Traitement Du Signal
November 17, 2022 | Author: Anonymous | Category: N/A
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^rd`tkakht nu s`ehdc @htronult`oh Ck trd`tkakht nu s`ehdc kst uhk n`sl`pc`hk qu‐kst nkvkhu `hlohtourhdbck nk hos gours, surtout kh tàcàloaauh`ldt`ohs. @c d pour obgkt c'àcdbordt`oh ou c'`htkrpràtdt`oh nks s`ehdux portkurs n'`hioradt`ohs. Ndhs lk lfdp`trk, dvdht d vdht n‐dvdhlkr ndhs ck sugkt proprkakht n`t, hous `htronu`s`ohs tous cks tkraks qu` skroht ut`c`sàs ndhs lk lours. 4 \`ehdc Oh dppkcck s`ehd s`ehdcc toutk erdhnkur pfys`quk tkhsor`kcck qu` vdr`k so`t loht`huakht (s`ehdux dhdcoe`quks) so`t n`slrætkakht (s`ehdux huaàr`quks) du lours nu tkaps kt qu` portk c‐`hioradt`oh ê trd`tkr. trd`tkr. Ou pcus s`apckakht 7 uh s`ehd s`ehdcc kst uhk vdr`dt`oh n‐uhk erdhnkur pfys`quk (ck pcus souvkht àckltr`quk, nàc`vràk pdr uh ldptkur) qu` portk nk c‐`hioradt`oh c‐`hioradt`oh (ndhs (ndhs hotrk lds) nk sd sourlk ê sd nkst`hdt`oh. C'àvocut`oh ndhs ck tkaps nk cd erdhnkur lohs`nàràk kst ràe`k pdr cd nyhda`quk spàl`i`quk nu s`ehdc. Lkttk erdhnkur pfys`quk pk ut çtrk c‐dapc`tunk, iràqukhlk ou pfdsk. Tukcquk io`s cd co` tkaporkcck ràe`ssdht ck pfàhoaæhk kst b`kh lohhuk oh pdrck nk s`ehdux nàtkra`h`stks kt n'dutrk io`s `c kst n`ii`l`ck, vo`rk `aposs`bck nk ck nàlr`rk àvocut`oh n‐uh s`ehdc, ndhs lk lds oh pdrck nk s`ehdux nk s`ehdux dcàdto`rks. Cks s`ehdux qu‐oh vd trd`tkr ndhs lk lours skroht ksskht`kcckakht cks s`ehdux àckltr`quk ê uhk n`akhs`oh lohtrd`rkakht dux `adeks. Nohl, oh pdrckrd`t kh eàhàrdc n‐uhk tkhs`oh àckltr`quk àckltr`quk kt n‐uh lourdht vdr`k so`t loht`hukcckakht ou so`t n`slrætkakht dvkl ck tkaps. 4.4 Aonks nk lcdss`i`ldt`oh nk s`ehdux 4.4.4 Lcdss`i`ldt`oh pfàhoaàhocoe`quk Oh akt d`hs` kh àv`nkhlk ck typk t ypk n‐àvocut`oh nu s`ehdc. @c pkut çtrk ê ldrdltærk prànàtkra`hà prànàtkr a`hà ou `c d uh loaportkakht hoh pràv`s`bck. [h s`ehdc nàtkra`h`stk kst uh s`ehdc noht c‐àvocut`oh kh iohlt`oh nu tkaps pkut çtrk pdrid`tkakht pràn`tk pdr uh aonæck adtfàadt`quk dppropr`à. Dutrkakht s` cks s`ehdux portkurs n‐`hioradt`oh pràskhtkht n‐`hioradt`oh pràskhtkht uhk lkrtd`hk `apràv`s`b`c`tà, `cs skroht aonàc`sàs loaak nks s`ehdux dcàdto`rks. Oh ck nàlr`t erâlk ê nks out`cs stdt`st`quks (nkhs`tà nk probdb`c`tàs, aoykhhk, vdr`dhlk, ktl.). 4.4.2 Lcdss`i`ldt`oh n`akhs`ohhkcck Oh pkut lcdsskr cks s`ehdux kh iohlt`oh nk ckur n`akhs`oh. D`hs`, cd pcupdrt nks s`ehdux quk hous `rohs trd`tkr soht rkpràskhtdbcks adtfàadt`qukakht pdr nks iohlt`ohs x(t) n‐uhk skuck vdr`dbck ràkcck. Oh cks dppkcck nohl s`ehdux aohon`akhs`ohhkcs ou uh`n`akhs`ohhkcs. Cks s`ehdux nks `adeks, du lohtrd`rk, soht b`n`akhs`ohhkc kt rkpràskhtà adtfàadt`qukakht pdr cd iohlt`oh `(x, y). . 4.4.< Lcdss`i`ldt`oh àhkreàt`quk Oh n`st`heuk `l` cks s`ehdux sdt`sid`sdht ê uhk lohn`t`oh n‐àhkre`k i`h`k ê lkux pràskhtdht uhk pu`ssdhlk aoykhhk aoykhhk i`h`k kt uhk àhkre`k `hi`h`k. Oh dppkcckrd àhkre`k n‐uh s`ehdc x(t) cd qudht`tà 7
−|| 3 ∘
kt pu`ssdhlk aoykhhk nk x(t) cd qudht`tà 7
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 4
4 c→` a −|| 3 ∘
Cd prka`ærk ldtàeor`k loaprkhn cks s`ehdux nk typk t ypk trdhs`to`rk qu‐`cs so`kht nàtkra`h`stks ou ou dcàdto`rks (kxkapck uhk `apucs`oh ldrrà ou eduss`khhk) kt cd nkux`æak ldtàeor`k khecobk cks s`ehdux nk typk pkradhkht, pàr`on`quk, nàtkra`h`stk kt cks s`ehdux dcàdto`rks pkradhkhts Rkadrquk7 - [h s`ehdc ê àhkre`k i`h`k d uhk pu`ssdhlk aoykhhk hucck ( Zx 1 :). - [h s`ehdc ê pu`ssdhlk aoykhhk i`h`k (hoh hucck) possænk uhk àhkre`k K x `hi`h`k. Oh nài`h`t, pdr d`cckurs7 - Cd pu`ssdhlk `hstdhtdhàk (n'`htkrdlt`oh).
∕ ∕
−∕ 4 4 − ∕
- Cd pu`ssdhlk aoykhhk (n'`htkrdlt`oh) sur uhk nuràk ^
- C‐ C‐àhkre`k aoykhhk (n'`htkrdlt`oh) sur uhk nuràk ^
4.4.> Lcdss`i`ldt`oh aorpfocoe`quk \kcoh quk ck s`ehdc x(t) ou cd vdr`dbck t kst loht`hu ou n`slrkt ( t h 1 h^ ) oh n`st`heuk qudtrk typks nk s`ehdux 7 Ck s`ehdc ê dapc`tunk kt tkaps loht`hus dppkcà lourdaakht s`ehdc dhdcoe`quk. dhdcoe`quk. Ck s`ehdc ê dapc`tunk n`slrkt kt tkaps loht`hu dppkcà s`ehdc qudht`i`à. qudht`i`à. àlfdht`ccohhà. Ck s`ehdc ê dapc`tunk loht`huk kt tkaps n`slrkt dppkcà s`ehdc àlfdht`ccohhà. huaàr`quk. Ck s`ehdc ê dapc`tunk n`slrkt kt tkaps n`slrkt dppkcà s`ehdc huaàr`quk. 4.4.; Lcdss`i`ldt`oh spkltrdck C'dhdcysk spkltrdck n'uh s`ehdc (ou cd ràpdrt`t`oh ràpdrt` t`oh àhkreàt`quk kh iohlt`oh nk cd iràqukhlk) lohnu`t ê uhk lcdss`i`ldt`oh 7 \`ehdux nk bdssks iràqukhlks. \`ehdux nk fdutks iràqukhlks. \`ehdux ê bdhnk àtro`tk. \`ehdux ê cdrek bdhnk. Cd cdrekur nk bdhnk B n'uh s`ehdc kst ck noad`hk pr`hl`pdck nks iràqukhlks ollupà pdr soh spkltrk. 4.2 \`ehdc nàtkra`h`stk kt dcàdto`rk @he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 2
[h s`ehdc nk cd lcdssk pfàhoaàhocoeuk kst eàhàrdckakht lcdssà lcdss à ndhs nkux erdhns eroupks pr`hl`pdcks ê sdvo`r 7 s`ehdc nàtkra`h`stk ou lkrtd`hs kt s`ehdc dcàdto`rk ou stolfdst`quk. 4.2.4 \`ehdc nàtkra`h`stk Ck s`ehdc nàtkra`h`stk kst uhk iohlt`oh qu` àvocuk pdr rdpport ê tkaps. Lkttk iohlt`oh pkut çtrk idl`ckakht aonàc`sàk pdr uhk àqudt`oh adtfàadt`quk pdrlk qu‐kcck kst pràv`s`bck. Nohl, uhk iohlt`oh nk cd vdr`dbck nu tkaps t pkrakt pkrakt nk lohhdètrk sd vdckur n dvkl lkrt`tunk qukc quk so`t ck tkaps t nohhà. nohhà. Ck s`ehdc nàtkra`h`stk kst duss` n`v`sà kh nkux erdhns eroupks 7 s`ehdc pàr`on`quk kt hoh pàr`on`quk. 4.2.4.4 \`ehdux pàr`on`quks [h s`ehdc x(t) pàr`on`quk kst uh s`ehdc noht cd vdckur `hstdhtdhàk sk ràpætk nk idåoh `nkht`quk ê kcck-açak, ê `htkrvdccks nk tkaps ràeuc`krs. Nohl, cd vdckur sk ràpætk dpræs uhk pàr`onk ^.
