Ndtis gdr Shkneas enj enj Systifs Wirshdn 5.<
[hasdn B. Pukm
Zmisi ndtis wiri jiviadpij gdr usi hn 90<.05:, Shkneas enj Systifs, Jipertfint dg Iaictrhcea enj Cdfputir Inkhniirhnk, Bdmns Mdpohns Ynhvirshty, dvir tmi pirhdj 0<<< – 0<<9. Es hnjhcetij `y tmi Ze`ai dg Cdntints, tmi ndtis cdvir trejhthdnea, hntrdjuctdry cdncipts hn tmi thfi jdfehn enj griquincy jdfehn eneayshs dg shkneas enj systifs. Ndt es cdfpaiti dr pdahsmij es e `ddo, tmdukm pirmeps su`bict td gurtmir jiviadpfint, tmisi ndtis eri dggirij dn en es hs dr usi et ydur dwn rhso `eshs. `eshs. \ririquhshtis gdr tmi fetirhea eri tmi erhtmfithc dg cdfpaix nuf`irs, jhggirinthea enj hntikrea ceacuaus, enj e cdursi hn iaictrhcea chrcuhts. (Chrcuhts eri usij es ixefpais hn tmi fetirhea, enj tmi aest sicthdn triets chrcuhts `y Aepaeci trensgdrf.) Cdncurrint stujy dg fuathverhe`ai ceacuaus hs miapgua, gdr dn dcceshdn e jdu`ai hntikrea dr perthea jirhvethvi eppiers. E cdursi hn jhggirinthea iquethdns hs ndt riquhrij, tmdukm sdfi viry shfpai jhggirinthea iquethdns eppier hn tmi fetirhea. Zmi fetirhea hncaujis ahnos td jifdnstrethdns jifdnstrethdns dg verhdus cdncipts. Zmisi enj dtmir jifdnstrethdns jifdnstrethdns cen `i gdunj et mttp2//www.bmu.iju/~shkneas/ . Ifeha cdffints td rukmLbmu.iju eri wiacdfi.
Cdpyrhkmt 0<<< -0<<9, Bdmns Mdpohns Ynhvirshty enj [hasdn B. Pukm, eaa rhkmts risirvij. Ysi dg tmhs fetirhea hs pirfhttij gdr pirsdnea dr ndn-prdght ndn-prdght ijucethdnea purpdsis purpdsis dnay. Ysi dg tmhs fetirhea gdr `ushniss dr cdffirchea purpdsis hs prdmh`htij.
Ndtis gdr Shkneas enj Systifs Ze`ai dg Cdntints
……………………………………………………………………………: Hntrdjucthdn <.5. Hntrdjuctdry Cdffints <.0. @ecokrdunj hn Cdfpaix Erhtmfithc <.3. Eneayshs @ecokrdunj Ixirchsis
<.
5.
Shkneas ……………………………………………………………………… ……………5<
5.5. Fetmifethcea Jighnhthdns dg Shkneas 5.0. Iaifintery Dpirethdns dn Shkneas 5.3. Iaifintery Dpirethdns dn tmi Hnjipinjint Werhe`ai 5.:. Inirky enj \dwir Caesshghcethdns 5.9. Syffitry-@esij Caesshghcethdns dg Shkneas 5.7. Ejjhthdnea Caesshghcethdns dg Shkneas 5.;. Jhscriti-Zhfi Shkneas2 Jighnhthdns, Caesshghcethdns, enj Dpirethdns Ixirchsis 0.
Cdnthnudus-Zhfi Shknea Caessis ………………………………………………………..03
0.5. Cdnthnudus-Zhfi Ixpdninthea Shkneas 0.0. Cdnthnudus-Zhfi Shnkuaerhty Shkneas 0.3. Kinireahzij Ceacuaus Ixirchsis 3.
Jhscriti-Zhfi Shknea Caessis ………………………………………………..…………..3;
3.5. Jhscriti-Zhfi Ixpdninthea Shkneas 3.0. Jhscriti-Zhfi Shnkuaerhty Shkneas Ixirchsis :.
Systifs ………………………………………………………..………………………….:3
:.5. Hntrdjucthdn td Systifs :.0. Systif \rdpirthis :.3. Hntircdnnicthdns dg Systifs Ixirchsis 9.
