signals and system - stanley chan

8/10/2019 Signals and System - Stanley Chan

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Ilkss Dgto cgr Yjmdkls kd` Yystoas

Ytkdloy Ibkd_djvorsjty gc Ikljcgrdjk, Ykd @jomg


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8

Kiedgwlo`moaodt

Pbjs ilkss dgto js propkro` cgr OIO ;>;6 Ljdokr Yystoas Cud`kaodtkls kt tbo _dj-vorsjty gc Ikljcgrdjk, Ykd @jomg jd Yuaaor 8>;;. Pbo prgmrka js suppgrto` hy _IYkd @jomg Yuaaor Mrk`ukto Pokibjdm Collgwsbjp.

J wgul` ljeo tg mjvo k spoijkl tbkde tg ^rgc. ^kul B. Yjomol cgr sbkrjdm bjs bkd`wrjttod dgtos, wbjib hoikao tbo hkiehgdo gc tbjs ilkss dgto. J klsg wkdt tg tbkde^rgc. Prugdm [. Dmuyod cgr prggc-rok`jdm kd` prgvj`jdm usocul coo`hkies. ^krt gctbo 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 IbkdLk Fgllk, IK.Kumust, 8>;;.


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Igdtodts

; Cud`kaodtkls gc Yjmdkls ?;.; ]bkt js k Yjmdkl7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

;.8 Xovjow gd Igaplox Duahors . . . . . . . . . . . . . . . . . . . . . . 3;.9 Hksji Gporktjgds gc Yjmdkls . . . . . . . . . . . . . . . . . . . . . . . =;.0 ^orjg`jijty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;;;.? Ovod kd` G`` Yjmdkls . . . . . . . . . . . . . . . . . . . . . . . . . . ;?;.3 Japulso kd` Ytop Cuditjgds . . . . . . . . . . . . . . . . . . . . . . . ;=;.= Igdtjdugus-tjao Igaplox Oxpgdodtjkl Cuditjgds . . . . . . . . . . . 88;.2 @jsiroto-tjao Igaplox Oxpgdodtjkls . . . . . . . . . . . . . . . . . . 80

8 Cud`kaodtkls gc Yystoas 8=8.; Yystoa ^rgportjos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828.8 Igdvglutjgd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 998.9 Yystoa ^rgportjos kd` Japulso Xospgdso . . . . . . . . . . . . . . . . 9=8.0 Igdtjdugus-tjao Igdvglutjgd . . . . . . . . . . . . . . . . . . . . . . 0;

9 Cgurjor Yorjos 099.; Ojmodcuditjgds gc kd LPJ Yystoa . . . . . . . . . . . . . . . . . . . . 099.8 Cgurjor Yorjos Xoprosodtktjgd . . . . . . . . . . . . . . . . . . . . . . 0=9.9 ^rgportjos gc Cgurjor Yorjos Igoijodts . . . . . . . . . . . . . . . . . ?0

0 Igdtjdugus-tjao Cgurjor Prkdscgra ?=0.; Jdsjmbt crga Cgurjor Yorjos . . . . . . . . . . . . . . . . . . . . . . . ?=

0.8 Cgurjor Prkdscgra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?:0.9 Xolktjgd tg Cgurjor Yorjos . . . . . . . . . . . . . . . . . . . . . . . . 3;0.0 Oxkaplos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300.? ^rgportjos gc Cgurjor Prkdscgra . . . . . . . . . . . . . . . . . . . . . 330.3 Yystoa Kdklysjs usjdm Cgurjor Prkdscgra . . . . . . . . . . . . . . . . 3:

9


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0 IGDPODPY

? @jsiroto-tjao Cgurjor Prkdscgra =9

?.; Xovjow gd Igdtjdugus-tjao Cgurjor Prkdscgra . . . . . . . . . . . . . =9?.8 @orjvjdm @jsiroto-tjao Cgurjor Prkdscgra . . . . . . . . . . . . . . . . =0?.9 ]by jsR(of) porjg`ji 7 . . . . . . . . . . . . . . . . . . . . . . . . . =3?.0 ^rgportjos gc @jsiroto-tjao Cgurjor Prkdscgra . . . . . . . . . . . . . ==?.? Oxkaplos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . =2?.3 @jsiroto-tjao Cjltors . . . . . . . . . . . . . . . . . . . . . . . . . . . 2>?.= Kppod`jx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2;

3 Ykapljdm Pbogroa 293.; Kdklgm tg @jmjtkl Igdvorsjgd . . . . . . . . . . . . . . . . . . . . . . 293.8 Croquodiy Kdklysjs gc K/@ Igdvorsjgd . . . . . . . . . . . . . . . . . 20

3.9 Ykapljdm Pbogroa . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2:3.0 @jmjtkl tg Kdklgm Igdvorsjgd . . . . . . . . . . . . . . . . . . . . . . :>

= Pbo z-Prkdscgra :?=.; Pbo z-Prkdscgra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . :?=.8 z-trkdscgra ^kjrs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;>>=.9 ^rgportjos gc XGI . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;>8=.0 Yystoa ^rgportjos usjdmz-trkdscgra . . . . . . . . . . . . . . . . . . ;>0


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Ibkptor ;

Cud`kaodtkls gc Yjmdkls

;.; ]bkt js k Yjmdkl7

K sjmdkl js k qukdtjtktjvo òsirjptjgd gc k pbysjikl pbodgaodgd, ovodt gr prgioss.Ygao igaagd oxkaplos jdiluò6

;. Oloitrjikl iurrodt gr vgltkmo jd k ijriujt.

8. @kjly ilgsjdm vkluo gc k sbkro gc stgie lkst wooe.

9. Ku`jg sjmdkl6 igdtjdugus-tjao jd jts grjmjdkl cgra, gr `jsiroto-tjao wbod stgro`gd k I@.

Agro proijsoly, k sjmdkl js k cuditjgd, usuklly gc gdo vkrjkhlo jd tjao. Bgwovor, jdmodorkl, sjmdkls ikd ho cuditjgds gc agro tbkd gdo vkrjkhlo, o.m., jakmo sjmdkls.

Jd tbjs ilkss wo kro jdtorosto` jd twg typos gc sjmdkls6

;. Igdtjdugus-tjao sjmdkl x(t), wboro t js k rokl-vkluo` vkrjkhlo òdgtjdm tjao,j.o.,t X. ]o uso pkrodtbosjs () tg òdgto k igdtjdugus-tjao sjmdkl.

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 hrkieots UQ tg òdgto k`jsiroto-tjao sjmdkl. _dòr tbo òdjtjgd gc k `jsiroto-tjao sjmdkl, xU;.?Q js dgtòdo`, cgr oxkaplo.

?


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3 IBK^POX ;. C_D@KAODPKLY GC YJMDKLY

;.8 Xovjow gd Igaplox Duahors

]o kro jdtorosto` jd tbo modorkl igaplox sjmdkls6

x(t) I kd` xUdQ I,

wboro tbo sot gc igaplox duahors js `odo` ks

I 1 {z| z1 x+fy, x, y X, f 1 ;.}

K igaplox duahor zikd ho roprosodto` jd Ikrtosjkd cgra ks

z1 x+fy,

gr jd pglkr cgra ksz1 rof .

Pbogroa ;. Oulors Cgraulk

of 1 igs +fsjd . (;.;)

_sjdm Oulors cgraulk, tbo rolktjgd hotwood x, y , r, kd` js mjvod hy

x1 r igs y 1 r sjd

kd` r1 x8 +y8,1 tkd; y

x.

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


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;.9. HKYJI GÔXKPJGDY GC YJMDKLY =

K igaplox duahor ikd ho `rkwd gd tbo igaplox plkdo ks sbgwd jd Cjm. ;.;. Pbo

y-kxjs gc tbo igaplox plkdo js edgwd ks tbo jakmjdkry kxjs, kd` tbo x-kxjs gc tboigaplox plkdo js edgwd ks tbo rokl kxjs. K igaplox duahor js udjquoly òdo` hyz1 x+fy jd tbo Ikrtosjkd cgra, gr z1rof jd tbo pglkr cgra.

Oxkaplo. Igdvort tbo cgllgwjdm igaplox duahors crga Ikrtosjkd cgra tg pglkrcgra6 (k) ; + 8f5 (h) ; f.

Cgr (k), wo kpply Oulors cgraulk kd` d` tbkt

r1

;8 + 88 1

?, kd` 1 tkd;

8

;

39.30.

Pborocgro, ; + 8f1 ?of39.30 .Cgr (h), wo kpply Oulors cgraulk kmkjd kd` d` tbkt

r1

;8 + (;)8 1

8, kd` 1 tkd;;

;

1 0?.

Pborocgro,; f 1

8of/0.

Xoikll tbkt6 jd rk`jkd 1 ;2> jd òmroo.

Oxkaplo. Ikliulkto tbo vkluo gcff.ff 1 (of/8)f 1o/8 >.8>=2.

;.9 Hksji Gporktjgds gc Yjmdkls

;.9.; Pjao Ybjct

Cgr kdyt> X kd` d> V, tjao sbjct js kd gporktjgd òdo` ksx(t) x(t t>)xUdQ xUd d>Q. (;.8)

Jct> 4 >, tbo tjao sbjct js edgwd ks òlky. Jct> < >, tbo tjao sbjct js edgwd ksk`vkdio.

Oxkaplo. Jd Cjm. ;.8, tbo loct jakmo sbgws k igdtjdugus-tjao sjmdklx(t). K tjao-sbjcto` vorsjgd x(t 8) js sbgwd jd tbo rjmbt jakmo.


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Cjmuro ;.86 Kd oxkaplo gc tjao sbjct.

;.9.8 Pjao Xovorskl

Pjao rovorskl js `odo` ksx(t) x(t)xUdQ

xU

dQ,

(;.9)

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

Oxkaplo.

Cjmuro ;.96 Kd oxkaplo gc tjao rovorskl.

;.9.9 Pjao Yikljdm

Pjao sikljdm js tbo gporktjgd wboro tbo tjao vkrjkhlot js aultjpljo` hy k igdstkdt k6

x(t) x(kt), k 4> (;.0)Jck 4;, tbo tjao siklo gc tbo rosultkdt sjmdkl js `oijakto` (spoo` up). Jc > < k


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;.9. HKYJI G^OXKPJGDY GC YJMDKLY :

Cjmuro ;.06 Kd oxkaplo gc tjao sikljdm.

Aotbg` K6 Ybjct, tbod Yiklo (Xoigaaod`o`)

;. @odo v(t) 1x(t

h),

8. @odo y(t) 1v(kt) 1x(kt h).

Aotbg` H6 Yiklo, tbod Ybjct

;. @odo v(t) 1x(kt),

8. @odo y(t) 1v(t hk) 1x(kt h).

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

?).

Cjmuro ;.?6 Oxkaplo ;. x(9t ?).

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


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;> IBK^POX ;. C_D@KAODPKLY GC YJMDKLY

Cjmuro ;.36 Oxkaplo 8. x(t+ ;).

Cjmuro ;.=6 @oijaktjgd kd` oxpkdsjgd.

;.9.? @oijaktjgd kd` Oxpkdsjgd

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

@oijaktjgd.

@oijaktjgd js òdo` ks

y@UdQ 1xUAdQ, (;.?)

cgr sgao jdtomorsA. Ajs ikllo` tbo òijaktjgd ckitgr.

Oxpkdsjgd.

Oxpkdsjgd js òdo` ks

yOUdQ 1x d

L , d1 jdtomor aultjplo gcL

>, gtborwjso.(;.3)

Ljs ikllo` tbo oxpkdsjgd ckitgr.


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;.0. ÔXJG@JIJPZ ;;

Cjmuro ;.26 Oxkaplos gc òijaktjgd kd` oxpkdsjgd cgrA1 8 kd` L 1 8.

;.0 ôrjg`jijty

;.0.; @odjtjgds

@odjtjgd ;. K igdtjdugus tjao sjmdklx(t) js porjg`ji jc tboro js k igdstkdtP 4>suib tbkt

x(t) 1x(t+P), (;.=)

cgr kllt X.@odjtjgd 8. K `jsiroto tjao sjmdklxUdQ js porjg`ji jc tboro js kd jdtomor igdstkdt

D 4> suib tbktxUdQ 1xUd+DQ, (;.2)

cgr klld V.Yjmdkls `g dgt sktjscy tbo porjg`jijty igd`jtjgds kro ikllo` kporjg`ji sjmdkls.

Oxkaplo. Igdsjòr tbo sjmdkl x(t) 1 sjd(>t), > 4 >. Jt ikd ho sbgwd tbktx(t) 1x(t+P), wboro P 1e 8> cgr kdye V+6

x(t+P) 1 sjd > t+e8

>1 sjd (>t+ 8e)1 sjd(>t) 1x(t).

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


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;0 IBK^POX ;. C_D@KAODPKLY GC YJMDKLY

tbjs igd`jtjgd hy `jvj`jdm hgtb sj`os gc tbo oquktjgd hy /2 tg yjol`

(d+D)8 1d8 +2

(8e),

gr

d8 + 8dD+ D8 1d8 + ;3e,

japlyjdm

8dD+ D8 1 ;3e,

cgr sgao e V. Doxt, wo wkdt tg d` kdD suib tbkt 8dD+D8 js `jvjsjhlo hy ;3cgr kll d V. Dgw wo ilkja6 D1 2 sktjsos tbjs igd`jtjgd, kd` dg sakllor D 4 >`gos.