Ndhs lk eroupk oh pkut duss` sous n`v`sà kh tro`s su`vdhtks 7 \`ehdux s`husoïndux \`ehdux pàr`on`quks loapos`tks kt \`ehdux pskuno “ pskuno “ dcàdto`rks dcàdto`rks 4.2.4.4.4 \`ehdux s`husoïndux [h s`ehdc s`husoïndc x(t) kst uh s`ehdc noht c‐dapc`tunk, obskrvàk ê uh khnro`t pràl`s, kst uhk iohlt`oh s`husoïndck nu tkaps.(4) [hk erdhnkur s`husoïndck x(t) kst rkpràskhtà pdr c'kxprkss`oh 7
dvkl - - - -
Dadx 1 dapc`tunk (qu` vdr`k nk +D adx ê “ ê “ Dadx) t 1 pàr`onk kh sklohnk rkpràskhtdht ck tkaps = w 1 cd pucsdt`oh kh rdn`dh pkr sklohnk kt 1 ck nàpfdsdek kh rdn`dh
4.2.4.4.2 \`ehdux pàr`on`quks loapos`tks Oh dppkcck s`ehdc loapos`tk, ck s`ehdc ràsuctdht nk cd soaak n'du ao`hs nkux loaposdhtks àcàakhtd`rks nk hdturk n`iiàrkhtk. Oh nàs`ehk pdr loaposdhtk àcàakhtd`rk, uh s`ehdc s`apck noht oh lohhdèt c'kxprkss`oh adtfàadt`quk, loaak pdr kxkapck ck s`ehdc loht`hu kt ck s`ehdc s`husoïndc. 4.2.4.< \`ehdux pskuno “ dcàdto`rks dcàdto`rks Cks s`ehdux pskuno “ pskuno “ dcàdto`rks dcàdto`rks iorakht uhk ldtàeor`k pdrt`luc`ærk nk s`ehdux pàr`on`quks noht ck loaportkakht rdppkcck lkcu` n‐uh s`ehdc dcàdto`rk. dcàdto`rk. 4.2.4.2 \`ehdux hoh pàr`on`quks [h s`ehdc x(t) kst n`t hoh pàr`on`quk corsquk cks vdr`dt`ohs nk sd vdckur (kh dapc`tunk) ê uhk pàr`onk kst pràv`s`bck ndhs ck tkaps. @l` sd `hlcus cks s`ehdux quds` “ quds` “ pàr`on`quks kt cks s`ehdux ^ trdhs`to`rks. @he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 <
4.2.4.2 4 \`ehdux quds` - pàr`on`quks [h s`ehdc x(t) kst n`t quds`- pàr`on`quk pàr`on`quk s‐`c kst uhk soaak nk s`husoïnks nk pàr`onks `hi`h`ks. `hi`h`ks. 4.2.4.2.2 \`ehdux trdhs`to`rks [h s`ehdc trdhs`to`rk kst uh s`ehdc vdr`dbck nk nuràk nài`h`k, qu` s‐àtdbc`t eàhàrdckakht khtrk nkux àtdts pkradhkhts 4.2.< Ldusdc “ Hoh Hoh ldusdc Khi`h, uh s`ehdc kst n`t ldusdc corsqu‐`c kst `nkht`qukakht huc pour tous cks `hstdhts cks `hstdhts hàedt`is. [h tkc s`ehdc ràsuctk souvkht n‐uhk dlt`oh pdrt`luc`ærk ê c‐`hstdht t 1:. 1:. @c kst n`t dht`ldusdc s‐`c kst huc pour tous cks `hstdhts pos`t`is. Ndhs ck lds eàhàrdc, `c kst hoh-ldusdc 4.2.2 \`ehdux dcàdto`rks Ck s`ehdc dcàdto`rk d uh loaportkakht `apràv`s`bck kt noht oh hk pkut pds pràn`rk cd vdckur ê uh tkaps t ldr `c nàpkhn nu fdsdrn. Oh id`rk eàhàrdckakht dppkc ê ckurs propr`àtàs stdt`st`quks pour cks nàlr`rk. \` ckurs propr`àtàs stdt`st`quks soht `hv `hvdr`dhtks dr`dhtks ndhs ck tkaps, oh n`t qu'`cs soht stdt`ohhd`rks. Loroccd`rkakht, s‐`cs h‐oht pds lkttk propr`àtà, oh pdrck nk hoh-stdt`ohhd`rk. hoh -stdt`ohhd`rk. Cks s`ehdux dcàdto`rks stdt`ohhd`rks soht sous n`v`sà kh nkux 7 kreon`quks kt hoh kreon`quk. Ndhs lk lours, hous `htàrksskrohs uh`qukakht dux s`ehdux nàtkra`h`stks Ck soaad`rk nks pdrderdpfks l`-nkssus pkut çtrk aohtrà ndhs cd i`eurk l`-nkssous.
2 ^rd`tkakht nu s`ehdc s`ehdc Ck trd`tkakht nu s`ehdc kst s`ehdc kst uhk n`sl`pc`hk qu` nàvkcoppk kt àtun`k cks tklfh`quks nk trd`tkakht, n'dhdcysk kt n'`htkrpràtdt`oh nks s`ehdux (4) 7 ^outks dppc`ldt`ohs nk aksurks, nk trd`tkakht n‐`hioradt`oh akttkht kh Ŝuvrk nks tklfh`quks nk nk trd`tkakht sur ck s`ehdc pour kxtrd`rk uh adx`aua n'`hioradt`ohs ut`cks nàs`ràk. 2.4. Zr`hl`pdcks iohlt`ohs nu trd`tkakht nk s`ehdc Cks pr`hl`pdcks iohlt`ohs nu trd`tkakht nk s`ehdc soht 7 - C‐dhdcysk C‐dhdcysk - Cd aksurk - Ck i`ctrdek - Cd ràeàhàrdt`oh - Cd nàtklt`oh - C‐`nkht`i`ldt`oh C‐`nkht`i`ldt`oh @he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 >
- Cd syhtfæsk - Ck londek - Cd aonucdt`oh kt ck lfdhekakht nk iràqukhlk 2.4.4 C‐dhdcysk 7 Oh lfkrlfk ê `sockr cks loaposdhtks ksskht`kccks n'uh s`ehdc nk iorak loapckxk, di`h n'kh a`kux loaprkhnrk cd hdturk kt or`e`hks. 2.4.2 Cd aksurk 7 aksurkr uh s`ehdc, kh pdrt`luc`kr dcàdto`rk, l'kst kssdykr n'kst`akr cd vdckur n'uhk erdhnkur ldrdltàr`st`quk qu` cu` kst dssol`àk dvkl uh lkrtd`h nkerà nk lohi`dhlk. 2.4.< Ck i`ctrdek 7 l'kst uhk iohlt`oh qu` lohs`stk ê àc`a`hkr n'uh s`ehdc lkrtd`h loaposdht `hnàs`rdbck. 2.4.> Cd ràeàhàrdt`oh 7 l'kst uhk opàrdt`oh pdr cdqukcck oh tkhtk nk rknohhkr sd iorak `h`t`dck 2.4.> ê uh s`ehdc dydht sub`s n`vkrsks n`stors`ohs. Lfdp`trk 2.4.; Cd nàtklt`oh 7 pdr lkttk opàrdt`oh oh tkhtk n'kxtrd`rk uh s`ehdc ut`ck nu bru`t nk iohn qu` cu` kst supkrposà. 2.4.9 C‐`nkht`i`ldt`oh 7 l'kst uh prolànà souvkht loapcàakhtd`rk qu` pkrakt n'kiikltukr uh lcdsskakht nu s`ehdc obskrvà. 2.4.5 Cd syhtfæsk 7 opàrdt`oh `hvkrsk nk c'dhdcysk, lohs`stk ê lràkr uh s`ehdc nk iorak dppropr`àk kh prolàndht, pdr kxkapck, ê uhk loab`hd`soh nk s`ehdux àcàakhtd`rks. 2.4.6 Ck londek 7 outrk sd iohlt`oh nk trdnult`oh kh cdhedek huaàr`quk, kst ut`c`sà so`t pour cuttkr lohtrk ck bru`t nk iohn, so`t pour tkhtkr nk ràdc`skr nks àlohoa`ks nk cdrekur nk bdhnk ou nk aàao`rk n'orn`hdtkur. 2.4.0 Cd aonucdt`oh kt ck lfdhekakht nk iràqukhlk 7 soht ksskht`kcckakht nks aoykhs pkrakttdht n'dndptkr uh s`ehdc dux ldrdltàr`st`quks iràqukht`kccks n'uhk vo`k nk trdhsa`ss`oh, n'uh i`ctrk n'dhdcysk ou n'uh rdpport n'khrke`strkakht. Cks pr`hl`pdcks iohlt`ohs nu trd`tkakht nk s`ehdc akht`ohhàs l`-nkssus soht dppc`ldbcks ê nks s`ehdux dhdcoeuks d`hs` quk huaàr`quks.
Cks s`ehdux ê trd`tkr pkuvkht çtrk nk sourlks n`vkrsks, ad`s pour hotrk lds, hous dccohs hous lohlkhtrkr sur cks s`ehdux àckltr`quks ou nkvkhus àckltr`quks ê c‐d`nk nk ldptkurs kt trdhsnultkurs. Oh n`st`heuk ksskht`kcckakht cks s`ehdux dhdcoe`quks, dhdcoe`quk-huaàr`quk kt huaàr`quks. @c idut rdppkckr quk ndhs lk lours oh vd lohlkhtrkr r`kh quk sur cks s`ehdux nàtkra`h`stks. @c kx`stk pcus`kurs brdhlfks pdrt`lucd`rks nu trd`tkakht nu s`ehdc, kh iohlt`oh nk cd hdturk nks s`ehdux lohs`nàràs. Ad`s cd i`eurk l`-nkssous hous nohhk dppropr`à cd eàhàrdt`oh kt ck trd`tkakht nu s`ehdc. Nàioradt`oh
\ourlk Kakttr`lk
A`c`ku nk trdhsa`ss`oh
Ldptkur
[h`tà nk trd`tkakht
I`eurk 4 \lfàad n'uh systæak nk eàhàrdt`oh kt nk trd`tkakht trd`t kakht nu s`ehdc @he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 ;
< Zr`hl`pdcks bdsks tfàor`quks Nkux tfàor`ks soht iohndakhtdcks pour cd ioraucdt`oh nks probcæaks nk trd`tkakht nu s`ehdc-7 Cd prka`ærk tfàor`k kst lkcck nks systæaks c`hd`rks `hvdr`dhts ndhs ck tkaps. Lkcck tfàor`k pkrakt nk pràvo`r cd ràpohsk r àpohsk n'uh systæak kh uh khtràk nohhkr. Kcck kst dppc`quàk dux s`ehdux nàtkra`h`stks ndhs uh systæak. Cd sklohnk tfàor`k kst c`k dux s`ehdux ê ldrdltærk dcàdto`rk. Lkttk cd tfàor`k nks probdb`c`tàs qu‐kst eàhàrdckakht ut`c`sàk. ut`c`sàk. Kcck pkrakt nk rkpràskhtkr lorrkltkakht kt n'kxtrd`rk du a`kux cks `hioradt`ohs iourh`ks pdr uh pfàhoaæhk dcàdto`rk. Lk typks nks s`ehdux skroht trd`tà ndhs cd toak nkux nu trd`tkakht nu s`ehdc. Lks nkux tfàor`ks ut`c`skht cks açaks opàrdtkurs adtfàadt`quks. Oh vd khtrkr kh nàtd`c ndhs ck lfdp`trk nks i`ctrks. .4 @apucs`oh nk N`rdl [hk `apucs`oh nk N`rdl kst uhk iohlt`oh dvkl uhk nuràk træs lourtk ou uhk iohlt`oh portk noht cd cdrekur ^ tkhnrd`t vkrs :, nohl c‐dapc`tunk kst suii`sdaakht erdhnk pour quk c‐kiikt so`t skhs`bck. skh s`bck. C‐d`rk nk cd iohlt`oh kst àedck à edck ê 4 4 7 ndhs lk lds oh pdrck n‐`apucs`oh uh`td`rk. uh`td`rk.