Jhscriti-Zhfi AZH Systifs ………………………………………….…………………..9<
9.5. JZ AZH Systifs enj Cdnvdauthdn 9.0. \rdpirthis dg Cdnvdauthdn - Hntircdnnicthdns dg JZ AZH Systifs 9.3. JZ AZH Systif \rdpirthis 9.:. Pispdnsi td Shnkuaerhty Shkneas 9.9. Pispdnsi td Ixpdnintheas (Ihkinguncthdn \rdpirthis) 9.7. JZ AZH Systifs Jiscrh`ij `y Ahnier Jhggirinci Iquethdns Ixirchsis 7.
Cdnthnudus-Zhfi AZH Systifs ……………………………………………….………….7=
7.5. CZ AZH Systifs enj Cdnvdauthdn
0
7.0. \rdpirthis dg Cdnvdauthdn - Hntircdnnicthdns dg JZ AZH Systifs 7.3. CZ AZH Systif \rdpirthis 7.:. Pispdnsi td Shnkuaerhty Shkneas 7.9. Pispdnsi td Ixpdnintheas (Ihkinguncthdn \rdpirthis) 7.7. CZ AZH Systifs Jiscrh`ij `y Ahnier Jhggirinci Iquethdns Ixirchsis ;.
Hntrdjucthdn td Shknea Piprisintethdn ……………………………………………………=0
;.5. Hntrdjucthdn td CZ Shknea Piprisintethdn ;.0. Drtmdkdneahty enj Fhnhfuf HSI Piprisintethdn ;.3. Cdfpaix @eshs Shkneas ;.:. JZ Shknea Piprisintethdn Ixirchsis =.
\irhdjhc CZ Shknea Piprisintethdn (Gdurhir Sirhis) …………………………………….80
=.5. CZ Gdurhir Sirhis =.0. Piea Gdrfs, Spictre, enj Cdnvirkinci =.3. Dpirethdns dn Shkneas =.:. CZ AZH Griquincy Pispdnsi enj Ghatirhnk Ixirchsis 8. \irhdjhc JZ Shknea Piprisintethdn (Gdurhir Sirhis) ……………………………….…….5<7
8.5. JZ Gdurhir Sirhis 8.0. Piea Gdrfs, Spictre, enj Cdnvirkinci 8.3. Dpirethdns dn Shkneas 8.:. JZ AZH Griquincy Pispdnsi enj Ghatirhnk Ixirchsis 5<. Gdurhir Zrensgdrf Piprisintethdn gdr CZ Shkneas …………………………..…………55=
5<.5. Hntrdjucthdn td CZ Gdurhir Zrensgdrf 5<.0. Gdurhir Zrensgdrf gdr \irhdjhc Shkneas 5<.3. \rdpirthis dg Gdurhir Zrensgdrf 5<.:. Cdnvdauthdn \rdpirty enj AZH Griquincy Pispdnsi 5<.9. Ejjhthdnea Gdurhir Zrensgdrf \rdpirthis 5<.7. Hnvirsi Gdurhir Zrensgdrf 5<.;. Gdurhir Zrensgdrf enj AZH Systifs Jiscrh`ij `y Jhggirinthea Iquethdns 5<.=. Gdurhir Zrensgdrf enj Hntircdnnicthdns dg AZH Systifs Ixirchsis 55. Ynhaetirea Aepaeci Zrensgdrf ……………………………………………………………5:3
55.5. Hntrdjucthdn 55.0. \rdpirthis dg tmi Aepaeci Zrensgdrf 55.3. Hnvirsi Zrensgdrf 55.:. Systifs Jiscrh`ij `y Jhggirinthea Iquethdns 55.9. Hntrdjucthdn td Aepaeci Zrensgdrf Eneayshs dg Systifs Ixirchsis 50. Eppahcethdn td Chrcuhts ………………………………………………………..………….597
50.5. Chrcuhts whtm Rird Hnhthea Cdnjhthdns 50.0. Chrcuhts whtm Ndnzird Hnhthea Cdnjhthdns Ixirchsis
3
Ndtis gdr Shkneas enj Systifs <.5 Hntrdjuctdry Cdffints
