E!)(&(& 3&'&> 3',& &' &!)&'!)(&!'))&%A&)'&*'3), $% )=&3A$&*4&"3(43&'&&!)&% )=&3A$&&"3(43&'&&!)&(3&'!'))% 49(" (UTO(&!*(,'> δuto(!()&*(,'> 5'(&*'>3&,5'(&*'>*&5'(&*'C55D>c3'c$> '3)&"c@)>c@))=&&> 3&3&,"&c'' &>*'38'3"c@*'> &()" & ')3>I&$'F','**,& 3''4,3, & ')3J% J. M. Dilhac Page 30

Filtrage Numérique Février 05 = -3 / 6 + '(& '(& &&'&, E ( &=& &(&,!2) 3 *' ( &$) 3&, 4UT C9(,'&()AD((&$)3&,=UTC9(,'&()&'D% &=&E!&3&,!(L,,&!)&!'!&&(@&" " &4 UT3'= UT&4. UT3'=. UT)'&4 UTP24. UT3'= UTP2=. UT&)&=&& )>23&,'&&> 4 / " &4UT3'=UT)'&4U / T3'=U / T9()9(&' / > "" &'(&$)4UT3A!)(3*3'(&$)=UT3A!)(3*> " &,(&)(&=&)9(A(&$)3&'=U / T&3!334UT!'('( / dc)a**!(!&!,3),(&d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ilhac Page 31 4UT =UTO4UT

Filtrage Numérique Février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δUT%'(&))'&3&(!!&3'C,9( @3@,)A9('3,(,D9()!'&!()&'))3,,'!))&=&" 3A(&&9(A))!3,),())!'&E(&$)9(),'9(UT% 4UT 4UT 5 KC M θd ' UT S =UT UT 5 C M θd QC M θd 5 UT&,(&)>'!(,9( UTOU/T% δutput% δutpu.t% δu.tp] )&=&&)@>!3*'>&!'&!(L)'&**,@,),() H))&)'&3)*' =UTOU/T% UTPUT% UTPU.T% U.TP] &'&'(&*','3& 9U T = @=/ =UT U@T%4U @T J. M. Dilhac Page 32

Filtrage Numérique Février 05 9(' & 4' / 3(&=&% ))! 3,),())!'& 3A( &=& E( 9(),'9(>,'&&&!'&!()&'))% )&=& ',(&))&2'&3&''3)A9('3,'@')('3PE)A*% ))(&>),'3'3&2)3A(&=&!(,L*'()" (&=&3&,&&2)&&!'&!()&'))&&'2)>&' 4U@T <, 6' + @= 3*'>) &*' 3 5'( 3&, C5D 3 UT,'&()!'& *9(, CM*D3A(&=&3&,%9(,'!)4!(L$! 9(3'(2)!&>!&''3()&'$(φ>'(&3(4*','3)*9(,*'(3)!()&'ω% &=&),'@(&$)3A&(&'f3)(&$)3L*9(,>&3' )A!)(3)! &3!33φ%,'@')('!'))()&!)(& (!'(),),() 3)&'3A(&=&>!(,L!),!(&!)!'3(3&)A&!,3&*9(,&% 4UT 4UT 5 KC M θd UT S =UT UT 5 C M θd QC M θd 5 =UT,? 6 @+ GD*'4!)&" L 9(!'()&&=&&)&,'(& ) 4& 3& @$& E()&)@2)! '3(!) &*' 3!),>' &'(@ L!'()&&=&& 3&,&)& E ()&)@2)F'3(!)_&*'F_% &*'FCkDKCFD3A(&$)4UT&3*!" /CJD = U T%J = &!'!&&&))&3)&*'F&'C)!9('3())3) &*''D" U T +#. U T / CJD +#/. CJD δu T δu T J U T J /CJD U T. U T / CJD%/. CJD J. M. Dilhac Page 33