, : :, ≪ : ∘
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 9
t 4
t
:
Rkadrquk 7 cks 4 adrquks sur cd icælfk pck`hk rkpràskhtk cd suridlk nk lkttk `apucs`oh kt hoh pdr c‐dapc`tunk. c‐dapc`tunk. >.4.4 Zropr`àtàs nk c‐`apucs`oh nk N`rdl lohvocut`oh lfdhekakht nk vdr`dbck
∕ ∕ ∕ 4||
`htàerdt`oh
∘
− 4
: − − 4 , 3 3 ∘
∘
pronu`t dvkl uh s`ehdc
t xt xt xtttt xt: tx: →− ξ
ξ ξ
ξ
>.4.2 Zk`ehk nk N`rdl [hk sullkss`oh pàr`on`quk sullkss`oh pàr`on`quk ^ nk c‐`apucs`oh nk N`rdl kst dppkcàk Zk`ehk nk N`rdl. N`rdl. ∘
∘
t -h^
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@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 5
Cd iohlt`oh l`-nkssus l`-nkssus kst pdrio`s dppkcàk trd`h n‐`apucs`oh. Kcck træs ut`c`sàk ndhs c‐àlfdht`ccohhdek. Oh vd khlork pdrckr nk lk trd`h n‐`apucs`oh ndhs ck trd`tkakht nu s`ehdc huaàr`quk. >.2 Iohlt`oh s`ehk
4, , 83 : \eh(t) 4
t
-
Ndhs bkduloup nks opàrdt`ohs, oh dnakt quk cd vdckur ê c‐ or`e`hk nk seh(t) 1 : pour t1:. t1:. uh`td`rk >..> Iohlt`oh Rdapk
−
r(t)
4 t 4
4,, 3 4 42 42 2 2 :,, 8 24
>.; Iohlt`oh Rkltdheucd`rk ou portk
dvkl ^ ck cdrekur nu rkltdheck.
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 6
rkl(t/^) 4
t
^/2
-^/2
Lkttk iohlt`oh kst duss` dppkcàk iohlt`oh portk kt skrt bkduloup ndhs ck ikhçtrdek àcàakhtd`rk àcàakhtd`rk kh trd`tkakht nu s`ehdc.
s`h
>.9 Iohlt`oh s`hus ldrn`hdc
ϊ
s`hl( t) 4
t -< -2
-4
:
4
2
<
Lkttk iohlt`oh gouk uh rþck `aportdht kh trd`tkakht nu s`ehdc. Ndhs lk c`vrk, oh vd pdrckr bkduloup sur lkttk iohlt`oh. Lkttk iohlt`oh d cks propr`àtàs su`vdhtks 7
− 4 ∘
∘
− 4
;. ;. Kxtrd nu rdppkc adtfàadt`quks ;.4 Hoabrk loapckxk
;.4.4 Nài`h`t`oh 7 [h hoabrk loapckxk s'àlr`t sous iorak dceàbr`quk 7 z 1 d + ` b oð d kt b soht nkux ràkcs kt ` ck hoabrk `ade`hd`rk tkc quk ` 2 1 -4 -4.. Ndhs n‐dutrks c`vrks oh ut`c`sk ck syabock g du c`ku nk ` . Nohl, `l` oh vd ut`c`skr g pour pour tous ck lours. C'khskabck nks hoabrks loapckxks kst hotà " L " " d " s'dppkcck cd pdrt`k ràkcck nk z kt sk hotk Rk (z) " b " s'dppkcck cd pdrt`k `ade`hd`rk nk z kt sk hotk @a (z) ;.4.2 Ck lohgueuà n'uh loapckxk \o`t z uh uh hoabrk loapckxk @he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 0
⟰̋ ∕ ⟰ ||
dvkl d kt b ràkcs. Dcors ck lohgueuà nk z kst cd su`vdhtk 7
Ck aonuck n‐uh loapckxk kst 7
;.4.2 \oaak nks hoabrks loapckxks \o`t nkux hoabrks loapckxks l`-nkssous, cd soaak kst 7
nohl cd soaak kst
;.4.< Zronu`t nks hoabrks loapckxks \o`t nkux hoabrks loapckxks l`-nkssous, cd soaak kst 7
Õ |Õ∐|̋|̋Õ|̋Õ|| ∕ ̋ | | Õ̋ Õ ∕ 4 ̋ |4|4 | | | | Õ dre dre dre dre dre dre dre dre
nohl ck pronu`t kst
;.4.> Tukcquks ioraucks ut`cks nks hoabrks loapckxks
;.2 C`kh nk g dvkl dvkl ck lkrlck tr`eohoaàtr`k Cks po`hts nu lkrlck tr`eohoaàtr`quk pkuvkht çtrk rkpàràs dvkl ckur dbsl`ssk kt ckur ornohhàk, ad`s oh vd pcus s`apckakht cks ckur rkpàrkr dvkl ckur aonuck kt ckur dreuakht. Oh dppkcck lkrlck tr`eohoaàtr`k, uh lkrlck or`khtà nk rdyoh 4. Zdr lohvkht`oh, oh lfo`s`t cd n`rklt`oh `hvkrsk nks d`eu`ccks n'uhk aohtrk loaak skhs n`rklt ou skhs pos`t`i. Ck lkrlck tr`eohoaàtr`quk oh ck aonuck nks po`hts nu lkrlck kst 4. @cs soht nohl nk cd iorak k g gζζ. Lkrtd`hs po`hts pdrt`luc`krs soht hotdaakht ê lohhdètrk 7
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 4:
;.2.4 Ioraucks pour cd soaak nk 2 dhecks 7
4 4 2 442 2222 2s` h los 2 22 4 22222 2 2 2 22 = 2 2 2 4 4 4,
;.2.2 Ioraucks nk nupc`ldt`oh Ndhs cks ioraucks nk pdrderdpfk l`-nkssus, s` d 1b, nohl oh durd`t 7
;.2.< \oaak nk 2 iohlt`ohs tr`eohoaàtr`quks
;.2.> Lfdhekakht nk vdr`dbck t 1 tdh (d/2) 7 Oh sd`t quk ndhs cd tr`eohoaàtr`k, pour d ≪ ϊ/2, oh d 7
kh n`v`sdht tous pdr
, oh d 7
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 44
`c v`kht dcors,
2 22 22 22 4 22 2 4 4 4 22(,(2)2) ≪ 4,2 4 4 2 2 4(2) 4 4 4 4 2((2) ) 4 2 4 2
\` hous posohs
oh obt`kht pour cks ioraucks l`-nkssus cks su`vdhtks 7
;.2.; Iorauck nk Ao`vrk Zour tout ràkc kt pour tout kht`kr rkcdt`i h 7
()
() ( )
;.2.; Ioraucks n‐Kuckr Cks ioraucks n'Kuckr rkc`kht cks iohlt`ohs tr`eohoaàtr`quks ê c'kxpohkht`kcck loapckxk. Zour tout ràkc , oh d 7
− − 2 2 − −− 2 ( )2 )2
Tukcquks lohsàqukhlks nu tfàoræak n‐Kuckr kst kst cks su`vdhtks 7 Ndhs cd iorauck nk Ao`vrk, oh d 7
;.< Rdppkc sur cks ldclucs n`iiàrkht`kcs
\o`t x(t) 1t.9.4 4kr lohstdtk pour h≫: oh d d
: − : −
pour h 8 : oh d
;.>.9.2 2æak lohstdtk pour a kt h hous dvohs dvohs
:
−− : − : − − )( ) 42 − (( )(
;.>.9.< .9.> >æak lohstdtk Zour a 1 h, kt a, h ≫4, oh d d
2 − 2 − 42 ( )( ) 42 4 422 4 4 4 2 2 2 − 4 4 − −4 2 −− > 22−− 2 > : 2
Zrkuvk pour a1h
Noht
Kh ut`c`sdht cks ioraucd`rks nkssus, hous pouvohs idl`ckakht nànu`rk cks su`vdhtks 7 \o`t ck tfàoræak nk Iour`kr su`vdht 7
Oh d 7
2 1 2 − 2 − , , 4 ≮ { 2 − ,, 4 ≮
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 46
@c idut hotkr quk ck `nkht`quks. `. k.
kst uhk lohvkht`oh lohvkht`oh dnoptà pour quk cks ioraucd`rks nks
4 − 4 − , , , ,44≮≮ {
,
kt
so`t
−
;.>.5 Rdppkcs sur cks su`tks dr`tfaàt`quks kt eàoaàtr`quks @l` hous dccohs rdppkckr cks n`iiàrkhts ràsuctdts sur cks su`tks nk hoabrks ràkcs qu` soht nks su`tks dr`tfaàt`quks ou nks su`tks eàoaàtr`quks [hk su`tk su`tk kst kst uhk « sullkss`oh ½ nk hoabrks su`tk.. [hk su`tk no`t dvo`r nk hoabrk ràkcs . ràkcs. Lks hoabrks ràkcs soht cks tkraks nk cd su`tk ;.>.5.4 \u`tks Dr`tfaàt`quks \o`t [h uhk su`tk, kcck kst uhk su`tk dr`tfaàt`quk s` lfdquk tkrak s‐obt`kht kh dgoutdht du
Õ +, , ∆ , ∆ + ≫ . ⟰ + ≫ :, + ≫ . ⟰ + ≮:, à
pràlànkht uh hoabrk ràkc lohstdht r dppkcà dppkcà rd`soh.
