Traitement du Signal Février 05

Transcription

1 Traitement du Signal Février 05!"##$%&'()'(&%*#+#,'(& -&'.//0.//1

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4 Signaux Aléatoires Février 05 E. Sicard Page 4

5 Signaux Aléatoires Février 05 )!&>'!))'&3&$(4!')&C,&')&!)(&,'(&D )A&!,)!&>'!)3&$(4&!'!')&C4")&'3&),'$9(&D &,,'(&'&&(4&$(4!')&%$)&'(4(&=!&&$(4!'&!&33**,()M(% 3**,&$)3&)'N '(*'','&!'3E(&$))'! ", )' δ!()&'33, π = (&,3) π + " δcd =N E. Sicard Page 5

6 Signaux Aléatoires Février 05! # *'()'3),'(236(&&&&!'!'(,,&)&&$(4)'&%) 2&>),'(2&3)*'".. = '(@)*'3)*',''2(O>','!)9((!()*'()'"... σ πσ!! " # $!% $%&'% *'')'3!33( &3%K&(@2))'>KCD&()&'% 4!)"K&(&$)!'(@@'A!'9())@)(/>C5$(.D% &!! '!( ) # "&@F@'(&$(&$))'N E. Sicard Page 6

7 Signaux Aléatoires Février 05 (" -)()'(*'!3/EC5',''( &)')&'()&D%!!! *+, E. Sicard Page 7

8 Signaux Aléatoires Février 05 ) *' +,-.. -)('=3A(&$)&((3(/ (/ - (/ / (&&,'=3A(&$)&((3(/C&4CD,'!)4D ( + /.. (/ $3A(&$) +. "),())@)('=>)!(&&,)8$3(&$)&(@". P.! / =. / 5 K &,'&&2' 5 K 3@&)'&9(43@&)A*>3@&/)'&9(43@&)A*. / # = / # / "*','3!'3(2(3)*$(. E. Sicard Page 8

9 Signaux Aléatoires Février 05 5 K C4D "" '!C4D'(* K K C4D '&!'3E(2'()3&@)(&3K% 3&3!'22)&),(()'&(@)A43&Q3)!&,3(&$)>'!C4D% &),&3A(2((*'3&2(3H%/E%/>!C4D&(*'3%/E%/% (# 01#! 2-,3-, ##! -2))'33&3!'22)$(&&%5','&'(&)2%A'2 *&)&'3!)(&(&2(&2),&C@'/!4!)D% (45!!!"(&'32(&2),&3@&(2(6(&&% '22)38L3σEPσ"7<R '22)38L3.σEP.σ"?1R '22)38L3σEPσ"??%;R E. Sicard Page 9

10 Signaux Aléatoires Février 05 "38(&(& *1 %/ ""% /" %/ / + 6CD = / CDS 6CD!!)" + CDS4CD = CτD4C τdτ (&'"(M(3I')'&3J>'),.3&%()&('&&')&!)(&&&&.N. E. Sicard Page 10

11 Signaux Aléatoires Février 05 + C/ D = U/ T = % / CD% 8!!E)@)('=% 4!)"UKT3(&(& (8&!,38(*','$38(@2))'Φ&,),()!)*'()" + U CφDT = CϕD% φcϕd% ϕ '!&" U % / T = % U/ T KQ&'3!3&" U/ % 6T = U/ T% U6T, 9(!3'(@)!'3,3A(&$)%))&**,,!'('(@ )A!)(3)*9(,3A(&$)'=3&)2(%@$",),()&!3,2&&(3& ()!),'&% '((&$)3&>)&8$3),'@')('3(&$)4CD@,4CD" γ 7 = + τ τ τ 089!!:; 4 ",)*','3A(',')'3(&'3)*$(0% E. Sicard Page 11

