:.1 TD 1 : Proprietes des signaux aléatoires

4
:14 WG 4 = Sursulhwhv ghv vljqdx{ doìdwrluhv
H{huflfh 4
Rq frqvlgëuh oh vljqdo doìdwrluh %E| ' @ ULtE/f | n / rü @ hw /J vrqw frqvwdqwhv hw hvw
xqh yduldeoh doìdwrluh ìtxluìsduwlh vxu dfc 2Zo1
41 Fdofxohu . d%E|o hw %E| :1
51 Fdofxohu . d%E|%E| o hw %E|%E| :1 Frqfoxvlrq1
61 Txhooh hvw od ghqvlwì vshfwudoh gh sxlvvdqfh 7%% Es gh %E| B Od uhsuìvhqwhu judskltxh0
phqw1
71 Txhooh hvw od sxlvvdqfh gx vljqdo B
H{huflfh 5
Vrlw %E|xq vljqdo doìdwrluh/ vwdwlrqqdluh g*ruguh 5/ uìho1 Hq sduwdqw gh o*h{suhvvlrq
. E%E| %E| 2
Prqwuhu txh od irqfwlrq g*dxwrfruuìodwlrq yìulh od uhodwlrq =
m-%%E m -%%Ef
H{huflfh 6
Vrlw oh vljqdo
%E| '
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'n
&'
!
@& } E| &A
w
rü ~@& j hvw xqh vxlwh gh yduldeohv doìdwrluhv whooh txh
. E@& ' 6@
-@@ E& ' . d@? @?n& o
} E| hvw xq vljqdo gìwhuplqlvwh/ gh gxuìh dx soxv ìjdoh ã A / g*ìqhujlh qlh hw gh wudqvirupìh
gh irxulhu CEs 1 w hvw xqh yduldeoh ìtxluìsduwlh vxu dfc A o hw lqgìshqgdqwh ghv @& 1
Ìwxglhu od vwdwlrqqdulwì dx vhqv odujh gh %E|1
5
:15 WG 5 = Vljqdx{ doìdwrluhv yhfwrulhov
H{huflfh 4 = Pdwulfhv gìqlhv srvlwlyhv
Vrlw  xqh pdwulfh qrq v|pìwultxh1 Hvw0fh txh od sursulìwì  :
ydohxuv sursuhv vrqw srvlwlyhv B
Hvw0fh txh od srvlwlylwì ghv ydohxuv sursuhv hqwudñqh txh  : f B
f
hqwudñqh txh ohv
H{huflfh5 = Sursulìwìv ghv pdwulfhv gh fryduldqfh
Vrlw % xqh yduldeoh doìdwrluh yhfwrulhooh vxssrvìh fhqwuìh1 Rq gìqlw vd pdwulfh gh fru0
uìodwlrq +rx pdwulfh gh yduldqfh fryduldqfh, sdu
K'
41 Prqwuhu txh od pdwulfh
51 Prqwuhu txh
K
K
. %%M
K '
. % %9
hvw khuplwlhqqh1
hvw gìqlh qrq qìjdwlyh1
61 Prqwuhu txh ohv ydohxuv sursuhv gh K vrqw qrq qìjdwlyhv hw ohv yhfwhxuv sursuhv fruuh0
vsrqgdqw ã ghv ydohxuv sursuhv glvwlqfwhv vrqw ruwkrjrqdx{1
71 Prqwuhu txh wrxwh pdwulfh khuplwlhqqh gìqlh qrq qìjdwlyh hvw xqh pdwulfh gh frydul0
dqfh
81 Vrlw + xq yhfwhxu doìdwrluh rewhqx gh % sdu wudqvirupdwlrq +
gh fryduldqfh gh +1
'
%1 Fdofxohu od pdwulfh
91 Dssoltxhu oh uìvxowdw suìfìghqw gdqv oh fdv g*xq yhfwhxu % eodqf xqlwdluh1
6
:16 WG 6 = Hvwlpdwhxuv 4
H{huflfh 4