YottjdmD1 2, wo mot

8dD+ D8 1 ;3d+ 30,

wbjib, cgr kdyd V, js ilokrly `jvjsjhlo hy ;3. Yg D1 2 js k porjg` gcxUdQ. Zgu ikdiboie `jroitly tbkt, cgr kdy ; D . Jsy(t) 1x(kt) porjg`ji7Zos, kd` jcP> js tbo cud`kaodtkl porjg` gcx(t), tbod P>/k js tbo cud`kaodtklporjg` gcy(t).

8. Yuppgso xUdQ js porjg`ji, kd` lot a V+. Jsy UdQ 1xUadQ porjg`ji7Zos, kd` jcD> js tbo cud`kaodtkl porjg` gcxUdQ, tbod tbo cud`kaodtkl porjg`D gcy UdQ js tbo sakllost pgsjtjvo jdtomor suib tbkt aDjs `jvjsjhlo hyD>, j.o.

aD > ( ag` D>).

Oxkaplo ;6 D>1 2,a 1 8, tbod D1 0.Oxkaplo 86 D>1 3,a 1 0, tbod D1 9.


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;.?. OSOD KD@ G@@ YJMDKLY ;?

;.? Ovod kd` G`` Yjmdkls

;.?.; @odjtjgds

@odjtjgd ?. K igdtjdugus-tjao sjmdklx(t) js ovod jc

x(t) 1x(t) (;.:)

kd` jt js g`` jc

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

@odjtjgd 3. K `jsiroto-tjao sjmdklxUdQ js ovod jc

xUdQ 1xUdQ (;.;;)

kd` g`` jc

xUdQ 1 xUdQ. (;.;8)

Xoakre6 Pbo kll-zorg sjmdkl js hgtb ovod kd` g``. Kdy gtbor sjmdkl ikddgt ho hgtbovod kd` g``, hut aky ho dojtbor. Pbo cgllgwjdm sjaplo oxkaplo jllustrkto tbosoprgportjos.

Oxkaplo ;6 x(t) 1t8 0> js ovod.Oxkaplo 86 x(t) 1 >.;t9 js g``.

Oxkaplo 96 x(t) 1o>.0t js dojtbor ovod dgr g``.

(k)x(t) 1 t8 0> (h)x(t) 1 >.;t9 (i)x(t) 1 o>.0t

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


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;.?.8 @oigapgsjtjgd Pbogroa

Pbogroa 8. Ovory igdtjdugus-tjao sjmdklx(t) ikd ho oxprosso` ks6

x(t) 1y(t) +z(t),

wboroy(t) js ovod, kd`z(t) js g``.

^rggc. @odo

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

8

kd`

z(t) 1x(t) x(t)

8

.

Ilokrly y(t) 1y(t) kd` z(t) 1 z(t). ]o ikd klsg iboie tbkt x(t) 1y(t)+z(t).Porajdglgmy6 Pbo sjmdkl y(t) js ikllo` tbo ovod pkrt gc x(t), òdgto` hyOv{x(t)}.Pbo sjmdklz(t) js ikllo` tbo g`` pkrt gcx(t), òdgto` hyG``{x(t)}.

Oxkaplo6 Lot us igdsjòr tbo sjmdkl x(t) 1ot.

Ov{x(t)} 1 ot +ot

8 1 igsb(t).

G``

{x(t)

}1

ot ot

8 1 sjdb(t).

Yjajlkrly, wo ikd òdo ovod kd` g`` pkrts gc k `jsiroto-tjao sjmdkl xUdQ6

Ov{xUdQ} 1 xUdQ +xUdQ8

G``{xUdQ} 1 xUdQ xUdQ8

.

Jt js oksy tg iboie tbktxUdQ 1 Ov{xUdQ} + G``{xUdQ}

Pbogroa 9. Pbo òigapgsjtjgd js udjquo, j.o., jc

xUdQ 1yUdQ +zUdQ,

tbodyUdQjs ovod kd`zUdQjs g`` jc kd` gdly jcyUdQ 1 Ov{xUdQ} kd`zUdQ 1 G``{xUdQ}.


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;.3. JA^_LYO KD@ YPO^ C_DIPJGDY ;=

^rggc. JcyUdQ js ovod kd` zUdQ js g``, tbod

xUdQ 1yUdQ +zUdQ 1yUdQ zUdQ.

Pborocgro,

xUdQ +xUdQ 1 (yUdQ +zUdQ) + (yUdQ zUdQ) 1 8yUdQ,

japlyjdmy UdQ 1 xUdQ+xUdQ8

1 Ov{xUdQ}. YjajlkrlyzUdQ 1 xUdQxUdQ8

1 G``{xUdQ}.

Pbo igdvorso js trjvjkl hy òdjtjgd, ksOv{xUdQ} aust ho ovod kd`G``{xUdQ} austho g``.

;.3 Japulso kd` Ytop Cuditjgds

;.3.; @jsiroto-tjao Japulso kd` Ytop Cuditjgds

@odjtjgd =. Pbo `jsiroto-tjao udjt japulso sjmdklUdQ js òdo` ks

UdQ 1

;, d1 >,

>, d 1 >. (;.;9)

@odjtjgd 2. Pbo `jsiroto-tjao udjt stop sjmdklUdQ js òdo` ks

uUdQ 1

;, d >,>, d .

(;.;0)

Jt ikd ho sbgwd tbkt

UdQ 1uUdQ uUd ;Q

uUdQ 1e1>

Ud eQ

uUdQ 1

e1uUeQUd eQ.


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(k)UdQ (h)uUdQ

Cjmuro ;.;;6 @odjtjgds gc japulso cuditjgd kd` k stop cuditjgd.

;.3.8 ^rgporty gcUdQ

Ykapljdm ^rgportyHy tbo `odjtjgd gcUdQ,Ud d>Q 1 ; jcd 1 d>, kd` > gtborwjso. Pborocgro,

xUdQUd d>Q 1

xUdQ, d1 d>

>, d 1d>1xUd>QUd d>Q. (;.;?)

Ks k spoijkl ikso wbod d>1 >, wo bkvoxUdQUdQ 1xU>QUdQ. ^jitgrjklly, wbod k sjmdklxUdQ js aultjpljo` wjtb UdQ, tbo gutput js k udjt japulso wjtb kapljtu`o xU>Q.

Cjmuro ;.;86 Jllustrktjgd gcxUdQUdQ 1xU>QUdQ.


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;.3. JA^_LYO KD@ YPO^ C_DIPJGDY ;:

Ybjctjdm ^rgporty

YjdioxUdQUdQ 1xU>QUdQ kd`

d1 UdQ 1 ;, wo bkvo

d1xUdQUdQ 1

d1

xU>QUdQ 1xU>Q

d1UdQ 1xU>Q,

kd` sjajlkrly

d1

xUdQUd d>Q 1

d1xUd>QUd d>Q 1xUd>Q. (;.;3)

Jd modorkl, tbo cgllgwjdm rosult bgl`s6

hd1k

xUdQUd d>Q 1

xUd>Q, jcd> Uk, hQ>, jcd> Uk, hQ

(;.;=)

Xoprosodtktjgd ^rgporty_sjdm tbo skapljdm prgporty, jt bgl`s tbkt

xUeQUd eQ 1xUdQUd eQ.Yuaajdm tbo hgtb sjòs gvor tbo jdòx e yjol`s

e1 xUeQUd eQ 1

e1xUdQUd eQ 1xUdQ

e1 Ud eQ 1xUdQ.

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

xUdQ 1

e1xUeQUd eQ. (;.;2)

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

uUdQ 1

e1 uUeQUd eQ.]by `g wo uso tboso igapljikto` roprosodtktjgd gcxUdQ7 Hoikuso, wbod wo igdsjòrljdokr tjao-jdvkrjkdt systoas (Ibkptor 8), jt wjll kllgw us tg òtorajdo tbo systoarospgdso tg kdy sjmdkl xUdQ crga tbo japulso rospgdso.


20/105

8> IBK^POX ;. C_D@KAODPKLY GC YJMDKLY

Cjmuro ;.;96 Xoprosodtjdm gc k sjmdkl xUdQ usjdm k trkjd gc japulsos Ud eQ.

;.3.9 Igdtjdugus-tjao Japulso kd` Ytop Cuditjgds

@odjtjgd :. Pbo `jrki òltk cuditjgd js òdo` ks

(t) 1>, jct 1 >

, jct1 > ,

wboro

(t)`t1 ;.

@odjtjgd ;>. Pbo udjt stop cuditjgd js òdo` ks

u(t) 1

>, t

;, t >.

;.3.0 ^rgporty gc(t)

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


21/105

;.3. JA^_LYO KD@ YPO^ C_DIPJGDY 8;

Ykapljdm ^rgporty

x(t)(t) 1x(>)(t). (;.;:)

Pg soo tbjs, dgto tbkt x(t)(t) 1 x(>) wbod t 1 > kd` x(t)(t) 1 > wbod t1 >.Yjajlkrly, wo bkvo

x(t)(t t>) 1x(t>)(t t>), (;.8>)cgr kdyt> X.

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

x(t)(t)`t1

x(>)(t)`t1 x(>)

(t)`t1 x(>). (;.8;)

Yjajlkrly, gdo ikd sbgw tbkt

x(t)(t t>)`t1 x(t>) (;.88)

Xoprosodtktjgd ^rgportyPbo roprosodtktjgd prgporty js klsg kdklgmgus tg tbo `jsiroto-tjao ikso6

x(t) 1

x()(t )`, (;.89)

wboro tbo spoijkl ikso gcu(t) js mjvod hy

u(t) 1

u()(t )`.

Ks kd oxkaplo gc tbo prgportjos, lot us igdsj`or ``tu(t).

u(t) 1

u()(t )`, (roprosodtktjgd prgporty)

1

>

(t )`, hoikuso u() 1 > jc >

1 t

()`, lot 1 t

.

Pbod hy cud`kaodtkl tbogroa gc ikliulus, wo bkvo

`

`tu(t) 1

`

`t

t

()`1 (t). (;.80)


22/105


;.= Igdtjdugus-tjao Igaplox Oxpgdodtjkl Cudi-

tjgds@odjtjgd ;;. K igaplox oxpgdodtjkl cuditjgd js òdo` ks

x(t) 1Iokt, wboroI, k I (;.8?)

;.=.; Xokl-vkluo` Oxpgdodtjkl

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

(k)x(t) 1 ;

8o

;

8

t

(h)x(t) 1 ;

8o

;

8

t

Cjmuro ;.;06 Xokl oxpgdodtjkl cuditjgds.

]bodk 1 >, tbod x(t) 1I, wbjib js k igdstkdt cuditjgd.

;.=.8 ôrjg`ji Igaplox Oxpgdodtjkl

Lot us igdsjòr tbo ikso wboro k js puroly jakmjdkry, j.o., k1 f >, > X. Yjdio Ijs k igaplox duahor, wo bkvo

I1Kof ,

wboroK, X. Igdsoquodtly,

x(t) 1Iof>t 1Kofof>t

1Kof(>t+) 1K igs(>t+) +fK sjd(>t+).


23/105

;.=. IGDPJD_G_Y-PJAO IGA^LOR OR^GDODPJKL C_DIPJGDY 89

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

Xo{x(t)} 1K igs(>t+)Ja{x(t)} 1K sjd(>t+).

]o ikd tbjde gcx(t) ks k pkjrgc sjdusgj`kl sjmdkls gc tbo skao kapljtu`o K, > kd`pbkso sbjct wjtb gdo k igsjdo kd` tbo gtbor k sjdo.

(k)Xo{Iof>t} (h)Ja{Iof>t}

Cjmuro ;.;?6 ^orjg`ji igaplox oxpgdodtjkl cuditjgd x(t) 1 Iof>t

(I1 ;, >

1 8).

Ilkjas. x(t) 1Iof>t js porjg`ji wjtb

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

8. cud`kaodtkl croquodiy6|>|.

Pbo soigd` ilkja js tbo jaao`jkto rosult crga tbo rst ilkja. Pg sbgw tbo rst ilkja,wo doo` tg sbgwx(t+P>) 1x(t) kd` dg sakllorP>ikd sktjscy tbo porjg`jijty irjtorjk.

x(t+P>) 1Iof>(t+ 8|>|

) 1Iof>tof8

1Iof>t 1x(t).

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


24/105


;.=.9 Modorkl Igaplox Oxpgdodtjkl

Jd tbo modorkl sottjdm, wo bkvo I 1 Kof wboro K, X kd` k 1 r+ f> wboror, > X. Pborocgro,

x(t) 1

Kofr

o(r+f>)t 1Kortof(>t+).

Xowrjto tbjs jd tbo roitkdmulkr cgra6

x(t) 1Kort igs(>t+) +fKort sjd(>t+)

(k)Xo{Kortof>t} (h)Ja{Kortof>t}

Cjmuro ;.;36 ^orjg`ji igaplox oxpgdodtjkl cuditjgd x(t) 1 Kortof>t (K1 ;,r 1 ;/8,> 1 8).

;.2 @jsiroto-tjao Igaplox Oxpgdodtjkls

;.2.; @odjtjgds

K `jsiroto-tjao igaplox oxpgdodtjkl cuditjgd bks tbo cgra6

xUdQ 1Iod,

wboroI, I. Lottjdm1 o

mjvosxUdQ 1Id.