Filtrage Numérique Février 05 '&9()&,'3'&)&&'())&>)&&(!'&&2)33*6 6 @CFD3A(&=&3&,3')A&4UT)&'=UT" ICJD = 6CJD /CJD *' $) 3 CFD & *,) '2( E! 3 )A9(' 3,(, 3' ) &*'3' &' 6CJD = # @ J @ /CJD + @ J @ 6CJD @=/ @= ICJD = @=/ ' @ J @ @=.GD(!'!" ) & 3 3!&& 3 CFD D CM*D!(&!)_F 4!CM.l*#5D_ 3!) &)(33&5k%,),()3)!'& '9(E!3)*','3&*F CFD3)A9('3,(,!(3',&*&!)>,&&!&!,()),),()3A(&*'35'(% 49(" )=>@,>&2)=>,(&)=>)@CD&=&&> &'$#3)=>33'#&(2&,'>()!),'2=,'&&> 9('(43**,&O3**,9('O,(,*'()>!()&*(,'>!()&&!'&>*9(,=,,&,&>&!&!'&> 9('3,'@')('O,'@')('&(% ' # @ J @ J. M. Dilhac Page 34

Filtrage Numérique Février 05 A 6 ' &6 + A'!'_*)$_ &)A'!')!)(&&'(@ )& (9( 3(&$)% *),'@(&$)3A4UT(&$)3&'=UT3')&!,&)3!2)E,)(34UT%&,'!'&&&!,)&&'(&'(&(!!&>&&L3,)& `,&&)A'!'3*)$&('!')% *)3&,!(L" ^6C5D'('C5D> ^E 6 C5D'( 6 CD> ^E '&A)!'&&3)'2(3,))()&3,&&&% $)>(*),(&*&E!'&!()&'))*>&&4,!'))!( LE!'&!()&'))*%*)3&,',(&*&'(M'(&E!'&!()&')) *%! &6+ GD9('3,(," &*)& 3&,& ',(&*&!'&&3!& 3 2'(,) 3,',' `)( 9(' 3,(,*3',!!Y(&()&'3&,>&'!'((*),(&)" B 9 [ ] = # [ ] =/ &(,(=!9(>3&@&)C'V&())@)(4UT&'&D&3',,'% & ' ' ' 3 3 3 3.GD!'&!()&'))"!!)'&9(A)&A$3)A4!&&'!,() UT3)&'=UT)'&9()A&)A!()&' (3&,δUT%)A9('3,(,>)!!Y3',9( 4 [ ] =# / δ [ ]+# δ [ ]+#. δ [. ]+%%% +# B δ [ B] B [ ] = # δ =/ *)',(&*&3','(M'(&&2)!(&9(&!'&!()&'))&E3(*") &' @ E F' ( 2'( 3 P, ))'&% &,'**,& ' ()& 3 UT &' )&,'**,&3)A9('3,(,% J. M. Dilhac Page 35

Filtrage Numérique Février 05 GD!'&3,))"!!)'& 9(A)&A$ 3)A4!&&'!,() 3UT 3)&' =UT)'&9()A &)A, )' (3&,(UT%'&'&W!'&*`'!()'&,9(" [@]= # / +# +%%% +# @!'(/gwd [@]= # / +# +%%% +# B = [ B]!'(gW (@9(3UT&()'(!&>)*)_!&&_)$,'(,'&'(!&% 0GD!'& '9(" ),()'&)&*'F3)A9('3,(,%)@" B 6( J) = # J /( J) I( J) = 6( J) B =/ /( J) = # J =/ *','3&*CFD!&9(3&F'&>,9(,'*),,'(M'(&&2) 3&*)&',(&*&%!'&*9(,&A'2&&3**,()!)\)@2)F!4!CM.l* D&'" IC>D = B =/ # >.π(!'&*9())&3',!'3()>!! &>!'39(3!'3$)E )*9(,3A, ))'$5 %&!,*,'3A(*)3&,&3','(M'(&3'&( )A@))*'3)U/g*g5 #.T% 1GD5)&E! &)" &'3&*)&)&9()&,'**,&2!&(&=%',&,'**,&&' (&& )& @)(& 3&, ))'& 3 )!'&!()&'))>,,,&9(!( L @&()&,',3&&'(& 3&),&'V)&,'**,&&' '(&!'&*&>&(@ 9(A)&&' '2!CE$(, D'(!CE3'D% $! & 3 CM*D@ )'&) *',' 3)*9(,*% )),,&9(,'!'3(4@$&%!&3!(3(,'!3(4C,&!D'(!&9( 3(4C,&!D3('23,'**,&'&&3A!&3()!),'C@'3&&D% 4UT T T T T T T T T + + + 2/ 2 2 2.. + =UT J. M. Dilhac Page 36

