Signals and System - Stanley Chan

Ilkss Dgto cgr Yjmdkls kd` Yystoas Ytkdloy Ibkd _djvorsjty gc Ikljcgrdjk, Ykd @jomg

8

Kiedgwlo`moaodt Pbjs ilkss dgto js propkro` propkro` cgr OIO ;>;6 Ljdokr Ljdokr Yystoas Cud`ka Cud`kaodt odtkls kls kt tbo _djvorsjt vorsjty y gc Ikljcgrdjk, Ikljcgrdjk, Ykd @jomg jd Yuaaor Yuaaor 8>;;. Pbo prgmrka js suppgrto` hy _I Ykd @jomg Yuaaor Mrkùkto Pokibjdm Collgwsbjp. J wgul` gul` lje ljeo tg mjvo mjvo k spoij spoijkl kl tbkd tbkde e tg ^rgc ^rgc.. ^kul kul B. Yjom Yjomol ol cgr sbkr sbkrjd jdmm bjs bjs bkd` bkd` wrjtto wrjttod d dgtos, dgtos, wbjib wbjib hoikao hoikao tbo hkiehgd hkiehgdoo gc tbjs tbjs ilkss ilkss dgto. dgto. J klsg klsg wk wkdt dt tg tbkde tbkde ^rgc. Prugdm [. Dmuyod Dmuyod cgr prggc-rok`jdm prggc-rok`jdm kd` prgvj`jd prgvj`jdmm usocul coo`hkie coo`hkies. s. ^krt ^krt gc tbo LKPORakdusirjpt js propkro` hy Ar. Lostor Loo-ekdm Lju kd` Ar. Fksgd Fukdm. Pbo toxthgge uso` cgr tbjs igurso js Gppodboja kd` ]jlsey, Yjmdkls kd` Yystoas, ^rodtjio Bkll. 8d` O`jtjgd. Ytkdloy Ibkd Lk Fgllk, IK. Kumust, Kumust, 8>;;.

Igdtodts ; Cud`kaodtkls gc Yjmdkls ;.; ]bkt js k Yjmdkl7 . . . . . . . . . . . . . . . . . . ;.8 Xovjow gd Igaplox Duah ors . . . . . . . . . . . ;.9 Hksji Gp orktjgds gc Yjmdkls . . . . . . . . . . . . ;.0 ôrjg `jijty . . . . . . . . . . . . . . . . . . . . . . ;.? Ovod kd` G`` Yjmdkls . . . . . . . . . . . . . . . ;.3 Japulso kd` Ytop Cuditjgds . . . . . . . . . . . . ;.= ;.= Ig Igd dtjd tjdugus ugus-t -tja jaoo Ig Igap aplo lox x Oxpgd Oxpgdod odtj tjkl kl Cud udit itjg jgds ds ;.2 @jsiroto-tjao Igaplox Oxpgd pgdodtjkls . . . . . . .

. . . . . . . .

? ? 3 = ;; ;? ;= 88 80

. . . .

8= 82 99 9= 0;

9 Cgurjor Yorjos 9.; Ojmodcuditjgds gc kd LPJ Yystoa . . . . . . . . . . . . . . . . . . . . 9.8 Cgurjor Yorjos Xoprosodtktjgd . . . . . . . . . . . . . . . . . . . . . . 9.9 ^rgpor portjos gc Cgurjor Yorjos Igo﬉ go﬉ijodts . . . . . . . . . . . . . . . . .

09 09 0= ?0

0 Igdtjdugus-tjao Cgurjor Prkdscgra 0.; Jdsjmbt crga Cgurjor Yorjos . . . . . . . 0.8 Cgurjor Prkdscgra . . . . . . . . . . . . . 0.9 Xolktjgd tg Cgurjor Yorjos . . . . . . . . 0.0 Oxkaplos . . . . . . . . . . . . . . . . . 0.? ^rgpor portjos gc Cgurjor Prkdscgra . . . . . 0.3 0.3 Yyst Yystooa Kd Kdkl kly ysjs sjs usjd usjdmm Cgur gurjor jor Prkds rkdsccgra gra

?= ?= ?: 3; 30 33 3:

8 Cud`kaodtkls gc Yystoas 8.; Yystoa ^rgp ortjos . . . . . . . . . . . . . 8.8 Igdvglutjgd . . . . . . . . . . . . . . . . 8.9 Yysto stoa ^rgpor portjos kd` Japulso Xospgd pgdso 8.0 Igdtjdugus-tjao Igdvglutjgd . . . . . .

9

. . . .

. . . . . .

. . . .

. . . . . .

. . . .

. . . . . .

. . . .

. . . . . .

. . . .

. . . . . .

. . . . . . . . . . . .

. . . . . .

. . . . . . . . . . . .

. . . . . .

. . . . . . . . . . . .

. . . . . .

. . . . . . . . . . . .

. . . . . .

. . . . . . . . . . . .

. . . . . .

. . . . . . . . . . . .

. . . . . .

. . . . . . . . . . . .

. . . . . .

. . . . . . . . . . . .

. . . . . .

. . . . . . . . . . . .

. . . . . .

. . . . . . . . . . . .

. . . . . .

. . . . . .

0

IGDPODPY

? @jsiroto-tjao Cgurjor Prkdscgra ?.; ?.; Xovj Xovjow ow gd Ig Igd dtjd tjdugus ugus-t -tja jaoo Cgurj gurjor or Prkds rkdscg cgra ra ?.8 ?.8 @orj @orjv vjdm jdm @jsi @jsiro roto to-t -tja jaoo Cgurj gurjoor Prkdsc kdscgr gra a. . . fϊ ?.9 ?.9 ]by ]by js R js R ((o ) porjg`ji 7 . . . . . . . . . . . . ?.0 ?.0 ^rgp ^rgpoortjo rtjoss gc @jsir jsirot otoo-tja -tjaoo Cgur gurjor jor Prkds rkdsccgra gra ?.? Oxkaplos . . . . . . . . . . . . . . . . . . . . ?.3 @jsiroto-tjao Cjltors . . . . . . . . . . . . . . ?.= Kppod`jx . . . . . . . . . . . . . . . . . . . . 3 Ykapljdm Pbogroa 3.; Kdklgm tg @jmjtkl Igdvorsjgd . . . . . 3.8 Croquodiy Kdklysjs gc K/@ Igdvorsjgd 3.9 Ykapljdm Pbogroa . . . . . . . . . . . 3.0 @jmjtkl tg Kdklgm Igdvorsjgd . . . . . = Pbo =.; =.8 =.9 =.0

z -Prkdscgra Pbo z z -Prkdscgra . . . . . . . . . . . z-trkdscgra ^kjrs . . . . . . . . . . . ^rgp ortjos gc XGI . . . . . . . . . . Yysto Yystoa a ^rgpo ^rgportj rtjos os usjdm usjdm z z --trkdscgra

. . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . .

=9 =9 =0 =3 == =2 2> 2;

. . . .

29 29 20 2: :>

. . . .

:? :? ;> > ;> 8 ;> 0

Ibkptor ; Cud`kaodtkls gc Yjmdkls ;.;

]bkt js k Yjmdk dkll7

K sjmdkl js k qukdtjtktjvo òsirjptjgd gc k pbysjikl pbodgaodgd, ovodt gr prgioss. Ygao igaagd oxkaplos oxkaplos jdiluò6 jdiluò6 ;. Oloitrjikl iurrodt gr vgltkmo jd k ijriujt. 8. @kjly ilgsjdm vkluo gc k sbkro gc stgie lkst wooe. 9. Ku`jg sjmdkl6 sjmdkl6 igdtjd igdtjdugus-t ugus-tjao jao jd jts grjmjdkl grjmjdkl cgra, gr `jsiroto`jsiroto-tjao tjao wbod stgro` stgro` gd k I@. Agro proijsoly, k sjmdkl js k cuditjgd, usuklly gc gdo vkrjkhlo jd tjao . Bgwo Bgwovo vor, r, jd modorkl, sjmdkls ikd ho cuditjgds gc agro tbkd gdo vkrjkhlo, o.m., jakmo sjmdkls. Jd tbjs ilkss wo kro jdtorosto` jd twg typos gc sjmdkls6 ;. Igdtjd Igdtjdugus-t ugus-tjao jao sjmdkl x(t), wboro t js k rokl-vkluo` vkrjkhlo òdgtjdm tjao, j.o., t j.o., t X. ]o uso pkrodtbosjs ( ) tg òdgto k igdtjdugus-tjao sjmdkl.

∂

½

8. @jsiroto-tjao sjmdkl xUdQ, wboro d js kd jdtomor-vkluo` vkrjkhlo òdgtjdm tbo `jsiroto skaplos gc tjao, j.o., d V. ]o uso squkro squkro hrkie hrkieots ots U Q tg òdgto k `jsiroto-tjao sjmdkl. _dòr tbo ò﬋djtjgd gc k `jsiroto-tjao sjmdkl, xU; x U;..?Q js dgt ò﬋do`, cgr oxkaplo.

∂

½

?

3

;.8 ;.8

IBK^POX IBK^POX ;. C_D@KAOD C_D@KAODP PKLY GC YJMDKLY YJMDKLY

Xovj Xovjow ow gd Igap Igaplo lox x Dua Duahor horss

]o kro jdtorosto` jd tbo modorkl igaplox sjmdkls6 x(t)

∂I

kd`

xUdQ

∂ I,

wboro tbo sot gc igaplox duahors js ò﬋do` ks I

∂ X, f 1 ∖ ∘;.}

1 z z 1 1 x + x + f fy, y, x, y

{ | |

K igaplox duahor z duahor z ikd ikd ho roprosodto` jd Ikrtosjkd cgra ks 1 x + + fy z 1 x fy , gr jd pglkr cgra ks 1 ro fβ . z 1

Pbogroa ;. Oulor‚s Cgraulk + f sjd sjd β. o fβ 1 igs β + f

(;.;)

_sjdm Oulor‚s cgraulk, tbo rolktjgd hotwood x, x , y, y , r, r , kd` β js mjvod hy



1 r igs β x 1 r 1 r sjd β y 1 r

kd`

 

r 1 x8 + y 8 , β 1 tkd∘; xy .

Cjmuro ;.;6 K igaplox duahor z ikd ho oxprosso` jd jts Ikrtosjkd cgra z 1 x 1 x + fy, fy , gr jd jts pglkr cgra z cgra z 1 ro 1 ro fβ .

=

;.9. HKYJI GÔXK GÔXKPJGD PJGDY Y GC YJMDKLY YJMDKLY

K igaplox igaplox duahor duahor ikd ho `rkwd `rkwd gd tbo igaplo igaplox x plkdo ks sbgwd sbgwd jd Cjm. Cjm. ;.;. Pbo jakmjdkry kxjs, kd` tbo x-kxjs gc tbo y-kxjs gc tbo igaplox plkdo js edgwd ks tbo jakmjdkry igaplox plkdo js edgwd ks tbo rokl tbo rokl kxjs. K igaplox duahor duahor js udjquoly udjquoly ò﬋do` hy fβ 1 x + + fy z 1 x fy jd tbo Ikrtosjkd cgra, gr z 1 ro jd tbo pglkr cgra. Igdvo vort rt tbo cgllgwjdm cgllgwjdm igaplox duahors duahors crga Ikrtosjkd Ikrtosjkd cgra tg pglkr Oxkaplo. Igd cgra6 cgra6 (k) ; + 8 f5 f 5 (h) ; f. f .

∘

Cgr (k), wo kpply Oulor‚s cgraulk kd` ﬋d` tbkt r 1

∖

;8 + 88 1

∖

?,

kd` β 1 tkd∘;

Pborocgro, ; + 8 f 8 f 1

∖

 ≂

r 1 Pborocgro,

◣

∖

;8 + (∘;)8 1 8, kd` β 1 tkd∘; ;

∘ f 1 f 1

39. 39.30◣ .

f 39..30 ?o f39 .

Cgr (h), wo kpply Oulor‚s cgraulk kmkjd kd` ﬋d` tbkt



8 ;

∘ 

; 1 ;

∘0?◣.

∖

8o∘ fώ /0 .

Xoikll tbkt6 ώ jd rk`jkd 1 ;2>◣ jd òmroo. gc f f . Oxkaplo. Ikliulkto tbo vkluo gc f fώ/ 8 f ώ/ 8 ) 1 o ∘ώ/8 >.8>=2. f f 1 (o fώ/8

≂

;.9 ;.9 ;.9. ;.9.; ;

Hksj Hk sjii Gpor Gporkt ktjg jgds ds gc gc Yjmd Yjmdkl klss Pjao Pjao Yb Ybjc jctt

Cgr kdy t>

∂ X kd` d ∂ V, tjao sbjct js kd gporktjgd ò﬋do` ks x(t) ∘↘ x(t ∘ t ) xUdQ ∘↘ xUd ∘ d Q. >

>

>

(;.8)

Jc t> 4 >, tbo tjao sbjct sbjct js edgwd edgwd ks “òlk “òlky‑. y‑. Jc t> < >, tbo tjao sbjct js edgwd ks “k`vkdio‑. sbgws k igdtjd igdtjdugus-t ugus-tjao jao sjmdkl x(t). K tjaotjaoOxkaplo. Jd Cjm. ;.8, tbo loct jakmo sbgws sbjcto` vorsjgd x vorsjgd x((t 8) js sbgwd jd tbo rjmbt jakmo.

∘

2


Cjmuro ;.86 Kd oxkaplo gc tjao sbjct.

;.9. ;.9.8 8

Pjao Pjao Xov Xovorsk orskll

Pjao rovorskl js ò﬋do` ks

x(t) xUdQ

∘↘ x(∘t) ∘↘ xU∘dQ,

(;.9)

wbjib ikd ho jdtorproto` ks tbo “﬈jp gvor tbo y-kxjs‑. y -kxjs‑.

Oxkaplo.

Cjmuro Cjmuro ;.96 Kd oxkaplo gc tjao rovorskl.

;.9. ;.9.9 9

Pjao Pjao Yi Yikl kljd jdm m

Pjao sikljdm js tbo gporktjgd wboro tbo tjao vkrjkhlo vkrjkhlo t t js aultjpljo` hy k igdstkdt k6 k 6 x(t)

∘↘ x(kt) kt),

k 4 >

(;.0)

Jc k k 4 ;, tbo tjao siklo gc tbo rosultkdt sjmdkl js “òijakto`‑ (spoo` up). Jc > < k < ;, tbo tjao siklo gc tbo rosultkdt sjmdkl js “oxpkdò`‑ (slgwo` `gwd).

;.9.0 ;.9.0

Igahjd Igahjdktj ktjgd gd gc Gporktj Gporktjgds gds

Jd modorkl, ljdokr gporktjgd (jd tjao) gd k sjmdkl x( ks y((t) 1 x( x(t) ikd ho oxprosso` ks y x (kt h), k, h X. Pboro kro twg aotbg`s tg òsirjho tbo gutput sjmdkl y( y (t) 1 x( x (kt h).

∂

∘

∘

:

;.9. HKYJI GÔXK GÔXKPJGD PJGDY Y GC YJMDKLY YJMDKLY

Cjmuro ;.06 Kd oxkaplo gc tjao sikljdm.

Aotbg` K6 “Ybjct, tbod Yiklo‑ (Xoigaaodò`) ;. @o﬋do v @o﬋do v((t) 1 x( x (t

∘ h),

8. @o﬋do y @o﬋do y((t) 1 v( v (kt) kt) 1 x( x (kt

∘ h).

Aotbg` H6 “Yiklo, tbod Ybjct‑ ;. @o﬋do v @o﬋do v((t) 1 x( x (kt), kt), 8. @o﬋do y @o﬋do y((t) 1 v( v (t

∘

h ) k

1 x( x (kt

∘ h).

Oxkaplo. Cgr tbo sjmdkl x sjmdkl x((t) sbgwd jd Cjm. ;.?, seotib x(9 x (9tt

∘ ?).

Cjmuro ;.?6 Oxkaplo ;. x(9t (9t ∘ ?).

Oxkaplo. Cgr tbo sjmdkl x sjmdkl x((t) sbgwd jd Cjm. ;.3, seotib x(; x (;

∘ t).

;>


Cjmuro ;.36 Oxkaplo 8. x(∘t + ;).

Cjmuro ;.=6 @oijaktjgd kd` oxpkdsjgd.

;.9.? ;.9.?

@oija @oijaktj ktjgd gd kd` Oxpkds Oxpkdsjgd jgd

@oijaktjgd @oijaktjgd kd` oxpkdsjgd oxpkdsjgd kro stkd`kr` stkd`kr` `jsiroto`jsiroto-tjao tjao sjmdkl prgiossjdm prgiossjdm gporktjgds. gporktjgds.

@oijaktjgd. @oijaktjgd js ò﬋do` ks y@ UdQ 1 xU x UA dQ,

(;.?)

gc L d 1 jdtomor aultjplo gc L gtborwjso. gtborwjso.

(;.3)

cgr sgao jdtomors A jdtomors A .. A js A js ikllo` tbo òijaktjgd ckitgr.

Oxpkdsjgd. Oxpkdsjgd js ò﬋do` ks yO UdQ 1

 x >,

L js ikllo` tbo oxpkdsjgd ckitgr.

d L

,

;;

;.0. ;.0. ÔXJG@ ÔXJG@JIJ JIJPZ PZ

Cjmuro ;.26 Oxkaplos Oxkaplos gc òijaktjgd òijaktjgd kd` oxpkdsjgd cgr A 1 A 1 8 kd` L kd` L 1 1 8.

;.0 ;.0

ôrjg` orjg`ji jijt jty y

;.0. ;.0.; ;

@o﬋d @o﬋djt jtjg jgds ds

@o﬋djtjgd ;. K igdtjdugus tjao sjmdkl x(t) js porjg`ji jc tboro js k igdstkdt P 4 > suib tbkt + P )), (;.=) x(t) 1 x( x (t + P cgr kll t

∂ X.

@o﬋djtjgd 8. K `jsiroto tjao sjmdkl xUdQ js porjg`ji jc tboro js kd jdtomor igdstkdt D 4 > suib tbkt + D QQ, (;.2) xUdQ 1 xU x Ud + D cgr kll d

∂ V.

Yjmdkls `g dgt sktjscy tbo porjg`jijty igd`jtjgds kro ikllo` kporjg`ji sjmdkls. kporjg`ji sjmdkls. sjd(ϊ> t), ϊ> 4 >. >. Jt ikd ikd ho sbg sbgwd tbkt tbkt Oxkaplo. Igdsjòr tbo sjmdkl x(t) 1 sjd(ϊ 8ώ + P ), ), wboro P wboro P 1 e ϊ cgr kdy e kdy e V+ 6 x(t) 1 x( x (t + P >

∂

  

8ώ ϊ> 1 sjd sjd (ϊ> t + 8ώe 8 ώe)) 1 sjd(ϊ sjd(ϊ> t) 1 x( x (t).

+ P )) 1 sjd ϊ> t + e + e x(t + P

Pborocgro, x Pborocgro, x((t) js k porjg`ji sjmdkl.

;8


@o﬋djtjgd 9. P > js ikllo` tbo cud`kaodtkl porjg` gc x(t) jc jt js tbo sakllost vkluo gc P 4 > sktjscy sktjscyjdm jdm tbo porjg`jij orjg`jijty ty igd`jt igd`jtjgd. jgd. Pbo duahor duahor ϊ> 1 8P ώ js ikllo` tbo cud`kaodtkl croqu croquodiy odiy gc x(t). >

@o﬋djtjgd 0. D > js ikllo` tbo cud`kaodtkl porjg` gc xUdQ jc jt js tbo sakllost vkluo 8ώ gc D 4 > wboro D V sktjscyjdm sktjscyjdm tbo porjg`jij porjg`jijty ty igd`jtjgd. Pbo duahor ℣> 1 D js ikllo` tbo cud`kaodtkl croquodiy gc xUdQ.

∂ ∂

>

f 9ώt/? ώt/ ? Oxkaplo. @otorajdo @otorajdo tbo cud`kaodt cud`kaodtkl kl porjg` gc tbo cgllgwjdm cgllgwjdm sjmdkls6 sjmdkls6 (k) o f9 5 f9 f 9ώd/? ώd/ ? (h) o (h) o . f 9ώt/? ώt/ ? Cgr (k), wo lot x( . Jc x( x (t) 1 o f9 x (t) js k porjg`ji sjmdkl, tbod tboro oxjsts P 4 > suib tbkt x tbkt x((t) 1 x( + P ). ). Pborocgro, x (t + P

⇘ ⇘ ⇘ ⇘

+ P )) x(t) 1 x( x (t + P f ώ t f ώ (t+P ) P ) 1 o o ώ ; 1 o f P ώ f 8eώ 1 o f P , cgr sgaoe sgaoe o f8 ;> P 1 9 . (e 1 ;) 9 ?

9 ?

9 ?

9 ?

+

∂ V

.

f 9ώd/? ώd/ ? Cgr (h), wo lot xU . Jc xU x UdQ 1 o f9 x UdQ js k porjg`ji sjmdkl, tbod tboro oxjsts kd jdtomor tbkt xUUdQ 1 xU + D Q. Q. Yg, D 4 > suib tbkt x x Ud + D

⇘ ⇘ ⇘ ⇘ ;.0. ;.0.8 8

+ D QQ xUdQ 1 xU x Ud + D ώ ώ D ) o f d 1 o f (d+D ) ώ f 8eώ 1 o f D , cgr sgaoe sgaoe o f8 ;>e ;>e D 1 9 ;>. (e 1 9). 9). D 1 ;>. 9 ?

9 ?

9 ?

∂ V

+

K Agr Agro o @j﬉ @j﬉iu iult lt Oxka Oxkapl plo o

Igdsjòr tbo cgllgwjdm twg sjmdkls

   

ώt8 x(t) 1 igs , 2 ώd8 xUdQ 1 igs . 2 ]o wjll sbgw tbkt x tbkt x((t) js kporjg`ji wboroks x wboroks xUUdQ js porjg`ji wjtb cud`kaodtkl porjg` D > 1 2.

;9

;.0. ;.0. ÔXJG@ ÔXJG@JIJ JIJPZ PZ

(k) igs

 ώt 8 2

(h) igs

Cjmuro ;.:6 @j﬎orodio hotwood x hotwood x((t) 1 igs wboroks x wboroks xUUdQ js porjg`ji.

 ώt8 2

kd` x kd` xUUdQ 1 igs

Cjm. ;.: plgts tbo sjmdkls x(t) 1 igs cgr

∘2 ≠ t ≠ 2 kd`

  ώd 8 2

 ώd 8 2

. Dgto tbkt x tbkt x((t) js kporjg`ji, kporjg`ji,

    ώt8 2

ώd 8 xUdQ 1 igs 2

cgr d 1 2, =, . . . , 2. Jt js ilokr ilokr tbkt tbkt x(t) js kporjg`ji, sjdio tbo vkluos gc t 4 > cgr wbjib x wbjib x((t) 1 > cgra k soquodio wbjib tbo `j﬎orodio hotwood igdsoiutjvo oloaodts js agdgtgdjiklly òiroksjdm.

∘ ∘

Gd tbo gtbor bkd`, xUdQ js porjg`ji, wjtb cud`kaodtkl porjg` D > 1 2. Pg soo tbjs, tbjs, igdsjòr tbo porjg`jijty igd`jtjgd xUdQ 1 xU + D Q, Q, wbjib hoigaos6 x Ud + D



  

igs ώ(d + D + D ))8 /2 1 igs ώd 8 /2 , cgr kll d kll d

∂ V. Pbjs aokds

+ D ))8 ώ(d + D ώd 8 1 + 8ώe 8ώe,, 2 2 cgr sgao e sgao e

∂ V, wboro e wboro e aky òpod` gd k pkrtjiulkr pkrtjiulkr tjao jdstkdt jdstkdt d. d. ]o ikd sjapljcy

;0


tbjs igd`jtjgd hy `jvj`jdm hgtb sjòs gc tbo oquktjgd hy ώ/2 ώ/2 tg yjol` 2 (d + D + D ))8 1 d8 + (8ώe (8ώe)), ώ gr 8dD + D + D 8 1 d 8 + ;3e, ;3e, d8 + 8dD japlyjdm 8dD + D + D 8 1 ;3e, ;3e, 8

cgr sgao e cgr kll d `gos.