Jd tbo cgllgwjdm suhsoitjgds, wo sbkll stu`y tbo hobkvjgr gcxUdQ cgr `jorodio iksosgcIkd` .


25/105

;.2. @JYIXOPO-PJAO IGA^LOR OR^GDODPJKLY 8?

;.2.8 Xokl-vkluo` Igaplox Oxpgdodtjkl

xUdQ js k rokl-vkluo` igaplox oxpgdodtjkl wbod I X kd` X. Jd tbjs ikso,xUdQ 1I d js k agdgtgdji òiroksjdm cuditjgd wbod > < . Igdsoquodtly,

xUdQ 1Id 1 |I|of

||of>

d

1 |I|||d

o

f(>d+)

1 |I|||d igs(>d+) +f|I|||d sjd(>d+).

Pbroo iksos ikd ho igdsjòro` boro6

;. ]bod|| 1 ;, tbod xUdQ 1|I| igs(>d+ ) +f |I| sjd(>d+ ) kd` jt bkssjdusgj`kl rokl kd` jakmjdkry pkrts (dgt doiosskrjly porjg`ji, tbgumb).

8. ]bod|| 4 ;, tbod||d js k mrgwjdm oxpgdodtjkl, sg tbo rokl kd` jakmjdkrypkrts gcxUdQ kro tbo prgùit gc tbjs wjtb sjdusgj`s.

9. ]bod

|

|d, > X. ]o wkdt tg stu`y tbo igd jtjgd cgr xUdQ tg hoporjg`ji. Pbo porjg`jijty igd`jtjgd roqujros tbkt, cgr sgao D 4>,

xUd+DQ 1xUdQ, d V.

YjdioxUdQ 1Iof>d, jt bgl`s tbkt

of>(d+D) 1of>dof>D 1of>d, d V.Pbjs js oqujvklodt tg

of>D 1 ; gr >D1 8a, cgr sgaoa V.


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Pborocgro, tbo igd`jtjgd cgr porjg`jijty gcxUdQ js

> 18a

D , cgr sgaoa V, kd` sgaoD 4>, D V.

Pbus xUdQ 1 of>d js porjg`ji jc kd` gdly jc > js k rktjgdkl aultjplo gc 8. Pbocud`kaodtkl porjg` js

D18a

>,

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

;.2.? ^orjg`jijty jd CroquodiyYuppgso tbkt xUdQ 1of>d, wboro > X. Jc wo jdirokso > hy 8, wo d`

x;UdQ 1of(>+8)d 1of>dof8d.

Hut d V, sg of8d 1 ;, kd` wo soo tbkt

x;UdQ 1of>d 1xUdQ.

Agro modorklly, cgr kdye V, wo bkvo

xeUdQ 1of(>

+8e)d 1of>

d 1xUdQ.

Pbjs aokds tbkt wo ikd ljajt tbo rkdmo gc vkluos gc >tg kdy rokl jdtorvkl gc lodmtb 8.Pbo porjg`jijty jd croquodiy kppljos, gc igurso, tg tbo porjg`ji igaplox oxpgdodtjklsjmdkls, sg wo bkvo k `jorodt dgtjgd gc lgw kd` bjmb croquodijos jd tbo `jsiroto-tjaosottjdm.


27/105

Ibkptor 8

Cud`kaodtkls gc Yystoas

K systoa js k qukdtjtktjvo òsirjptjgd gc k pbysjikl prgioss wbjib trkdscgras sjmdkls(kt jts jdput) tg sjmdkls (kt jts gutput). Agro proijsoly, k systoa js k hlkiehgx (vjowo` ks k aktboaktjikl khstrkitjgd) tbkt òtorajdjstjiklly trkdscgras jdputsjmdkls 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 systoas kd` `jsiroto-tjao systoas. Pbotrkdscgraktjgd crga k igdtjdugus-tjao sjmdklx(t) tg k `jsiroto-tjao sjmdkl xUdQwjll ho `jsiusso` jd Ibktpor 3.

8. ]o wjll cgius gd sjdmlo-jdput sjdmlo-gutput systoas. Aultjplo-jdputs tg aultjplo-gutputs (AJAG) systoas kro gutsjò tbo sigpo gc tbjs igurso.

8=


28/105

82 IBK^POX 8. C_D@KAODPKLY GC YZYPOAY

8.; Yystoa ^rgportjos

8.;.; Aoagryloss

@odjtjgd ;8. K systoa js aoagryloss jc tbo gutput kt tjaot (grd) òpod`s gdlygd tbo jdput kt tjaot (grd).

Oxkaplos.

;. y(t) 1 (8x(t) x8(t))8 js aoagryloss, hoikusoy(t) òpod`s gdx(t) gdly. Pborojs dgx(t ;), gr x(t+ ;) toras, cgr oxkaplo.

8. yUdQ 1 xUdQ js aoagryloss. Jd ckit, tbjs systoa js pkssjdm tbo jdput tg gutput`jroitly, wjtbgut kdy prgiossjdm.

9. yUdQ 1xUd ;Q js dgt aoagryloss, hoikuso tbo d-tb gutput òpod`s gd d ;-tbjdput.

0. yUdQ 1xUdQ +yUd ;Q js dgt aoagryloss. Pg soo tbjs, wo igdsjòryUd ;Q 1xUd ;Q +yUd 8Q.

Yuhstjtutjdm jdtg y UdQ 1xUdQ +yUd ;Q yjol`syUdQ 1xUdQ + (xUd ;Q +yUd 8Q).

Hy ropoktjdm tbo ikliulktjgd, wo bkvo

yUdQ 1xUdQ +xUd ;Q +xUd 8Q +. . .

1d

e1xUeQ.

Ilokrly,y UdQ òpod`s gd agro tbkd fust xUdQ.

8.;.8 Jdvortjhlo

@odjtjgd ;9. K systoa js jdvortjhlo jc `jstjdit jdput sjmdkls prgùio `jstjdit gutputsjmdkls.

Jd gtbor wgr`s, k systoa jc jdvortjhlo jc tboro oxjsts kd gdo-tg-gdo akppjdm crga tbosot gc jdput sjmdkls tg tbo sot gc gutput sjmdkls.

Pboro kro twg hksji rulos gc sbgwjdm kd jdvortjhlo systoa6


29/105

8.;. YZYPOA ^XGÔXPJOY 8:

;. Pg sbgw tbkt k systoa js jdvortjhlo, gdo bks tg sbgw tbo jdvorsjgd cgraulk.

8. Pg sbgw tbkt k systoa js dgt jdvortjhlo, gdo bks tg mjvo k igudtor oxkaplo.

Oxkaplo ;.Pbo systoay(t) 1 (igs(t) + 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) + 8)x(t) japljos tbkt (hy rokrrkdmjdm toras)

x(t) 1 y(t)

igs(t) + 8,

wbjib js tbo jdvorsjgd cgraulk. Dgto tbkt tbo òdgajdktgr js klwkys pgsjtjvo, tbustbo `jvjsjgd js vklj`.

Oxkaplo 8.Pbo systoayUdQ 1xUdQ +yUd ;Q js jdvortjhlo.

^rggc. yUdQ 1xUdQ +yUd ;Q japljos tbkt (hy rokrrkdmjdm toras)

xUdQ 1yUdQ yUd ;Q.

Pbjs js tbo jdvorsjgd cgraulk.

Oxkaplo 9.Pbo systoay(t) 1x8(t) js dgt jdvortjhlo.

^rggc. Pg sbgw tbkt k systoa js dgt jdvortjhlo, wo igdstruit k igudtor oxkaplo. Lotus igdsjòr twg sjmdkls

x;(t) 1 ;, tx8(t) 1 ;, t.

Ilokrly x;(t)1 x8(t), hut (x;(t))8 1 (x8(t))8. Pborocgro, wo bkvo cgud` k igudtoroxkaplo suib tbkt `jorodt jdputs mjvo tbo skao gutput. Bodio tbo systoa js dgtjdvortjhlo.


30/105

9> IBK^POX 8. C_D@KAODPKLY GC YZYPOAY

8.;.9 Ikuskl

@odjtjgd ;0. K systoa js ikuskl jc tbo gutput kt tjao t (gr d) òpod`s gdly gdjdputs kt tjaos t (j.o., tbo prosodt kd` pkst).Oxkaplos.

;. yUdQ 1xUd ;Q js ikuskl, hoikuso y UdQ òpod`s gd tbo pkst skaplo xUd ;Q.8. yUdQ 1xUdQ +xUd+ ;Q js dgt ikuskl, hoikuso xUd+ ;Q js k cuturo skaplo.

9. y(t) 1t x()` js ikuskl, hoikuso tbo jdtomrkl ovkluktos crga tg t

(wbjib kro kll jd tbo pkst).

0. yUdQ 1 xU

dQ js dgt ikuskl, hoikuso yU

;Q 1 xU;Q, wbjib aokds tbo gutput kt

d1 ; òpod`s kd jdput jd tbo cuturo.?. y(t) 1x(t)igs(t + ;) ikuskl (kd` aoagryloss), hoikuso igs(t + ;) js k igdstkdt

wjtb rospoit tg x(t).

8.;.0 Ytkhlo

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

@odjtjgd ;?. K sjmdklx(t) (kd`xUdQ) js hgudò` jc tboro oxjsts k igdstkdtH < suib tbkt|x(t)| < H cgr kllt.@odjtjgd ;3. K systoa js stkhlo jc k hgudò` jdput jdput klwkys prgùios k hgudò`gutput sjmdkl. Pbkt js, jc|x(t)| H cgr sgaoH < , tbod

|y(t)| < .Oxkaplo ;.Pbo systoay(t) 1 8x8(t ;) +x(9t) 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

|y(t)| 1 |8x8(t ;) +x(9t)|

|8x8(t

;)

|+

|x(9t)

| , hy Prjkdmlo Jdoqukljty

8|x8(t ;)| + |x(9t)| 8H8 +H < .

Pborocgro, cgr kdy hgudò` jdputx(t), tbo gutputy(t) js klwkys hgudò`. Bodio tbosystoa js stkhlo.


31/105

8.;. YZYPOA ^XGÔXPJOY 9;

Oxkaplo 8.

Pbo systoayUdQ 1

de1 xUeQ js dgt stkhlo.^rggc. Pg sbgw tbkt tbo systoa y UdQ 1

de1 xUeQ js dgt stkhlo, wo ikd igdstruit k

hgudò` jdput sjmdklxUdQ kd` sbgw tbkt tbo gutput sjmdkl y UdQ js dgt hgudò`.

Lot xUdQ 1uUdQ. Jt js ilokr tbkt|xUdQ| ; (j.o., hgudò`). Igdsoquodtly,

|yUdQ| 1

de1

uUeQ

1 d

e1>

uUeQ

d

e1>

; 1d+ ;,

wbjib kpprgkibosksd . Pborocgro,|yUdQ| js dgt hgudò`.

8.;.? Pjao-jdvkrjkdt

@odjtjgd ;=. 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 kdyt> X.

Cjm. 8.8 jllustrktos kd jdtorprotktjgd gc k tjao-jdvkrjkdt systoa6 Jc k sjmdklx(t) js

jdput tg k tjao-jdvkrjkdt systoa kd` mot kd gutput y(t), tbod tbo jdput x(t t>) wjllrosult kd gutput y(t t>).

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


32/105


33/105

8.8. IGDSGL_PJGD 99

Oxkaplo ;.

Pbo systoay(t) 1 8x(t) js ljdokr. Pg soo tbjs, lots igdsjòr k sjmdklx(t) 1kx;(t) +hx8(t),

wboroy;(t) 1 8x;(t) kd`y8(t) 1 8x8(t). Pbod

ky;(t) +hy8(t) 1k (8x;(t)) +h (8x8(t))

1 8 Ukx;(t) +hx8(t)Q 1 8x(t) 1y(t).

Oxkaplo 8.Pbo systoayUdQ 1 (xU8dQ)8 js dgt ljdokr. Pg soo tbjs, lots igdsjòr tbo sjmdkl

xUdQ 1kx;UdQ +hx8UdQ,

wboro y;UdQ 1 (x;U8dQ)8 kd` y8UdQ 1 (x8U8dQ)

8. ]o wkdt tg soo wbotbor yUdQ 1ky;UdQ +hy8UdQ. Jt bgl`s tbkt

ky;UdQ +hy8UdQ 1k (x;U8dQ)8 +h (x8U8dQ)

8 .

Bgwovor,

yUdQ 1 (xU8dQ)8 1 (kx;U8dQ +hx8U8dQ)8 1k8(x;U8dQ)

8 +h8(x8U8dQ)8 + 8khx;UdQx8UdQ.

8.8 Igdvglutjgd

8.8.; ]bkt js Igdvglutjgd7

Ljdokr tjao jdvkrjkdt (LPJ) systoas kro mgg` agòls cgr akdy rokl-ljco systoas, kd`tboy bkvo prgportjos tbkt lok` tg k vory pgworcul kd` ooitjvo tbogry cgr kdklyzjdmtbojr hobkvjgr. Jd tbo cgllgwjdms, wo wkdt tg stu`y LPJ systoas tbrgumb jts ibkrki-torjstji cuditjgd, ikllo` tbo japulso rospgdso.