Filtrage Numérique Février 05 3(4@$&3!&'3*)&*'&3A'33&&$(4*)&%**>)!& 3!'!$'3$'(!&3*! [ ]? I ( > ) τ =.π!& 3!'!$' 3 $'(! & 3',,'&> 9()) 9(&')*9(, 3&&$(4,'&3&" 3(4&(&'f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ilhac Page 37

Filtrage Numérique Février 05 & ' ' ' ' ' ' 3 3 3 3 *'>)),'9(>,''&)&!),&'% & ' ' ' 3 3 3.GD!'&!()&'))" (*,&!,()> UTA&!&$'((&$)EF'(3)E3A(,$>,',A& ),&!'()&*)&E!'&!()&'))*C*)&',(&*&D%$)>','&(!'&!()&'))E3(*% '(*'&>&)*)&&2)>4 /%&@)(&3&,'**,& 3)&2)%& [ ],'3'&3&2)!(@L',&E!3A(4!&&')=9(3 UT% GD!'& '9(" @,) L 3, 9(!'()& *)& ',(&*&> ) &!'&&2) 3A2))*',' 3 &*F"6 J B + J 6 J ( ) = # J /( J) =/ K = ( ) I( J)= 6 J / J ( ) ( ) = B =/ K # J = J )),!'&&3 3&!m)& 3& F'&% &,'3'& 3 &2)!(@ L ',&,'&3)!'&'3&!m)&@&E@&3(,,)3='(",(4,3'@LE)4 3='(C@'4,,&D% '9(,,3&2)3**&&2)3 &'9(@))'$9(>&A!!)9()((4*','&3&*&=2')9(&C()&)@2) 3!),'!'(&D% '&9( '(&)& F'&&' E)A( 3(,,) 3 ='(>)&=& & 3 E *`)&)'&3A@&&2)% J. M. Dilhac Page 38

Filtrage Numérique Février 05,'(*'&>)!'& *9(,&A'2&& 3**,()!)\)@2) F! 4!CM.l* D&'"I ( > )= 6 J / J ( ) ( ) = B =/ K # = >.π >.π!'&*9()) & 3', E '(@(! '3()>!! &>!'39( 3!'3$)E)*9(,3A, ))'$5 %&!,*,'3A(*)3&,&3',2 '(M'(&3'))(&&&()A@))*'3)U/g*g5 #.T%, 66 &6+ &,& 3, '4&' &(& 3&) 2)( 3'!)(&)'% )& 3'@ L,'(& 3(,',!(>,!& 3&(!',3( ('& 3,',!'> ))& 9(,))& 3,&(,!&(@% & 5 (,( ',(&@)!)(&&'(@,(&@(9( 5','3&* (9(3&F'&!m)&F'& 2) *)'(M'(&&2)!(L&2)!'&*9(,"$!'&9(),'9(!'&&2) *)&&!)&&()!'&*9(,"! &! &)!'&&2) 3,)(&!'&&2)& '!)4!)(&*2)9(5 (*,'!&3A**3 **&'(M'(&!&& 49(" 5O5C*!()&&!'&D> OC*!()&&!'&D>,(&@b',(&@> *9(,3,'(!(O,('***9(,=> )! &,,&,>!&&23>&'!23>&'23c > *)'=(O'@$@$*)> '&,))'O!!)> )'c!&&> $!&&>23!&&>23M,% J. M. Dilhac Page 39