12 Signaux Aléatoires Février 05 *&! 9!# < )!!! * '((&$))'" γ CD UCD% C τdt '!&"!>4(EO/% E. Sicard Page 12

13 Signaux Aléatoires Février 05 0 / 1 '3A'3.C@)(9(39('=D. +.. C/ D = U/ T = % C / D% 6)&' + C/ D = U/ T = % / CD% &- = / = 9 = / /!',&&(&)'&&'&)&!'!&C'=>,=!D&'3!3&3 )A'$3&!&%!',&&(&&$'39(&)&'=&&(!)(&(&)&'&&'9(@)&E3& '=&3A(&()!(@>&((!&&(**&)'$% 3&&!,)3!(&&,3&),&3A(&$))'4CD,'&!'3E)&(3)!(&&,))'$3)A43&*9(,&% $&()&'4 CD34CD3((3(,),())&*'35'(K C*D34 CD ( (. > π ( = /( = ( / *'3&),&)'. / ( &. =! ( + ( '((&$))'>)C*O/D&$)E)!(&&,3),'!'&,'(>&' &. = / /. E. Sicard Page 13

14 Signaux Aléatoires Février 05 &3&&$(4 σ &.?. π + ϕ = δ + / + δ / 0!3(2(2),",'&%!"3(2((42'&3'(&&,% 9(),'!&''(@,'2(E(&'))'%&(*','(@&))3)&&, 3)!(%A$3, 9(),'@(" 8. -O@&&3)A),'3&)A43)&&, O&&3)A),' WO%<4/. '()#GX>,'&3:')3F O!(3((CGXD &. = 0@(+ O&&,3(,'3(,(C D O//GX>)(42'&3A(&&,3/X' &3)A'33/ 7 -. #F%2( 9(&(&'%&*'35'(&A4!-%&%!(&&,-. >& -. #F% / 0 -. #F / 1 / 7 / ; 59(, XF /X //XF $ /$ //$ A!!! ## ) A!)(33(2( 9(&((3&2('$(&&C)(&(&))'&32(&2),& )&3, 9(),'D% E. Sicard Page 14

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

16 Filtrage Numérique Février 05-34' &, Y3A,9(&','!3" (,!( (,'3'(C&3(,'>!)*,'&')>*)!)D (, (,'(% )*2)> )!'3(,2)3&,,(&3> )!'&&2)3&,'!)4&> ),'Z% &3*(&&'" =&9(&)'$9([D> )A2&,3,,(&!&&*&> 49(",!(O&&'>, ))'(#2)'9((O&!)3 ')3,,(>,'(O,('> =&!( '3'(, ))'( 3&$(4 #2)'9(( '$9(3,'3 **, $,'( µ P '& '!'))& J. M. Dilhac Page 16

17 Filtrage Numérique Février 05 / '& ' 8'!' 3,'@&' (9( )'$9( &*'( ' 2 3 2& ( )' 9(8( '!' 3 *)$!( * 3&!Y 3 ) *' 38'3 3( &$)> )& 3&,'(& &&)9(*,'% A2@' 2 &()& *\&> C$) ' )'$ '@&'D $)&% & &=2')&&(@&&'!)'=&3&)&&, &2)',&% 5 A'!'3!(L&(!)3&&,3&&'(&9(@L4!),3&)!$! &(@% 6 + GD9('&,,&9(&" '=)&$)&()3),'@&'>))'=O*CD,'&())'3,'@&'%)),!(L(*''('C@',!9(*,'D%&),&38()'(*'>38(,'3 2()>'!(," =O U. 3 P.. 3. P]P. 3 P. / 3 T 'V ' 6 >! =&9(4!@')&C'(!&D>3 )23!'3& *'>3 )23!'3&*2)%!&)!)(&!@'!'&&2)3=>&')!'3&3(23!'3&*2)3 % J. M. Dilhac Page 17

18 Filtrage Numérique Février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`3&),&38(,'@&'!'39(>)&@)(&3&' '3*&$()!(3&!'&*3(9(3(&$)%!'3&!!)!'338, ))'$%&3'2(4,&3*$(>)3&!'&*33(&$)CaD &'2)&!&!,!'(!!)9(E)A3()$3(2E,'@% '(9(C($&D&@3_!'_% (!'3@(3)&'>)=(&9(E()&3,)&'3(>E,(&3A)&!'&&2)& )'& 3&, $& 3 ) &'% '( )& = ( *',' &9()> )A(&$ 3, & 2 &Z!'&&2)% ' )'$9( C(, ))'( 2)'9((D&3',,,)% J. M. Dilhac Page 18