Oh n`rk qu'uhk su`tk [h kst lro`ssdhtk s` pour tout kht`kr h,
Loroccd`rkakht oh n`rk qu'uhk su`tk ([h) kst nàlro`ssdhtk s` pour tout kht`kr h, [ h+4 ≮ [h.
8, ,, 3, à ⋯ − 2 2 −, 22 4, 2 ⋯−,
Oh pkut pkut idl` idl`cka ckak kht ttrr ouvk ouvkr quk s` cd rrd`soh d`soh r 7
;.>.5.4.4 Cd soaak nks tkraks lohsàlut`is n‐uhk su`tk Dr`tfaàt`quk \o`t uhk su`tk dr`tfaàt`quk [h, cd soaak nks tkraks lohsàlut`is cd su`tk kst 7
Dvkl cà h`æak tkrak 1 (h-4) Ou c‐àqudt`oh dctkrhdt`vk kst 7 kst 7
;.>.5.4.2 Cks hoabrks nks tkraks n‐uhk su`tk Dr`tfaàt`quk
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 40
Dvkl ck h
`æak
tkrak àedck ê 7
−
;.>.5.2 \u`tks eàoaàtr`quks \o`t [h uhk su`tk, su`tk, kcck kst uhk su`tk eàoaàtr`quk s` lfdquk tkrak s‐obt`kht kh auct`pc`dht du pràlànkht uh hoabrk ràkc lohstdht q dppkcà rd`soh.
+ Õ , ∆
\o`t q uh hoabrk ràkc nohhà. Dcors ck skhs nk vdr`dt`oh nk cd su`tk eàoaàtr`quk (qh) nk rd`soh q kt nk prka`kr tkrak 4 kst nohhà pdr 7 Cd su`tk ( qh) kst lohstdhtk s` kt skuckakht s`7 q1 4. Cd su`tk (qh) kst lro`ssdhtk s` kt skuckakht s`7 q 8 4. Cd su`tk (qh) kst nàlro`ssdhtk s` kt skuckakht s`7 : 3 q 3 4. Cd su`tk (qh) h'kst h` lro`ssdhtk, h` nàlro`ssdhtk s` kt skuckakht s`7 q 3 :. ;.>.5.2 .4 cd soaak nks tkraks lohsàlut`is n‐uhk su`tk eàoaàtr`quk \o`t uhk su`tk eàoaàtr`quk eàoaàtr`quk [h, cd soaak nks tkraks lohsàlut`is cd su`tk kst 7
⋯ − 4 44 4
;.>.5.2 .2 cd lohvkrekhlk ou cd soaak n‐uhk su`tk eàoaàtr`quk d `hi`h`k (h 1 ∘) \o`t q uh hoabrk ràkc nohhà. 4º) \` q 8 4, dcors auct`pc`kr pdr uh hoabrk supàr`kur ê 4 lorrkspohn ê uh derdhn`sskakht. Nohl, cks tkraks nk cd su`tk eàoaàtr`quk (q h) dueakhtkht `hnài`h`akht corsquk h tkhn vkrs +∘ +∘ kt nàpdsskht tout hoabrk lfo`s` du nàpdrt ê pdrt`r pdrt` r n'uh lkrtd`h rdhe. Oh àlr`t7
c→`a ∘
Oh n`t quk « cd c`a`tk c` a`tk n nk k qh cor corsquk squk h tkhn tkhn vk vkrr s +∘, kst àedck ê +∘ ½. 2º) \` : 3 q 3 4 , dcors auct`pc`kr pdr uh hoabrk loapr`s loapr`s khtrk : kt 4 lorrkspohn ê uhk rànult`oh. Nohl, cks tkraks nk cd su`tk eàoaàtr`quk (q ( qh) n`a`hukht `hnài`h`akht corsquk h tkhn vkrs +∘ +∘.. Oh n`t quk « cd c`a`tk nk qh corsquk h tkhn vkrs +∘, ∘, kst àedck ê : ½. : ½. Oh àlr`t 7
c`a : → 4
Ndhs lk lds cd soaak nk cd c d su`tk \ qudhn h tkhn vkrs `hi`h`k kst 7
\o`t ([ ([ h) uhk su`tk nk hoabrks ràkcs. \` ([ ([ h) tkhn vkrs uhk c`a`tk i`h`k, i`h`k, oh n`t qu'kcck kst lohvkrekhtk. \` ([ ([ h) tkhn vkrs c'`hi`h` ou h'dnakt pds nk c`a`tk, c`a`tk, oh n`t qu'kcck kst n`vkrekhtk.
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 2:
Kxkrl`lks 4. Nàtkra`hkr s` cks s`ehdux l`-nkssous soht cks s`ehdux so`t ê àhkre`ks i`h`s, so`t ê pu`ssdhlk i`h`s ou so`t hoh. d) b) l) \ocut`oh d) Loaak ck s`ehdc h‐kst pds pàr`on`quk, oh vd lfkrlfkr s` ck s`ehdc kst ê àhkre`ks i`h`s
−, 8 || 3∘|−| − − − − − 4 − 4 − 4 2 2 2 3 ∘
Loaak ck s`ehdc kst ldusdc cd iorauck l`-nkssus nkv`khhk
Nohl ck s`ehdc x(t) kst uh s`ehdc ê àhkre`k i`h`s. b) Ck s`ehdc x(t) kst pàr`on`quk nohl hous pouvohs trouvkr cd pu`ssdhlk aoykhhk cd pàr`onk7
−||2 →3 ∘ 2 c→` a∭2 c→` a 4 −|| → 4 −| | 4 2 2 >s`h22> 2 3 ∘ 44los >22: los : c→` a −|| c→` a −| | c→` a c→` a >< >0 >25 ⋯ >520 > > > > ⋯ > > 4 4 4 4 ⋯ 4 \>444< 45204>
4 xt −Vw 2ϊ − Vwkg nw
Oh vd pdrckr sur cd iorauck l`-nkssus kh pku pcus tdrn ndhs ck lfdp`trk. Lohn`t`oh n‐kx`stkhlk nk cd ^rdhsioraàk nk Iour`kr Nài`h`t`oh 7 \o`t 7 \o`t uh s`ehdc dvkl cd iohlt`oh x(t) dnakt pour cd trdhsioraàk nk Iour`kr 7 • s` cd iohlt`oh x(t) kst borhàk `.k. possænk uh hoabrk i`h` nk n`sloht`hu`tàs sur tout `htkrvdcck i`h`. • Cks n`sloht`hu`tàs nk cd iohlt`oh x(t) soht kh hoabrk i`h`. ∘ l‐kst “ ê-n`rk ê-n`rk V(w) kx`stk. • \` cd iohlt`oh x(t) kst `htàerdbck 1 o \` x(t) kst `htàerdbck, dcors 7
∭−|| 3 ∘ − − 3 −|| 3 ∘ ∘
∘
•
Loroccd`rkakht 7
∘
−|| 3 ∘
@c idut hotkr quk ck s`ehdc nkssus kst ê àhkre`k i`h`k, nohl cks s`ehdux n‐àhkre`k i`h`k oht cks trdhsioraàks nk Iour`kr. Zropr`àtàs nk cd ^rdhsioraàk nk Iour`kr \o`t x(t) kt y(t) nkux s`ehdux nk trdhsioraàk nk Iour`kr rkspklt`vks V(i) kt P(i) kt δ kt β nkux lohstdhtks. Cd trdhsioraàk nk Iour`kr nk δx(t)+ βy(t) β y(t) kst nohhàk pdr c‐àqudt`oh 77
δ kt β cks lohstdhts `.
⟷ =⟷
C`hàdr`tà 7 Auct`pc`kr uh s`ehdc pdr uh ed`h δ ndhs ck noad`hk tkaporkc rkv`kht ê auct`pc`kr sd trdhsioraàk nk Iour`kr pdr uh ed`h δ ndhs ck noad`hk iràqukht`kc. Dnn`t`ohhkr nkux s`ehdux ndhs ck noad`hk tkaporkc rkv`kht ê dnn`t`ohhkr ckurs
⟷ ⟷ 2− 6
xt −Vw 2ϊ4 − Vwkg nw g 4 − x − Vwknt 22 (22 ) 2
v``.
N`iiàrkht`dt`oh
()⟷2 )⟷2 − −. ′
• pdr rdpport du tkaps-
Zrkuvk 7 so`t ck s`ehdc x(t) d sd trdhsioraàk nk Iour`kr qu` kst nohhàk pdr 7 ∘
− gi − à à () ⟰ − VwkginwVwk nw − − Vw kgi nw V− wkgi nw 2
Nohl
)⟷2 ()⟷2 () ⟷ 2 2 ⟷ ( ) − −
Rkadrquk. Zcus eàhàrdckakht, kt sous ràskrvk n‐kx`stkhlk n‐kx`stkhlk nk cd nàr`vàk lohs`nàràk kt nk sd trdhsioraàk nk Iour`kr 7
′
• pdr rdpport ê cd iràqukhlk-7
Zrkuvk 7 so`t cd trdhsioraàk nk Iour`kr nu s`ehdc x(t) kst V(w) qu` kst nohhà pdr cd io iorauck rauck su`vdhtk 7 ∘
∘
− − ( ( )
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 >0
( ) 2 ) − (−) 2− − 2 2 2 ⟷ ( ) 22 ⟷ (( ) − ⟷ 4 : ( )2 ) )⟷2 2 ⟷2 ( () ()2 ) 2− − 2 − 2 ∘
∘
Loroccd`rkakht
7
v```.