Filtrage Numérique Février 05 B? ' + C9*" "+ &, 2)', 38(*)>!&!)(& 9(&' 9(' 3,(,> *!!Y)8'3 3& )9())&'!'&)&C33'>()!),'>&'D')(%,'3!(L)&&( &3)'&9()&,'&3!&3,'Z&'&@&% ')84!)38(*)3')&, 2)',&3',,'%'9('3,(,&" =UTO4UTP. 4UTP =UT 3&,!'!)(&!,& 3 &' *',' & $)!'!'&` ))!,& )8, Y3&'!'&)&>&E)2&3)8!3!'$')$$, % 4UT. P =UT 4UT =UT 4UT =UT '&' 3&3'& x > 3UT 3UT x > '&'!' 3&3'&!'& 9(&. '&' 3&,'**,& 4UT =UT 4UT =UT :(**&#!& J. M. Dilhac Page 40

Filtrage Numérique Février 05 4UT P. (!' 3@( 3) ',( 3& &()& 3&>) ' I!'J (! 4!))*,()3&',W3&'&2&!)(&)'$&9()'&33% &, 2)',()&!'(,!&','&!'3 E, 9( '(& @'&!!) I*' 3,J9(>'(&)&@'&>8&!&,'9(%,'''(@)I*'3,.J3( L*)>&9()&'!'&)&,'&!'3&9(>')9(>&', '23(% 3UT 3UT P 3UT =UT '&' 3&3'& 3UT 3UT 3UT '&'!' 3&3'& x > x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ilhac Page 41

Filtrage Numérique Février 05 & *(,,( (9('!&!'() )&' 3&'!'& 3 2& 3( (9( 3(&$)% ) @))>&(@)& '3)&> @$()*4'(*)'> ) )'$((3&'&(9(&!(L@2)% 2D(,(!,()&3&"C,9(&(&'(**3&4,!'&D ^ )&!(@ )& )A9(@) 3A( C( '&D &(,','!) C),( P 3,'3$ P 4,('P$D(&()!'338 ')'$>)&&(),&9(!&!'(9(&(,'&3**&C@'3&&D%!,()&!$@,)&!',&&(& C3(,3&(,''!(&D%!*', & 3(!'&&2) $j, E (,,( I@3J 9( @ ),'*)!9(4&&'(@(3&:'C(),(($&()&D` )'&)'&'$&('&3(4!)&"(!'()&&(,'&((!'()& 3'&%4&3',('&9(:383&&&33'&#&(,'&%8&)&(,( 3I J %,,(8&'(*'&!&)!,"))&!&3*(&)&()3)8&(,' &,&&(3,'3$3)8&(,'CPD>'(2&(&(,'3'))(%&'& &(,'&&(@& C@' 3&&D,'(& 384,(' &'!)(& )'& M(&*&>'@3 CI' *)(& JD)'&)!!)%!& &(,', &(,'P % &(,'P. &(,'P 3&&& 3'&,' 3%, % 4,%,' 3%, % $ % 4,%,' 3%, % $ % 4,%,' 3%, % 3&&& $ % 4,%,' 3%, % &(,'& $ % 4,%,' 3%, % $ % 4,%,' 3%, % ^ )&!'&&3 3&,,(()(& 3(& '2 3 2&>!'(!& '9()& &()& 3,),()&3&% ^ )&!'&&3 3& I& 3'c $&&J" '(&)& $&&&' 3'(2)&> )8 3(!( L &(@$3)'&3)8@38((!')'$,))'()))'&38((&$!&)% $&!'>,)!',3(3'))38L&((!',&&./EE/,=,)& 38 ')'$% ^ )&&!'&&3 3& $&& IE 2))J!'( 3,))@$())'& 3 ()!),'& @$() *)'%,D:)" 8,,(&&!,*9(3&!3$3&$&3!&&!(!&&@E'(>!3 && @$&!'(,&!!),'& C32, & ()!)&> '2(&& (!'&]D%)&&2),',(,&!)&9(*'$3(&$3',,!'(,,))& 4,('&> &)8@$ 3 3& & 9(>!'()&!!),'&!& )> )(&!*',& &' I3&&J C9(*2)&D,8& 2,) 9(,'!>!!!'E3&9(@())&'C)'D!)(&!3&>3&,&,&3*$(!(!@&2)&!'%!'()&>,&'3&$&&I&J9(&'!)'=&% J. M. Dilhac Page 42

