Sayısal İşaret İşleme - Sakarya Üniversitesi Çağlar GÜL Ders Notları FULL Slaytlar

8/9/2019 Sayısal İşaret İşleme - Sakarya Üniversitesi Çağlar GÜL Ders Notları FULL Slaytlar

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Ŀŗhrftdfr vf Ŀŗhrft Ŀŗdfbf

• Ŀŗhrftdfr môedô` lhyhtĸbĸzah úefbdk ckr rgd gyehr.

• Ckr kŗhrft zhbhe, uzh`dĸ`, `geub, sĸnh`dĸ` vf chėĸbsĸz afėkŗ`fedfrke ckr jge`skygeuaur.

• @hrŗĸdhŗtĸėĸbĸz ígėu kŗhrft agėhd gdhrh` ôrftkdkr.

• Henh`, ckr kŗhrft yhphy gdhrh` yh ah ckr ckdmkshyhr hôrftkdfckdkr.

Qk k` k d c ú `d ė a kd k k


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Qk k` k td c ú `d ė a kd k tk

Qkpk` Ŀŗhrftdfrf Úref`dfr

• ^fs vf bôzk` kŗhrftdfrk „ lhvh chsĸenĸeĸ uzhyah zhbheĸe ckr jge`skygeu gdhrh` tfbskd fafr

• —K dk`f akmkthd skmehd prgnfsskem‒ sfs kŗhrftkeke ahdmmústfrkdbkŗtkr.


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• Fdf`trg`hrakygmrhjk (F@M) kŗhrftk „ `hdcke fdf`trk`stfbskd fafr.

• Qkpk` ckr F@M kŗhrftk hŗhėĸah mústfrkdbkŗtkr.


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• Fdf`trgfesfdfjgmrhb (FFM) kŗhrftdfrk „ cfykeaf`eúrgeue rhstmfdf uyhrĸdbhsĸydh gduŗhe fdf`trk`

tfbskd fafr.


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• ^ksbk` kŗhrftdfr „ ckr afprfb, ckr vgd`hek` phtdhbhhdtĸ phtdhbhsĸeahe `hyeh`dhehe `hyh lhrf`ftdfrkydf

• [fr lhrf`ftk, lhrf`ftke `hyehėĸeahe chŗdhyĸp`htbhedhrĸeahe tôb yúedfraf kdfrdfyfe ôí tôr gduŗturur.


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• Qkpk` ckr sksbgmrhj `hyaĸ hŗhėĸah vfrkdbkŗtkr.


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• Sfe`skz múrôetô - ĸŗĸ` ŗkaaftkek k`k uzhbshd `gjge`skygeu gdhrh` tfbskd fafr.


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• Zkafg kŗhrftdfrk„ífrífvf gdhrh` hadheaĸrĸdhe múrôetgduŗur vf > afėkŗ`feke ckr jge`skygeuaur9 3 uzhbsh

zhbhe.


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Ŀŗhrftdfr vf Ŀŗhrft Ŀŗdfbf

• Ckr kŗhrft ckdmk thŗĸbh`thaĸr.

• Ŀŗhrft kŗdfbfeke hbhnĸ kŗhrftke thŗĸaĸėĸ jhyíĸ`hrtbh`tĸr.

• Ckdmk íĸ`hrtbh yúetfbk, kŗhrftke tôrôef vf kŗh ckdmkeke agėhsĸeh chėdĸaĸr.

• Cu afrs, kŗhrftdfrke hyrĸ`-zhbhe mústfrkdkbk vf

kŗdfbfskek fdf hdbh`thaĸr.


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Ŀŗhrftdfrke Qfbskdk vf ^ĸeĸjdheaĸrĸdb

•

Ŀŗhrft tôrdfrk, chėĸbsĸz afėkŗ`fedfrke ef gdauėutheĸbdhyhe jge`skygeue hdaĸėĸ afėfrf chėdĸaĸr.

• Úrefėke, chėĸbsĸz afėkŗ`fedfr sôrf`dk vfyh hyrĸ` gd

• Cfezfr ŗf`kdaf, kŗhrft chėĸbsĸz afėkŗ`fedfrke sôrf` ckr jge`skygeu gdhckdkr.


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•

Hyrĸnh, kŗhrft mfrífd vfyh `hrbhŗĸ` afėfrdk ckr jge`s

• Qf` ckr `hyeh`the ôrftkdfe kŗhrftf ^@HDFS ĿŕHSFQ

• Ŀ`k vfyh ahlh jhzdh `hyeh`the ôrftkdfe kŗhrftf ZFÍG@ @HEHDDK kŗhrft afekr.


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• Ckr cgyutdu (8-A) ckr kŗhrft, tf` ckr chėĸbsĸz a

jge`skygeuaur

• Íg` cgyutdu (B-A) ckr kŗhrft, ckrafe jhzdh chėĸbs ckr jge`skygeuaur.

• ^fs kŗhrftk, chėĸbsĸz afėkŗ`feke zhbhe gdauėu 8-A

• Ckr múrôetô kŗhrftk, chėĸbsĸz afėkŗ`fedfrke uzhbsh

gdauėu 3-A ckr kŗhrftf úref`tkr.


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• Sfe`dk ckr múrôetô vf múrôetôeôe ôí rfe` ckdmústfrkdbkŗtkr.


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• Sfe`skz ckr shyĸshd vkafg kŗhrftkeke lfr ckr ífrífvfszhbheĸe hyrĸ` hedhrĸeah bfyaheh mfdbf` ôzfrf hy

afėkŗ`feke ckr jge`skygeu gdhe 3-A ckr múrôetô kŗhrf

• Cu efafedf, rfe`skz ckr vkafg kŗhrftk > chėĸbsĸuzhbshd `ggrakeht vf zhbhe gdbh` ôzfrf, >-A c

úref` gdhrh` aôŗôeôdfckdkr.

• Sfe`dk ckr vkafg kŗhrftk, > ckrkenkd rfemk (`ĸrbĸzĸSMC‛yk tfbskd fafe > haft >-A kŗhrfttfe gduŗhekŗhrfttkr.


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• 8-A ckr kŗhrft kíke, chėĸbsĸz afėkŗ`fe mfefdaf zhadheaĸrĸdĸr.

• Chėĸbsĸz afėkŗ`fe sôrf`dkysf, kŗhrftf ^ÔSF@DĿ-Yafekr.

• Chėĸbsĸz afėkŗ`fe hyrĸ`sh, kŗhrftf H[SK@-YHBHE

• Ckr sôrf`dk-zhbhe kŗhrft, zhbheĸe lfr heĸeah thhyrĸ`-zhbhe kŗhrft, zhbheĸe cfdkrdk hedhrĸeah theĸbd

akzkskakr.


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Ŀŗhrftdfrke Qfbskdk vf ^ĸeĸjdheaĸrĸdb• ^gedu shyĸah rh`hbdh tfbskd fakdfe hyrĸ`-mfedk`dk ck

kŗhrftkef ^H[K^HD kŗhrft afekdkr.

• ^hyĸshd ckr kŗhrftf úref`, ckr AZA‛yf shyĸshddhŗtĸrĸdbĸŗ buzk` kŗhrftkakr.

• ^ôrf`dk mfedk`dk ckr hyrĸ`-zhbhe kŗhrftkef ÚSEF@Dafekr.

• G lhdaf, shyĸshd ckr kŗhrft `uhethdhebĸŗ úref`dfebkŗ

• ^ge gdhrh` hyrĸ` mfedk`dk sôrf`dk zhbhe kŗh


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• Ckr kŗhrftke bhtfbhtk`sfd tfbskdkeaf`k jge`skygemfefdaf híĸ` ckr ŗf`kdaf mústfrkdkr.

• ^ôrf`dk-zhbhe 8-A ckr kŗhrft kíke, sôrf`dk chėĸbmfefdaf t kdf mústfrkdkr. Úrefėke u(t ), sôrf`dk-zhbhe tfbskd fafr.

• Hyrĸ`-zhbhe 8-A ckr kŗhrft kíke, hyrĸ` chėĸbsĸz afėe kdf mústfrkdkr. Úrefėke {vPeR} hyrĸ`-zhbhe 8-A ckrfafr.

Ú


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• Ígėu uymudhbhah, ckr hyrĸ`-zhbhe kŗhrft ckr kŗhrftke zhbheah aôzmôe hrhdĸ`dhrdh úref`dfebfskyd

• Hyrĸ`-zhbhe kŗhrftke theĸbdheaĸėĸ zhbhedhr aôzm chėĸbsĸz hyrĸ` afėkŗ`fe e, thbshyĸ afėfrdfr hegrbhddfŗtkrkdfckdkr.

• ^ôrf`dk-zhbhe 3-A ckr kŗhrft aurubueah, 3 chėĸbmfefdaf x vf y kdf mústfrkdfe uzhbshd `ggrakehtdhraĸr

• Úrefėke, rfe`skz ckr múrôetôeôe ( x, y) `geubuea


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• ^hyĸshddhŗtĸrĸdbĸŗ ckr múrôetô 3-A hyrĸ`-zhbhe ckr chėĸbsĸz afėkŗ`fek mfefdaf b vf e kdf cfdkrtkdfe hy

uzhbshd afėkŗ`fedfrakr. G lhdaf, shyĸshddhŗtĸrĸdbĸvPb,eR gdhrh` kjhaf fakdfckdkr.

• Sfe`skz ckr vkafg kŗhrftk >-A ckr kŗhrfttkr vf u( x

tfbskd fakdfckdkr.

• Sfe`dk ckr vkafg kŗhrftk, ckrkenkd rfe`dfrk tfbskd fagduŗhe ckr vf`túr kŗhrfttkr9


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• Bhtfbhtk`sfd ckr kjhaf yh ah `urhd, vfyh thcdgahe g

theĸbdhebĸŗ ckr kŗdfbdf thb gdhrh` cfdkrdfefckdfAFQFSBĿEĿ^QĿ@ ckr kŗhrft afekr.

• Shstmfdf ckr ŗf`kdaf ôrftkdfe vf úenfafe `fstkrkdfbf

SH^QMFDF ckr kŗhrft afekr.


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Qfbfd Yhbhe-uzhyĸ Ŀŗdfbdfrk

• Fe tfbfd ôí zhbhe-uzhyĸ kŗdfbk údíf`dfbf,

tgpdhbhaĸr.

• ÚDÍF@DFBF, ckr kŗhrftke pgzktkj vfyh efmhtkíhrpĸdbhsĸaĸr. Hehdgm kŗhrftdfr aurubueah, `hzheí

shcktkeke mfedkėk 8‛afe côyô`sf kŗdfbf @VZZFQh`sk lhdaf YH[KJDHQBH afekr.

• x(t ) kŗhrftk kdf údíf`dfebkŗŗf, kŗdfb sgeunueah

kŗhrftk gduŗur.


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• Akėfr tfbfd k`k kŗdfb, ĿEQFMSHD vf QÔSFZ hdbh

• x(t ) hehdgm kŗhrftkeke ketfmrhdk

kŗhrftkek gduŗturur.

• x(t ) hehdgm kŗhrftk eke tôrfvk


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• ÚQFDFBF kŗdfbk, grkikehd kŗhrftke útfdfebkŗ ckr `g

kŗhrft gduŗturur.

• x(t ) hehdgm kŗhrftk kíke, y(t ) 2 x(t -t ?), x(t )‛eke pvhrshyĸdhe t ? `hahr sôrfydf útfdfebfskydf fdaf fakdfe

• t ? sôrfsk efmhtkjsf, kŗdfbf ĿDFSDFQBF afekr.


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• Ígėu uymudhbh, yfek ckr kŗhrft gduŗturbh` hbhn

ahlh jhzdh kŗhrft kífrfe kŗdfbdfr mfrf`tkrkr. Úrefėke x3(t ) + x>(t ) kdf vfrkdfe kŗhrft x8(t ), x3(t ) vfkŗhrftdfrkeke tgpdhebhsĸydh gduŗturudur.

•

x8(t ) vf x3(t ) kŗhrftdfrkeke íhrpĸbĸ y(t ) 2 x8(t ) gduŗturur.

• ŕkbakyf `hahr thrtĸŗĸdhe tfbfd kŗdfbdfr, hyrĸ`-zh

ôzfrkeaf af uymudheĸr.


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Jkdtrfdfbf

• JĿDQSFDFBF, fe sĸ` `uddheĸdhe `hrbhŗĸ`

yúetfbdfrkeafe ckrkskakr. Cu kŗdfbk mfríf`dfŗtkrfeJĿDQSF afekr.

• Ckr jkdtrf, cfdkrdk jrf`hes ckdfŗfedfrkek cgzueubs

akėfr jrf`hes ckdfŗfedfrkek chstĸrĸr.