@htàerdt`oh
Zrkuvk 7 7 cd propr`àtà nk c‐`htàerdt`oh nk cd trdhsioraàk nk Iour`kr kst træs `htr`quàk pour y drr`vàk. Hous dv`ohs vuk vuk pour cd propr`àtà propr`àtà nk cd n`iiàrkht`dt`oh loaak su`t 7
Hous pouvohs rààlr`rk cd iorauck nkssus loaak su`t 7
Zrkhohs
, dcors cd iorauck nkv`kht 7
Nohl 7
@c idut hotkr quk c‐àqudt`oh nkssus kst vrd` pdrlk q quk uk hous dvohs dvohs pr`à cd propr`àtà nk cd trdhsioraàk nk Iour`kr nk cd n`iiàrkht`dt`oh n‐uh s`ehdc y(t). Noht, hous dvohs oubc`à cd lohstdhtk qu` pkut çtrk uhk pdrt`k nk cd iohlt`oh. Hous sdvohs duss` quk cd n`iiàrkht`dt`oh nk x(t) kst àedck ê lkcck nk y(t) + d. Ndhs lk lds, c‐àqudt`oh nkssus quk hous dvohs prouvà kst lorrkltk s` kt skuckakht s` 7
, 2 − −
\` hous dvohs uhk lohstdhtk dcors cd trdhsioraàk nk Iour`kr nkv`kht loaak su`t 7
− 2 2 , −
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 ;:
− −
\kcoh cd propr`àtà nk cd trdhsioraàk nk Iour`kr n‐uhk lohstdhtk, oh dd 7 7
dvkl d 1 V(:) Nohl
`x.
− ⟷ 4 :
^rdhsioraàk nk Iour`kr nk qukcquks s`ehdux usukcs d. @apucs`oh nk N`rdl Loaak nài`h` ndhs ck lfdp`trk (), uhk `apucs`oh nk N`rdl kst uhk iohlt`oh dvkl uhk nuràk træs lourtk ou uhk iohlt`oh portk noht cd cdrekur ^ tkhnrd`t vkrs :, nohl c‐dapc`tunk kst suii`sdaakht erdhnk pour erdhnk pour quk c‐kiikt c‐ kiikt so`t skhs`bck. C‐d`rk nk cd iohlt`oh kst àedck ê 4 7 ndhs lk lds oh pdrck n‐`apucs`oh uh`td`rk. uh`td`rk.
, : :, ≪ : ∘
t
4
t
:
Rkadrquk 7 cks 4 adrquks sur cd icælfk pck`hk rkpràskhtk cd suridlk nk lkttk `apucs`oh kt hoh pdr c‐dapc`tunk. c‐dapc`tunk. Lkttk `apucs`oh h‐kst pds kxdltkakht uh s`ehdc pkr sk, ad`s kcck kst lohs`nàrà loaak tkcck ê ldusk n‐kiikt sur uh systæak. systæak. Cd trdhsioraàk nk Iour`kr hk s‐dppc`q s‐dppc`quk uk str`ltkakht qu‐dux qu‐dux s`ehdux s`ehdux qu` vàr`i`kht cks lohn`t`ohs nk N`r`lfckt. @c skrd`t deràdbck n‐àtkhnrk n‐àtkhnrk ck ck ioradc`sak di`h nk pouvo`r nài`h`r uhk trdhsioraàk nk Iour`kr pour cks s`ehdux nk pu`ssdhlk aoykhhk i`h`k, kt nk rktrouvkr cd sàr`k nk Iour`kr loaak lds pdrt`luc`kr nk cd trdhsioraàk nk Iour`kr. Lkttk kxtkhs`oh kst poss`bck kh ut`c`sdht cd tfàor`k nks n`str`but`ohs, kt kh pdrt`luc`kr cd n`str`but`oh nk N`rdl. Cd n`str`but`oh nk N`rdl kst uhk n`str`but`oh, kt hous nkvr`ohs id`rk dcors dppkc dux ràsuctdts nk cd tfàor`k nks n`str`but`ohs. Lkl` sort cdrekakht nu ldnrk nk lk lours, kt hous hous lohtkhtkrohs `l` n‐uhk n‐uhk dpprolfk dpprolfk fkur`st`quk.
∘:,, ≪ ::
Oh dppkcck `apucs`oh nk N`rdl cd iohlt`oh ∂(t) ∂(t)
kt tkcck quk
−+ 4
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 ;4
Kh ràdc`tà c‐`apucs`oh nk N`rdl gouk ck rþck n‐uhk iohlt`oh `hn`ldtr`lk corsqu‐kcck corsq u‐kcck `htkrv`kht ndhs uhk `htàerdt`oh. Kh kiikt, c‐`apucs`oh nk N`rdl kst hucck sdui corsquk soh dreuakht kst huc, duqukc lds, soh dapc`tunk kst `hi`h`k, ad`s soh « d`rk ½ uh`tà.
Zropr`àtàs nk c‐`apucs`oh nk N`rdl lohvocut`oh
∕∕
4||
lfdhekakht nk vdr`dbck
`htàerdt`oh
4 − − : − 4 , 3 3 x t t x : t x: xt t t xt t t xt ∘
∘
∘
pronu`t dvkl uh s`ehdc
ξ
ξ
ξ ξ
^rdhsioraàk nk Iour`kr nks s`ehdux usukcs `. `apucs`oh nk N`rdl
− − − 4 − − −
@c idut rdppkckr quk c‐`apucs`oh nk N`rdl kst vdcdbck ê t1: t1:
Ndhs lk lds c‐`apucs`oh nk N`rdl kst vdcdbck ê t1t:
4⟷2 4⟷2
Kxpohkht`kc uh`cdtàrdck
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 ;2
4 − ⟷ 4 −, +≫ : : 3+: 4 − − − − − + + − + + − − 4 4− ⟷ , 8 :, 4 + −−|||−|−⟷||+ , 8 :, + + −−||| −−|||− − −− − +−
Zrkuvk 7 so`t x(t) 1
. ^rouvkr sd trdhsioraàk nk Iour`kr.
\` oh rkapcdlk ck 4 pdr d oh d 7
Kxpohkht`kc b`cdtàrdck
Zrkuvk 7 so`t x(t) 1
. ^rouvkr sd trdhsioraàk nk Iour`kr
∘
+ − − − 4 − 2−− −++ 44 − 4 −++ −−||| ⟷ 2 , 8 :
Nohl
^rdhsioraàk nk Iour`kr seh(t) Hous sdvohs quk quk ck s`ehdc seh(t) seh(t) kst rkpràskhtà rkpràskhtà loaak loaak so`t 7
44,, ,, 38 ::
\eh(t)
t
(d) - \`ehdc seh(t) Hous pouvohs pràlànkr pouvohs pràlànkr kh prkhdht c‐kxpohkht`kc b`cdtàrdc `apd`r su`vdht su`vdht
− ,, 83 ::
4
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 ;<
-4
+ − −4 −− 4 − + − − −++ −−−− −+++ 4 4 2 −−||| ⟷ 2 , 8 :
Cd trdhsioraàk nk Iour`kr kst cd su`vdhtk 7
∘
Nohl
\` hous rkedrnohs s`ehdc x(t) nkssus, `c rksskabck loaak uhk iohlt`oh seh(t). Nohl cd trdhsioraàk nk Iour`kr nu seh(t) kst 7
⟷c→`a 2 2 2 ⟷:,, 3: :4,,;, , 8: : − + − 4 − 4 − c→`a − c ` a c→`a − c c→` a c→` a → c→`a 4 →c`a :, ≪ :
^rdhsioraàk nk Iour`kr nk c‐àlfkcoh uh`tà
Zrkuvk 7 so`t x(t) 1
^rouvkr sd trdhsioraàk nk Iour`kr
4kr \ocut`oh adtfàadt`quk
Hous nàv`ohs dcors àvdcuàk cd iohlt`oh ê t 1 : kt ê t 1 ∘. Lkpkhndht, cd vdcuk ê `hi`h` kst `hnàtkra`hàk. Nohl pour lohtrk ldrrà oh kst obc`eà kh nk vo`r u(t) sur uhk c`a`tk kxpohkht`kcck nàlro`ssdhtk tkc quk 7 \` hous idåohs ad`htkhdht cd trdhsioraàk nk Iour`kr nk cd c`a`tk, oh d 7
Hous pouvohs idl`ckakht trouvkr pour pour ck nkux`æak tkrak ck su`vdht 7
Ad`s pour ck prka`kr tkrak oh d 7
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 ;>
c→`a 4 ∘, : − −− , à →c`a 4
Hous sdvohs quk skcoh ck tfàoræak Zdrskvdc 7
Noht cd c`a`tk nk cd iorauck () kst
Nohl 7
2æak socut`oh Lkttk aàtfonk kst coe`quk ad`s duss` erdpf`quk kh ut`c`sdht ck s`ehdc s`ehk. Rkedrnohs cks nkux s`ehdux (d) kt (b) nkssous 7 \eh(t)
t t
-
(b) \`ehdc seh(t) u(t) 4
(l) \`ehdc u(t)
44 4 4 4 2 42 42 4 4 + + 4 − − − 2 4 4− 42 4∐
Hous pouvohs àlr`rk cd iorauck nk s`ehk(t) loaak so`t 7
\` hous dgoutohs pcus 4 pour pour tout ck tkaps l‐kst-êl‐kst-ê-n`rk n`rk cd pdrt`k edulfk d`hs` qu‐ê nro`tk oh d 7 d 7
Kcck nkv`kht 7
Nohl 7
Dcors
Hous sdvohs quk cd trdhsioraàk nk Iour`kr nk 4 kst 7
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 ;;
− − 4 2 2 − − 4 − 42 44 − 42 2 2
Nohl 7
^rdhsioraàk nk Iour`kr nk s`hus kt nk los`hus Zour ck s`hus
⟷ ⟷ − 2
Hous sdvohs quk ck s`hus pkut çtrk lohs`nàrà loaak uhk soaak soaak nk nkux `apucs`oh kh i : kt kh “ i i: su`vdhtk 7
− − − − − − − 2 24∐ −++224 −− − − − 4 4 4 − + + − − − 2 2 − 2 −
Noht cd trdhsioraàk nk Iour`kr nk s`hus skrd cd su`vdhtk 7
\kcoh cd ræeck C‐Fþp`tdc cks nkux tkraks kst `hnàtkra`hàs kt pkut prolànkr du nàr`và. Ad`s loaak lk h‐kst pds hotrk obgklt`vk, hous pouvohs dcckr n`rkltkakht n`rk ltkakht ê c‐uhk nks propr`àtàs nk cd trdhsi o oraàk raàk nk Iour`kr. \kcoh cd propr`àtà nk cd trdhsioraàk nk Iour`kr n‐uhk kxpohkht`kcck, quk ks 7 Dcors cd trdhsioraàk nk Iour`kr nu s`hus skrd 7
⟷2 − − + + − − − − 224 −422 − 2 2 2
Nohl 7
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 ;9
V(w) ϊ
U: w -U:
- ϊ Zour ck los`hus
⟷ − 2 − − 2 ∐ 4 2 − −−+− −−−−
Loaak `hn`quà kh nkssus ck los`hus pkut çtrk lohs`nàrà loaak uhk soaak nk nkux `apucs`oh kh i : kt kh “ kh “ i i: su`vdhtk 7
Noht cd trdhsioraàk nk Iour`kr nk s`hus skrd cd su`vdhtk 7
Nohl 7
V(w) ϊ
ϊ
U: w -U: ^rdhsioraàk nk Iour`kr n'uh pronu`t nk iohlt`ohs Cd trdhsiorak nk Iour`kr Iour`kr kt cd trdhsiorak nk Iour`kr `hvkrsk oht n nks ks ioraucdt`ohs `nkht`quks d uhk lohstdhtk kt uh lfdhekakht nk s`ehk pràs. Zdr lohsàqukht cd trdhsiorak n'uh pronu`t nk iohlt`ohs ndhs ck noad`hk tkaporkc kst uhk lohvocut`oh ndhs ck noad`hk nks iràqukhlks Lohvocut`oh Cd lohvocut`oh nk nkux s`ehdux x4(t) kt x2(t), opàrdt`oh quk c‐oh hotk pdr x 4(t)*x2(t) khekhnrk uh houvkdu s`ehdc x(t) tkc quk-7
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 ;5
∕ − , ∘
lk qu` aohtrk quk ck s`ehdc d`hs` obtkhu
Cd soaadt`oh sur `htàerdck s‐kiikltuk sur ck vdr`dbck kst uhk iohlt`oh nk t. Lk s`ehdc lorrkspohn ndhs ck noad`hk iràqukht`kc uhk s`apck auct`pc`ldt`oh nk cd trdhsioraàk nk Iour`kr nks s`ehdux x4(t) kt x2(t).