∂ V. Doxt, Doxt, wo wkdt wkdt tg ﬋d` kd D suib 8dD + D js `jvjsjhlo hy ;3 D suib tbkt 8dD ∂ V. Dgw Dgw wo ilkja6 ilkja6 D 1 D 1 2 sktjs﬋os tbjs igd`jtjgd, kd` dg sakllor D 4 >

Yottjdm D Yottjdm D 1 1 2, wo mot 8dD + D + D 8 1 ;3d ;3d + 30, 30, wbjib, cgr kdy d kdy d V, js ilokrly `jvjsjhlo hy ;3. Yg D Yg D 1 1 2 js k porjg` gc xU x UdQ. Zgu ikd iboie `jroitly tbkt, cgr kdy ; D < 2, tboro js k vkluo d V suib tbkt 8dD 8 dD + + D 8 js dgt `jvjsjhlo hy ;3. Cgr oxkaplo jc wo igdsjòr D 1 D 1 0, wo mot

∂

≠

∂

8dD + D 8 1 2d + ;3, ;3, wbjib, cgr d cgr d 1 1 ;, js dgt `jvjsjhlo hy ;3. Yg D Yg D 1 1 0 js dgt k porjg` gc xUdQ.

;.0.9 ;.0.9

ôrjg`j orjg`jijt ijty y kd` Yik Yikljd ljdm m

;. Yuppgso x Yuppgso x((t) js porjg`ji, kd` lot k lot k 4 >. Js y (t) 1 x( x (kt) kt) porjg`ji7 Zos, kd` jc P > js tbo cud`kaodtkl porjg` gc x( x (t), tbod P >/k js /k js tbo cud`kaodtkl porjg` gc y y (t). 8. Yuppgso x Yuppgso xUUdQ js porjg`ji, kd` lot a lot a V+ . Js y Js y UdQ 1 xU x UadQ adQ porjg`ji7 Zos, kd` jc D > js tbo cud`kaodtkl porjg` gc xUdQ, tbod tbo cud`kaodtkl porjg` D gc yU y UdQ js tbo sakllost pgsjtjvo jdtomor suib tbkt aD js aD js `jvjsjhlo hy D > , j.o.

∂

≫ > ≫

aD

( ag` D >).

Oxkaplo Oxkaplo ;6 D > 1 2, a 2, a 1 1 8, tbod D tbod D 1 0. Oxkaplo Oxkaplo 86 D > 1 3, a 3, a 1 1 0, tbod D tbod D 1 9.

;?

;.?. ;.?. OSO OSOD D KD@ G@@ G@@ YJMDK YJMDKLY LY

;.? ;.?

Ovod Ovod kd kd` ` G`` G`` Yjmd Yjmdkl klss

;.?. ;.?.; ;

@o﬋d @o﬋djt jtjg jgds ds

@o﬋djtjgd ?. K igdtjdugus-tjao sjmdkl x(t) js js ovod jc x( t) 1 x( x (t)

(;.:)

∘

kd` jt js g`` jc x( t) 1

∘

∘x(t).

(;.;>)

js ovod jc @o﬋djtjgd 3. K `jsiroto-tjao sjmdkl xUdQ js xU dQ 1 xU x UdQ

(;.;;)

∘

kd` g`` jc xU dQ 1

∘

∘xUdQ.

(;.;8)

Xoakre6 Pbo kll-zorg sjmdkl js hgtb ovod kd` g``. Kdy gtbor sjmdkl ikddgt ho hgtb ovod ovod kd` g``, hut aky ho dojtbo dojtbor. r. Pbo cgllgw cgllgwjdm jdm sjaplo sjaplo oxkapl oxkaploo jllust jllustrkt rktoo tboso tboso prgportjos. Oxkaplo ;6 x(t) 1 t 8 0> js ovod. Oxkaplo 86 x(t) 1 >.;t9 js g``. Oxkaplo 96 x(t) 1 o >.0t js dojtbor ovod dgr g``.

∘

(k) x (k) x((t) 1 t 8

∘ 0>

(h) x( x (t) 1 >.;t9

(i) x (i) x((t) 1 o 1 o >.0t

Cjmuro ;.;>6 Jllustrktjgds gc g`` kd` ovod cuditjgds. (k) Ovod5 (h) G``5 (i) Dojtbor.

;3

;.?.8 ;.?.8


@oiga @oigapgsj pgsjtjg tjgd d Pbogro Pbogroa a

Pbogroa 8. Ovory igdtjdugus-tjao sjmdkl x(t) ikd ho oxprosso` ks6 + z (t), x(t) 1 y( y (t) + z wboro y (t) js ovod, kd` z (t) js g``. ^rggc. @o﬋do y(t) 1 kd` z (t) 1 Ilokrly y ( t) 1 y( y (t) kd` z ( t) 1

∘

∘

+ x(( t) x(t) + x 8

∘

x(t)

∘ x(∘t) .

8 iboie tbkt x(t) 1 y( )+z (t). z (t). ]o ikd klsg iboie y (t)+z

∘

Porajdglgmy6 orajdglgmy6 Pbo sjmdkl y (t) js ikllo` tbo ovod pkrt gc x(t), òdgto` hy Pbo sjmdkl z sjmdkl z ((t) js ikllo` tbo g`` pkrt gc x gc x((t), òdgto` hy `` x(t) .

G { }

O v{x(t)}.

Oxkaplo6 Lot us igdsjòr tbo sjmdkl x( x (t) 1 o t . ot + o∘t 1 igsb(t igsb(t). v x(t) 1 8 ot o∘t 1 sjdb(t sjdb(t). `` x(t) 1 8

O { } G { }

∘

Yjajlkrly, wo ikd ò﬋do ovod kd` g`` pkrts gc k `jsiroto-tjao sjmdkl xUdQ6 + xUU∘dQ O v{xUdQ} 1 xUdQ + x 8 G``{xUdQ} 1 xUdQ ∘8xU∘dQ . Jt js oksy tg iboie tbkt xUdQ 1 v xUdQ +

O { } G``{xUdQ}

Pbogroa 9. Pbo òigapgsjtjgd js udjquo, j.o., jc + z UdQ, xUdQ 1 yU y UdQ + z tbod yU y UdQ js ovod kd` z z UdQ js g`` jc kd` gdly jc yU y UdQ 1 v xUdQ kd` z z UdQ 1

O { }

G``{xUdQ}.

;=

;.3. JA^_LYO JA^_LYO KD@ KD@ YPO^ YPO^ C_DIPJ C_DIPJGDY GDY ^rggc. Jc y UdQ js ovod kd` z kd` z UUdQ js g``, tbod + z U dQ 1 yU xU dQ 1 yU y U dQ + z y UdQ

∘

∘

∘

∘ z UdQ.

Pborocgro, + xUU dQ 1 (yUdQ + z + z UdQ) + (y (yUdQ xUdQ + x

∘ z UdQ) 1 8y 8y UdQ, ∘ 1 O v{xUdQ}. Yjajlkrly z UdQ 1 ∘ ∘ 1 G``{xUdQ}. japlyjdm y japlyjdm yUUdQ 1 Pbo igdvorso js trjvjkl hy ò﬋djtjgd, ks O v {xUdQ} aust ho ovod kd` G ``{xUdQ} aust ∘

xUdQ+x Q+xU dQ 8

xUdQ xU dQ 8

ho g``.

;.3 ;.3 ;.3.;

Japul Japulso so kd kd` ` Yto Ytop p Cud Cudit itjg jgds ds @jsiroto@jsiroto-tjao tjao Japulso Japulso kd` Ytop Cuditjgds uditjgds

@o﬋djtjgd =. Pbo `jsiroto-tjao udjt japulso sjmdkl ΰ UdQ js ò﬋do` ks ΰ UdQ 1



;, >,

d 1 >, d 1 >.



(;.;9)

@o﬋djtjgd 2. Pbo `jsiroto-tjao udjt stop sjmdkl ΰ UdQ js ò﬋do` ks



;, >,

d >, > . d < >.

ΰ UdQ 1 uU u UdQ

∘ uUd ∘ ;Q ΰ Ud ∘ e Q

uUdQ 1

≯

Jt ikd ho sbgwd tbkt

• • •

uUdQ 1

∔

  e1>

uUdQ 1

∔

e1

∘∔

uUe Qΰ Ud

∘ eQ.

(;.;0)

;2


(k) ΰ (k) ΰ UUdQ

(h) uU u UdQ

Cjmuro ;.;;6 @o﬋djtjgds gc japulso cuditjgd kd` k stop cuditjgd.

;.3. ;.3.8 8

^rgp ^rgpor ortty gc ΰ UdQ

Ykapljdm ^rgporty Hy tbo ò﬋djtjgd gc ΰ Q, ΰ UUd ΰ UdQ, ΰ

∘ d Q 1 ; jc d 1 d 1 d d , kd` > gtborwjso. Pborocgro,

xUdQΰ Ud

>

>

∘d Q1 >



xUdQ, >,

1 xU x Ud> Qΰ Ud

1 d> d 1 d d 1 d>



∘ d Q. >

(;.;?)

Ks k spoijkl ikso wbod d > 1 >, wo bkvo x bkvo xUUdQΰ UdQ 1 xU>Q ^jitgrjklly,, wbod k sjmdkl x U>Qΰ ΰ UdQ. ^jitgrjklly xUdQ js aultjpljo` wjtb ΰ U ΰ UdQ, tbo gutput js k udjt japulso wjtb kapljtuò xU>Q. x U>Q.

Cjmuro ;.;86 Jllustrktjgd gc xU x UdQΰ UdQ 1 xU>Q x U>Qΰ ΰ UdQ.

;:

;.3. JA^_LYO JA^_LYO KD@ KD@ YPO^ YPO^ C_DIPJ C_DIPJGDY GDY

Ybjctjdm ^rgporty

  ∔

Yjdio x Yjdio xUUdQΰ UdQ 1 xU>Q x U>Qΰ ΰ UdQ kd`

d1

∔

 

xUdQΰ UdQ 1

d1

kd` sjajlkrly

∔

d1

∘∔ ∔ d1

∘∔

xUdQΰ Ud

∘∔

ΰ UdQ 1 ;, wo bkvo U>Qΰ UdQ 1 xU>Q xU>Qΰ x U>Q

∘∔

∔

∘d Q1 >

d1k

 

xUd> Qΰ Ud

d1

∘∔

h

xUdQΰ Ud

ΰ UdQ 1 xU>Q x U>Q,,

d1

∘∔

Jd modorkl, tbo cgllgwjdm rosult bgl`s6



∔



∘d Q1 >

xUd> Q, >,

∘ d Q 1 xU x Ud Q. >

jc d d > jc d d >

>

∂ Uk, hQ ∂ Uk, hQ

(;.;3)

(;.;=)

Xoprosodtktjgd ^rgporty _sjdm tbo skapljdm prgporty, jt bgl`s tbkt xUe Qΰ Ud

∘ eQ 1 xU x UdQΰ Ud ∘ e Q.

Yuaajdm tbo hgtb sjòs gvor tbo jdòx e yjol`s

∔



e1

∘∔

xUe Qΰ Ud

∘ eQ 1

∔



e1

xUdQΰ Ud

∘∔

∘ eQ 1 xU x UdQ

∔



e1

∘∔

ΰ Ud

∘ eQ 1 xU x UdQ.

Pbjs rosult sbgws tbkt ovory `jsiroto-tjao sjmdkl xUdQ ikd ho roprosodto` ks k ljdokr igahjdktjgd gc sbjcto` udjt japulsos xUdQ 1

∔



xUe Qΰ Ud

e1

∘∔

∘ eQ.

(;.;2)

Cgr oxkaplo, tbo udjt stop cuditjgd ikd ho oxprosso` ks uUdQ 1

∔



e1

∘∔

uUe Qΰ Ud

∘ eQ.

]by `g wo uso tboso igapljikto` roprosodtktjgd gc x gc xUUdQ7 Hoikuso, wbod wo igdsjòr ljdokr tjao-jdvkrjkdt systoas (Ibkptor 8), jt wjll kllgw us tg òtorajdo tbo systoa rospgdso tg kdy sjmdkl x sjmdkl xUUdQ crga tbo japulso rospgdso.

8>


Cjmuro ;.;96 Xoprosodtjdm gc k sjmdkl xU x UdQ usjdm k trkjd gc japulsos ΰ U ΰ Ud ∘ eQ.

;.3.9

Igdtjd Igdtjdugu ugus-tja s-tjao o Japulso Japulso kd` kd` Ytop Ytop Cudit Cuditjgds jgds

@o﬋djtjgd :. Pbo `jrki òltk cuditjgd js ò﬋do` ks

 ∔   >,

ΰ (t) 1 wboro

,

∔

jc t 1 > , jc t 1 >



ΰ (t)`t 1 `t 1 ;.

∘∔

@o﬋djtjgd ;>. Pbo udjt stop cuditjgd js ò﬋do` ks u(t) 1

;.3. ;.3.0 0

>, ;,

t < > t >.

≯

^rgp ^rgpor ortty gc ΰ (t)

Pbo prgportjos gc ΰ ( ΰ (t) kro kdklgmgus tg tbo `jsiroto-tjao ikso.

8;

;.3. JA^_LYO JA^_LYO KD@ KD@ YPO^ YPO^ C_DIPJ C_DIPJGDY GDY

Ykapljdm ^rgporty x(t)ΰ (t) 1 x(>) x(>)ΰ ΰ (t).

(;.;:)

Pg soo tbjs, dgto tbkt x(t)ΰ (t) 1 x(>) wbod t 1 > kd` x(t)ΰ (t) 1 > wbod t 1 >. Yjajlkrly, wo bkvo (;.8>) x(t)ΰ (t t> ) 1 x( x (t> )ΰ (t t> ),



∘

cgr kdy t kdy t >

∂ X.

∘

Ybjctjdm ^rgporty Pbo sbjctjdm prgporty cgllgws crga tbo skapljdm prgporty. Jdtomrktjdm x( x (t)ΰ (t) yjol`s



∔

 

∔

x(t)ΰ (t)`t 1 `t 1

∘∔

(>)ΰ (t)`t 1 (>) x(>)ΰ `t 1 x x(>)

∘∔

Yjajlkrly, gdo ikd sbgw tbkt

∔



∔

(>).. ΰ (t)`t 1 `t 1 x x(>)

(;.8;)

∘∔

x(t)ΰ (t

∘ t )`t 1 `t 1 x x((t ) >

∘∔

(;.88)

>

Xoprosodtktjgd ^rgporty Pbo roprosodtktjgd prgporty js klsg kdklgmgus tg tbo `jsiroto-tjao ikso6 x(t) 1

 

∔

x(π ) π )ΰ (t

∘∔

∘ π ) π )`π,

(;.89)

wboro tbo spoijkl ikso gc u(t) js mjvod hy u(t) 1

∔

u(π ) π )ΰ (t

∘ π ) π )`π.

Ks kd oxkaplo gc tbo prgportjos, lot us igdsjòr

` ( ). `t u t

∘∔

u(t) 1 1

  

∔

∘∔ ∔ >

u(π ) π )ΰ (t ΰ (t

∘ π ) π )`π,

∘ π ) π )`π,

(roprosodtktjgd (roprosodtktjgd prgporty)

hoikuso u hoikuso u((π ) π ) 1 > jc π π

≠ > ≠

t

1

ΰ (ω)`ω,

lot ω lot ω 1 t 1 t

∘∔

∘ π.

Pbod hy cud`kaodtkl tbogroa gc ikliulus, wo bkvo



` ` t u(t) 1 ΰ (ω)`ω 1 `ω 1 ΰ ΰ (t). `t `t ∘∔

(;.80)

88

;.=


Igdtjd Igdtjdugu ugus-t s-tjao jao Igapl Igaplox ox Oxpgdod Oxpgdodtjkl tjkl Cudi Cudi-tjgds

@o﬋djtjgd ;;. K igaplox oxpgdodtjkl cuditjgd js ò﬋do` ks x(t) 1 I okt , wboro I, k

;.=.; ;.=.;

∂I

(;.8?)

Xokl-v Xokl-vklu kluo` o` Oxpgdo Oxpgdod dtjkl tjkl

]o ﬋rst igdsjòr tbo ikso gc rokl-vkluo` oxpgdodtjkl cuditjgds, j.o., I

∂ ∂ X kd` kd` k k ∂ X.

(k) x (k) x((t) 1 8; o∘

; 8

(h) x (h) x((t) 1 8; o

t

; 8

t

Cjmuro ;.;06 Xokl oxpgdodtjkl cuditjgds. ]bod k ]bod k 1 1 >, tbod x tbod x((t) 1 I , I , wbjib js k igdstkdt cuditjgd.

;.=.8 ;.=.8

ôrjg`j orjg`jii Igap Igaplox lox Oxpgdo Oxpgdod dtjkl tjkl

Lot us igdsjòr tbo ikso wboro k js puroly jakmjdkry, j.o., k 1 f 1 f ϊ> , ϊ> js k igaplox duahor, wo bkvo I 1 Ko fβ ,

∂ X. Yjdio I

wboro K, wboro K, β

∂ X. Igdsoquodtly, x(t) 1 I o fϊ t 1 Ko fβ o fϊ >

f (ϊ 1 Ko f(

>

>

t

t+β )

1 K igs(ϊ igs(ϊ> t + β + β)) + fK + fK sjd(ϊ sjd(ϊ> t + β + β)).

;.=. IGDPJD_G IGDPJD_G_Y-PJ _Y-PJAO AO IGA^LOR IGA^LOR OR^GDODP OR^GDODPJKL JKL C_DIPJGDY C_DIPJGDY

89

Pbo rokl kd` jakmjdkry pkrts gc x( x (t) kro mjvod hy

Xo{x(t)} 1 K igs(ϊ igs(ϊ t + β + β)) J a{x(t)} 1 K sjd(ϊ sjd(ϊ t + β + β)). >

>

]o ikd tbjde gc x( k pkjr gc gc sjdusgj`kl sjmdkls gc tbo skao kapljtuò K kapljtuò K,, ϊ > kd` x (t) ks k pkjr pbkso sbjct β sbjct β wjtb gdo k igsjdo kd` tbo gtbor k sjdo.

fϊ > t

(k)

(h) a Io fϊ

Xo{Io }

J {

>

t

}

Cjmuro ;.;?6 ôrjg`ji igaplox oxpgdodtjkl cuditjgd x( x (t) 1 Io I ofϊ t (I 1 1 ;, ϊ ;, ϊ > 1 8ώ). >

Ilkjas. x(t) 1 I o fϊ t js porjg`ji wjtb >

;. cud`kaodtkl porjg`6 P > 1 |8ϊώ| . >

8. cud`kaodtkl croquodiy6 ϊ> .

| |

Pbo soigd` ilkja js tbo jaao`jkto rosult crga tbo ﬋rst ilkja. Pg sbgw tbo ﬋rst ilkja, wo doo` tg sbgw x sbgw x((t + P > ) 1 x( sakllor P > ikd sktjscy tbo porjg`jijty irjtorjk. x (t) kd` dg sakllor P 8ώ fϊ > (t+ |ϊ ) |

+ P > ) 1 I o x(t + P

>

f 8ώ 1 I o fϊ t o» f8

1 I o fϊ t 1 x( x (t). >

Jt js oksy tg sbgw tbkt P > js tbo sakllost porjg`!

>

80


;.=.9 ;.=.9

Modor Modorkl kl Igaplo Igaplox x Oxpgd Oxpgdod odtjk tjkll

Jd tbo modorkl sottjdm, wo bkvo I 1 Ko fβ wboro K, β r, ϊ> X. Pborocgro,

∂

 

x(t) 1 Ko fr o(r+ fϊ

>

)t

∂ X kd` k 1 r + f + fϊ ϊ wboro >

f (ϊ 1 Ko rt o f(

>

t+β )

.

Xowrjto tbjs jd tbo roitkdmulkr cgra6

igs(ϊ> t + β + β)) + fKo + fKo rt sjd(ϊ sjd(ϊ> t + β + β)) x(t) 1 Ko rt igs(ϊ

(k)

rt fϊ > t

Xo{Ko

o

(h) a Kort ofϊ

}

J {

>

t

}

Cjmuro ;.;36 ôrjg`ji igaplox oxpgdodtjkl cuditjgd x( x (t) 1 Ko rt ofϊ t (K 1 ;, r ;, r 1 ∘;/8, ϊ 8, ϊ > 1 8ώ ). >

;.2 ;.2. ;.2.; ;

@jsiro @jsirototo-tja tjao o Igaplo Igaplox x Oxpgdod Oxpgdodtjk tjkls ls @o﬋d @o﬋djt jtjg jgds ds

K `jsiroto`jsiroto-tjao tjao igaplox oxpgdodtj oxpgdodtjkl kl cuditjgd cuditjgd bks tbo cgra6 cgra6 xUdQ 1 I oθd , wboro I wboro I,, θ

∂ ∂ I. Lottjdm λ 1 o 1 o

θ

mjvos xUdQ 1 I λd .

Jd tbo cgllgwjdm suhsoitjgds, wo sbkll stu`y tbo hobkvjgr gc xUdQ cgr `j﬎orodio iksos gc I kd` λ I kd` λ..

8?

;.2. @JYIXOPO-P @JYIXOPO-PJAO JAO IGA^LOR IGA^LOR OR^GDOD OR^GDODPJKLY PJKLY

;.2.8

Xokl-v Xokl-vkluo` kluo` Igaplox Igaplox Oxpgdod Oxpgdodtjkl tjkl

X. Jd tbj tbjss ikso ikso,, xUdQ js k rokl-vkluo` igaplox oxpgdodtjkl wbod I X kd` λ d xUdQ 1 I λ js k agdgtgdji òiroksjdm cuditjgd wbod > < λ < ;, kd` js k agdgtgdji jdiroksjdm cuditjgd wbod λ wbod λ 4 ;.

∂

;.2.9

∂

Igaplox-v Igaplox-vkluo kluo` ` Igaplox Igaplox Oxpgdodt Oxpgdodtjkl jkl

xUdQ js k igaplox-vkluo` igaplox oxpgdodtjkl wbod I, λ I. Jd tbjs ikso, I kd` λ f ℣ ikd ho wrjttod ks I 1 I o fβ , kd` λ kd` λ 1 λ o f℣ . Igdsoquodtly,

| |

||

∂

>

 

f ℣ xUdQ 1 I λd 1 I o fβ λ o f℣

d

| | || ||λ| o 1 |I || ||λ| igs(℣ d + β ||λ| 1 |I || + β)) + f + f |I || >

d f(℣ f (℣> d+β ) d

d

>

sjd(℣> d + β + β)).

Pbroo iksos ikd ho igdsjòro` boro6 ;. ]bod λ 1 ;, tbod xUdQ 1 I igs(℣> d + β) β ) + f I sjd(℣> d + β) β ) kd` jt bks sjdusgj`kl rokl kd` jakmjdkry pkrts (dgt doiosskrjly porjg`ji, tbgumb).

| |

| |

| |

8. ]bod λ 4 ;, tbod λ d js k mrgwjdm oxpgdodtjkl, sg tbo rokl kd` jakmjdkry pkrts gc xU x UdQ kro tbo prgùit gc tbjs wjtb sjdusgj`s.

| |

| |

9. ]bod λ < ;, tbod tbo rokl kd` jakmjdkry pkrts gc xU x UdQ kro sjdusgj`s soklo` hy k òikyjdm oxpgdodtjkl.

||

;.2.0 ;.2.0

ôrjg`j orjg`jii Igap Igaplox lox Oxpgdo Oxpgdod dtjkls tjkls

f ℣ d X. ]o wkdt Igdsjòr xUdQ 1 I o f℣ , ℣> wkdt tg stu`y stu`y tbo igd`jtj igd`jtjgd gd cgr xUdQ tg ho porjg`ji. Pbo porjg`jijty igd`jtjgd roqujros tbkt, cgr sgao D 4 >,

∂

>

+ D QQ 1 xU xUd + D x UdQ,

∎d ∂ V.

f ℣ d Yjdio x Yjdio xUUdQ 1 I o f℣ , jt bgl`s tbkt >

f ℣ o f℣

>

(d+D ) D )

f ℣ d f℣ 1 o f℣ o f ℣ >

>

D

f ℣ d 1 o f℣ , >

∎d ∂ V.

Pbjs js oqujvklodt tg f ℣ o f℣

>

D

1;

gr

℣> D 1 8ώa,

cgr sgao a sgao a

∂ V.

83


Pborocgro Pborocgro,, tbo igd`jtjgd cgr porjg`jijty porjg`jijty gc x gc xUUdQ js ℣> 1

8ώa , D

cgr sgao a sgao a

∂ V, kd` sgao D 4 >, ∂ V. > , D ∂

f ℣ d Pbus xUdQ 1 o f℣ js porjg`ji jc kd` gdly jc ℣ > js k rktjgdkl aultjplo gc 8ώ 8 ώ. Pbo cud`kaod cud`kaodtkl tkl porjg` js 8ώa D 1 , ℣> >

wboro wo kssuao tbkt a kd` D mi`( a, d) 1 ;, j.o., D kro rolktjvoly prjao, mi`(a, roùio` cgra.