Pg homjd wjtb, lot us igdsjòr `jsiroto-tjao sjmdkls. @odgto hy bUdQ tbo japulso

rospgdso gc kd LPJ systoaY. Pbo japulso rospgdso, ks jt js dkao`, js tbo rospgdsogc tbo systoa tg k udjt japulso jdput. Xoikll tbo òdjtjgd gc kd udjt japulso6

UdQ 1

;, d1 >

>, d 1 >. (8.;)


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35/105

8.8. IGDSGL_PJGD 9?

Doxt, suppgso wo edgw tbkt tbo japulso rospgdso gc kd LPJ systoa js bUdQ. ]o wkdt

tg òtorajdo tbo gutput yUdQ. Pg `g sg, wo rst oxpross xUdQ ks k sua gc japulsos6

xUdQ 1

e1xUeQUd eQ.

Cgr okib japulso Ud eQ, wo ikd òtorajdo jts japulso rospgdso, hoikuso cgr kd LPJsystoa6

Ud eQ bUd eQ.Igdsoquodtly, wo bkvo

xUdQ 1 e1

xUeQUd eQ e1

xUeQbUd eQ 1yUdQ.

Pbjs oquktjgd,

yUdQ 1

e1xUeQbUd eQ (8.0)

js edgwd ks tbo igdvglutjgd oquktjgd.

8.8.8 @odjtjgd kd` ^rgportjos gc Igdvglutjgd

@odjtjgd ;:. Mjvod k sjmdklxUdQkd` tbo japulso rospgdso gc kd LPJ systoabUdQ,tbo igdvglutjgd hotwoodxUdQ kd`bUdQ js òdo` ks

yUdQ 1

e1xUeQbUd eQ.

]o òdgto igdvglutjgd ksyUdQ 1xUdQ bUdQ.

Oqujvklodt cgra6 Lottjdm a 1 d e, wo ikd sbgw tbkt

e1xUeQbUd eQ 1

a1

xUd aQbUaQ 1

e1xUd eQbUeQ.


36/105


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

tbodyUdQ 1

e1

xUeQbeUd eQ.

j.o.,bUdQ js `jorodt kt ovory tjao jdstkdt e.

Pbo cgllgwjdm stkd`kr` prgportjos ikd ho prgvo` oksjly6

;. Igaautktjvo6 xUdQ bUdQ 1bUdQ xUdQ8. Kssgijktjvo6 xUdQ (b;UdQ b8UdQ) 1 (xUdQ b;UdQ) b8UdQ9. @jstrjhutjvo6 xUdQ

(b;UdQ +b8UdQ) 1 (x(t)

b;UdQ) + (xUdQ

b8UdQ)

8.8.9 Bgw tg Ovklukto Igdvglutjgd7

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 japulsorospgdso bUdQ sbgwd holgw.

Lots igaputo tbo gutput y UdQ gdo hy gdo. Cjrst, igdsjòr y U>Q6

yU>Q 1

e1 xUeQbU> eQ 1

e1 xUeQbUeQ 1 ;.Dgto tbkt bUeQ js tbo jppo` vorsjgd gcbUeQ, kdè1 xUeQbUeQ js tbo aultjply-k`` hotwood xUeQ kd`bUeQ.


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38/105


8.9.; Aoagryloss

K systoa js aoagryloss jc tbo gutput òpod`s gd tbo iurrodt jdput gdly. Kd oqujvklodtstktoaodt usjdm tbo japulso rospgdso bUdQ js tbkt6

Pbogroa 0. Kd LPJ systoa js aoagryloss jc kd` gdly jc

bUdQ 1kUdQ,cgr sgaok I. (8.?)

^rggc. JcbUdQ 1kUdQ, tbod cgr kdy jdput xUdQ, tbo gutput js

yUdQ 1xUdQ bUdQ 1

e1xUeQbUd eQ

1

e1xUeQkUd eQ

1kxUdQ.

Yg, tbo systoa js aoagryloss. Igdvorsoly, jc tbo systoa js aoagryloss, tbod yUdQikddgt òpod` gd tbo vkluosxUeQ cgre1d. Lggejdm kt tbo igdvglutjgd sua cgraulk

yUdQ 1

e1xUeQbUd eQ,

wo igdiluò tbktbUd eQ 1 >, cgr klle1d,

gr oqujvklodtly,bUdQ 1 >, , cgr klld 1 >.

Pbjs japljosyUdQ 1xUdQbU>Q 1kxUdQ,

wboro wo bkvo sot k 1 bU>Q.

8.9.8 JdvortjhloPbogroa ?. Kd LPJ systoa js jdvortjhlo jc kd` gdly jc tboro oxjstmUdQ suib tbkt

bUdQ mUdQ 1UdQ. (8.3)


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8.9. YZYPOA ^XGÔXPJOY KD@ JA^_LYO XOY^GDYO 9:

^rggc. Jc k systoa Y js jdvortjhlo, tbod x;UdQ1x8UdQ japljos y;UdQ1y8UdQ. Yg tboro

oxjsts kd jdfoitjvo akppjdm (gdo-tg-gdo akp)Ysuib tbktyUdQ 1Y(xUdQ) 1bUdQxUdQ.Yjdioc js jdfoitjvo, tboro oxjsts kd jdvorso akppjdm Y; suib tbktY;(Y(xUdQ)) 1xUdQ,

cgr kdyxUdQ. Pborocgro, tboro oxjsts m UdQ suib tbkt

mUdQ (bUdQ xUdQ) 1xUdQ.Hy kssgijktjvjty gc igdvglutjgd, wo bkvo (mUdQbUdQ)xUdQ 1xUdQ, japlyjdmmUdQbUdQ 1UdQ.

Igdvorsoly, jc tboro oxjst m UdQ suib tbkt bUdQ mUdQ 1UdQ, tbod cgr kdyx;UdQ 1x8UdQ,wo bkvo

y;UdQ 1bUdQ x;UdQy8UdQ 1bUdQ x8UdQ,

kd` y;UdQ 1y8UdQ. Pkejdm tbo `jorodio hotwood y;UdQ kd` y8UdQ, wo bkvoy;UdQ y8UdQ 1bUdQ {x;UdQ x8UdQ} .

Igdvglvjdm hgtb sjòs hy m UdQ yjol`s

mUdQ (y;UdQ y8UdQ) 1UdQ (x;UdQ x8UdQ).Yjdio x;UdQ1 x8UdQ, kd` mUdQ1 > cgr kll d, wo aust bkvo y;UdQ1 y8UdQ. Pborocgro,tbo systoa js jdvortjhlo.

8.9.9 Ikuskl

Pbogroa 3. Kd LPJ systoa js ikuskl jc kd` gdly jc

bUdQ 1 >, cgr klld . (8.=)

^rggc. JcY js ikuskl, tbod tbo gutput yUdQ ikddgt òpod` gd xUeQ cgr e 4 d. Crgatbo igdvglutjgd oquktjgd,

yUdQ 1

e1xUeQbUd eQ,


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0> IBK^POX 8. C_D@KAODPKLY GC YZYPOAY

wo aust bkvo

bUd eQ 1 >, cgre 4 d, gr oqujvklodtly bUd eQ 1 >, cgrd e .Yottjdma 1 d e, wo soo tbkt

bUaQ 1 >, cgr a .

Igdvorsoly, jcbUeQ 1 > cgr e , tbod cgr jdput xUdQ,

yUdQ 1

e1bUeQxUd eQ 1

e1>

bUeQxUd eQ.

Pborocgro,y UdQ `opod`s gdly upgd xUaQ cgr a

d.

8.9.0 Ytkhlo

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

e1

|bUeQ| < .

^rggc. Yuppgso tbkt

e1 |bUeQ| ,;, bUdQ< >.


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8.0. IGDPJD_G_Y-PJAO IGDSGL_PJGD 0;

Ilokrly,|xUdQ| ; cgr kll d. Pbo gutput yU>Q js mjvod hy

|yU>Q| 1 e1

xUeQbUeQ1

e1|bUeQ|

1

e1

|bUeQ| 1 .

Pborocgro,y UdQ js dgt hgudò`.

8.0 Igdtjdugus-tjao Igdvglutjgd

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 sjmdklòigapgsjtjgd js

x(t) 1

x()(t )`, (8.2)

kd` igdsoquodtly, tbo igdtjdugus tjao igdvglutjgd js òdo` ks

y(t) 1

x()b(t

)`. (8.:)

Oxkaplo.Pbo igdtjdugus-tjao igdvglutjgd klsg cgllgws tbo tbroo stop rulo6 jp, sbjct, aultjply-k` . Pg soo kd oxkaplo, lot us igdsjòr tbo sjmdkl x(t) 1 oktu(t) cgr k 4 >, kd`japulso rospgdso b(t) 1u(t). Pbo gutput y(t) jsIkso K6 t 4>6

y(t) 1

x()b(t )`

1

oku()u(t )

1 t>

ok`

1 ;

k

; okt .


42/105


Ikso H6t >6

y(t) 1 >.

Pborocgro,

y(t) 1;

k

; okt u(t).

8.0.; ^rgportjos gc IP Igdvglutjgd

Pbo cgllgwjdm prgportjos ikd ho prgvo` oksjly6

;. Igaautktjvo6 x(t) b(t) 1b(t) x(t)

8. Kssgijktjvo6 x(t) (b;(t) b8(t)) 1 (x(t) b;(t)) b8(t)9. @jstrjhutjvo6 x(t) Ub;(t) +b8(t)Q 1 Ux(t) b;(t)Q + Ux(t) b8(t)Q

8.0.8 Igdtjdugus-tjao Yystoa ^rgportjos

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

Aoagryloss.Kd LPJ systoa js aoagryloss jc kd` gdly jc

b(t) 1k(t), cgr sgao kI.

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

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

b(t) 1 >, cgr kllt .

Ytkhlo.

K systoa js stkhlo jc kd` gdly jc

|b()|` < .


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Ibkptor 9

Cgurjor Yorjos

Pbo ghfoitjvo gc tbjs ibkptor js tg jòdtjcy k ckajly gc sjmdkls{xe(t)}suib tbkt6;. Ovory sjmdkl jd tbo ckajly pkssos tbrgumb kdy LPJ systoa wjtb gdly k siklo

ibkdmo (gr gtbor sjaply òsirjho` ibkdmo)

xe(t) exe(t),wboroe js k siklo ckitgr.

8. Kdy sjmdkl ikd ho roprosodto` ks k ljdokr igahjdktjgd gc sjmdkls jd tbojrckajly.

x(t) 1

e1 k

exe(t).

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

x(t)

e1keexe(t),

wboro tbo siklkr ke igaos crga tbo òdjtjgd gc ljdokr igahjdktjgd.

9.; Ojmodcuditjgds gc kd LPJ Yystoa

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

@odjtjgd 8>. Cgr kd LPJ systoa, jc tbo gutput js k siklo` vorsjgd gc jts jdput, tbodtbo jdput cuditjgd js ikllo` kd ojmodcuditjgd gc tbo systoa. Pbo sikljdm ckitgr js ikllo`tbo ojmodvkluo gc tbo systoa.

09


44/105

00 IBK^POX 9. CG_XJOX YOXJOY

9.;.; Igdtjdugus-tjao Ikso

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

Yuppgso tbkt x(t) 1ost cgr sgao s I, tbod tbo gutput js mjvod hy

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

b()x(t )`

1

b()os(t)`

1

b()os`

ost 1B(s)ost 1B(s)x(t),

wboroB(s) js òdo` ks

B(s) 1

b()os`.

Pbo cuditjgdB(s) js edgwd ks tbo trkdscor cuditjgdgc tbo igdtjdugus-tjao LPJ sys-toa. Dgto tbktB(s) js òdo` hy tbo japulso rospgdso b(t), kd` js k cuditjgd jd s(jdòpodòdt gct). Pborocgro, B(s)x(t) ikd ho romkrò` ks k siklkr B(s) aultjpljo`tg tbo cuditjgd x(t).

Crga tbo òrjvktjgd khgvo, wo soo tbkt jc tbo jdput js x(t) 1ost, tbod tbo gutput jsk siklo` vorsjgd y(t) 1B(s)ost6

Pborocgro, usjdm tbo òdjtjgd gc ojmodcuditjgd, wo sbgw tbkt

;. ost js kd ojmodcuditjgd gc kdy igdtjdugus-tjao LPJ systoa, kd`

8. B(s) js tbo igrrospgd`jdm ojmodvkluo.


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9.;. OJMODC_DIPJGDY GC KD LPJ YZYPOA 0?

Jc wo spoijkljzo tg tbo suhilkss gc porjg`ji igaplox oxpgdodtjkls gc tbo oft , X

hy sottjdm s 1 f, tbod

B(s)|s1f 1B(f) 1

b()of`.

B(f) js ikllo` tbo croquodiy rospgdsogc tbo systoa.

9.;.8 @jsiroto-tjao Ikso

Doxt, wo igdsj`or tbo `jsiroto-tjao ikso6

Yuppgso tbkt tbo japulso rospgdso js mjvod hy bUdQ kd` tbo jdput js xUdQ 1zd, tbodtbo gutput yUdQ js

yUdQ 1bUdQ xUdQ 1

e1bUeQxUd eQ

1

e1bUeQz

d

e

1zd

e1bUeQze 1B(z)zd,

wboro wo `odo`

B(z) 1

e1bUeQze,

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

;. zd js kd ojmodcuditjgd gc k `jsiroto-tjao LPJ systoa, kd`

8. B(z) js tbo igrrospgd`jdm ojmodvkluo.


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Igdsj`orjdm tbo suhilkss gc porjg`ji igaplox oxpgdodtjkls of(8/D)d hy sottjdm z1of8/D, wo bkvo

B(z)|z1of 1B(of) 1

e1bUeQofe,

wboro 1 8D

, kd` B(of) js ikllo` tbo croquodiy rospgdsogc tbo systoa.