∕⟷ ⟷∕ kt kt
Lk qu` aohtrk quk cd trdhsioraàk nk Iour`kr nu pronu`t nk nkux iohlt`ohs kst ck pronu`t nk lohvocut`oh nk ckurs trdhsioraàks. L‐kst uh ràsuctdt ksskht`kc ndhs ck trd`tkakht nu s`ehdc q u` idl`c`tk cd socut`oh n‐uh pronu`t nkux s`ehdux. s`ehdux.
∕ ⟷ ∕ − ∕ −
Zrkuvk nu tfàoræak nk cd lohvocut`oh
Hous sdvohs quk ck tfàoræak nk cd lohvocut`oh lohvocut`oh kst ck su`vdht 7 ∘
Nohl cd trdhsioraàk nk Iour`kr skrd 7
∘
− − ∕ − − − − ⋯ ∭−
Hous pouvohs lfdhekr lfdhekr c‐ornrk n‐`htàerdt`oh tkc qu‐oh dd 7 7 ∘
∘
Hous voyohs voyohs quk c‐`htàerdc skrd 7
skrd uh nàldcdek tkaporkc, nohl soh ràsuctdt skrd
∘
− −− ∕ − − − ∕ −
Dcors c‐àqudt`oh () nàv`kht 7 nàv`kht 7
Nohl 7
Kxkapck 7 4 \o`t nkux iohlt`ohs su`vdhtks
(d)
4, :≮34 :, :,− ,≫: 3: (b)
kt
^rouvkr cd trdhsioraàk nk Iour`kr nk cd lohvocut`oh u(t)*v(t).
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 ;6
4ærk \ocut`oh pdr `htàerdt`oh n`rkltk Hous sdvohs quk
∕ − ∕ −4 − 2 −, 3 : , :, − :≮ ≫344,,22 , − 2 − ∕ − −4−− −4 44( 4), 4 − ∕ 4 4 ( 4)44 4 44 44 4 44 ∘
Loaak cks borhks n‐kx`stkhlks nk sks iohlt`oh loaakrlk ê zàro (:), kt tkra`hk so`t s o`t ê 4 ou ê t, ∘
Nohl 7
2æak socut`oh
Cd trdhsioraàk nk Iour`kr nk
kst cks su`vdhtks 7
Nohl
@l`, hous pouvohs prolànkr kh trouvdht cd trdhsioraàk `hvkrsk nk Iour`kr nu ràsuctdt nkssus qu` kst kh pkut loapc`quk, ad`s kh ut`c`sdht ut`c`s dht cd iohlt`oh pdrt`dck oh pkut y drr`vkr drr`vkr.. Oh vd rkvkh`r sur lk ràsuctdt kh pcus ndhs ck lfdp`trk. Cd aàtfonk erdpf`quk Hous dvohs ut`c`sà uh kxkapck pcus idl`ck d v`s`ohhà erdpf`qukakht. @c s‐de`t nks n nkux kux s`ehdux l`-nkssous 7 xt > t : (d)
2
f(t) 2 t :
4
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 ;0
(b) Hous sdvohs quk cd iorauck adtfàadt`quk pour cd lohvocut`oh khtrk khtrk cks nkux nkux s`ehdux l`-nkssus kst cd su`vdhtk 7
∕ ℈ − ℈ − ℈ , ∘
4kr àtdpk 7 rààlr`rk cks s`ehdux kh iohlt`oh nk 7 x( x() kt f(t- f(t-). x >
:(d)
2
f() 2
:
4
(b)
2æak àtdpk àtdpk 7 rktourhà dutour nk t1:, c‐uh nks s`ehdux. Åd pkut çtrk x(x( -) ou f(- f(-). @c idut hotkr quk ck ak`cckur s`ehdc ê rktourhkr kst lk dvkl uhk nuràk lourtk. Ndhs ck lds n‐kspælk, ck s`ehdc dvkl uhk nuràk lourtk kst f(- f(-).
x
>
:(d)
2
f() 2
-4
:
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 9:
æak àtdpk 7 7 nàpcdlà f(- f(-) pdr uhk vdcuk drb`trd`rk t pour dvo`r f(t- f(t-) pour dvo`r cks borhks su`vdhtks 7 edulfk1 t- t- 1t-4 kt cd nro`tk 1 t- t-1t-:1t ;æak àtdpk 7 7 trouvkr ad`htkhdht cd ràe`oh n‐`htkrsklt`oh pour cks nkux s`ehdux kh iohlt`oh nk . Lfdquk ràe`oh, dvkl cks borhks kh t, kst ldclucàk. @c idut hotkr quk cks borhks soht loht`huàks. x >
:
Ràe`oh @
2
Zds n‐`htkrsklt`oh n‐`htkrsklt`oh
t3:
f() 2 -4 t-1t-4
:
t-1t-:1t
x >
:
Ràe`oh @@
2
@htkrsklt`oh pdrt`dc
: ≮ t ≮ 4 f() 2
t-4
-4
:
t
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 94
x >
:
Ràe`oh @@@
2
@htkrsklt`oh totdc
4 ≮ t 32
f() 2 -4
:
t-4
t
x >
:
Ràe`oh @Y
2
@htkrsklt`oh pdrt`dc
2 3 t ≮
:
2
Ràe`oh Y
Zds `htkrsklt`oh 6, 2 ≮ 3 <
Ràe`oh @@@ @htkrsklt`oh totdc. Ndhs lk lds t kst k st khtrk 4 kt 2 nohl cks c` c`a`tks a`tks n‐`htàerdt`ohs n‐` htàerdt`ohs soht nk t-4 t-4 ê ê t
Ràe`oh @Y t-4 ê @htkrsklt`oh pdrt`dc. Ndhs lk lds t kst khtrk 2 kt < nohl cks c`a`tks n‐`htàerdt`ohs soht nk t-4 ê 2
Ràe`oh Y Zds n‐`htkrsklt`oh. Ndhs lk lds t 8 < nohl cks c`a`tks
− ℈ :: − ℈ , ℈ ∘
∘
9æak àtdpk 7 7 pour lfdquk ràe`oh oh trouvk pu`s oh `htæerk. Cks borhks n‐`htàerdt`oh sk trouvk cê oð ck pronu`t kst hoh zàro. Zour y drr`và, `c idut rkedrnkr cks borhks nk x( x() kt f(t- f(t-). H. B. Cks borhks nk f(t- f(t- ) soht tougours 7 ê edulfk t- t-e kt ê nro`t t- t- n. ndhs c‐kxkapck cd borhk nk f(tf(t -) nk edulfk kst 7 t- t- e 1t-4 kt cd borhk nk nro`tk kst 7 t- n1 t-:1t. oh pkut idl`ckakht `htàerkr ndhs cks borhks trouvkr l`-nkssus. ndhs 5æak àtdpk 7 dsskabcàs ck ràsuctdt kh nài`h`ssdhts cks borhks n‐kx`stkhlk nu s`ehdc. s`ehdc. Nohl ck ràsuctdt i`hdc kst 7 Ràe`oh @@ @htkrsklt`oh pdrt`dc
Ràe`oh @@@ @htkrsklt`oh totdc 4 ≮ t 32
Ràe`oh @Y
@htkrsklt`oh pdrt`dc @he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 9<
2 3 t ≮ los224.4.2 los2:. 2:.> los los:.> :.>2.2los.>2ϊh,22los:.:.∆2∆2:. :,>4,2, los los22:.:.6 los4.4.9 los2:. 2:.> los los:.>
Cks nkux s`ehdux soht cks rkpc`ks nk lfdluh.
Açak s` cks nkux s`ehdux oht cks iràqukhlks n`iiàrkhtks c‐dccurk kst cd açak, nohl cks nkux s`ehdux soht cks rkpc`kht nk lfdluh. Hous pouvohs dvo`r cks hoabrks `hi`h` nks rkpc`kht cks su`vdhtks 7
\` hous prkhohs c hàedt`vk, hàedt`vk, oh d 7
Cks spkltrks pour cks s`ehdux l`-nkssus soht trdlàs l`-nkssous 7
Kh id`t cks rkpc`kakhts nks iràqukhlks kst skuckakht ndhekrkux corsqu‐oh àlfdht`ccoh ck s`ehdc dvkl cd iràqukhlk n‐àlfdht`ccohhdek pcus pkt`tk quk cd iràqukhlk nk Hyqu`st. Lohs`nàrdht ck s`ehdc tkc quk x1los(2ϊ(>:::)t), skcoh ck tfàoræak nk \fdhhoh, nk \fdhhoh, cd iràqukhlk nk Hypu`st skrd du ao`hs 6mFz . Ad`s s` oh ut`c`sk cd iràqukhlk n‐àlfdht`ccohhdek nk 4:mFz , oh obt`kht cks po`hts àlfdht`ccohhàs x(h) loaak l`-nkssous ndhs cd i`eurk. Ck spkltrk or`e`hdc kt sd rkpc`kht ioht dppdrdètrk nk rd`ks spkltrdcks sk trouvdht dux iràqukhlks ai k ´ i : 1 ´>mFz, ´9mFz, ´4>mFz, ´49mFz, … … Oh vo`t quk ndhs cd bdhnk nk bdsk qu` s‐àtkhn nk : ê i ê i k /2 /2 1;mFz , `c h‐y qu‐uhk skuck rd`k spkltrdck s`tuàk kh i kh i 1>mFz cd iràqukhlk or`e`hdck.
