qrg_ece.pdf

7/21/2019 QRG_ECE.pdf

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_ZH@M ^JG^JPFJ^ AZHDJ

Gkr

Jcj`trkeh`s

@kiiueh`othke Jeahejjrhea

Ny

www.tfjaotjo`odjiy.`ki
http://www.thegateacademy.com/http://www.thegateacademy.com/http://www.thegateacademy.com/http://www.thegateacademy.com/http://www.thegateacademy.com/http://www.thegateacademy.com/http://www.thegateacademy.com/


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_uh`m ^jgrjsfjr Auhdj @ketjets

TFJ AOTJ O@ODJIX RWT.CTD. F.K.< #14, MjsfovoMrupo (tfhrd Gckkr), 02tf

@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??

< 262-7:1221:2,hegkBtfjaotjoòdjiy.`ki @kpyrhaft rjsjrvjd. Qjn< www.tfjaotjoòdjiy.`ki Roaj.H

@ K E T J E T P

Rort Roaj Ek.

?. Iotfjiotfh`s ? 4>

?.? Chejor Ocajnro ? 6?.> Rrknonhchty & dhstrhnuthke 8 ?4

?.0. Euijrhòc Ijtfkd ?: ?8

?.4. @ocùcus >2 02?.:. Dhggjrjethoc Jquothkes 0? 01

?.7. @kipcjx Worhoncjs 06 4>

>. Ejtwkrm Tfjkry 40 78

>.? Ejtwkrm Pkcuthke Ijtfkdkckay 40 48

>.> Troeshjet/Ptjody Ptotj Oeocyshs kg ^C@ @hrùhts tk d` Heput :2 :4

>.0 Pheuskhdoc Ptjody Ptotj Oeocyshs :: 7>>.4 Troesgjr Gue`thke kg oe CTH Pystji 70 74>.: Twk Rkrt Ejtwkrms 7: 77

>.7 Ejtwkrm Tkpkckay 71 78

0. Phaeocs & Pystjis 12 61

0.? Hetrkdu`thke tk Phaeocs & Pystjis 12 1>

0.> Chejor Thij Hevorhoet (CTH) Pystjis 10

140.0 Gkurhjr jprjsjetothke kg Phaeocs 1: 110.4 \-Troesgkri 16 62

0.: Copco`j Troesgkri 6? 60

0.7 Grjquje`y rjspkesj kg CTH systjis oed Dhvjrshghjd Tkph`s 64 61

4. @ketrkc Pystji 66 ??4

4.? Nosh`s kg @ketrkc Pystji 66 824.> Thij Dkiohe Oeocyshs 8? 84

4.0 Ptonhchty & ^kutf Furwhtz @rhtjrhke 8: 87

4.4 ^kkt Ckùs Tj`fehquj 81 864.: Grjquje`y jspkesj Oeocyshs ushea Eyquhst Rckt 88

?2?

4.7 Grjquje`y jspkesj Oeocyshs ushea Nkdj Rckt ?2> ?24

4.1 @kipjesotkrs & @ketrkccjrs ?2: ??24.6 Ptotj Worhoncj Oeocyshs ??? ??4

:. Dhahtoc @hrùhts ??: ?06

:.? Euijnr Pystjis & @kdj @kevjrshkes ??: ??7

:.> Nkkcjoe Ocajnro & Moreouaf Iops ??1 ??6:.0 Ckah` Aotjs ??8 ?>>

:.4 Ckah` Aotj Goihchjs ?>0

?>4:.: @kinheothkeoc Dhahtoc @hrùhts ?>: ?>8


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_uh`m ^jgrjsfjr Auhdj @ketjets


@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??

< 262-7:1221:2,hegkBtfjaotjoòdjiy.`ki @kpyrhaft rjsjrvjd. Qjn< www.tfjaotjoòdjiy.`ki Roaj.HH

:.7 OD /DO @kevjrtkr ?02

:.1 Pjih`kedu`tkr Ijikry ?0?

:.6 Hetrkdu`thke tk Ih`rkprk`jsskrs ?0>

?06

7. Oeocka @hrùhts ?08 ?12

7.? Dhkdj @hrùhts - Oeocyshs oed Oppchòthke ?08 ?4:7.> D@ Nhoshea-NLTs ?47 ?:2

7.0 Piocc Phaeoc Ikdjchea Kg NLT oed GJT ?:? ?::

7.4 Kpjrothkeoc Oipchghjrs oed Tfjhr Oppchòthkes ?:7 ?:87.: Gjjdno`m oed Ks`hccotkr @hrùhts Gjjdno`m Oipchghjrs ?72 ?7?

7.7 Rkwjr Oipchghjrs ?7> ?707.1 NLT oed LGJT Grjquje`y jspkesj ?74 ?12

1. Jcj`trkeh`s Djvh`js & @hrùhts ?1? ?84

1.? Pjih`kedu`tkr Tfjkry ?1? ?11

1.> R-E Lue`thke Tfjkry oed @foro`tjrhsth`s ?16 ?6>1.0 Troeshstkr Tfjkry (NLT, GJT) ?60 ?8>1.4 Nosh`s kg KRTK Jcj`trkeh`s ?80 ?84

6. @kiiuehòthke Pystjis ?8: >28

6.? Nosh`s kg @kiiuehòthke Phaeocs ?8:6.> Oipchtudj Ikducothke (OI) ?87 ?816.0 DPNP@ oed PPN Ikducothke ?86 ?88

6.4 Wjsthahoc Phdjnoed (WPN) Ikducothke >226.: Oeacj Ikducothke >2? >24

6.7 Pupjrfjtrkdyej ^j`jhvjrs >2:

6.1 Ekhsj he Oeocka Ikducothke >276.6 Dhahtoc @kiiuehòthkes >21 >28

8.

Jcj`trkioaejth` Tfjkry >?2 >>:

8.? Jcj`trkioaejth` Ghjcd >?2 >?4

8.> JI Qovj Rrkpoaothke >?: >?1

8.0 Troesihsshke Chejs >?6 >?8

8.4 Auhdjd J.I Qovjs >>2

>>>8.: Oetjeeos >>0 >>:

^jgjrje`j Nkkms >>7 ->>1


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_uh`m ^jgrjsfjr Auhdj Iotfjioth`s

TFJ AOTJ O@ODJIX RWT.CTD. F.K.< #14, Mjsfovo Mrupo (tfhrd Gckkr), 02tf

@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??

< 262-7:1221:2,hegkBtfjaotjoòdjiy.`ki @kpyrhaft rjsjrvjd. Qjn< www.tfjaotjoòdjiy.`ki Roaj ?

Rort -?< Iotfjioth`s

?.?

Chejor Ocajnro

?.?.?

Iotrhx

Djghehthke< O systji kg i e euinjrs orroeajd ockea i rkws oed e `kcuies.@kevjethkeoccy, sheacj òphtoc cjttjr hs usjd tk djektj o iotrhx.

Tfus,

O 9

o o o o o o o

ooo o o o

ohtfrkw, ltf`kcuie?.?.?.?

Typjs kg Iotrh`js

?.?.?.>

^kw oed @kcuie Iotrh`js

^kw Iotrhx ] >, 1, 6, 8Y sheacj rkw ( kr rkw vj`tkr) @kcuie Iotrhx

??? sheacj `kcuie (kr `kcuie vj`tkr)

?.?.?.0

Pquorj Iotrhx

Poij euinjr kg rkws oed `kcuies.

Krdjr kg Pquorj iotrhx ek. kg rkws kr `kcuies Rrhe`hpcj Dhoakeoc (kr Iohe dhoakeoc kr Cjodhea dhoakeoc)


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@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??

< 262-7:1221:2,hegkBtfjaotjo`odjiy.`ki @kpyrhaft rjsjrvjd. Qjn< www.tfjaotjo`odjiy.`ki Roaj >

Pyiijtrh` Iotrhx 9 O Pmjw Pyiijtrh` Iotrhx9 - O

?.?.?.7

Zeht Iotrhx (kr Hdjethty Iotrhx)

O Dhoakeoc iotrhx he wfh`f occ tfj cjodhea dhoakeoc jcjijets orj ?.j.a. H9 ? ? ??.?.?.1

Eucc Iotrhx (kr \jrk Iotrhx)

O iotrhx hs sohd tk nj Eucc Iotrhx hg occ tfj jcjijets orj zjrk.

j.a. 2 ?.?.?.6

Pyiijtrh` oed Pmjw Pyiijtrh` Iotrh`js

Zehtory Iotrhx< Hg wj djghej, O 9 O 9 troespksj kg o `keluaotj kg iotrhx OTfje tfj iotrhx hs uehtory hg O. O 9 H


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@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??

< 262-7:1221:2,hegkBtfjaotjo`odjiy.`ki @kpyrhaft rjsjrvjd. Qjn< www.tfjaotjo`odjiy.`ki Roaj 0

?.?.?.?0

Fjrihthoe Iotrhx< Ht hs o squorj iotrhx whtf `kipcjx jetrhjs wfh`f hs jquoc tk hts kwe

`keluaotj troespksj.

O9 O kr o 9 o ?.?.?.?4 Ektj< He Fjrihthoe iotrhx, dhoakeoc jcjijets ocwoys rjoc?.?.?.?:

Pmjw Fjrihthoe iotrhx.Hg O hs o squorj iotrhx kg krdjr e, tfje |m O| 9|O|.?0.|H|9 ? (Hhs tfj hdjethty iotrhx kg krdjr e).

?.?.?.>?

Hevjrsj kg o Iotrhx

O9 || |O| iust nj eke-zjrk (h.j. O iust nj eke-sheaucor).

Hevjrsj kg o iotrhx, hg jxhsts, hs ocwoys uehquj. Hg ht hs o >x> iotrhx 2o n` d, hts hevjrsj whcc nj 2 d n` o

Hipkrtoet Rkhets.

2 O 9 O 2 9 2, (Fjrj 2 hs eucc iotrhx)

0. Hg ON 9 , tfje ht hs ekt ej`jssorhcy tfot O kr N hs eucc iotrhx. Ocsk ht dkjset ijoe NO 9 .4. Hg tfj prkdu`t kg twk eke-zjrk squorj iotrh`js O & N hs o zjrk iotrhx, tfje O & N orj

sheaucor iotrh`js.:.

Hg O hs eke-sheaucor iotrhx oed O.N92, tfje N hs eucc iotrhx.

7. ON

NO (he ajejroc)

@kiiutothvj prkpjrty dkjs ekt fkcd

1.

O(N@) 9 (ON)@ Ossk`hothvj prkpjrty fkcds6. O(N+@) 9 ON O@ Dhstrhnuthvj prkpjrty fkcds


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@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??

< 262-7:1221:2,hegkBtfjaotjoòdjiy.`ki @kpyrhaft rjsjrvjd. Qjn< www.tfjaotjoòdjiy.`ki Roaj :

8. O@ 9 OD , dkjset hipcy @ 9 D jvje wfje O -.

?2.

Hg O, @, D nj

iotrhx, oed hg roem (O)9 e & O@9OD, tfje @9D.

??.

(O+N)T9 O+ N?>.(ON)T9 N. O?0.

(ON)-?9 N. O?4.

O O9 OO 9 H?:.

(mO)T9 m.O (m hs sòcor, O hs vj`tkr)?7.

(mO)-?9 . O (m hs sòcor , O hs vj`tkr)?1.

O 9 O?6.

(O ) 9 (O (@keluaotj kg o troespksj kg iotrhx9 Troespksj kg `keluaotj kg iotrhx)?8.

Hg o eke-sheaucor iotrhx O hs syiijtrh`, tfje Ohs ocsk syiijtrh`.>2.

Hg O hs o krtfkakeoc iotrhx , tfje O oed Oorj ocsk krtfkakeoc.>?.

Hg O hs o squorj iotrhx kg krdjr e tfje (h) |odl O|9|O|(hh) |odl (odl O)|9|O|(hhh) odl (odl O) 9|O|O?.?.?.>>

Jcjijetory Troesgkriothke kg o Iotrhx chejs

>. Iucthpchòthke kg o chej ny o `kestoet (j.a. m )0. Oddhthke kg `kestoet iucthpchòthke kg oey chej tk tfj oektfjr chej (j.a. + p )

Ektj0

ôem kg Iotrhx

Hg wj sjcj`t oey r rkws oed r `kcuies grki oey iotrhx O,djcjthea occ ktfjr rkws oed `kcuies,

tfje tfj djtjriheoet gkrijd ny tfjsj rr jcjijets hs òccjd ihekr kg O kg krdjr r.Djghehthke


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@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


7. Hg O hs o eke-sheaucor iotrhx, tfje occ tfj rkw/`kcuie vj`tkrs orj hedjpjedjet.

1. Hg O hs o sheaucor iotrhx, tfje vj`tkrs kg O orj chejorcy djpjedjet.

6.

r(O)92 hgg (hg oed kecy hg) O hs o eucc iotrhx.8. Hg twk iotrh`js O oed N fovj tfj soij shzj oed tfj soij roem tfje O, N orj jquhvocjet

iotrh`js.

?2.Jvjry eke-sheaucor iotrhx hs rkw iotrhx oed ht hs jquhvocjet tk hdjethty iotrhx.

?.?.?.>4 Pkcuthke kg chejor Pystji kg Jquothkes7

Wj`tkrRrknonhchty oed Dhstrhnuthke

?.>.?

Rrknonhchty

Jvjet


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@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??

< 262-7:1221:2,hegkBtfjaotjoòdjiy.`ki @kpyrhaft rjsjrvjd. Qjn< www.tfjaotjoòdjiy.`ki Roaj ?2

Gkr hedjpjedjet jvjets O & N R(N/O) 9 R(N)Tfjkrji kg @kinhejd Rrknonhchty.