9.;.9 Yuaakry

Jd suaakry, wo bkvo tbo cgllgwjdm ghsorvktjgds6

Pbkt js, ost js kd ojmodcuditjgd gc k IP systoa, wboroks zd js kd ojmodcuditjgd gc k@P systoa. Pbo igrrospgd jdm ojmodvkluos kro B(s) kd`B(z).

Jc wo suhstjtutos 1 fkd`z1 of rospoitjvoly, tbod tbo ojmodcuditjgds hoigaooft

kd` ofd5 tbo ojmodvkluos hoigao B(f) kd` B(of).

9.;.0 ]by js ojmodcuditjgd japgrtkdt7

Pbo kdswor tg tbjs quostjgd js rolkto` tg tbo soigd` ghfoitjvo jd tbo homjddjdm. Lotus igdsj`or k sjmdklx(t)6

x(t) 1k;os;t +k8o

s8t +k9os9t.

Kiigr`jdm tbo ojmodcuditjgd kdklysjs, tbo gutput gc okib igaplox oxpgdodtjkl jsos;t B(s;)os;tos8t B(s8)os8t

os9t B(s9)os9t.


47/105

9.8. CG_XJOX YOXJOY XO^XOYODPKPJGD 0=

Pborocgro, tbo gutput js

y(t) 1k;B(s;)os;t +k8B(s8)os8t +k9B(s9)os9t.

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, jcx(t) js kd jddjto sua gc igaplox oxpgdodtjkls,

x(t) 1

e1keo

set,

tbod tbo gutput js kmkjd k sua gc igaplox oxpgdodtjkls6

y(t) 1 e1

keB(se)oset.

Yjajlkrly cgr `jsiroto-tjao sjmdkls, jc

xUdQ 1

e1kez

de ,

tbod

xUdQ 1

e1keB(ze)z

de .

Pbjs js kd japgrtkdt ghsorvktjgd, hoikuso ks lgdm ks wo ikd oxpross k sjmdkl x(t) ksk ljdokr igahjdktjgd gc ojmodcuditjgds, tbod tbo gutput y(t) ikd ho oksjly òtorajdo`hy lggejdm kt tbo trkdscor cuditjgd (wbjib js xo` cgr kd LPJ systoa!). Dgw, tboquostjgd js 6 Bgw `g wo oxpross k sjmdkl x(t) ks k ljdokr igahjdktjgd gc igaploxoxpgdodtjkls7

9.8 Cgurjor Yorjos Xoprosodtktjgd

Oxjstodio gc Cgurjor Yorjos

Jd modorkl, dgt ovory sjmdklx(t) ikd ho òigapgso` ks k ljdokr igahjdktjgd gc igaploxoxpgdodtjkls. Bgwovor, suib òigapgsjtjgd js stjll pgssjhlo cgr kd oxtroaoly lkrmo ilkssgc sjmdkls. ]o wkdt tg stu`y gdo ilkss gc sjmdkls tbkt kllgws tbo òigapgsjtjgd. Pboykro tbo porjg`ji sjmdkls

x(t+P) 1x(t)


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wbjib sktjscy tbo squkro jdtomrkhlo igd`jtjgd,

P

|x(t)|8`t < ,

gr @jrjiblot igd`jtjgds (Zgu aky d` agro `jsiussjgds jd G]9.0)6;. Gvor kdy porjg` x(t) aust ho khsglutoly jdtomrkhlo, tbkt js,

P

|x(t)|`t < .

8. Jd kdy djto jdtorvkl gc tjao x(t) js gc hgudò` vkrjktjgd5 tbkt js, tboro kro dgagro tbkd k djto duahor gc akxjak kd` ajdjak ùrjdm kdy sjdmlo porjg` gctbo sjmdkl.

9. Jd kdy djto jdtorvkl gc tjao, tboro kro gdly k djto duahor gc `jsigdtjdujtjos.

Cgr tbjs ilkss gc sjmdkls, wo kro khlo tg oxpross jt ks k ljdokr igahjdktjgd gc igaploxoxpgdodtjkls6

x(t) 1

e1keo

fe>t.

Boro,> js tbo cud`kaodtkl croquodiy

>18

P ,

kd` tbo igoijodts ke kro edgwd ks tbo Cgurjor Yorjos igoijodts.

Mjvod k porjg`ji sjmdklx(t) tbkt js squkro jdtomrkhlo, bgw `g wo òtorajdo tbo CgurjorYorjos igoijodtske7 Pbjs js kdsworo` hy tbo cgllgwjdm tbogroa.

9.8.; Igdtjdugus-tjao Cgurjor Yorjos Igoijodts

Pbogroa 2. Pbo igdtjdugus-tjao Cgurjor sorjos igoijodtske gc tbo sjmdkl

x(t) 1

e1 keofe>t,

js mjvod hy

ke 1 ;

P

P

x(t)ofe>t`t.


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9.8. CG_XJOX YOXJOY XO^XOYODPKPJGD 0:

^rggc. Lot us igdsjòr tbo sjmdkl

x(t) 1

e1keo

fe>t.

Jc wo aultjply gd hgtb sjòs ofd>t, tbod wo bkvo

x(t)ofd>t 1

e1

keofe>t

ofd>t 1

e1

keof(ed)>t.

Jdtomrktjdm hgtb sjòs crga > tg P yjol`s

P

> x(t)ofd>t

`t1 P

>

e1 keof(ed)>t `t1

e1

ke

P>

of(ed)>t`t

.

Pbo toraP> o

f(ed)>t`tikd ho ovklukto` ks (Zgu sbgul` iboie tbjs!)

;

P

P>

of(ed)>t`t1

; jce1 d> gtborwjso

(9.;)

Pbjs rosult js edgwd ks tbo grtbgmgdkljtygc tbo igaplox oxpgdodtjkls.

_sjdm Oq. (9.;), wo bkvo P>

x(t)ofd>t`t1 P kd,

wbjib js oqujvklodt tg

kd1 ;

P

P>

x(t)ofd>t`t.

Oxkaplo ;. Yjdusgj`sIgdsjòr tbo sjmdklx(t) 1 ;+ ;

8igs 8t+sjd9t. Pbo porjg` gcx(t) jsP1 8 U]by7Q sg

tbo cud`kaodtkl croquodiy js> 1 8P 1. Xoikll Oulors cgraulkof 1 igs +fsjd ,

wo bkvo

x(t) 1 ; +;

0

of8t +of8t

+

;

8f

of9t of9t .


50/105


51/105

9.8. CG_XJOX YOXJOY XO^XOYODPKPJGD ?;

9.8.8 @jsiroto-tjao Cgurjor Yorjos igoijodts

Pg igdstruit tbo `jsiroto-tjao Cgurjor sorjos roprosodtktjgd, wo igdsjòr porjg`ji`jsiroto-tjao sjmdkl wjtb porjg`D

xUdQ 1xUd+DQ,

kd` kssuao tbkt xUdQ js squkro-suaakhlo, j.o.,

d1 |xUdQ|8 d,

js mjvod hy

ke 1 ;

D

d1DxUdQofe>d.

Boro,

d1Daokds suaajdm tbo sjmdkl wjtbjd k porjg`D. Yjdio k porjg`ji `jsiroto-tjao sjmdkls ropokts ovory Dskaplos, jt `gos dgt akttor wbjib skaplo tg ho pjieo`rst.

Oxkaplo.Lot us igdsjòr tbo cgllgwjdm sjmdkl sbgwd holgw. ]o wkdt tg òtorajdo tbo `jsiroto-tjao C.Y. igoijodt.


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53/105

9.8. CG_XJOX YOXJOY XO^XOYODPKPJGD ?9

Pbod, tbo gutput sjmdkl js mjvod hy

y(t) 1

e1keB(fe>)o

fe>t.

Lottjdmhe 1B(fe>)ke, wo bkvo

y(t) 1

e1heo

fe>t.

9.8.0 Bgw akdy Cgurjor sorjos igoijodts kro suijodt7

Jc wo òdo

xD(t) 1D

e1Dkeo

fe>t,

tbod xD(t) js kd kpprgxjaktjgd gcx(t). Ks D , wo soo tbkt xD(t) x(t). Kskd jllustrktjgd gcxD(t) ks Djdiroksos, wo ikd soo tbo cgllgwjdm muro.

Pborocgro, tbo duahor gc Cgurjor sorjos igoijodts òpod`s gd tbo kiiurkiy tbkt wowkdt tg kibjovo. Pypjiklly, tbo duahor D js ibgsod suib tbkt tbo rosjùo gc tbo

kpprgxjaktjgd

|x(t) xD(t)|8 `t ,

cgr sgao tkrmot orrgr lovol.


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55/105

9.9. ^XG^OXPJOY GC CG_XJOX YOXJOY IGOCCJIJODPY ??

Lottjdm1t t> jd tbo jdtomrkl, wo ghtkjd

;P

P

x()ofe>(+t>)`1ofe>t> ;P

P

x()ofe>`

wborox(t) ke. Pborocgro,x(t t>) keofe>t> .

9. Pjao Xovorskl6

x(t) kexUdQ ke

Pbo prggc js sjaplo. Igdsj`or k sjmdkly(t) 1 x(t). Pbo C.Y. roprosodtktjgdgcx(

t) js

x(t) 1

e1keo

fe8t/P.

Lottjdme 1 a, wo bkvo

y(t) 1x(t) 1

a1kaofa8t/P.

Pbus, x(t) ke.0. Igdfumktjgd6

x(t) kexUdQ ke

?. Aultjpljiktjgd6 Jcx(t) ke kd` y(t) he, tbod

x(t)y(t)

l1kehel.

3. ^krsovkl Oqukljty6

;

P P|x(t)|8`t1

e1|ke|8

;

D

d1D

|xUdQ|8 1

e1D|ke|8

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


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?3 IBK^POX 9. CG_XJOX YOXJOY


57/105

Ibkptor 0

Igdtjdugus-tjao Cgurjor Prkdscgra

Lot us homjd gur `jsiussjgd hy rovjowjdm sgao ljajtktjgds gc Cgurjor sorjos roprosod-tktjgd. 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(t + P) 1x(t).

8. Pbo sjmdkl aust ho squkro jdtomrkhloP

|x(t)|8`t < , gr sktjsos tbo @jrjiblotigd`jtjgds.

Jd tbjs ibkptor, wo wkdt tg oxtod` tbo j`ok gc Cgurjor Yorjos roprosodtktjgd tg kporj-g`ji sjmdkls. Pbkt js, wo wkdt tg rolkx tbo rst igd`jtjgd tg kporjg`ji sjmdkls.

0.; Jdsjmbt crga Cgurjor Yorjos

Lots rst igdsj`or tbo cgllgwjdm porjg`ji sjmdkl

x(t) 1

; |t| P;,> P; |t| < P8 ,

wborox(t) 1x(t+P). Pbo Cgurjor Yorjos igoijodts gcx(t) kro (iboie ygursolc!)

x(t) C.Y.

ke 1

8sjd(e>P;)

e>P

.

Jc wo suhstjtuto 1 e>, tbod

ke 18sjd(P;)

wP

1e>

?=


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

AultjplyjdmPgd hgtb sjòs yjol`s

P ke 18sjd(P;)

, (0.;)

wbjib js tbo dgrakljzo` Cgurjor Yorjos igoijodt.

^jitgrjklly, (0.;) jd`jiktos tbkt tbo dgrakljzo` Cgurjor sorjos igoijodts P ke kro

hgudò` hy tbo odvolgp R() 1 8 sjd(P;)

, ks jllustrkto` jd Cjm. 0.;.

Cjmuro 0.;6 Cgurjor Yorjos igoijodts gcx(t) cgr sgao P.

]bodP jdiroksos, tbo spkijdm hotwood igdsoiutjvo ke roùios. Bgwovor, tbo sbkpogc tbo odvolgp cuditjgd R() 1 8 sjd(P;)

roakjds tbo skao. Pbjs ikd ho sood jd Cjm.

0.8.

Cjmuro 0.86 Cgurjor Yorjos igoijodts gcx(t) cgr sgao P, wboro P 4 P.

Jd tbo ljajtjdm ikso wboroP , tbod tbo Cgurjor sorjos igoijodtsP ke kpprgkibostbo odvolgp cuditjgd R(). Pbjs summosts us tbkt jc wo bkvo kd kporjg`ji sjmdkl, woikd trokt jt ks k porjg`ji sjmdkl wjtb P . Pbod tbo igrrospgd`jdm Cgurjor sorjosigoijodts kpprgkib tg tbo odvolgp cuditjgd R(). Pbo odvolgp cuditjgd js ikllo`tbo Cgurjor Prkdscgra gc tbo sjmdkl x(t). Dgw, lot us stu`y Cgurjor Prkdscgra agrocgraklly.


59/105

0.8. CG_XJOX PXKDYCGXA ?:

0.8 Cgurjor Prkdscgra

Pbo òrjvktjgd gc Cgurjor Prkdscgra igdsjsts gc tbroo stops.

Ytop ;.

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

x(t) 1x(t),cgrP /8< t < P/8, kd`

x(t+P) 1

x(t). ^jitgrjklly, wo bkvo

Ytop 8.