[met hs ‑Shkneas enj Systifs4‖ Iesy, `ut pirmeps unmiapgua enswirs, hncauji • tmi ε enj tmi ψ , ψ , • tmi quisthdn enj tmi enswir, • tmi givir enj tmi curi, • ceacuaus enj cdfpaix erhtmfithc gdr gun enj prdght, Fdri sirhdusay, shkneas eri guncthdns dg thfi (cdnthnudus-thfi shkneas) dr siquincis hn thfi (jhscriti-thfi shkneas) tmet prisufe`ay riprisint quenththis dg hntirist. Systifs eri dpiretdrs tmet eccipt e khvin shknea (tmi hnput shknea) enj prdjuci e niw shknea (tmi (t mi dutput shknea). Dg cdursi, tmhs hs en e`strecthdn dg tmi prdcisshnk dg e shknea. Grdf e fdri kinirea vhiwpdhnt, systifs eri shfpay guncthdns tmet mevi jdfehn enj renki tmet eri sits dg guncthdns dg thfi (dr siquincis hn thfi). Ht hs trejhthdnea td usi e genchir tirf sucm es dr fepphnk hn paeci dg guncthdn, td jiscrh`i sucm e shtuethdn. Mdwivir wi whaa ndt `i sd dpiretdr dr gdrfea whtm dur vhiwpdhnts dr tirfhndadkhis. Shfpay rifif`ir tmet shkneas eri e`strecthdns dg thfi-veryhnk quenththis dg hntirist, enj systifs eri e`strecthdns dg prdcissis tmet fdjhgy tmisi quenththis td prdjuci niw thfi-veryhnk quenththis dg hntirist. Zmisi ndtis eri e`dut tmi fetmifethcea riprisintethdn dg shkneas enj systifs. Zmi fdst hfpdrtent riprisintethdns wi hntrdjuci hnvdavi tmi griquincy jdfehn – e jhggirint wey dg addohnk et shkneas enj systifs, enj e cdfpaifint td tmi thfi-jdfehn vhiwpdhnt. Hnjiij inkhniirs enj schinthsts dgtin tmhno dg shkneas hn tirfs dg griquincy cdntint, enj systifs hn tirfs dg tmihr iggict dn tmi griquincy cdntint dg tmi hnput shknea. Sdfi dg tmi essdchetij fetmifethcea cdncipts enj fenhpuaethdns hnvdavij eri cmeaainkhnk, `ut tmi fetmifethcs aiejs td e niw wey dg addohnk et tmi wdraj! <.0 @ecokrdunj hn Cdfpaix Erhtmfithc Erhtmfithc
[i essufi iesy gefhaherhty whtm tmi erhtmfithc dg cdfpaix nuf`irs. Hn perthcuaer, tmi pdaer gdrf dg e cdfpaix nuf`ir c , wrhttin es b c c 6| c | i ∬ hs fdst cdnvinhint gdr fuathpahcethdn enj jhvhshdn, i.k., b∬c b∬c0 6 | c5 | | c0 | i b (∬c5 + ∬c0 ) c5 c0 6 | c5 | i 5 | c0 | i Zmi rictenkuaer gdrf gdr c , wrhttin c 6 e + b` wmiri e enj ` eri riea nuf`irs, hs fdst cdnvinhint gdr ejjhthdn enj su`trecthdn, i.k., c5 + c 0 6 e5 + b`5 + e0 + b`0 6 ( e5 + e0 ) + b(`5 + `0 ) Dg cdursi, cdnnicthdns `itwiin tmi twd gdrfs dg e cdfpaix nuf`ir c hncauji | c | 6 | e + b` | 6 e 0 + `0 , ∬c 6 ∬( e + b`) 6 ten ∐5( ` / e) enj, tmi dtmir wey rdunj,
:
e 6 Pi{c} 6 | c | cds(∬c) , ` 6 Hf{c} 6 | c | shn( ∬c) Ndti ispicheaay tmet tmi quejrent ef`hkuhty dg tmi hnvirsi tenkint fust `i risdavij hn feohnk tmisi cdfputethdns. Gdr ixefpai, ∬(5 ∐ b) 6 ten ∐5( ∐5/ 5 / 5) 6 ∐ ό / : wmhai ∬( ∐5 + b ) 6 ten ∐5(5 /( ∐5)) 6 3ό / : Ht hs hfpdrtent td `i e`ai td finteaay cdfputi tmi shni, cdshni, enj tenkint dg enkais tmet eri hntikir fuathpais dg ό / : , shnci feny prd`aifs whaa `i sit up tmhs wey td evdhj tmi jhstrecthdn dg ceacuaetdrs.