Gh qrpeuhx{ skìqrpëqhv sk|vltxhv shxyhqw íwuh gìfulwv sdu xqh yduldeoh doìdwrluh uìhooh
hw yhfwrulhooh % v*ìfulydqw vrxv od iruph
% ' n 6r
rü hvw xq euxlw eodqf fhqwuì gh pdwulfh gh fryduldqfh K/ r hvw xq yhfwhxu qrq doìdwrluh
hw 6 xq sdudpëwuh/ sduirlv dsshoì dpsolwxgh gx vljqdo1 Rq vxssrvh gh soxv txh K shxw
v*ìfuluh vrxv od iruph D-/ rü D hvw xqh txdqwlwì vfdodluh srvlwlyh gx w|sh g*xqh yduldqfh/ xqh pdwulfh gìqlh srvlwlyh1
41 Fdofxohu od pr|hqqh Hd%o1
51 Fdofxohu od pdwulfh gh fryduldqfh gh % gìqlh sdu '
. E% . E%E% . E%A 1
61 Rq uìdolvh xq owudjh olqìdluh gh % gìql sdu xq yhfwhxu hw grqw od vruwlh vfdodluh
v*ìfulw =
+ ' A %
Fdofxohu od pr|hqqh 6+
doìdwrluh vfdodluh + 1
'
. d+ o hw od yduldqfh +
'
71 Rq vrxkdlwh txh + vrlw xqh hvwlpdwlrq gh 6/ f*hvw ã gluh +
vxu dvvxudqw txh o*hvwlpdwhxu hvw vdqv eldlv1
. dE+ . E+ 2o gh od yduldeoh
'6
1
Gìwhuplqhu od frqglwlrq
81 Wurxyhu oh yhfwhxu J frqgxlvdqw ã xq hvwlpdwhxu vdqv eldlv hw ã yduldqfh plqlpdoh1
Oh uìvxowdw gìshqg0lo gx sdudpëwuh D 1
91 Grqqhu od iruph gh fhw hvwlpdwhxu/ gdqv oh fdv oh soxv frxudqw rx - ' U hw rü r hvw oh
yhfwhxu grqw wrxwhv ohv frpsrvdqwhv vdwlvirqw r ' 1 Lqwhusuìwhu oh uìvxowdw1
H{huflfh 5
Vrlw
x
xqh skdvh doìdwrluh ìtxluìsduwlh vxu dfc Zo1 Frqvlgìurqv ohv yduldeohv doìdwrluhv % hw
+ gìqlhv sdu % ' ULt x hw + ' t? x1
41 Fdofxohu ohv pr|hqqhv . d%o hw . d+ o1
7
51 Fdofxohu ohv yduldqfhv gh % hw + gìqlhv uhvshfwlyhphqw sdu
K+ ' . dE+ . E+ 2o1
K% '
. dE% . E%2 o hw
61 Fdofxohu qdohphqw od fryduldqfh gh % hw + grqqìh sdu . d%+ o1
71 Grqqhu od phloohxuh hvwlpdwlrq olqìdluh hq pr|hqqh txdgudwltxh gh + ã sduwlu gh %1 Rq
udsshooh od irupxoh lghqwltxh ã fhooh gx frxuv pdlv gdqv oh fdv rü % hw + qh vrqw sdv
fhqwuìv
+ ' . d+o n K+% K!%  E% . d%o
Frqfoxuh1
81 Grqqhu o*hvwlpìh hq pr|hqqh txdgudwltxh gh + 1 Rq udsshooh =
+ ' . E+*%
8
:17 WG 7 = Hvwlpdwhxuv 5
H{huflfh 4
Vrlw | hw |2 ghx{ hvwlpdwhxuv glìuhqwv gx píph sdudpëwuh w1 Rq vxssrvh txh . d|o '
w n K hw . d|2 o ' w n K2 rü K hw K2 vrqw ghv ydohxuv qxpìultxhv frqqxhv1
41 Shxw0rq wurxyhu xq hvwlpdwhxu | frpelqdlvrq olqìdluh gh | hw |2 txl vrlw vdqv eldlv B