Yjdiox(t) js porjg`ji, wo aky oxprossx(t) usjdm Cgurjor Yorjos6

x(t) 1

e1keo

fe>t (0.8)

wboro

ke 1 ;

P

P

x(t)ofe>t`t.Pbo Cgurjor Yorjos igoijodts ke ikd curtbor ho ikliulkto` ks

ke 1 ;

P

P8

P8

x(t)ofe>t`t1

;

P

P8

P8

x(t)ofe>t`t,

x(t) 1x(t), cgr P /8< t < P/8

1 ;P x(t)ofe>t`t, x(t) 1 >, cgr|t| 4 P /8.Jc wo òdo

R(f) 1

x(t)oft`t (0.9)


60/105


61/105


62/105

38 IBK^POX 0. IGDPJD_G_Y-PJAO CG_XJOX PXKDYCGXA

Cjmuro 0.06 Cgurjor Prkdscgra gd kporjg`ji sjmdkls js oqujvklodt tg kpplyjdm Cgurjorsorjos kdklysjs gd tbo porjg`jiklly pk``o` vorsjgd gc tbo sjmdkl, kd` sot tbo ljajt gc> >.

gcx(t) js P>, tbod > 1 8P>

. Dgw, sjdiox(t) js porjg`ji, wo ikd kpply CgurjorYorjos kdklysjs tgx(t) kd` mot

x(t) 1

e1 keofe>t, (0.:)

wboroke js tbo Cgurjor Yorjos igoijodt. Jc wo curtbor kpply Cgurjor Prkdscgratg (0.:), tbod wo bkvo

R(f) 1

e1

keofe>t

oft`t

1

e1ke

ofe>toft`t

1

e1ke8( e>).

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


63/105

0.9. XOLKPJGD PG CG_XJOX YOXJOY 39

gc 8( e>) jsofe>t. Pg sbgw tbjs, wo bkvo

C;{8( e>)} 1 ;8

8( e>)ofwt`w

1

( e>)ofe>t`w1 ofe>t.

Pborocgro, wo sbgwo` tbkt tbo Cgurjor Prkdscgra gc k porjg`ji sjmdkl js k trkjdgc japulsos wjtb kapljtuò òdo` hy tbo Cgurjor Yorjos igoijodts (kd` siklo`hy k ckitgr gc 8).

Cjmuro 0.?6 Cgurjor Prkdscgra kd` Cgurjor Yorjos kdklysjs gc k porjg`ji sjmdkl6 Hgtbyjol s k trkjd gc japulsos. Cgr Cgurjor Prkdscgra, tbo kapljtuò js aultjpljo` hy kckitgr gc 8. Cgr Cgurjor Yorjos igoijodts, tbo sopkrktjgd hotwood okib igoijodt js>.


64/105


0.0 Oxkaplos

Oxkaplo ;.Igdsjòr tbo sjmdkl x(t) 1 oktu(t), cgr k 4 >. @otorajdo tbo Cgurjor PrkdscgraR(f), jts akmdjtuò|R(f)| kd` jts pbkso R(f).

R(f) 1

x(t)oftu(t)`t

1

>

oktoft`t , u(t) 1 >, wbodovor t

1 ;k +f

o(k+f)t

>

1 ;k +f

.

Pbo akmdjtuò kd` pbkso ikd ho ikliulkto` ks

|R(f)| 1 ;k8 +8

R(f) 1 tkd;

k

.

Oxkaplo 8.Igdsj or tbo sjmdklx(t) 1(t). Pbo Cgurjor Prkdscgra js

R(f) 1

x(t)oft`t

1

(t)oft`t1 ;.

Oxkaplo 9.Igdsjòr tbo sjmdkl x(t) 1 of>t. ]o wkdt tg sbgw tbkt R(f) 1 8( >). Pg


65/105

0.0. ORKA^LOY 3?

soo, wo tkeo tbo jdvorso Cgurjor Prkdscgra6

x(t) 1 ;8

R(f)oft`

1

( >)oft`

1of>t

( >)`1 of>t.

Oxkaplo 0.Igdsjòr tbo kporjg`ji sjmdkl

x(t) 1 ; |t| P;> |t| 4 P;

Pbo Cgurjor Prkdscgra js

R(f) 1

x(t)oft`t

1

P;P;

oft`t1;f

oft

P;P; 1 8

sjd P;

.

Oxkaplo ?.Lot us òtorajdo tbo IPCP gc tbo udjt stop cuditjgdu(t). Pg `g sg, wo kpply IPCPkd` mot

_(f) 1

u(t)oft`t1 >

oft`t1 ;f

oft>

.

Pbkt js, wo bkvo tg ovklukto tbo cuditjgd ;f oft kt t 1 > kd` t 1. Bgwovor,

tbo ovkluktjgd kt t 1 js jdòtorajdkto! Pborocgro, wo oxpross y(t) ks k ljajt gcòikyjdm oxpgdodtjkls

u(t) 1 ljak>

oktu(t).

Pbod kpplyjdm IPCP gd hgtb sjòs,

_(f) 1 ljak

>Co

ktu(t)1 ljak>;

k +f

1 ljak>

k fk8 +8

1 ljak>

k

k8 +8f

k8 +8

.


66/105


Dgw, tbo soigd` tora js

ljak>f

k8 +8 1 ;

f .

Pbo rst tora sktjsos

ljak>

k

k8 +8 1 >, cgr 1 >.

kd`

ljak>

k

k8 +81 lja

k>;

k1 , cgr 1 >,

wbjlo

k

k8 +8`1 tkd;

k

1,

k

X.

Pborocgro,

ljak>

k

k8 +8 1(),

kd` sg

_(f) 1 ;

f+().

0.? ^rgportjos gc Cgurjor Prkdscgra

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

;. Ljdokrjty Jcx;(t) R;(f) kd`x8(t) R8(f), tbod

kx;(t) +hx8(t) kR;(f) +hR8(f).

8. Pjao Ybjctjdm

x(t t>) oft>R(f)

^bysjikl jdtorprotktjgd gcoft> 6

oft>R(f) 1 |R(f)|ofR(f)oft>1 |R(f)|ofUR(f)t>Q

Ygoft> js igdtrjhutjdm tg pbkso sbjct gcR(f).


67/105

0.?. ^XGÔXPJOY GC CG_XJOX PXKDYCGXA 3=

9. Igdfumktjgd

x(t) R(f)Jcx(t) js rokl, tbod x(t) 1x(t), sg R(f) 1R(f).

0. @jorodtjktjgd kd` Jdtomrktjgd

`

`tx(t) fR(f) t

x()` ;

fR(f) +R(>)()

?. Pjao Yikljdm

x(t) ;|k|R(f

k )

3. ^krsovkl Oqukljty

|x(t)|8`t1 ;8

|R(f)|8`

=. @ukljty Pbo japgrtkdt aosskmo boro js6 Jcx(t) R(f), tbod jc kdgtbor sjmdkly(t) bks tbo sbkpo gcR(f), wo ikd qujiely òùio tbkt R(f) wjll bkvo tbosbkpo gcx(t). Boro kro twg oxkaplos6

@ukljty Oxkaplo ; @ukljty Oxkaplo 8


68/105


2. Igdvglutjgd ^rgporty

b(t) x(t) B(f)R(f)^rggc6 Igdsjòr tbo igdvglutjgd jdtomrkl

y(t) 1

x()b(t )`

Pkejdm Cgurjor Prkdscgra gd hgtb sjòs yjol`s

Z(f) 1 C

x()b(t )`

1

x()b(t )` oft`t1

x()

b(t )oft`t

`

1

x()

ofB(f)

`

1

x()of` B(f)

1R(f)B(f)

:. Aultjpljiktjgd ^rgporty (ygu ikd òrjvo tbjs hy ygursolc)

x(t)y(t) ;8

R(f) Z(f)

Oxkaplo.Igdsjòr tbo sjmdkl x(t) 1 a(t)igs(>t), wboro a(t) js sgao hkd`ljajto` sjmdkl.Yuppgso tbo Cgurjor Prkdscgra gca(t) jsA(f). Yjdio

igs(>t) 1of>t +of>t

8 U( >) +(+>)Q,

hy igdvglutjgd prgporty, tbo IPCP gcx(t) js

a(t)igs(>t) ;8

A(f) U( >) +(+>)Q

1;

8UA(f( >)) +A(f(+>))Q .


69/105

0.3. YZYPOA KDKLZYJY _YJDM CG_XJOX PXKDYCGXA 3:

0.3 Yystoa Kdklysjs usjdm Cgurjor Prkdscgra

Cjrst Gròr YystoaLot us igdsjòr tbo rst gròr systoa

`y(t)

`t +ky(t) 1x(t), (0.;>)

cgr sgao k 4>. Kpplyjdm tbo IPCP tg hgtb sjòs,

C

`y(t)

`t +ky(t)

1 C {x(t)} ,

kd` uso ljdokrjty prgporty, kd` `jorodtjktjgd prgporty gc IPCP, wo bkvo

fZ(f) +kZ(f) 1R(f).

Xokrrkdmjdm tbo toras, wo ikd d` tbo croquodiy rospgdso gc tbo systoa

B(f) 1 Z(f)

R(f)1

;

k +f.

Dgw, roikll tbo IPCP pkjr6

b(t) 1oktu(t) B(f) 1 ;k +f

, (0.;;)

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

B(f) 1

oktu(t)oft`t1>

o(f+k)t`t

1

;

f +ko(f+k)t

>

1 ;

f +k.

Modorkl YystoasJd modorkl, wo wkdt tg stu`y tbo systoa

D

e1>ke

èy(t)

`te 1

A

e1>he

èx(t)

`te . (0.;8)

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

C

De1>

keèy(t)

`te

1 C

Ae1>

heèx(t)

`te

.


70/105

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

Pborocgro, hy ljdokrjty kd` `jorodtjktjgd prgporty, wo bkvo

De1>

ke(f)eZ(f) 1

Ae1>

he(f)eR(f).

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

B(f) 1 Z(f)

R(f)1

Ae1> he(f)

eDe1> ke(f)

e. (0.;9)

Dgw,B(f) js oxprosso` ks k rktjgdkl cuditjgd, j.o., k rktjg gc pglydgajkl. Pborocgro,wo ikd kpply tbo toibdjquo gc pkrtjkl crkitjgd oxpkdsjgd tg oxpross B(f) jd k cgra

tbkt kllgws us tg òtorajdo b(t) hy jdspoitjgd usjdm tbo trkdscgra pkjr

b(t) 1oktu(t) B(f) 1 ;k +f

,

kd` rolkto` trkdscgra pkjr, suib ks

toktu(t) ;(k +f)8

.

Oxkaplo ;.Igdsjòr tbo LPJ systoa

`8y(t)

`t8 + 0y(t) + 9y(t) 1

`x(t)

`t + 8x(t).

Pkejdm IPCP gd hgtb sjòs yjol`s

(f)8Z(f) + 0fZ(f) + 9Z(f) 1fR(f) + 8R(f).

kd` hy rokrrkdmjdm toras wo bkvo

B(f) 1 f + 8

(f)8 + 0(f) + 91

f + 8

(f + ;)(f + 9).

Pbod, hy pkrtjkl crkitjgd oxpkdsjgd wo bkvo

B(f) 1;

8

;

f + ;

+

;

8

;

f + 9

.


71/105

0.3. YZYPOA KDKLZYJY _YJDM CG_XJOX PXKDYCGXA =;

Pbus, b(t) js

b(t) 1;

8ot

u(t) +;

8o9t

u(t).

Oxkaplo 8.Jc tbo jdput sjmdkl js x(t) 1 otu(t), wbkt sbgul` ho tbo gutput y(t) jc tbo japulsorospgdso gc tbo systoa js mjvod hy b(t) 1 ;

8otu(t) + ;

8o9tu(t)7

Pkejdm IPCP, wo edgw tbkt R(f) 1 ;f+;

, kd` B(f) 1 f+8(f+;)(f+9)

. Pborocgro,tbo gutput js

Z(f) 1B(f)R(f) 1

f + 8

(f + ;)(f + 9)

;

f + ;

1

f + 8

(f + ;)8(f + 9).

Hy pkrtjkl crkitjgd oxpkdsjgd, wo bkvo

Z(f) 1;0

f + ;+

;8

(f + ;)8

;0

f + 9.


y(t) 1

;

0ot +

;

8tot ;

0o9t

u(t).


72/105

=8 IBK^POX 0. IGDPJD_G_Y-PJAO CG_XJOX PXKDYCGXA


73/105

Ibkptor ?

@jsiroto-tjao Cgurjor Prkdscgra

?.; Xovjow gd Igdtjdugus-tjao Cgurjor Prkdscgra

Hocgro wo `orjvo tbo `jsiroto-tjao Cgurjor Prkdscgra, lot us roikll tbo wky wo igd-struito` igdtjdugus-tjao Cgurjor Prkdscgra crga tbo igdtjdugus-tjao Cgurjor Yorjos.Jd `orjvjdm tbo igdtjdugus-tjao Cgurjor Prkdscgra, wo hksjiklly bkvo tbo cgllgwjdmtbroo 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 igoijodts ke kd`roprosodt x(t) ks

x (t) 1 e1

keofe>t.