;.2.? ;.2.?

jd

ôrjg`j orjg`jijt ijty y jd Croquo roquodiy diy

f ℣ d Yuppgso tbkt x tbkt xUUdQ 1 o f℣ , wboro ℣ > >

∂ X. Jc wo jdirokso ℣

f (℣ x; UdQ 1 o f(℣

>

Hut d

a js D

f8 f 8ώd

∂ V, sg o sg o

+8ώ +8ώ )d

>

hy 8ώ, wo ﬋d`

f ℣ d f8 1 o f℣ o f 8ώd . >

1 ;, kd` wo soo tbkt f ℣ x; UdQ 1 o f℣

>

Agro modorklly, cgr kdy e

d

1 xU x UdQ.

∂ V, wo bkvo f (℣ xe UdQ 1 o f(℣

>

+8ώe +8ώe))d

f ℣ 1 o f℣

>

d

1 xU x UdQ.

Pbjs aokds tbkt wo ikd ljajt tbo rkdmo gc vkluos gc ℣ > tg kdy rokl jdtorvkl gc lodmtb 8ώ 8 ώ. Pbo porjg`jijty jd croquodiy kppljos, gc igurso, tg tbo porjg`ji igaplox oxpgdodtjkl sjmdkls, sg wo bkvo k `j﬎orodt dgtjgd gc lgw kd` bjmb croquodijos jd tbo `jsiroto-tjao sottjdm.

Ibkptor 8 Cud`kaodtkls gc Yystoas K systoa js k qukdtjtktjvo òsirjptjgd gc k pbysjikl prgioss wbjib trkdscgras sjmdkls (kt jts “jdput‑ “jdput‑)) tg sjmdkl sjmdklss (kt jts “gutpu “gutput‑) t‑).. Agro Agro proijs proijsoly oly,, k systoa systoa js k “hlki “hlkie e hgx‑ (vjowo` ks k aktboaktjikl khstrkitjgd) tbkt òtorajdjstjiklly trkdscgras jdput sjmdkls jdtg gutput sjmdkls. Jd tbjs ibkptor, wo wjll stu`y tbo prgportjos gc systoas.

Cjmuro 8.;6 Igdtjdugus-tjao kd` `jsiroto-tjao systoas. Xoakres6 ;. ]o wjll igdsjòr hgtb igdtjdugus-tjao igdtjdugus-tjao systoas kd` `jsiroto-tjao `jsiroto-tjao systoas. Pbo trkdscgraktjgd crga k igdtjdugus-tjao sjmdkl x sjmdkl x((t) tg k `jsiroto-tjao sjmdkl xU x UdQ wjll ho `jsiusso` jd Ibktpor 3. 8. ]o wjll cgius gd sjdmlo-jdput sjdmlo-gutput systoas. Aultjplo-jdputs tg aultjplogutputs (AJAG) systoas kro gutsjò tbo sigpo gc tbjs igurso.

8=

82

IBK^POX IBK^POX 8. C_D@KAOD C_D@KAODP PKLY GC YZYPOAY YZYPOAY

8.; 8.;

Yyst Yystoa oa ^rgpo ^rgport rtjo joss

8.;. 8.;.; ;

Aoag Aoagry rylo loss ss

@o﬋djtjgd ;8. K systoa js aoagryloss jc tbo gutput kt tjao t (gr d) òpod`s gdly gd tbo jdput kt tjao t (gr d). Oxkaplos. ;. y (t) 1 (8x (8x(t) x8 (t))8 js aoagryloss, hoikuso y hoikuso y((t) òpod`s gd x gd x((t) gdly. Pboro js dg x dg x((t ;), gr x(t + ;) toras, cgr oxkaplo.

∘

∘

8. y UdQ 1 xUdQ js aoagry aoagrylos loss. s. Jd ckit, ckit, tbjs tbjs systoa systoa js pkssjd pkssjdmm tbo jdput jdput tg gutput gutput `jroitly, wjtbgut kdy prgiossjdm. 9. y UdQ 1 xU x Ud jdput.

∘ ;Q js dgt aoagryloss, hoikuso tbo d-tb gd d ∘ ;-tb d -tb gutput òpod`s gd d

0. y UdQ 1 xU + yUUd x UdQ + y

∘ ;Q js dgt aoagryloss. Pg soo tbjs, wo igdsjòr + yUUd ∘ 8Q. 8Q. yUd ∘ ;Q 1 xU x Ud ∘ ;Q + y Yuhstjtutjdm jdtg y jdtg yUUdQ 1 xU + yUUd ∘ ;Q yjol`s x UdQ + y ( xUd ∘ ;Q + y + yUUd ∘ 8Q). 8Q). yUdQ 1 xU x UdQ + (x Hy ropoktjdm tbo ikliulktjgd, wo bkvo + xUUd yUdQ 1 xU x UdQ + x d

1



∘ ;Q + x + xUUd ∘ 8Q + . + . . .

xUe Q.

e1

∘∔

Ilokrly, y Ilokrly, yUUdQ òpod`s gd agro tbkd fust x fust xUUdQ.

8.;. 8.;.8 8

Jdv Jd vortj ortjhl hlo o

@o﬋djtjgd ;9. K systoa js jdvortjhlo jc `jstjdit jdput sjmdkls prgùio `jstjdit gutput sjmdkls. Jd gtbor wgr`s, k systoa jc jdvortjhlo jc tboro oxjsts kd gdo-tg-gdo akppjdm crga tbo sot gc jdput sjmdkls tg tbo sot gc gutput sjmdkls. Pboro kro twg hksji rulos gc sbgwjdm kd jdvortjhlo systoa6

8:

8.;. YZYPOA YZYPOA ^XGÔX ^XGÔXPJOY PJOY ;. Pg sbgw tbkt k systoa systoa js jdvortj jdvortjhlo, hlo, gdo bks tg sbgw tbo jdvorsjg jdvorsjgd d cgraulk. 8. Pg sbgw tbkt k systoa js dgt jdvortjhlo, gdo bks tg mjvo k igudtor oxkaplo.

Oxkaplo ;. Pbo systoa y (t) 1 (igs(t (igs(t) + 8)x 8)x(t) js jdvortjhlo. ^rggc. Pg sbgw tbkt tbo systoa js jdvortjhlo, wo doo` tg ﬋d` kd jdvorsjgd cgraulk. Pbjs js oksy6 y (t) 1 (igs(t (igs(t) + 8)x 8)x(t) japljos tbkt (hy rokrrkdmjdm toras) x(t) 1

y (t) , igs(t igs(t) + 8

wbjib wbjib js tbo jdvorsjg jdvorsjgd d cgraulk. cgraulk. Dgto tbkt tbo òdgajdktgr òdgajdktgr js klwkys klwkys pgsjtjvo, pgsjtjvo, tbus tbus tbo `jvjsjgd js vklj`.

Oxkaplo 8. Pbo systoa y UdQ 1 xU + yUUd x UdQ + y ^rggc. y UdQ 1 xU + yUUd x UdQ + y

∘ ;Q js jdvortjhlo.

∘ ;Q japljos tbkt (hy rokrrkdmjdm toras) ;Q. xUdQ 1 yU y UdQ ∘ yUd ∘ ;Q.

Pbjs js tbo jdvorsjgd cgraulk.

Oxkaplo 9. Pbo systoa y (t) 1 x 8 (t) js dgt jdvortjhlo. ^rggc. Pg sbgw tbkt k systoa js dgt jdvortjhlo, wo igdstruit k igudtor oxkaplo. Lot us igdsjòr twg sjmdkls x;(t) 1 ;, t x8(t) 1 ;, t.

∘

∎

∎

Ilokrly x; (t) 1 x8 (t), hut (x (x; (t))8 1 (x8 (t))8 . Pboroc Pborocgro gro,, wo bkvo bkvo cgud` cgud` k igudt igudtor or oxkapl oxkaploo suib suib tbkt tbkt `j﬎oro `j﬎orodt dt jdputs jdputs mjvo mjvo tbo skao gutput. gutput. Bodio Bodio tbo systo systoa a js dgt jdvortjhlo.



9>


8.; 8.;.9

Iku Ikuskl skl

@o﬋djtjgd ;0. K systoa js ikuskl jc tbo gutput kt tjao t (gr d) òpod`s gdly gd jdputs kt tjao s t (j.o., tbo prosodt kd` pkst).

≠

Oxkaplos. ;. y UdQ 1 xU x Ud

∘ ;Q js ikuskl, hoikuso y hoikuso yUUdQ òpod`s gd tbo pkst skaplo x skaplo xUUd ∘ ;Q.

8. y UdQ 1 xU + xUUd + ;Q js dgt ikuskl, hoikuso x hoikuso xUUd + ;Q js k cuturo skaplo. x UdQ + x

 t

9. y (t) 1 ∘∔ x(π ) π )`π `π js ikuskl, hoikuso tbo jdtomrkl ovkluktos π π crga (wbjib kro kll jd tbo pkst).

∘∔ tg t

0. y UdQ 1 xU dQ js dgt ikuskl, hoikuso y U ;Q 1 xU;Q, wbjib aokds tbo gutput kt d 1 ; òpod`s kd jdput jd tbo cuturo.

∘

∘

∘

?. y (t) 1 x( )igs(t + ;) ikuskl (kd` aoagryloss) aoagryloss),, hoikuso igs(t igs(t + ;) js k igdstkd igdstkdtt x (t)igs(t wjtb rospoit tg x( x (t).

8.;. 8.;.0 0

Ytkh Ytkhlo lo

Pg òsirjho k stkhlo systoa, wo ﬋rst doo` tg ò﬋do tbo hgudò`doss gc k sjmdkl.

@o﬋djtjgd ;?. K sjmdkl x( x (t) (kd` xU x UdQ) js hgudò` jc tboro oxjsts k igdstkdt H < suib tbkt x(t) < H cgr kll t.

| | |

∔

@o﬋djtjgd ;3. K systoa js stkhlo jc k hgudò` jdput jdput klwkys prgùios k hgudò` gutput sjmdkl. Pbkt js, jc x(t) H cgr sgao H < , tbod

| | | ≠

Oxkaplo ;. Pbo systoa y (t) 1 8x8 (t

∔

|y(t)| < ∔. ∘ ;) + x + x(9 (9tt) js stkhlo.

^rggc. Pg sbgw tbo systoa js stkhlo, lot us igdsjòr k hgudò` sjmdkl x(t), tbkt js, x(t) H cgr sgao H < . Pbod

| |≠

∔

8

|y(t)| 1 |8x (t ∘ ;) + x + x(9 (9tt)| ≠ |8x (t ∘ ;)| + |x(9t (9t)| ≠ 8|x (t ∘ ;)| + |x(9t (9t)| ≠ 8H + H < ∔. 8

, hy Prjkdmlo Jdoqukljty

8

8

Pborocgro, cgr kdy hgudò` jdput x jdput x((t), tbo gutput y gutput y((t) js klwkys hgudò`. Bodio tbo systoa js stkhlo.

9;

8.;. YZYPOA YZYPOA ^XGÔX ^XGÔXPJOY PJOY

Oxkaplo 8.

 d

Pbo systoa y UdQ 1

e1

xUeQ js dgt stkhlo.

∘∔



^rggc. Pg sbgw tbkt tbo systoa yU y UdQ 1 ed1∘∔ xUe Q js dgt stkhlo, wo ikd igdstruit k hgudò` jdput sjmdkl x sjmdkl xUUdQ kd` sbgw tbkt tbo gutput sjmdkl yU y UdQ js dgt hgudò`. Lot x Lot xUUdQ 1 uU u UdQ. Jt js ilokr tbkt xUdQ

| | ≠ ; (j.o., hgudò`). Igdsoquodtly,

  |    ≠ d

|yUdQ

1

uUe Q

e1 d

∘∔

1

 

uUe Q

e1> d

; 1 d + ;, d + ;,

e1>

wbjib kpprgkibos

8.;.? 8.;.?

∔ ks d ↘ ∔. Pborocgro, |yUdQ| js dgt hgudò`.

Pjao-j Pjao-jd dvkrjkd krjkdtt

@o﬋djtjgd ;=. K systoa js tjao-jdvkrjkdt jc k tjao-sbjct gc tbo jdput sjmdkl rosults jd tbo skao tjao-sbjct gc tbo gutput sjmdkl. Pbkt js, jc x(t)

∘↘ y(t),

tbod tbo systoa js tjao-jdvkrjkdt jc x(t

∘ t ) ∘↘ y(t ∘ t ), >

>

cgr kdy t>

∂ X.

Cjm. 8.8 jllustrktos jllustrktos kd jdtorprotkt jdtorprotktjgd jgd gc k tjao-jdv tjao-jdvkrjkdt krjkdt systoa6 Jc k sjmdkl x(t) js jdput tg k tjao-jdvkrjkdt systoa kd` mot kd gutput y( jdput x((t t> ) wjll y (t), tbod tbo jdput x rosult kd gutput y gutput y((t t> ).

∘

Oxkaplo ;. Pbo systoa y (t) 1 sjdUx sjdUx(t)Q js tjao-jdvkrjkdt.

∘

98


Cjmuro 8.86 Jllustrktjgd gc k tjao-jdvkrjkdt systoa. ^rggc. Lot us igdsjòr k tjao-sbjcto` sjmdkl x; (t) 1 x(t lot y lot y ; (t) ho tbo gutput gc x x ; (t). Pborocgro, sjdUx; (t)Q 1 sjdUx sjdUx(t y; (t) 1 sjdUx Dgw, wo bkvo tg iboie wbotbor y ; (t) 1 y( y (t

∘ t ). >

Igrrospgd`jdm Igrrospgd`jdmly ly,, wo

∘ t )Q. )Q. >

∘ t ). Pg sbgw tbjs, wo dgto tbkt sjdUx(t ∘ t )Q, )Q, y(t ∘ t ) 1 sjdUx >

>

>

wbjib js tbo skao ks y ; (t). Pborocgro, tbo systoa js tjao-jdvkrjkdt.

Oxkaplo 8. Pbo systoa y UdQ 1 dxU dx UdQ js dgt tjao-jdvkrjkdt. ^rggc. Pg sbgw tbkt tbo systoa jd dgt tjao-jdvkrjkdt, wo ikd igdstruit k igudtor oxkapl oxkaplo. o. Lot xUdQ 1 ΰ UdQ, tbod y UdQ 1 dΰ UdQ 1 >, d (]b (]by7) y7).. Dgw, Dgw, lot x; UdQ 1 xUd ;Q 1 ΰ Ud ;Q. Jc y y ; UdQ js tbo gutput prgùio` hy x ; UdQ, jt js oksy tg sbgw tbkt

∘

∎

∘

y; UdQ 1 dx ; UdQ 1 dΰ Ud ;Q 1 ΰ Ud ;Q. ;Q. (]by7)

∘ ∘  yU Bgwovor, y Bgwovor, yUUd ∘ ;Q 1 (d (d ∘ ;)x ;)xUd ∘ ;Q 1 (d (d ∘ ;)ΰ ;)ΰ Ud ∘ ;Q 1 > cgr kll d kll d.. Yg y Yg y UdQ 1 y Ud ∘ ;Q. Jd gtbor wgr`s, wo bkvo igdstruito` kd oxkaplo suib tbkt yU y Ud ∘ ;Q js dgt tbo gutput gc xU x Ud ∘ ;Q. ;

8.;. 8.;.3 3

Ljdo Ljdokr kr

@o﬋djtjgd ;2. K systoa js ljdokr jc jt js k``jtjvo kd` siklkhlo. Pbkt js, + hx8 (t) kx; (t) + hx cgr kll k, h

∂ I.

∘↘ ky (t) + hy + hy (t), ;

8

99

8.8. IGDSGL_P IGDSGL_PJGD JGD

Oxkaplo ;. Pbo systoa y (t) 1 8ώx( ώx(t) js ljdokr. Pg soo tbjs, lot‚s igdsjòr k sjmdkl + hx8 (t), x(t) 1 kx ; (t) + hx wboro y wboro y ; (t) 1 8ώx; (t) kd` y kd` y 8(t) 1 8ώx8 (t). Pbod + hy8 (t) 1 k (8ώx (8ώx; (t)) + h + h (8ώx (8ώx8 (t)) ky; (t) + hy 1 8ώ Ukx; (t) + hx + hx8 (t)Q 1 8ώx 8ώx((t) 1 y( y (t).

Oxkaplo 8. Pbo systoa y UdQ 1 (xU8d U8dQ)8 js dgt ljdokr. Pg soo tbjs, lot‚s igdsjòr tbo sjmdkl + hx8 UdQ, xUdQ 1 kx; UdQ + hx wboro y; UdQ 1 (x; U8d U8dQ)8 kd` y8 UdQ 1 (x8 U8d U8dQ)8 . ]o wkdt wkdt tg soo soo wbotb wbotbor or y UdQ 1 + hy8 UdQ. Jt bgl`s tbkt ky; UdQ + hy + hy8 UdQ 1 k (x; U8d U8dQ)8 + h (x8 U8d U8dQ)8 . ky; UdQ + hy Bgwovor, U8dQ)8 1 (kx; U8d U8dQ + hx + hx8 U8d U8dQ)8 1 k 8(x; U8d U8dQ)8 + h8 (x8 U8d U8dQ)8 + 8khx 8khx; UdQx8 UdQ. yUdQ 1 (xU8d

8.8 8.8 8.8.; 8.8.;

Igd Igdvglut glutjg jgd d ]bkt ]bkt js Igdv Igdvglutjg glutjgd7 d7

Ljdokr tjao jdvkrjkdt (LPJ) systoas kro mgg` agòls cgr akdy rokl-ljco systoas, kd` tboy bkvo prgportjos tbkt lok` tg k vory pgworcul kd` o﬎oitjvo tbogry cgr kdklyzjdm tbojr hobkvjgr. hobkvjgr. Jd tbo cgllgwjdms cgllgwjdms,, wo wkdt wkdt tg stu`y LPJ systoas tbrgumb jts ibkrkiibkrkitorjstji cuditjgd, ikllo` tbo japulso rospgdso. Pg homjd homjd wjtb, wjtb, lot us igdsj` igdsjòr or `jsiro `jsirototo-tja tjaoo sjmdkl sjmdkls. s. @odgto @odgto hy bUdQ tbo “japulso rospgdso‑ gc kd LPJ systoa Y systoa Y .. Pbo japulso rospgdso, ks jt js dkao`, js tbo rospgdso gc tbo systoa tg k udjt japulso jdput. Xoikll tbo ò﬋djtjgd gc kd udjt japulso6 ΰ UdQ 1



;, >,

d 1 > d 1 >.



(8.;)

90


Cjmuro 8.96 @o﬋djtjgd gc kd japulso rospgdso ]o bkvo sbgwd tbkt xUdQΰ Ud

∘ d Q 1 xU x Ud Qΰ Ud ∘ d Q. >

>

>

(8.8)

_sjdm tbjs ckit, wo mot tbo cgllgwjdm oqukljtjos6 xUdQΰ UdQ xUdQΰ Ud ;Q xUdQΰ Ud 8Q .. .

1 xU>Qΰ U>Qΰ UdQ 1 xU;Qΰ U;Qΰ Ud ;Q 1 xU8Qΰ U8Qΰ Ud 8Q .. .

∘ ∘

∘ ∘

(d> 1 >) (d> 1 ;) (d> 1 8)

          ∘  1xUdQ

∔

1

ΰ Ud eQ

e1∘∔

∘

∔

xUeQΰ Ud eQ

∘

e1∘∔

Pbo sua gd tbo loct bkd` sjò js xUdQ

∔

e1

hoikuso

 ∔

e1

∘∔

ΰ Ud

ΰ Ud

1 xUUdQ, e Q 1 x

∘∔

∘ eQ 1 ; cgr kll d. Pbo sua gd tbo rjmbt bkd` sjò js ∔



e1

xUe Qΰ Ud

∘∔

∘ eQ

Pborocgro, oquktjdm tbo loct bkd` sjò kd` rjmbt bkd` sjò yjol`s xUdQ 1

∔



e1

∘∔

xUe Qΰ Ud

∘ eQ

(8.9)

Jd gtbor wgr`s, cgr kdy sjmdkl x sjmdkl xUUdQ, wo ikd klwkys oxpross jt ks k sua gc japulsos!

9?

8.8. IGDSGL_P IGDSGL_PJGD JGD

Doxt, suppgso wo edgw tbkt tbo japulso rospgdso gc kd LPJ systoa js bU b UdQ. ]o wkdt tg òtorajdo tbo gutput y UdQ. Pg `g sg, wo ﬋rst oxpross xU x UdQ ks k sua gc japulsos6

∔



xUdQ 1

xUe Qΰ Ud

e1

∘∔

Cgr okib japulso ΰ japulso ΰ Ud systoa6

∘ eQ.

∘ eQ, wo ikd òtorajdo jts japulso rospgdso, hoikuso cgr kd LPJ ΰ Ud ∘ e Q ∘↘ bUd ∘ e Q.

Igdsoquodtly, wo bkvo xUdQ 1

∔



e1

xUe Qΰ Ud

∘∔

∘ eQ

∔

 ∘↘ e1

xUe QbUd

∘∔

∘ eQ 1 yU y UdQ.

Pbjs oquktjgd,

∔



yUdQ 1

xUe QbUd

e1

∘∔

∘ eQ

(8.0)

js edgwd ks tbo igdvglutjgd oquktjgd.

8.8.8 8.8.8

@o﬋d @o﬋djtj jtjgd gd kd` kd` ^rgport ^rgportjos jos gc Igd Igdvglutjg glutjgd d

systoa bUUdQ, @o﬋djtjgd ;:. Mjvod k sjmdkl xU x UdQ kd` tbo japulso rospgdso gc kd LPJ systoa b tbo igdvglutjgd hotwood xUdQ kd` bUdQ js ò﬋do` ks yUdQ 1

∔



xUe QbUd

e1

∘∔

∘ eQ.

]o òdgto igdvglutjgd ks y UdQ 1 xU x UdQ bUdQ.

∝ • Oqujvklodt cgra6 Lottjdm a 1 a 1 d d ∘ e, wo ikd sbgw tbkt ∔



e1

∘∔

xUe QbUd

∘ eQ 1

∔



a1

∘∔

xUd

∘ aQbUaQ 1

∔



e1

∘∔

xUd

∘ eQbUeQ.

93


• Igdvglutjgd js truo gdly wbod tbo systoa js LPJ. Jc tbo systoa js tjao-vkryjdm, tbod

∔



yUdQ 1

xUe Qbe Ud

e1

∘∔

∘ eQ.

j.o., b j.o., bUUdQ js `j﬎orodt kt ovory tjao jdstkdt e. Pbo cgllgwjdm “stkd`kr`‑ prgportjos ikd ho prgvo` oksjly6 ;. Igaautktjvo6 Igaautktjvo6 xUdQ bUdQ 1 bU b UdQ xUdQ

∝

∝

8. Kssgijktjvo Kssgijktjvo66 xUdQ (b; UdQ b8 UdQ) 1 (x (xUdQ b; UdQ) b8 UdQ

∝ ∝ ∝ ∝ 9. @jstrjhutj @jstrjhutjvo6 vo6 xUdQ ∝ (b UdQ + b + b UdQ) 1 (x (x(t) ∝ b UdQ) + (x (xUdQ ∝ b UdQ) ;

8.8.9 8.8.9

8

;

8

Bgw Bgw tg Ovklu Ovklukto kto Igd Igdv vglutjg glutjgd7 d7

Pg ovklukto igdvglutjgd, tboro kro tbroo hksji stops6 ;. Cljp 8. Ybjct 9. Aultjply kd` K``

Oxkaplo ;. (Yoo Ilkss @oagdstrktjgd) Igdsjòr tbo sjmdkl xUdQ kd` tbo japulso rospgdso bUdQ sbgwd holgw.

Lot‚s igaputo tbo gutput yU y UdQ gdo hy gdo. Cjrst, igdsjòr yU>Q6 y U>Q6

∔

yU>Q 1



e1

∘∔

xUe QbU>

∘ eQ 1

∔



∘

∘

e1

∘∔

Dgto tbkt b tbkt bUU e Q js tbo ﬈jppo` vorsjgd gc b gc bUUeQ, kd` k`` hotwood x hotwood xUUe Q kd` b kd` bUU e Q.

∘

xUe QbU e Q 1 ;.



∔

e1

∘∔ xUeQbU∘eQ js tbo aultjply-

9=

8.9. YZYPOA YZYPOA ^XGÔX ^XGÔXPJOY PJOY KD@ JA^_LYO JA^_LYO XOY^GDYO XOY^GDYO

Pg ikliulkto y ikliulkto yU;Q, U;Q, wo ﬈jp b ﬈jp bUUeQ tg mot b mot bUU eQ, sbjct b sbjct bUU eQ mg mot b mot bU; U; eQ, kd` aultjply-k`` ∔ tg mot e1∘∔ xUe QbU; eQ. Pborocgro,



yU;Q 1

∔



e1

∘∔

∘

∘

xUe QbU;

∘ eQ 1

∘

∔



xUe QbU;

e1

∘∔

∘

∘ eQ 1 ; Ý ; + 8 Ý ; 1 9.