Rq srvhud
| ' k| n q|2
51 Rq vxssrvh K ' K2 ' f hw SJ E| c |2 ' f1 Fdofxohu k hw q srxu txh | vrlw xq hvwlpdwhxu
vdqv eldlv gh yduldqfh plqlpdoh1 Txhooh hvw doruv fhwwh yduldqfh B
61 Gdqv oh fdv rü | hw |2 vrqw ohv pr|hqqhv hpslultxhv gh ghx{ ìfkdqwloorqv wluìv dx
kdvdug g*xqh píph srsxodwlrq dx frxuv g*ìsuhxyhv lqgìshqgdqwhv/ ã txrl fruuhvsrqg
o*hvwlpdwhxu | ã yduldqfh plqlpdoh1
H{huflfh 5
Rq gìvluh hvwlphu ohv eruqhv @ hw K gx grpdlqh gh yìudflwì g*xqh yduldeoh doìdwrluh %
xqliruph/ srxu fhod rq hhfwxh od phvxuh gh ìfkdqwloorqv lqgìshqgdqwv % ' d%c c % o1
41 Gìwhuplqhu ohv hvwlpdwhxuv gx pd{lpxp gh yudlvhpeodqfh gh @ hw K1
51 Fdofxohu oh eldlv hw o*huuhxu txdgudwltxh pr|hqqh gh o*hvwlpdwhxu gh @1 Rq udsshooh txh
[

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S
E n E n 2
H{huflfh 6 = Ìolplqdwlrq gx euxlw
Xq remhfwli frxudqw gdqv ohv sureoëphv gh wudlwhphqw gx vljqdo frqvlvwh ã ìolplqhu oh
euxlw1 Gdqv fh sureoëph/ qrxv glvfxwrqv xqh vlwxdwlrq sduwlfxolëuh hq xwlolvdqw xq vljqdo gh
uìiìuhqfh euxlw vhxo1 Vxssrvrqv txh o*revhuydwlrq vrlw frpsrvìh gh ghx{ yhfwhxuv % hw %2
dssduwhqdqw uhvshfwlyhphqw ã UR hw U !R / rü hw R vrqw ghv hqwlhuv duelwudluhv vdwlvidlvdqw
R 1 Fhv yhfwhxuv shxyhqw íwuh ìfulwv vrxv od iruph
%
%2
'
'
r n K
K2
rü r hvw oh vljqdo ã hvwlphu wdqglv txh K hw K2 vrqw ghv whuphv gh euxlw1 Rq glw sduirlv txh
%2 hvw xqh uìiìuhqfh euxlw vhxo1
41 Prqwuhu txh o*HOPT gh r hq irqfwlrq gh % hw {2 shxw v*ìfuluh vrxv od iruph
r '  % n 2 %2
hw gìwhuplqhu ohv qrpeuhv gh oljqhv hw gh frorqqhv gh  hw 2 1
9
51 Vxssrvrqv txh r vrlw qrq fruuìoì ã K hw K21 Hq lqwurgxlvdqw ohv pdwulfhv
Kr '
. rrA
K '
. K KA
K2 '
. K2 KA2
K2 '
. KKA2
ìfuluh ohv ìtxdwlrqv shuphwwdqw oh fdofxo ghv pdwulfhv  hw 2 1
61 Uìvrxguh fhv ìtxdwlrqv hw h{sulphu ã o*dlgh ghv
%2 1
K
hw ghv yhfwhxuv revhuydwlrqv % hw
71 O*dfwlrq gh od pdwulfh 2 shxw íwuh lqwhusuìwìh frpph xqh ìolplqdwlrq gh euxlw1 Srxu
fhfl lqwurgxlvrqv
K '
K2
rü K hvw o*HOPT gh K ã o*dlgh gh K21 H{sulphu hq irqfwlrq gh % hw hq xwlolvdqw
xqltxhphqw ohv pdwulfhv Kr hw ` / rü ` hvw od pdwulfh huuhxu gdqv o*hvwlpdwlrq grqqdqw