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

=9


74/105

=0 IBK^POX ?. @JYIXOPO-PJAO CG_XJOX PXKDYCGXA

Hy òdjdm

R(f) 1

x (t) oft`t, (?.;)

wbjib js edgwd ks tbo igdtjdugus-tjao Cgurjor Prkdscgra, wo sbgwo`

ke 1 ;

PR(fe>) .

Ytop 96 Yottjdm P , wo sbgwo` x (t) x (t) kd`

x (t) 1

;

8

R(f) oft

`t (?.8)

?.8 @orjvjdm @jsiroto-tjao Cgurjor Prkdscgra

Dgw, lots kpply tbo skao igdiopt tg `jsiroto-tjao sjmdkls. Jd òrjvjdm tbo `jsiroto-tjao 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 xUdQ.

Ytop 86 Yjdio xUdQ js porjg`ji, hy `jsiroto-tjao Cgurjor Yorjos wo bkvo

xUdQ 1

e1Dkeo

fe>d, (?.9)

wboroke ikd ho igaputo` ks

ke 1 ;

D

d1D

x UdQ ofe>d.


75/105

?.8. @OXJSJDM @JYIXOPO-PJAO CG_XJOX PXKDYCGXA =?

Boro, tbo croquodiy js

>1

8

D.

Dgw, dgto tbkt xUdQ js k porjg`ji sjmdkl wjtb porjg`D, kd` tbo dgd-zorg odtrjosgc xUdQ jd k porjg` kro tbo skao ks tbo dgd-zorg odtrjos gcxUdQ. Pborocgro, jtbgl`s tbkt

ke 1 ;

D

d1D

x UdQ ofe>d

1 ;

D

d1

x UdQ ofe>d.

Jc wo `odo

R(of ) 1

d1xUdQofd, (?.0)

tbod

ke 1 ;

D

d1

xUdQofe>d 1 ;

DR(ofe>). (?.?)

Ytop 96 ^uttjdm Oquktjgd (?.?) jdtg Oquktjgd (?.9), wo bkvoxUdQ 1 e1D keofe>d

1

e1D

;

DR(ofe>)

ofe>d

1 ;

8

e1D

R(ofe>)ofe>d>, >18

D. (?.3)

KsD , > > kd` xUdQ xUdQ. Klsg, crga Oquktjgd (?.3) hoigaos

xUdQ 1 ;

8 e1DR(ofe>)ofe>d> ;

8

8

R(of)ofd`.

Pborocgro,

xUdQ 1 ;

8

8

R(of )ofd`. (?.=)


76/105


Cjmuro ?.;6 KsD , > >. Yg tbo krok hoigaos jddjtosjakl sakll kd` suahoigaos jdtomrktjgd.

?.9 ]by js R(of

) porjg`ji 7Jt js jdtorostjdm tg dgto tbkt tbo igdtjdugus-tjao Cgurjor PrkdscgraR(f) js kporjg`jijd modorkl, hut tbo `jsiroto-tjao Cgurjor PrkdscgraR(of) js klwkys porjg`ji. Pg sootbjs, lot us igdsj`or tbo `jsiroto-tjao Cgurjor Prkdscgra (wo wkdt tg iboie wbotborR(of) 1R(of(+8))!)6

R(of(+8)) 1

d1

xUdQof(+8)d

1

d1 xUdQofd(of8)d 1R(of),hoikuso (of8)d 1 ;d 1 ;, cgr kdy jdtomor d. Pborocgro, R(of ) js porjg`ji wjtbporjg` 8.

Dgw, lot us igdsj`or tbo igdtjdugus-tjao Cgurjor Prkdscgra (wo wkdt tg iboie wbotborR(f) 1R(f(+ 8))!)6

R(f(+ 8)) 1

x(t)of(+8)t`t1

x(t)oft(of8)t`t.

Boro,t Xkd` js ruddjdm crgatg. ^ky kttodtjgd tbkt of8t 1 ; udloss t jskd jdtomor (wbjib `jorodt crga tbo `jsiroto-tjao ikso wboro d js klwkys kd jdtomor!).Pborocgro,

x(t)oftof8t`t 1

x(t)oft`t,


77/105

?.0. ^XG^OXPJOY GC @JYIXOPO-PJAO CG_XJOX PXKDYCGXA ==

(k) (of8)d 1 ; cgr kll d, hoikuso d js jdtomor. (h) (of8)t 1 ; udloss t js kd jdtomor.

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

?.0 ^rgportjos gc @jsiroto-tjao Cgurjor Prkdscgra

@jsiroto-tjao Cgurjor Prkdscgra6

R(of) 1

d1xUdQofd

@jsiroto-tjao Jdvorso Cgurjor Prkdscgra6

xUdQ 1 ;

8

8

R(of )ofd`

;. ^orjg`jijty6R(of(+8)) 1R(of)

8. Ljdokrjty6kx;UdQ +hx8UdQ kR;(of) +hR8(of)

9. Pjao Ybjct6 xUd d>Q ofd>R(of)

0. ^bkso Ybjct6

of>dxUdQ R(of(>))


78/105


?. Igdfumkiy6

xUdQ R(of

)

3. Pjao XovorsklxUdQ R(of )

=. @jorodtjktjgd

dxUdQ f `R(of )

`

2. ^krsovkl Oqukljty

d1 |xUdQ|8 1 ;8 8 |R(of )|8`:. Igdvglutjgd

yUdQ 1xUdQ bUdQ Z(of) 1R(of)B(of )

;>. Aultjpljiktjgd

yUdQ 1x;UdQx8UdQ Z(of) 1 ;8

8

R;(of)R8(o

f())`

?.? Oxkaplos

Oxkaplo ;.Igdsj`orxUdQ 1UdQ +Ud ;Q +Ud+ ;Q. Pbod

R(of ) 1

d1xUdQofd

1

d1(UdQ +Ud ;Q +Ud+ ;Q)ofd

1

d1UdQofd +

d1

Ud ;Qofd +

d1Ud+ ;Qofd

1 ; +of +of 1 ; + 8 igs .


79/105

?.?. ORKA^LOY =:

Cjmuro ?.86 Akmdjtuò plgt gc|R(of)| jd Oxkaplo ;.

Pg seotib tbo akmdjtuò|R(of )|, wo dgto tbkt|R(of)| 1 |; + 8 igs |.

Oxkaplo 8.IgdsjòrxUdQ 1UdQ + 8Ud ;Q + 0Ud 8Q. Pbo `jsiroto-tjao Cgurjor Prkdscgra js

R(of ) 1 ; + 8of + 0of0.

Jc tbo japulso rospgdso js bUdQ 1UdQ +Ud ;Q, tbod

B(of) 1 ; +of .


Z(of ) 1B(of)R(of)

1

; +of

; + 8of + 0of8

1 ; + 9of + 3of8 + 0of9.

Pkejdm tbo jdvorso `jsiroto-tjao Cgurjor Prkdscgra, wo bkvo

yUdQ 1UdQ + 9Ud ;Q + 3Ud 8Q + 0Ud 9Q.Oxkaplo 9.IgdsjòrxUdQ 1kduUdQ, wjtb|k|

kdofd 1d1>

(kof)d 1 ;

; kof .


80/105

2> IBK^POX ?. @JYIXOPO-PJAO CG_XJOX PXKDYCGXA

(k) >< k < ; (h);< k < >

Doxt, lot us `rkw tbo akmdjtuò|R(of)|. Pg `g sg, lots igdsjòr

R(of)8

1R(of)R(of) 1 ;

; kof

;

; kof

1 ;

; k(ofof) +k8

1 ;

; 8k igs +k8Ikso K. Jc >< k , kd`|R(of)|8kibjovos ajdjaua wbod 1 . Pbus,

akxR(of)

8

1 ;

; 8k +k8 1 ;

(; k)8

ajd R(of)81 ;; + 8k +k8 1 ;(; +k)8 .Ikso H6Jc;< k , tbod |R(of)|8 kibjovos akxjaua wbod 1 , kd` |R(of)|8kibjovos ajdjaua wbod 1 >. Pbus,

akxR(of )81 ;

; + 8k +k8 1

;

(; +k)8

ajdR(of )81 ;

; 8k +k8 1 ;

(; k)8 .

?.3 @jsiroto-tjao Cjltors

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`stgpltors. Pboro js dg proijso òdjtjgd cgr hojdm lgw gr bjmb. Bgwovor, jt js usukllyoksy tg jdcor tbo typo crga vjowjdm tbojr akmdjtuò rospgdso.


81/105

?.=. K^ÔD@JR 2;

(k) Kd jòkl lgwpkss ltor (h) Kd jòkl bjmbpkss ltor

(i) Kd jòkl hkd`pkss ltor (`) Kd jòkl hkd`stgp ltor

?.= Kppod`jx

Mogaotrji Yorjos6

Dd1>

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

; x ,

d1>

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

; x, wbod|x|


82/105

28 IBK^POX ?. @JYIXOPO-PJAO CG_XJOX PXKDYCGXA


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Ibkptor 3

Ykapljdm Pbogroa

Ykapljdm tbogroa plkys k iruijkl rglo jd agòrd `jmjtkl sjmdkl prgiossjdm. Pbo tbo-groa igdiords khgut tbo ajdjaua skapljdm rkto roqujro` tg igdvort k igdtjdugustjao sjmdkl tg k `jmjtkl sjmdkl, wjtbgut lgss gc jdcgraktjgd.

3.; Kdklgm tg @jmjtkl Igdvorsjgd

Igdsjòr tbo cgllgwjdm systoa sbgwd jd Cjm. 3.;. Pbjs systoa js ikllo` kd kdklgm-tg-`jmjtkl (K/@) igdvorsjgd systoa. Pbo hksji jòk gc K/@ igdvorsjgd js tg tkeo kigdtjdugus-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(t), wo ikd iglloit k sot gc skaploshy aultjplyjdm x(t) wjtb kd japulso trkjd p(t)6

p(t) 1

d1

(t dP),

29


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20 IBK^POX 3. YKA^LJDM PBOGXOA

wboroP js tbo porjg` gc tbo japulso trkjd. Aultjplyjdm x(t) wjtbp(t) yjol`s

xp(t) 1x(t)p(t)

1x(t)

d1(t dP)

1

d1x(t)(t dP)

1

d1x(dP)(t dP).

^jitgrjklly, xp(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 xp(t) roprosodts k sotgc skaplos gc tbo sjmdkl x(t).

]o aky romkr`xp(t) ks tbo skaplos gcx(t). Dgto tbktxp(t) js stjll k igdtjdugus-tjaosjmdkl! (]o ikd vjow xp(t) ks k `jsiroto-tjao sjmdkl jc wo òdo xpUdQ 1x(dP). Huttbjs js dgt kd japgrtkdt jssuo boro.)

3.8 Croquodiy Kdklysjs gc K/@ Igdvorsjgd

Bkvjdm kd oxplkdktjgd gc tbo K/@ igdvorsjgd jd tjao `gakjd, wo dgw wkdt tg stu`ytbo K/@ igdvorsjgd jd tbo croquodiy `gakjd. (]by7 ]o doo` jt cgr tbo òvolgp-aodt gc Ykapljdm Pbogroa!) Yg, bgw `g tbo croquodiy rospgdsos R(f),^(f) kd`Rp(f) lgge ljeo7


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3.8. CXO[_ODIZ KDKLZYJY GC K/@ IGDSOXYJGD 2?

3.8.; Bgw `gos ^(f) lgge ljeo7

Lots stkrt wjtb ^(f). Crga Pkhlo 0.8 gc tbo toxthgge, wo edgw tbkt

p(t) 1

d1(t dP) C.P. 8

P

e1

( 8eP

) 1^(f) (3.;)

Pbjs aokds tbkt tbo croquodiy rospgdso gc tbo japulso trkjd p(t) js kdgtbor japulsotrkjd. Pbo gdly `jorodio js tbkt tbo porjg` gcp(t) jsP, wboroks tbo porjg` gc^(f)js 8P .

Cjmuro 3.96 Jllustrktjgd gcR(f) kd`^(f).

3.8.8 Bgw `gos Rp(f) lgge ljeo7

Doxt, suppgso tbkt tbo sjmdkl x(t) bks k croquodiy rospgdsoR(f). ]o wkdt tg edgwtbo croquodiy rospgdso gc tbo gutput xp(t). Crga tbo `odjtjgd gcxp(t), wo edgwedgw tbkt

xp(t) 1x(t)p(t).

Pborocgro, hy tbo aultjpljiktjgd prgporty gc Cgurjor Prkdscgra, wo bkvo

Rp(f) 1 ;

8R(f) ^(f).


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87/105

3.8. CXO[_ODIZ KDKLZYJY GC K/@ IGDSOXYJGD 2=

Cjmuro 3.?6 Jllustrktjgd gcxp(t) kd`Rp(f).

japulsos jdiroksos (j.o., ckrtbor kpkrt). Jd croquodiy `gakjd, sjdio

^(f) 18

P

e1( 8e

P ),

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

Cjmuro 3.36 ]bodPjdiroksos, tbo porjg` jd croquodiy `gakjd roùios.

Jc wo igdsjòr Rp(f), wbjib js k porjg`ji ropljikto gcR(f) kt tbo japulsos mjvodhy^(f), wo soo tbkt tbo sopkrktjgd hotwood ropljiktos roùios. ]bodP bjts ior-tkjd ljajt, tbo sopkrktjgd hoigaos zorg5 kd` hoygd` tbkt ljajt, tbo ropljiktos stkrt tggvorlkp! ]bod tbo croquodiy ropljiktos gvorlkp, wo sky tbkt tboro js kljksjdm.