^jitgrjklly, tbo ikliulktjgd js sbgwd jd tbo ﬋muro holgw.

Oxkaplo 8. (Yoo Ilkss @oagdstrktjgd)

Oxkaplo 9. (Yoo Ilkss @oagdstrktjgd) xUdQ 1 kd`

 ; 8

d

uUdQ,

+ ΰ Ud bUdQ 1 ΰ UdQ + ΰ

8.9 8.9

∘ ;Q. ;Q.

Yysto Yystoa a ^rgpor ^rgportj tjos os kd` kd` JJap apul ulso so Xos Xospgds pgdso o

]jtb tbo dgtjgd gc igdvglutjgd, wo ikd dgw prgioo` tg `jsiuss tbo systoa prgportjos jd toras gc japulso rospgdsos.

92

8.9. 8.9.; ;


Aoag Aoagry rylo loss ss

K systoa js aoagryloss jc tbo gutput òpod`s gd tbo iurrodt jdput gdly. gdly. Kd oqujvklodt oqujvklodt stktoaod stktoaodtt usjdm tbo japulso japulso rospgdso rospgdso bU b UdQ js tbkt6

Pbogroa 0. Kd LPJ systoa js aoagryloss jc kd` gdly jc bUdQ 1 kΰ UdQ, cgr sgao k

∂ I.

(8.?)

^rggc. Jc bUdQ 1 kΰ UdQ, tbod cgr kdy jdput xU x UdQ, tbo gutput js

∔



y UdQ 1 xU x UdQ bUdQ 1

∝

1

∔



e1

xUe QbUd

e1

∘∔

xUe Qkΰ Ud

∘∔

∘ eQ

∘ eQ

1 kxU kxUdQ. Yg, tbo systoa systoa js aoagry aoagrylos loss. s. Igd Igdvo vorso rsoly ly,, jc tbo systoa systoa js aoagry aoagrylos loss, s, tbod tbod y UdQ ikddgt òpod` gd tbo vkluos vkluos x cgr e 1 d. xUUeQ cgr e d . Lggejdm kt tbo igdvglutjgd sua cgraulk



yUdQ 1

∔



e1

∘∔

xUe QbUd

∘ eQ,

wo igdiluò tbkt bUd

∘ eQ 1 >,

cgr kll e kll e 1 d,



gr oqujvklodtly, bUdQ 1 >, ,

cgr kll d kll d 1 >.



Pbjs japljos 1 kxUUdQ, yUdQ 1 xU x UdQbU>Q 1 kx wboro wo bkvo sot k 1 U>Q. k 1 b bU>Q.

8.9. 8.9.8 8

Jdv Jd vortj ortjhl hlo o

Pbogroa ?. Kd LPJ systoa js jdvortjhlo jc kd` gdly jc tboro oxjst mUdQ suib tbkt bUdQ mUdQ 1 ΰ UdQ.

∝

(8.3)

9:

8.9. YZYPOA YZYPOA ^XGÔX ^XGÔXPJOY PJOY KD@ JA^_LYO JA^_LYO XOY^GDYO XOY^GDYO

^rggc. Jc k systoa Y js jdvortjhlo, tbod x; UdQ 1 x 8 UdQ japljos y; UdQ 1 y 8 UdQ. Yg tboro tboro oxjsts kd jdfoitjvo akppjdm (gdo-tg-gdo akp) Y akp) Y suib suib tbkt y tbkt yUUdQ 1 Y (xUdQ) 1 bU bUdQ xUdQ. ; ∘ Yjdio c Yjdio c js js jdfoitjvo, tboro oxjsts kd jdvorso akppjdm Y suib tbkt





∝

1 xUUdQ, Y ∘; (Y (xUdQ)) 1 x cgr kdy x kdy xUUdQ. Pborocgro, tboro oxjsts mU m UdQ suib tbkt m UdQ (bUdQ xUdQ) 1 xU x UdQ.

∝

∝

Hy kssgijktjvjty kssg ijktjvjty gc igdvglutjgd, igdvglutjgd, wo bkvo (m ( m UdQ bUdQ) xUdQ 1 xU japlyjdm mUUdQ bUdQ 1 x UdQ, japlyjdm m ΰ UdQ.

∝

∝

∝

Igdvorsoly, jc tboro oxjst mU tbkt bUUdQ m UdQ 1 ΰ UdQ, tbod cgr kdy x kdy x ; UdQ 1 x 8 UdQ, m UdQ suib tbkt b wo bkvo

∝



y; UdQ 1 bU b UdQ x; UdQ y8 UdQ 1 bU b UdQ x8 UdQ,

∝ ∝

kd` y kd` kd` y 8 UdQ, wo bkvo y ; UdQ 1 y 8 UdQ. Pkejdm tbo `j﬎orodio hotwood y ; UdQ kd` y



y; UdQ

∘ y UdQ 1 bU b UdQ ∝ {x UdQ ∘ x UdQ} . 8

;

8

Igdvglvjdm hgtb sjòs hy m hy mUUdQ yjol`s mUdQ (y; UdQ

∘ y UdQ) 1 ΰ UdQ ∝ (x UdQ ∘ x UdQ). Q). Yjdio x UdQ  1 x UdQ, kd` mUdQ  1 > cgr kll d, wo aust bkvo y UdQ  1 y UdQ. ∝

;

8

;

8

8

;

8

Pborocgr Pborocgro, o,

tbo systoa js jdvortjhlo.

8.9 8.9.9

Iku Ikuskl skl

Pbogroa 3. Kd LPJ systoa js ikuskl jc kd` gdly jc cgr kll d < >. > .

bUdQ 1 >,

(8.=)

^rggc. Jc Y Y js ikuskl, tbod tbo gutput y UdQ ikddgt òpod` gd xUeQ cgr e 4 d. Crga tbo igdvglutjgd oquktjgd, yUdQ 1

∔



e1

∘∔

xUe QbUd

∘ eQ,

0>


wo aust bkvo bUd

∘ eQ 1 >, cgr e cgr e 4 d, Yottjdm a Yottjdm a 1 1 d d ∘ e, wo soo tbkt

gr oqujvklodtly oqujvklodtly bUd

∘ eQ 1 >,

cgr d cgr d

∘ e < >. > .

cgr a < >. > . bUaQ 1 >, cgr a Igdvorsoly, jc bUeQ 1 > cgr e cgr e < >, tbod cgr jdput x jdput xUUdQ,

∔



yUdQ 1

bUe QxUd

∘ eQ 1

e1

∘∔

Pborocgro, y Pborocgro, yUUdQ òpod`s gdly upgd x upgd xUUaQ cgr a cgr a

8.9. 8.9.0 0

∔



bUe QxUd

e1>

∘ eQ.

≠ d.

Ytkh Ytkhlo lo

Pbogroa =. Kd LPJ systoa js stkhlo jc kd` gdly jc

∔

|

bUe Q <

| ∔.

e1

^rggc. Yuppgso tbkt gutput js



∘∔

∔

e1

hgudò` sjmdkl sjmdkl | xUdQ| ∘∔ |bUeQ| < ∔ . Cgr kdy hgudò`

≠ H, tbo

   | ≠  ∘   | |½| ∘ |  ≠ ½ | ∘ | ∔

|yUdQ

xUe QbUd

eQ

e1

∘∔ ∔

1

e1

∘∔

H

xUe Q

∔

e1

bUd

bUd

eQ

eQ .

∘∔

Pborocgro, y Pborocgro, yUUdQ js hgudò`. Igdvorsoly Igdvor soly,, suppgso tbkt hgudò`. @o﬋do k sjmdkl



∔

∘∔ |bUeQ| 1

e1

xUdQ 1



;,

∘;,

∔.

]o wkd wkdt tg sbg sbgw tbkt tbkt y UdQ js dgt

bU dQ 4 > 4 >,, bU dQ < > < >..

∘ ∘

0;

8.0. IGDPJD_G IGDPJD_G_Y-PJ _Y-PJAO AO IGDSGL_P IGDSGL_PJGD JGD Ilokrly, xUdQ

| | ≠ ; cgr kll d kll d.. Pbo gutput yU>Q js mjvod hy |yU>Q| 1 1

  ∘       | ∘ | | ∘ | ∔ ∔

xUeQbU eQ

e1

∘∔ ∔

bU e Q

e1

1

∘∔ ∔

e1

bU e Q 1

.

∘∔

Pborocgro, y Pborocgro, yUUdQ js dgt hgudò`.

8.0

Igdtjd Igdtjdugu ugus-t s-tjao jao Igdv Igdvglutjg glutjgd d

Pbus ckr wo bkvo hood cgiusjdm gd tbo `jsiroto-tjao ikso. Pbo igdtjdugus-tjao ikso, jd ckit, js kdklgmgus tg tbo `jsiroto-tjao ikso. Jd igdtjdugus-tjao sjmdkls, tbo sjmdkl òigapgsjtjgd js x(t) 1

 

∔

x(π ) π )ΰ (t

∘∔

∘ π ) π )`π,

(8.2)

(8.:)

kd` igdsoquodtly, tbo igdtjdugus tjao igdvglutjgd js ò﬋do` ks y (t) 1

∔

x(π ) π )b(t

∘∔

∘ π ) π )`π.

Oxkaplo. Pbo igdtjdugus-tjao igdvglutjgd klsg cgllgws tbo tbroo stop rulo6 ﬈jp, sbjct, aultjplyk``.. Pg soo kd oxkapl k`` oxkaplo, o, lot us igdsj` igdsjòr or tbo sjmdkl x(t) 1 o∘kt u(t) cgr k 4 >, kd` japulso rospgdso b rospgdso b((t) 1 u( gutput y((t) js u(t). Pbo gutput y Ikso K6 t K6 t 4 >6 y (t) 1 1

  

∔

∘∔ ∔

x(π ) π )b(t

∘ π ) π )`π

o∘kπ u(π ) π )u(t

∘∔ t

1

o∘kπ `π

>

1

  ∘ ∘ ; ; k

o∘kt .

∘ π ) π )

08


Ikso H6 t

≠ >6 y(t) 1 >.

Pborocgro, y (t) 1

8.0.; 8.0.;

∘ 

; ; k

o∘kt u(t).

^rgpor ^rgportjo tjoss gc gc IP Igdvg Igdvglut lutjgd jgd

Pbo cgllgwjdm prgportjos ikd ho prgvo` oksjly6 ;. Igaautktjvo6 Igaautktjvo6 x(t) b(t) 1 b( b (t) x(t)

∝

∝

8. Kssgijktjvo Kssgijktjvo66 x(t) (b;(t) b8(t)) 1 (x (x(t) b; (t)) b8 (t)

∝ ∝ ∝ ∝ 9. @jstrjhutj @jstrjhutjvo6 vo6 x(t) ∝ Ub (t) + b + b (t)Q 1 Ux Ux(t) ∝ b (t)Q + Ux Ux(t) ∝ b (t)Q ;

8.0.8

8

;

8

Igdtjd Igdtjdugu ugus-tja s-tjao o Yystoa Yystoa ^rgportjos ^rgportjos

Pbo cgllgwjdm rosults kro kdklgmgus tg tbo `jsiroto-tjao ikso.

Aoagryloss. Kd LPJ systoa js aoagryloss jc kd` gdly jc sgao k b(t) 1 kΰ (t), cgr sgao k

∂ I.

Jdvortjhlo. Kd LPJ systoa js jdvortjhlo jc kd` gdly jc tboro oxjst m( m (t) suib tbkt b(t) m (t) 1 ΰ (t).

∝

Ikuskl. K systoa js ikuskl jc kd` gdly jc b(t) 1 >,

cgr kll t kll t < >. > .

Ytkhlo. K systoa js stkhlo jc kd` gdly jc

 | ∔

∘∔

b(π ) π ) `π <

|

∔.

Ibkptor 9 Cgurjor Yorjos Pbo ghfoitjvo gc tbjs ibkptor js tg jòdtjcy k ckajly gc sjmdkls

{x (t)} suib tbkt6 e

;. Ovory sjmdkl jd tbo ckajly pkssos tbrgumb kdy LPJ systoa wjtb gdly k siklo ibkdmo ibkdmo (gr gtbor sjaply òsirjho` òsirjho` ibkdmo) ibkdmo) xe (t)

∘↘ ε x (t), e e

wboro ε wboro ε e js k siklo ckitgr. 8. “Kdy‑ sjmdkl ikd ho roprosodto` ks k “ljdokr igahjdktjgd‑ gc sjmdkls jd tbojr ckajly.

∔

  ∘↘

x(t) 1

e1

ke xe (t).

∘∔

Pbjs wgul` kllgw us tg òtorajdo tbo gutput modorkto` hy x( x (t)6 x(t)

∔

ke εe xe (t),

e1

∘∔

wboro tbo siklkr ke igaos crga tbo ò﬋djtjgd gc ljdokr igahjdktjgd.

9.; 9.;

Ojmo Ojmodc dcudi uditj tjgd gdss gc kd LPJ LPJ Yys Yysto toa a

Pg kdswor tbo ﬋rst quostjgd, wo doo` tbo dgtjgd gc ojmodcuditjgd gc kd LPJ systoa.

@o﬋djtjgd 8>. Cgr kd LPJ systoa, jc tbo gutput js k siklo` vorsjgd gc jts jdput, tbod tbo jdput cuditjgd js ikllo` kd ojmodcuditjgd gc tbo systoa. Pbo sikljdm ckitgr js ikllo` tbo ojmodvkluo gc tbo systoa. 09

00

IBK^POX IBK^POX 9. CG_XJOX CG_XJOX YOXJOY YOXJOY

9.;.; 9.;.;

Igdtjd Igdtjdug ugusus-tja tjao o Ikso Ikso

Igdsjòr kd LPJ systoa wjtb japulso rospgdso b(t) kd` jdput sjmdkl x sjmdkl x((t)6

Yuppgso tbkt x tbkt x((t) 1 o st cgr sgao s y (t) 1 b( b (t) x(t) 1

∝

1 1

   ∔

∘∔ ∔

∂ I, tbod tbo gutput js mjvod hy b(π ) π )x(t

∘ π ) π )`π

π ) b(π ) π )os(t∘π ) `π

∘∔ ∔



b(π ) π )o∘sπ `π ost 1 B ( B (s)ost 1 B ( B (s)x(t),

∘∔

wboro B wboro B ((s) js ò﬋do` ks B (s) 1



∔

b(π ) π )o∘sπ `π.

∘∔

Pbo cuditjgd B cuditjgd B ((s) js edgwd ks tbo trkdscor cuditjgd gc cuditjgd gc tbo igdtjdugus-tjao LPJ systoa. toa. Dgto Dgto tbkt B (s) js ò﬋do` hy tbo japulso rospgdso b(t), kd` js k cuditjgd jd s (jdòpodòdt gc t). t ). Pborocgro, B (s)x(t) ikd ho romkrò` ks k siklkr B ( B (s) aultjpljo` tg tbo cuditjgd x cuditjgd x((t). Crga tbo òrjvktjgd khgvo, wo soo tbkt jc tbo jdput js x(t) 1 o st , tbod tbo gutput js k siklo` vorsjgd y vorsjgd y (t) 1 B ( B (s)ost 6

Pborocgro, usjdm tbo ò﬋djtjgd gc ojmodcuditjgd, wo sbgw tbkt ;. ost js kd ojmodcuditjgd gc kdy igdtjdugus-tjao LPJ systoa, kd` 8. B (s) js tbo igrrospgd`jdm ojmodvkluo.

0?

9.;. OJMODC_DI OJMODC_DIPJGD PJGDY Y GC KD LPJ LPJ YZYPOA YZYPOA Jc wo spoijkljzo tg tbo suhilkss gc porjg`ji igaplox oxpgdodtjkls gc tbo o fϊt , ϊ hy sottjdm s sottjdm s 1 1 f fϊ ϊ, tbod



∔

B (s) s1 fϊ 1 B ( B ( fϊ) fϊ ) 1

|

∂ X

b(π ) π )o∘ fϊπ `π.

∘∔

tbo croquodiy rospgdso gc rospgdso gc tbo systoa. B ( fϊ) fϊ ) js ikllo` tbo croquodiy

9.;.8 9.;.8

@jsiro @jsirototo-tja tjao o Ikso Ikso

Doxt, wo igdsjòr tbo `jsiroto-tjao ikso6

Yuppgso tbkt tbo japulso rospgdso js mjvod hy bUdQ kd` tbo jdput js xUdQ 1 z d , tbod tbo gutput gutput y UdQ js yUdQ 1 bU b UdQ xUdQ 1

∝

∔

   

bUe QxUd

e1

1

∘∔ ∔

e1

1 z d wboro wo ò﬋do` B (z ) 1

bUe Qz d∘e

∘∔ ∔ e1

∘ eQ

bUe Qz ∘e 1 B ( B (z )z d ,

∘∔

∔

bUeQz ∘e ,

e1

∘∔

kd` B kd` B ((z ) js edgwd ks tbo trkdscor cuditjgd gc tbo `jsiroto-tjao LPJ systoa. Yjajlkr tg tbo igdtjdugus-tjao ikso, tbjs rosult jd`jiktos tbkt ;. z d js kd ojmodcuditjgd gc k `jsiroto-tjao LPJ systoa, kd` 8. B (z ) js tbo igrrospgd`jdm ojmodvkluo.

03


f (8ώ/D ώ/D )d Igdsjòrjdm tbo suhilkss gc porjg`ji igaplox oxpgdodtjkls o∘ f(8 hy sottjdm z 1 f8 f 8ώ/D , wo bkvo o

∔



f℣ f ℣

B (z ) z1of 1 B ( B (o ) 1

|

wboro ℣ 1

9.;. 9.;.9 9

8ώ , D

℣

e1

f ℣e bUe Qo∘ f℣ ,

∘∔

f ℣ kd` B kd` B ((o f℣ ) js ikllo` tbo croquodiy tbo croquodiy rospgdso gc rospgdso gc tbo systoa.

Yuaa Yu aakr kry y

Jd suaakry, wo bkvo tbo cgllgwjdm ghsorvktjgds6

Pbkt js, ost js kd ojmodcuditjgd gc k IP systoa, wboroks z d js kd ojmodcuditjgd gc k @P systoa. systoa. Pbo igrrospgd`jdm igrrospgd`jdm ojmodv ojmodvkluos kro B ( ). B (s) kd` B (z ). f ℣ Jc wo suhstjtuto s suhstjtuto s 1 1 f kd` z 1 o f℣ rospoitjvoly, tbod tbo ojmodcuditjgds hoigao o fϊt fϊ ϊ kd` z 1 f ℣d f ℣ kd` o kd` 5 tbo ojmodvkluos hoigao B ( kd` B ((o f℣ ). o f℣ B ( fϊ) fϊ ) kd` B

9.;.0 9.;.0

]by ]by js ojmo ojmodcu dcudit ditjgd jgd japgrtk japgrtkd dt7

Pbo kdswor tg tbjs quostjgd js rolkto` tg tbo soigd` ghfoitjvo jd tbo homjddjdm. Lot us igdsjòr k sjmdkl x(t)6 x(t) 1 k; os t + k8 os t + k9 os t . ;

8

9

Kiigr`jdm tbo ojmodcuditjgd kdklysjs, tbo gutput gc okib igaplox oxpgdodtjkl js os t os t os t ;

8

9

s; t

∘↘ B (s )o ∘↘ B (s )o ∘↘ B (s )o ; 8

9

s8 t

s9 t

.

0=

9.8. CG_XJOX YOXJOY XO^XOYODPK XO^XOYODPKPJGD Pborocgro, tbo gutput js y(t) 1 k ; B (s; )os t + k8 B (s8 )os t + k9 B (s9 )os t . ;

8

9

Pbo rosult japljos tbkt jc tbo jdput js k ljdokr igahjdktjgd gc igaplox oxpgdodtjkls, tbo gutput gc kd LPJ systoa js klsg k ljdokr igahjdktjgd gc igaplox oxpgdodtjkls. Agro modorklly, jc x jc x((t) js kd jd﬋djto sua gc igaplox oxpgdodtjkls,

∔



x(t) 1

ke ose t ,

e1

∘∔

tbod tbo gutput js kmkjd k sua gc igaplox oxpgdodtjkls6

∔

y (t) 1



ke B (se )ose t .

e1

∘∔

Yjajlkrly cgr `jsiroto-tjao sjmdkls, jc

∔

 

xUdQ 1

ke z ed ,

e1

∘∔

tbod xUdQ 1

∔

e1

ke B (z e )z ed .

∘∔

Pbjs js kd japgrtkdt ghsorvktjgd, hoikuso ks lgdm ks wo ikd oxpross k sjmdkl x( x (t) ks k ljdokr igahjdktjgd gc ojmodcuditjgds, tbod tbo gutput y( y (t) ikd ho oksjly òtorajdo` hy lggejdm lggejdm kt tbo trkdsc trkdscor or cuditjgd cuditjgd (wbjib (wbjib js ﬋xo` ﬋xo` cgr kd LPJ systoa!) systoa!).. Dgw, Dgw, tbo quos quostj tjgd gd js 6 Bgw Bgw `g wo oxpr oxpros osss k sjmd sjmdkl kl x(t) ks k ljdokr igahjdktjgd gc igaplox oxpgdodtjkls7

9.8

Cgurjor gurjor Yorjos Yorjos Xoproso Xoprosod dtktjgd tktjgd

Oxjstodio gc Cgurjor Yorjos Jd modorkl, dgt ovory sjmdkl x sjmdkl x((t) ikd ho òigapgso` ks k ljdokr igahjdktjgd gc igaplox oxpgdodtjkls. Bgwovor, suib òigapgsjtjgd js stjll pgssjhlo cgr kd oxtroaoly lkrmo ilkss gc sjmdkls. ]o wkdt tg stu`y gdo ilkss gc sjmdkls tbkt kllgws kllgws tbo òigapgsjtjgd. òigapgsjtjgd. Pboy kro tbo porjg`ji sjmdkls + P )) 1 x( x(t + P x (t)

02


wbjib sktjscy tbo squkro jdtomrkhlo igd`jtjgd,

 |

x(t) 8 `t <

|

P

∔,

gr @jrjiblot igd`jtjgds (Zgu aky ﬋d` agro `jsiussjgds jd G] 9.0)6

§

;. Gvor kdy porjg` x( ho khsglutoly jdtomrkhlo , tbkt js, x (t) aust ho khsglutoly

 |

x(t) `t <

|

P

∔.

8. Jd kdy ﬋djto jdtorvkl gc tjao x( x (t) js gc hgudò` vkrjktjgd5 tbkt js, tboro kro dg agro tbkd k ﬋djto duahor gc akxjak kd` ajdjak ùrjdm kdy sjdmlo porjg` gc tbo sjmdkl. 9. Jd kdy ﬋djto jdtorv jdtorvkl kl gc tjao, tboro kro gdly k ﬋djto duahor duahor gc `jsigdtjd `jsigdtjdujtjo ujtjos. s. Cgr tbjs ilkss gc sjmdkls, wo kro khlo tg oxpross jt ks k ljdokr igahjdktjgd gc igaplox oxpgdodtjkls6 x(t) 1

∔



ke o feϊ t . >

e1

∘∔

Boro, ϊ Boro, ϊ > js tbo cud`kaodtkl croquodiy

8ώ , P kd` tbo igo﬉ijodts k e kro edgwd ks tbo Cgurjor Yorjos igo﬉ijodts. ϊ> 1

Mjvod k porjg`ji sjmdkl x sjmdkl x((t) tbkt js squkro jdtomrkhlo, bgw `g wo òtorajdo tbo Cgurjor Yorjos igo﬉ijodts ke 7 Pbjs js kdsworo` hy tbo cgllgwjdm tbogroa.

9.8.;

Igdtjd Igdtjdugu ugus-tja s-tjao o Cgurj Cgurjor or Yorjos Yorjos Igo﬉ijod Igo﬉ijodts ts

Pbogroa 2. Pbo igdtjdugus-tjao Cgurjor sorjos igo﬉ijodts ke gc tbo sjmdkl

∔

 

x(t) 1

ke o feϊ t , >

e1

∘∔

js mjvod hy ke 1

; P

x(t)o∘ feϊ t `t.