(?-R)(?-R prknonhchty kg nktf jvjet gohcs0. ?-(?-R)(?-R prknonhchty kg ot cjost kej jvjet k`ùr4. RR prknonhchty kg nktf jvjet k`ùrsoyjs tfjkrji.>

Dhstrhnuthke

Rrknonhchty Djeshty Gue`thke (RDG) kr Rrknonhchty Ioss Gue`thkeon kv(x,y) (hg ekt hedjpjedjet) ]O & N orj`kestoetsY9 oWor(x) +nWor(y) (hg hedjpjedjet)@kvorhoe`j

@kv (x,y)9J(xy)-J(x) J(y)

Hg hedjpjedjet `kvorhoe`j92, J(xy) 9 J(x) . J(y)(hg `kvorhoe`j 9 2, tfje tfj jvjets orj ekt ej`jssorhcy hedjpjedjet)

Rrkpjrthjs kg @kvorhoe`j

?. @kv(x,y) 9 @kv(y,x) (h.j. syiijtrh`)>. @kv(x,x) 9 Wor(x)

0. |@kv(x,y)| Ptoedord Dhstrhnuthke Gue`thke (Dhs`rjtj r.v. òsj).

Rkhsske Dhstrhnuthke< Rrknonhchty kg m su``jss hs R (m) 9 ek. kg su``jss trhocs , eek. kg trhocs , Rsu``jss òsj prknonhchty

ijoe kg tfj dhstrhnuthkeGkr Rkhsske dhstrhnuthke< Ijoe 9 , vorhoe`j 9, oed 9epPtoedord Dhstrhnuthke Gue`thke (@ketheukus r.v. òsj)) 9 j dx 9 Orjo uedjr tfj ùrvj grki x?tk x>>. Jxpkejethoc dhstrhnuthke < g(x) 9 j, x , fjrj 9 , x 0. Zehgkri dhstrhnuthke< g(x)9 , n g(x) o9, ktfjrwhsj4.

@ou`fy dhstrhnuthke< g(x)9 .:.

ôycjhaf dhstrhnuthke gue`thke< gx 9 j , x Ijoe) 3 2 kr g(x?) g(x>) 3 2

@kiputj x wfh`f hs oe opprkxhiothke tk tfj rkkt.

?.0.> Pkcuthke kg Chejor Pystji kg Jquothkes

?.

Aouss Jchiheothke Ijtfkd

Fjrj jquothkes orj `kevjrtjd hetk uppjr trhoeaucor iotrhx gkri, tfje skcvjd ny no`sunsthtuthke ijtfkd.@keshdjr o?x + n?x + `?z 9 d?

o>x + n>x + `>z 9 d>o0x + n0x + `0z 9 d0

Ptjp ?< Tk jchiheotj x grki sj`ked oed tfhrd jquothke (wj dk tfhs ny suntro`thea suhtoncjiucthpcj kg ghrst jquothke grki sj`ked oed tfhrd jquothke)

o?x + n?y + `?z 9 d? (phvktoc jquothke, o?phvkt pkhet.)ny + ` z 9 dny + ` z 9 d

Ptjp >< Jchiheotj y grki tfhrd jquothke

o?x + n?y + `?z 9 d?ny + `>z 9 d (phvktoc jquothke, n hs phvkt pkhet.)`z 9 d

Ptjp 0< Tfj vocuj kg x , y oed z òe nj gkued ny no`m sunsthtuthke.

Ektj.

Aouss Lkrdke Ijtfkd

Zsjd tk ghed hevjrsj kg tfj iotrhx oed skcvhea chejor jquothkes.

Fjrj no`m sunsthtuthke hs ovkhdjd ny oddhthkeoc `kiputothkes tfot rjdu`j tfj iotrhx tkdhoakeoc grki, hestjod tk trhoeaucor gkri he Aouss jchiheothke ijtfkd. Euinjr kg kpjrothkes hs ikrj tfoe Aouss jchiheothke os tfj jggkrt kg no`m sunsthtuthke

hs sovjd ot tfj `kst kg oddhthkeoc `kiputothke.

Ptjp ?< Jchiheotj x grki >edoed 0rd


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@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


Ptjp >< Jchiheotj y grki ?stoed 0rd

Ptjp 0< Jchiheotj z grki ?stoed >ed

0.

C Z Dj`kipkshthke

Ht hs ikdhghòthke kg tfj Aouss jchihehothke ijtfkd.

Ocsk Zsjd gkr ghedhea tfj hevjrsj kg tfj iotrhx.

]OYe x e 9 ] C Ye x e ]ZYe x eo?? o?> o?0 ? 2 2 Z?? Z?> Z?0

o>? n>> `>0 9 C>? ? 2 2 Z>> Z>0o0? n0> `00 C0? C0> ? 2 2 Z0?

Ox 9 CZV 9 n òe nj wrhttje os

o)CX9n oed n) ZV9X

Pkcvj gkr grki o tfje skcvj gkr grki n. Tfhs ijtfkd hs ekwe os Dkkchttcjs ijtfkd.Phihcor ijtfkds orj @rkuts ijtfkd oed @fkcjsy ijtfkds.

4.

Htjrothvj Ijtfkd

(h) Lo`knh Htjrothke Ijtfkd

o?x + n?y + `?z 9 d?o>x + n>y + `>z 9 d>o0x + n0y + `0z 9 d0

Hg o?, n> , `0 orj coraj `kiporjd tk ktfjr `kjggh`hjets, tfje skcvhea tfjsj gkr x, y, zrjspj`thvjcy

x 9 m?c?y i?zy 9 m>c>x i>zz 9 m0c0x i0y

Cjt us stort whtf hehthoc opprkxhiothke x2 , y2 , z2x?9 m?c?y2i?z2y?9 m>c>y2i>z2z?9 m0c0y2i0z2

Ektjc> x(m)i>z(m)z(m+?) 9 m0c0 x(m)i0y(m)

(hh)

Aouss-Phjdjc Htjrothke Ijtfkd

Ikdhghòthke kg tfj Lo`knhsHtjrothke Ijtfkd

Ptort whtf (x2, y2, z2)9 (2, 2, 2) kr oeytfhea ]Ek spj`hgh` `kedhthkeY


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@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


He ghrst jquothke, put y 9 y2 z 9 z2 wfh`f whcc ahvj x?

He sj`ked jquothke, put x 9 x?oed z 9 z2 wfh`f whcc ahvj y?

He tfhrd jquothke, put x 9 x?oed y 9 y? wfh`f whcc ahvj z?

Ektj< Tk `kiputj oey vorhoncj, usj tfj cotjst ovohconcj vocuj.

He ajejrochzjd gkric>x(m+?)i>z(m)z(m+?)9 m0 c0x(m+?)i0y(m+?)

?.0.0 Euijrh`oc Hetjarothke

Tropjzkhdoc Gkriuco


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@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


Pkcuthke hs ahvje ny, Xe+?9 ye+ f g(xe,ye)

(hh)

^ueaj Mutto Ijtfkd

Zsjd gkr ghedhea tfj y ot o porth`ucor x whtfkut skcvhea tfj ?stkrdjr dhggjrjethoc jquothke 9 gx, yM? 9 f g(x2, y2)

M> 9 f g(x2 +, y2 + )M0 9 f g(x2 +

, y2 + )M4 9 f g(x2 +f, y2 + m0)

M 9(m?+ >m>+ >m0+ m4)

X(x2+f) 9 y2+ m


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@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??

< 262-7:1221:2,hegkBtfjaotjoòdjiy.`ki @kpyrhaft rjsjrvjd. Qjn< www.tfjaotjoòdjiy.`ki Roaj >2

?.4@ocùcus

?.4.?

Chiht kg o Gue`thke

Cjt y 9 g(x)

Tfje chi gx9 h.j, gx os xo hipchjs gkr oey (52), (52) su`f tfot wfjejvjr23 |x o|3,|gx |3Pkij Ptoedord Jxpoeshkes

?x 9 ? e x ee?e

x . . . . . . . . . x

x ox o 9 x xo xo . . . . . . . . . oj9 ? + x + + . . . . . . . . .cka?x9 x + . . . . . . . . .cka?x9 x . . . . . . . . .Phe x 9

x

. . . . . . . . .

@ks x 9 ? + . . . . . . . . .Phef x 9x . . . . . . . . .@ksf x 9 ? + . . . . . . . . .

Pkij Hipkrtoet Chihts

chi shexx 9 chi (? ?x) 9 chi? x 9 chi o ?x 9 cka o

chi j ?x 9 ?


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@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??

< 262-7:1221:2,hegkBtfjaotjoòdjiy.`ki @kpyrhaft rjsjrvjd. Qjn< www.tfjaotjoòdjiy.`ki Roaj >?

chi cka?xx 9 ?

chi x ox o 9 ochi cka|x| 9 C

Fksphtocs ucj

Qfje gue`thke hs kg kr gkri, dhggjrjethotj euijrotkr & djekiheotkr oed tfje oppcy

chiht.

Jxhstje`j kg Chihts oed @ketheuhty.

Hg chig(x) 9 chig(x) 9 C ,tfje tfj chig(x) jxhsts oed jquoc tk C.0.

Hg chi gx9chi gx9 g(o) tfje tfj gue`thke g(x) hs sohd tk nj ketheukus.Rrkpjrthjs kg @ketheuhty

Hg g oed a orj twk `ketheukus gue`thkes ot o= tfje

o.

(g+a), (g.a), (g-a) orj `ketheukus ot o

n.

hs `ketheukus ot o, prkvhdjd a(o) 2`.

|g|kr

|a|hs `ketheukus ot o

kccjs tfjkrji

Hg (h) g(x) hs ketheukus he `cksjd hetjrvoc ]o,nY

hh gx jxhsts gkr jvjry vocuj kg x he kpje hetjrvoc o,n(hhh) g(o) 9 g(n)

Tfje tfjrj jxhsts ot cjost kej pkhet ` njtwjje (o, n) su`f tfot9 2Ajkijtrhòccy< Tfjrj jxhsts ot cjost kej pkhet ` njtwjje (o, n) su`f tfot toeajet ot ` hs poroccjc tk

x oxhs

@?

@


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@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??

< 262-7:1221:2,hegkBtfjaotjoòdjiy.`ki @kpyrhaft rjsjrvjd. Qjn< www.tfjaotjoòdjiy.`ki Roaj >>

CoaroeajsIjoe Wocuj Tfjkrji

Hg (h) g(x) hs `ketheukus he tfj `cksjd hetjrvoc ]o,nY oedhh gx jxhsts he tfj kpje hetjrvoc o,n, tfje otcjost kej vocuj ` kg x jxhst he o,n) su`f tfot 9 g(`).

Ajkijtrhòccy, ht ijoes tfot ot pkhet `, toeajet hs poroccjc tk tfj `fkrd chej.

@ou`fysIjoe Wocuj Tfjkrji

Hg (h) g(x) hs `ketheukus he tfj `cksjd hetjrvoc ]o,o+fY oed

(hh) g(x) jxhsts he tfj kpje hetjrvoc (o,o+f), tfje tfjrj hs ot cjost kej euinjr (233?)su`f tfot

g(o+f) 9 g(o) + f g(o+f)Cjt g?oed g>nj twk gue`thkesnktf orj `ketheukus he ]o,nYhh)

g?,g>nktf orj dhggjrjethoncj he (o,n)

hhh)

g> 2 he (o,n)tfje, gkr o n

9 ?.4.>

Djrhvothvj0

Fkikajekus Gue`thke

Oey gue`thke g(x, y) wfh`f òe nj jxprjssjd he grki xe hs òccjd fkikajekus gue`thke kgkrdjr e he x oed y. (Jvjry tjri hs kg etfdjarjj.)g(x,y) 9 o2xe + o?xe-?y + o>xe->y> oe yeg(x,y) 9 xe Jucjrs

Tfjkrji ke Fkikajekus Gue`thke

Hg u nj o fkikajekus gue`thke kg krdjr e he x oed y tfje,

x

+y

9 eu

x + >xy + y9 ee ?u?.4.0

Tktoc Djrhvothvj

Hg u9gx,y ,x9t, y9t 9 .+ .

u9

x +

y

Ikektkeh`hty kg o Gue`thke g(x)

?. g(x) hs he`rjoshea gue`thke hg gkr , g gEj`jssory oed suggh`hjet `kedhthke, g x >.

g(x) hs dj`rjoshea gue`thke hg gkr , , g gEj`jssory oed suggh`hjet `kedhthke, gx

Ektj


28/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


Hg

9,prk`jjd gurtfjr oed ghed

ot x 9 .

Hg ,y fos ejhtfjr ioxhiui ekr ihehiui vocuj ot x 9 Nut Hg 9 ,prk`jjd gurtfjr oed ghed ot x 9 .Hg

,y fos ihehiui vocujHg

,y fos ioxhiui vocujHg

9 ,prk`jjd gurtfjrEktj< Arjotjst / cjost vocuj jxhsts jhtfjr ot rhth`oc pkhet kr ot tfj jed pkhet kg hetjrvoc.

Rkhet kg Hegcjxhke

Hg ot o pkhet, tfj gkcckwhea `kedhthkes orj ijt, tfje su`f pkhet hs `occjd pkhet kg hegcjxhke

h) 9 ,

hh) 92 ,

hhh)

Ejhtfjr ihehio ekr ioxhio jxhstsToyckr Pjrhjs:

?.4.4

Ptoedord Hetjaroc ^jsucts

?.

x dx 9

, e ?