Xdu smduaj easd `i gefhaher whtm Iuair‘s gdrfuae, i bν 6 cds(ν ) + b shn(ν ) enj tmi cdfpaix ixpdninthea riprisintethdn gdr trhkdndfitrhc guncthdns2 bν bν ∐ bν ∐ bν i +i i ∐i cds(ν ) 6 , shn(ν ) 6 0 0 b
Ndthdns dg cdfpaix nuf`irs ixtinj td ndthdns dg cdfpaix-veauij guncthdns (dg e riea verhe`ai) hn tmi d`vhdus wey. Gdr ixefpai, wi cen tmhno dg e cdfpaix-veauij guncthdn dg thfi, x(t ) , hn tmi rictenkuaer gdrf x(t ) 6 Pi { x(t )} + b Hf { x(t )} Hn e shfpair ndtethdn tmhs cen `i wrhttin es x(t ) 6 x P (t ) + b xH (t ) wmiri x P (t ) enj x H (t ) eri riea-veauij guncthdns dg t . Dr wi cen cdnshjir pdaer gdrf, b ∬x (t ) x(t ) 6| x(t ) | i wmiri | x(t ) | enj ∬ x(t ) eri riea-veauij guncthdns dg t (whtm, dg cdursi, | x(t ) | ndnnikethvi gdr eaa t ). Hn tirfs dg tmisi gdrfs, fuathpahcethdn enj ejjhthdn dg cdfpaix guncthdns cen `i cerrhij dut hn tmi d`vhdus wey, whtm pdaer gdrf fdst cdnvinhint gdr fuathpahcethdn enj rictenkuaer gdrf fdst cdnvinhint gdr ejjhthdn. Hn eaa cesis, shkneas wi incduntir eri guncthdns dg tmi riea verhe`ai t . Zmet hs, wmhai shkneas tmet eri cdfpaix-veauij guncthdns dg t , dr sdfi dtmir riea verhe`ai, whaa erhsi es fetmifethcea cdnvinhincis, wi whaa ndt jiea whtm guncthdns dg e cdfpaix verhe`ai untha nier tmi inj dg tmi cdursi. <.3 Eneayshs @ecokrdunj
[i whaa usi tmi ndtethdn x_ nV gdr e riea dr cdfpaix-veauij siquinci (jhscriti-thfi shknea) jighnij gdr hntikir veauis dg n. Zmhs ndtethdn hs hntinjij td ifpmeshzi tmi shfhaerhty dg dur trietfint dg guncthdns dg e cdnthnudus verhe`ai (thfi) enj dur trietfint dg siquincis (hn thfi). @ut usi dg tmi squeri `recoits hs hntinjij td rifhnj us tmet tmi shfhaerhty smduaj ndt `i dvirjdni! Suffethdn ndtethdn, gdr ixefpai,
9
3
∕ x_o V 6 x_5V + x_0V + x_3V o 65
hs ixtinshviay usij. Dg cdursi, ejjhthdn hs cdffutethvi, enj sd wi cdncauji tmet 3
5
∕ x_o V 6 ∕ x_ o V o 65 o 6 3 Ceri fust `i ixirchsij hn cdnsuathnk dtmir rigirincis shnci sdfi usi tmi cdnvinthdn tmet e suffethdn hs zird hg tmi uppir ahfht hs aiss tmen tmi adwir ahfht. Enj dg cdursi tmhs suffethdn ahfht rivirsea hs ndt td `i cdngusij whtm tmi hntikrea ahfht rivirsea gdrfuae2 3
5
5
3
∯ x(t ) jt 6 ∐ ∯ x(t ) jt
Ht hs hfpdrtent td feneki suffethdn hnjhcis td evdhj cdaahshdns. Gdr ixefpai, z_ o V
3
∕ x_ o V o 65
hs ndt tmi sefi tmhnk es 3
∕ z_o V x_ o V o 65
@ut ht hs tmi sefi tmhnk es 3
∕ z_o V x_ bV b 65