P

>

0:

9.8. CG_XJOX YOXJOY XO^XOYODPK XO^XOYODPKPJGD ^rggc. Lot us igdsjòr tbo sjmdkl

∔

                x(t) 1

ke o feϊ t . >

e1

∘∔

Jc wo aultjply gd hgtb sjòs o sjòs o ∘ fdϊ t , tbod wo bkvo >

∔

x(t)o∘ fdϊ t 1 >

e1

feϊ > t

ke o

∔

o∘ fdϊ t 1 >

f (e∘d)ϊ t ke o f( . >

e1

∘∔

∘∔

Jdtomrktjdm hgtb sjòs crga > tg P P yjol`s P

∔

P

x(t)o∘ fdϊ t `t 1 `t 1 >

>

>

e1

∔

1

∘∔

∘

; P

>

t

`t

f (e∘d)ϊ t o f( `t . >

>

∘∔

P P f( f (e d)ϊ> t `t ikd `t ikd > o

P

ke

e1

Pbo tora

f (e∘d)ϊ ke o f(

ho ovklukto` ks (Zgu sbgul` iboie tbjs!)

P

; >

f (e∘d)ϊ t o f( `t 1 `t 1 >

>

jc e 1 d 1 d gtbo gtborw rwjs jsoo

(9.;)

Pbjs rosult js edgwd ks tbo tbo grtbgmgdkljty gc grtbgmgdkljty gc tbo igaplox oxpgdodtjkls. _sjdm Oq. (9.;), wo bkvo



P

x(t)o∘ fdϊ t `t 1 `t 1 P P kd , >

>

wbjib js oqujvklodt tg

; kd 1 P



P

x(t)o∘ fdϊ t `t. >

>

Oxkaplo Oxkaplo ;. Yjdusgj`s Yjdusgj`s Igdsjòr tbo sjmdkl x sjmdkl x((t) 1 ;+ ;8 igs igs 8ώt+sjd9 js P 1 1 8 U]by7Q sg ώt +sjd9ώt ώt.. Pbo porjg` gc x( x(t) js P 8ώ fβ tbo cud`kaodtkl croquodiy js ϊ js ϊ> 1 P 1 ώ. cgraulk o 1 igs β + f sjd ώ . Xoikll Oulor‚s cgraulk o f sjd β, wo bkvo ; f 8ώt ; f9 f 8ώt f 9ώt + o∘ f8 + x(t) 1 ; + o f8 o f 9ώt o∘ f9 . 0 8 f



 

∘



?>


Pborocgro, tbo Cgurjor sorjos igo﬉ijodts kro (fust “rok` g﬎‑ crga tbjs oquktjgd!)6 k> 1 ;,

k; 1 k∘; 1 >,

k8 1 k∘8 1

; , 0

k9 1

; , 8 f

k∘9 1

∘ 8; f ,

kd` k kd` k e 1 > gtborwjso.

Oxkaplo 8. ôrjg`ji Xoitkdmulkr ]kv ]kvo o

Lot us òtorajdo tbo Cgurjor sorjos igo﬉ijodts gc tbo cgllgwjdm sjmdkl x(t) 1



; >

|t| < P , P < |t| < ;

;

P 8

.

Pbo Cgurjor sorjos igo﬉ijodts kro (e ( e 1 >)6







; P /8 ; P ∘ feϊ t feϊ t ∘ ke 1 x(t)o `t 1 `t 1 o `t P ∘P /8 P ∘P ; P 1 o∘ feϊ t ∘P feϊ fe ϊ> P 8 8sjd(eϊ 8sjd(eϊ> P ; ) o feϊ P o∘ feϊ P 1 1 . 8 f eϊ> P eϊ> P

∘

>

   ∘    ∘  >

>

;

;

;

>

Jc e e 1 >, tbod

;

;

>

;

; P 8P ; k> 1 `t 1 `t 1 . P ∘P P Oxkaplo Oxkaplo 9. ôrjg`ji ôrjg`ji Japulso Japulso Prkjd Igdsjòr tbo sjmdkl x(t) 1 e∔1∘∔ ΰ (t eP ). cud`kaod odtkl tkl porjg` gc x(t) js P eP ). Pbo cud`ka U]by7Q. Pbo C.Y. igo﬉ijodts kro ;

;

; P /8 ; ke 1 ΰ (t)`t 1 `t 1 , P ∘P /8 P

cgr kdy e kdy e..

?;

9.8. CG_XJOX YOXJOY XO^XOYODPK XO^XOYODPKPJGD

9.8.8

@jsiroto@jsiroto-tjao tjao Cgurjor gurjor Yorjos Yorjos igo﬉ijo igo﬉ijodt dtss

Pg igdstruit igdstruit tbo `jsiroto`jsiroto-tjao tjao Cgurjor gurjor sorjos sorjos roprosod roprosodtktjg tktjgd, d, wo igdsjòr igdsjòr porjg`ji `jsiroto`jsiroto-tjao tjao sjmdkl wjtb porjg` p orjg` D D + D QQ, xUdQ 1 xU x Ud + D kd` kssuao tbkt xUdQ js squkro-suaakhlo, j.o., tbo @jrjiblot igd`jtjgds. Jd tbjs ikso, wo bkvo



∔

d1

8 ∘∔ |xUdQ| < ∔, gr xUdQ sktjs﬋os

Pbogroa :. Pbo `jsiroto-tjao Cgurjor sorjos igo﬉ijodts ke gc tbo sjmdkl xUdQ 1

∔

 

fe ℣ d ke o fe℣ , >

e1

∘∔

js mjvod hy ke 1

; D

fe ℣ d xUdQo∘ fe℣ . >

d1 D

 



Boro, d1D  aokds suaajdm tbo sjmdkl wjtbjd k porjg` D porjg` D .. Yjdio k porjg`ji `jsirot `jsirotootjao sjmdkls ropokts ovory D skaplos, D skaplos, jt `gos dgt akttor wbjib skaplo tg ho pjieo` ﬋rst.

Oxkaplo. Lot us igdsjòr tbo cgllgwjdm sjmdkl sbgwd holgw. ]o wkdt tg òtorajdo tbo `jsirototjao C.Y. igo﬉ijodt.

?8


Cgr e gr e 1 >, D, 8D , . . ., ., wo bkvo

 » »

; ke 1 D

 

; fe ℣ d 1 o∘ fe℣ >

 

8D ;

; fe ℣ 1 o∘ fe℣ D a1>

>

8D ;



a1>

jt cgllgws tbkt

∘

;

;

(a 1 d 1 d + + D D > )

,

fe ℣ a o∘ fe℣ . >

a1>

fe ℣ (8D (8D +;) ; ∘ o∘ fe℣ fe℣ fe ℣ a ∘ 1 o , fe ℣ ; ∘ o∘ fe℣ >

;

>

>

∘

fe ℣ (8D (8D +;) ; fe℣ ; o∘ fe℣ fe ℣ D ke 1 o fe ℣ ; o∘ fe℣ D fe (8ώ/ ώ/8 8D ) D ) fe8 +;/8)/D 8)/D ; o∘ fe(8 Uo fe 8ώ(D +;/ 1 fe (8ώ/ ώ/8 8D ) D ) Uo fe(8 fe (8ώ/ ώ/8 8D ) D o∘ fe(8 ; sjdU8ώe sjdU8ώe((D ; + ;/ ; /8)/D 8)/D Q 1 . D sjd( ώe ) D >

d

8D ;



> D ;

Yjdio

(a D ; )

fe ℣ o∘ fe℣

>

D d1∘D

d1 D

; fe ℣ 1 o fe℣ D

D ;



>

∘

;

>

;



(℣> 1 8ώ/D )

,

fe8 fe 8ώ (D ; +;/ +;/8)/D 8)/D

∘ o∘ ∘ o∘

Q

fe(8 fe (8ώ/ ώ/8 8D ) D ) Q

Cgr e gr e 1 >, D, 8D , . . ., ., wo bkvo

» »

ke 1

9.8.9 9.8.9

8D ; + ; . D

Bgw Bgw `g wo uso Cgu Cgurjo rjorr sorjos sorjos roproso roprosod dtktjgd tktjgd7 7

Cgurjor sorjos roprosodtktjgd skys tbkt kdy porjg`ji squkro jdtomrkhlo sjmdkls (gr sjmdkls tbkt sktjscy @jrjiblot igd`jtjgds) ikd ho oxprosso` ks k ljdokr igahjdktjgd gc igaplox igaplox oxpgdodtjkls. oxpgdodtjkls. Yjdio igaplox igaplox oxpgdodtjkls oxpgdodtjkls kro ojmodcudi ojmodcuditjgds tjgds tg LPJ systoas, tbo gutput sjmdkl aust ho k ljdokr igahjdktjgd gc igaplox oxpgdodtjkls. Pbkt js, cgr kdy sjmdkl x sjmdkl x((t) wo roprosodt jt ks x(t) 1

∔



e1

∘∔

ke o feϊ t . >

?9

9.8. CG_XJOX YOXJOY XO^XOYODPK XO^XOYODPKPJGD

Pbod, tbo gutput sjmdkl js mjvod hy y(t) 1

∔



ke B ( feϊ fe ϊ> )o feϊ t . >

e1

∘∔

Lottjdm h Lottjdm h e 1 B ( B ( feϊ fe ϊ> )ke , wo bkvo

∔



y(t) 1

e1

9.8.0

he o feϊ t . >

∘∔

Bgw Bgw akdy akdy Cgurj Cgurjor or sorjos sorjos igo﬉ijod igo﬉ijodts ts kro su﬉ijo su﬉ijodt dt7 7

Jc wo ò﬋do D

xD (t) 1



ke o feϊ t , >

e1 D

∘

tbod xD (t) js kd kpprgxjaktjgd gc x(t). Ks D , wo soo tbkt xD (t) x( x (t). Ks kd jllustrktjgd gc xD (t) ks D ks D jdiroksos, jdiroksos, wo ikd soo tbo cgllgwjdm ﬋muro. Pborocgro, tbo duahor gc Cgurjor sorjos igo﬉ijodts òpod`s gd tbo kiiurkiy tbkt wo wkdtt tg kibjov wkd kibjovo. o. Pypjikll Pypjiklly y, tbo duahor duahor D D js ibgsod suib tbkt tbo rosjùo gc tbo kpprgxjaktjgd

↘ ↘ ∔

 | ∔

∘∔

cgr sgao tkrmot orrgr lovol ο.

x(t)

D (t)

∘x

8

| `t ≠ ο,

↘

?0


9.9

^rgportj ^rgportjos os gc gc Cgur Cgurjor jor Yorjos Yorjos Igo﬉i Igo﬉ijod jodts ts

Pboro kro k duahor gc Cgurjor sorjos prgportjos tbkt wo odigurkmo ygu tg rok` tbo toxt. Pbo cgllgwjdm js k qujie suaakry gc tboso prgportjos. ;. Ljdokrjty6 Jc x x ; (t)

↞↘ h , tbod + Hx (t) ↞↘ Kk + Hh + Hh . Kx (t) + Hx Cgr @P ikso, wo bkvo jc x jc x UdQ ↞↘ k kd` kd` x x UdQ ↞↘ h , tbod + Hx UdQ ↞↘ Kk + Hh + Hh . Kx UdQ + Hx ↞↘ k

e

kd` x kd` x 8 (t)

;

e

8

;

e

;

e

e

8

e

8

e

e

8. Pjao Ybjct6 feϊ > t>

∘ t ) ↞↘ k o∘ xUd ∘ d Q ↞↘ k o∘ x(t

>

e

>

e

fe℣ fe ℣> d>

Pg sbgw tbo tjao sbjctjdm prgporty, lot us igdsjòr tbo C.Y. igo﬉ijodt h e gc tbo sjmdkl y sjmdkl y((t) 1 x( x (t t> ).

∘

; he 1 P



x(t

P

fϊ > t

∘ t )o∘ >

`t.

??

9.9. ^XGÔX ^XGÔXPJOY PJOY GC CG_XJOX CG_XJOX YOXJOY IGOCCJIJO IGOCCJIJODPY DPY Lottjdm π Lottjdm π 1 t

∘ t jd tbo jdtomrkl, wo ghtkjd >

; P wboro x wboro x((t)



x(π ) π )o∘ feϊ

>

(π + π +t> )

`π 1 o ∘ feϊ

; P

t

> >

P

↞↘ k . Pborocgro,



x(π ) π )o∘ feϊ π `π >

P

e

feϊ > t>

∘ t ) ↞↘ k o∘

x(t

>

e

.

9. Pjao Xovorskl Xovorskl66 x( t) xU dQ

∘ ↞↘ k∘ ∘ ↞↘ k∘

e e

Pbo prggc js sjaplo sjaplo.. Igd Igdsj` sjòr or k sjmdkl sjmdkl y (t) 1 x( t). Pbo C.Y. roprosod roprosodtktjg tktjgd d gc x( x ( t) js

∘

x( t) 1

∘

Lottjdm e Lottjdm e 1

∔

 

fe 8ώt/P ke o∘ fe8 .

e1

∘∔

∘a, wo bkvo

∔

y (t) 1 x( x ( t) 1

∘

Pbus, x Pbus, x(( t)

∘

∘ ↞↘ k∘ .

a1

∘∔

fa 8ώt/P k∘a o fa8 .

e

0. Igdfumktjgd Igdfumktjgd66 x∝ (t) x∝ UdQ

↞↘ k∝∘ ↞↘ k∝∘ ?. Aultjpljiktjgd6 Jc x( kd` y x (t) ↞↘ k kd` y((t) ↞↘ h , tbod ∔ x(t)y(t) ↞↘ k h ∘. e e

e

e

  | |  | | | | | | e e l

l1

3. ^krsovkl ^krsovkl Oqukljty6 ; P ; D

∘∔

8

x(t) `t 1 `t 1

P

∔

e1

 

Zgu kro roqujro` tg rok` Pkhlo 9.; kd` 9.8.

8

ke

8

∘∔

xUdQ 8 1

d1 D

ke

e1 D

 

?3


Ibkptor 0 Igdtjdugus-tjao Cgurjor Prkdscgra Lot us homjd gur `jsiussjgd hy rovjowjdm sgao ljajtktjgds gc Cgurjor sorjos roprosodtktjgd. Jd Cgurjor sorjos kdklysjs, twg igd`jtjgds gd tbo sjmdkls kro roqujro`6 ;. Pbo sjmdkl aust ho porjg`ji, j.o., tboro oxjst k P 4 > suib tbkt x tbkt x((t + P ) P ) 1 x( x (t). 8. Pbo sjmdkl aust aust ho squkro squkro jdtomrkhlo jdtomrkhlo igd`jtjgds.

 | P

x(t) 8 `t <

|

∔, gr sktjs﬋os tbo @jrjiblot

Jd tbjs ibkptor, wo wkdt tg oxtod` tbo jòk gc Cgurjor Yorjos roprosodtktjgd tg kporjg`ji sjmdkls. Pbkt js, wo wkdt tg rolkx tbo ﬋rst igd`jtjgd tg kporjg`ji sjmdkls.

0.; 0.;

Jdsjm Jdsjmb bt crg crga a Cgu Cgurj rjor or Yor Yorjo joss

Lot‚s ﬋rst igdsjòr tbo cgllgwjdm porjg`ji sjmdkl x(t) 1



; >

|t| ≠ P , P ≠ |t| < ;

;

P 8,

wboro x wboro x((t) 1 x( + P ). ). Pbo Cgurjor Yorjos igo﬉ijodts gc x gc x((t) kro (iboie ygursolc!) x (t + P x(t)

C.Y.

↞↘ k

e

1

8sjd(eϊ 8sjd(eϊ> P ; ) . eϊ> P

Jc wo suhstjtuto ϊ suhstjtuto ϊ 1 e 1 eϊ ϊ> , tbod 8sjd(ϊP 8sjd(ϊP ; ) ke 1 wP ?=



ϊ 1eϊ>

?2

IBK^POX 0. IGDPJD_G_Y-PJAO IGDPJD_G_Y-PJAO CG_XJOX PXKDYCGXA PXKDYCGXA

Aultjplyjdm P Aultjplyjdm P gd gd hgtb sjòs yjol`s P ke 1

8sjd(ϊP 8sjd(ϊP ; ) , ϊ

(0.;)

wbjib js tbo dgrakljzo` Cgurjor Yorjos igo﬉ijodt. ^jitgrjkll ^jitgrjklly y, (0.;) jd`jiktos jd`jiktos tbkt tbo dgrakljzo` dgrakljzo` Cgurjor gurjor sorjos sorjos igo﬉ijodt igo﬉ijodtss P ke kro 8 sjd( sjd(ϊP ϊP ) hgudò` hy tbo odvolgp R ( , ks jllustrkto jllustrkto` ` jd jd Cjm. Cjm. 0.;. R (ϊ ) 1 ϊ ;

Cjmuro Cjmuro 0.;6 Cgurjor gurjor Yorjos Yorjos igo﬉ijodts igo﬉ijodts gc x gc x((t) cgr sgao P sgao P .. ]bod P roùios. Bgwovo Bgwovor, r, tbo sbkpo P jdiroksos, tbo spkijdm hotwood igdsoiutjvo ke roùios. 8 sjd( sjd(ϊP ϊP ) gc tbo odvolgp cuditjgd R (ϊ ) 1 roakjd roakjdss tbo tbo skao. skao. Pbjs Pbjs ikd ikd ho sood sood jd jd Cjm. Cjm. ϊ 0.8. ;

Cjmuro Cjmuro 0.86 Cgurjor gurjor Yorjos Yorjos igo﬉ijodt igo﬉ijodtss gc x( sgao P  , wboro P wboro P  4 P . x (t) cgr sgao P P . Jd tbo ljajtjdm ikso wboro P wboro P , tbod tbo Cgurjor sorjos igo﬉ijodts P ke kpprgkibos tbo odvolgp cuditjgd R (ϊ ). Pbjs summosts summosts us tbkt jc wo bkvo kd kporjg`ji sjmdkl, sjmdkl, wo ikd trokt jt ks k porjg`ji sjmdkl wjtb P . Pbod tbo igrrospgd`jdm Cgurjor sorjos igo﬉ijodts kpprgkib tg tbo odvolgp cuditjgd R (ϊ). Pbo odvo odvolgp lgp cuditjg cuditjgd d js ikllo` ikllo` tbo Cgurjor Prkdscgra gc tbo sjmdkl x(t). Dgw, Dgw, lot us stu`y Cgurjo Cgurjorr Prkdscgra rkdscgra agro cgraklly.

↘ ∔ ↘

↘ ∔ ↘

?:

0.8. CG_XJOX CG_XJOX PXKDYCG PXKDYCGXA XA

0.8

Cgurjor gurjor Prkdscgr rkdscgra a

Pbo òrjvktjgd gc Cgurjor Prkdscgra igdsjsts gc tbroo stops.

Ytop ;. ]o kssuao tbkt kd kporjg`ji sjmdkl x sjmdkl x((t) bks ﬋djto ùrktjgd, j.o., x j.o., x((t) 1 > cgr t 4 P /8, cgr sgao P sgao P .. Yjdio x Yjdio x((t) js kporjg`ji, wo ﬋rst igdstruit k porjg`ji sjmdkl x(t)6



x(t) 1 x( x (t), cgr

 

||

∘P /8 < t < P/8, + P )) 1 x(t). ^jitgrjklly, wo bkvo P/ 8, kd` x(t + P



Ytop 8. Yjdio x(t) js porjg`ji, wo aky oxpross x(t) usjdm Cgurjor Yorjos6



x(t) 1



wboro

 ∔

ke o feϊ

>

t

(0.8)

e1

∘∔

 

; ke 1 x(t)o∘ feϊ t `t. P P Pbo Cgurjor Yorjos igo﬉ijodts k e ikd curtbor ho ikliulkto` ks ; ke 1 P

   

>

P 8

∘P

x(t)o∘ feϊ t `t >

8

P

; 1 x(t)o∘ feϊ t `t, P ∘P ; ∔ 1 x(t)o∘ feϊ t `t, P ∘∔ 8

>

8

>

Jc wo ò﬋do R ( fϊ) fϊ ) 1

x(t) 1 x( x (t), cgr

 

∘ P /8 < t < P/8 P/ 8

x(t) 1 >, cgr t 4 P /8.

∔

∘∔

| |

x(t)o∘ fϊt `t

(0.9)

3>


tbod jt bgl`s tbkt

; ke 1 R ( feϊ fe ϊ>). P Igdsoquodtly, suhstjtutjdm (0.0) jdtg (0.8) yjol`s x(t) 1

Ytop 9.

∔ ;

 

feϊ > t

R ( feϊ fe ϊ> )o P e1∘∔

1

∔



(0.0)

; R ( feϊ fe ϊ> )o feϊ t ϊ> . 8ώ e1∘∔

>

(0.?)

Dgw, dgto tbkt x(t) js tbo porjg`ji pk`ò` vorsjgd gc x gc x((t). ]bod tbo porjg` P porjg` P tbo porjg`ji sjmdkl x(t) kpprgkibos x kpprgkibos x((t). Pborocgro,

↘ ∔, ↘



x(t)



ks P ks P

↘ ∔. ↘

∘↘ x(t),

Agrogvor, wbod P wbod P kd jdtomrkl6

(0.3)

↘ ∔, gr oqujvklodtly ϊ ↘ oqujvklodtly ϊ ↘ >, tbo ljajt gc tbo sua jd (0.?) hoigaos >

∔





∔ ; ; feϊ t lja R ( feϊ fe ϊ> )o ϊ> 1 R ( fϊ) fϊ )o fwt `w. ϊ ↘> 8ώ ∘∔ 8ώ e1∘∔ >

>

Mrkpbjiklly, tbjs ikd ho sood jd Cjm. 0.9.

Cjmuro 0.96 Jllustrktjgd gc (0.=).

(0.=)

3;

0.9. XOLKPJGD XOLKPJGD PG PG CG_XJOX CG_XJOX YOXJOY YOXJOY Igahjdjdm (0.=) kd` (0.3), wo bkvo



∔ ; x(t) 1 R ( fϊ) fϊ )o fwt `w 8ώ ∘∔

(0.2)

Pbo twg oquktjgds (0.9) kd` (0.2) kro edgwd ks tbo Cgurjor Prkdscgra pkjr. (0.9) js ikllo` tbo Kdklysjs tbo Kdklysjs Oquktjgd (hoikuso (hoikuso wo kro kdklyzjdm tbo tjao sjmdkl jd tbo Cgurjor `gakjd) kd` (0.2) js ikllo` tbo Yydtbosjs Oquktjgd Oquktjgd (hoikuso wo kro mktborjdm tbo Cgurjor `gakjd jdcgraktjgd kd` roigdstruit tbo tjao sjmdkl). Pg suaakrjzo wo bkvo

Pbogroa ;>. Pbo Cgurjor Prkdscgra R ( fϊ) fϊ ) gc k sjmdkl x(t) js mjvod hy

 

R ( fϊ) fϊ ) 1

∔

x(t)o∘ fϊt `t,

∘∔

kd` tbo jdvorso Cgurjor Prkdscgra js mjvod hy

∔ ; x(t) 1 R ( fϊ) fϊ )o fwt `w. 8ώ ∘∔

0.9 0.9

Xolk Xolktj tjgd gd tg Cgurj gurjor or Yorj Yorjos os

Kt tbjs pgjdt p gjdt ygu aky wgdòr6 wgdòr6 ]bkt js tbo `j﬎orodio `j﬎orodio hotwood hotwood Cgurjor Cgurjor Yorjos kd` Cgurjor gurjor Prkdscgra7 rkdscgra7 Pg kdswor kdswor tbjs quostjgd, quostjgd, lot us kpply Cgurjo Cgurjorr Prkdscgra rkdscgra tg tbo cgllgwjdm twg typos gc sjmdkls. ;. Kporjg`ji Yjmdkl6 Ks wo `jsiusso` `jsiusso` jd tbo òrjvktjgd òrjvktjgd gc Cgurjor gurjor Prkdscgr Prkdscgra, a, tbo Cgurjor Prkdscgra gc kd kporjg`ji sjmdkl js tbo ljajtjdm ikso (wbod ϊ> >) gc kpplyjdm Cgurjor Yorjos kdklysjs gd tbo porjg`jiklly pk`ò` vorsjgd gc tbo kporjg`ji kporjg`ji sjmdkl sjmdkl.. Cgurjor gurjor Prkds Prkdscgr cgra a ikd ho kpp kppljo ljo` ` tg hgtb hgtb porjg`ji kd` kporjg`ji sjmdkls, wboroks Cgurjor Yorjos kdklysjs ikd gdly ho kppljo` tg porjg`ji sjmdkls. Yoo Cjm. 0.0.