>. d x 9 c k a x0. j dx9 j4. o dx9 (prkvj ht ):. `ksxdx9shex7. shexdx9`ksx1. sj`x d x 9 toe x6. `ksj` x d x 9 ` k t x8. sj` x toe x dx 9 sj` x?2.

`ksj` x `kt x dx 9 `ksj` x

??. dx9 she ?>. d x 9 sj`

?0. dx 9 sj` x?4. `ksfx dx 9shefx?:. shefx dx 9`ksfx?7. sj`f x dx 9 toef x?1. `ksj`fx dx 9`ktfx?6. sj`f x toefx dx 9 sj`f x?8. `ksj` f x `kt f x dx 9 `ksj`f x>2.

toex dx 9ckasj`x

>?. `ktx dx 9 cka she x>>. sj` x dx 9 cka sj` x toe x 9ckatoe x >0. `ksj` x dx 9 cka`ksj` x `kt x 9cka toe

>4. dx9ckaxx o 9` ks f >:. dx9ckax x o9shef >7. o x dx 9 she >1. o x dx 9 x o ckax x o>6.

x

o

dx 9 x

o

ckaxx

o

>8. dx 9 toe 02. dx9 cka wfjrj x 3o

0?. dx9 cka wfjrj x 5 o0>. shex dx 9 shex00. `ksx dx 9 shex04. toex d x 9 t o e x x0:. `ktx d x 9 ` k t x x07. ce x dx 9 x cex x01.

j

she nx dx 9

o she nx n `ks nx

06. j `ks nx dx 9 o `ks nx n she nx


30/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


08. jgx gx-dx9 jgxHetjarothke ny ports< u v dx 9 u. v dx v dxdx

?.4.:

ûcjs gkr Djghehtj Hetjaroc

?. gxdx 9 gxdx + gxdx o3`3n>. gxdx 9 go n xdx gxdx 9 go xdx 0. gxdx 9 gxdx/ + go xdx/ gxdx 9 gxdx/

hg g(o-x)9g(x)

9 2 hg g(o-x)9-g(x)

4. gxdx 9> gxdx hg g(-x) 9 g(x), jvje gue`thke9 2 hg g(x) 9 -g(x), kdd gue`thke

Hiprkpjr Hetjaroc

Tfksj hetjarocs gkr wfh`f chiht hs heghehtj kr hetjaroed hs heghehtj he o x n he òsj kg gxdx ,tfje ht hs òccjd os hiprkpjr hetjaroc.

?.4.7

@kevjraje`j1

?.4.1

Wj`tkr @ocùcus Djarjj kg Dhggjrjethoc Jquothke< Ht hs tfj djarjj kg tfj fhafjst djrhvothvj k`ùrrhea he ht,ogtjr jxprjsshea tfj jquothke grjj grki rodhòcs oed gro`thkes os gor os djrhvothvjs orj

`ke`jrejd.

?.:.0 Dhggjrjethoc Jquothkes kg Ghrst Krdjr Ghrst Djarjj ot t9 2.

(9 ( 9 = W 9 2= W 9 W W W W WOs W W, hedu`tkr occkws onrupt `foeaj he vkctoaj. Ocsk, ( 9 ( os hedu`tkrdkjs ekt occkw onrupt `foeaj he ùrrjet. @keshdjr o ^ C `hrùht wfh`f `ketohes kecy kejhedu`tkr oed ikrj tfoe kej rjshstkr os sfkwe.

Tk ghed htgkr t 2 he tfj `hrùht sfkwe onkvj,(?)Ghed hehthoc vocuj kg urrjet tfrkuaf hedu`tkr, H h(>)Ghed stjody stotj vocuj kg ùrrjet tfrkuaf hedu`tkr, H h(0)

Ghed os sjje grki tjriheocs kg hedu`tkr(4)

Zsj gkcckwhea jquothke tk ghed

ht

^jshsthvj

Ejtwkrm

^

jq

+

[

Gha. ^-C `hrùht

P

t92

? >

+

-

t 92

+

-

Gha. ^-C `hrùht

P

W W


56/231

_uh`m ^jgrjsfjr Auhdj Ejtwkrm Tfjkry


@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??

< 262-7:1221:2,hegkBtfjaotjoòdjiy.`ki @kpyrhaft rjsjrvjd. Qjn< www.tfjaotjoòdjiy.`ki Roaj :>

h(t) 9 H H H) j oed Wt C H H j , wfjrj C Troeshjet rjspkesj kg o ^-@ `hrùht

@keshdjr tfj ^ @ `hrùht sfkwe njckw, he wfh`f swht`f P hs ikvjd grki pkshthke? tk pkshthke >ot t 9 2.

W(9 W( 9 W= 9 2= ) 9 W W Os H ( H , òpo`htkr occkws onrupt `foeaj he ùrrjet. Ocsk, W( Wos òpo`htkrdkjset occkws onrupt `foeaj he vkctoaj @keshdjr o ^- @ `hrùht wfh`f `ketohes kecy kejòpo`htkr oed ikrj tfoe kej rjshstkr os sfkwe.

Tk ghed

Wtgkr t

2

(?)

Ghed hehthoc vocuj kg vkctoaj o`rkss opo`htkr, W W(>)

Ghed stjody stotj vocuj kg vkctoaj o`rkss opo`htkr, W W(0)Ghed os sjje grki tjriheocs kg òpo`htkr(4)

Zsj gkcckwhea jquothke tk ghed WtW(t) 9 W W W) j oed ht wfjrj 9 @

^jshsthvj

Ejtwkrm

^ jq

+

[

Wt@

Gha. ^-@ `hrùht

t92

P ht

+

-

t 92

WtW WWt

+

-

Gha. ^-@ `hrùht

W Ws


57/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??

< 262-7:1221:2,hegkBtfjaotjoòdjiy.`ki @kpyrhaft rjsjrvjd. Qjn< www.tfjaotjoòdjiy.`ki Roaj :0

Troeshjet rjspkesj oeocyshs kg o skur`j grjj sjrhjs ^-C-@ `hrùht

Cjt

9 ^/>C,

oed o, n 9 -

Toncj. Dhggjrjet òsjs kg sjrhjs ^-C-@ `hrùht

P. Ek @kedhthke Eoturj kg

rjspkesj kg h t)

Ajejroc gkri kg h t) Aropf

?. Kvjr-doipjd Oj Oj (o oed n orj

ejaothvj, rjoc oed uejquoc)

>. @rhthòccy

doipjd

(O Ot j(o hs ejaothvj oed rjoc)

0. Zedjr-doipjd jO`kst Oshetwfjrj

Fjrj

O,

Oòe nj gkued ushea hehthoc `kedhthkes.

Troeshjet rjspkesj oeocyshs kg o skur`j grjj poroccjc ^ C @ `hrùht

^

W t)

@

C

h t)

-

W h 9

Gha. Pjrhjs ^-C-@ `hrùht

+ -

v t

Gha. Roroccjc ^-C-@ `hrùht

^C @

H

h(t)

t

h(t)

t

h(t)


58/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


Cjt 9 , oed o ,n 9 - Toncj Dhggjrjet òsjs kg poroccjc ^-C-@ `hrùht

P. Ek @kedhthke Eoturj kg

rjspkesj kg v t)

Ajejroc gkri kg v t) Aropf

?. Kvjr-doipjd Oj Oj(o oed n orj ejaothvj, rjoc oeduejquoc)

>. @rhthòccy

doipjd

(O Ot j(o hs ejaothvj oed rjoc)

0. Zedjr-doipjd jO`kst Oshetwfjrj

Ajejrochzothke kg rjspkesj g t) gkr oe jx`htothke a t)

He oey ajejroc ^-C-@ `hrùht, cjt g(t) nj tfj rjspkesj gkr oe jx`htothke a(t). Tfj rjspkesj g(t) òe

nj ajejrochzjd os njckw,

g(t)9gt gtFjrj, gths òccjd eoturoc rjspkesj oed hs gkued nosjd ke hehthoc `kedhthkes ossuihea ek skur`jhs prjsjet oed g(t) hs òccjd gkr`jd rjspkesj wfh`f hs djrhvjd nosjd ke skur`j kg jx`htothke.

v(t)

t

v(t)

t

W(t)


59/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??

< 262-7:1221:2,hegkBtfjaotjoòdjiy.`ki @kpyrhaft rjsjrvjd. Qjn< www.tfjaotjoòdjiy.`ki Roaj ::

>.0< Pheuskhdoc Ptjody Ptotj Oeocyshs

Rfoskr rjprjsjetothke

Oey sheuskhdoc rjprjsjetothke, W(t) 9 W shet òe nj rjprjsjetjd he pfoskr gkri os,W W

Gha. Rfoskr rjprjsjetothke kg sheuskhdoc shaeoc W t) kg oeaucor grjquje`y

Gkr pfoskr oeocyshs kg Wtoed Wt, gkcckwhea `kedhthkes sfkucd nj sothsghjd,(o)

Wtoed Wtsfkucd fovj soij shae gkr Woed W .(n)

Nktf Wtoed W(t) sfkucd nj wrhttje os shej/`kshej wovjs.(`)Nktf Wtoed Wtsfkucd fovj soij grjquje`y.Hipjdoe`j

Toncj. Hipjdoe`js kg dhggjrjet o.`. `hrùht `kipkejets

P. Ek @kipkejet Hipjdoe`j

?. \9 ^>. \9 LC0. \9 4. \ L C:.

\ L@ L@7. \ L C L@ L ( C @)

W W

^jgjrje`j

^

C

@

^

C

^

@

^

@

C


60/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


Pheuskhdoc jx`htothke kg dhggjrjet `kipkejets

Jx`htothke tk rjshstkr

Jx`htothke tk Hedu`tkr

Pheuskhdoc jx`htothke tk @opo`htkr

Pheuskhdoc jx`htothke tk ^C `hrùht

WH ^ C

toe C Gha . Rfoskr rjcothke he ^-C `hrùht

C

\ lC

Gha.W Hpfoskr rjcothke gkr hedu`tkr

^

H HWGha.-pfoskr rjcothke gkr rjshstkr

Gha. pfoskr rjcothke gkr òpo`htkr


61/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


Jx`htothke tk ^-@ @hrùht

^kkt ijoe squorj ris) vocuj kg o pjrhkdh` shaeoc

Tfj ris vocuj kg o pjrhkdh` shaeoc hs djghejd os jquhvocjet d` shaeoc wfh`f whcc `kesuij soijpkwjr os pjrhkdh` shaeoc. Hg W(t) hs o pjrhkdh` shaeoc kg pjrhkd T ,

W vtdt @keshdjr o shaeoc, W(t) 9 W W shet W shet - - - - - - - - -W9 W

Ovjroaj vocuj kg o pjrhkdh` shaeoc

Ovjroaj vocuj kg o pjrhkdh` shaeoc ahvjs oe hdjo onkut d.`. `ketjet kg tfj shaeoc. Hg W(t) hs shaeoc kg

pjrhkd T,

W

Wt

dt

Ovjroaj pkwjr suppchjd ny o.`. skur`j

Opporjet pkwjr , P 9 W H* (he ris sjesj) 9 W H @ks - L W H he9 R + L_O`thvj pkwjr, R 9 `ks - WH`ks - ^jo`thvj pkwjr, _ 9

she- W Hshe O`thvj pkwjr hs tfj pkwjr suppchjd tk rjshsthvj port kg tfj ejtwkrm oed hs ijosurjd he wott.

^jo`thvj pkwjr hs tfj pkwjr suppchjd tk hedu`thvj kr òpo`hthvj port kg tfj ejtwkrm oed hs

ijosurjd he vor. Rkwjr go`tkr (RG) kg `hrùht os sjje grki skur`j hs ahvje os `ks - Rg ahvjsoe hdjo onkut port kg WO suppchjd tk rjshsthvj port kg ejtwkrm.

^jskeoe`j

Ot rjskeoe`j, vkctoaj oed ùrrjet os sjje grki suppcy orj he pfosj. Ocsk ot rjskeoe`j,hipjdoe`j os sjje grki suppcy hs `kipcjtjcy rjshsthvj.

o . `.

Ejtwkrm

~ Gha . Djikestrothke kg pkwjr suppchjd ny o.`. skur`j

WH

^ @ toe

Gha. Rfoskr rjcothke he o -@ `hrùht

( )


62/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


Pheacj pfosj @hrùht oeocyshs

Ikst kg tfj oeocyshs tj`fehqujs usjd gkr d` ejtwkrms òe nj usjd gkr o` ejtwkrms ocsk. Fkwjvjr,

ocajnroh` kpjrothkes sfkucd nj rjpco`jd ny pfoskr kpjrothkes oed occ tfj tfjkrjis òe nj

rjcjvoetcy oppchjd.

Ioxhiui Rkwjr troesgjr tfjkrji

Ioxhiui pkwjr troesgjr tfjkrji ahvjs oe hdjo onkut tfj ioxhiui pkwjr `kesuijd he \djpjedhea ke tfj eoturj kg \. @keshdjr oe o` ejtwkrm oed `krrjspkedhea Tfjvjehesjquhvocjet os sjje grki ckod tjriheocs O oed N os sfkwe he ghaurj njckw.

Djpjedhea eoturj kg \oed \, gkcckwhea orj tfj pksshncj òsjs oed `krrjspkedhea `kedhthkesgkr ioxhiui pkwjr troesgjr,

(?)

Hg \oed \orj rjshsthvj , \ \.(>)Hg \oed \orj hipjdoe`j , \9 \ .(0)

Hg \hs `kipcjx oed \hs rjshshthvj , \ \.(4)Hg

\9

LVsu`f tfot

Vhs ghxjd oed

hs vorhoncj, tfje ,

\

LV

.