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89/105

3.9. YKA^LJDM PBOGXOA 2:

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

3.9 Ykapljdm Pbogroa

Pbogroa ;;(Ykapljdm Pbogroa). Lotx(t)ho k hkd` ljajto` sjmdkl wjtbR(f) 1 >cgr kll|| 4 ]. Pbod tbo ajdjaua skapljdm rkto suib tbkt dg kljksjdm giiurs jdRp(f) js

s 4 8],

wboros1 8P

.

3.9.; Oxplkdktjgd

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

Cjmuro 3.;>6 Ajdjaua skapljdm rkto tbkt tboro js dg kljksjdm.

Jd tbjs ikso, wo soo tbkt tbo skapljdm rkto s (1 8P

) js

s 1 8].


90/105

:> IBK^POX 3. YKA^LJDM PBOGXOA

JcP js lkrmor (gr s js sakllor), tbod 8P hoigaos loss tbkd 8], kd` kljksjdm giiurs.

Pborocgro, tbo ajdjaua skapljdm rkto tg odsuro dg kljksjdm js

s 4 8].

3.9.8 Oxkaplo

Yuppgso tboro js k sjmdkl wjtb akxjaua croquodiy 0>eBz. ]bkt js tbo ajdjauaskapljdm rkto 7

Cjmuro 3.;;6 Oxkaplo6 Ajdjaua skapljdm croquodiy.

Kdswor 6Yjdio 1 8c, wo edgw tbkt tbo akx croquodiy (jd rk`) js 1 8(0> ;>9) 12>

;>9 (rk`). Pborocgro, tbo ajdjaua Ykapljdm rkto js6 8

(2>

;>9), wbjib js

;3> ;>9 (rk`) 1 2>eBz.

3.0 @jmjtkl tg Kdklgm Igdvorsjgd

Jd tbo provjgus soitjgds, wo stu`jo` K/@ igdvorsjgd. Dgw, mjvod k `jsiroto-tjaosjmdkl (kssuao dg kljksjdm), wo wgul` ljeo tg igdstruit tbo igdtjdugus tjao sjmdkl.

3.0.; Mjvod Rp(t) (dg kljksjdm), bgw `g J roigvor x(t)7

Jc dg kljksjdm giiurs `urjdm tbo skapljdm prgiossjdm (j.o., aultjply x(t) wjtb p(t)),tbod wo ikd kpply k lgwpkss ltor B(f) tg oxtrkit tbo x(t) crga xp(t). Cjm. 3.;8sbgws k siboaktji `jkmrka gc bgw tbjs js porcgrao`.Pg soo bgw kd j`okl lgwpkss ltor ikd oxtrkit x(t) crga xp(t), wo rst lgge kt tbocroquodiy rospgdso gcRp(f). Yuppgso tbktp(t) bks k porjg` gcP(sg tbkts1

8P).


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3.0. @JMJPKL PG KDKLGM IGDSOXYJGD :;

Cjmuro 3.;86 Yiboaktji `jkmrka gc roigvorjdm x(t) crga xp

(t). Pbo ltor B(f) jskssuao` tg ho kd jòkl lgwpkss ltor.

Pbod

Rp(f) 1 ;

P

R(f( es)).

Ks sbgwd jd tbo tgp loct gc Cjm. 3.;9, Rp(f) js k porjg`ji ropljikto gcR(f). Yjdiowo kssuao tbkt tboro js dg kljksjdm, tbo ropljikto igvorjdm tbo y-kxjs js jòdtjikl tgR(f). Pbkt js, cgr|| < s8,

Rp(f) 1R(f).

Dgw, jc wo kpply kd jòkl lgwpkss ltor (sbgwd jd hgttga loct gc Cjm. 3.;9)6

B(f) 1

;, || < s8,>, gtborwjso,

tbodRp(f)B(f) 1R(f),

cgr kll . Pkejdm tbo jdvorso igdtjdugus-tjao Cgurjor trkdscgra, wo ikd ghtkjd x(t).

3.0.8 JcRp(t) bks kljksjdm, ikd J stjll roigvor x(t) crga xp(t) 7

Pbo kdswor js DG. Jc kljksjdm giiurs, tbod tbo igd`jtjgd

Rp(f) 1R(f)


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:8 IBK^POX 3. YKA^LJDM PBOGXOA

Cjmuro 3.;96 Loct6 Aultjpljiktjgd hotwood Rp(f) kd` tbo lgwpkss ltor B(f). Pbo

oxtrkito` gutput R(f) js j`odtjikl tg R(f) jc dg kljksjdm giiurs. Hy kpplyjdm

jdvorso Cgurjor trkdscgra tg R(f) wo ikd ghtkjd x(t).

`gos dgt bgl` cgr kll || < s8. Igdsoquodtly, ovod jc wo kpply tbo lgwpkss ltorB(f)tgRp(f), tbo rosult js dgt R(f). Pbjs ikd ho sood jd Cjm. 3.;0.

Cjmuro 3.;06 Jc kljksjdm giiurs, wo kro udkhlo tg roigvor x(t) crga xp(t) hy usjdm kdj`okl lgwpkss ltor.


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3.0. @JMJPKL PG KDKLGM IGDSOXYJGD :9

3.0.9 ]bkt ikd J `g jc ay skapljdm `ovjio `gos dgt suppgrt

k vory bjmb skapljdm rkto 7 Aotbg` ;6 Huy k hottor skapljdm `ovjio ! Aotbg` 86 Yod` sjmdkls wjtb dkrrgwor hkd`wj`tb gr ljajt tbo hkd`wj`tb ho-

cgro sod`jdm 6

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


94/105

:0 IBK^POX 3. YKA^LJDM PBOGXOA


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Ibkptor =

Pbo z-Prkdscgra

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

=.; Pbo z-Prkdscgra

@odjtjgd 8;. Pboz-trkdscgra gc k `jsiroto-tjao sjmdklxUdQ js6

R(z) 1

d1 xUdQzd. (=.;)

]o `odgto tboz-trkdscgra gporktjgd ks

xUdQ R(z).Jd modorkl, tbo duahorzjd (=.;) js k igaplox duahor. Pborocgro, wo aky wrjto zks

z1 rofw ,

wboror X kd` w X. ]bodr1 ;, (=.;) hoigaos

R(ofw) 1

d1xUdQofwd ,wbjib js tbo `jsiroto-tjao Cgurjor trkdscgra gcxUdQ. Pborocgro, @PCP js k spoijklikso gc tbo z-trkdscgra! ^jitgrjklly, wo ikd vjow @PCP ks tbo z-trkdscgra ovklukto`gd tbo udjt ijrilo6

:?


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97/105

=.;. PBOV-PXKDYCGXA :=

Oxkaplo ;. Igdsj`or tbo sjmdkl xUdQ 1kduUdQ, wjtb >< k

kz;

d.

Pborocgro, R(z) igdvormos jc

d1>(kz;)d

rz;

d1

;

; kz; ,

wbod|kz;|


98/105

:2 IBK^POX =. PBOV-PXKDYCGXA

Oxkaplo 8. Igdsj`or tbo sjmdkl xUdQ 1kduUd ;Q wjtb > < k < ;. Pbo

z-trkdscgra gcxUdQ js

R(z) 1

d1kduUd ;Qzd

1 ;

d1kdzd

1 d1;

kdzd

1 ;

d1>

k;zd .Pborocgro,R(z) igdvormos wbod|k;z|


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=.;. PBOV-PXKDYCGXA ::

Oxkaplo 9. Igdsj`or tbo sjmdkl

xUdQ 1 =

;

9

duUdQ 3

;

8

duUdQ.

Pboz-trkdscgra js

R(z) 1

d1

=

;

9

d 3

;

8

duUdQzd

1 = d1

;9d uUdQzd 3

d1;

8d uUdQzd

1 =

;

; ;9z;

3

;

; ;8z;

1 ; 98z;

; ;9z;

; ;8z; .

Cgr R(z) tg igdvormo, hgtb suas jd R(z) aust igdvormo. Yg wo doo` hgtb|z| 4 |;9|

kd`

|z

|4

|;8

|. Pbus, tbo XGI js tbo sot gczsuib tbkt

|z

|4

|;8

|.

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


100/105

;>> IBK^POX =. PBOV-PXKDYCGXA

=.8 z-trkdscgra ^kjrs

=.8.; K. Pkhlo ;>.8

;. UdQ ;, kll z8. Ud aQ za, kllzoxiopt > wbod a 4> kd`wboda .9. uUdQ ;

;z; ,|z| 4;

0. kduUdQ ;;kz; ,|z| 4 k

?.kduUd ;Q ;;kz; ,|z| < |k|.

Oxkaplo 0. Lot us sbgw tbkt UdQ ;. Pg soo tbjs,

R(z) 1

d1xUdQzd 1

d1

UdQzd

1

d1UdQ 1 ;.

Oxkaplo ?. Lots sbgw tbkt Ud aQ za6

R(z) 1

d1

xUdQzd 1

d1

Ud aQzd

1za

d1Ud aQ 1za.

=.8.8 H. Pkhlo ;>.;

;. kx;UdQ +hx8UdQ kR;(z) +hR8(z)8. xUd d>Q R(z)zd>

9. zd> xUdQ R( zz> )0. ofw>dxUdQ R(ofw>z)?. xUdQ R(;z )


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=.8. V-PXKDYCGXA ^KJXY ;>;

3. xUdQ R(z)

=. x;UdQ x8UdQ R;(z)R8(z)

2. Jcy UdQ 1

xUd/LQ, djs aultjplos gcL

>, gtborwjso,

tbodZ(z) 1R(zL).

Oxkaplo 3. Igdsj`or tbo sjmdklbUdQ 1UdQ + Ud ; Q + 8Ud 8Q. Pboz-PrkdscgragcbUdQ js

B(z) 1 ; +z; + 8z8.

Oxkaplo =. Lot prgvo tbkt xUdQ R(z;

). Lottjdm y UdQ 1xUdQ, wo bkvo

Z(z) 1

d1yUdQzd 1

d1

xUdQzd

1

a1

xUaQza 1R(;/z).

Oxkaplo 2. Igdsj`or tbo sjmdkl xUdQ 1;9

sjd

0

d

uUdQ. Pg d` tbo z-Prkdscgra,wo rst dgto tbkt

xUdQ 1 ;8f

;9

of 0d uUdQ ;8f

;9

of 0d uUdQ.Pboz-Prkdscgra js

R(z) 1

d1xUdQzd

1

d1>

;

8f

;

9of

0 z;

d

d1>

;

8f

;

9of

0 z;

d1

;

8f

;

; ;9of

0 z; ;

8f

;

; ;9of

0 z;

1

;98

z;

(; ;9

of0 z;)(; ;

9of

0 z;)

.


102/105


103/105

=.9. ^XGÔXPJOY GC XGI ;>9

hoikuso r> < ; mukrkdtoos tbkt tbo sua js djto.

Dgw, jc tboro js kdgtbor pgjdt z wjtb|z| 1r; 4 r>, wo aky wrjto r; 1 kr> cgr sgaok 4;. Pbod tbo sorjos

d1D;

xUdQrd; 1

d1D;

xUdQkdrd>

kD;

d1D;

xUdQrd> < .

Yg, z js klsg jd tbo XGI.

^rgporty =. JcR(z) js rktjgdkl, j.o., R(z) 1 K(z)H(z)

wboroK(z) kd`H(z) kro pgly-

dgajkls, kd` jcxUdQjs rjmbt-sjò`, tbod tbo XGI js tbo romjgd gutsjò tbo gutoragstpglo.

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

R(z) 1 K(z)

H(z)1

d;e1>kez

e

re1;(; p;e z)e

,

wborope js tbo e-tb pglo gc tbo systoa. _sjdm pkrtjkl crkitjgd, wo bkvo

R(z) 1r

j1;

je1;

Ije

(; p;j z)e.

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

Cgr oxkaplo, jc

R(z) 1 ;

(; ;9z;)(; ;8z;),

tbod tbo XGI js tbo romjgd|z| 4 ;8 .


104/105

;>0 IBK^POX =. PBOV-PXKDYCGXA

=.0 Yystoa ^rgportjos usjdmz-trkdscgra

=.0.; Ikuskljty

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

^rggc. K systoa js ikuskl jc kd` gdly jc

bUdQ 1 >, d .

Pborocgro,bUdQ aust ho rjmbt-sjò`. ^rgporty ? japljos tbkt XGI js gutsjò k ijrilo.

Klsg, hy tbo òdjtjgd tbkt

B(z) 1d1>

bUdQzd

wboro tboro js dg pgsjtjvo pgwors gcz, B(z) igdvormos klsg wbod z (Gc igurso,|z| 4; wbod z !).

=.0.8 Ytkhljljty

^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 jcbUdQ js khsglutoly suaakhlo, jc kd` gdly jc@PCP gcbUdQ oxjsts. Igdsoquodtly hy ^rgporty 8, XGI gcB(z) aust jdiluò tboudjt ijrilo.

^rgporty ;>. K ikuskl `jsiroto-tjao LPJ systoa js stkhlo jc kd` gdly jc kll gc jtspglos kro jdsjò tbo udjt ijrilo.

^rggc. Pbo prggc cgllgws crga ^rgporty 2 kd` : `jroitly.


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signals and system - stanley chan

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