↘

8. ôrjg`ji Yjmdkl6 Jc tbo sjmdkl x sjmdkl x((t) js porjg`ji, tbod wo `g dgt doo` tg igdstruit sot ϊ> >. Jd ckit, ϊ ckit, ϊ > js ﬋xo` hy tbo porjg` gc tbo sjmdkl6 Jc tbo porjg` x(t) kd` sot ϊ



↘

38


Cjmuro Cjmuro 0.06 Cgurjor gurjor Prkdscgra rkdscgra gd kporjg`ji sjmdkls js oqujvklod oqujvklodtt tg kpplyjdm kpplyjdm Cgurjor gurjor sorjos kdklysjs gd tbo porjg`jiklly pk`ò` vorsjgd gc tbo sjmdkl, kd` sot tbo ljajt gc >. ϊ>

↘

gc x(t) js P > , tbod ϊ> 1 8P ώ . Dgw, Dgw, sjdio sjdio x(t) js porjg`ji, wo ikd kpply Cgurjor Yorjos kdklysjs tg x(t) kd` mot >

x(t) 1

∔



ke o feϊ t ,

>

(0.:)

e1

∘∔

wboro k wboro k e js tbo Cgurjor Yorjos igo﬉ijodt. Jc wo curtbor kpply Cgurjor Prkdscgra tg (0.:), tbod wo bkvo

R ( fϊ) fϊ ) 1 1

        ∘ ∘∔ ∔

e1

1

∔

∔

∘∔ ∔

e1

ke

ke o feϊ

∘∔

∔

>

t

o∘ fϊ t `t

o feϊ t o∘ fϊ t `t >

∘∔

ke 8ώΰ (ϊ

eϊ> ).

e1

∘∔

Boro, tbo lkst oqukljty js ostkhljsbo` hy tbo ckit tbkt jdvorso Cgurjor Prkdscgra

39

0.9. XOLKPJGD XOLKPJGD PG PG CG_XJOX CG_XJOX YOXJOY YOXJOY gc 8ώΰ (ϊ

feϊ > t

∘ eϊ ) js o js o >

. Pg sbgw tbjs, wo bkvo

C ∘;{8ώΰ (ϊ ∘ eϊ>)} 1 ;



∔

8ώ ∘∔

1



∔

∘∔

8ώΰ (ϊ

ΰ (ϊ

>

feϊ > t

∘ eϊ )o >

fwt

∘ eϊ )o

`w

`w 1 `w 1 o o feϊ t . >

Pborocgro, wo sbgwo` tbkt tbo Cgurjor Prkdscgra gc k porjg`ji sjmdkl js k trkjd gc japulsos wjtb kapljtuò ò﬋do` hy tbo Cgurjor Yorjos igo﬉ijodts (kd` siklo` hy k ckitgr gc 8ώ 8 ώ).

Cjmuro Cjmuro 0.?6 Cgurjor gurjor Prkdscgra rkdscgra kd` Cgurjor gurjor Yorjos Yorjos kdklysjs kdklysjs gc k porjg`ji sjmdkl6 Hgtb yjol`s yjol`s k trkjd gc japulsos. japulsos. Cgr Cgurjor Cgurjor Prkdsc Prkdscgra, gra, tbo kapljtuò kapljtuò js aultjpljo aultjpljo` ` hy k ckitgr gc 8ώ 8ώ. Cgr Cgurjor Yorjos igo﬉ijodts, tbo sopkrktjgd hotwood okib igo﬉ijodt js ϊ> .

30

0.0


Oxkap kaplos

Oxkaplo ;. Igdsjòr tbo sjmdkl x(t) 1 o∘kt u(t), cgr k 4 >. @otorajdo @otorajdo tbo Cgurjor Cgurjor Prkdsc Prkdscgra gra R ( fϊ), fϊ ), jts akmdjtuò R ( fϊ) fϊ ) kd` jts pbkso R ( fϊ). fϊ ).

|

R ( fϊ) fϊ ) 1 1

 

∔

∘∔ ∔

|

x(t)o∘ fϊ t u(t)`t o∘kt o∘ fϊt `t

>

1

wbodovor t < > , u(t) 1 >, wbodovor t

 

∘;

k + fϊ ; 1 . k + fϊ

o∘(k+ fϊ )t

∔

>

Pbo akmdjtuò kd` pbkso ikd ho ikliulkto` ks



; k8 + ϊ 8

ϊ R ( fϊ) fϊ ) 1 ∘ tkd∘; .

|R ( fϊ) fϊ )| 1 ∖

k

Oxkaplo 8. Igdsjòr Igdsjòr tbo sjmdkl x(t) 1 ΰ (t). Pbo Cgurjor Prkdscgra js R ( fϊ) fϊ ) 1 1

 

∔

∘∔ ∔

x(t)o∘ fϊt `t ΰ (t)o∘ fϊ t `t 1 `t 1 ;.

∘∔

Oxkaplo 9. Igdsjòr tbo sjmdkl x(t) 1 o fϊ t . ]o wkdt wkdt tg sbgw tbkt tbkt R ( fϊ) fϊ ) 1 8ώΰ (ϊ >

∘ ϊ ). >

Pg

3?

0.0. 0.0. ORKA^ ORKA^LOY LOY soo, wo tkeo tbo jdvorso Cgurjor Prkdscgra6

  ∘ 

∔ ; x(t) 1 R ( fϊ) fϊ )o fϊ t `ϊ 8ώ ∘∔ 1

∔

ΰ (ϊ

∘∔

∔

fϊ > t

1 o

ϊ> )o fϊ t `ϊ

∘ ϊ )`ϊ 1 `ϊ 1 o o

; >

|t| ≠ P |t| 4 P

∘∔

Oxkaplo 0. Igdsjòr tbo kporjg`ji sjmdkl x(t) 1 Pbo Cgurjor Prkdscgra js R ( fϊ) fϊ ) 1

 

∔



1

>

.

;

;

x(t)o∘ fϊt `t

∘∔ P ;

fϊ > t

ΰ (ϊ

o∘ fϊ t `t 1 `t 1

∘P

;

 

∘;

P ;

o∘ fϊ t ∘P 1 8 fϊ ;

sjd ϊP ; . ϊ

Oxkaplo ?. Lot us òtorajdo tbo IPCP gc tbo udjt stop cuditjgd u( u(t). Pg `g sg, wo kpply IPCP kd` mot _ ( fϊ) fϊ ) 1



∔

u(t)o∘ fϊ t `t 1 `t 1



∔

fϊt

o

`t 1 `t 1

>

∘∔

∘  ; fϊ t o fϊ

∔

.

>

Pbkt js, wo bkvo tg ovklukto tbo cuditjgd fϊ; o fϊt kt t 1 > kd` t 1 . Bgw Bgwovor, ovor, tbo ovkluktjgd kt t 1 js jdòtorajdkt jdòtorajdkto! o! Pborocgr Pborocgro, o, wo oxpross oxpross y (t) ks k ljajt gc òikyjdm oxpgdodtjkls u(t) 1 lja o∘kt u(t).

∔

∔

k

Pbod kpplyjdm IPCP gd hgtb sjòs,

↘>

  kt

C o∘ u(t) ↘ k ∘ fϊ 1 lja

_ ( fϊ) fϊ ) 1 lja k

>

↘> k8 + ϊ 8

k

1 lja k

↘>



k k8 + ϊ 8

∘

; k↘> k + fϊ f ϊ

1 lja

ϊ f 8 k + ϊ 8



.

33


Dgw, tbo soigd` tora js lja f

k

↘>

∘

; ϊ 1 . k8 + ϊ 8 fϊ

Pbo ﬋rst tora sktjs﬋os k 1 >, k↘> k8 + ϊ 8 lja

kd`

; k 1 lja 1 k↘> k8 + ϊ 8 k↘> k lja

wbjlo





cgr ϊ 1 >,

∔,



∔

k ∘; ϊ ∔ 1 ώ, 1 tkd `ϊ 1 `ϊ ώ , 8 + ϊ 8 ∘∔ k k ∘∔

Pborocgro,

∎ k ∂ X.

k 1 ώΰ ώ ΰ (ϊ ), k↘> k8 + ϊ 8 lja

kd` sg _ ( _ ( fϊ) fϊ ) 1

0.?

cgr ϊ 1 >.

; + ώΰ (ϊ ). fϊ

^rgportj ^rgportjos os gc Cgurjor gurjor Prkdscgr rkdscgra a

Pbo prgportjos gc Cgurjor Prkdscgra js vory sjajlkr tg tbgso gc Cgurjor Yorjos.

;. Ljdokrjt Ljdokrjty y Jc x x ; (t)

↞↘ R ( fϊ) fϊ ) kd` x (t) ↞↘ R ( fϊ), fϊ ), tbod + hx (t) ↞↘ kR ( fϊ) + hR ( fϊ) kx (t) + hx fϊ ) + hR fϊ ). ;

8

;

8

8

;

8

8. Pjao Ybjctjdm x(t

fϊ t>

∘ t ) ↞↘ o∘ >

R ( fϊ) fϊ )

^bysjikl jdtorprotktjgd gc o o ∘ fϊ t 6 >

o∘ fϊt R ( fϊ) fϊ ) 1 R ( fϊ) fϊ ) o f R ( fϊ ) o∘ fϊ t >

| | 1 |R ( fϊ) fϊ )|o

>

fU f UR ( fϊ) fϊ ) ϊt> Q

Yg o∘ fϊ t js igdtrjhutjdm tg pbkso sbjct gc R ( fϊ). fϊ ). >

∘

3=

0.?. ^XGÔX ^XGÔXPJOY PJOY GC CG_XJOX CG_XJOX PXKDYCG PXKDYCGXA XA

9. Igdfumktjgd Igdfumktjgd x∝ (t)

↞↘ R ∝(∘ fϊ) fϊ ) Jc x( sg R (( fϊ) x (t) js rokl, tbod x ∝ (t) 1 x( x (t), sg R fϊ ) 1 R ∝ (∘ fϊ). fϊ ). 0. @j﬎orodtjktjgd @j﬎orodtjktjgd kd` Jdtomrktjgd



` x(t) `t

t

∘∔

↞↘ fϊR fϊ R ( fϊ) fϊ ) ↞↘ fϊ; R ( fϊ) + ώR (>) (>)ΰ x(π ) π )`π ↞↘ fϊ ) + ώR ΰ (ϊ )

?. Pjao Yikljdm x(t)

↞↘ |k;| R ( fϊk )

3. ^krsovkl ^krsovkl Oqukljty

 | ∔

 |

∔ ; x(t) `t 1 `t 1 R ( fϊ) fϊ ) 8 `ϊ 8ώ ∘∔ ∘∔ 8

|

|

=. @ukljty @ukljty Pbo japgrtkdt aosskmo boro js6 Jc x( x(t) R ( fϊ), fϊ ), tbod jc kdgtbor sjmdkl y(t) bks tbo sbkpo gc R ( fϊ), fϊ ), wo ikd qujiely òùio tbkt R ( fϊ) fϊ ) wjll bkvo tbo sbkpo gc x(t). Boro kro twg oxkaplos6

↞↘

@ukljty Oxkaplo ;

@ukljty Oxkaplo 8

32


2. Igdvglutjgd Igdvglutjgd ^rgporty b(t) x(t)

∝

↞↘ B ( fϊ) fϊ )R ( fϊ) fϊ )

^rggc6 Igdsjòr tbo igdvglutjgd jdtomrkl

 ∘ C  ∘    ∘    ∘      ∔

y(t) 1

x(π ) π )b(t

π ) π )`π

∘∔

Pkejdm Cgurjor Prkdscgra gd hgtb sjòs yjol`s Z ( Z ( fϊ) fϊ ) 1 1 1 1 1

∔

∔

∘∔ ∔ ∘∔ ∔ ∘∔ ∔

∘∔

x(π ) π )b(t

∔

x(π ) π )b(t

∘∔

∔

x(π ) π )

b(t

π ) π )`π

π ) π )`π o∘ fϊ t `t π ) π )o∘ fϊ t `t `π

∘∔ x(π ) π ) o∘ fϊπ B ( fϊ) fϊ ) `π x(π ) π )o∘ fϊ π `π B ( fϊ) fϊ )

∘∔

1 R ( R ( fϊ) fϊ )B ( fϊ) fϊ )

:. Aultjplji Aultjpljiktjgd ktjgd ^rgporty ^rgporty (ygu ikd òrjvo tbjs hy ygursolc) x(t)y(t)

↞↘ 8;ώ R ( fϊ) fϊ ) ∝ Z ( Z ( fϊ) fϊ )

Oxkaplo. Igdsjòr tbo sjmdkl x(t) 1 a(t)igs(ϊ )igs(ϊ> t), wboro wboro a(t) js sgao hkd`ljajto` sjmdkl. Yuppgso tbo Cgurjor Prkdscgra gc a gc a((t) js A js A (( fϊ). fϊ ). Yjdio o fϊ t + o∘ fϊ igs(ϊ igs(ϊ> t) 1 8 >

>

t

↞↘ ώUΰ (ϊ ∘ ϊ ) + ΰ + ΰ (ϊ + ϊ + ϊ )Q, )Q, >

>

hy igdvglutjgd prgporty, tbo IPCP gc x gc x((t) js )igs(ϊ> t) a(t)igs(ϊ

↞↘ 8;ώ A ( fϊ) + ΰ (ϊ + ϊ + ϊ )Q fϊ ) ∝ ώUΰ (ϊ ∘ ϊ ) + ΰ ; 1 UA U A ( f( + A (( f( + ϊ ))Q . f (ϊ ∘ ϊ )) + A f (ϊ + ϊ 8 >

>

>

>

3:

0.3. YZYPOA YZYPOA KDKLZYJY KDKLZYJY _YJDM CG_XJOX CG_XJOX PXKDYC PXKDYCGXA GXA

0.3

Yystoa Yystoa Kdklys Kdklysjs js usjdm usjdm Cgur Cgurjor jor Prkds Prkdscgr cgra a

Cjrst Gròr Yystoa Lot us igdsjòr tbo ﬋rst gròr systoa `y( `y(t) + ky( ky (t) 1 x( x (t), `t cgr sgao k sgao k 4 >. Kpplyjdm tbo IPCP tg hgtb sjòs,

C



`y( `y(t) + ky( ky (t) 1 `t

(0.;>)

C {x(t)} ,

kd` uso ljdokrjty prgporty, kd` `j﬎orodtjktjgd prgporty gc IPCP, wo bkvo + kZ (( fϊ) fϊZ fϊ Z (( fϊ) fϊ ) + kZ fϊ ) 1 R ( R ( fϊ) fϊ ). Xokrrkdmjdm tbo toras, wo ikd ﬋d` tbo croquodiy rospgdso gc tbo systoa B ( fϊ) fϊ ) 1

; Z ( Z ( fϊ) fϊ ) 1 . R ( fϊ) fϊ ) k + fϊ

Dgw, roikll tbo IPCP pkjr6 b(t) 1 o ∘kt u(t)

; ⇞⇘ B ( fϊ) fϊ ) 1 , k + fϊ

(0.;;)

b(t) ikd ho òùio`. Fust ks qujie òrjvktjgd gc tbjs oquktjgd, wo dgto tbkt B ( fϊ) fϊ ) 1

 ∘

∔

o∘kt u(t)o∘ fϊt `t 1 `t 1

∘∔

1

; o∘( fϊ +k)t fϊ + fϊ + k k



∔



∔

o∘( fϊ +k)t `t

>

1

>

; . fϊ + fϊ + k k

Modorkl Yystoas Jd modorkl, wo wkdt tg stu`y tbo systoa D



A

  C 

è y(t) 1 ke e `t e1>

è x(t) he . e `t e1>

Gur ghfoitjvo js tg òtorajdo b( kd` B (( fϊ). b (t) kd` B fϊ ). Kpplyjdm IPCP gd hgtb sjòs6

C  D

e1>

e

ke

` y(t) 1 `te

A

e1>

e

he

` x(t) `te



.

(0.;8)

=>


Pborocgro, hy ljdokrjty kd` `j﬎orodtjktjgd prgporty, wo bkvo D



A

e

ke ( fϊ) fϊ ) Z ( Z ( fϊ) fϊ ) 1

e1>



he ( fϊ) fϊ )e R ( fϊ) fϊ ).

e1>

Pbo igdvglutjgd prgporty mjvos Z ( Z ( fϊ) fϊ ) 1 R ( R ( fϊ) fϊ )B ( fϊ), fϊ ), sg Z ( Z ( fϊ) fϊ ) 1 B ( fϊ) fϊ ) 1 R ( fϊ) fϊ )



A fϊ )e e1> he ( fϊ) . D e ( ) k fϊ) fϊ e1> e

(0.;9)

Dgw, B Dgw, B (( fϊ) oxprosso` ks k rktjgdkl rktjgdkl cuditjgd, cuditjgd, j.o., k rktjg gc pglydgajkl. pglydgajkl. Pborocgr Pborocgro, o, fϊ ) js oxprosso` wo ikd kpply tbo toibdjquo gc pkrtjkl crkitjgd oxpkdsjgd tg oxpross B ( fϊ) fϊ ) jd k cgra tbkt kllgws us tg òtorajdo b( b (t) hy jdspoitjgd usjdm tbo trkdscgra pkjr b(t) 1 o ∘kt u(t)

; ⇞⇘ B ( fϊ) fϊ ) 1 , k + fϊ

kd` rolkto` trkdscgra pkjr, suib ks to∘kt u(t)

⇞⇘ (k +; fϊ . f ϊ) 8

Oxkaplo ;. Igdsjòr tbo LPJ systoa `8 y(t) `x( `x(t) + 0y 0 ( ) + 9y 9 ( ) 1 + 8x 8x(t). y t y t `t8 `t Pkejdm IPCP gd hgtb sjòs yjol`s ( fϊ) 0 fϊZ 9 Z (( fϊ) 8 R ( fϊ) fϊ )8 Z ( Z ( fϊ) fϊ ) + 0 fϊ Z (( fϊ) fϊ ) + 9Z fϊ ) 1 fϊR f ϊR (( fϊ) fϊ ) + 8R fϊ ). kd` hy rokrrkdmjdm toras wo bkvo B ( fϊ) fϊ ) 1

fϊ + fϊ + 8 fϊ + fϊ + 8 1 . ( fϊ) 0( fϊ)) + 9 ( fϊ + fϊ )8 + 0( fϊ fϊ + ;)( fϊ + fϊ + 9)

Pbod, hy pkrtjkl crkitjgd oxpkdsjgd wo bkvo

   

; ; B ( fϊ) fϊ ) 1 8 fϊ + fϊ + ;

+

; ; 8 fϊ + fϊ + 9

.

=;

0.3. YZYPOA YZYPOA KDKLZYJY KDKLZYJY _YJDM CG_XJOX CG_XJOX PXKDYC PXKDYCGXA GXA Pbus, b(t) js

; ; b(t) 1 o∘t u(t) + o∘9t u(t). 8 8

Oxkaplo 8. Jc tbo jdput sjmdkl js x(t) 1 o∘t u(t), wbkt sbgul` ho tbo gutput y (t) jc tbo japulso rospgdso gc tbo systoa js mjvod hy b( b (t) 1 ;8 o∘t u(t) + ;8 o∘9t u(t)7 Pkejdm IPCP, wo edgw tbkt R ( fϊ) fϊ ) 1 tbo gutput js

; , fϊ +;

kd` B ( fϊ) fϊ ) 1



fϊ + fϊ + 8 Z ( Z ( fϊ) fϊ ) 1 B ( B ( fϊ) fϊ )R ( fϊ) fϊ ) 1 ( fϊ + fϊ + ;)( fϊ + fϊ + 9)

fϊ+8 fϊ +8 . ( fϊ+;)( fϊ +;)( fϊ fϊ+9) +9)

 

; fϊ + fϊ + 8 1 . ( fϊ + fϊ + fϊ + ; fϊ + ;) 8 ( fϊ + fϊ + 9)

Hy pkrtjkl crkitjgd oxpkdsjgd, wo bkvo Z ( Z ( fϊ) fϊ ) 1

; 0

fϊ + fϊ + ;

+

; 8

; 0

( fϊ + fϊ + ;) 8

∘ fϊ + . fϊ + 9

Pborocgro, tbo gutput js



; ; y (t) 1 o∘t + to∘t 0 8

∘

Pboro Pborocgr cgro, o,



; ∘9t o u(t). 0

=8


Ibkptor ? @jsiroto-tjao Cgurjor Prkdscgra ?.;

Xovjow gd Igdtjd Igdtjdugus-t ugus-tjao jao Cgurjor Prkdscgra rkdscgra

Hocgro wo òrjvo tbo `jsiroto-tjao Cgurjor Prkdscgra, lot us roikll tbo wky wo igdstruito` igdtjdugus-tjao Cgurjor Prkdscgra crga tbo igdtjdugus-tjao Cgurjor Yorjos. Jd òrjvjdm tbo igdtjdugus-tjao Cgurjor Prkdscgra, wo hksjiklly bkvo tbo cgllgwjdm tbroo stops6

• Ytop ;6 ^k` tbo kporjg`ji sjmdkl

;

x(t) tg igdstruit k porjg`ji ropljikto ˜x(t)

• Ytop 86 Yjdio x˜ (t) js porjg`ji, wo ﬋d` tbo Cgurjor sorjos igo﬉ijodts

ke kd`

roprosodt x ˜ (t) ks

x˜ (t) 1

∔



e1 ;

ke o feϊ t . >

∘∔

]o kro jdtorosto` jd kporjg`ji sjmdkls, hoikuso porjg`ji sjmdkls ikd ho bkd`lo` usjdm igdtjdugustjao Cgurjor Yorjos!

=9

=0

IBK^POX IBK^POX ?. @JYIXOPO@JYIXOPO-PJAO PJAO CG_XJOX CG_XJOX PXKDYCG PXKDYCGXA XA Hy ò﬋djdm ò﬋djdm ( fϊ)) 1 R ( fϊ



∔

x (t) o fϊ t `t,

(?.;)

∘∔

wbjib js edgwd ks tbo igdtjdugus-tjao Cgurjor Prkdscgra, wo sbgwo` ; ( feϊ ke 1 R ( fe ϊ> ) . P

• Ytop 96 Yottjdm P ↘ ∔, wo sbgwo` x˜ (t) ↘ x (t) kd` Yottjdm P ↘



∔ ; ( fϊ)) o fϊ t `t x (t) 1 R ( fϊ 8ώ ∘∔

?.8

(?.8)

@orjvj @orjvjdm dm @jsir @jsiroto oto-tj -tjao ao Cgur Cgurjor jor Prkds Prkdscgr cgra a

Dgw, Dgw, lot‚s kpply tbo skao igdiopt tg `jsiroto`jsiroto-tjao tjao sjmdkls. Jd òrjvjdm tbo `jsiroto`jsirototjao Cgurjor Prkdscgra, wo klsg bkvo tbroo eoy stops.

• Ytop ;6 Igdsjòr kd kporjg`ji `jsiroto-tjao sjmdkl xUdQ.

]o pk` xUdQ tg igd-

struit k porjg`ji sjmdkl x ˜ UdQ.

• Ytop 86 Yjdio x˜UdQ js porjg`ji, hy `jsiroto-tjao Cgurjor Yorjos wo bkvo x˜UdQ 1



ke o feϊ d ,



x˜ UdQ o feϊ d .

>

e1 D

 

wboro k wboro k e ikd ho igaputo` ks ; ke 1 D

d1 D

 

>

(?.9)

=?

?.8. @OXJSJDM @OXJSJDM @JYIXOPO-PJA @JYIXOPO-PJAO O CG_XJOX CG_XJOX PXKDYCGXA PXKDYCGXA Boro, tbo croquodiy js

8ώ . D Dgw, dgto tbkt x ˜ UdQ js k porjg`ji sjmdkl wjtb porjg` D porjg` D ,, kd` tbo dgd-zorg odtrjos gc x ˜UdQ jd k porjg` kro tbo skao ks tbo dgd-zorg odtrjos gc xUdQ. Pboroc Pborocgro gro,, jt bgl`s tbkt ; ke 1 x˜ UdQ o feϊ d D ϊ> 1

  

>

d1 D

  ∔

; x UdQ o feϊ d . D d1∘∔

1 Jc wo ò﬋do

R (o fϊ ) 1

∔

d1

tbod

>

xUdQo∘ fϊ d ,

(?.0)

∘∔

∔



; ke 1 xUdQo feϊ D d1∘∔

>

d

1

; R (o feϊ ). D

>

(?.?)

• Ytop 96 ûttjdm Oquktjgd (?.?) jdtg Oquktjgd (?.9), wo bkvo

    ↘ ↘   ∘↘ ∘ ↘  ke o feϊ

x˜UdQ 1

>

d

e1 D

 

1

>

e1 D

 

1

Ks D Ks D

↘ ∔, ϊ ↘

>

x˜UdQ 1 Pborocgro,

; R (o feϊ ) o feϊ D

; 8ώ

d

R (o feϊ )o feϊ d ϊ> , >

ϊ> 1

>

e1 D

 

> kd` x˜UdQ

; 8ώ

>

(?.3)

xUdQ. Klsg, crga Oquktjgd (?.3) hoigaos

R (o feϊ )o feϊ d ϊ> >

>

e1 D

 

xUdQ 1

8ώ . D

; 8ώ

; 8ώ

8ώ

R (o fϊ )o fϊ d `ϊ.