• Jkdtrfafe mfíbfyf kzke vfrkdfe jrf`hes hrhdĸėĸe cheaĸ, jkdtrfeke chstĸraĸėĸ jrf`hes hrhdĸėĸeh ah

cheaĸ afekr.


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Jkdtrfdfbf

• x(t ) mkrkŗ kŗhrftk, y(t ) íĸ`ĸŗ kŗhrftk vf l(t ) jkdtrfeke

gdbh` ôzfrf, agėrushd ckr hehdgm jkdtrfdfbf kŗdfbk

fŗktdkėkydf vfrkdfe `gevgdôsyge ketfmrhdkydf theĸbdhe

• Aúrt tôr tfbfd jkdtrf bfvnut gdup theĸbdhrĸ hŗhėĸah y


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Jkdtrfdfbf• HDÍH@ MFÍĿSFE ckr jkdtrf, @F^ĿB JSF@HE^K

ckr jrf`hes j n‛afe `ôíô` jrf`hesdhrĸ mfíkrkr, j

jrf`hesdhrĸ chstĸrĸr.

• [Ô@^F@ MFÍĿSFE ckr jkdtrf, cfdkrdk ckr `frskb `ôíô` jrf`hesdhrĸ chstĸrĸr, j n‛afe‛afe côyô` jrf`hes

• CHEA MFÍĿSFE ckr jkdtrf, j n84 j n3 gdbh` ôzfjrf`hesĸ j n8vf j n3 hrhsĸeah`k jrf`hesdhrĸ mfíkrkr, j n8‛

j n3‛afe côyô` jrf`hesdhrĸ chstĸrĸr.

• CHEA ^ÚEAÔSFE ckr jkdtrf j 4 j gdbh` ôzf


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Jkdtrfdfbf• Hŗhėĸah`k ŗf`kd, 6? Lz, 88? Lz vf 33? Lz jrf`

skeôzgkahd > kŗhrfttfe gduŗhe ckr mkrkŗ kŗhrftkeke

jkdtrfdfebfskydf gduŗhe sgeunu mústfrbf`tfakr.


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Jkdtrfdfbf• Hŗhėĸah`k ŗf`kd, hyeĸ kŗhrftke yô`sf` vf c

jkdtrfdfebfskydf fdaf fakdfe sgeuídhrĸ mústfrbf`ftak


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Jkdtrfdfbf

• [u`hrĸah vfrkdfe aúrt tôr jkdtrfafe chŗ`h tôt jkdtrfdfr

• Qf` ckr jrf`hes ckdfŗfekek chstĸrhe jkdtrfyf ÍFEQĿ@

• ÍG@ CHEADK ckr jkdtrfeke ckrafe jhzdh mfíkrbf

cheaĸ vhraĸr.

• QHSH@ jkdtrf, ckr aôŗô` jrf`hesĸe thbshyĸ `htshyĸdh


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@hrbhŗĸ` Ŀŗhrftdfrke Ôrftkdbfs

• Ckr kŗhrft mfrífd vf `hrbhŗĸ` afėfrdk gdhckdkr. Ŀ

kŗhrftdfr MFSÍFD, k`kenk aurubah`k kŗhrftdfrf kŗhrft afekr.

• LĿDCFSQ AÚEÔŕQÔSÔNÔ^Ô `uddheĸdhrh`, mfríf

`hrbhŗĸ` ckr kŗhrft ôrftkdfckdkr.

• Lkdcfrt aúeôŗtôrônôsôeôe kbpuds yheĸtĸ hŗhėĸah`k f


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• ^ôrf`dk-zhbhe Jgurkfr aúeôŗôbô (NQJQ) \ ( i ) hŗmfrífd ckr kŗhrft x(t )‛yk fdf hdhdĸb.

• \ ( i )‛yh x(t )‛eke ^_F@QSVBV afekr. Mfrífd ckr kspf`trubu íkjt, jhz spf`trubu ksf tf` ckr kŗhrfttkŗhrftke spf`trubu pgzktkj vf efmhtkj jrf`hesdhr

\ p( i ) vf \ e( i ), \ ( i )‛eĸe sĸrhsĸydh pgzktkj vf ehrhdĸėĸeĸ `hpshyhe `ĸsĸbdhrĸ gdbh` ôzfrf, hŗhpĸah yhzĸdhckdkr9


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@hrbhŗĸ` Ŀŗhrftdfrke Ôrftkdbfs• x(t ) ckr Lkdcfrt aúeôŗtôrônôsôef uymudheĸrsh íĸ`ĸŗth

• y(t )‛eke spf`trubu [ ( i ), lLQ(t ) vf x(t )‛eke spíhrpĸbĸeh fŗkttkr. lLQ(t )‛eke spf`trubu

gdauėueahe [ ( i ) hŗhėĸah vfrkdfe fŗktdk`tfe fdaf fak


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• y(t )‛af mfrífd ckr kŗhrfttkr. m(t ) 2 x(t ) + i y(t ) kdf vfr

kŗhrftk fdf hdhdĸb. m(t )‛eke spf`trubu hŗhėĸah`k fŗktdk

• Akėfr ckr afykŗdf, HEHDĿQĿ@ ckr kŗhrft hadheaĸrkŗhrft m(t ) shafnf pgzktkj jrf`hes ckdfŗfedfrkef shlkpt


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Bgaôdhsyge vf Afbgaôdhsyge

• Aôŗô` jrf`hesdĸ ckr kŗhrftke ckr lhcfrdfŗbf `hehdĸ ô

ckr ŗf`kdaf kdftkdfckdbfsk kíke ckr bgaôdhsyge kŗdfbkŗhrftke yô`sf` jrf`hesdĸ ckr kŗhrftf aúeôŗtôrôdbfsk

• Hdĸnĸ thrhjth, bgaôdhsygedu yô`sf` jrf`h

afbgaôdhsyge kŗdfbk uymudhehrh` kdftkdfe aôŗô` jrmfrk fdaf fakdkr.

• Hehdgm kŗhrftdfrke bgaôdhsygeu kíke aúrt yútfb vhr

(8) Mfedk` bgaôdhsygeu(3) Jrf`hes bgaôdhsygeu


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Mfedk` Bgaôdhsygeu• Mfedk` bgaôdhsygeueah QHŕK[KNK afefe yô`

skeôzgkahd ckr kŗhrft Hngs( ?t )‛eke mfedkėk, CĿDMĿaôŗô` jrf`hesdĸ ckr x(t ) kŗhrftkydf afėkŗtkrkdkrBGAÔDH^[GEDV kŗhrft afefe yô`sf` jrf`hegduŗturur9 y(t ) 2 Hngs( ?t ) x(t ).

• Jrf`hes útfdfbf úzfddkėkek mústfrbf` kíke, 8 4

ôzfrf, x(t ) 2 ngs( 8t ) gdsue. y(t ) vf y(t )‛eke spf`trufŗktdk`dfrafe lfshpdhehckdkr9


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Mfedk` Bgaôdhsygeu• Ckdmk kŗhrftk vf bggaôdhsygedu kŗhrftke spf`tr

vfrkdbkŗtkr9


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Mfedk` Bgaôdhsygeu

• ŕf`kdafe múrôdaôėô mkck, bgaôdhsygedu kŗhrft y(t )

jrf`hesĸ gdhe 3

b cheamfekŗdk`dk cheasĸeĸrdĸ yô`sfkŗhrfttkr. Bgaôdhsygedu kŗhrftke ? kdf ? + hrhsĸeah `hdhe `ĸsbĸeh Ô^Q [HE CHEA, ? jrf`hesdhrĸ hrhsĸeah `hdhe `ĸsbĸeh ksf HDQ [HE CH

• Ŀ`k yhe chea gduŗtuėueahe vf thŗĸyĸnĸ bfvnut gdbyúetfbf dktfrhtôraf QHŕK[KNK^K CH^QKSKDBKŕCHEADK (^N-A^C) bgaôdhsyge afebkŗtkr.


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Mfedk` Bgaôdhsygeu

• Ckdmk kŗhrftkek mfrk fdaf ftbf` yhpĸdhe afbgaôdhsyhaĸbahe gduŗur9 Ŀd` úenf, y(t ) thŗĸyĸnĸ kdf hyeĸ jrf`he

ckr kŗhrftdf íhrpĸdĸr9

• r (t )‛eke spf`trubu hŗhėĸah vfrkdbkŗtkr9


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Mfedk` Bgaôdhsygeu

• Ahlh sgerh, r (t )‛afe x(t )‛yk fdaf ftbf` kíke r (t ), `

n ,

b 4

n 4 3

? -

b fŗktskzdk`dfrkek shėdhyhe ckr jkdtrfafe mfíkrkdkr.

• Bgaôdhsyge vf afbgaôdhsyge kŗdfbdfrkeke cd

mústfrkdkbdfrk hŗhėĸah vfrkdbkŗtkr.


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Mfedk` Bgaôdhsygeu

• _rhtk`tf, afbgaôdhsygeah `uddheĸdhe skeôzgk

jrf`hesĸeĸe thŗĸyĸnĸ jrf`hesĸeh fŗkt gdbhsĸeĸ shėdhb prgcdfbke ôstfskeafe mfdbf` kíke, bgaôdhbgaôdhsygedu kŗhrft thŗĸyĸnĸyĸ ah kífrfnf` ŗf`kdaf af

•

Cu kŗdfb, b `htshyĸsĸ tôb t afėfrdfrk kíke P8+bx(tŗf`kdaf sfíkdfe ckr shyĸ gdbh` ôzfrf mfedk` bgaôdhhŗhėĸah`k mkck yfekafe theĸbdhehrh` yhpĸdĸr9


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Mfedk` Bgaôdhsygeu

• Hŗhėĸah`k ŗf`kdaf, 3? Lz jrf`hesdĸ skeôzgkahd ckr c

b 2?.6 kíke A^C yúetfbk `uddheĸdhrh` fdaf fakdfe 1jrf`hesdĸ mfedk` bgaôdhsygedu kŗhrft mústfrkdbkŗtkr.


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Mfedk` Bgaôdhsygeu

• Mfdfef`sfd A^C mfedk` bgaôdhsygeu aurubueah, c cheamfekŗdkėk b k`fe bgaôdhsygedu kŗhrftke`k 3

`hphsktfskek hrttĸrbh` hbhnĸydh bgaôdhsygedu kŗhrfvfyh ôst yhe cheaĸ kdftkdkr.

• @hrŗĸdĸ` mfdfe kŗdfbf QF@ [HE CHEA (^^C)

afekr vf ehsĸd yhpĸdhckdfnfėk hŗhėĸah mústfrkdbkŗtkr.


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Mfedk` Bgaôdhsygeu

• ^^C bgaôdhsyge yúetfbkeaf kdmkdk kŗhrftdfrke hŗhėĸah vfrkdbkŗtkr.


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A^_‛eke Ôstôedô`dfrk

• Jkdtrf `hrh`tfrkstk`dfrkeaf shpbhdhrĸe gduŗbh`htshyĸdhrĸ lhjĸzhah sh`dhehe k`kdk thcheah `htshyĸdhlhrknk ífvrf vf sĸnh`dĸ` mkck lhrknk phrhbftrfdfrafe f

• Ŀykdfŗtkrkdbkŗ `hdktf sfvkyfsk9 Ŀŗdfbfeke `hdktfsk sha`ĸsĸtdhrahe ft`kdfekr. Zfrkyk/`htshyĸdhrĸ tfbskd ftbf

ckt shyĸsĸ hrttĸrĸdhrh` kstfekdfe `hdktf fdaf fakdfckdkr. Q

hrttĸrbh` ^ES‛ah : aC kykdfŗbfyf efafe gdur.

• Hyeĸ sgeunue tf`rhr fdaf fakdfckdbfsk9 _hrhbftrskstfb pfrjgrbhesĸeĸ ft`kdfbfz. Nklhzĸe úb

`hdkcrhsygeh mfrf` yg`tur.


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A^_‛eke Ôstôedô`dfrk

• Ígėuddhbh9 Hyeĸ aôzfef` ckr`hí kŗhrft thrhjĸeahe

agdhyĸsĸydh f`fgegbk` `hzheĸb shėdheĸr.

• Bgaôdfrdk`9 Mfríf`dfŗtkrbf kíke stheahra shyùddheĸdĸr.