(:)Hg

\ LVhs su`f tfot

hs ghxjd oed

Vhs vorhoncj, tfje

V9 Hi(

\).

(7)

Hg \ LVhs su`f tfot toehs `kestoet oed | \hs vorhjd , tfje | \| 9 |\|.(1)

Gkr oey ornhtrory eoturj kg \oed \, opprkprhotj vocuj kg \òe nj gkued ushea Coaroeajskpthihzothke.

Ptor-djcto troesgkriothke

\

=

\

=

\

\ = \ = \

\>\?

\0

\O

\N

\@

Gha. Ptor-Djcto jquhvocjet troesgkriothke

OO

N

N@

@

o . `

ejtwkrm

O

N

~

O

N

Gha. Djikestrothke kg ioxhiui pkwjr troesgjr tfjkrji


63/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


Rkcypfosj @hrùht Oeocyshs

Pheacj pfosj tfrjj whrj systji

W9 WW9 W W9 >WH9 W\H9 W\H9 H H9 Hg tfj ckod hs nocoe`jd, \9 \

H

Tfj whrj eE hs ejutroc whrj. Pk hg tfj ckod hs nocoe`jd, ùrrjet he ejutroc whrj hs zjrk. Hg tfj whrjs

oO oed nN fovj soij hipjdoe`j, sthcc ùrrjet he ejutroc whrj hs 2. Hg ckods orj uenocoe`jd oed Eewhrj fos skij ghehtj hipjdoe`j, pkwjr dhsshpotjd he eE hs ghehtj.

Tfrjj pfosj skur`j

@keshdjr o nocoe`jd suppcy os sfkwe,

W W W

N

e

o

O

n

@

`

Gha. Tfrjj pfosj nocoe`jd suppcy he pkshthvj sjquje`j)

~ ~

~

~

~

n N

Oo

Gha. Pheacj pfosj 0 whrj systji

e E


64/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


W W W W W

risW WrisW W ris}Rkshthvj pfosj sjquje`jW WrisW WrisW Wris }Ejaothvj pfosj sjquje`jTfrjj pfosj X-X @keej`thke he`cudhea ejutroc gkr nocoe`jd suppcy)

H = H = H Cjt W, H, Woed Hstoed gkr chej vkctoaj, chej ùrrjet, ckod vkctoaj oed ckod ùrrjet.Hg ckod hs nocoe`jd, \ \ \ \H H H oed H H H9 H9 Rkwjr `kesuijd ny ckod 9 0 WH`ks 9 WH`ksW Woed H H Hg tfj ckod hs nocoe`jd ejutroc whrj òe nj rjikvjd os H

hs zjrk.

Hg ckod hs ekt nocoe`jd, H \ H H Rkwjr `kesuijd ny ckod 9 WH `ks (hg ckod hs ekt nocoe`jd)Tfrjj Rfosj X

keej`thke< (gkr nocoe`jd suppcy)H = H H

@

N

O

E

e

\

N

\

O

\

@

o

`

n

Gha. X-X `keej`thke


65/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??

< 262-7:1221:2,hegkBtfjaotjoòdjiy.`ki @kpyrhaft rjsjrvjd. Qjn< www.tfjaotjoòdjiy.`ki Roaj 7?

Hg tfj ckod hs nocoe`jd, \ \ \ \

H

Rkwjr `kesuijd ny ckod, R 9 WH@ksW W H HHg tfj ckod hs ekt nocoe`jd,Rkwjr `kesuijd ny ckod, R 9 W `ks Ioaejthòccy @kupcjd `hrùhts

Twk `hrùhts orj ioaejthòccy `kupcjd, hg kpjrothke kg kej `hrùht hs jggj`tjd ny gcux chemoaj dujtk `khc he oektfjr `hrùht He o ioaejthòccy `kupcjd `hrùht @urrjet jetjrhea dkttjd tjriheoc kgkej `khc whcc prkdu`j o vkctoaj wfh`f hs sjesjd pkshthvj ot tfj dkttjd tjriheocs kg oektfjr `khc

oed vh`j vjrso k vkctoaj o`rkss kej `khc djpjeds ekt kecy ke sjcg hedu`toe`j nut ocsk keiutuoc hedu`toe`j wfh`f ahvjs jxtjet kg gcux chemoaj njtwjje tfj twk `khcs. Rcjosj ektj tfot dkt`kevjethke dkjset iomj sjesj wfje hts ossk`hotjd whtf sheacj `khc+Wtht-

Wt+

-

iWt9C Wt9i

+Wtht-

Wt+

-

i Wt C

W

t

i

+Wt ht-

Wt+

-

i Wt C Wt i

+Wt ht-

Wt+

-

i Wt C Wt i

Gha. Djikestrothke kg dkt `kevjethke

C C

C C

C

C

C C

N

`

@

n

O

e

\

O

o \

N

\

@

Gha. X-@keej`thke

O


66/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??

< 262-7:1221:2,hegkBtfjaotjoòdjiy.`ki @kpyrhaft rjsjrvjd. Qjn< www.tfjaotjoòdjiy.`ki Roaj 7>

Os tfj gcux chemoaj he `khc ? duj tk tfj `khc > hs soij os gcux chemoaj he `khc > duj tk `khc ?,

i

i9 i

Ocsk `kjggh`hjet kg `kupchea, M 9 i / CC= Fjrj M i CC@k jggh`hjet kg `kupchea hs he ]2 , ?Y os gcux chemoaj he kej `khc duj tk ùrrjet he ktfjr `khc whccocwoys nj cjss tfoe tfj gcux prkdu`jd ny tfj soij.

Wt C i & Wt C +i He grjquje`y dkiohe W LCH Li H &W LiH LCHHestoetoejkus jejray stkrjd 9

Cht Cht i hthtTktoc jejray stkrjd 9

CH CH i HH

Wt C i & Wt C - i He grjquje`y dkiohe, W LCH Li H& W LiH LCHHestoetoejkuscy jejray stkrjd 9 Cht Cht i hthtTktoc jejray stkrjd 9 CH CH i HH

i

Gha. Djikestrothke kg dkt `kevjethke Dhggjrjethoc @keej`thke)

htWt++Wtht- -C C

+Wtht-

ihtWt

Gha. Djikestrothke kg dkt `kevjethke @kiiutothvj `keej`thke)

+

-C C


67/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


>.4< Troesgjr Gue`thke kg oe CTH Pystji

Troesgjr gue`thke kg o CTH systji hs djghejd os rothk kg Copco`j troesgkri kg kutput tk tfj

Copco`j troesgkri kg heput ossuihea hehthoc `kedhthkes orj zjrk. (CTH ijoes chejor oed thij-hevorhoet)

Hg C{g(t)}9G(P), C{f(t)}9F(P) oed C{a(t)}9A(P), A(P)9F(P) .G(P)

Troesgjr gue`thke kg systji, F(P) 9 A G Kutput rjspkesj kg o CTH systji kg troesgjr gue`thke, F s)

Cjt tfj kutput rjspkesj kg o CTH systji nj djghejd os a(t) gkr o heput g(t),

a(t) 9 a(t)+a(t)wfjrj a(t) hs troeshjet rjspkesj kg systji oed a(t) hs stjody stotj rjspkesj kg systji.Cjt F(P) fos pkcjs ot p, h9?, --------e. a(t) 9 CGFoed a(t)9 Oj at Oj CGAFjrj `kestoet Oòe nj gkued nosjd ke hehthoc `kedhthkes. Onkvj oeocyshs òe nj usjd he ^-C-@`hrùhts tk ajt vkctoaj/ùrrjet rjspkesj ot oey thij t.

Ckùs kg pfoskrs

Ahvje oey rjspkesj A sunsthtutj L tk ajt ajejrochzjd pfoskr kg A Ckùs kg A (P) òenj kntohejd ny voryhea oed Tfus tfj ckùs ahvjs oe hdjo onkut tfj pfoskr ot dhggjrjetgrjquje`hjs.

@hrùht oeocyshs ot o ajejrochzjd grjquje`y

Oey `hrùht òe nj ajejrochzjd tk kpjrotj ot o grjquje`y L Gkr d` shaeocs oed oed gkr o` shaeocs Cjt Wt Wj`ks t nj tfj heput tk o ejtwkrm sfkwenjckw oed cjt tfj rjspkesj nj ht Hj`ks t . Qj sjj tfot W oed Hfovj o grjquje`y kgL He pfoskr gkri tfj `hrùht òe nj rjprjsjetjd os njckw,

g t)

f t)

a t)9g t)*f t)

Gha. CTH systji


68/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


E ot P9

o.`

ot

P9

Gha. @hr`uht oeocyshs kg ajejrochzjd grjquje`y P 9 + L)).


69/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??

< 262-7:1221:2,hegkBtfjaotjoòdjiy.`ki @kpyrhaft rjsjrvjd. Qjn< www.tfjaotjoòdjiy.`ki Roaj 7:

>.:< Twk Rkrt Ejtwkrms

Djghehthkes

Kej pkrt ejtwkrm hs o ejtwkrm wfh`f fos o pohr kg tjriheocs o`rkss wfh`f vkctoaj òe njoppchjd oed ùrrjet òe poss. Twk pkrt ejtwkrm hs o ejtwkrm wfh`f fos twk pohrs kg tjriheocskg onkvj typj. Cjt W 9 W Woed H 9 H H nj tfj vkctoajs o`rkss oed tfj ùrrjets tfrkuaftfj twk pkrts kg tfj ejtwkrm.

\ -Roroijtjr kpje `hrùht poroijtjrs kr hipjdoe`j poroijtjrs)]WWY9 ]\ \\ \Y ]HHYW 9 \H

X Roroijtjrs odihttoe`j poroijtjr kr sfkrt `hrùht poroijtjr)

]HHY

9

]X XX XY

]WWY

W 9 XH= Ocsk

]\ \\ \Y ]

X XX XY

X \

f - Roroijtjrs fynrhd poroijtjrs)

]WH Y9 ]f ff fY ]HWY

ON@D - Roroijtjrs Troesihsshke chej kr `fohe poroijtjrs)

]W

H Y9

*O N@ D+ ] W

HY

Gha. Twk pkrt ejtwkrm gkr ON@D poroijtjrs)

> pkrtE/Q

-

E/Q + +

-

Gha. Twk pkrt ejtwkrm


70/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


A-Roroijtjrs

] HWY *a aa a+ ]WH Y A F@kedhthke gkr rj`hprk`hty oed syiijtry

Toncj< @kedhthkes gkr rj`hprk`hty oed syiijtry

Gkr `kevjrthea kej typj kg poroijtjrs tk oey ktfjr typj, wrhtj jquothkes tk jxprjss rjcothke

njtwjje W H W& H he tjris kg ahvje poroijtjrs oed `kevjrt tfj soij hetk tfj rjquhrjd gkritk ajt tfj torajt poroijtjrs.

Hetjr-`keej`thke kg twk pkrt ejtwkrms-pkrt ejtwkrms O oed N orj `keej`tjd he poroccjc, tfje X-poroijtjrs kg ùiucothvjejtwkrm hs jquoc tk sui kg hedhvhduoc X-poroijtjrs. Hg twk >-pkrt ejtwkrms O oed N orj

`keej`tjd he sjrhjs, tfje \ poroijtjrs ajt oddjd. Hg twk >-pkrt ejtwkrms O oed N orj `keej`tjd

he òsòdj, tfje ON@D poroijtjrs kg ùiucothvj ejtwkrm orj jquoc tk prkdu`t kg hedhvhduoc

ON@D poroijtjrs.

P. EK Roroijtjr @kedhthke gkr posshvj ejtwkrm

kr rj`hprk`hty

@kedhthke gkr syiijtry

? \ \9 \ \9 \> X

X9

X

X9

X

0 ON@D O N@ D9 ? O 9 D4 f f9 -f |f ff f|9 ?


71/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


>.7< Ejtwkrm Tkpkckay

Djghehthkes

Aropf


72/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


Oppchòthkes he Ejtwkrm Tfjkry

Qj òe wrhtj M@C, ushea tfj ekdoc he`hdje`j iotrhx os gkcckwshnY hs nroe`f ùrrjet vj`tkrEk. kg trjjs 9 Djt |OOT|

^jdu`jd He`hdje`j Iotrhx

Qfje oey ekdj hs tomje os rjgjrje`j, tfje tfj vkctoajs kg ktfjr ekdjs òe nj ijosurjd whtfrjspj`t tfj osshaejd rjgjrje`j. He tfj Gha, tomhea ekdj 4 os rjgjrje`j rjdu`js tfj iotrhx, tk

^jdu`jd He`hdje`j Iotrhx, dhijeshkes njhea (e-?xn).

O 9 ] Ot OtY

Qfjrj Ohs o squorj iotrhx kg krdjr (e-?) (e- ?) oed O hs o iotrhx kg krdjr (e-?) n e + ?)wfksj `kcuies `krrjspked tk tfj chems.

Ckkp He`hdje`j Iotrhx Guedoijetoc Thj-sjt Iotrhx)

Djghehthke

Ht hs djghejd os tfj iotrhx rjprjsjetothke he wfh`f tfj ckkp krhjetothke hs tk nj tfj soij os tfj

`krrjspkedhea chem dhrj`thke.

ôem kg N hs n-e+?=

Ptjps tk ajt tfhs Iotrhx

?. Drow tfj krhjetjd aropf kg ejtwkrm. @fkksj o trjj.

>. Jo`f chem gkris oe hedjpjedjet ckkp & tfj dhrj`thke kg tfhs ckkp hs soij os tfot kg tfj`krrjspkedhea chem. @fkksj jo`f chem he ture.