Eaa tmisi d`sirvethdns eri hnvdavij hn cmenkis dg verhe`ais dg suffethdn. E typhcea cesi hs 3
∕ x_n ∐ o V o 65
Ait b 6 n ∐ o (riayhnk dn cdntixt td jhsthnkuhsm tmi niw hnjix grdf tmi hfekhnery unht b ) td riwrhti tmi suf es n ∐3 n ∐5 6 x b _ V ∕ ∕ x_ bV b 6 n ∐5 b 6 n ∐3 Sdfithfis wi whaa incduntir fuathpai suffethdns, dgtin es e risuat dg e prdjuct dg suffethdns, gdr ixefpai, 9 : ⎞ : 9 ⎟ : ⎞⎟ 9 x o z b x o z b 6 6 ⎑ ∕ _ V ⎔ ⎑⎑ ∕ _ V ⎔⎔ ∕ ∕ _ V _ V ∕ ∕ x_ oV z_ bV b 6 < o 65 ⎖ o 65 ⎬ ⎖ b 6< ⎬ o 65 b 6<
Zmi drjir dg suffethdns miri hs hffetirhea. @ut, ekehn, addo emiej td `i suri td evdhj hnjix cdaahshdns `y cmenkhnk hnjix nefis wmin niijij. Gdr ixefpai, wrhti
⎞ ⎟ : ⎞⎟ 9 ⎞ ⎟ : ⎞⎟ 9 x o z o x o z b 6 _ V _ V _ V _ V ⎑ ⎔ ∕ ∕ ∕ ∕ ⎑ ⎔⎑ ⎔ ⎑ ⎔⎑ ⎔ ⎖ o 65 ⎬ ⎖ o 6< ⎬ ⎖ o 65 ⎬ ⎖ b 6< ⎬ `igdri prdciijhnk es e`dvi. Zmisi cdnshjirethdns easd erhsi, hn sahkmtay jhggirint gdrf, wmin hntikrea ixprisshdns eri fenhpuaetij. Gdr ixefpai, cmenkhnk tmi verhe`ai dg hntikrethdn hn tmi ixprisshdn
7
t
∯ x(t ∐ ϊ ) j ϊ <
td σ 6 t ∐ ϊ khvis <
t
∯ x(σ ) ( ∐jσ ) 6 ∯ x(σ ) j σ
<
t
[i incduntir fuathpai hntikreas dn reri dcceshdns, usueaay es e risuat dg e prdjuct dg hntikreas, enj cdaahshdns dg hntikrethdn verhe`ais fust `i evdhjij `y rinefhnk. Gdr ixefpai,
⎟3 ⎞⎟ 3 ⎞ ⎟3 ⎞⎟ 3 ⎞ ⎑⎑ ∯ x(t ) jt ⎔⎔ ⎑⎑ ∯ z (t ) jt ⎔⎔ 6 ⎑⎑ ∯ x(t ) jt ⎔⎔ ⎑⎑ ∯ z(ϊ ) j ϊ ⎔⎔ ⎖< ⎬ ⎖ ∐5 ⎬ ⎖< ⎬ ⎖ ∐5 ⎬ 33
6 ∯ ∯ x(t ) z(ϊ ) jt j ϊ < ∐5
Zmi Gunjefintea Zmidrif dg Ceacuaus erhsis griquintay2 j t
∯ x(ϊ ) jϊ 6 x(t ) jt ∐∞ Gdr ghnhti sufs, dr hntikreas dg wiaa-`imevij (i.k. cdnthnudus) guncthdns whtm ghnhti hntikrethdn ahfhts, tmiri eri nd perthcuaer ticmnhcea cdncirns e`dut ixhstinci dg tmi suf dr hntikrea, dr hntircmenki dg drjir dg hntikrethdn dr suffethdn. Mdwivir, gdr hnghnhti sufs dr hfprdpir hntikreas (dvir en hnghnhti renki) wi smduaj `i cdncirnij e`dut cdnvirkinci enj tmin e`dut verhdus fenhpuaethdns hnvdavhnk cmenki dg drjir dg dpirethdns. Mdwivir, wi whaa `i e `ht ceveahir e`dut tmhs. Gdr suffethdns sucm es ∞ ∕ x_o V o 6∐∞ e retmir d`vhdus nicissery cdnjhthdn gdr cdnvirkinci hs tmet | x_o V | ← < es o ← ← µ ∞ . Zyphceaay wi whaa ndt wdrry e`dut kinirea sugghchint cdnjhthdns, retmir wi aievi cdnshjirethdn dg cdnvirkinci td perthcuaer cesis. Gdr hntikreas sucm es ∞
∯ x(t ) jt ∐∞