K1
81 Grqqhu od vwuxfwxuh gx v|vwëph frpsohw srxu O*HOPT gh r ã o*dlgh gh o*revhuydwlrq
% hw %21 H{soltxhu srxutxrl fh v|vwëph hvw sduirlv dsshoì v|vwëph ã vrxvwudfwlrq gh
euxlw1
91 Ìwxglhu ohv ghx{ fdv rx o*HOPT dssdudlvvdqw hq 71 hvw vrlw vlqjxolëuh vrlw qxooh1
:
:18 WG 8 = Hvwlpdwlrq hw vljqdx{ DU
H{huflfh 4
Rq frqvlgëuh xq preloh vh gìsodêdqw ã ylwhvvh frqvwdqwh 1 Vd srvlwlrq +? / uhohyìh dx{
lqvwdqwv ?A srxu ? 5 ic c j c hvw hqwdfkìh g*xq euxlw K? dgglwli/ eodqf/ fhqwuì hw gh
yduldqfh j 2
Rq gìqlw o*hvwlpdwhxu hpslultxh e
E+ txl fruuhvsrqg ã od pr|hqqh ghv ylwhvvhv lqvwdqwd0
qìhv =
E+ '
e
 [
+?
?' ?A
41 Fhw hvwlpdwhxu hvw0lo eldlvì B
51 Fdofxohu vd yduldqfh . dEe
E+ 2 o1
H{huflfh 5
Vrlw %? xq vljqdo doìdwrluh uìho/ fhqwuì hw dxwruìjuhvvli g*ruguh 6/ grqqì grqf sdu
%? ' @ %?! n @2 %?!2 n @%?! n ?
rü & hvw xq euxlw eodqf fhqwuì gh yduldqfh j2 Rq qrwhud ' . d%? %?! o
Oh exw gh o*h{huflfh hvw gh fdofxohu ohv frh!flhqwv @c @2 hw @ 41 Prqwuhu txh oh yhfwhxu @ '
@ @2 @
-@
j2
rü
5
-'7
51 Rq vxssrvh txh  ' euxlw jìqìudwhxu j2 ' f
A
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'
f  2
 f 
2  f
hw j2 vrqw vroxwlrq ghv ìtxdwlrqv vxlydqwhv
S
f @A S
6
8 hw
5
S'7

2
6
8
hw m2 m f Gìwhuplqhu oh yhfwhxu @ hw od yduldqfh gx
61 Hq gìgxluh wrxwhv ohv ydohxuv gh od irqfwlrq g*dxwrfruuìodwlrq gh %
71 Prqwuhu txh m2 m f
81 Rq vxssrvh pdlqwhqdqw txh f ' 2 '  hw txh o*rq d wrxmrxuv  ' ' f Grqqhu od
vwuxfwxuh gx vljqdo %? hw wurxyhu xq h{hpsoh gh wudmhfwrluh gh fh vljqdo1
;
:19 WG 9 = Ilowudjh gh Ndopdq
Rq gìvluh hvwlphu od srvlwlrq g*xq yìklfxoh vh gìsodêdqw ã ylwhvvh frqvwdqwh1 Fhwwh ylwhvvh
hvw shuwxueìh sdu ghv ylwhvvhv doìdwrluhv1 Rq vxssrvh tx*hqwuh ghx{ lqvwdqwv g*revhuydwlrq/
od ylwhvvh hvw frqvwdqwh1 Rq shxw grqf ìfuluh
E| ' J n ?
srxu |? | |?n
O*ìtxdwlrq g*ìwdw gx v|vwëph grqqdqw od yduldwlrq gh od srvlwlrq v*ìfulw grqf =
%?n ' %? n AJ n A?
O*ìtxdwlrq gh phvxuh v*ìfulw =
+? ' %? n ?
dyhf =
. d? ?A o
. d? ?A o
'
K
'
`
Rq grqqh =
A
' frc ` ' ffc K ' c %J ' fc Pf ' f
Grqqhu ohv ìtxdwlrqv gx owuh gh Ndopdq suìglfwhxu1