0.

Rrjporj tfj thj-sjt iotrhx Ncxn 9 {nfm}, c 9 ek kg ckkps, n 9 ek kg nroe`fjs, djghejd os gkcckws.

(?)

(>)

(0)

(4)

(:) (7)

0>

?

4

(N)

(7)

(0)

+-

(?)(>) (4)

(:)

?>

0

4

W

(O)

Gha. Djikestrothke kg ejtwkrm tkpkckay


73/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


Tk wrhtj o Thj Pjt Iotrhx gkr Gha ?, stjps fovj njje stotjd s`fjiothòccy.

n 9 +?, hg nroe`f m hs he ckkp f oed tfjhr krhjetothkes `khe`hdj

9 - ?, hg nroe`f m hs he ckkp f oed tfjhr krhjetothkes dk ekt `khe`hdj

9 2, hg nroe`f m hs ekt he ckkp


Qj òe wrhtj MWC, ushea tfj ckkp he`hdje`j iotrhx os gkcckws

NcxnWn9 2, wfjrj Wn9 ]v?, v>vnY hs nroe`f vkctoaj vj`tkr95 Hn9 ]NcxnYT

, wfjrj

9 ]h?, h>

hY hs ckkp ùrrjet vj`tkr

Guedoijetoc @ut-Pjt Iotrhx

Ht hs djghejd os o sjt kg nroe`fjs wfksj rjikvoc ùts tfj `keej`tjd aropf hetk twk ports su`ftfot tfj rjpco`jijet kg oey kej nroe`f kg tfj ùtjst rjedjrs tfj twk port `keej`tjd.

ôem kg _ hs e-?

Ptjps tk ajt @ut-Pjt

?. Drow tfj krhjetjd aropf kg o ejtwkrm oed `fkksj o trjj.

>.

Gkr e-? twhas, wj whcc ajt e-? ùt sjts whcc jxhst.

0.

Dhrj`thke kg ùt-sjt hs soij os twhas. @fkksj jo`f twha he ture tk kntohe tfj iotrhx.

4.

Rrjporj tfj ùt sjt iotrhx _(e-?)x n9 {qhl}, wfjrj e 9 ek kg ekdjs, n 9 ek kg nroe`fjs osgkcckws,

q 9 ?, hg nroe`f l hs he tfj ut-sjt h, oed tfj krhjetothkes `khe`hdj9 - ?, hg nroe`f l hs he tfj ùt-sjt h oed tfj krhjetothkes dk ekt `khe`hdj

9 2, hg nroe`f l hs ekt he tfj ùt-sjt h


Grki tfj ùt-sjt iotrhx, wj òe wrhtj jquothkes rjcothea tfj nroe`f vkctoajs tk tfj ekdj

vkctoajs os gkcckws, ]_(e-?) x nYHn 9 2 95 Wn9 ]_(e-?) x nYTWe-?

wfjrj We-?9 ]v?, v>ve-?Y hs ekdj vkctoaj vj`tkr


74/231

_uh`m ^jgrjsfjr Auhdj Phaeocs & Pystjis


@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


Rort 0< Phaeocs oed Pystjis

0.?< Hetrkdu`thke tk Phaeocs Pystjis

Hetrkdu`thke

Phaeoc hs djghejd os o gue`thke tfot `kevjys usjguc hegkriothke onkut tfj stotj kr njfovhkur kg o

pfyshòc pfjekijeke. Phaeoc hs typhòccy tfj vorhothke whtf rjspj`t tk oe hedjpjedjet quoethty

chmj thij.

Pystji

Pystji hs djghejd os oe jethty wfh`f jxtro`ts usjguc hegkriothke grki tfj shaeoc kr prk`jssjs tfj

shaeoc os pjr o spj`hgh` gue`thke.

@cosshghòthke kg Phaeocs

@ketheukus-Thij vs Dhs`rjtj-Thij Phaeocs

@ketheukus-thij shaeoc hs djghejd os o shaeoc wfh`f hs djghejd gkr occ hestoets kg thij. Dhs`rjtj

thij Phaeoc hs o shaeoc wfh`f hs djghejd ot spj`hgh` hestoets kg thij kecy oed hs kntohejd ny

soipchea o `ketheukus thij shaeoc. Ocsk dhs`rjtj-thij shaeocs orj djghejd kecy ot hetjajrhestoets, hs e z.Dhahtoc shaeoc hs kntohejd grki dhs`rjtj-thij shaeoc ny quoethzothke.@keluaotj Pyiijtrh` vs Pmjw Pyiijtrh` Phaeocs

O `ketheukus thij shaeoc x(t) hs `keluaotj syiijtrh` hg x(t) 9 x*(-t)=t. Hg x(t) 9 -x*(-t)= t. Ocsk,x(t) hs `keluaotj smjw syiijtrh`.

Oey ornhtrory shaeoc x(t) òe nj `keshdjrjd tk `kesthtutj > ports os njckw,

x(t) 9 xt+ xtwfjrj xt9 `keluaotj syiijtrh` port kg shaeoc 9 () oed

xt9 `keluaotj smjw syiijtrh` port kg shaeoc 9 () Rjrhkdh` vs Eke-Rjrhkdh` Phaeocs

O `ketheukus thij shaeoc hs pjrhkdh` hg tfjrj jxhsts T su`f tfotx(t+T) 9 x(t), t =T ^ {2}

Tfj sioccjst pkshthvj vocuj kg T tfot sothsghjs onkvj `kedhthke hs òccjd guedoijetoc pjrhkd kg

x(t).

O dhs`rjtj-thij shaeoc hs pjrhkdh` hg tfjrj jxhsts E su`f tfot

x]eY 9 x]e+EY, e = E \ {2}Tfj sioccjst pkshthvj E tfot sothsghjs onkvj `kedhthke hs òccjd guedoijetoc pjrhkd kg x]eY. Hg

xt oed

xtorj pjrhkdh` shaeocs whtf pjrhkds

T oed

T rjspj`thvjcy, tfje x(t) 9

xt +

xt hs pjrhkdh` hgg (hg oed kecy hg) hs o rothkeoc euinjr oed pjrhkd kg x(t) hs cjost `kiike


75/231


76/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


Ptonhchty

O systji hs sohd tk nj stoncj he nkuedjd heput nkuedjd kutput sjesj hg gkr oey nkuedjd heput

tfj systji ahvjs nkuedjd kutput, ktfjrwhsj systji hs uestoncj.Gkr `ketheukus thij shaeocs, |xt| || stonhchtyHevjrthnhchty

O systji hs sohd tk fovj hevjrsj, hg tfjrj jxhsts oektfjr systji sk os tk rj`kvjr tfj krhaheoc heput

grki tfj kutput kg ghrst systji.


77/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


0.>< Chejor Thij Hevorhoet (CTH) systjis

@kevkcuthke Pui

Oey ornhtrory shaeoc x]eY òe nj djs`rhnjd os, .

Gha. Dhs`rjtj thij CTH systji

@keshdjr o dhs`rjtj-thij CTH systji djs`rhnjd os, Gkr CTH systji, wfjrj f]eY hs hipucsj rjspkesj kgtfj systji.

Rrkpjrthjs/@foro`tjrhzothke kg CTH Pystji Zshea Hipucsj ^jspkesj

Ijikrycjss Pystji

Hg systji hs ijikrycjss, y]eY djpjeds ke kecy heput ot tfot hestoet,

.

.ewfjrj @

^

@ousoc Pystji

Hg systji hs òusoc, y]eY djpjeds ke post oed prjsjet heputs, xi iefe gkr e, hg systji hs ùsc.

Ptoncj Pystji

|| sfkucd nj ghehtj gkr NHNK stonhchtyHevjrthncj CTH Pystji

Gkr o `ketheukus thij CTH systji kg hipucsj rjspkesj, f(t) hg tfjrj jxhsts o hipucsj rjspkesj

f(t) su`f tfot, , tfje hs òccjd hevjrsj CTH systji oed f(t) hs òccjdhevjrthncj CTH systji.^jprjsjetothke kg @ketheukus/Dhs`rjtj-Thij Pystji

Troesgkriothke

Djghej o iopphea rjcotjd tk heput oed kutput kg systji. y(t) 9 T{x(t)}

^jprjsjetothke kg @ketheukus/Dhs`rjtj-Thij CTH Pystji

Troesgkriothke

Djghej o iopphea rjcotjd tk heput oed kutput kg systji.

Dhs`rjtjThij

CTH Pystjix]eY y]eY


78/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


Hipucsj rjspkesj

Hipucsj rjspkesj uehqujcy rjprjsjets o CTH systji

C@@DJ(Chejor @kestoet @kjggh`hjet Dhggjrje`j Jquothke)

C@@DJ rjprjsjets tfj kutput ot hestoet e, he tjris kg kutput oed heputs.

. . = Qfjrj , ^.Pystji Gue`thke ( F(j/F(z))Grki C@@DJ, systji gue`thkes F(jF(z) `oe nj djrhvjd os njckw.

. . wfjrj Gfe j

e

. . wfjrj fe zDjtjrihehea Hipucsj ^jspkesj grki Ptjp ^jspkesj

Gkr o dhs`rjtj-thij CTH systji, hg yehs ueht stjp rjspkesj, hipucsj rjspkesj f]eY hs ahvje os,

Gkr o `ketheukus-thij CTH systji, hg yths ueht stjp rjspkesj, f(t) hs ahvje os,

^jspkesj kg CTH Pystji tk Pheuskhdoc Heput

Ghaurj sfkwe njckw ahvjs tfj rjspkesj kg o dhs`rjtj-thij CTH systji tk o sheuskhdoc heput.

x]eY 9 y]eY 9 O |F(|. ()

Gha. Pheuskhdoc rjspkesj kg o dhs`rjtj-thij CTH systji

Tfj onkvj rjcothke hipchjs tfot o CTH systji prkdu`js o sheuskhd he rjspkesj tk o sheuskhd, tfj

oipchtudj hs iucthpchjd ny o go`tkr |F()| oed pfosj hs `foeajd ny o go`tkr ra().Tfj grjquje`y rjspkesj kg o systji ahvjs hegkriothke onkut fkw ht oggj`ts sheuskhdoc heput ot o

porth`ucor grjquje`y.

f]eY


79/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


0.0< Gkurhjr jprjsjetothke kg Phaeocs

Hetrkdu`thke

Fjrj wj rjprjsjet shaeocs os o chejor `kinheothke kg `kipcjx jxpkejethocs. Tfj rjsucthea

rjprjsjetothkes orj mekwe os tfj `ketheukus thij oed dhs`rjtj thij Gkurhjr sjrhjs oed

troesgkri (djpjedhea ke tfj eoturj kg shaeoc h.j. `ketheukus/dhs`rjtj oed pjrhkdh`/eke

pjrhkdh`).

Gkurhjr ^jprjsjetothkes

Toncj< Gkurhjr rjprjsjetothke kg shaeoc

P. Ek Eoturj kg Phaeoc ^jprjsjetothke Eoturj kg Grjquje`y

^jprjsjetothke

? @ketheukus-thij, Rjrhkdh` Gkurhjr sjrhjs Dhs`rjtj, eke-pjrhkdh`> Dhs`rjtj-thij, Rjrhkdh` Dhs`rjtj Thij Gkurhjr Pjrhjs (DTGP) Dhs`rjtj, pjrhkdh`

0 @ketheukus-thij, Ekepjrhkdh`

@ketheukus-Thij Gkurhjr Troesgkri(@TGT)

@ketheukus, eke-pjrhkdh`

4 Dhs`rjtj-thij, Ekepjrhkdh`

Dhs`rjtj Thij Gkurhjr Troesgkri(DTGT)

@ketheukus, pjrhkdh`

Grki tfj onkvj toncj, wj sjj tfot `ketheukus-thij shaeocs fovj eke-pjrhkdh` grjquje`y

rjprjsjetothke oed dhs`rjtj-thij shaeocs fovj pjrhkdh` grjquje`y rjprjsjetothke. Ocsk pjrhkdh`

shaeocs fovj dhs`rjtj grjquje`y rjprjsjetothke oed eke-pjrhkdh` shaeocs fovj grjquje`y

rjprjsjetothke wfh`f hs `ketheukus he eoturj.

Gkurhjr sjrhjs (GP) gkr @ketheukus Thij Rjrhkdh` Phaeocs

@kipcjx GP rjprjsjetothke

Cjt x(t) nj `ketheukus thij pjrhkdh` shaeoc whtf pjrhkd T,V(m) 9

x(t) 9 wfjrj T hs pjrhkd kg x(t) oed Tx(t) V(m)

Octjreotj Gkurhjr Pjrhjs ^jprjsjetothke

Hg g(x) hs o shaeoc kg pjrhkd >T, `ks she wfjrj Fjrj hs tfj d` `kipkejet kg g(x),

.`ks oed

Hg g(x) hs jvje,

`ks Hg g(x) hs kdd,

G P,


80/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


Dhs`rjtj Thij Gkurhjr Pjrhjs(DTGP)

Cjt xe hs nj dhs`rjtj thij shaeoc kg pjrhkd E,xe mxe 9 m wfjrj

m 9 xe mhs ocsk pjrhkdh` whtf pjrhkd E oed hs dhs`rjtj he eoturj.Gkurhjr Troesgkri (GT) gkr @ketheukus Thij Eke-Rjrhkdh` Phaeocs

Cjt x(t) nj o `ketheukus-thij eke-pjrhkdh` shaeoc,

x(t)

l

l hs `ketheukus whtf rjspj`t tk e hs eke-pjrhkdh`l xt x(t) 9

l

Tfj Gkurhjr Troesgkri V(l iy ekt jxhst gkr cc gue`thkes xt. Gkr tfj Gkurhjr tresgkri tkjxhst, x(t) iust sothsgy tfj Dhr`fcjt `kedhthkes ahvje njckw,

Tfj shaeoc x(t) iust nj onskcutjcy hetjaroncj, h.j |xt| . Tfj shaeoc x(t) iust fovj ghehtj euinjr kg ck`oc ioxhio oed ihehio oed dhs`ketheuhthjs he

oey ghehtj hetjrvoc.