• ZD^K vf VD^K tfègdgikdfrk ùddheĸdhrh` tf` íkp mfrí

• [ô`dfbf ft`kskeke gduŗbhbhsĸ


50/200

A^_‛eke F`sk`dk`dfrk

• Ahlh hz môvfekdkrdk`9 shyĸshd skstfbdfr h`tkj nklhzdh

ahlh jhzdh môí `uddheĸrdhr vf ahlh hz môvfekdkrakrdfr

• ^ĸeĸrdĸ íhdĸŗbh jrf`hesĸ hrhdĸėĸ9 Jrf`hmfríf`dfŗtkrkdfckdfnf` vf `uddheĸdhckdfnf` bh`skbu

`hphsktfskef `hrŗĸdĸ` mfdfe afėfrdfrf sĸeĸrdĸaĸr.

• Hehdgm kŗhrftdfrke kŗdfebfskeaf kdhvf `hrbhŗĸ`dhehdgm-shyĸshd (H/A) vf shyĸshd-hehdgm (A/H) a

tgpdhb skstfb `hrbhŗĸ`dĸėĸeĸ hrttĸrĸr.


51/200

A^_ Vymudhbh Úref`dfrk

• Nfp tfdfjgeu

• Hyrĸ` íg`du tge kdftkb

• ^hyĸshd jgtgėrhj bh`kefsk vf `hbfrh

• ^hyĸshd sfs sfetfzk

• Ŀŗhrft `gadhbh vf sĸ`ĸŗtĸrbh

• Ŀŗhrft Ŀykdfŗtkrbf


52/200

Nfp Qfdfjgeu Cdg` Akyhmrhbĸ


53/200

Ckr Nlkp Ôzfrkeaf`k Nfp Qfdfjgeu

• 8??-3?? BLz

• H^KN dgik`

• [gėue lhjĸzh

• Hehdgm hrhyôz


54/200

^ôrf`dk-zhbhe Ŀŗhrftdfrke Hyrĸ` Ŀŗdfebf

• Ckr sôrf`dk-zhbhe kŗhrftke shyĸshd kŗdfebfsk ôí haĸbahe gd

8. ^ôrf`dk-zhbhe kŗhrftkeke ckr hyrĸ`-zhbhe kŗhrftkef aúeô3. Hyrĸ`-zhbhe kŗhrftke kŗdfebfsk>. Ŀŗdfebkŗ hyrĸ`-zhbhe kŗhrftke tf`rhr ckr sôrf

ífvrkdbfsk

• Ckr sôrf`dk-zhbhe kŗhrftke shyĸshd lhdf ífvrkdbfsk HE(H/A) aúeôŗtôrônôydf yhpĸdĸr.

• ^hyĸshd ckr kŗhrftke hehdgm ckr kŗhrftf aúeôŗtôrôdbf^H[K^HD HEHDGM (A/H) aúeôŗtôrônôydf yhpĸdĸr


55/200


• H/A aúeôŗtôrbf kŗdfbk sgedu ckr zhbhe hdaĸ

thbhbdheĸenhyh `hahr H/A aúeôŗtôrônôeôe mkrkŗkeaf`k shckt `hdbhsĸeĸ mhrhetkdfykp shyĸshd tfbskdkeaf`k lhthyĸ hbhnĸydh ckr ÚSEF@DF-vf -QVQ (^/L) afvrfsk ùddheĸdĸr

• Úrtôŗbfyk úedfbf` hbhnĸydh ^/L afvrfskeafe úenf ckr

ÚEDF[ĿNĿ jkdtrf ùddheĸdĸr.

• A/H aúeôŗtôrônôeôe chshbh` ŗf`dkeaf`k ahdmhŗf`dkyubuŗhtbh` hhbhnĸydh ckr hehdgm [FEĿAFE GDV

ùddheĸdĸr.


56/200


• Hedhtĸdhedhr ĸŗĸėĸeah tgpdhb cdg` akyhmrhb mústfrkdkbk yu`

• Úrtôŗbf úedfyknk vf yfekafe gduŗturbh jkdtrfdfrk hehgdauėueahe, shyĸshd jkdtrf thshrĸbĸeahe úenf hehdgm hthshrĸbĸeĸ fdf hdhnhėĸz.

• Hyrĸnh, fe sĸ` `uddheĸdhe KKS shyĸshd jkdtrf thshrĸb yúe


57/200


• Úenfafe cfdkrtkdakėk mkck, ígėu uymudhbhah hyrĸ`-zhbhzhbhe kŗhrtftdfrke úref`dfebfskydf gduŗturudur.

• Afėkŗk` ckrafe jhzdh sôrf`dk-zhbhe kŗhrftke úref`dfebfskŗhrftkek vfrfckdkr. Hsdĸeah, úref`dfeakėkeaf hyeĸ hyrĸ`-zhsgesuz shyĸah sôrf`dk-zhbhe kŗhrftk vhraĸr.

• Henh`, chzĸ `gŗuddhr hdtĸeah vfrkdfe ckr hyrĸ`-zhbhe kŗhrtf` ckr sôrf`dk-zhbhe kŗhrftk cudbh` bôb`ôeaôr.

• Cu `gŗuddhr shėdheaĸėĸeah, úref`dfebkŗ kŗhrfttfe grkik

kŗhrftkek yfekafe gduŗturbh bôb`ôeaôr. ŕkbak, cu kdkŗ`ky` dd c dk d ėk


58/200

Úref`dfbfeke Jrf`hes Vzhyĸeah`k Ft`k

• m h(t ), t 2 eQ hedhrĸeah úref`dfekp m PeR 2 m h(eQ ) akzkskek gdzhbhe kŗhrftk gdsue. Curhah, Q ÚSEF@DFBFmústfrbf`tfakr.

• Q ‛eke tfrskef ÚSEF@DFBF JSF@HE^K afekr vf J Q kdf m

• m h(t )‛eke jrf`hes uzhyĸ mústfrkdkbk sôrf`dk-zhbhe Jgu

(NQJQ) vfrkdkr9

• Cfezfr ŗf`kdaf, m PeR ‛eke jrf`hes uzhyĸ mústfrkdkbk hyaúeôŗôbôydf (AQJQ) vfrkdkr9


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• Hbhnĸbĸz Mh( i ) kdf M(f i) hrhsĸeah`k kdkŗ`kyk cudb

úref`dfbf kŗdfbkek bhtfbhtk`sfd gdhrh` m h(t ) kdf

fŗktdkėkydf theĸbdhehe pfrkygak` kbpuds akzksk p(t )‛eke hdĸrĸz.

• p(t ), hŗhėĸah mústfrkdakėk mkck Q hrhdĸ`dĸ kafhd kbpudsdhrahe


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• Íhrpbh kŗdfbk sgeunueah m p(t ) kdf mústfrkdfe ckr kbpuds ak

• m p(t ), t 2 eQ hedhrĸeah kbpudsdhrahe gduŗhe ckr sôrf`Ŀbpudsdhrĸe mfedkėkek, úref`dfefnf` hehdgm kŗhrft m h(t )‛e m h(eQ ) cfdkrdfr. Hŗhėĸah ckr hehdgm kŗhrft m h(t ) vf cueh `h

akzksk m p(t ) mústfrkdbkŗtkr.


61/200


• M p( i )‛eĸe fŗafėfr k`k kjhafsk vhraĸr. Ckrkenksk, (t -eQ )‛ekdf vfrkdkr9

• Ŀ`kenk kjhaf, _GK^^GE jgrbôdôeafe yhrhdheĸdhrh` fdaf fakíke _gkssge jgrbôdô, Q 2 3Ϗ /Q vf ( i ) kŗhrftke NQJQ‛s

kdkŗ`kskydf vfrkdkr. _gkssge jgrbôdô, t 2 ? kíke hŗhėĸah vfrkdf


62/200


• NQJQ‛eke jrf`hes útfdfbf úzfddkėkeafe m h(t ) f - i t ‛eke NQ

gdhnh`tĸr. t 2 ? kíke vfrkdfe _gkssge jgrbôdôeaf (t ) 2 mhŗhėĸah vfrkdfe kdkŗ`kyk fdaf fafrkz9

• [u`hrĸah`k kjhafaf yfrkef yhzĸdĸrsh M p( i )‛eĸe

kdf vfrkdfe k`kenk mústfrkdkbk fdaf fakdkr.

• M p( i )‛eĸe kjhafskeafe cu kŗhrftke pfrkygak` ckr


63/200


• M p( i ) fŗktdkėkeke shė thrhjĸeah`k tgpdhbhah ` 2? kíM p( i )‛eĸe QFBFDCHEA `ĸsbĸ, akėfr tôb ` afėfrdfrk kíksf M p( i )‛eĸe jrf`hes útfdfebkŗ `ĸsĸbdhrĸaĸr.

• - Q /3 ≡ ≡

Q /3 jrf`hes hrhdĸėĸeh QFBFDCHEA vfy

afekr.• m h(t )‛eke cheasĸeĸrdĸ ckr kŗhrft gdauėueu vhrshyhdĸb. Q 2 3

hŗhėĸah úref` ckr Mh( i ) vf pfrkygak` kbpuds akzksk p(t )‛íkzkdbkŗtkr.


64/200


Q ‛ eke jhr`dĸ k`k afėfrk kíke M p( i ) hŗhėĸah íkzkdbkŗtkr. ŕf`mkck, Q 5 3 b aurubueah, M p( i )‛yĸ gduŗturhe útfdfebkŗ`fskŗbf yg`tur. Akėfr yheahe, Q 4 3 b aurubueah, Mútfdfebkŗ `gpyhdhr hrhsĸeah `fskŗbf vhraĸr. Cu gdhyh ÚSQÔŕ


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• Hŗhėĸah mústfrkdakėk mkck Q 5 3b ksf, úref`dfebkŗ kŗhrvf b 4 n 4 Q -b fŗktskzdk`dfrkek shėdhyhe n `fskbhdíh` mfíkrfe jkdtrfafe mfíkrkdfrf` hehdgm kŗhrft m h(t ) l

yfekafe gduŗturudhckdkr.

• Akėfr yheahe, Q 4 3b ksf, Mh( i )‛eĸe `fskŗfe útfdfe

agdhyĸ jkdtrfdfbf hrhnĸdĸėĸydh hehdgm kŗhrft m h(t ) lhthsĸz c


66/200


[fekafe gduŗturbh jkdtrfskeke vf kdmkdk kŗhrftdfrke spfvfrkdbkŗtkr.


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Úref`dfbf Qfgrfbk9 m h(t ), NQJQ‛sk || 5 |b| kíke Mh( i ) 2 ckr kŗhrft gdsue. Q 2 3Ϗ /Q gdbh` ôzfrf, Q 5 3b ksf, m m h(eQ )‛afe tf` gdhrh` cfdkrdfefckdkr. { m h(eQ )} vfrkdakėkeaf,

kdf vfrkdfe ckr kbpuds akzksk gduŗturup akzkyk Q `hzheídĸ vf fŗktskzdk`dfrkek shėdhyhe n `fskb jrf`hesdĸ kafhd ckr hd

L r ( i )‛afe mfíkrfrf` m h(t )‛yk lhthsĸz gdhrh` yfekafe gduŗturh

[grubdhr98. Q 5 3b `gŗudueh E[XVK^Q `gŗudu afekr.

3. Q /3 jrf`hesĸeh @HQDHBH jrf`hesĸ afekr.> m (t)‛af bkekbub úref`dfbf jrf`hesĸeĸ cfdkrdfyfe fe


68/200


• ^hyĸshd tfdfjgeah >.< `Lz kŗhrft cheamfekŗdkėk `hcuefafedf, shyĸshd tfdfjge lhcfrdfŗbfskeaf sfs kŗhrftkeke c

`htĸeahe ahlh côyô` gdhe < `Lz úref`dfbf jrf`hesĸ `uddh

• [ô`sf` `hdktfdk hehdgm bôzk` kŗhrftk kŗdfbfaf ksf 3? `Lyftfrdk gdauėu tfspkt fakdbkŗtkr. @gbph`t aks` (NA) bkŗhrft cheamfekŗdkėkeke k`k `htĸeahe ckrhz ahlh côyô

úref`dfbf jrf`hesĸ `uddheĸdbh`thaĸr.


69/200


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Cu kŗhrftdfr, Q 23?Ϗ rha/s úref`dfbf jrf`hesĸeah (Q 2?.8s)Úref`dfbf sgeunueah, vfrkdfe spf`trubh shlkp ôí haft sôrf`dk-zhbhe kbpuds akzkskgdŗur. Akzkdfrke spf`trubdhrĸ hŗhėĸah vfrkdbkŗtkr.


71/200


• m 8(t ) aurubueah, úref`dfbf jrf`ehsĸ Eyqukst `gŗuduúrtôŗbf gduŗbhbĸŗtĸr. Hyrĸnh, hdíh` mfíkrfe jkdtrf íĸ`ĸŗĸe

sôrf`dk-zhbhe kŗhrftf fŗkttkr.