← µ ∞ , `ut ekehn en d`vhdus nicissery cdnjhthdn gdr cdnvirkinci hs tmet | x (t ) | ← < es t ← gurtmir jitehas whaa `i hkndrij. [i ispicheaay whaa hkndri cdnjhthdns unjir wmhcm tmi drjir dg e jdu`ai (hnghnhti) suffethdn cen `i hntircmenkij, h ntircmenkij, dr tmi drjir dg e jdu`ai (hfprdpir) hntikrea cen `i hntircmenkij. Hnjiij, feny dg tmi fetmifethcea fekhc trhcos tmet eppier hn dur su`bict eri ixpaehne`ai dnay `y teohnk e viry rhkdrdus vhiw dg tmisi hssuis. Sucm rhkdr hs `iydnj dur scdpi. Gdr cdfpaix-veauij guncthdns dg thfi, dpirethdns sucm es jhggirinthethdn enj hntikrethdn eri cerrhij dut hn tmi usuea gesmhdn whtm b vhiwij es e cdnstent. Ht sdfithfis miaps td tmhno dg tmi guncthdn hn rictenkuaer gdrf td busthgy tmhs vhiw2 gdr ixefpai, hg x(t ) 6 x P (t ) + b xH (t ) , tmin
;
t
t
∯ x(ϊ ) jϊ 6 ∯ ∐∞
∐∞
x P (ϊ ) jϊ + b
t
∯
∐∞
xH (ϊ ) j ϊ
Shfhaer cdffints eppay td cdfpaix suffethdns enj siquincis. \etmdadkhis tmet sdfithfis erhsi hn tmi ceacuaus, sucm es ivirywmiri cdnthnudus `ut ndwmiri jhggirinthe`ai guncthdns (shkneas), eri dg nd hntirist td us! Dn tmi dtmir menj, cirtehn kinireahzij ndthdns dg guncthdns, perthcuaeray tmi hfpuasi guncthdn, whaa `i viry usigua gdr riprisinthnk spichea typis dg shkneas enj systifs. @iceusi wi jd ndt prdvhji e cerigua fetmifethcea `ecokrdunj gdr kinireahzij guncthdns, wi whaa teoi e viry gdrfuaehc epprdecm td wdrohnk whtm tmif. Hfpuasi guncthdns eshji, gussy fettirs sucm es shkneas tmet mevi hncdnvinhint veauis et hsdaetij pdhnts whaa `i menjaij hngdrfeaay `y shfpay ejbusthnk veauis td ecmhivi cdnvinhinci. Ixefpai Cdnshjir tmi guncthdn
⎫5, t 6 < ⎨ <, iasi
x(t ) 6 ⎭
Cirtehnay tmi hntikrea dg x(t ) `itwiin eny twd ahfhts, hs zird – tmiri `ihnk nd erie unjir e shnkai pdhnt. Zmi jirhvethvi dg x(t ) hs zird gdr eny t ≬ ≬ < , `ut tmi jirhvethvi hs unjighnij et t 6 <, tmiri `ihnk nd riesdne`ai ndthdn dg ‑sadpi.‖ Mdw jd wi jiea whtm tmhs4 Zmi enswir hs td t d vhiw x(t ) es iquhveaint td tmi hjinthceaay-zird guncthdn. Hnjiij, wi whaa mepphay ejbust tmi veaui dg e guncthdn et hsdaetij veauis dg t gdr gdr purpdsis dg cdnvinhinci enj shfpahchty. Hn e shfhaer gesmhdn, cdnshjir
⎫5, ⎨<,
u (t ) 6 ⎭
t > < t 1 <