19 Filtrage Numérique Février 05 ))(&> )'$9( (9( CD 4!')( 38'3 F'> 3&) &('V 3(4 && 38, ))'$> ),'&)8!)(3 3 =% =CD & 3',(*',' &,)")& 3&,'(& 3'CD&)!&'(9(%,'&('!)(&*3,&$))'$9(!( L)&!)8M'(38(*))'$9(>3=!!&&2&>3I*)3)&&$J% A@'!9(3A(&3',,',)(&')&(@% *+ µ, () & % &% & % &% & % ' % -"% "!! (",'@&'(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

20 Filtrage Numérique Février 05 5 &!&!,!)&3)8'!'&'" )8, ))'$> )9(*,'!)(3> ),'3$% A& E)A&&( 3,& '&'!'& 9()A!)(3 )'$9( 3(&$) ( &(**& _&*'_!'(L,'! &2)!(&=&(9(CaD% &% #$%" & % /0%"-"%"1"%0 & 2 % & &!&!,!(43@L, '&&)'&3)&B(@&'" )*9(,38, ))'$> )=!39(*,'C'3'(',(D> )!&39(*,'dOe)'232&3,'3$> )(3(,'3(9(% * + A& )A'!' 9(! 3!&>! ( &(,,&&' 3A, ))'&> ( ))'$&!'39(>3*&&&(!'3( *9(,53A, ))'$% &% & % ' *\'@3>)&()3()!)E)A4,&)'23A, ))'&!(3!&")&,,(&@)))'&()@C3A'V(,'Z)@D)a&@'&(, $3@)!)A@3A('!$3'23A, ))'&%)&!,''((&&@39(A('2 (3A, ))'&&,&&!'(,',!&)&$)%!(3'9()*9(,(3A, ))'$5 &)E)!)(&$3 *9(,* 4!&3&)&!,3(&$))'$9(E,'@!)*'()C '3 'D" 5 O.* 4 J. M. Dilhac Page 20 %

21 Filtrage Numérique Février 05 '* 4 A&&'(@!&())3*C**3(2(3*'3>'(&!,3(&$)!'&&3!&3&(!!'2'D>(*)!&&2&&()&!'(*4* 4 ",*)&)*)!)% $ 6 + A&($)3,'&!'3,)'2*3@)(&!'&&2)&!'(4 CD)'2 *3@)(&!&&!'(4 W CD)(*9(A(3),'@&'('223 )'$((*!')A*''% 9(*,'!(L! C@'3$&E$(, D'()!)(&&'(@>,)&,,(&&'!)(&&!)&>! C@'3$&E3'D%&'(&)&,&>( ' 6 ε W C9(F''&D&,'&`)),'&!'3E(!3A*'' 3'L&!), '43A(!&39(*,'&(**&*2)" ε W O4 W CD4 CD 4 W C D 4 W C D 10 '3 10 ',( C D 1 4 C D ε ε= 4 W 4 ε /2 4 C D C D /2!CεD # #!CεD ε ε /2 / /2 / 3&&!,)3!(&&, 3(2(3,'@&'!'3CO. #.5 D 3&&!,)3!(&&, 3(&$)4 *CFD 5 #. / 5 #. J. M. Dilhac Page 21

22 Filtrage Numérique Février 05 )'&9()A!)(3)'$9(3A3A(!!', 3)*9(,3A, ))'$D)A!)(3)'$9(3&) 3!,'&& E **, 3A!)(3 9(* C '2*D( _9(_(9(2>$)2()%&),&3,'@&&(&2!')&> (2,'3)&$3)A!)(3%,1 7/ '+ &!&!,!(43@L, '&&)'&3)&B(@C!!'@,), 3&, $&D&'" )*9(,38, ))'$> )=!39(*,'C'3'(',(D> )!&39(*,'dOe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µ 49(",'@&' )'$9((9( O )'$ ' 3$),'@&'> > I,'@$ )'$ &$)'2=,'3J>, ))'$O&!)$>&!)3 ')3,,(>&!)$*9(,=C&!)$D>&!)$ 'C '>=9(&D>!)O)&$>)&$*)> 9(*,'O9(F'>!&39(*,'O&!&F> ',(O(,'>'3O'(3$% J. M. Dilhac Page 22