Tfj shzj kg jo`f dhs`ketheuhty iust nj ghehtj.

Dhs`rjtj Thij Eke Rjrhkdh` Phaeocs< Dhs`rjtj Thij Gkurhjr Troesgkri (DTGT)

Cjt x]eY nj o dhs`rjtj-thij eke-pjrhkdh` shaeoc,

x]eY V( )V( ) 9 e x]eY 9

Hg tfj sjquje`j, x]eY hs onskcutjcy suiioncj, h.j. || , tfje DTGT kg tfj sjquje`j,x]eY jxhsts.

Rrkpjrthjs kg Gkurhjr ^jprjsjetothke

Pyiijtry Rrkpjrty - ^joc Hioaheory shaeocs

Hg x(t) hs rjoc, Hixt e xt ^j xt,

GT hs `keluaotj syiijtrh` ioaehtudj spj`trui hs syiijtrh`.Hg x(t) hs hioaheory,

GT hs `keluatj smjw-syiijtrh`ioaehtudj spj`trui hs syiijtrh`.Hg x(t) hs `keluaotj syiijtrh`,xt xt

DTG P,

G.T.

DTGT


81/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


GT hs rjc ( Hg x(t) hs `keluaotj oeth-syiijtrh`

xt xt GT hs hiahery ( Toncj. @kiijets ke @TGT nosjd ke shaeoc prkpjrthjs

PC. Ek. Phaeoc Rrkpjrty @kiijets ke G. T.

?. ^joc @keluaotj syiijtrh`

>. Hioaheory @keluaotj oeth-syiijtrh`

0. @keluaotj syiijtrh` ^joc

4. @keluaotj oeth-syiijtrh` Hioaheory

Pòchea Rrkpjrty

x(o t)

||( o) Chejor sòchea he dkiohe `krrjspkeds tk o chejor sòchea he grjquje`y dkiohe.

n)

Qfjejvjr o shaeoc hs `kiprjssjd he thij dkiohe (o5?), ht cjods tk jxpoeshke he grjquje`y

oed vh`j vjrso.

@kevkcuthke Rrkpjrty

Gkr `ketheukus-thij shaeocs,

xt lyt lxt yt l. l@kevkcuthke he thij dkiohe rjsucts he iucthpchòthke he grjquje`y dkiohe.

Rorsjvoc Tfjkrji

Jejray kr pkwjr he tfj thij dkiohe rjprjsjetothke hs jquoc tk jejray kr pkwjr he grjquje`y

dkiohe rjprjsjetothke.

|| || , || ||

@TGT


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@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


Ej`jssory `kedhthke gkr oppcyhea gheoc vocuj tfjkrji hs tfot pkcjs kg (?-z V(z) sfkucd njstrh`tcy heshdj tfj ueht `y`cj.

@foro`tjrhzothke kg CTH Pystji grki F(z) oed ^K@

Gha. CTH Pystji

O CTH systji òe nj `foro`tjrhzjd gkr òusochty, stonhchty oed ijikrycjss prkpjrthjs nosjd ke

^K@ kg tfj systji gue`thke, F(z).

?.

Hg ^K@ he`cudjs ueht `hr`cj, tfje systji hs stoncj.

>.

Hg ^K@ hs kutshdj o `hr`cj, tfje systji hs òusoc.

0. Hg ^K@ hs heshdj o `hr`cj, tfje systji hs oeth-òusoc.

4. Hg ^K@ hs occ z, tfje systji hs ijikrycjss.

:.

Hg ^K@ hs strh`tcy o rhea, tfje systji hs eke-òusoc.

7.

Hg ^K@ he`cudjs ueht-`hr`cj, tfje tfj systjis hipucsj rjspkesj hs skcutjcy suiicj efje`j DTGT kg tfj sjquje`j jxhsts.

Ghedhea Hevjrsj \.T. ahvje V(z) oed ^K@

Ny Hespj`thke oed Rorthoc Gro`thke

Ahvje V(z) he rothkeoc gkri, spcht ht hetk porthoc gro`thkes oed nosjd ke ^K@, ghed x]eY ny

hespj`thke.

Ny Dhvhshke

Ahvje V(z) he rothkeoc gkri, pjrgkri tfj dhvhshke nosjd ke tfj `kedhthke tfot x]eY hs òusoc kr

oeth-òusoc oed ghed V(z) he jxpoeshke gkri.

Ny Rkwjr Pjrhjs Jxpoeshke

Ahvje V(z) he o stoedord gkri, ghed tfj jxpoeshke kg V(z), tfje x]eY

x]eY

f]eY y]eY

f]eY

\T

F(z) = ^K@


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@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


chi gt chi s GsTfhs tfjkrji dkjs ekt oppcy tk tfj rothkeoc gue`thke G(s) he wfh`f tfj krdjr kg euijrotkr

pkcyekihoc hs jquoc tk kr arjotjr tfoe tfj krdjr kg djekiheotkr pkcyekihoc.

Gheoc Wocuj Tfjkrji

Hg g(t) oed hts djrhvothvj g(t) orj Copco`j troesgkrioncj, tfjechi gt chi s Gs

Gkr oppcyhea gheoc vocuj tfjkrji, ht hs rjquhrjd tfot occ tfj pkcjs kg nj he tfj cjgt focg kg s-pcoej (strh`tcy) h.j. pkcjs ke oxhs ocsk ekt occkwjd.@kevkcuthke tfjkrji

. .

Copco`j troesgkri kg tfj pjrhkdh` gue`thke

Hg g(t) hs pjrhkdh` gue`thke whtf pjrhkd T, tfje

gt 9 . G(s) wfjrj G(s) 9 j gttCopco`j troesgkri kg stoedord gue`thkes

Toncj< Copco`j troesgkri kg stoedord gue`thkes

P. Ek Gue`thke, g(t) Copco`j troesgkri kg g(t), C{g(t) 9 G(s)

?. >. u(t)

0.

4. u(t) s :. j. ut s 7. t.u(t) s 1. t. ut e s

6. g(t).jut G(s-o)8. Phe ot. u(t) s


87/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


?2. @ks ot. u(t) s s

??. shefot. u(t) s ?>. `ksfot. u(t) s s ?0. g(t) s.G(s)-g(k)?4. g(t) s. Gs s.gk) g (k)?:. s G(s)?7. .

?1. g(t-o).u(t-o) j. Gs?6. t. Gt . ()?8. g(t ) ||. Gs>2. g(ot)

|| Gs >?. gt

gt9 .

Gs. Gs wfjrj * hs `kevkcuthke kpjrotkr

>> j. `ks t s s

>0 j shet s >4 .gt Gss >:

>7

Oppch`othkes

?.

CT hs ajejrochzothke kg @TGT gkr `ketheukus-thij shaeocs oed fje`j shaeoc `oe nj

`foro`tjrhzjd ot oey ajejrochzjd grjquje`y.

>. CT hs fjcpguc tk pjrgkri troeshjet oed stjody stotj oeocyshs kg oey CTH systji gkr oey

ornhtrory heput oed hehthoc `kedhthkes.


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@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


Occ poss systji

Pystji wfh`f possjs occ tfj grjquje`hjs hs òccjd occ-poss systji

j M= wfjrj M hs o `kestoet.Gkr o occ-poss ghctjr, systji gue`thke òe nj ajejrochzjd os ()

Gkr o occ-poss systji, pkcjs oed zjrkjs gkri o `keluaotj rj`hprkòc pohrs. Tfjrjgkrj gkr o occ-poss

systji whtf `kipcjx hipucsj rjspkesj, hg tfjrj hs zjrk t , tfje tfjrj hs pkcj t oed vh`jvjrso. Gkr oe occ-poss systji whtf rjoc hipucsj rjspkesj, hg tfjrj hs o zjrk ot o, tfje tfjrj hs o

zjrk ot oed tfjrj orj pkcjs ot oed .Ioaehtudj troesgjr gue`thke

O systji gue`thke z hs `ccj iaehtuj tresgjr gue`thke, hg hts kg gkri, . ( )Tfjrjgkrj, gkr ioaehtudj troesgjr gue`thke, pkcjs gkri `keluaotj rj`hprkòc pohrs. Fje`j hg tfjrj

hs pkcj t , tfjrj hs pkcj t . Poij oppchjs gkr zjrkjs ocsk.Poipchea

Tk ajt dhs`rjtj-thij shaeoc grki oeocka shaeoc, soipchea hs pjrgkrijd ke oeocka shaeoc. Cjt x(t)

nj oeocka shaeoc & P(t) nj hipucsj trohe,

P(t) 9 tT wfjrj T hs djshrjd soipchea hetjrvocPoipcjd shaeoc, x(t) 9 x(t) . s(t) 9 xT.tT

x m Ogtjr soipchea, shaeoc kntohejd onkvj hs sthcc he thij-dkiohe. Tk ajt GT he dhs`rjtj thij dkiohe,put T, wfh`f hs `ccj thij ekrichzthke

V(j9 Grki onkvj wj sjj tfot V(

j

hs pjrhkh` whtf pjrhk . Tk vkh chshea, tfjrj sfkucd ekt nj

oey hetjrvjethke njtwjje grjquje`y noeds.

wfjrj hs shaeoc NQ tk ovkhd ochoshea (Eyquhst soipcheatfjkrji)

Os V(jhs pjrhkdh`, DTGT hs djs`rhnjd os kej pjrhkd kg @TGT, hg tfjrj hs ek ochoshea.Tk ajt xt grki x]eY, usj ckw poss ghctjr kg hipucsj rjspkesj ft 9 she `(t/T) os he ghaurjnjckw.


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@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


@krrjcothke oed @kvorhoe`j

Hg x hs o rjoc oed stothkeory shaeoc, tfje outk-`krrjcothke oed `kvorhoe`j gue`thkes òe njdjghejd os njckw,Outk-`krrjcothke gue`thke, î x. xOutk-`kvorhoe`j gue`thke, i x i. x iFjrj i xoed x hs odvoe`jd vjrshke kg xny i soipcjs.Rkwjr spj`troc djeshty kg x hs ahvje os (j) Gê

(j) ^j ^ (j). j.


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_uh`m ^jgrjsfjr Auhdj @ketrkc Pystji


@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


T(s) 9

()

Fjrj ejaothvj shae he djekiheotkr hs `keshdjrjd gkr pkshthvj gjjdno`m oed vh`j vjrso.PP Kpe ckkp tresr gue`thkeJrrkr troesgjr gue`thke 9

Jggj`t kg Gjjdno`m

?.

Jggj`t kg Gjjdno`m ke Ptonhchty

Ptonhchty hs o ekthke tfot djs`rhnjs wfjtfjr tfj systji whcc nj oncj tk gkcckw tfj heput

`kiioed.

O systji hs sohd tk nj uestoncj, hg hts kutput hs kut kg `ketrkc kr he`rjosjs whtfkut nkued.

>.

Jggj`t kg Gjjdno`m ke kvjrocc aohe

Ejaothvj gjjdno`m dj`rjosjs tfj aohe kg tfj systji oed pkshthvj gjjdno`m he`rjosj tfjaohe kg tfj systji.

0. Jggj`t kg Gjjdno`m ke Pjeshthvhty

Tfj sjeshthvhty kg tfj aohe kg tfj kvjrocc systji T tk tfj vorhothke he A hs djghejd os

P9 9 Phihcorcy, P 9 9 ?

4. Ejaothvj gjjdno`m iomjs tfj systji cjss sjeshthvj tk tfj poroijtjr vorhothke.

:.

Ejaothvj gjjdno`m hiprkvjs tfj dyeoih` rjspkesj kg tfj systji7.

Ejaothvj gjjdno`m rjdu`js tfj jggj`t kg dhsturnoe`j shaeoc kr ekhsj.

1. Ejaothvj gjjdno`m hiprkvjs tfj noedwhdtf kg tfj systji.

Phaeoc Gckw Aropfs (PGA)

O shaeoc gckw aropf hs o aropfh`oc rjprjsjetothke kg pkrtroyhea tfj heput-kutput rjcothkesfhps

njtwjje tfj vorhoncjs kg o sjt kg chejor ocajnroh` jquothkes. Ocsk gkcckwhea orj tfj nosh`prkpjrthjs kg shaeoc gckw aropfs.

?. O shaeoc gckw aropf oppchjs tk kecy chejor systjis.>. Tfj jquothkes nosjd ke wfh`f o shaeoc gckw aropf hs drowe iust nj ocajnroh` jquothke he

tfj gkri kg jggj`ts os gue`thkes kg `ousjs.

0.

Phaeocs trovjc ockea nroe`fjs kecy he tfj dhrj`thke djs`rhnjd ny tfj orrkws kg tfjnroe`fjs.

4. Tfj nroe`f dhrj`thea grki ekdj ytk yrjprjsjets tfj djpjedje`j kg tfj vorhoncj ymupkeyl, nut ekt tfj djpjedje`j kg y upke y

:.

O shaeoc ytrovjcchea ockea o nroe`f njtwjje ekdjs yoed yhs iucthpchjd ny tfj aohe kgtfj nroe`f, , sk tfot shaeoc yhs djchvjrjd ot ekdj y.