• Akėfr k`k aurubah, úref`dfbf jrf`ehsĸ Eyqukst `gŗudueuLfr k`k aurubah ah hdíh` mfíkrfe jkdtrf íĸ`ĸŗĸ ngs(:Ϗt)‛yf f

• M3p( i )‛ah 2 :Ϗ‛af gduŗhe kbpuds, M3( i )‛ah 2 -úrtôŗbfskeafe `hyeh`dhebĸŗtĸr.

• Cfezfr ŗf`kdaf, M>p( i )‛ah 2 :Ϗ‛af gduŗhe kbpuds, M>( i

kbpudsue úrtôŗbfskeafe `hyeh`dhebĸŗtĸr.


72/200


• ŕkbak, m PeR‛eke AQJQ‛sk kdf m p(t )‛eke NQJQ‛sk hrhsĸeah`kĿ`k spf`trubue theĸbĸ hŗhėĸah vfrkdbkŗtkr.

• [u`hrĸah`k afe`dfbdfraf, m PeR 2 m h(eQ ) kdkŗ`ksk `uddheĸdĸrs

fdaf fakdkr. ̂ ge gdhrh`, hŗhėĸah`k sgeuí cudueur


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• Ckr úenf`k yhesĸah vfrkdfe spf`trub hdtfrehtkj gdhrh`

ŗf`dkeaf kjhaf fakdfckdkr. cudueaurudursh ŗu sgeuí íĸ`hr9

M p( i)‛ah 2 /Q yhzĸdĸrsh, M(f i) fdaf fakdkr.

• M p( i), Q 2 3Ϗ /Q pfrkygadu pfrkygak` ckr jge`skygeaur.

• 2 /Q kdf vfrkdfe aúeôŗôbafe agdhyĸ, M(f i)‛ah pfrkyd k a 3 ‛ak


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Hehdgm Ŀŗhrftke [fekafe Gduŗturudbhs

• ŕkbak, kafhd yfekafe gduŗturbh jkdtrfskeke íĸ`ĸŗĸeah`k^geuí, jkdtrfeke kbpuds yheĸtĸ kdf mkrkŗ hrhsĸeah`k `gevfŗkttkr.

• Jkdtrfeke kbpuds yheĸtĸ, Lr(i)‛eĸe tfrs AQJQ‛sk hdĸehrh` c

• Jkdtrfeke mkrkŗkeke gdauėueu• G lhdaf, jkdtrf íĸ`ĸŗĸ


75/200


[u`hrĸah vfrkdfe afe`dfb, cheasĸeĸrdĸ ckr hrhafėfrdfafe`dfbaf`k skeôzgkahd jge`skygedhrĸe hehdgm kŗhrftk ehsĸd g

vfrkdfe ŗf`kdaf híĸ`dhebhyh íhdĸŗĸdbĸŗtĸr.


76/200


• n 2 Q /3 ksf, lr (?) 2 8 vf e ≯ ? kíke lr (eQ ) 2 ? gdauėumúzdfb ketfrpgdhsyge afe`dfbkeaf `uddheĸdaĸėĸeah - ∕ 4

tôb thbshyĸdhr kíke

sgeunu íĸ`hr.

• [u`hrĸah`k sgeuí, úref`dfbf tfgrfbkeaf vfrkdfe `gŗgdbhaĸėĸeahe chėĸbsĸz gdhrh` agėruaur.

• Henh`, úref`dfbf jrf`hesĸ Q úref`dfbf tfgrfbkeaf`kka d t d k t a ú `d d ` t


77/200

Chea Mfíkrfe Ŀŗhrftdfrke Úref`dfebfs

• ŕkbakyf `hahr yhpĸdhe thrtĸŗbhah sôrf`dk-zhbhe kŗhrftke cheasĸeĸrdĸ ckr kŗhrft gdauėu vhrshyĸdaĸ. Cúydf sôrf`dHDÍH@ MFÍĿSFE kŗhrft afekr.

• Ŀŗhrftke ahlh yô`sf` ckr Ϋ D4 |Ϋ| 4 Ϋ L jrf`hes hrhdĸėĸehuymudhbhdhr vhraĸr. Cúydf kŗhrftdfrf CHEA MFÍĿSFE kŗ

• Cheamfíkrfe kŗhrftdfr úref`dfekr`fe ΫQ ≦ 3 Ϋ L sfíkdfrf` úHenh` cu yh`dhŗĸbĸe úefbdk k`k sĸeĸrdhbhsĸ vhraĸr9 (k) fakdfe hyrĸ`-zhbhe kŗhrftke spf`trubueah spf`trhd cgŗduΫ D côyô` ckr afėfrf shlkpsf mfrf`dk úref`dfbf jrf`he

chzĸ uymudhbhdhrah prhtk` gdbhyhckdkr.


78/200


• ΑΫ 2 Ϋ D - Ϋ L chea mfíkrfe kŗhrftke cheamfekŗdkėkek theĸb• Ŀŗhrfttf`k fe yô`sf` jrf`hes Ϋ L ‛ ĸe cheamfekŗkdkėkeke

gdauėueu vhrshyhdĸb. [hek, B ckr thbshyĸ gdbh` ôzfrf Ϋ L

• Úref`dfbf jrf`hesĸeĸ ΫQ 2 3(ΑΫ) 2 3Ϋ L /B gdhrh` sjrf`hesĸeĸe Eyqukst jrf`hesĸeĸe B ‛af ckrkef fŗkt gdauėueh

• Úref`dfbf jrf`hesĸeĸe cu afėfrk M p( iΫ) kíke ahlh úenfŗktdk`tf yfrkef `geudursh M p( iΫ) hŗhėĸah`k ŗf`kdaf fdaf fa

• Hdíh` mfíkrfe kŗhrftdfrke úref`dfebfskeaf gdauėu mkck,

cheamfíkrfe kŗhrftke spf`trubueue údíf`dfebkŗ vf útfdftgpdhbĸeahe gduŗbh`thaĸr


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Hŗhėĸah`k ŗf`kd thrtĸŗbhyh híĸ`dĸ` mftkrbf`tfakr.

•

Hehdgm kŗhrft m h(t ), m p(t )‛eke mfíkrbf cheaĸ Ϋ D4 |Ϋ| 4 Ϋ L cheamfíkrfe ckr jkdtrfafe mfíkrkdbfskydf fdaf fakdfckdkr


80/200

Hehdgm Hdíh` Mfíkrfe Jkdtrf @hrh`tfrkstk`dfrk

Hehdgm hdíh` mfíkrfe ckr jkdtrfeke mfedk` yheĸtĸ |Lh(iΫ)| hŗhėĸah mústfvfrkdfckdkr.

Ϋ p9 Mfíkrbf cheaĸ `fehr jrf`hesĸΫs9 ^úeaôrbf cheaĸ `fehr jrf`hesĸα p9 Mfíkrbf cheaĸ bh`skbub ahdmhdhebhsĸαs9 ^úeaôrbf cheaĸ b h`skbub ahdmhdhebhsĸ

Ahdmhdhebhdhr aC nkeskeafe af vfrkdfckdkr9

Bh`skbub mfíkrbf cheaĸ ahdmhdhebhsĸ9


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? ≡ Ϋ ≡ Ϋ p jrf`hes hrhdĸėĸ MFÍĿSBF CHEAK gdhrh` theĸbdheĸr.

Mfíkrbf cheaĸeah |Lh(iΫ)| 8‛afe αp `hahr shphckdkr. [hek

Ϋs ≡ Ϋ ≡ ∕ jrf`hes hrhdĸėĸ ^ÚEAÔSBF CHEAK gdhrh` theĸbdheĸr.

^úeaôrbf cheaĸeah |Lh(iΫ)| 8‛afe αs `hahr shphckdkr. [hek


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Mfedk` `hrh`tfrkstk`dfrk hŗhėĸah mústfrkdakėk mkck mfedkėke mfíkrbf cheaĸeafėfrk 8 gdhnh` ŗf`kdaf af vfrkdfckdkr.

Bh`skbub mfíkrbf cheaĸ shpbhsĸ

Bh`skbub súaôrbf cheaĸ

mfedkėk

Mfefdaf k`k phrhbftrf ahlh theĸbdheĸr9

Mfíkŗ grheĸ9 Hyĸrtfaknkdk` phrhbftrfsk9


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Cuttfrwgrtl [h`dhŗĸ`dĸėĸ

E. afrfnfafe hehdgm Cuttfrwgrtl hdíh` mfíkrfe jkdtrfskeke mfedk` yhehŗhėĸah`k fŗktdk`df vfrkdkr9

Ϋ 2 ?‛ ah |Lh(iΫ)|3 eĸe kd` 3E‟8 tôrfvk sĸj ĸrh fŗkttkr.

aC nkeskeafe @HYHEÍ

gdhrh` vfrkdkr.

Ϋn >‟ac @F^ĿB JSF@HE^K gdhrh` hadheaĸrĸdĸr. Cu hadheaĸrbheĸe efaf`hzhenĸe yh`dhŗĸ` gdhrh` ‟>aC gdbhsĸaĸr9


84/200


Ϋn 2 8 aurubueah ífŗktdk E afėfrdfrk kíke Cuttfrwgrtl yh`dhŗĸ`dĸėĸ kdfyheĸtdhrĸ hŗhėĸah vfrkdbkŗtkr9


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Cuttfrwgrtl hdíh` mfíkrfe jkdtrfskek theĸbdhyhe k`k phrhbftrf afrfnjrf`hesĸ Ϋn akr.

Cu phrhbftrfdfr, cfdkrtkdfe `fehr jrf`hesdhrĸ Ϋ p vf Ϋs kdf mfíkrbf c

mfedkėk vf bh`skbub súaôrbf cheaĸ

ahdmhdhebhsĸ

eahe hŗhėĸ

ah`k a`uddheĸdhrh` lfshpdhehckdkr9

Ŀ`k afe`dfb E kíke íúzôdôrsf


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Afrfnf thbshyĸ gdauėueahe, lfshpdhehe E afėfrk fe yh`ĸe thbshyĸyh

E afėfrk lfrlhemk ckr afe`dfbaf yfrkef `geudhrh` >‟aC `fskb jrf`hes

Hehdgm Cuttfrwgrtl hdíh` mfíkrfe jkdtrfskeke trhesjfr jge`skygeu

ŗf`dkeaf vfrkdkr. Afe`dfbaf

d a dk A (s)‛ E A a C tt tl _ dk a


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Úref`9 8‟aC `fskb jrf`hesĸ 8 `Lz‛af vf 6 `Lz‛af`k zhyĸjdhtbhsĸ 1? aC afrfnfdk Cuttfrwgrtl hdíh` mfíkrfe cfdkrdfykekz.

Zfrkdfedfrafe

yhzhrĸz. Ŀ`k afe`dfbk íúzfrf`

cuduruz. Afrfnf lfshcĸ kíke mfrf`dk gdhe 8/` vf 8/`8 phrhbftrfdfrk lf

fdaf fakdkr. G lhdaf, afrfnf


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Nlfcyslfv [h`dhŗĸ`dĸėĸ

E. afrfnfafe hehdgm 8. tôr Nlfcyslfv hdíh` mfíkrfe jkdtrfskeke mfedk` yhŗhėĸah`k fŗktdk`df vfrkdkr9

_hyahah`k Q E(Ϋ)‛yf E. afrfnfafe Nlfcyslfv _gdkegbu afekr vf

fŗktdkėkydf theĸbdheĸr.


89/200


Ífŗktdk E afėfrdfrk kíke 8. tôr Nlfcyslfv yh`dhŗĸ`dĸėĸ kdf fdaf fakdfe hŗhėĸah vfrkdbkŗtkr9


90/200


Ϋ 2 Ϋs af mfedk` 8/ H gdauėueahe

yhzĸdhckdkr. [u`hrĸah`k afe`dfb E kíke íúzôdôrsf

cudueur. Jkdtrfeke afrfnfsk thbshyĸ gdbh` zgrueah gdauėueahe, lfshpthbshyĸyh yuvhrdheĸr.


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E. afrfnfafe hehdgm 3. tôr Nlfcyslfv hdíh` mfíkrfe jkdtrfskeke mfedk`hŗhėĸah`k fŗktdk`df vfrkdkr9

_hyahah`k Q E(Ϋ) tfrkbk E. afrfnfafe Nlfcyslfv pgdkegbuaur. 8. tôryh`dhŗĸ`dĸėĸeah gdauėu mkck, 3. tôr Nlfcyslfv hdíh` mfíkrfe jkdtrfskeke

fŗktdkėk `uddheĸdhrh` lfshpdhehe shyĸ fe yh`ĸe thbshyĸyh yuvhrdhehrh`


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Ífŗktdk E afėfrdfrk kíke 3. tôr Nlfcyslfv yh`dhŗĸ`dĸėĸ kdf fdaf fakdfe mfhŗhėĸah vfrkdbkŗtkr9


93/200


Úref`9 8‟aC `fskb jrf`hesĸ 8 `Lz‛af vf 6 `Lz‛af`k zhyĸjdhtbhsĸ 1? aC gmfíkrfe jkdtrfeke fe `ôíô` afrfnfskek cfdkrdfyfdkb.