8ώ

R (o fϊ )o fϊd `ϊ.

(?.=)

=3

IBK^POX IBK^POX ?. @JYIXOPO@JYIXOPO-PJAO PJAO CG_XJOX CG_XJOX PXKDYCG PXKDYCGXA XA

Cjmuro Cjmuro ?.;6 ?.;6 Ks D hoigaos jdtomrktjgd.

↘ ∔, ϊ ↘ >.

?.9

>

Yg tbo krok krok hoigaos hoigaos jd﬋djto jd﬋djtosja sjakl kl sakll kd` sua

]by js R (o fϊ ) porjg`ji 7

Jt js jdtorostjdm tg dgto tbkt tbo igdtjdugus-tjao igdtjdugus-tjao Cgurjor Cgurjor Prkdscgra R Prkdscgra R ( fϊ) fϊ ) js kporjg`ji fϊ jd modorkl, hut tbo `jsiroto-tjao Cgurjor Prkdscgra R ( R (o ) js klwkys porjg`ji. Pg soo tbjs, lot us igdsjòr tbo `jsiroto-tjao Cgurjor Prkdscgra (wo wkdt tg iboie wbotbor f (ϊ +8ώ +8ώ ) )!)6 R (o fϊ ) 1 R ( R (o f( f( f (ϊ +8ώ +8ώ )

R (o

)1

∔

 

f (ϊ +8ώ +8ώ )d xUdQo∘ f(

d1

1

∘∔ ∔

f 8ώ d ) 1 R ( xUdQo∘ fϊd (o∘ f8 R (o fϊ ),

d1

∘∔

f 8ώ d hoikuso (o (o∘ f8 ) 1 ;d 1 ;, cgr kdy jdtomor d. Pboroc Pborocgro gro,, R (o fϊ ) js porjg`ji wjtb porjg` 8ώ 8ώ .

Dgw, lot us igdsjòr tbo igdtjdugus-tjao Cgurjor Prkdscgra (wo wkdt tg iboie wbotbor 8 ώ))!)6 R ( fϊ) fϊ ) 1 R ( R ( f( f (ϊ + 8ώ 8 ώ)) 1 R ( f( f (ϊ + 8ώ



∔

f (ϊ +8ώ +8ώ )t x(t)o∘ f( `t 1 `t 1

∘∔



∔

f 8ώ t ) `t. x(t)o∘ fϊ t (o∘ f8

∘∔

f 8ώt Boro, t Boro, t X kd` js ruddjdm crga tg . ^ky kttodtjgd tbkt o ∘ f8 1 ; udloss t udloss t js kd jdtomor (wbjib `j﬎orodt crga tbo `jsiroto-tjao ikso wboro d js d js klwkys kd jdtomor!). Pborocgro,

∂

∘∔ ∔



∔

∘∔

f 8ώt  x(t)o∘ fϊ t o∘ f8 `t 1





∔

∘∔

x(t)o∘ fϊ t `t,

?.0. ^XGÔXPJOY ^XGÔXPJOY GC @JYIXOPO-PJAO @JYIXOPO-PJAO CG_XJOX CG_XJOX PXKDYCGXA PXKDYCGXA

==

f 8ώ d f 8ώ t (k) (o (o f8 ) 1 ; cgr kll d kll d,, hoikuso d hoikuso d js jdto jdtomo mor. r. (h) (h) ( o f8 ) 1 ; udloss t udloss t js kd jdtomor.



kd` igdsoquodtly, igdsoquodtly, 8 ώ)) 1 R ( R ( f( f (ϊ + 8ώ R ( fϊ) fϊ ).



?.0

^rgportjos ^rgportjos gc @jsiroto-t @jsiroto-tjao jao Cgurjor gurjor Prkdscgra rkdscgra

@jsiroto-tjao Cgurjor Prkdscgra6 fϊ

∔

 

R (o ) 1

d1

xUdQo∘ fϊ d

∘∔

@jsiroto-tjao Jdvorso Jdvorso Cgurjor Prkdscgra6 xUdQ 1

; 8ώ

R (o fϊ )o fϊ d `ϊ

8ώ

;. ôrjg`jijty6 ôrjg`jijty6 f (ϊ +8ώ +8ώ ) ) 1 R ( R (o f( R (o fϊ )

8. Ljdokrjt Ljdokrjty6 y6 + hx8 UdQ kx; UdQ + hx

fϊ

↞↘ kR (o ;

) + hR + hR 8 (o fϊ )

9. Pjao Ybjct6 xUd

fϊ d>

∘ d Q ↞↘ o∘ >

R (o fϊ )

0. ^bkso Ybjct6 o fϊ d xUdQ >

f( f (ϊ ϊ> )

↞↘ R (o

∘ )

=2


?. Igdfumkiy6 x∝ UdQ

fϊ

)

fϊ

)

↞↘ R ∝(o∘

3. Pjao Xovorskl Xovorskl

∘ ↞↘ R (o∘

xU dQ

=. @j﬎orodtjktjgd @j﬎orodtjktjgd dxU dxUdQ

↞↘

`R (o fϊ ) f `ϊ

2. ^krsovkl ^krsovkl Oqukljty

∔

|

; xUdQ 1 8ώ d1∘∔

|

8

 |

R (o fϊ ) 8`ϊ

|

8ώ

:. Igdvglutjgd Igdvglutjgd y UdQ 1 xU x UdQ bUdQ

fϊ

∝

↞↘ Z ( Z (o

↞↘

; Z ( Z (o fϊ ) 1 8ώ

) 1 R ( R (o fϊ )B (o fϊ )

;>. Aultjplji Aultjpljiktjgd ktjgd yUdQ 1 x ; UdQx8 UdQ

?.?



f (ϊ ∘β ) )`β R ;(o fϊ )R 8 (o f(

8ώ

Oxkap kaplos

Oxkaplo ;. Igdsjòr x Igdsjòr xUUdQ 1 ΰ UdQ + ΰ + ΰ Ud fϊ

R (o ) 1

∔

  

∘ ;Q + ΰ + ΰ Ud + ;Q. Pbod

xUdQo∘ fϊ d

d1

1

∘∔ ∔

d1

1

(ΰ UdQ + ΰ + ΰ Ud

∘∔ ∔

d1

fϊd

∘ ;Q + ΰ + ΰ Ud + ;Q)o ;Q)o∘

ΰ UdQo∘ fϊd +

∔



;Qo∘ fϊ d + ΰ Ud ∘ ;Qo

d1∘∔ ∘∔ ∘ fϊ fϊ 1 ; + o + o + o 1 ; + 8 igs igs ϊ.

∔



d1

∘∔

;Qo∘ fϊ d ΰ Ud + ;Qo

=:

?.?. ?.?. ORKA^ ORKA^LOY LOY

Cjmuro ?.86 Akmdjtuò plgt gc R (o fϊ ) jd Oxkaplo ;.

| |

|

Pg seotib tbo akmdjtuò R (o fϊ ) , wo dgto tbkt R (o fϊ ) 1 ; + 8 igs igs ϊ .

|

|

Oxkaplo 8. Igdsjòr x Igdsjòr xUUdQ 1 ΰ UdQ + 8ΰ 8 ΰ Ud

|

| |

|

∘ ;Q + 0ΰ 0ΰ Ud ∘ 8Q. Pbo `jsiroto-tjao Cgurjor Prkdscgra js f 0ϊ 8o∘ fϊ + 0o 0o∘ f0 R (o fϊ ) 1 ; + 8o .

Jc tbo japulso rospgdso js b js bUUdQ 1 ΰ UdQ + ΰ + ΰ Ud

∘ ;Q, tbod

+ o∘ fϊ . B (o fϊ ) 1 ; + o Pborocgro, tbo gutput js Z ( Z (o fϊ ) 1 B ( B (o fϊ )R (o fϊ )

 

1 ; + o + o∘ fϊ

f 8ϊ ; + 8o 8 o∘ fϊ + 0o 0o∘ f8

f 8ϊ f 9ϊ 1 ; + 9o 9o∘ fϊ + 3o 3o∘ f8 + 0o 0o∘ f9 .



Pkejdm tbo jdvorso `jsiroto-tjao Cgurjor Prkdscgra, wo bkvo 9 ΰ Ud yUdQ 1 ΰ UdQ + 9ΰ

∘ ;Q + 3ΰ 3ΰ Ud ∘ 8Q + 0ΰ 0ΰ Ud ∘ 9Q. 9Q.

Oxkaplo 9. Igdsjòr x Igdsjòr xUUdQ 1 kd uUdQ, wjtb k < ;. Pbo `jsiroto-tjao Cgurjor Prkdscgra js

| |

fϊ

R (o ) 1

∔

 d1>

d

k o∘ fϊ d 1

∔

 d1>

(ko∘ fϊ )d 1

;

∘

; . ko∘ fϊ

2>


(k) > < > < k < ;

(h)

∘; < k < >

Doxt, lot us `rkw tbo akmdjtuò R (o fϊ ) . Pg `g sg, lot‚s igdsjòr ; ; 8 R R (o fϊ ) 1 R ( R (o fϊ )R ∝ (o fϊ ) 1 ; ko∘ fϊ ; ko fϊ ; 1 ; k(o∘ fϊ o fϊ ) + k + k8 ; 1 ; 8k igs ϊ + k + k8

|

 

|

∘

∘ ∘

½ ∘

8

Ikso K. Jc > < k < ;, tbod R (o fϊ ) kibjovos akxjaua wbod ϊ wbod ϊ 1 >, kd` R (o fϊ ) kibjovos ajdjaua wbod ϊ wbod ϊ 1 ώ 1 ώ.. Pbus, ; ; 8 akx R 1 R (o fϊ ) 1 ; 8k + k8 (; k)8 ; ; 8 ajd R 1 R (o fϊ ) 1 . ; + 8k 8 k + k8 (; + k + k))8

|

  |  

|

  |  

|

∘

|

∘

8

kibjovos akxjaua wbod ϊ wbod ϊ 1 1 ώ Ikso H6 Jc ; < k < >, tbod R (o fϊ ) kibjovos ώ,, kd` R (o fϊ ) kibjovos ajdjaua wbod ϊ wbod ϊ 1 >. Pbus, ; ; 8 akx R 1 R (o fϊ ) 1 ; + 8k 8k + k8 (; + k + k))8 ; ; 8 ajd R 1 R (o fϊ ) 1 . ; 8k + k8 (; k)8

∘

?.3 ?.3

|

∘

8

|

8

∘

@jsi @jsiro roto to-t -tja jao o Cjlt Cjltor orss

Jd `jmjtkl sjmdkl prgiossjdm, tboro kro modorklly cgur typos gc ﬋ltors tbkt wo gctod uso. Dkaoly, tboy kro tbo lgwpkss ﬋ltors, bjmbpkss ﬋ltors, hkd`pkss ﬋ltors kd` hkd`stgp ﬋ltors. ﬋ltors. Pboro Pboro js dg proijso ò﬋djtjgd ò﬋djtjgd cgr hojdm “lgw‑ gr “bjmb‑. Bgwov Bgwovor, or, jt js usuklly usuklly oksy tg jdcor tbo typo crga vjowjdm tbojr akmdjtuò rospgdso.

2;

?.=. ?.=. K^^ K^ÔD@ OD@JR JR

?.=

(k) Kd jòkl lgwpkss ﬋ltor

(h) Kd jòkl bjmbpkss ﬋ltor

(i) Kd jòkl hkd`pkss ﬋ltor

(`) Kd jòkl hkd`stgp ﬋ltor

Kppod pod`jx

Mogaotrji Yorjos6 D



+ x + + x + xd 1 xd 1 ; + x x8 + . . . + x

d1>

∔

 d1>

+ x + + x xd 1 ; + x x8 + . . . 1

;

;

;

D +; D +;

∘x ;∘x

,

; . ∘ x , wbod |x| < ;.

28


Ibkptor 3 Ykapljdm Pbogroa Ykapljdm Ykapljdm tbogroa plkys plkys k iruijkl iruijkl rglo jd agòrd `jmjtkl `jmjtkl sjmdkl prgiossjdm. Pbo tbogroa igdiords khgut tbo ajdjaua skapljdm rkto roqujro` tg igdvort k igdtjdugus tjao sjmdkl tg k `jmjtkl sjmdkl, wjtbgut lgss gc jdcgraktjgd.

3.; 3.;

Kdkl Kdklgm gm tg @jmj @jmjtk tkll Igd Igdv vorsjg orsjgd d

Igdsjòr Igdsj` or tbo cgllgwjd cgllgwjdmm systoa systoa sbgwd sbgwd jd Cjm. Cjm. 3.;. Pbjs Pbjs systoa systoa js ikllo` ikllo` kd kdklgm kdklgm-tg-`jmjtkl tg-`jmjtkl (K/@) igdvorsjg igdvorsjgd d systoa. Pbo hksji jòk gc K/@ igdvorsjg igdvorsjgd d js tg tkeo k igdtjdugus-tjao sjmdkl, kd` igdvort jt tg k `jsiroto-tjao sjmdkl.

Cjmuro 3.;6 Kd kdklgm tg `jmjtkl (K/@) igdvorsjgd systoa. Aktboaktjiklly, jc tbo igdtjdugus-tjao sjmdkl js x( x (t), wo ikd iglloit k sot gc skaplos hy aultjplyjdm x aultjplyjdm x((t) wjtb kd japulso trkjd p( p (t)6 p( p(t) 1

∔



ΰ (t

d1

∘∔ 29

∘ dP ) dP ),

20

IBK^POX IBK^POX 3. YKA^LJDM YKA^LJDM PBOGXOA PBOGXOA

wboro P wboro P js js tbo porjg` gc tbo japulso trkjd. Aultjplyjdm x( x (t) wjtb p(t) yjol`s x p (t) 1 x( x (t) p( p(t)

∔

1 x( x (t)



ΰ (t

∘ dP ) dP )

x(t)ΰ (t

∘ dP ) dP )

d1

1

∔

 

d1

1

∘∔ ∔

d1

∘∔

∘∔

x(dP ) dP )ΰ (t

∘ dP ) dP ).

^jitgrjklly, x p (t) js k sot gc japulsos hgudò` hy tbo odvolgp x(t) ks sbgwd jd Cjm. 3.8.

Cjmuro 3.86 Kd oxkaplo gc K/@ igdvorsjgd. Pbo gutput sjmdkl x p (t) roprosodts k sot gc skaplos gc tbo sjmdkl x sjmdkl x((t). ]o aky romkr` x romkr` x p (t) ks tbo skaplos gc x( tbkt x p (t) js stjll k igdtjdugus-tjao x (t). Dgto tbkt x sjmdkl! sjmdkl! (]o (]o ikd vjow x p (t) ks k `jsiroto-tjao sjmdkl jc wo ò﬋do x p UdQ 1 x( x (dP ). dP ). Hut tbjs js dgt kd japgrtkdt jssuo boro.)

3.8

Croquodi roquodiy y Kdkly Kdklysjs sjs gc K/@ K/@ Igd Igdv vorsjgd orsjgd

Bkvjdm kd oxplkdktjgd gc tbo K/@ igdvorsjgd jd tjao `gakjd, wo dgw wkdt tg stu`y tbo K/@ igdvor igdvorsjgd sjgd jd tbo croquod croquodiy iy `gakjd. `gakjd. (]by7 (]by7 ]o doo` doo` jt cgr tbo òvolg òvolgppaodt gc Ykapljdm Pbogroa!) Yg, bgw `g tbo croquodiy rospgdsos R ( fϊ), fϊ ), ^ ^ (( fϊ) fϊ ) kd` R p ( fϊ) fϊ ) lgge ljeo7

3.8. CXO[_ODI CXO[_ODIZ Z KDKLZYJY KDKLZYJY GC K/@ IGDSOXYJG IGDSOXYJGD D

3.8 3.8.;

2?

Bgw `gos gos ^ ( fϊ f ϊ ) lgge ljeo7

Lot‚s stkrt wjtb ^ ( ^ ( fϊ). fϊ ). Crga Pkhlo 0.8 gc tbo toxthgge, wo edgw tbkt p( p(t) 1

∔



d1

∘∔

ΰ (t

C.P.

∘ dP ) dP ) ↞↘

∔



8ώ ΰ (ϊ P e1∘∔

8 ώe ∘ 8ώe ) 1 ^ ( ^ ( fϊ) fϊ ) P

(3.;)

Pbjs aokds tbkt tbo croquodiy rospgdso gc tbo japulso trkjd p(t) js kdgtbor japulso trkjd. Pbo gdly `j﬎orodio js tbkt tbo porjg` gc p gc p((t) js P js P ,, wboroks tbo porjg` gc ^ ( ^ ( fϊ) fϊ ) 8ώ js P .

Cjmuro 3.96 Jllustrktjgd gc R ( kd` ^ (( fϊ). R ( fϊ) fϊ ) kd` ^ fϊ ).

3.8 3.8.8

Bgw `gos gos R p ( fϊ f ϊ ) lgge ljeo7

Doxt, suppgso tbkt tbo sjmdkl x sjmdkl x((t) bks k croquodiy rospgdso R rospgdso R (( fϊ). fϊ ). ]o wkdt tg edgw tbo croquodiy rospgdso gc tbo gutput x p (t). Crga tbo ò﬋djt ò﬋djtjgd jgd gc x p (t), wo edgw edgw tbkt x p (t) 1 x( x (t) p( p(t). Pborocgro, hy tbo aultjpljiktjgd prgporty gc Cgurjor Prkdscgra, wo bkvo R p ( fϊ) fϊ ) 1

; R ( fϊ) fϊ ) ^ ( ^ ( fϊ) fϊ ). 8ώ

∝

23


Ybgwd jd Cjm. Ybgwd Cjm. 3.9 kro tbo croquo croquodiy diy rospgdso rospgdso gc R ( fϊ) rospoitjvoly. Pg fϊ ) kd` ^ ( ^ ( fϊ) fϊ ) rospoitjvoly. porcgra porcgra tbo igdvglutjg igdvglutjgd d jd croquodiy croquodiy `gakjd, wo ﬋rst dgto tbkt ^ tbkt ^ (( fϊ) fϊ ) js kd japulso trkjd. trkjd. Pborocgro Pborocgro,, igdvglv igdvglvjdm jdm R ( fϊ) fϊ ) wjtb ^ ( ^ ( fϊ) fϊ ) js hksjiklly prgùijdm ropljiktos kt 8ώ ovory P . Pbo rosult js sbgwd jd Cjm. 3.0.

Cjmuro Cjmuro 3.06 3.06 Igd Igdvg vglut lutjgd jgd hotwo hotwood od R ( fϊ) yjol`s porjg`ji porjg`ji roplji ropljikto ktoss gc fϊ ) kd` ^ ( ^ ( fϊ) fϊ ) yjol`s R ( fϊ). fϊ ). Aktboaktjiklly, tbo gutput R p ( fϊ) fϊ ) js mjvod hy



∔ ; ; ))`β R p ( fϊ) fϊ ) 1 R ( fϊ) fϊ ) ^ ( ^ ( fϊ) fϊ ) 1 R ( fβ) fβ )^ ( ^ ( f( f (ϊ β))`β 8ώ 8ώ ∘∔ ∔ ∔ ; 8ώ 8ώe 8 ώe 1 ) `β R ( fβ) fβ ) ΰ (ϊ β 8ώ ∘∔ P e1∘∔ P

∝

   ∘ ∘    ∘ ∘    ∘  ∔

; 1 P e1∘∔

∔

R ( fβ) fβ )ΰ (ϊ

β

∘∔

∔

; 1 R f( f (ϊ P e1∘∔

∘

8ώe 8 ώe )`β P

8ώe 8 ώe ) . P

Pbo rosult js jllustrkto` jd Cjm. 3.?.

3.8. 3.8.9 9

]bk bktt bk bkpp ppod odss jc P hoigaos lkrmor kd` lkrmor 7

Jc P P hoigaos lkrmor kd` lkrmor (j.o., wo tkeo cowor kd` cowor skaplos), wo edgw crga tbo ò﬋djtjgd gc p( p (t) tbkt tbo porjg` (jd tjao `gakjd) hotwood twg igdsoiutjvo

3.8. CXO[_ODI CXO[_ODIZ Z KDKLZYJY KDKLZYJY GC K/@ IGDSOXYJG IGDSOXYJGD D

2=

Cjmuro 3.?6 Jllustrktjgd gc x p (t) kd` R kd` R p p ( fϊ). fϊ ). japulsos jdiroksos (j.o., ckrtbor kpkrt). Jd croquodiy `gakjd, sjdio

∔



8ώ ^ ( ^ ( fϊ) fϊ ) 1 ΰ (ϊ P e1∘∔

8 ώe ∘ 8ώe ), P

tbo porjg` 8P ώ roùios! roùios! Jd gtbor wgr`s, wgr`s, tbo japulsos kro agro pkieo` pkieo` jd croquodi croquodiy y `gakjd wbod P wbod P jdiroksos. jdiroksos. Cjm. 3.3 jllustrktos tbjs jòk.

Cjmuro 3.36 ]bod P jdiroksos, P jdiroksos, tbo porjg` jd croquodiy `gakjd roùios. Jc wo igdsjòr R p ( fϊ), fϊ ), wbjib js k porjg`ji ropljikto gc R ( fϊ) fϊ ) kt tbo japulsos mjvod hy ^ ( sopkrktjgd h hotw otwood ood ropljiktos ropljiktos roùios. roùios. ]bod P ^ ( fϊ), fϊ ), wo soo tbkt tbo sopkrktjgd P bjts iortkjd ljajt, tbo sopkrktjgd hoigaos zorg5 kd` hoygd` tbkt ljajt, tbo ropljiktos stkrt tg gvorlkp gvorlkp!! ]bod tbo croquodiy croquodiy ropljiktos ropljiktos gvorlkp, gvorlkp, wo sky tbkt tboro js kljksjdm .

22


Cjmuro Cjmuro 3.=6 3.=6 ]bod ]bod P js P js su﬉ijodtly lkrmo, tboro wjll ho gvorlkp hotwood igdsoiutjvo ropljiktos.

Pborocgro, jd gròr tg kvgj` kljksjdm, P ikddgt P ikddgt ho tgg lkrmo. Jc wo ò﬋do tbo skapljdm rkto tg ho ϊs 1

8ώ , P

tbod sakllor sakllor P bjmbor ϊ s . Jd gtbor wgr`s, tboro js k ajdjaua skapljdm rkto P japljos bjmbor ϊ suib tbkt dg kljksjdm giiurs.

Cjmuro 3.26 Aokdjdms gc bjmb skapljdm rkto v.s. lgw skapljdm rkto.

3.8.0 3.8.0

]bkt ]bkt js tbo ajdja ajdjaua ua skaplj skapljdm dm rkto rkto suib suib tbkt tbor tboro o js dg kljksjdm7

Boro, lot us kssuao tbkt tbo sjmdkl x sjmdkl x((t) js hkd`-ljajto`. Pbkt js, wo kssuao R kssuao R ( fϊ) fϊ ) 1 > cgr kll ϊ 4 ] , wboro ] js js edgwd ks tbo hkd`-wj`tb. ] , wboro ]

| |

Pg kdswor tbjs quostjgd, wo doo` tbo Ykapljdm Pbogroa.

2:

3.9. YKA^LJDM YKA^LJDM PBOGXOA PBOGXOA

Cjmuro 3.:6 Loct6 K hkd` ljajto` sjmdkl (sjdio R ( fϊ) fϊ ) 1 > cgr kll ϊ 4 ] .) Xjmbt6 K hkd` dgd-ljajto` sjmdkl.

| |

3.9 3.9

Ykap Yk aplj ljdm dm Pbog Pbogro roa a

Pbogroa ;; (Ykapljdm Pbogroa). Lot x( x (t) ho k hkd` ljajto` sjmdkl wjtb R ( R ( fϊ) fϊ ) 1 > cgr kll ϊ 4 ] . Pbod d tbo ajdjaua ajdjaua skapljd skapljdmm rkto suib tbkt dg kljksj kljksjdm dm giiurs iurs jd ] . Pbo R p ( fϊ) fϊ ) js

| | |

ϊs 4 8 4 8], ], wboro ϊs 1

3.9. 3.9.; ;

8ώ . P

Oxpl Oxplkd kdkt ktjg jgd d

Yuppgso x Yuppgso x((t) bks hkd`wj`tb ] hkd`wj`tb ] .. Pbo tjmbtost krrkdmoaodt tbkt dg kljksjdm giiurs js sbgwd jd Cjm. 3.;>

Cjmuro 3.;>6 Ajdjaua skapljdm rkto tbkt tboro js dg kljksjdm. Jd tbjs ikso, wo soo tbkt tbo skapljdm rkto ϊ s (1 ϊs 1 8].

