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peopaoae ype y y(t) opeeec paee y(i) = y(t)e-itdt, pe o oc yco, o y(t) = 0 p t < 0 y(t)dt cyecye.

Cpaa peopaoa aaca ype, o, o opao oo oe oyeo peopaoa aaca poco aeo s a i, o -a opoo yco peopaoae ype oec oee opaeoo acca y. ae ypae (4.9) s a i, oyae:

(an(i)n + an-1(i)n-1 +... + a1(i) + a0 )y(i) = = (bm (i)m + bm-1(i)m-1 +... + b1(i) + b0 )x(i), oya x(i) bm (i)m + bm-1(i)m-1 +... + b1(i) + bW (i) = =. (4.8) y(i) an (i)n + an-1(i)n-1 +... + a1(i) + apoo aa pae (4.8), oo aca, o B() + i B1() M ()ei () W (i) = = A() + i A1() M ()eiH () M () cea o: ayo-acoa xapaepca M () = ec eo M () ye; ao-acoa xapaepca () = () () eeo ye;

eecea acoa xapaepca Re() eo ye; a acoa xapaepca Im() eeo ye (pc. 4.6 4.7).

a) M Re ) Pc. 4.6 Coco eoc acox xapaepc:

a AX; BX ) Im a) Pc. 4.7 Coco eeoc acox xapaepc:

a X; MX Ayo-aoa xapaepca ae oe paccapac a opaee ype o ecoo y:

-it W (i) = (4.9) w(t)e dt.

Ta a e-it = cost - isin t, o (4.9) oy oye opy opeee eeceo o xapaepc:

W (i) = w(t){cost - i sin t}dt, , ceoaeo, Re() = cos tdt, (4.10) w(t) Im() = - (4.11) w(t)sin tdt.

ocex opy ceye, o Re() = Re(-), Im() = - Im(-), (4.12) i Im() < = Re() >Pc. 4.8 oopa AX a o ceecye o o, o AX p opaex acoax ec epa oopaee AX ooex aco ooceo eeceo oc (pc. 4.8).

p paecx paceax oo opaac ocpoee AX oo ooex aco. coy opyy opaoo peopaoa ype, oo o AX oy ecoy xapaepcy:

w(t) = (i)eitd. (4.13) W e-s pep 4.1 yc aaa epeaoa y oea W (s) =, peyec s2 + 2s + opee acoe xapaepc.

ae s a i, acae paee AX:

e-i e-i W (i) = =.

(i)2 + 2(i) + 3 (3 - 2) + 2i Ta a paccapae oe ee caoape, o, pe p cyepo, ee:

AX (pc. 4.9, a) M () = ;

(3 - 2)2 + X (pc. 4.9, ) () = - - arctg.

3 - oopa ayo-aoo xapaepc opae a pc. 4.9, .

a) ) M 1/1/0 1 2 i Im() ) 1/ = Re() Pc. 4.9 pa acox xapaepc:

a AX; X; AX Beecey y acoe xapaepc oo oya yoee ce aeae a paee, copeoe aeae:

e-i cos - i sin (3 - 2 ) - 2i W (i) = = = (3 - 2 ) + 2i (3 - 2 ) + 2i (3 - 2 ) - 2i (3 - 2 ) cos - 2sin - i (3 - 2 ) sin + 2 cos, = (3 - 2 )2 + oya eeceo-acoa xapaepca:

(3 - 2)cos - 2sin Re() = ;

(3 - 2)2 + a acoa xapaepca:

(3 - 2)sin + 2cos Im() =.

(3 - 2 )2 + 4.4 CB EPEHAHOO PABHEH C ACTOTHM XAPATEPCTAM Peee epeaoo ypae (3.36, a) ee y(t) = yc(t) + y (t), (4.14) e y(t) yeoe ee, ocaeoe ac peee; yc(t) cooe e, ocaee o peee oopooo ypae.

ycaoe c ey AX epea ypaee paccapac yee e p xoo apoeco oec a: x(t) = 2A cost, oopoe oo peca o opye epa x(t) = Aeit + Ae-it paccapa a cyy xox cao, .e..

x(t) = x1(t) + x2(t) B o cyae acoe peee epeaoo ypae cy pa cyepo ae pecaec e cy y(t) = y1 (t) + y2 (t), e y1 (t) y2 (t) opeec cooeceo o x1(t) x2(t). B c c pee yy cac e y1 (t) = AW (i)eit ; y2 (t) = AW (-i)e-it, e W(i), W(-i) eoope eece y, e ace o t, oeae opeee.

axoe W(i) y1 (t) epepyec n pa, a x1(t) - m pa ocac cxooe epeaoe ypaee, peyae oya AW(i)eit[an(i)n + an-1(i)n-1 +... + a1(i) + a0] = (4.15) = Aeit[bm(i)m + bm-1(i)m-1 +... + b1(i) + b0].

oyeoe paee (4.15) ooc coaae c oye paee paee (4.8) AX ee pa oepae o a, o ayo-aoa xapaepca oe oyea poco aeo epeeo s a i.

y W(i) oyaec aao opao o opye (4.15) aeo i a ( i).

aca oyee pae oecx y W(i) W(i) oaaeo ope W (i) = M ()ei(); W (-i) = M ()e-i(), acoe peee ypae (4.7) peopayec y y (t) = AM ()[ei()eit + e-i()e-it ] = 2AM () cos[t + ()].