Zfrkdfe afėfrdfrafe afrfnf

gdhrh` lfshpdheĸr. ^geuí yuvhrdheĸrsh E 2 > gdur.


94/200


95/200

Fddkptk` [h`dhŗĸ`dĸėĸ

Úref`9 8‟aC `fskb jrf`hesĸ 8 `Lz‛af vf 6 `Lz‛af`k zhyĸjdhtbhsĸ 1? aC gmfíkrfe jkdtrfeke fe `ôíô` afrfnfskek cfdkrdfyfdkb.

` 2 ?.3 vf 8/` 8 2 8;:.68>1 gdhrh` cuduebuŗtu. Cu afėfrdfrk afrfnf jg

`gyhrsh`

cuduruz. ^geuí gdhrh` afrfnf E2 3.3>>? sfí


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Fddkptk` [h`dhŗĸ`dĸ`

Ϋ p 2 8 aurubueah ífŗktdk E afėfrdfrk kíke fddkptk` yh`dhŗĸ` kdf fdaf fahŗhėĸah vfrkdbkŗtkr9


97/200

BHQDHC @uddheĸdhrh` Hehdgm Hdíh` Mfíkrfe Jkdtrf Qh

Úref`9 8‟aC `fskb jrf`hesĸ 8 `Lz‛af vf 6 `Lz‛af`k zhyĸjdhtbhsĸ 1? aCmfíkrfe jkdtrfyk BHQDHC `uddhehrh` thshrdhyhdĸb.

Mfrf`dk `gbutdhr

ŗf`dkeafakr. @gbutdhr yhzĸdĸr`fe ]p 2 3Ϗ8???, ]s 2 3Ϗ6???, Sp 2 8,`hzheí mrhjkėk hŗhėĸah vfrkdfe jkdtrf thshrdhebĸŗ gdur.


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Hehdgm [ô`sf` Mfíkrfe, Chea Mfíkrfe vf Chea ^úeaôrfe J

Haĸb 89 Vymue ckr jrf`hes aúeôŗôbô `uddheĸdhrh` mfrf`dk jkdtrf trhesjprgtgtkp ckr hdíh` mfíkrfe trhesjfr jge`skygeu LD_(s)‛f aúeôŗtôrôdôr.

Haĸb 39 _rgtgtkp hdíh` mfíkrfe jkdtrf thshrdheĸr.

Haĸb >9 Mfrf`dk jkdtrf trhesjfr jge`skygeu LA(s), LD_(s)‛yf jrf`hes aúeuymudhehrh` fdaf fakdkr.

_rgtgtkp hdíh` mfíkrfe trhesjfr jge`skygeu LD_(s)‛eke Dhpdhnf aúeôŗô

mfrf`dk jkdtrf trhesjfr jge`skygeu LA(Œ )‛eke`k ksf Œ kdf cfdkrtkdske. s‟uhzyaúeôŗôb tfrsk bfvnut gdhe s JŒ jge`skygeu kdf vfrkdkr. G lhda


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Hehdgm [ô`sf` Mfíkrfe Jkdtrf Qhshrĸbĸ

Mfrf`dk spf`trhd aúeôŗôb9 LD_(s)‛ke mfíkrbf cheaĸ `fehr jrf`hesĸ Ϋ mfíkrbf cheaĸ `fehr jrf`hesĸ Ϋ p

U gbh` ôzfrf

kdf vfrkdkr. s 2 i, Œ iŶ yhzĸdhrh` jrf`hesdhr hrhsĸeah`k kdkŗ`k

gdur. Jrf`hes aúeôŗôbô hŗhėĸah mrhjk`sfd gdhrh` mústfrkdbkŗtkr.


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Hehdgm [ô`sf` Mfíkrfe Jkdtrf Qhshrĸbĸ

Úref`9 Hŗhėĸah`k `hrh`tfrkstk`dfrf shlkp ckr hehdgm Cuttfrwgrtl yô`sfthshrdhyhdĸb9 Ŷp 1 `Lz, Ŷs 8 `Lz, θp ?.8 aC, θs 1? aC.

Ϋ p 2 8 sfífdkb.

G lhdaf, prgtgtkp hdíh` mfíkrfe jkdtrfeke `hrh`tfrkstk`dfrk ŗúydfakr9 Ϋ p 2

θs 1? aC. Hŗhėĸah BHQDHC `gbutdhrĸ vf prgtgtkp hdíh` mfíkrfe j

yô`sf` mfíkrfe jkdtrfeke `hzheídhrĸ íkzkdbkŗtkr.


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Hehdgm Chea Mfíkrfe Jkdtrf Qhshrĸbĸ

Mfrf`dk spf`trhd aúeôŗôb9 LD_(s)‛ke mfíkrbf cheaĸ `fehr jrf`hesĸ Ϋ p cheaĸ `fehr jrf`hesdhrĸ Ϋ p8

U vf Ϋ p3U gdbh` ôzfrf


gdur. C w

cheamfíkrfe jkdtrfeke mfíkrbf cheaĸ mfekŗdkėk, Ϋ?

U ksf mfíkrjrf`hesĸaĸr. Jrf`hes aúeôŗôbô hŗhėĸah mrhjk`sfd gdhrh` mústfrkdbkŗtkr.

hdíh` mfíkrfe


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Hŗhėĸah vfrkdfe `gŗud shėdhebhdĸaĸr9

Cu `gŗud shėdhebĸygrsh, `gŗud shėdhehnh` ŗf`kdaf jrf`hesdhrahe ckrksk

Ŀ`k aurubdh `hrŗĸdhŗbh` bôb`ôeaôr9

Aurub 89

Aurub 39

Lfr k`k aurubah fŗktdkėke ehsĸd shėdhehckdfnfėk hŗhėĸah thrtĸŗĸdbĸŗtĸr.


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Aurub 89

Hŗhėĸah mústfrkdakėk mkck súeaôrbf cheaĸ `fehr jrf`hesdhrĸeahe ckrksk hmfíkrbf cheaĸ `fehr jrf`hesdhrĸeahe ckrksk hzhdtĸdhckdkr.

Úrefėke, Ϋ p8U , afėfrkef aôŗôrôdfckdkr. Cu aurubah, mfíkr

mfekŗdfyfnf` vf sgdah`k mfíkŗ cheaĸ ahrhdhnh`tĸr. Cfezfr ŗf`kdaf, Ϋs8U ,

afėfrkef yô`sfdtkdfckdkr. Cu `fz, mfíkrbfcheaĸ mfekŗdkėk afėkŗbfz`fe sg

ahrhdhnh`tĸ

r.


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Aurub 39

Hŗhėĸah mústfrkdakėk mkck súeaôrbf cheaĸ `fehr jrf`hesdhrĸeahe ckrksk hmfíkrbf cheaĸ `fehr jrf`hesdhrĸeahe ckrksk hrttĸrĸdhckdkr.

Úrefėke, Ϋ p3U , afėfrkef yô`sfdtkdfckdkr. Cu aurubah, mfík

mfekŗdfyfnf` vf shėah`k mfíkŗ cheaĸ ahrhdhnh`tĸr. Cfezfr ŗf`kdaf, Ϋs3U ,

afėfrkef hzhdtĸdhckdkr. Cu `fz, mfíkrbfcheaĸ mfekŗdkėk afėkŗbfz`fe shėaahrhdhnh`tĸr.


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Úref`9 @hrh`tfrkstk`dfrk hŗhėĸah vfrkdfe hehdgm ckr fddkptk` chea mfíkrfŶ p8 1 `Lz, Ŷ p3 = `Lz, Ŷ s8 > `Lz, Ŷ s3 < `Lz θ p 8 aC, θ s

Ŷ p8 Ŷ p3 3.1=863< `Lz

afėfrkef aôŗôrôdôr. _rgtgtkp hdíh` mfíkrfe jkdtrf kíke Ϋ p 2 8 sfífdkbjgrbôdôeafe súeaôrbf cheaĸ `fehr jrf`hesĸ

gdhrh` lfshpdheĸr. G lhdaf, prgtgtkp hdíh` mfíkrfe jkdtrfeke `hrh`tfrkst

Ϋ 2 8, Ϋ 2 8.1, θ 8 aC, θ 33 aC.


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Mfrf`dk BHQDHC `gadhrĸ vf fdaf fakdfe fėrkdfr hŗhėĸah vfrkdbkŗtkr.


107/200

Hehdgm Chea ^úeaôrfe Jkdtrf Qhshrĸbĸ

Mfrf`dk spf`trhd aúeôŗôb9 LD_(s)‛ke súaôrbf cheaĸ `fehr jrf`hesĸ Ϋssúeaôrbf cheaĸ `fehr jrf`hesdhrĸ Ϋs8

U vf Ϋs3U gdbh` ôzfrf


C w

cheasúeaôrfe jkdtrfeke súeaôrbf cheaĸ mfekŗdkėk, Ϋ?U ksf súaô

jrf`hesĸaĸr. Jrf`hes aúeôŗôbô hŗhėĸah mrhjk`sfd gdhrh` mústfrkdbkŗtkr.

H


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Hŗhėĸah vfrkdfe `gŗud shėdhebhdĸaĸr9

Cu `gŗud shėdhebĸygrsh, `gŗud shėdhehnh` ŗf`kdaf jrf`hesdhrahe ckrksk

Ŀ`k aurubdh `hrŗĸdhŗbh` bôb`ôeaôr9

Aurub 89

Aurub 39

Lfr k`k aurubah fŗktdkėke ehsĸd shėdhehckdfnfėk hŗhėĸah thrtĸŗĸdbĸŗtĸr.


109/200


Aurub 89

Hŗhėĸah mústfrkdakėk mkck súeaôrbf cheaĸ `fehr jrf`hesdhrĸeahe ckrksk hmfíkrbf cheaĸ `fehr jrf`hesdhrĸeahe ckrksk hzhdtĸdhckdkr.

Úrefėke, Ϋ p3U , afėfrkef aôŗôrôdfckdkr. Cu aurubah, mfíkr

mfekŗdfyfnf` vf sėah`k mfíkŗ cheaĸ ahrhdhnh`tĸr. Cfezfr ŗf`kdaf, Ϋs3U ,

yô`sfdtkdfckdkr. Cu `fz, mfíkrbfcheaĸ mfekŗdkėk afėkŗbfz`fe shėah`k mfí

Egt9 Fŗktdk` `gŗudu Ϋp8U hzhdtĸdhrh` vfyh Ϋs8U hrttĸrĸdhrh` ah shėdhehckdkr


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Hyrĸ`‟Jgurkfr Aúeôŗôbô

Qheĸb9 ? ≡ e ≡ E‟8 hrhdĸėĸeah theĸbdĸ E‟uzuedu`du ckr akzk x PeR‛eke H[SK@AÚEÔŕÔBÔ (AJQ), hyrĸ`‟zhbhe Jgurkfr aúeôŗôbô (AQJQ) \ (f iϋ)‛eĸe ?ϋ` 2 3Ϗ` /E , ` 2 ?,8, …, E‟8 eg`thdhrĸeah úref`dfebfskydf fdaf fakdkr.

AQJQ‛eke theĸbĸeahe AQJQ kdf AJQ hrhsĸeah`k kdkŗ`k

ŗf`dkeaf gdhnh`tĸr. ] E 2 f‟ i 3ϏE egthsygeu `uddheĸdhrh` AJQ mfefdaf


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Qfrs AJQ jgrbôdôeôe agėruduėueu mústfrbf` kíke fŗktdkėke lfr k`k yheĸ e afėfrdfrk ôzfrkeafe tgpdheĸrsh

cudueur.

kdkŗ`kskeafe tgpdhbĸeah`k sĸj ĸrahe jhr`dĸ tf` tfrkb ` 2 d k`f


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Úref`9 Hŗhėĸah vfrkdfe akzkeke AJQ‛skek lfshpdhyhdĸb.

Úref`9 Hŗhėĸ

ah vfrkdfe E‟uzuedu`du akzkeke AJQ‛skek lfshpdhyhdĸ

b.