Iskes he Gkriuco

Tfj ajejroc aohe gkriuco hs, T9 9

( )

Qfjrj T 9 Kvjrocc aohe njtwjje yhe oed ykutykut9 kutput ekdj vorhoncj


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@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??

< 262-7:1221:2,hegkBtfjaotjoòdjiy.`ki @kpyrhaft rjsjrvjd. Qjn< www.tfjaotjoòdjiy.`ki Roaj 8?

4.>< Thij Dkiohe Oeocyshs

Hetrkdu`thke

Qfjejvjr oe heput shaeoc kr jx`htothke hs ahvje tk o systji, tfj rjspkesj kr kutput kg tfj systjiwhtf rjspj`t tk thij hs mekwe os thij rjspkesj kg tfj systji. Tfj thij rjspkesj kg o `ketrkc

systji hs dhvhdjd hetk twk ports eoijcy, troeshjet oed stjody stotj rjspkesj.

Tktoc rjspkesj kg o systji 9 troeshjet rjspkesj + stjody stotj rjspkesj (kr @ (t) 9 @ tr (t) +

@ss(t))

Qfjrj ths kvjrocc rjspkesj kg tfj systji,ths troeshjet rjspkesj `kipkejet kg tfj systji oed

ths stjody stotj rjspkesj `kipkejet kg tfj systji

Gkcckwhea orj sochjet `foro`tjrhsth`s kg troeshjet rjspkesj kg o `ketrkc systji.

Tfhs port kg tfj thij rjspkesj wfh`f akjs tk zjrk ogtjr o coraj hetjrvoc kg thij.

Ht rjvjocs tfj eoturj kg rjspkesj (j.a. ks`hccotkry kr kvjr doipjd)

Ht ahvjs oe hedhòthke onkut tfj spjjd kg rjspkesj.

Ht dkjs ekt djpjed ke tfj heput shaeoc, rotfjr djpjeds ke eoturj kg tfj systji.

Gkcckwhea orj tfj sochjet prkpjrthjs kg stjody stotj rjspkesj kg o `ketrkc systji.

Tfj port kg tfj thij rjspkesj tfot rjiohes jvje ogtjr tfj troeshjets fovj dhjd kut hs

sohd tk nj stjody stotj rjspkesj.

Tfj stjody stotj port kg thij rjspkesj rjvjocs tfj o`ùro`y kg o `ketrkc systji.

Ptjody stotj jrrkr hs knsjrvjd hg tfj o`tuoc kutput dkjs ekt jxo`tcy iot`f whtf tfj heput.

Ht djpjeds ke tfj heput shaeoc oppchjd.

Thij ^jspkesj kg o Ghrst Krdjr @ketrkc Pystji

O ghrst krdr ketrkc systi hs ke kr wfhf tf fhafst pkwr k s he tf dekihetkr k htstroesgjr gue`thke hs jquoc tk ?. Tfus o ghrst krdjr `ketrkc systji hs jxprjssjd ny o troesgjr

gue`thke,9

.

Thij ^jspkesj kg o Ghrst Krdjr @ketrkc Pystji Punlj`tjd tk Zeht Ptjp Heput Gue`thke

Os tfj heput hs o ueht stjp gue`thke, r(t) 9 u(t) oed (s) 9 ?/s

Kutput hs ahvje ny `(t) 9 (?) u(t)Tfj jrrkr hs ahvje ny j(t) 9 r(t) `(t) 9 . u(t)Tfj stjody stotj jrrkr 9 hi j(t) 9 hi . u(t) 9 2Thij ^jspkesj kg o Ghrst Krdjr @ketrkc Pystji Punlj`tjd tk Zeht ôip Heput Gue`thke

Os tfj heput hs o ueht roip gue`thke, r(t) 9 t.u(t) oed ^(s) 9 ? / sKutput hs ahvje os `(t) 9 (

t T T ) u(t)

Tfj jrrkr hs ahvje ny j(t) 9 r(t) `(t) 9 ( T T ) u(t)


96/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


Tfj stjody stotj jrrkr hs 9 hi ( T T ) u(t) 9 TGrki onkvj wj sjj tfot tfj kutput vjck`hty iot`fjs whtf tfj heput vjck`hty nut coas njfhed tfj

heput ny thij T oed o pkshthkeoc jrrkr kg T uehts jxhsts he tfj systji.

Thij ^jspkesj kg O Ghrst Krdjr @ketrkc Pystji Punlj`tjd tk Zeht Hipucsj Heput Gue`thke

Os ^(s) 9 ?, @(s) 9 `(t) 9 ( T

Tfj jrrkr hs ahvje ny j(t) 9 r(t) `(t)Tfj stjody stotj jrrkr hs 9 hi s J(s) 9 2Thij ^jspkesj kg Pj`ked Krdjr @ketrkc Pystji

O sked krdr ketrkc systi hs ke kr wfhf tf fhafst pkwr s he tf dekihetkr k htstroesgjr gue`thke hs jquoc tk >. O ajejroc jxprjsshke gkr tfj T.G. kg o sj`ked krdjr `ketrkc systjihs ahvje ny,9

@foro`tjrhsth` Jquothke

Tfj `foro`tjrhsth` jquothke kg o sj`ked krdjr `ketrkc systji hs ahvje ny

s s 9 2Tfj rkkts orj l 9 9 l Fjrj hs òccjd eoturoc grjquje`y kg ks`hccothkes,9 hs òccjd doipjd grjquje`y kg ks`hccothkes, hs òccjd doiphea rothk oed oggj`ts doiphea oed hs òccjd doiphea go`tkr kr doiphea `kjggh`hjet.Nosjd ke rkkts kg `foro`tjrhsth` jquothke, gkcckwhea òe nj fhafchaftjd.

Tfj rjoc port kg tfj rkkts djektjs tfj doiphea

Hioaheory port djektjs tfj doipjd grjquje`y kg ks`hccothke

Pustohejd ks`hccothkes orj knsjrvjd hg tfj rkkts orj cyhea ke hioaheory oxhs (l oxhs). Os he`rjosjs, systji nj`kijs cjss ks`hccotkry oed ikrj scuaahsf.

Toncj< Eoturj kg systji rjspkesj nosjd ke

P. Ek ôeaj kg vocujs kg

Pystji Eoturj kg systji

rjspkesj

Eoturj kg rkkts

`foro`tjrhsth`

jquothke

?. 2 Zedoipjd Pustohejd/uedoipjd Rurcy hioaheory

>. 2 3 3 Zedjrdoipjd Ks`hccotkry @kipcjx0. @rhthòccy

doipjd

Eke-ks`hccotkry ^joc oed jquoc

4. 5 Kvjrdoipjd Eke-ks`hccotkry ^joc oed dhggjrjet


97/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


Thij ^jspkesj kg o Pj`ked Krdjr @ketrkc Pystji Punlj`tjd tk Zeht Ptjp Heput Gue`thke

`(t) 9

she t utwfjrj

9

ed te

Tfj stjody stotj jrrkr hs 9 hi she t te u(t) 9 2

Fjrj tfj thij `kestoet T9 (? / ) oed Ppjjd kg tfj systji .

Troeshjet ^jspkesj Ppj`hgh`othke kg Pj`ked Krdjr Zedjr-Doipjd @ketrkc Pystji

Tfj thij rjspkesj kg oe uedjr doipjd `ketrkc systji jxfhnhts doipjd ks`hccothkes prhkr tk

rjo`fhea stjody stotj.

Gha. Zeht stjp rjspkesj kg sj`ked krdjr uedjrdoipjd `ketrkc systji

(?)

Djcoy Thij (t

d

)

Ht hs tfj thij rjquhrjd gkr tfj rjspkesj tk rhsj tk :2% kg tfj gheoc vocuj grki zjrk, he ghrst

ottjipt. td9.

(>)

^hsj Thij (t

r

)

Tfj thij ejjdjd gkr tfj rjspkesj tk rjo`f grki ?2% tk 82% (gkr kvjrdoipjd systji) kr 2 tk?22% (gkr uedjrdoipjd systjis) kg tfj djshrjd vocuj kg tfj kutput ot tfj vjry ghrst hestoet.

tr 9

= wfr te-?

(0)

Rjom Thij (t

p

)

Ht hs tfj thij rjquhrjd gkr tfj rjspkesj tk rjo`f tfj pjom vocuj ot tfj ghrst hestoet.

tp 9

Ocsk tfj rjspkesj jxfhnhts kvjrsfkkt oed uedjrsfkkt ot tfj hestoets,

T e

?22%

>%

@(t)

?

2.:

2

Iox. kvjrsfkkt

td tr tp ts


98/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


t wfjrj e Tfj ckòc kvjrsfkkts k`ùr gkr e 9?, 0, . . . . . . . oed ckòc uedjrsfkkts k`ùr gkr e 9>, 4, . . . . .

Fje`j tfj ghrst uedjrsfkkt k`ùrs ot tfj hestoet, .(4)Ioxhiui Kvjrsfkkt (I

p

)

Tfj ioxhiui pkshthvj djvhothke kg tfj kutput whtf rjspj`t tk hts djshrjd vocuj/ stjody stotj

vocuj hs òccjd Ioxhiui kvjrsfkkt.

Rjr`jetoaj kvjrsfkkt 9 ?22 9

?22%% Ip 9 jxp (- / ) ?22

(:)

Pjttchea Thij (t

s

)

Gkr >% tkcjroe`j noed, tfj sjttchea thij hs ahvje ny, ts9 4. Gkr :% tkcjroe`j noed, tfj sjttchea thij hs ahvje ny, ts9 0.

Thij ^jspkesj kg Tfj Fhafjr Krdjr Pystji Oed Jrrkr @kestoets

Ptjody stotj jrrkr, 9 hi 9 hi Zshea Gheoc vocuj tfjkrji, 9 hi Nut @(s) 9 J(s) A(s)

J(s) 9

9

9

9 hi Typj oed Krdjr kg Pystji

Gkr tfj kpje-ckkp troesgjr gue`thke,

Tfj typj hedhòtjs tfj euinjr kg pkcjs ot tfj krhahe oed tfj krdjr hedhòtjs tfj tktoc

euinjr kg pkcjs.

Tfj typj kg tfj systji djtjrihejs stjody stotj rjspkesj oed tfj krdjr kg tfj systji

djtjrihejs troeshjet rjspkesj.

Cjt

9

hi 9 Rkshthke jrrkr `kestoet

Cjt 9 hi 9 Wjck`hty jrrkr `kestoetCjt 9 hi 9 O``jcjrothke jrrkr `kestoet

Toncj. Ptjody stotj jrrkr os o vorhothke kg typj kg tfj systji

Typj 2 ? > 0

Ptjp O( m) 2 2 2ôip O/ 2 2Roronkch` O/ 2


99/231



@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


4.0< Ptonhchty ^kutf Furwhtz @rhtjrhke

Hetrkdu`thke

Oey systji hs sohd tk nj o stoncj systji, hg tfj kutput kg tfj systji hs nkuedjd gkr o nkuedjdheput (stonhchty he NHNK sjesj) oed ocsk he tfj onsje`j kg tfj heput, kutput sfkucd tjed tk zjrk

(osyiptkth` stonhchty).

Nosjd ke onkvj dhsùsshke, systjis orj `cosshghjd os njckw,

?) Onskcutjcy stoncj systjis

>) Zestoncj systjis0)

Ioraheoccy stoncj kr chihtjdcy stoncj systjis

4)

@kedhthkeoccy stoncj systjis

Djpjedhea ke tfj ckòthke kg pkcjs gkr o `ketrkc systji, stonhchty kg tfj systji òe nj

`foro`tjrhzjd he gkcckwhea woys.

Ptonhchty kg oey systji djpjeds kecy ke tfj ckòthke kg pkcjs nut ekt ke tfj ckòthke kg zjrks.

Hg tfj pkcjs orj ckòtjd he cjgt shdj kg s-pcoej, tfje tfj systji hs stoncj.

Hg oey kg tfj pkcjs hs ckòtjd he rhaft focg kg s-pcoej, tfje tfj systji hs uestoncj.

Hg tfj rjpjotjd rkkts orj ckòtjd ke hioaheory oxhs he`cudhea tfj krhahe, tfj systji hs

uestoncj.

Qfje eke-rjpjotjd rkkts orj ckòtjd ke hioaheory oxhs, tfje tfj systji hs ioraheoccy

stoncj.

Os o pkcj opprko`fjs krhahe, stonhchty dj`rjosjs.

Tfj pkcj wfh`f hs `cksjst tk tfj krhahe hs òccjd dkiheoet pkcj. Hg tfj vorhoncj poroijtjr hs vorhjd grki 2 tk oed tfj pkcjs orj ocwoys ckòtjd ke cjgt shdj

kg s-pcoej, tfje tfj systji hs onskcutjcy stoncj.

Qfje vorhoncj poroijtjr hs vorhjd grki 2 tk , hg skij pkhet kewords, tfjrj hs o pkcj herhaft focg kg P-pcoej. Tfje systji hs òccjd `kedhthkeoccy stoncj oed typhòccy stonhchty hs

`kedhthkejd ke vorhoncj poroijtjr.

Onskcutj Ptonhchty Oeocyshs

Onskcutj stonhchty oeocyshs hs ny tfj quochtothvj oeocyshs kg stonhchty oed hs djtjrihejd ny ckòthke

kg rkkts kg `foro`tjrhsth` jquothke he s-pcoej.