23 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( & $3( &,'3&> *&!!Y 3)$( $) E)!'3 3A, ))'$>(&$) 3&, A 3* 9(A(4 &&3A, ))'$%&$)&)'&'4U T>@,)*`)&,', &UT!!)))(3&,3(&$)% & ' & %'! ' 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

24 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%

25 Filtrage Numérique Février 05 )A$)&!),!(&'3&,> /ω&(*',','!)4>,'(>!'39(>3!'3.π# > >ω( '(/C Mθ Oω(!'(!!),!'3,% >$)P# π# >!!) #.%/ω&)&!,34ut> '((&$)4UT)>U/ωT&!U/ωT&!% KC MωD KC MθD 4 UT ^ ^ ^ ^ ^ ^ ^ ^ i# / i# / i / i U ( T /C >θ D 5 #. U T ω θ 5 #. * >θ /C >θ D δu T U T. U T / C >θ D. /. C >θ D U T+#. U T / U >θ T +#/. U >θ T U T.. U T / C >θ D /.C >θ D &) 2)(>' '(@)&!'!& &&))& 3) 5>!,())!'! 3 )%!'3( 3,'@')(' 3&$(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

26 Filtrage Numérique Février 05 )'3('23A, ))'&,'&3&!'(),),())'(*'&)&!,,),()" )!'3( 4UT%cUT & 3(& 3& )A&!, 3& *9(,&! )!'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'($3*&&3!&' &,&& 3 '2(4,),()&>!(L )@> (,'> 3' ( &()!3 3A*'' (> ) ( &!,, ))'!!& 3A'2 ) &$) '$)% ((&>) ' 3 '> 3ME,'!'()A, ))'$!')>!'&&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 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 π B

27 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

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

29 Filtrage Numérique Février 05 2D &*''35'(!3" E3()'23A'!'&,&&&!'(),),()3) 5%&&()&)&!)(&,'(&&',(43&'')=' (W=C?71D>,'(&&'(&) '3A)$' 3A),!')%'(&))'&3',)!,!% &(!!'&9()'23A, ))'&&(!(&&,33(4>&'" O. >()% 4UT!'&&3)'&())('2!3A, ))'&>)5!(&A," B. >0 B /U@ ωt = U. T + U. +T > 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

30 Filtrage Numérique Février 05 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

31 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% '()A&&)>(&=&3&,)&,,&>(, '4>!&'9('(43**,& C9(@) E )A9(' 3**)) 3& &=&& )&,'(&D>! &&!'&&!()&'))& UT '9(&CM*D>!&*','3&*FCFDC@'!)(&)') 3*' 3,& &D% 9(!!',!),),() 3)!'& 3(&=& E( 9(),'9( C)=&D> & )A()&' 3!)(&(&!!', & & $),&&!'(,',@'(&=&E!3A(, 3&, $&C&= &D% 4UT 9('(43**,& UT CM*D CFD ' =UT GD'3(,'" &&=&&3&,&)&&')&2)&!&&2)$3'&'!'&32&")A33'> ) ()!),'!(,'& ) & '% &'!'&*$( 3&)& 2)',& *',')& )&()&& 3&)&&, & 2)',&!&(&=&% 3&&'(&!&(,'&>&=2')&)!'33A, ))'$% 4. UT 4 UT 4UT > =UTO4 UTP4. UT =UTO4UT J. M. Dilhac Page 31 4UT =UTO4UT

32 Filtrage Numérique Février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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 J. M. Dilhac Page 32

33 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,? GD*'4!)&" L 9(!'()&&=&&)&,'(& ) 4& 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

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

35 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',,'% & ' ' ' 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