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Úref`9 Hŗhėĸah vfrkdfe ? ≡ e ≡ E‟8 hrhdĸėĸeah theĸbdĸ E‟uzuedu`du akzkelfshpdhyhdĸb. (? ≡ r ≡ E‟8)

Fudfr kdkŗ`ksk `uddheĸdhrh` mPeR hŗhėĸah`k ŗf`kdaf yfekafe yhzĸdhckdkr9

G lhdaf, mPeR‛eke E‟eg`th AJQ‛sk

gdur. kdkŗ`kskek `uddh


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AJQ‛eke Zf`túr‟Bhtrks Egthsygeueah Mústfrkdbfsk

AJQ, vf`túr bhtrks egthsygeueah hŗhėĸah`k ŗf`kdaf kjhaf fakdfckdkr9


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AJQ‛eke Zf`túr‟Bhtrks Egthsygeueah Mústfrkdbfsk

Cfezfr ŗf`kdaf, tfrs AJQ vf`túr bhtrks egthsygeueah hŗhėĸah`k ŗf`kdaf

AE‟8 bhtrkskef, QFS^ AJQ‟BHQSĿ^Ŀ afekr. Ŀ`k bhtrks hrhsĸeah hŗhėĸah vf


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BHQDHC @uddheĸdhrh` AJQ Lfshpdhebhsĸ

• BHQDHC‛af AJQ vf tfrs AJQ lfshpdhbh` kíke bfvnut `gbutdhr9 jjt

• Cu jge`skygedhr, agėruahe lfshpdhbh yfrkef ft`ke lfshp yô`ôeô ghdmgrktbhsĸeĸ `uddhebh`thaĸr.

• Hŗhėĸah x PeR 2 ngs(:Ϗe/8:), ? ≡ e ≡ 86 akzkskeke AJQ vf AQJQ‛skeke


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AQJQ‛eke AJQ‛afe Hrhafėfrdfbfydf Lfshpdhebhs

E‟uzuedu`du x PeR akzkeke AJQ‛sk \ P` R, akzkeke AQJQ‛sk \ (f iϋ)‛eĸe aôzmôe h` 2 ?,8, …, E‟8 eg`thdhrĸeah`k jrf`hes úref`dfrkakr. G lhdaf, E‟uzueduAJQ‛sk \ P` R vfrkdakėkeaf akzkeke AQJQ‛sk \ (f iϋ) \ P` R‛ahe cfdkrdfefckdkr.

r hŗhėĸah`k ŗf`kdaf theĸbdhesĸe.


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AQJQ‛eke AJQ‛afe Hrhafėfrdfbfydf Lfshpdhebhs

[u`hrĸah`k theĸbdhrĸ `uddhehrh`

vfyh fŗafėfr gdhrh`

yhzhckdkrkz. G lhdaf,

^geuí gdhrh` AQJQ kdf AJQ hrhsĸeah`k kdkŗ`k ŗúydfakr9


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AQJQ‛eke Úref`dfebfsk

AQJQ‛sk \ (f iϋ) gdhe x PeR akzkskek fdf hdhdĸb. \ (f iϋ)‛yĸ aôzmôe hrhdĸ`dĸ ϋ` 2E‟8 eg`thdhrĸeah úref`dfyfrf` E haft \ (f iϋ`) jrf`hes úrefėk fdaf fafjrf`hes úrefėk E‟eg`th tfrs AJQ‛sk E‟uzuedu`du y PeR akzksk gdhe E‟eg`thaôŗôeôdfckdkr. Hŗhėĸah \ (f iϋ) vf [ P` R vfrkdbkŗtkr.

Hbhnĸbĸz, y PeR kdf x PeR hrhsĸeah`k kdkŗ`kyf lfshpdhbh`tĸr. [ P` R‛yh tfrs Afdaf fakdkr9


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y PeR kdf x PeR hrhsĸeah`k kdkŗ`kyk cudbh` kíke hŗhėĸah fŗktdk`tfe yhrhdhnhėĸz

Cu sgeuí, úenf`k shyah`k `úŗfdk phrhetfz kíkeaf`k tfrkb yfrkef `geudurscudueur9

^geuí9 y PeR akzksk x PeR akzkskeke sgesuz shyĸah útfdfebkŗ `gpydhrĸeĸe tgpakzkskeke E‛eke thbshyĸ `htdhrĸ `hahr útfdfebfskydf fdaf fakdkr vf tgpdhbhrhdĸėĸeah ak``htf hdĸeĸr.

Jgrbôdô sgedu uzuedu`du akzkdfrf uymudhbh` kíke, cfdkrtkdfe hrhdĸ` aĸŗĸeagdauėueu vhrshyhrĸz G lhdaf B ≡ E gdhnh` ŗf`kdaf xPeR B‟uzuedu`du c


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Henh`, B 5 E gdauėueah, úrtôŗbf bfyaheh mfdkr vf x PeR y PeR‛afe fdaf

Úref`9 Hŗhėĸah vfrkdfe akzkyk fdf hdhdĸb9

Akzkeke AQJQ‛sk \ (f iϋ)‛yĸ ϋ` 2 3Ϗ` /1 , ` 2 ?,8,3,> eg`thdhrĸeah úref`dfyfcu akzkyf 1‟eg`th afrs AJQ uymudhyhrh`

cuduruz. [hek, { x PeR} {y PeR}‛afe lfshpdhehbhz.


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AJQ @uddheĸdhrh` AQJQ Lfshpdhebhsĸ

B 55 E gdbh` ôzfrf, \ (f iϋ)‛yĸ ϋ` 2 3Ϗ` /B , ` 2 ?,8,…,B‟8 jrf`hesdhrĸekstkygruz.

x fPeR akzksk ŗúydf theĸbdhesĸe9

\ (f iϋ)‛yĸ x fPeR akzksk nkeskeafe yhzbh` bôb`ôeaôr9


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125/200

AJQ Úzfddk`dfrk


126/200

AJQ Úzfddk`dfrk


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Ckr Akzkeke Ahkrfsfd Útfdfbfsk

? ≡ e ≡ E‟8 hrhdĸėĸeah theĸbdĸ E‟uzuedu`du ckr akzkyk fdf hdhdĸb. x

útfdfebkŗ akzk x 8PeR 2 x Pe‟e?R ? ≡ e ≡ E‟8 hrhdĸėĸeah theĸbdĸ gútfdfebkŗ akzk ahkbh ? ≡ e ≡ E‟8 hrhdĸėĸeah theĸbdĸ gdhnh` theĸbdhbhbĸz mfrf`dkakr. AHĿSF^FD ÚQFDFBF gdhrh` hadheaĸrĸdhegpfrhtúrô `uddheĸdhrh`

ŗf`dkeaf theĸbdheĸr. e? 5 ? ksf (shė thrhjh ahkrfsfd útfdfbf), yu`hrĸah`

hedhbĸeh mfdkr. E‟uzuedu`du akzk ckr ahkrf ôzfrkeaf fŗkt hrhdĸ`dĸ E eg`ahkrfsfd útfdfbf gpfrhtúrô akzkeke shht yúeôeaf vfyh shht yúeôe


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Ckr Akzkeke Ahkrfsfd Útfdfbfsk


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Ahkrfsfd @gevgdôsyge

E‟ uzuedu`du mPeR vf lPeR akzkdfrkek fdf hdhdĸb. Ŀ`k akzkeke agėrushd `gevuzuedu`du y DPeR akzkskek vfrkr9

y DPeR akzksk lfshpdheĸr`fe k`k akzkeke uzuedu`dhrĸ (3E‟8) gdhnh` ŗf`dfeakėk vhrshyĸdbĸŗtĸr. y DPeR akzksk, lPeR akzkskeke tfrs ífvrkdbfsk vf gduŗturdur. E‟uzuedu`du ckr akzk vfrfe `gevgdôsygeh cfezfr ckr kŗahkrfsfd útfdfbf `uddhebh` mfrf`dkakr. ^geuídhehe kŗdfbf AHĿSF^FD hŗhėĸah`k fŗktdk`df theĸbdheĸr9

Ŀŗdfbf, E‟

uzuedu`du k`k akzk kífrakėkeafe E‟eg`th ahkrsfd `gevgdôsyge


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Úref`9 1‟uzuedu`du hŗhėĸah`k k`k akzkeke ahkrfsfd `gevgdôsygeueu lfsh

? ≡ e ≡ >


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Ahkrfsfd `gevgdôsyge AJQ `uddheĸdhrh` ah lfshpdhehckdkr. [u`hrĸah vfr`uddhehrh` íúzfdkb.


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Úref`9 [u`hrĸah vfrkdfe k`k akzkyk sgedhrĸeh sĸj ĸr f`dfyfrf` = uzuedu`du mmfPeR vf lfPeR gduŗturhdĸb vf mfekŗdftkdbkŗ akzkdfrke = uzuedu`du ahkrfsfdlfshpdhyhdĸb.


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AJQ kdf Agėrushd @gevgdôsygeue Lfshpdhebhsĸ

E vf B uzuedu`du mPeR vf lPeR akzkdfrkek fdf hdhdĸb. D 2 B+E‟8 sgedhrĸeh sĸj ĸr f`dfyfrf` D uzuedu`du mfekŗdftkdbkŗ akzkdfr mfPeR vf lfPe

G lhdaf,

Cdg` akyhmrhb mústfrkdkbk hŗhėĸah vfrkdbkŗtkr9


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E‟eg`th ahkrfsfd `gevgdôsyge vf`túr‟bhtrks egthsygeueah hŗhėĸah`k

Bhtrkske lfrlhemk ckr `úŗfmfekeaf`k tôb fdfbhedhrĸe fŗkt gdauėueh aCúydf bhtrksdfrf AHĿS^FD BHQSĿ^ afekr.


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Ahkrfsfd @gevgdôsyge Qhcdg [úetfbkydf Lfshpdheb

[úetfb, 1‟uzuedu`du k`k akzkeke ahkrfsfd `gevgdôsygeu kíke ckr úref`dfúenf akzkdfrke úref`dfrk hŗhėĸah mústfrkdakėk mkck íhrpĸdĸr.

d d cd d d


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Ahkrfsfd @gevgdôsyge Qhcdg [úetfbkydf Lfshpdheb

Ahkrfsfd útfdfbfafe sgerh afėkŗtkrkdbkŗ thcdg ŗu ŗf`kdaf gdur9

{y nPeR} akzkskeke fdfbhedhrĸ, úrefėf `hrŗĸdĸ` mfdfe sôtueah`k 1 íhrpĸb tg

M d Ak kd k AJQ‛ k k L d


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Mfrífd Akzkdfrke AJQ‛skeke Lfshpdhebhsĸ

_rhtk` ígėu uymudhbhah, kdmkdfekdfe kŗhrftdfr mfrífdakr. Cúydf kŗhrftdfúzfddk̀ dfrkeafe yhrhrdheĸdhrh` AJQ‛eke lfshpdhebhsĸ `gdhydhŗtĸrĸuzuedu`du mPeR vf lPeR akzkdfrkeke E‟eg`th AJQ‛sk hŗhėĸah mústfrkdakėkAJQ kdf lfshpdhehckdkr. Ŀd` úenf `hrbhŗĸ` x PeR 2 mPeR + i lPeR akzkskek the

mPeR 2 Sf{ x PeR}, lPeR 2 Kb{ x PeR}

AJQ‛eke `hrbhŗĸ` kŗhrftdfr kíke skbftrk úzfddk`dfrkeafe mfrf`dk AJQ‛dflfshpdhehckdkr9

? ≡ ` ≡E‟8 kíke gdauėueh ak``ht fa

M d Ak kd k AJQ‛ k k L d


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Mfrífd Akzkdfrke AJQ‛skeke Lfshpdhebhsĸ

Úref`9 Hŗhėĸah`k akzkdfrke 1‟eg`th AJQ‛dfrkek lfshpdhyhdĸb.

M d Ck Ak k k 3E `t AJQ‛ k k E `t AJQ kd L


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Mfrífd Ckr Akzkeke 3E‟eg`th AJQ‛skeke E‟eg`th AJQ kdf L

3E‟uzuedu`du v PeR akzkskeke 3E‟eg`th AJQ‛skek lfshpdhbh` kstkygruz. Evfrkdfe k`k akzk theĸbdhyhdĸb9

mPeR 2 v P3eR, lPeR 2 v P3e+8R , ? ≡ e ≡ E

M d Ck Ak k k 3E `t AJQ‛ k k E `t AJQ kd L


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Úref`9 Hŗhėĸah vfrkdfe


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^hyĸshd Jkdtrf Qhshrĸbĸ


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•

^hyĸshd jkdtrf thshrĸbĸeah hbhí, vfrkdfe ckr jrf`hes yheĸtĸeĸ yh`dhŗĸ` gdhrh` shėdhyhe mfríf`dfefckdkr ckr trhesjfr jge`skygeuM( z ) fdaf ftbf`tkr.