Cpaee y(t), ocaeo ycaoec oea a xoe oea, c xo cao x(t) oaae, o ooee ay xox xox oea 2AM () pao = M (), a o a pa ec ayo-acoa xapaepca; paoc 2A a [t + ()]- t = () - ao-acoa xapaepca.

C eee aco oea ayo- ao-acoe xapaepc ec o opeeeoy aoy acoc o ecx coc oea.

Oao ce peae ece cce oaa o o coco, oopoe aaec o, o p yee aco xox oea e eoopoo peea (aco cpea) cp oe paec e peapye a oea, .e.

aya xox oea paa y. Ta opao, oo peaoo oea lim M () = 0.

4.5 EC CMC ACTOTHX XAPATEPCT ec cc acox xapaepc ycaaaec p x cepeao opeee.

yc a xo eoo oea oaec apoec ca a x(t) = Asint. Ha xoe oea ycaoec pee (coceoe ee pepaoc) cy pa cyepo ye aac ae apoec ca c acoo, pao acoe xox oea, cy ooceo x o ae pyo ay (pc.

4.10), .e. y(t) = Bsin(t + ).

Cee pa ey apaepa xox xox apoecx cao e ac o ay a xooo caa, a opeeec oo aec coca caoo oea acoo oea, ooy aece aecx xapaepc oea ec coyc paccopee e acoe xapaepc. oye ocex cepea ye pooc p oo, oopx coyec aapaypa cocae eepaopa apoecx oea c peypyeo acoo ycpoca epe ay a oea.

B peyae poeex cepeo acoe xapaepc opeec cey opao.

Ayo-acoa xapaepca (AX) - ooee ay xox oea aye xooo caa:

B M () =. (4.16) A ao-acoa xapaepca (X) - paoc a xox xox oea:

() = x x (4.17) t 2, () = T e t - pe ca.

Ta opao, ayo-aoa xapaepca (AX) oe opeeea a oeca y, oopo AX ec oye, a X apyeo. ocee coooe a pa opee ec cc acox xapaepc.

e coe pacope ayo-aoy xapaepcy, cy cepeao, xoo ca, oo aca xoo ca. Hapep, AX aaa oopao (pc. 4.11), a xo oaec ca x(t) = 2 sin0,5t + 3 cos0,1t 0,8 sin10t.

a) y(t) x(t) Oe ) ) x x 0 t t T1 = 2/T2 = 2/x(t) = A2 sin(2t) x(t) = A1 sin(1t) ) yx yx ) t t tTt2 Ty(t) = B1 sin(1t + 1) y(t) = B2 sin(2t + 2) Pc. 4.10 cepeaoe opeeee acox xapaepc:

a oe; xoo ca aco 1; xoo ca aco 2;

xoo ca aco 1; xoo ca aco i Im() / 4 / Re() M = 1,M = = - = 0, = 0,Pc. 4.11 oopa AX Bxoo ca y(t) paccapaeo cyae oo aca, coy p cyepo, a cyy pex cao y1(t) = 22sin(0,5t /2);

y2(t) = 33sin(0,1t + /2 /4);

y3(t)= 1,50,8sin(10t 3/2);

y(t) = 4sin(0,5t /2) + 9 sin(0,1t /4) 1,2 sin(10t (3/2)).

4.6 MHMAHO-AOBE CCTEM Ayo-aoy xapaepcy cce oo aca e e (4.8), a, ocooac eopeo ey, a A A B B m ) (i - q j j=W (i) = k, (4.18) n ) (i - s j j=e qj y, a sj - oc epeaoo y.

ce y (4.18) pecae coo poeee cooee (i qj ).

eoepec a paoc ec eopo, aao oopoo e oe qj, a oe a o oc oe i (pc. 4.12). Cpaee yx eopo(i qj) (i qj), o oopx qj e eo oyococ xapaepyec ao, a pyo qj pao oyococ xapaepyec ao, oaae, o p oo o e oye cea <, .e. eopa, eaeo eo oyococ, aa ee.

i Im() i Im() ) a) i i q i qj q qj Re() Re() Pc. 4.12 opeee ao-aox cce Cce (e), ce y oca epeaox y oopx ea eo oyococ (ecea ac ye oco ec opaeo eo Re qj < 0; Re sj < 0), aac uuao-aou.