8ώ ) P

js

:>


Jc P 8] ,, kd` kljksjdm giiurs. P js lkrmor (gr ϊs js sakllor), tbod 8P ώ hoigaos loss tbkd 8] Pborocgro, tbo ajdjaua skapljdm rkto tg odsuro dg kljksjdm js ϊs 4 8 4 8]. ].

3.9. 3.9.8 8

Oxka Oxkapl plo o

Yuppgso tboro Yuppgso tboro js k sjmdkl sjmdkl wjtb akxjaua akxjaua croquo croquodiy diy 0>eBz. 0>eBz. ]bkt ]bkt js tbo ajdjau ajdjaua a skapljdm rkto 7

Cjmuro 3.;;6 Oxkaplo6 Ajdjaua skapljdm croquodiy.

Kdswor 6 Yjdio ϊ 1 8ώc , ώc , wo edgw tbkt tbo akx croquodiy (jd rk`) js ϊ 1 8ώ(0> ;>9 ) 1 2> ;>9 ώ (rk`). Pborocgro, tbo ajdjaua Ykapljdm rkto js6 8 (2> ;>9 ώ), wbjib js ;3> ;>9 ώ (rk`) 1 2>eBz.

Ý Ý

3.0 3.0

Ý

Ý

Ý

@jmj @jmjtk tkll tg tg Kdk Kdklg lgm m Igd Igdv vorsjg orsjgd d

Jd tbo provjgus provjgus soitjg soitjgds, ds, wo stu`jo` stu`jo` K/@ igdvo igdvorsj rsjgd. gd. Dgw, Dgw, mjvod mjvod k `jsiro `jsirototo-tja tjaoo sjmdkl (kssuao dg kljksjdm), wo wgul` ljeo tg igdstruit tbo igdtjdugus tjao sjmdkl.

3.0 3.0.;

Mjvo jvod R p(t) (dg kljksjdm), bgw `g J roigvor x(t)7

Jc dg kljksjdm giiurs ùrjdm tbo skapljdm prgiossjdm (j.o., aultjply x(t) wjtb p(t)), tbod wo ikd kpply k lgwpkss ﬋ltor B ( fϊ) Cjm. 3.;8 3.;8 fϊ ) tg oxtrkit tbo x(t) crga x p (t). Cjm. sbgws k siboaktji `jkmrka gc bgw tbjs js porcgrao`. Pg soo bgw kd jòkl lgwpkss ﬋ltor ikd oxtrkit x(t) crga x p (t), wo ﬋rst lgge kt tbo croquodiy rospgdso gc R tbkt p((t) bks k porjg` gc P (sg tbkt ϊ s 1 8P ώ ). R p p ( fϊ). fϊ ). Yuppgso tbkt p P (sg tbkt ϊ

:;

3.0. @JMJPKL @JMJPKL PG KDKLGM KDKLGM IGDSOX IGDSOXYJGD YJGD

Cjmuro Cjmuro 3.;86 Yiboaktj Yiboaktjii `jkmrka `jkmrka gc roigvorjdm roigvorjdm x(t) crga x p (t). Pb Pboo ﬋lto ﬋ltorr B ( fϊ) fϊ ) js kssuao` tg ho kd jòkl lgwpkss ﬋ltor. Pbod

∔



; R p ( fϊ) fϊ ) 1 R ( f( f (ϊ P ∘∔

∘ eϊ )). )). s

Ks sbgwd jd tbo tgp loct gc Cjm. 3.;9, R p p ( fϊ) gc R (( fϊ). fϊ ) js k porjg`ji ropljikto gc R fϊ ). Yjdio wo kssuao tbkt tboro js dg kljksjdm, tbo ropljikto igvorjdm tbo y-kxjs js jòdtjikl tg R ( fϊ). fϊ ). Pbkt js, cgr ϊ < ϊ8s ,

| |

R p ( fϊ) fϊ ) 1 R ( R ( fϊ) fϊ ). Dgw, jc wo kpply kd jòkl lgwpkss ﬋ltor (sbgwd jd hgttga loct gc Cjm. 3.;9)6 B ( fϊ) fϊ ) 1



;, ϊ < ϊ8s , >, gtborwjso, gtborwjso,

||

tbod R p ( fϊ) fϊ )B ( fϊ) fϊ ) 1 R ( R ( fϊ) fϊ ), cgr kll ϊ kll ϊ.. Pkejdm tbo jdvorso igdtjdugus-tjao Cgurjor trkdscgra, wo ikd ghtkjd x(t).

3.0.8

Jc R p(t) bks kljksjdm, ikd J stjll roigvor x(t) crga x p(t) 7

Pbo kdswor js DG. Jc kljksjdm giiurs, tbod tbo igd`jtjgd R p ( fϊ) fϊ ) 1 R ( R ( fϊ) fϊ )

:8


Cjmuro 3.;96 Loct6 Aultjpljiktjgd hotwood R p p ( fϊ) ﬋ltor B (( fϊ). fϊ ) kd` tbo lgwpkss ﬋ltor B fϊ ). Pbo ˃ ( fϊ) oxtrkito` gutput R kljksjd jdmm giiur giiurs. s. Hy kpply kpplyjd jdmm fϊ ) js jòdtjikl tg R ( fϊ) fϊ ) jc dg kljks ˃ jdvorso Cgurjor trkdscgra tg R ( fϊ) fϊ ) wo ikd ghtkjd x( x (t).

`gos dgt bgl` cgr kll ϊ < ϊ8s . Igdsoquodtly, ovod jc wo kpply tbo lgwpkss ﬋ltor B ( fϊ) fϊ ) tg R p ( fϊ), fϊ ), tbo rosult js dgt R ( R ( fϊ). fϊ ). Pbjs ikd ho sood jd Cjm. 3.;0.

||

Cjmuro Cjmuro 3.;06 Jc kljksjdm giiurs, wo kro udkhlo tg roigvor roigvor x(t) crga x p (t) hy usjdm kd jòkl lgwpkss ﬋ltor.

3.0. @JMJPKL @JMJPKL PG KDKLGM KDKLGM IGDSOX IGDSOXYJGD YJGD

3.0.9 3.0.9

:9

]bkt ]bkt ikd ikd J `g jc ay ay skaplj skapljdm dm òvji òvjio o `gos dgt dgt suppgr suppgrtt k vory bjmb skapljdm rkto 7

• Aotbg` ;6 Huy k hottor skapljdm òvjio ! • Aotbg` 86 Yod` sjmdkls wjtb dkrrgwor hkd`wj`tb gr ljajt tbo hkd`wj`tb hocgro sod`jdm 6

• Aotbg` 96 Mg tg mrk` sibggl kd` lokrd agro iggl aotbg`s !!

:0


Ibkptor = Pbo z -Prkdscgra -Prkdscgra Pbo z Pbo z -trkdscgra -trkdscgra js k modorkljzktjgd gc tbo `jsiroto-tjao Cgurjor trkdscgra wo lokrdo` jd Ibkptor ?. Ks wo wjll soo, z -trkdscgra z -trkdscgra kllgws us tg stu`y sgao systoa prgportjos tbkt @PCP ikddgt `g.

=.;

Pbo z -Prkdscgra -Prkdscgra

@o﬋djtjgd 8;. Pbo z -trkdscgra -trkdscgra gc k `jsiroto-tjao sjmdkl xUdQ js6 R (z ) 1

∔



d1

xUdQz ∘d .

(=.;)

∘∔

]o òdgto tbo z -trkdscgra -trkdscgra gporktjgd ks xUdQ

↞↘ R (z ).

Jd modorkl, tbo duahor z duahor z jd jd (=.;) js k igaplox duahor. Pborocgro, wo aky wrjto z ks 1 ro fw , z 1 wboro r wboro r

∂ X kd` w ∂ X. ]bod r 1 ;, (=.;) hoigaos fw

R (o ) 1

∔



d1

xUdQo∘ fwd ,

∘∔

wbjib js tbo `jsiroto-tjao Cgurjor trkdscgra gc xUdQ. Pboroc Pborocgro gro,, @PCP @PCP js k spoijkl spoijkl ikso gc tbo z tbo z -trkdscgra! -trkdscgra! ^jitgrjklly, wo ikd vjow @PCP ks tbo z -trkdscgra -trkdscgra ovklukto` gd tbo udjt ijrilo6 :?

:3

IBK^PO IBK^POX X =. PBO V -PXKDYCGXA V -PXKDYCGXA

Cjmuro Cjmuro =.;6 Igaplox Igaplox z -plkdo -plkdo.. Pbo z -trkdscgra -trkdscgra roùios tg @PCP cgr vkluos gc z gd tbo udjt ijrilo. ]bod r ]bod r 1 ;, tbo z -trkdscgra -trkdscgra js oqujvklodt tg



fw

R (ro ) 1 1 1

∔

      C  xUdQ ro fw

∘∔ ∔ ∘∔

∘d

r∘d xUdQ o∘ fwd

r∘d xUdQ ,

wbjib js tbo @PCP gc tbo sjmdkl r∘d xUdQ. Bgwov Bgwovor, or, crga tbo òvolgpaod òvolgpaodtt gc @PCP wo edgw edgw tbkt @PCP `gos dgt klwkys klwkys oxjst. oxjst. Jt oxjsts gdly wbod tbo sjmdkl js squkro suaakhlo, suaakhlo, gr sktjs﬋os sktjs﬋os tbo @jrjiblot @jrjiblot igd`jtjgds igd`jtjgds.. Pborocgr Pborocgro, o, R (z ) `gos dgt klwkys igdvormo. Jt igdvormos gdly cgr sgao vkluos gc r gc r.. Pbjs rkdmo gc r r js ikllo` tbo romjgd gc igdvormodio. -trkdscgra js tbo sot gc @o﬋djtjgd 88. Pbo Xomjgd gc Igdvormodio (XGI) gc tbo z -trkdscgra suib tbkt R (z ) igdvormos, j.o., z suib

∔

|

d1

∘∔

xUdQ r∘d <

|

∔.

:=

=.;. =.;. PB PBO O V -PXKDYCGXA V -PXKDYCGXA

Oxkaplo ;. Igdsjòr tbo sjmdkl x sjmdkl xUUdQ 1 k d uUdQ, wjtb > < > < k < ;. Pbo z -trkdscgra -trkdscgra gc xUdQ js

∔

  

R (z ) 1

kd uUdQz ∘d

∘∔ ∔

1

kz ∘;

d

.

d1>

Pborocgro, R (z ) igdvormos jc tbkt

   ∔

∘; )d < ∔ . Crga mogaotrj mogaotrjii sorjos sorjos,, wo edgw edgw

(kz d1> (kz

∔

rz ∘;

d

1

d1>

;

∘

; , kz ∘;

wbod kz ∘; < ;, gr oqujvklodtly z 4 k . Yg,

|

|

| | | | R (z ) 1

;

; , kx∘;

∘ wjtb XGI hojdm tbo sot gc z suib z suib tbkt |z | 4 |k|.

Cjmuro =.86 ^glo-zorg plgt kd` XGI gc Oxkaplo ;.

:2


Oxkaplo Oxkaplo 8. Igdsjòr tbo sjmdkl xUdQ 1 -trkdscgra gc xU z -trkdscgra x UdQ js

∘k uU∘d ∘ ;Q wjtb >

R (z ) 1

∔

d

 ∘∘ ∘  ∘  ∘  ∘   kd uU d

< k < ;. Pbo

;Qz ;Qz ∘d

d1

1

∘∔ ∘;

kd z ∘d

d1

1

∘∔ ∔ k∘d z d

d1;

∔

1;

d

k∘; z .

d1>

Pborocgro, R Pborocgro, R ((z ) igdvormos wbod k∘; z < ;, gr oqujvklodtly z < k . Jd tbjs ikso,

|

R (z ) 1 ;

|

| | | |

∘ ; ∘ ;k∘ z 1 ; ∘ ;kz ∘ , ;

;

wjtb XGI hojdm tbo sot gc z suib Dgto tbkt tbkt tbo tbo z -trkdscgra -trkdscgra js tbo z suib tbkt z < k . Dgto skao skao ks tbkt tbkt gc Oxkapl Oxkaploo ;. Pbo gdly `j﬎oro `j﬎orodio dio js tbo XGI. Jd ckit, Oxkaplo Oxkaplo 8 js fust tbo loct-sjò` vorsjgd gc Oxkaplo ;!

| | | |

Cjmuro =.96 ^glo-zorg plgt kd` XGI gc Oxkaplo 8.

::

=.;. =.;. PB PBO O V -PXKDYCGXA V -PXKDYCGXA

Oxkaplo 9. Igdsjòr tbo sjmdkl



; xUdQ 1 = 9

d

uUdQ

  ∘ ; 3 8

d

uUdQ.

Pbo z -trkdscgra -trkdscgra js

   ∘    ∘   ∘   ∘ ∘  ∘ ∘  ∘  ∔

d

d

; ; = 3 R (z ) 1 uUdQz ∘d 9 8 d1∘∔ ∔ ∔ d ; ; d ∘ 1= 3 uUdQz 9 8 d1∘∔ d1∘∔ ;

1= 1

;

; ; ; 9 z ; 98 z ; ; ; ; ;8 z ; 9 z

∘

;

∘

uUdQz ∘d

;

3

∘

d

; ; 8 z

∘

∘ .

Cgr R (z ) tg igdvormo, hgtb suas jd R (z ) aust igdvormo igdvormo.. Yg wo doo` hgtb z 4 ; kd` z 4 8 . Pbus, tbo XGI js tbo sot gc z suib suib tbkt z 4 ;8 .

| | | |

| | | |

Cjmuro =.06 ^glo-zorg plgt kd` XGI gc Oxkaplo 9.

; 9

| | | |

;>>


=.8 =.8

z-tr ztrkd kdsc scgr gra a ^kjrs kjrs

=.8. =.8.; ;

K. Pkh khlo lo ;>.8 ;>.8

;. ΰ UdQ

↞↘ ;, kll z kll z 8. ΰ Ud ∘ aQ ↞↘ z ∘ , kll z kll z oxiopt oxiopt > wbod a 4 > kd` ∔ wbod a wbod a < >. 9. uUdQ ↞↘ ∘ , |z | 4 ; 0. k uUdQ ↞↘ ∘ , |z | 4 k ?. ∘k uU∘d ∘ ;Q ↞↘ ∘ , |z | < |k|. Oxkaplo 0. Lot us sbgw tbkt ΰ UdQ ↞↘ ;. Pg soo tbjs, a

; ; z ∘;

; ; kz ∘;

d

; ; kz ∘;

d

∔

 

R (z ) 1

∔



xUdQz ∘d 1

d1

∘∔ ∔

1

d1

ΰ UdQz ∘d

∘∔

ΰ UdQ 1 ;.

d1

∘∔

Oxkaplo ?. Lot‚s sbgw tbkt ΰ Ud

∘ aQ ↞↘ z ∘

∔



R (z ) 1

d1

xUdQz ∘d 1

∘∔

1 z ∘a



ΰ Ud

∘∔

↞↘ kR (z ) + hR + hR (z ) ;

d>

∘ d Q ↞↘ R (z )z ∘ 9. z xUdQ ↞↘ R ( ) 0. o xUdQ ↞↘ R (o∘ z ) ?. xU∘dQ ↞↘ R ( ) 8. xUd d >

>

z z>

fw > d

fw >

; z

∔



ΰ Ud

∘∔

H. Pkh khlo lo ;>.; ;>.;

;. kx; UdQ + hx + hx8 UdQ

6

d1

∔

d1

=.8. =.8.8 8

a

8

∘ aQz ∘

a

∘ aQ 1 z ∘

.

d

;>;

=.8. V-PXKDYC V-PXKDYCGXA GXA ^KJXY 3. x∝ UdQ

↞↘ R ∝(z ∝) =. x UdQ ∝ x UdQ ↞↘ R (z )R (z ) ;

8

;

8



xUd/LQ d/LQ, d js aultjplos gc L L >, gtborwjso, gtborwjso, tbod Z tbod Z ((z ) 1 R ( R (z L ).

2. Jc yU y UdQ 1

sjmdkl bUUdQ 1 ΰ UdQ + ΰ Ud ; Q + 8ΰ 8ΰ Ud Oxkaplo 3. Igdsjòr tbo sjmdkl b gc bU b UdQ js + z ∘; + 8z 8z ∘8 . B (z ) 1 ; + z

∘

∘ 8Q. Pbo z Pbo z -Prkdscgra -Prkdscgra

;

∘ ↞↘ R (z ∘ ). Lottjdm y Lottjdm y UdQ 1 xU x U∘dQ, wo bkvo

Oxkaplo =. Lot prgvo tbkt xU dQ Z ( Z (z ) 1

∔

 

∔



yUdQz ∘d 1

d1

1

∘∔ ∔

a1

∘

d1

∘∔

xUaQz a 1 R (; R (;/z /z ).

∘∔

     ∘   ; 9

sjd

uUdQ

; 8 f

Oxkaplo 2. Igdsjòr tbo sjmdkl xUdQ 1 wo ﬋rst dgto tbkt ; xUdQ 1 8 f

xU dQz ∘d

d

; f ώ o 9 0

ώ d 0

-Prkdscgra, uUdQ. Pg ﬋d` tbo z -Prkdscgra,

; ∘ f ώ o 9 0

d

uUdQ.

Pbo z -Prkdscgra -Prkdscgra js R (z ) 1

∔

    ∘  xUdQz ∘d

d1

1

∘∔ ∔

d1>

1 1

; 8 f

; 8 f ;

(;

∘

∘

; f ώ ∘; o z 9

∔

d

0

d1>

; ; ώ ; f o z ∘; 8 f ; 9 ; ∖ z ∘; 9 8 0

∘

; f ώ0 o z ; )(; 9

∘

∘

∘

; 8 f

; ; ∘ f ώ ∘; o z 9 0

∘ .

; f ώ0 o z ;) 9

∘

; ∘ f ώ ∘; o z 9 0



d

;>8

=.9 =.9


^rg ^rgportj portjos os gc XGI

-plkdo iodtor kt grjmjd. ^rgporty ;. Pbo XGI js k rjdm gr `jse jd tbo z -plkdo

^rgporty 8. @PCP gc xUdQ oxjsts jc kd` gdly jc XGI jdiluòs tbo udjt ijrilo. ^rggc. Hy ò﬋djtjgd, XGI js tbo sot gc z suib suib tbkt R tbkt R ((z ) igdvormos. @PCP js tbo z z trkdscgra ovklukto` ovklukto` gd tbo udjt ijrilo. Pborocgro, jc XGI jdiluòs tbo udjt ijrilo, tbod gd tbo udjt ijrilo. Pbkt js, @PCP igdvormos. R (z ) igdvormos cgr kdy vkluo gc z gd

^rgporty 9. Pbo XGI igdtkjds dg pglos.

^rgporty 0. Jc xUdQ js k ﬋djto japulso rospgdso (CJX), tbod tbo XGI js tbo odtjro -plkdo. z -plkdo. ^rgporty ?. Jc xUdQ js k rjmbt-sjò` soquodio, tbod XGI oxtod`s gutwkr` crga tbo gutoragst pglo. ^rgporty ^rgporty 3. Jc xUdQ js k loct-sjò` soquodio, tbod XGI oxtod`s jdwkr` crga tbo jddoragst pglo. ^rggc. Lot‚s igdsjòr tbo rjmbt-sjò` ikso. Dgto tbkt jt js su﬉ijodt tg sbgw tbkt jc k igaplox duahor z duahor z wjtb wjtb akmdjtuò z 1 r > js jdsjò tbo XGI, tbod kdy gtbor igaplox  duahor z duahor z wjtb akmdjtuò z  1 r ; 4 r> wjll klsg ho jd tbo XGI. Dgw, suppgso xUdQ js k rjmbt-sjò` rjmbt-sjò` soquodio soquodio.. Yg, xUdQ js zorg prjgr tg sgao vkluos gc sky D ;. Pbkt js d, sky D xUdQ 1 >, d D ; .

| |

||

≠

Igdsjòr k pgjdt z pgjdt z wjtb wjtb z 1 r > , kd` r> < ;. < ;. Pbod

| |

R (z ) 1

∔

 

xUdQz ∘d

d1

1

∘∔ ∔

d1D ;

xUdQr>∘d <

∔,

;>9

=.9. ^XGÔX ^XGÔXPJOY PJOY GC GC XGI XGI

hoikuso r hoikuso r > < ; < ; mukrkdtoos tbkt tbo sua js ﬋djto. Dgw, jc tboro js kdgtbor pgjdt z  wjtb z  1 r ; 4 r> , wo aky wrjto r wrjto r ; 1 kr> cgr sgao k 4 ;. Pbod tbo sorjos

| |

∔



xUdQr∘d 1 ;

d1D ;

∔



xUdQk∘d r>∘d

d1D ; D ;

≠k

∔



xUdQr>∘d <

d1D ;

∔.

Yg, z Yg, z  js klsg jd tbo XGI.

K(z ) wboro K(z ) kd` H (z ) kro pgly^rgporty =. Jc R (z ) js rktjgdkl, j.o., R (z ) 1 H (z ) dgajkls, kd` jc xU x UdQ js rjmbt-sjò`, tbod tbo XGI js tbo romjgd gutsjò tbo gutoragst pglo.

^rggc. Jc R (z ) js rktjgdkl, tbod hy (Kppod`jx, K.?=) gc tbo toxthgge K(z ) 1 R (z ) 1 H (z )

 

d ; e e1> k e z r ; ωe , (; p e z ) e1;

∘

∘

∘

wboro p wboro p e js tbo e-tb pglo gc tbo systoa. _sjdm pkrtjkl crkitjgd, wo bkvo r

R (z ) 1

ωj

 j1; e1;

I je je ; e. (; p∘ j z )

∘

Okib gc tbo tora jd tbo pkrtjkl crkitjgd bks kd XGI hojdm tbo sot gc z z suib tbkt rjmbt-sjò`). `). Jd gròr tg bkvo bkvo R (z ) igdvormodt, tbo XGI z 4 p pj (hoikuso xUdQ js rjmbt-sjò aust ho tbo jdtorsoi jdtorsoitjgd tjgd gc kll jd`jvjùkl jd`jvjùkl XGIs. XGIs. Pborocgro Pborocgro,, tbo XGI js tbo romjgd gutsjò tbo gutoragst pglo.

| | | |

Cgr oxkaplo, jc R (z ) 1

(;

tbod tbo XGI js tbo romjgd z 4 ;8 .

| |

∘

; ; ∘; )(; 9 z

∘

, ; ;) 8 z

∘

;>0

=.0 =.0 =.0. =.0.; ;


Yyst Yystoa oa ^rgpo ^rgport rtjo joss usj usjdm dm z -trkdscgra -trkdscgra Ikus Ikuskl kljt jty y

^rgporty 2. K `jsiroto-tjao LPJ systoa js ikuskl jc kd` gdly jc XGI js tbo oxtorjgr gc k ijrilo (jdilu`jdm ).

∔ ∔

^rggc. K systoa js ikuskl jc kd` gdly jc bUdQ 1 >,

> . d < >.

Pborocgro, b Pborocgro, bUUdQ aust ho rjmbt-sjò`. ^rgporty ? japljos tbkt XGI js gutsjò k ijrilo. Klsg, hy tbo ò﬋djtjgd tbkt B (z ) 1

∔



bUdQz ∘d

d1>

wboro tboro js dg pgsjtjvo pgwors gc z gc z ,, B ( B (z ) igdvormos klsg wbod z wbod z !). z 4 ; wbod z

||

=.0. =.0.8 8

↘ ∔ ↘

↘ ∔ (Gc igurso, ↘

Ytkh Ytkhlj ljlj ljtty

^rgporty ^rgporty :. K `jsiroto-tjao LPJ systoa js stkhlo jc kd` gdly jc XGI gc B (z ) jdiluòs tbo udjt ijrilo. ^rggc. K systoa js stkhlo jc kd` gdly jc bUdQ js khsglutoly suaakhlo, jc kd` gdly jc @PCP gc bUdQ oxjsts. Igdsoquod Igdsoquodtly tly hy ^rgporty ^rgporty 8, XGI gc B (z ) aust jdiluò tbo udjt ijrilo.

^rgporty ;>. K ikuskl `jsiroto-tjao LPJ systoa js stkhlo jc kd` gdly jc kll gc jts pglos kro jdsjò tbo udjt ijrilo. ^rggc. Pbo prggc cgllgws crga ^rgporty 2 kd` : `jroitly.

=.0. YZYPOA YZYPOA ^XGÔX ^XGÔXPJOY PJOY _YJDM _YJDM V V -PXKDYCGXA -PXKDYCGXA

Oxkaplos.

;>?

Signals and System - Stanley Chan

Recommend Documents