• Ígėu uymudhbhah shyĸshd jkdtrfeke thshrĸbĸ kíke mfedk` vf/vfyh

jhz yheĸtĸ cfdkrtkdkr. Chzĸ aurubdhrah, kbpuds vfyh chshbh` yheĸtĸ cfdkrtkdfckdkr.

• _rhtk` ígėu uymudhbhah, vfrkdfe ckr mfedk` yheĸtĸeĸ yh`dhŗĸ` gdhrh` shėdhyhe mfríf`dfefckdkr ckr trhesjfr jge`skygeu fdafftbf` kstfekr. Cu efafedf, cu afrstf ckz shafnf mfedk`yh`dhŗĸ`dĸėĸeĸ fdf hdhnhėĸz.

Jkdtrf @hrh`tfrkstk`dfrkeke Cfdkrtkdbfsk


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•

Aúrt tôr kafhd jkdtrfyf `hrŗĸdĸ` mfdfe kbpuds yheĸtdhrĸeĸe efafesfdgdbhaĸ`dhrĸeahe vf sgesuz uzuedu`du gdau`dhrĸeahe kafhd jkdtrfdfrmfríf`dfefbfz.

• _rhtk`tf, shyĸshd ckr jkdtrfeke mfedk` yheĸtĸ mfíkrbf vf súeaôrbf

cheaĸeah `hcud fakdfckdkr tgdfrhesdhrdh cfdkrtkdkr. Hyrĸnh, mfíkrbf vf súeaôrbf cheadhrĸ hrhsĸeah ckr mfíkŗ cheaĸ vhraĸr .

• Úrefėke, shyĸshd hdíh` mfíkrfe ckr jkdtrfeke mfedk` yheĸtĸ |M(f i)|hŗhėĸah mústfrkdbkŗtkr .



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p

9 mfíkrbf cheaĸ `fehr jrf`hesĸ s 9 súeaôrbf cheaĸ `fehr jrf`hesĸ p 9 mfíkrbf cheaĸeah`k bh`skbub ahdmhdhebh s 9 súeaôrbh cheaĸeah`k bkekbub ahdmhdhebh



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•

@hrh`tfrkstk`dfr mfefdaf aC gdhrh`

H() 2 -3?dgm8? |M(f i)|

kdf theĸbdhehe `hyĸp jge`skygeu nkeskeafe vfrkdkr.

• Cfezfr ŗf`kdaf, aC nkeskeafe mfíkrbf cheaĸ bh`skbubahdmhdhebhsĸ p vf súeaôrbf cheaĸ bkekbub zhyĸjdhtbhsĸ s

p 2 -3?dgm8? (8- p)

s 2 -3?dgm8? ( s)

gdhrh` lfshpdheĸr .



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Jkdtrf `hrh`tfrkstk`dfrk hdtfrehtkj gdhrh` hŗhėĸah`k mústfrkdakėk mkck cfdkrtkdfckdkr. Hdtfrehtkj mústfrkdkbaf, mfedkėke mfíkrbf cheaĸeah`k bh`skbub afėfrkeke 8 gdauėu vhrshyĸdr .



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• _rhtk`tf, mfíkrbf cheaĸ `fehr jrf`hesĸ J p vf súeaôrbf cheaĸ

`fehr jrf`hesĸ J s Lz nkeskeafe cfdkrtkdkr.

• ^hyĸshd jkdtrf thshrĸb jgrbôddfrkeaf mfíkrbf vf súaôrbf cheaĸ `fehr jrf`hesdhrĸ rhayhe nkeskeafe gdauėu vhrshyĸdaĸėĸeahe Lznkeskeafe vfrkdfe jrf`hesdhrĸe rhayhe nkeskeafe fŗafėfrdfrk

lfshpdhebhdĸaĸr . Úref`dfbf jrf`hesĸeĸe J Q gdauėu vhrshyĸdĸrsh, `fehr jrf`hesdhrĸ hŗhėĸah`k fŗktdk`dfr `uddheĸdhrh` lfshpdhehckdkr9

Jkdtrf Qôrôeôe ^fíkdbfsk


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• Cfdkrtkdfe jrf`hes yheĸtĸ úzfddk`dfrkek shėdhyhe trhesjfrjge`skygeu L ( z ) efafesfd ckr trhesjfr jge`skygeu gdbhdĸaĸr .

• ^gesuz kbpuds yheĸtdĸ (KKS) jkdtrf aurubueah trhesjfr jge`skygeu

ŗf`dkeaf mfrífd ckr rhsygefd jge`skygeaur. Cu aurubah, L ( z )`hrhrdĸ gdbheĸe yheĸeah lfshp yô`ôeô fe hzh keakrbf` kíke

`ôíô` afrfnfyf ( E ) shlkp gdbhdĸaĸr .

Jkdtrf Qôrôeôe ^fíkdbfsk


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• ^gedu kbpuds yheĸtdĸ (JKS) jkdtrf aurubueah trhesjfr jge`skygeu

ŗf`dkeaf mfrífd `htshyĸdĸ ckr pgdkegbaur.

• Lfshp `hrbhŗĸ`dĸėĸeĸe hz gdbhsĸ kíke trhesjfr jge`skygeueueafrfnfsk ( E ) bôb`ôe gdauėu `hahr `ôíô` gdbhdĸaĸr . L ( z ) `utupkífrbfakėkeafe JKS jkdtrfdfrke `hrhrdĸdĸ` prgcdfbk yg`tur.

•

Agėrushd jhz kstfekygrsh, jkdtrf `htshyĸdhrĸeĸe lPeR 2 ∞ l PE -e Rkdkŗ`kskek shėdhbhsĸ mfrf`dkakr.

^hyĸshd Jkdtrf Qhshrĸbĸ9 Qfbfd [h`dhŗĸbdhr


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• Fe sĸ` `uddheĸdhe KKS jkdtrf thshrĸb yúetfbk hŗhėĸah`k haĸbdhrahe

gduŗur 9

8.^hyĸshd jkdtrf `hrh`tfrkstk`dfrk prgtgtkp ckr hehdgm hdíh` mfíkrfe jkdtrf `hrh`tfrkstk`dfrkef aúeôŗtôrôdôr .

3.Hehdgm hdíh` mfíkrfe jkdtrf thrhesjfr jge`skygeu L h( s) cfdkrdfekr.

>.L h( s), mfrf`dk shyĸshd trhesjfr jge`skygeu M(z)‛yf aúeôŗtôrôdôr .

• Cu yh`dhŗĸbĸe `uddheĸdbhsĸeĸe efafedfrk ŗúydf sĸrhdhehckdkr 9hehdgm jkdtrf thshrĸb yúetfbdfrk gdau`íh mfdkŗbkŗ gdup mfefdafhehdktk` íúzôbdf sgeuídheĸrdhr . Cu efafedf, hehdgm jkdtrf thshrĸbĸ kíke thcdgdhr bfvnuttur. Ŀdhvf gdhrh`, ígėu uymudhbh hehdgmskstfbdfrke shyĸshd skbôdhsygeueu mfrf`tkrbf`tfakr.



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• —h‒ hehdgm uzhyĸ cfdkrtbf` ôzfrf, hehdgm trhesjfr jge`skygeu

gdhrh` cfdkrtkdfnf`tkr.

• L h( s)‛afe tôrftkdfe shyĸshd trhesjfr jge`skygeu ah hŗhėĸah`k mkcktfbskd fakdfnf`tkr9

• L h( s), M( z )‛yf aúeôŗtôrbf`, hehdgm jrf`hes yheĸtĸeĸe tfbfd`hrh`tfrkstk`dfrk `gruehnh` ŗf`kdaf s-uzhyĸeahe z -uzhyĸeh ckraúeôŗôb uymudhbh`tĸr . G lhdaf, aúeôŗôb `hrhrdĸ ckr hehdgmtrhesjfr jge`skygeueu `hrhrdĸ ckr shyĸshd trhesjfr jge`skygeuehaúeôŗtôrbfdkakr .



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• JKS jkdtrf thshrĸbĸ, cfdkrtkdfe mfedk` yheĸtĸeĸe agėruahe yh`dhŗĸ`dĸėĸeh ahyhdĸaĸr . Hyrĸnh, mfefdaf jkdtrfeke agėrushd jhzhshlkp gdbhsĸ kstfekr.

• E . afrfnfafe ckr JKS jkdtrfeke thshrĸbĸ, yh ( E +8)-uzuedu`dukbpuds yheĸtĸ `htshyĸdhrĸ {lPeR}, yh ah jrf`hes yheĸtĸ |M(f i)|‛eĸe ( E +8) úrefėk cuduehrh` yhpĸdhckdkr . Fe sĸ` `uddheĸdhe JKS jkdtrfthshrĸb yúetfbdfrk ŗúydfakr 9

8. _fenfrfdfebkŗ Jgurkfr sfrksk yh`dhŗĸbĸ 3. Jrf`hes úref`dfbf yh`dhŗĸbĸ >. Ckdmkshyhr thchedĸ gptkbkzhsyge yúetfbdfrk

KKS ^hyĸshd Jkdtrf Qhshrĸbĸ9 Íkjt Agėrushd Aúeôŗôb


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• Íkjt agėrushd aúeôŗôb (ÍAA) s-uzhyĸeah`k ckr eg`thyĸ z -

uzhyĸeah`k ckr eg`thyh aúeôŗtôrôr vf hŗhėĸah`k fŗktdk`df vfrkdkr9

• G lhdaf, M( z ) kdf L h( s) hrhsĸeah`k kdkŗ`k ŗúydf gdur9

• ^hyĸshd jkdtrf thshrĸbĸ ôí haĸbahe gduŗur 9 (k) M( z )‛eke `hrh`tfrkstk`dfrkef tfrs ÍAA uymudheĸp L

h

( s)‛eke `hrh`tfrkstk`dfrkfdaf fakdkr, (kk) L h( s) cfdkrdfekr, (kkk) L h( s)‛yf ÍAA uymudheĸp M( z )

cfdkrdfekr. Aúeôŗôb jgrbôdôeaf`k Q phrhbftrfskeke ft`kskgdbhaĸėĸeahe, mfefdaf Q 2 3 sfíkdkr .



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• Q 2 3 kíke tfrs ÍAA jgrbôdô `gdhydĸ`dh fdaf fakdfckdkr9

• s 2 ?+ i? yhzĸp, s kdf z hrhsĸeah yu`hrĸah vfrkdfe fŗktdk`tfe

fdaf fakdkr. ?‛ĸe jhr`dĸ afėfrdfrk kíke z ‛eke mfedk`dfrk hŗhėĸah vfrkdbkŗtkr .

KKS ^hyĸshd Jkdtrf Qhshrĸbĸ9 Íkjt Agėrushd Aúeôŗôb • Hŗhėĸah mústfrkdakėk mkck, sgd yhrĸ s-aôzdfbk, `hrbhŗĸ` z -


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ė m ė m yaôzdfbkeaf ckrkb ífbcfrke kíkef, shė yhrĸ s-aôzdfbk ckrkb

ífbcfrke aĸŗĸeh, i-f`sfek af ckrkb ífbcfrf aúeôŗbôŗtôr .

• s-aôzdfbkeaf `hrhrdĸdĸ` `gŗudu, `utupdhrĸe sgd yhrĸ s-aôzdfbkeaf, z -aôzdfbkeaf `hrhrdĸdĸ` `gŗudu ksf `utupdhrĸe ckrkb ífbcfr kíkeaf gdbhsĸaĸr . G lhdaf, ÍAA `hrhrdĸ ckr hehdgm trhesjfr jge`skygeueu`hrhrdĸ ckr shyĸshd trhesjfr jge`skygeueh aúeôŗtôrbf`tfakr .


ŕkbak af hehdgm jrf`hes kdf shyĸshd jrf`hes hrhsĸeah`k kdkŗ`kyk


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ŕkbak af hehdgm jrf`hes kdf shyĸshd jrf`hes hrhsĸeah`k kdkŗ`kyk cfdkrdfyfdkb. ÍAA kdkŗ`kskeaf (Q 23 kíke) s2 i, z 2f i yhzĸdĸrsh

cudueur. Cu kjhaf aôzfedfekrsf, 2 the (/3) fdaf fakdkr vf hŗhėĸah mústfrkdakėk mkck hrhdhrĸeah agėrushd gdbhyhe ckr kdkŗ`k vhraĸr



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Hehdgm jrf`hes kdf shyĸshd jrf`hes hrhsĸeah`k agėrushd gdbhyhe kdkŗ`k jrf`hes f`sfekeaf JSF@HE^ CÔ@BF^Ŀ afefe ckr

cgzueub gduŗturur . Jrf`hes cô`bfskeke ft`ksk hŗhėĸah mústfrkdbkŗtkr 9


^ d jkdt t a `k a d ú d ú td ckdk


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• ^hyĸshd jkdtrf thshrĸbĸeah`k haĸbdhr ŗúydf úzftdfefckdkr 9

8. ( p, s) jrf`hesdhrĸeh úe cô`bf kŗdfbk uymudhehrh` (tfrsÍAA `uddhehrh`) hehdgm `hrŗĸdĸ`dhrĸ ( p, s) cudueur.