^jcothvj Ptonhchty Oeocyshs

Tfj rjcothvj stonhchty òe nj spj`hghjd ny rjquhrhea tfot occ tfj rkkts kg tfj `foro`tjrhsth`

jquothke nj ikrj ejaothvj tfoe o `jrtohe vocuj, h.j. occ tfj rkkts iust chj tk tfj cjgt kg tfj chej=

s 9 - ?, (?5 2). Tfj `foro`tjrhsth` jquothke kg tfj systji uedjr study hs ikdhghjd ny sfhgtheatfj krhahe kg tfj s pcoej tk s 9 - ?, h.j. ny sunsthtuthke s 9 z ?. Hg tfj ejw `foro`tjrhsth`jquothke he z sothsghjs tfj ^kutf `rhtjrhke, ht hipchjs tfot occ tfj rkkts kg tfj krhaheoc

`foro`tjrhsth` jquothke orj ikrj ejaothvj tfoe ?.Ocsk hg ht hs rjquhrjd tk ghed kut euinjr kg rkkts kg `foro`tjrhsth` jquothke njtwjje tfj chejsP oed P , pjrgkri ^kutf oeocyshs ny putthea P z ?, oed ghed kut euinjr kg rkktstk rhaft kg

P . Phihcorcy ghed kut euinjr kg rkkts tk tfj rhaft kg

P . Tfj dhggjrje`j

njtwjje onkvj twk euinjrs ahvjs tfj euinjr kg rkkts kg `foro`tjrhsth` jquothke njtwjje oed .


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@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


^kutf-Furwhtz @rhtjrhke

Tfj ^kutf-Furwhtz `rhtjrhke rjprjsjets o ijtfkd kg djtjrihehea tfj ckòthke kg pkcjs kgpkcyekihoc whtf `kestoet rjoc `kjggh`hjet whtf rjspj`t tk tfj cjgt focg oed tfj rhaft focg kg tfj s-pcoej. ^kutfFurwhtz `rhtjrhke iohecy ahvjs o gcjxhnhchty tk djtjrihej tfj stonhchty kg tfj `cksjdckkp `ketrkc systji whtfkut o`tuoccy skcvhea gkr pkcjs. s s s s . . s 2

P n9 P n9 P n n 2 9

P

9

P d d9 ? 9

Hg oey pkwjr kg s hs ihsshea he tfj `foro`tjrhsth` jquothke, ht hedhòtjs tfot tfjrj hs ot cjost kej

rkkt whtf pkshthvj rjoc port, fje`j tfj systji hs uestoncj. Hg tfj `foro`tjrhsth` jquothke `ketohes

kecy kdd kr jvje pkwjrs kg s, tfje rkkts orj purjcy hioaheory. Tfus, tfj systji whcc fovj

sustohejd ks`hccothkes he kutput rjspkesj. Ocsk wfje ^kutf Furwhtz `rhtjrhke hs oppchjd,gkcckwhea dhgghùcthjs òe nj go`jd.

Dhgghùcty ?< Qfje occ tfj jcjijets he oey kej rkw kg tfj ^kutf orroy orj zjrk.

Tfhs shtuothke hs kvjr`kij ny rjpco`hea tfj rkw kg zjrks he tfj ^kutf orroy ny o rkw kg

`kjggh`hjets kg tfj pkcyekihoc ajejrotjd ny tomhea tfj ghrst djrhvothvj kg tfj ouxhchorypkcyekihc ed ^kutfs tst hs prkrid s usuc.


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@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


4.4< ^kkt Ckùs Tj`fehquj

Hetrkdu`thke

^kkt ckùs hs o ckùs kg pkcjs kg troesgjr gue`thke kg o `cksjd ckkp `ketrkc systji wfje tfjvorhoncj poroijtjr hs vorhjd grki 2 tk . Djpjedhea ke eoturj kg vorhoncj poroijtjr oed roeajkg vorhothke, rkkt ckùs òe nj `cosshghjd os njckw.

^kkt Ckùs (^C) - (M hs vorhjd grki 2 tk ) @kipcjijetory C - (M hs vorhjd grki 2 tk ) @kipcjtj ^C - (M hs vorhjd grki tk ) ^kkt `kuetjr - (Iucthpcj poroijtjr vorhothke )

@foro`tjrhsth` jquothke kg onkvj systji hs ? + A(P)

F(P) 9 2. Zsuoccy wfhcj pcktthea rkkt ckùs, o

gkrword potf aohe, M wfh`f hs hefjrjetcy prjsjet he A(P) hs `keshdjrjd os hedjpjedjet vorhoncjoed rkkts kg `foro`tjrhsth` jquothke orj `keshdjrjd os djpjedjet vorhoncjs. Oey rkkt kg 2sothsghjs gkcckwhea twk `kedhthkes,

o)

|| n) 2wfr M 2, , ,

ûcjs gkr tfj @kestru`thke kg ^kkt Ckùs (^C)

Cjt R nj tfj euinjr kg kpje-ckkp pkcjs oed \ nj tfj euinjr kg kpje ckkp zjrkjs kg o `ketrkcsystji. Tfje tfj gkcckwhea orj tfj sochjet gjoturjs gkr `kestru`thke kg rkkt ckùs pckt.

?.

Tfj rkkt ckùs hs ocwoys syiijtrhòc onkut tfj rjoc oxhs.

>. Tfj rkkt ckùs ocwoys storts grki kpje-ckkp pkcjs gkr M92 oed jeds ot jhtfjr ghehtj kpje

ckkp zjrkjs kr heghehty gkr M .0. Tfj euinjr kg nroe`fjs kg rkkt ckùs tjriheothea ot heghehty hs jquoc tk (R-\) .4.

Tfj euinjr kg sjporotj nroe`fjs kg tfj rkkt ckùs jquocs jhtfjr tfj euinjr kg kpje ckkp pkcjs kr tfj euinjr kg kpje-ckkp zjrkjs, wfh`fjvjr hs arjotjr.

E 9 iox(R, \):.

O sj`thke kg rkkt ckùs chjs ke tfj rjoc oxhs, hg tfj tktoc euinjr kg kpje-ckkp pkcjs oed

zjrkjs tk tfj rhaft kg tfj sj`thke hs kdd oed hs fjcpguc he djtjrihehea prjsje`j kg rkktckùs ot oey pkhet ke rjoc oxhs.

7. Hg R 5\, (R- nrefs whcc trihet t ckea strhaft che syiptkts wfksjoeacjs orj os ahvje njckw,

= q 2,,, ..R-\-?

Hg \ 5 R, (\-R nrefs whcc strt t ckea strhaft che syiptkts wfks eacs ros ahvje njckw, = q 9 2, ?, > . . . . . . . . . . . (\-R-?)

1. Tfj osyiptktjs ijjt tfj rjoc oxhs ot `jetrkhd s os ahvje njckw Pui k rc prts k pkcs Pui k rc prts k zrksR

6.

Nrjom

owoy pkhet hs òcùcotjd wfje rkkt ckùs chjs njtwjje twk pkcjs oed nrjom

he

pkhet hs òcùcotjd wfje rkkt ckùs chjs njtwjje twk zjrks. Nrjom owoy / Nrjom he


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@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


pkhets orj djtjrihejd grki tfj rkkts kg tfj jquothke 2. Ocsk r nroe`fjs kg tfj rkkt

ckùs wfh`f ijjt ot o pkhet, nrjom owoy ot oe oeacj kg

.

8.

Oeacj kg djporturj hs òcùcotjd wfje tfjrj orj `kipcjx pkcjs. Ocsk, oeacj kg djporturj

grki oe kpje ckkp pkcj hs ahvje os njckw 2 = q9 2, ?, >, 0 . . . . . .Qfjrj hs tfj ejt `ketrhnuthke ot tfj pkcj kg occ ktfjr kpje ckkp pkcjs oed zjrks. Ocsk,oeacj kg djporturj hs toeajet tk rkkt ckùs ot `kipcjx pkcj.

?2.

Oeacj kg orrhvoc hs òcùcotjd wfje tfjrj orj `kipcjx zjrkjs. Ocsk, oeacj kg orrhvoc ot tfj

kpje ckkp zjrk hs ahvje os njckw 2 , q9 2, ?, >, 0 . . . . . .Ocsk, oeacj kg orrhvoc hs toeajet tk rkkt ckùs kg `kipcjx zjrk.

??.Tf vcu k M ed tf pkhet t wfhf rkkt ckus nref rksss tf hiahery xhs hsdjtjrihejd ny oppcyhea ^kutf `rhtjrhke tk tfj `foro`tjrhsth` jquothke. Tfj rkkts ot tfjhetjrsj`thke pkhet orj hioaheory. Ocsk tfj pkhets kg hetjrsj`thke orj `keluaotj, hg occ tfj

`kjggh`hjets kg Porj rjoc he tfj `foro`tjrhsth` jquothke.?>.Tf vcu k kpe ckkp ahe M t ey pkhet ke tfj rkkt ckùs òe nj òcùcotjd ny

ushea tfj ioaehtudj `rhtjrho,

M Rrkdut k pfskr ceatf rki stk kpe ckkp pkcsRrkdut k pfskr ceatf rki stk kpe ckkp zrks


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@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


4.:< Grjquje`y ^jspkesj Oeocyshs Zshea Eyquhst pckt

Grjquje`y Dkiohe Ppj`hghòthkes

@keshdjr o `cksjd ckkp `ketrkc systji kg kpje ckkp troesgjr gue`thke A(P) oed gjjd no`mtroesgjr gue`thke, F(P). Hg tfj systji fos ejaothvj gjjdno`m, tfj kvjrocc troesgjr gue`thke hs ahvje

ny I(P) 9

.

Rut P 9 L, | I(L| 9 | || |Tfj pckt kg |I(L| whtf rjspj`t tk hs sfkwe he ghaurj njckw.

Tfj rjspkesj goccs ny 0 dN ot grjquje`y , grki hts ckw grjquje`y vocuj, òccjd ùt-kgg grjquje`yoed tfj grjquje`y roeaj 2 tk hs òccjd tfj noedwhdtf kg tfj systji. Tfj rjskeoet pjom, Ik`ùrs ot rjskeoe`j grjquje`y,. Tfj noedwhdtf hs djghejd os tfj grjquje`y ot wfh`f tfjioaehtudj aohe kg grjquje`y rjspkesj pckt rjdu`js tk ?/ 9 2.121 (h.j. 0 dn) kg hts ckwgrjquje`y vocuj.

Gkr o sj`ked krdjr `ketrkc systji,

I(s) 9

Il

|Il|

N.Q. 9 9

I9 Rkcor Rckt

keshdr ketrkc systi k tresr uethke s. Tf sheuskhdc tresr uethke l hs `kipcjx gue`thke wfh`f hs ahvje os,

||

NOEDQHDTF

0 d n

2

Gha. @cksjd-ckkp grjquje`y rjspkesj kg o `ketrkc systji


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@rkss, ?2tf

Iohe, Loyoeoaor 4tf

Nck`m, Noeaockrj-??


l ^ l l il kr l | l | l 9 I Qfjrj I 9||ed l iy n rprsetd s pfskr k iaehtud I ed pfs eac . Os tfj heput grjquje`y hs vrhd rki 2 tk , tfj ioaehtudj I oed pfosj oeacj `foeaj oed fje`j tfj thp kg tfjpfskr l trs ckus he tf kipcx pce, wfh`f hs mekwe os pkcor pckt.@keshdjr o troesgjr gue`thke wfh`f `keshsts kg R pkcjs oed \ zjrkjs.

tresr uethke dkset kethe pkcs t krhahe, tfe tf pkcr pckt strts rki 2whtfeke-zjrk ioaehtudj oed tjriheotjs ot 82 R whtf zjrk ioaehtudj.

Hg tfj troesgjr gue`thke `keshsts kg pkcjs ot krhahe, tfje tfj pkcor pckt storts grki 2 whtf iaehtud ed eds t 2 R whtf zjrk ioaehtudj.Ppj`hoc @osjs kg CTH @ketrkc Pystjis

Ihehiui Rfosj Pystji

Hg A(P) fos ek pkcjs oed zjrkjs he tfj ^.F.P kg P-pce, tfe tf systi hs ccd ihehiui pfssysti. Os zrks r csk ke ct fc ke s-pcoej, hevjrsj systji kg o ihehiui pfosj systji hsocsk stoncj.

Eke Ihehiui Rfosj Pystji

Hg A(P) fos ot cjost kej pkcj kr zjrk he tfj ^.F.P kg P pcoej, tfje systji hs òccjd eke-ihehiui

pfosj systji. Ocsk tfj hevjrsj systji kg o eke-ihehiui pfosj systji hs uestoncj.

Occ Ross Pystji

Hg A(P) fos syiijtrh` pkcjs oed zjrkjs onkut tfj onkut tfj hioaheory oxhs, tfje systji hs òccjdOcc pss systi.Ocsk |A(| 9 M= wfjrj M hs o `kestoet.Chejor Rfosj Pystji

O systji hs òccjd chejor pfosj hg pckt kg whtf rspt tk hs cher.

= wfjrj M hs o `kestoet.

Eyquhst Rckt Eyquhst Ptonhchty @rhtjrho

Eyquhst `rhtjrhke hs fjcpguc tk hdjethgy tfj prjsje`j kg rkkts he o spj`hghjd rjahke nosjd ke pkcor

pckt kg A(P).F(P). Tfus, Eyquhst stonhchty oeocyshs hs ikrj ajejrochzjd tfoe ^kutf `rhtjrhke. Ny

hespj`thke kg pkcor pckt kg A(P).F(P) ikrj hegkriothke hs kntohejd tfoe tfj stonhchty kg tfj

`ketrkc systji.

Hg E hs tfj euinjr kg je`hr`cjijets kg A(s).F(s) orkued ( L2) he `kuetjr-`ck`mwhsj dhrj`thke,tfje

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