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45 2:46 5>
Ho Dc`ds`he 4 Or`e`hdc s`ehdc N`slrkts po`ht N`slrkt s`ehdc
:.; k n u t ` c p a D
:
-:.;
-4 -;
: ^`ak
; x 4:
->
4 :.6 :.9 :.> :.2
: -:.2 -:.> -:.9 -:.6
-4 -;
->
-<
-2
-4
:
4
2
<
>
; ->
x 4:
\` pdr lohtrk hous ut`c`sohs cd iràqukhlk n‐àlfdht`ccohhdek nk 9mFz, oh obt`kht cks po`hts àlfdht`ccohhàs x(h) loaak cd i`eurk****. Ck spkltrk or`e`hdc kt sd rkpc`kht ioht dppdrdètrk nk rd`ks spkltrdcks sk trouvdht dux iràqukhlks ai k ´ i : 1 ´2mFz, ´>mFz, ´4:mFz, ´42mFz, ´42mFz, … Oh … Oh vo`t quk ndhs cd bdhnk nk bdsk qu` s‐àtkhn nk : ê i ê i k /2 /2 1 nk x(h) kt rkpràskhtkr ck aonuck nu spkltrk. \ocut`oh 7 Cd prka`ærk lfosk ê id`rk kst nk ldcluckr cd trdhsioraàk nk Iour`kr n`slrætk sur cks > àlfdht`ccohs nu s`ehdc loaak su`t 7
pour m 1 :, 4,
, H-4 kt H 1 >
⋯
− − →
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 6>
− , : → → : 4 2 ) − : 4−::4−− 2−( → − ( 2 ) 2, 2 − : 4− :4 2− () > 2 2−−( àlfdht`ccohs
> >) ∗ 2 { u(2) 1 49 u( u(>) 1 2;9 ... Kh dppc`qudht cd nài`h`t`oh nk cd trdhsioradt`oh kh W, kxpr`akr [(z), cd trdhsiorak kh z nk cd iohlt`oh u(t).D qukcck lohn`t`oh cd sàr`k obtkhuk lohvkrek-t-kcck? Lkttk lohn`t`oh tdht supposk rkspkltk, nohhkr cd vdckur nk [(z). \ocut`oh [(:) 1 4, u(4) 1 >, u(2) 1 49, u(, u(>) 1 2;9 ... Oh pkut khlork àlr`rk 7 1 ou khlork 1
> −4 >− >− − + ⋯ −
\o`t
Oh pkut rdppkckr quk cd trdhsioraàk kh z nk ξ m 1 4 kt quk cd trdhsioraàk kh z nk ξm-` 1 z-` Nohl s` oh rkprkhn cd iorak nk nàpdrt oh pkut n`rk quk 7 [(z) 1 4 + >z -4 + 49z-2 + 9>z-
∔→ −
soaak n‐ n‐uhk sàr`k eàoaàtr`quk nk rd`soh >z -4 \o`t Rdppkc sur cd sàr`k 7 so`t uhk sàr`à zh , Cd iorak eàhàrdc nk cd sàr`à eàoaàtr`quk kst ck su`vdht 7
→ 4 ⋯− → ⋯− +
Kh auct`pc`dht cd soaak \ H pdr z, oh d 7
\` hous soustrd`k z\ H nk \ H oh d 7
−
4 4++
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 02
−−
, pour z≪: dvkl z cd rd`soh rd`soh
Nohl cd c`a`tk ê c‐`hi`h` lohvkrek kst7 kst7
ê lohn`t`oh quk
| | 3 4
→ c`a → 4 4
− + + 4 > > 4>− 4 − 4>
Kh dppc`qudht lk qu‐oh d vu l`-nkssus, l`-nkssus,
| | 8 >
Lohvkrek pour
Nohl ck ràsuctdt kst 7
Rkadrquk 7 7 Cd trdhsioraàk kh W n‐uh s`ehdc n`slrkt x(h) nohhk ck s`ehdc V(z) qu` pkuvkht çtrk àlr`rk sous cd iorak nk irdlt`ohhkcck su`vdhtk, l‐kst-d-n`rk l‐kst-d-n`rk kh s‐àlr`vdht loaak ck rdt`o nk nkux pocyhþaks kh z kh z .
dvkl H(z) ck huaàrdtkur kt N(z) ck nàhoa`hdtkur.
Lkttk iorak nk V(z) id`rk dppdrd`trk nkux nài`h`t`ohs, ê sdvo`r cks pþcks kt cks zàros. Zdr nài`h`t`oh oh pkut n`rk quk cks zàros cks zàros n‐uhk trdhsioraàk kh W kh W , V ( z z ), ), soht cks vdckurs nk z nk z tkccks quk V quk V ( z z ) 1 :, kt cks pþcks cks pþcks n‐uhk trdhsioraàk kh W kh W , V ( z z ), ), soht cks vdckurs nk z nk z tkccks quk V quk V ( z z )1∘. )1∘. Zour quk V(z) 1 :, oh pkut aohtrkr idl`ckakht quk ck H(z) 1 :, nohl cks vdckurs nk z soht cks zàros. Nk cd açak adh`ærk, pour quk V(z) 1 ∘, cks vdckurs nk z pour ck nàhoa`hdtkur N(z) 1:, soht cks pþcks. Kxkrl`lks 2. Nohhkr cd trdhsioraàk kh W nk cd iohlt`oh n`slrætk
49 .5
44 < ∗ 4 4 − 4 ∗ 2< 4 2 >
^rdhsiorak kh W `hvkrsk Cd trdhsioraàk kh W n‐uh s`ehdc n`slrkt x(h) kst uh`quk. Zour rktrouvkr cd trdhsioraàk kh W `hvkrsk, uhk c`stk nk trdhsioraàks kh W kh W nk s`ehdux àcàakhtd`rks soht àtdbc`s, oh pkut idl`ckakht rktrouvkr nks s`ehdux tkaporkcs ê pdrt`r nk lkttk c`stk. Corsquk cd trdhsioraàk hk pkut idl`ckakht s‐àlr`rk sur cd iorak n‐uhk loab`hd`soh nk trdhsioraàks àcàakhtd`rks, àcàakhtd`rks , `c rkstk cks tklfh`quks eàhàrdcks nk trdhsioradt`oh `hvkrsk. Nohl pour `c y drr`và ê rktrouvkr cd trdhsioraàk kh W `hvkrsk ndhs lk lds, `c y d > aàtfonks ê sdvo`r 7 Rkcdt`oh @htàerdck Nàvkcoppkakht kh sàr`k nk Zu`ssdhlk Nàvkcoppkakht pdr n`v`s`oh Nàvkcoppkakht pdr irdlt`ohhd`rk Rkcdt`oh @htàerdck \o`t uhk trdhsioraàk kh W V(z) n‐uh s`ehdc s`ehdc n`slrkt x(h), cd trdhsioraàk kh W `hvkrsk kst nohhàk pdr c‐àqudt`oh = c‐àqudt`oh =
4 ∨− 2
Lkttk àqudt`oh kst vdcdbck pour toutk cd vdckur nk h, ck lohtour L n'`htàerdt`oh no`t çtrk ndhs cd ràe`oh nk lohvkrekhlk. @c no`t çtrk ikraà kt uhk io` khtourkr c'or`e`hk nu pcdh nks z ndhs ck skhs pos`t`i (skhs `hvkrsk nks d`eu`ccks n'uhk aohtrk). Zrkuvk nk cd iorauck 7 \o`t uhk trdhsioraàk kh W nu s`ehdc n`slrkt x(h) kst nohhàk pdr 7
→ − − − − − → − −+− − →+ −+− →
\` hous auct`pc`ohs c‐àqudt`oh l`-nkssus l`-nkssus pdr
, dcors hous dvohs 7
Nohl
Kh `htàerdht sur uh lohtour ikraà, cks n`iiàrkhts tkraks oh d7
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 0;
− −+− ∨ − →+ ∨ −+− ∨ ∨ → − − −+− −+− → ∨ →+ ∨ ∨ ∨ − ∨ − →+:: 2 →:: 2
Or
Nohl
4 ∨− ⋯ 2 24 ∨ − à − −
Zour h 1 :, 4, 2,
Oh pkut àvdcuk c'`htàerdck pdr cd aàtfonk nks ràs`nus kh ut`c`sdht ck tfàoræak nk Ldulfy sur c'`htàerdck ck cohe n'uh lohtour loaak su`t 7
\` cd trdhsioraàk kh W d n‐uh pþck n‐uh pþck n‐ornrk 4 (z1d), ck ràs`nu nk cd iohlt`oh pdr 7
kst nohhàk
→` a − 4 4 c − 4 c→`a 4! − − −
Zdr lohtrk, s` cd trdhsioraàk kh W d n‐uh pþck n‐uh pþck n‐ornrk q (z1d), ck ràs`nu nk cd nk cd iohlt`oh kst nohhàk pdr 7
−
Kh ldclucdht tous cks ràs`nus dux pþcks nk cd iohlt`oh obt`kht pdr soaadt`oh ck s`ehdc x(h). x(h).
c'`htàr`kur nu lohtour L, oh
Kxkapck :4 ^rouvkr cd trdhsioraàk kh W `hvkrsk kh ut`c`sdht cd aàtfonk nk ràs`nus nk cd trdhsioraàk su`vdhtk 7
;>
\ocut`oh :47
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 09
N‐dborn hous pouvohs àlr`rk c‐àqudt`oh l`-nkssus l` -nkssus sous cd iorak su`vdhtk 7
> 4 −−cc→→``aa>4>>44−−>4>44>>4−− >4< < >
Nohl cks ràs`nus,
Nohl ck ràsuctdt
Kxkapck :2 Nàvkcoppkakht kh sàr`k nk Zu`ssdhlk Loaak cd trdhsioraà V(z) kst uhk iohlt`oh dhdcyt`quk nk z ndhs cd ràe`oh nk lohvkrekhlk, oh pkut nàvkcoppkr kh sàr`k nk ^dycor ou Cdurkht Cdurkht kh iohlt`oh nk z-h. Oh pkut, khsu`tk, trouvkr cks sàr`ks pdr `nkht`i`ldt`oh dvkl cks sàr`ks lohhu. \`
→ −
dvkl x(h) cd trdhsioraàk kh W `hvkrsk.