3. Hehdgm jkdtrf thshrdhehrh` `hrŗĸdĸ` mfdfe trhesjfr jge`skygeu L h( s) fdaf fakdkr.

>. L h( s)‛yf ÍAA umudhehrh` shyĸshd jkdtrfyf `hrŗĸdĸ` mfdfetrhesjfr jge`skygeu M( z ) cfdkrdfekr.

• ÍAA, shafnf phríhdĸ shckt afėfrdk mfedk` yheĸtdĸ shyĸshd jkdtrfthshrĸbĸeah `uddheĸdhckdkr .

• Aúeôŗôb, hehdgm jkdtrfeke jhz yheĸtĸeĸ `grubhz. Akėfr ckrafykŗdf, hehdgm jkdtrfeke jhz yheĸtĸ aúeôŗôb sgeueah cgzudhckdkr.


Ú ` H ė a kd d ` k C tt tl d


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• Úref` 9 Hŗhėĸah vfrkdfe hdíh` mfíkrfe Cuttfrwgrtl hehdgmtrhesjfr jge`skygeueu fdf hdhdĸb9

• L h( s)‛yf ÍAA uymudheĸrsh hdíh` mfíkrfe Cuttfrwgrtl shyĸshd trhesjfr jge`skygeu fdaf fakdkr9

• Ŀjhaf yfekafe aôzfedfekrsf hŗhėĸah`k kjhaf fdaf fakdkr9


• Úref`9 |L (i )| ? |L (i?)| |L (i )| 8 d ` ô ė a


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• Úref` 9 | L h( i?)|2?, | L h ( i?)|2 | L h ( i∕)|28 gdbh` ôzfrf, hŗhėĸah vfrkdfe 3. afrfnf hehdgm ífetk` trhesjfr jge`skygeueu fdf hdhdĸb9

• ?‛h ÍFEQĿ@ JSF@HE^K afekr. | L h( i3)| 2 | L h( i8)| 28/3 ksf,

C2 3- 8‛yf >-aC ÍFEQĿ@ CHEAMFEĿŕDĿĕĿ afekr. L h( s)‛yf ÍAA uymudhehrh` `hrŗĸdĸ` mfdfe shyĸshd ífetk` trhesjfrjge`skygeu hŗhėĸah`k ŗf`kdaf fdaf fakdkr9


• Úref` 9 Ífetk` jrf`hesĸ :? Lz, >-aC ífetk` cheamfekŗdkėk Lz


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Í , í m ŗ ėgdhe vf 1?? Lz úref`dfbf jrf`hesĸeah íhdĸŗhe 3. afrfnf shyĸshd

ífetk` jkdtrf thshrdhyhdĸb.• ?23Ϗ(:?/1??)2?.>Ϗ, Cw23Ϗ(:/1??)2?.?>Ϗ fdaf fakdkr. vf γ

lfshpdheĸrsh 2?.;?;;>, γ2?.6


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ŗ ė p í m íCuttfrwgrtl shyĸshd jkdtrf thshrdhyhdĸb9 p2?.36Ϗ, s2?.66Ϗ,

p2?.6 aC, s286 aC.

• Zfrkdfedfrafe 32?.833?88.:33=== cudueur. Qfrs ÍAA `uddheĸdhrh` `hrŗĸdĸ` mfdfe hehdgm jrf`hesdhr

gdhrh` fdaf fakdkr. 8/` vf 8/` 8

ŗf`dkeaf fdaf fakdkr. Afrfnf lfshpdheĸrsh cudueur. E 2> sfíkdkr .


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L h( s)‛yf ÍAA uymudheĸrsh mfrf`dk shyĸshd trhesjfr jge`skygeu fdaf


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h ) y ym yfakdkr.

M( z )‛eke mfedk` vf `hzheí yheĸtdhrĸ hŗhėĸah íkzkdbkŗktkr .

[M, CM vf C^ KKS ^hyĸshd Jkdtrf Qhshrĸbĸ

Cu hbhídh k`k yh`dhŗĸb bfvnuttur. Ŀ`k yh`dhŗĸbĸe ah afthydhrĸ


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hŗhėĸah vfrkdbkŗtkr 9

• Ckrkenk yh`dhŗĸb

8. Mfrf`dk shyĸshd jkdtrf M A( z )‛eke jrf`hes `hrh`tfrkstk`dfrkef úe cô`bf uymudhehrh` hyeĸ tôr hehdgm jkdtrf L A( s)‛eke jrf`hes

`hrh`tfrkstk`dfrk cfdkrdfekr.3. L A( s)‛eke `hrh`tfrkstk`dfrk uymue ckr jrf`hes aúeôŗôbôydf

prgtgtkp hdíh` mfíkrfe jkdtrf L D_ ( s)‛yf aúeôŗtôrôdôr .>. Hehdgm hdíh` mfíkrfe jkdtrf L D_ ( s) thshrdheĸr .1. Ŀ`kenk haĸbah `uddheĸdhe jrf`hes aúeôŗôbôeôe tfrsk

`uddheĸdhrh` L D_ ( s), L A( s)‛yf aúeôŗtôrôdôr .6. L A( s)‛yf ÍAA uymudhehrh` mfrf`dk shyĸshd jkdtrf M A( z ) fdaf

fakdkr.

(QFS^ ÍAA)


Ŀ`k k `d


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• Ŀ`kenk yh`dhŗĸb

8. Mfrf`dk shyĸshd jkdtrf M A( z )‛eke jrf`hes `hrh`tfrkstk`dfrkef úe cô`bf uymudhehrh` hyeĸ tôr hehdgm jkdtrf L A( s)‛eke jrf`hes`hrh`tfrkstk`dfrk cfdkrdfekr.

3. L A( s)‛eke `hrh`tfrkstk`dfrk uymue ckr jrf`hes aúeôŗôbôydf

prgtgtkp hdíh` mfíkrfe jkdtrf L D_ ( s)‛yf aúeôŗtôrôdôr .>. Hehdgm hdíh` mfíkrfe jkdtrf L D_ ( s) thshrdheĸr .1. L D_ ( s)‛yf ÍAA aúeôŗôb uymudhehrh` shyĸshd hdíh` mfíkrfe

trhesjfr jge`skygeu M D_ ( z ) fdaf fakdkr.

6. ^hyĸshd uzhyah uymue ckr jrf`hes aúeôŗôbô `uddheĸdhrh`

M D_ ( z ), mfrf`dk shyĸshd jkdtrf M A( z )‛yf aúeôŗtôrôdôr .

(QFS^ ÍAA)


• Úref` 9 8. yh`dhŗĸbĸ `uddhehrh` hŗhėĸah`k `hrh`tfrkstk`dfrf shlkp


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8. tôr Nlfcyslfv KKS shyĸshd yô`sf` mfíkrfe jkdtrf thshrdhyhdĸb9

J p2=?? Lz, J s26?? Lz, p28aC, s2>3 aC, J Q 23 `Lz.• Ŀd` úenf, rhayhe nkeskeafe chea `fehr jrf`hesdhrĸ lfshpdheĸr 9

• ^gerh, úe cô`bfydf `hrŗĸdĸ` mfdfe hehdgm jrf`hesdhr cudueur9

• _rgtgtkp hdíh` mfíkrfe hehdgm jkdtrf kíke Ϋ p

2 8 sfíkdkp kdkŗ`kskeafe Ϋ s 2 8.;:38?6 cudueur.

• G lhdaf, hehdgm hdíh` mfíkrfe jkdtrf `hrh`tfrkstk`dfrk ŗúydf fdaffakdbkŗ gdau9 Ϋ p 2 8, Ϋ s 2 8.;:38?6, p28aC, s2>3 aC.


Jkdt t a ` dd d BHQDHC ` td ` td


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• Jkdtrf thshrĸbĸeah `uddheĸdhe BHQDHC `gbutdhrĸ vf `gbutdhr

íhdĸŗtĸrĸdhrh` fdaf fakdfe `hzheí mrhjkėk hŗhėĸah vfrkdbkŗtkr .


• Úref` 9 Hŗhėĸah`k `hrh`tfrkstk`dfrf shlkp Cuttfrwgrtl KKS shyĸshd


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chea mfíkrfe jkdtrf thshrdhyhdĸb9 ϋ p82?.16Ϗ, ϋ p32?.:6Ϗ, ϋ s82?.>Ϗ

ϋ s32?.=6Ϗ , p28aC, s21? aC.

• Úe cô`bfydf `hrŗĸdĸ` mfdfe hehdgm jrf`hesdhr lfshpdheĸr 9

• Cheamfekŗdkėk gdauėueahe

gdup íhrpĸbdhr fŗkt gdhnh` ŗf`kdaf chea `fehrdhrĸeĸe afėkŗtkrkdbfsk mfrf`dkakr. ^fíkdkrsf íhrpĸbdhr fŗkt gdur.


• _rgtgtkp hdíh` mfíkrfe hehdgm jkdtrf kíke Ϋ p 2 8 sfíkdkp


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kdkŗ`kskeafe

• G lhdaf, hehdgm hdíh` mfíkrfe jkdtrf `hrh`tfrkstk`dfrk ŗúydf fdaffakdbkŗ gdau9 Ϋ p 2 8, Ϋ s 2 3.>:8=:3=, p28aC, s21? aC.

• Jkdtrf thshrĸbĸeah `uddheĸdhe BHQDHC `gbutdhrĸ vf `gbutdhríhdĸŗtĸrĸdhrh` fdaf fakdfe `hzheí mrhjkėk hŗhėĸah vfrkdbkŗtkr .



176/200


• Úref` 9 Hŗhėĸah`k `hrh`tfrkstk`dfrf shlkp fddkptk` KKS shyĸshd chea


177/200

súeaôrfe jkdtrf thshrdhyhdĸb9 ϋ p82?.>Ϗ, ϋ p32?.=6Ϗ, ϋ s82?.16Ϗ

ϋ s32?.:6Ϗ , p28aC, s21? aC.

• Úe cô`bfydf `hrŗĸdĸ` mfdfe hehdgm jrf`hesdhr lfshpdheĸr 9

• Cheamfekŗdkėk gdauėueahe

gdup íhrpĸbdhr fŗkt gdhnh` ŗf`kdaf chea `fehrdhrĸeĸe afėkŗtkrkdbfsk mfrf`dkakr. ^fíkdkrsf íhrpĸbdhr fŗkt gdur.


• _rgtgtkp hdíh` mfíkrfe hehdgm jkdtrf kíke Ϋ s 2 8 sfíkdkp


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kdkŗ`kskeafe

• G lhdaf, hehdgm hdíh` mfíkrfe jkdtrf `hrh`tfrkstk`dfrk ŗúydf fdaffakdbkŗ gdau9 Ϋ s 2 8, Ϋ p 2 ?.13>183:, p28aC, s21? aC.

• Jkdtrf thshrĸbĸeah `uddheĸdhe BHQDHC `gbutdhrĸ vf `gbutdhríhdĸŗtĸrĸdhrh` fdaf fakdfe `hzheí mrhjkėk hŗhėĸah vfrkdbkŗtkr .

?.6==>?> gdhnh`



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KKS ^hyĸshd Jkdtrfdfrke ^pf`trhd Aúeôŗôbdfrk

• Hbhí9 Zfrkdfe ckr hdíh` mfíkrfe shyĸshd trhesjfr jge`skygeuM ( )‛ k d ` ô` ` c a k c a ú aô ck


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M D( z )‛yk hdíh`, yô`sf`, cheamfíkrfe vfyh cheasúeaôrfe ckr

jkdtrfyf `hrŗĸdĸ` mfdfe akėfr ckr trhesjfr jge`skygeu yfaúeôŗtôrbf` .

• @hrĸŗĸ`dĸėĸ úedfbf` kíke prgtgtkp hdíh` mfíkrfe jkdtrfeke trhesjfrjge`skygeueah chėĸbsĸz afėkŗ`fe kíke z , mfrf`dk jkdtrfeke`k kíke ksf `uddheĸdbĸŗtĸr .

• Ŀ`k uzhyah`k ckrkb ífbcfrdfr vf uzhydhr hrh

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