Filtrage Numérique Février 05 &'(@))&,,(&,'!'&&>!,'>)$)&3(4=!&3,,(&"!4!) (I,B(J,''3')8,@3!)(&(&% (2(,'!',&&(&&&',E(,'!',&&(E(&!,*9(>,3!2)&'!'$2)% ''&*9(,',)&>,(4,','\(&!'(L!'$&&&2)(` )&,'!)(&&'!!(&&(%8@&()(!'()&%&,'!)(&!'(> &'3',,'3ME3'&!*'&9(8(!'$'&&2)(%)&&''(*'&!'$&!`'!(L(M'(38 ($3(,'3&&2)(3,E!38( &, 2)',&*',')&% & ),3 3( 3@)'!! 38(!!),'> (, '4,(,) & $),)( 38(!',&&(@$()*4'(*)'%@$()*4(!',&&(@(/5>(!',&&( @$()*)' 3 L $'@(3 /// 5[ &!& 3 3@)'!!&'!,' 9()@3&)!!'@&% ',)('&'(&!,&'(@$'"E!(&,3,),()9(@)>)!(&&, ),9(3&&!!(&2'39(!'((a,)&&9(%,,&9(3' (@$&&)(4!'()&!!),'&!'2)&% & $3&*'(&&(& 3 &'> (&> 4& &(&> (,> ''')> )'$ @,&% 49(" '3!()&'))Oc3'c$3&$ '3> &*''2)O2)&*'> )'$((3'&*O*c'3)$ **,>,'@&'3&,'**,&3(*)%%%O,'**,9(F''> '33&&()&3&O'(3'**'&% J. M. Dilhac Page 43

Travaux Dirigés Février 05 ) / E. Sicard, J.M Dilhac Page 44

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Travaux Dirigés Février 05 ) E2CF! % /" G H 'θ(@2))'(*'!π#.pπ#.% 'QOCθD D,)3&3!'22)* θ C4D.D,QCθD D ),()5 Q C4D&'(&*'3A($) 0D 33(* Q C4D 1D ),()UQT H '(@2))'K=(3&3!'22)* K C4D3'!" CD = %4!C D% / %())&))'N &)&(3)A4,,>'&(!!'&O% %.')A))(3* K C4D %')A))(3)*','3!'5 K C4D %0'(4!)3)&'3K*','3(!& %1(@))A&!,3K>)@,)A,=!N E. Sicard, J.M Dilhac Page 46

Travaux Dirigés Février 05 )) E2C)F % 6 " G H,),()3)*','3A(',')'3A(&$)(9()'/'(>9(!'22)>, $E, 9(!&()!)3C*$(E$(, D>3')&()&(@C5$(E 3'D% 33()!$@Cd/D ),())@)(3: γ 44 / :!&!& &!! '! E! H& 4 6 " '4CDO@,O@,(!'22)!>O@,(!'22)9 D ())&))'!9N 2D '.4!)&3)&',D &&* C4D 3D (@(),'**,3A(',')'N!',&&(&&)&'(&&)$N D '9(U4T&3**3)'=34C',&&(&'$'39(D E. Sicard, J.M Dilhac Page 47

Travaux Dirigés Février 05 )0 E2C0F "G, Y&&'#,!'(9(*','&)')!,!3)*$(,3&&'(&% &&'>(I/J9(@(E1->(IJE1-%*''3=!I/J'(IJ&@'=!)8('(&)&&%'&3)8&&'>(2(2),6(&&CO/->σO.-D@3 &(!!'&(&$)>9(!(2),!'% (,!( 1#! GD&*''&I/JIJ9(!'22)&>3&&()&'3(2(2), $(&&()&'3(&$)(@(3(,!(.GD(!!'&9()8(@'(2I/J>9(@()3&3!'22)3(&$)E),!'N GD(@()!'22)38(3&&&'3(2I/JN 0GD(!!'&9()8(@'(2IJ>9(@()3&3!'22)3(&$)E),!'N 1GD(@()!'22)38(3&&&'3(2IJN 7GD),())!'22)38($)'2) '"38!&)&2)&3)*','3!'3&)'&36(&&>''(@". 4!C. π. 4!C. π... D = /%?;;. D = /%??<71 E. Sicard, J.M Dilhac Page 48