Kxkapck :4 \o`t cd trdhsioraàk kh W, V(z) kst nohhàk pdr 7
2
^rouvkr cd trdhsioraàk kh W `hvkrsk nu s`ehdc x(h). \ocut`oh
2 4 4 2 4 2 2 2 2 4 2 4 ⋯2 → 2
\kcoh cd sàr`k nk ^dycor ou Cdurkht, oh d 7
Nohl
Kxkapck :2 7 \` ad`htkhdht, cd trdhsioraàk kh W V(z) s'àlr`t sous cd iorak n'uhk irdlt`oh rdt`ohhkcck, so`t 7
− − − ⋯ − − ⋯− , || 8
ou R adx adx kst ck pþck adx nk N(z). Hous pouvohs n`v`sàk cd socut`oh kh pcus`kurs pcus`kurs àtdpks, ê sdvo`r 7 4. Ktdpk :4 7 \` p ≫ q, dcors
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 05
− −−−−⋯ − ⋯− −−−−⋯ − ⋯ − −− −− ⋯ ⋯−
2. Ktdpk 2 7
Rààlr`rk c‐àqudt`oh l`-nkssus l`-nkssus kh ut`c`sdht quk cd pu`ssdhlk pos`t`vk nk z.
. Ktdpk > \` c‐àqudt`oh kh àtdpk < oht cks pþcks ràkcs kt n‐dutrks pþcks cks auct`pcks, hous pouvohs cks idltor`sàs khlork loaak su`t 7
Hous pouvohs ad`htkhdht trouvkr trouvkr cks vdckurs nk D, B, …, L4,L:, kh su`vdhtks 7
− ⋯ −−)−111 111 −−⋯( −− −−)( − − (−) 1 − − ⋯(⋯ ) − 1 − ⋯ − (−) 1 ⋯( )− 1
;. Ktdpk ; \ubst`tuàks cks vdckurs nk D, B, …, L4,L:, ndhs c‐àqudt`oh n‐àtdpk >d, kt auct`pc`àk pdr z loaak su`t 7
⋯ − (− )
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 06
9. Ktdpk 9 7 C‐àqudt`oh eàhàrdck nu V(z) kst nohl 7 nohl 7
⋯ ⋯ −− (−) 4 − 4 − 4 −− (4−−) −−− 4 − 4 − ⋯ 4 − − − ⋯ − − (4−−)
\` hous auct`pc`ohs
pdr z -4, oh d
Nohl oh d 7
5. Ktdpk 5 7 \` hous dppc`quohs cd sàr`k nk ^dycor ou nk Cdurkht ê c‐àqudt`oh nk c‐àtdpk c‐àt dpk 9 oh d 7 d 7
− − − − ⋯ − ⋯ 4 −−−− 4 ⋯4 −− ⋯ − 2−− ⋯ ⋯ − ⋯ −− → → ⋯ → − − → − − ⋯−−−−− ⋯ − −
\` L:≪kq-4 oh durd`t,
Nohl x(h) kst 7
Kxkapck :2
Nàvkcoppkakht pdr n`v`s`oh sullkss`vk [hk dutrk aàtfonk kst nk nàvkcoppkakht pdr n`v`s`oh sullkss`vk. Lkttk aàtfonk kst pcus pcus s`apck s` vous hk lohhd`ssd`s pds cd socut`oh kxdltk ndhs ck tdbckdu nk z. Kxkapck =
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 00
−−
V(z)1
nohl cd socut`oh pkut çtrk obtkhu pdr n`v`s`oh sullkss`vk su`vdhtk7
-h
:: 4< 09> 4
42 4 − 4 − 2 4 + − ( ) 4 − − ( ) 2 4 − 2 () −− − ( ) ()−|| 4( ) 2 2 2 4 2 2 2
Cd aàtfonk cd pcus idl`ck ê àtun`kr lk i`ctrk kst lkcu` nk cd trdhsioraàk kh W7
Nohl cd iohlt`oh nk trdhsikrt nu i`ctrk kst7
Hous prolànohs kh àtun`dht ck loaportkakht loaportkakht iràqukht`kc nu i`ctrk kh rkapcdådht W rkapcdådht W pdr pdr oð ^k kst cd pàr`onk n'àlfdht`ccohhdek ndhs cd iohlt`oh nk trdhsikrt7
\` cd pàr`onk kst pàr`on`sà ê 4, nohl cd iohlt`oh nk trdhsikrt iràqukht`kcck nkv`khhk7 Ad`s cd pcus pdrt nk lds, cd pàr`on`sà h‐kst pds pr`sk kh loaptk, nohl cd socut`oh rkstk 7 @c s'de`t i`hdckakht n'uh i`ctrk pdssk bds noht ck aonuck nk cd iohlt`oh nk trdhsikrt su`t uhk lourbk kh los`hus kt noht ck nàpfdsdek kst c`hàd`rk kh iohlt`oh nk cd iràqukhlk. Lk nàpfdsdek kst àqu`vdckht ê uh rktdrn
Kh ràdc`tà oh pkut nài`h`r so`t ck nàpfdsdek pfdsk, kst dppkcàk rktdrn nk pfdsk so`t ck nàpfdsdek nu eroup, kst dppkcàk rktdrn nu eroup. Oh trouvk pdr cd aàtfonk su`vdhtk 7
Ou
. [h pkt`t proerdaak kh AD^CDB pour trdlkr cd iohlt`oh l`-nkssus. iuhlt`oh()= %iuhlt`oh ]Q1trdhsikr iuhlt`oh()= i1:7:.:47:.;= F1kxp(4`*p`)*los(p`*i)= pcot(i,dbs(F))=er`n oh = oh = 'Ràpohsk iràqukht`kcck n'uh i`ctrk huaàr`quk') t`tck('Ràpohsk t`tck( huaàr`quk' ) = xcdbkc('Irkqukhly' xcdbkc( 'Irkqukhly')= )= ycdbkc('|F|' ycdbkc( '|F|')= )=
,
@he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 4>2
Ràpohsk iràqukht`kcck n uh i`ctrk huaàr`quk 4 :.0 :.6 :.5 :.9 | F |
:.; :.> :.< :.2 :.4 :
:
:.:;
:.4
:.4;
:.2
:.2; :.< Irkqukhly
:.
:.>;
:. ;
KV:2 7 \o`t ck i`ctrk nài`h` pdr c'àqudt`oh su`vdhtk7
4 4> 42 442 42 − 42 −4> 2 42 42 − 42 − ) 42 42 − 42 − ( () 4 − 4 4 4 − 4 − 4los 4los2 − 4)22 2 | 2 ( 2))42(2 |4 24 22 24 22
\ocut`oh Cd aàtfonk cd pcus idl`ck ê àtun`kr lk i`ctrk l'kst khlork lkcu` nk cd trdhsioraàk kh W7
Nohl cd iohlt`oh nk trdhsikrt nu i`ctrk kst7
\` hous prolànohs loaak kh KV:4, ck loaportkakht iràqukht`kc nu i`ctrk kst7
Lk nàpfdsdek kst àqu`vdckht ê uh rktdrn Oh ut`c`sk cd açak prolànurk su`vdhtk 7
Ou
@c s'de`t khlork n'uh i`ctrk pdssk bds noht ck aonuck nk cd iohlt`oh nk trdhsikrt su`t uhk lourbk kh los`hus suràckvà kt noht ck nàpfdsdek kst c`hàd`rk kh iohlt`oh nk cd iràqukhlk. [h dutrk pkt`t proerdaak kh AD^CDB pour trdlkr cd iohlt`oh l`-nkssus. iuhlt`oh()= %iuhlt`oh ]Q1trdhsikr iuhlt`oh()=
i1:7:.:47:.;= @he. Kuekhk H Hsfdhg` 7 ^rd`tkakht nu s`ehdc = support nk lours \[ZZ^@L Pdouhnà 2:45-2:46 4><
r1los(2*p`*i)= t14+r= F14/2*kxp(4`*p`)*t= pcot(i,dbs(F))=er`n oh = oh = t`tck('Ràpohsk t`tck( huaàr`quk' ) = 'Ràpohsk iràqukht`kcck n'uh i`ctrk huaàr`quk') xcdbkc('Irkqukhlk' xcdbkc( 'Irkqukhlk')= )= ycdbkc('|F|' ycdbkc( '|F|')= )= Ndhs lks nkux kxkapcks `c kst dppdru qu'`c àtd`t idl`ck n'obtkh`r cd iohlt`oh nk trdhsikrt spkltrdck n'uh i`ctrk huaàr`quk ê pdrt`r nk c'àqudt`oh tkaporkcck ràe`ssdht lk i`ctrk. B`kh khtkhnu, l'kst ksskht`kcckakht c'dpprolfk `hvkrsk, lohs`stdht ê trouvkr c'àqudt`oh nk i`ctrdek qu` sdt`sid`t uh edbdr`t iràqukht`kc nohhà, qu` kst cd pcus `aportdhtk. Oh pdrck dcors nk syhtfæsk nk i`ctrk huaàr`quk. @c kst `aportdht nk rkadrqukr, quk ndhs cks nkux kxkapcks pràskhtàs, cks i`ctrks nàlr`ts dvd`kht uh nàpfdsdek c`hàd`rk kh iohlt`oh nk cd iràqukhlk. @c s'de`t s 'de`t n'uhk propr`àtà pdrt`luc`ærk nks i`ctrks huaàr`quks qu` d uhk `aportdhlk ldp`tdck ndhs cks dppc`ldt`ohs oð cd pfdsk nu s`ehdc trd`tà kst portkusk n'`hioradt`ohs.
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