Cce (e), y oopx xo o y oc epeaoo y e pao oyococ (ecea ac ye, oco ec ooeo eo Re qj > 0; Re sj > 0), aac euuao-aou.

Moo oaa, o ao-aox ee cyecy acoc:

1 Im() Re() = - du;

u - 1 Re() Im() = - du; (4.19) u - 1 dL () = - cth d, d - u e L(u) = ln M(u); = ln ; u - epeea eppoa.

acoc oaa, o ayo-aoa xapaepca aoaoo cce (ea) ooc opeeec ee BX, MX AX. o ooe aeo ypoc aa aaa cea paccapaex cce, opaac yee x BX AX.

Heao-aoy ccey pocee cyae oo peca e oceoaeoo coee ao-aoo cce ea, eeo o y pao oyococ , cooeceo, xapaepyeoc AX:

i - q q - i j W (i) = = e. (4.20) i + q q + i Ayo-acoa xapaepca oo ea M() = 1, a ao-acoa () = - arctg. Ta opao, paccapaeoe eo coxpae ayy xooo q apoecoo caa pao aye xooo caa p o acoe, aa e p ee aco o 0 o eec epae o o 0, .e. ee ea c AX W(i) po oae ooeoo ca a (), oop p i pae yeaec p opaca aco.

ooe e a pae coyc oppepoa aox xapaepc ee, oe ycooc ..

4.7 OHTE O OAPMECX ACTOTHX XAPATEPCTAX poe paccapaex e acox xapaepc, oa coy, a aaee, oapece acoe xapaepc (X). x oye paee AX (4.15) acaec e bm (i)m +... +b0 bW (i) = = k0M0()ei() a0 an (i)n +... +a oappyec lgW (i) = lg k0 + lg M0() + i()lg e.

oe ooe yx e coyec oapeca ea ee.

C ey co ee S eoop co N aec opyo S = 20lg N = LmN.

Xapaepca L() = Lm[k0M0()] = Lmk0 + LmM0() = 20lg M () (4.21) aaec oapeco ayo acoo xapaepco (AX).

p ocpoe oapecx acox xapaepc o oc accc oaaec acoa oapeco acae lg, ooy oapeca aya acoa xapaepca cpoc oopaax L(); lg, oapeca aoa acoa xapaepca (X) - (); lg (pc. 4.13). oapece acoe xapaepc aa ae apaa oe.

L a) ) 20 lg k 0 lg lg /Pc. 4.13 oapece acoe xapaepc:

a AX; X 4.8 BAMOCB HAMECX XAPATEPCT Ocoo aeco xapaepco oea cce ec epeaoe ypaee. poe eo oy pec:

1) epeaoa y;

2) acoe xapaepc: ayo-acoa, ao-acoa, ayoaoa;

3) epexoe xapaepc: epexoa y, ecoa y.

a x xapaepc oe opeeea, ec eco epeaoe ypaee oea. Ho, ecop a o, ceye ee pa ocaoc a x aoc.

B aece pepa paccop aoc ey epexoo ye py xapaepca.

Ec eca epexoa y h(t), o o opye (3.39) opeeec epeaoa y oea W(s) = s h(s), aeo s = i oopo, co oepe, oy oye acoe xapaepc: W(i) = (i) h(i).

Ta a (t) ec pooo o eo cyeao y, o ex cce ecoa y ec pooo o epexoo y, .e. w(t) = h(t).

epeaoe ypaee o cepeao co po paoa oya c oo pax eo, oox opee eo oe.

C ey oco xapaepca peea a. 4.1.

Taa 4.Bae cooec aecx xapaepc epeaoe an y n (t) + an-1y n-1 (t) +... + a1y (t) + a0 y(t) = ypaee = bmx m (t) + bm-1x m-1 (t) +... + b1x (t) + b0x(t) p yex A(s) Y(s) = B(s) X(s) aax ycox A(s) = ansn + an-1sn-1 +...+ a1s + a0; B(s) = bmsm + bm-1sm-1 +...+ b1s + bcxoe epeae aoe W(s) h(t) w(t) Xapaypaee epca B(s) epeaoa W (s) = W(s) = s h(s) W(s) = w(s) y W(s) A(s) W (i) = B(i) ocaa W (i) = AX W(i) W(i) = i h(i) -it A(i) s = i = dt w(t)e t W (s) Peee .

epexoa h(t) = L-h(t) = ypae p s w()d y h(t) x(t) = 1(t) Becoa y Peee .

w(t) = L-1{W (s)} w(t) = h(t) w(t) ypae p p aae aecx xapaepc o oax opoco ec opeeee oea yce oea, o oop oa ooee xoo epeeo xoo ycaoec pee:

y() K =, (4.22) A o, a a y() = lim y(t), o t lim y(t) t K =.

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