Travaux Dirigés Février 05 )=!2'E2CF*. / G (!()'$9(X&,'3!(&$)!'39(CD>&E)8, ))'$ 38(&$)4CDC@')*$(D% GD'4CDO,'&C.π*D>5 O1*O/>1`!&4 CD!'((!'334CD%.GDL9(&'!'(),,(3)*$(.% GD*9(,38, ))'$, '&!)(& (&!,))) '3 'N(&* )!'&% CD α X 4CD P CD 4CD X* X'(@ 5$("4CD&($(3&'!*5 O# % / 4CD P X CD 4CD 5$(."4CD&($(3&'!*% &$)_)@&'_)'$9(&,'3!(>!'((&&&'!(&=& (9(%&!,3(&$))'$9(&A)3/E7F%&$)(!')&9(*! '3% &'( 9()A(39(*,'3!&&M&R3)!), ))% GD())3'L)*9(,)3A, ))'$N.GD()&>3&,,&>)32(9(3'!'&&3)&=&3&&&'(9( C2&#&DN & "' 6 + GD@)(4(3(!!'&$)&(2(39(*,'3A(E.2&&3;>;7 3:%!(&&,3(&$)&3/n%())&)'&)!(&&,3(2(N.GD @)))! ',('(! '3>!'(, 9( & 3) 9(*,'!)(3N'(9('N E. Sicard, J.M Dilhac Page 49

Travaux Dirigés Février 05 )A!2'E2CF 6! G ),())53&&$(4" GD4UTOδUT.GD4UTOδUCDT!!)))A9('3)&*'35'(3&&$(43&,&C5D" + /CD = [ ] >.π % C5 *9(,3A, ))'$D% = '3A(!)&$)"UTOδUTP.%δUTPδUPT% GD!&UT%.GD),()C*D>53UT%, C D 3&)A@))o5%!!)))A9('3)&*'35'(3&,C5DE!'&" /U@% T = > @.π [ ] % B CW> *O5#D% GD),()UW *T>5E!'&3UT%, U@ T 3&)A@))o5% 0GD),()AUW *T>5E1!'&3UT%, A U@ T 3&)A@))o5% 1GD *!&& ) &$) UT> @, O #5>!( )@( 3*9(, 3A, ))'$C5D>3!!'3A)@'O.%'=UAT)&$)3&'3)A5%, =UT%),()QC*D>53=UAT>*','3*!% 7GD9()9(&! &&,)&2@&>,',)(&()A&2)3,@)% %:"&(,,(3&,&3&9(&'&0>'&(!!'&),'(23)9(&'.% )$' 3 55 E /.0!'& &()&!& ) \'&& 3'& 38( C'@&&()'$9((9(D@))E(*9(,38, ))'$3.1>7WF% GD9()!&3,=,)C,),()38(33'38(()!),'!)C$)$) ',&&'D, '&!'()84,('3)8)$' D@'(&,'F@'(&N.GD(&!&&)&)!&3,=,)&!)(&$39(,@)(N GD8)&!)(&,'(N E. Sicard, J.M Dilhac Page 50

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Travaux Dirigés Février 05 )I!2'E2C0F!G '(&(,(,'9(3(*),3&&'(&% 4UT P =UT '(&(,(3*)&@&)(*),3&&'(&% 4UT 2 P, 3 P =UT ')*)3*!)A9('(43**,&" =UTO / p4ut4utq GD),()!&$! 9(&&!'&&!()&'))>3,)) '9(!'( / O#.%.GD!3&!'& '9(>3&(% GDL9(&'!'()A9('(43**,&" =UTO / p4ut4u.tq> / O#.% 0GDL9(&'!'()A9('(43**,&" =UTO / p4utp4utq> / O#.% E. Sicard, J.M Dilhac Page 52

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