wmhcm prd`e`ay hs gefhaher es tmi unht-stip guncthdn. [met veaui smduaj wi esshkn td u (<) 4 Ekehn, tmi enswir hs tmet wi cmddsi u (<) gdr cdnvinhinci. Gdr sdfi purpdsis, sitthnk u (<) 6 5 / 0 hs fdst suhte`ai, gdr dtmir purpdsis u (<) 6 5 hs `ist. @ut hn iviry hnstenci wi griiay cmddsi tmi veaui dg u (<) td ght tmi purpdsi et menj. Zmi jirhvethvi dg u (t ) hs zird gdr eaa t ≬ ≬ < , `ut hs unjighnij hn tmi usuea ceacuaus sinsi et t 6 6 < . Mdwivir tmiri hs en hntuhthvi ndthdn tmet e bufp upwerj mes hnghnhti sadpi (enj e bufp jdwnwerj mes sadpi ∐∞ ). [i whaa cepturi tmhs ndthdn ushnk kinireahzij guncthdns enj e ndthdn dg kinireahzij ceacuaus hn tmi siquia. @y cdfperhsdn, tmi shknea x(t ) hn tmi ixefpai e`dvi iggicthviay ixmh`hts twd shfuatenidus bufps, enj tmiri hs ahttai eatirnethvi tmen td shfpahgy x(t ) td tmi zird shknea. Ixcipt gdr kinireahzij guncthdns, td `i jhscussij hn tmi siquia, wi typhceaay wdro hn tmi cdntixt dg phiciwhsi-cdnthnudus guncthdns, enj pirfht dnay shfpai, ghnhti bufps es jhscdnthnuhthis. Ixirchsis b (5+ b )
5. Cdfputi tmi pdaer gdrf dg tmi cdfpaix nuf`irs i
0. Cdfputi tmi rictenkuaer gdrf dg tmi cdfpaix nuf`irs
=
enj (5 + b )i∐ bό / 0 .
0 i b 9ό / : enj i∐ bό + i b 7ό .
| ( 0 ∐ b 0 )3 | enj tmi enkai ∬ ( ∐5 ∐ b ) 0 .
3. Iveaueti, tmi iesy wey, tmi feknhtuji bν
:. Yshnk Iuair's riaethdn, i
6 cds ν + b shn ν , jirhvi tmi ixprisshdn cdsν 6 50 i bν + 50 i ∐ bν
9. Hg z5 enj z0 eri cdfpaix nuf`irs, enj e ster jindtis cdfpaix cdnbuketi, ixpriss tmi
gdaadwhnk quenththis hn tirfs dg tmi riea enj hfekhnery perts dg z5 enj z0 2
Pi_ z5 ∐ z5∛ V ,
Hf_ z5z0 V ,
Pi_ z5 / z0 V
7. [met hs tmi riaethdnsmhp efdnk tmi tmrii ixprisshdns `iadw4
∞
∞
∯ x(∐σ ) jσ , 0 ∯ x(0σ ) j σ
∯ x(σ ) jσ , ∐∞
∞
∐∞
∐∞
;. Shfpahgy tmi tmrii ixprisshdns `iadw. t j x(σ ) jσ jt
∯
<
,
j jt
<
∯ x(σ ) jσ ,
∐t
8
j jσ
<
∯ x(σ ) j σ
t
Ndtis gdr Shkneas enj Systifs 5.5 Fetmifethcea Jighnhthdns dg Shkneas
E cdnthnudus-thfi shknea hs e quenthty dg hntirist tmet jipinjs dn en hnjipinjint verhe`ai, wmiri wi usueaay tmhno dg tmi hnjipinjint verhe`ai es thfi. Zwd ixefpais eri tmi vdateki et e perthcuaer ndji hn en iaictrhcea chrcuht enj tmi rddf tifpireturi et e perthcuaer spdt, `dtm es guncthdns dg thfi. E fdri prichsi, fetmifethcea jighnhthdn hs tmi gdaadwhnk. E cdnthnudus-thfi jighnij gdr ∐∞ 1 . E cruji ∐∞ 1 t 1 ∞ . cdnthnudus-thfi shknea hs e guncthdn x(t ) dg tmi riea verhe`ai t jighnij t 1 ∞ riprisintethdn dg sucm e shknea hs e soitcm, es smdwn.
Dn paenit iertm, pmyshcea quenththis teoi dn riea nufirhcea veauis, tmdukm ht turns dut tmet sdfithfis ht hs fetmifethceaay cdnvinhint td cdnshjir cdfpaix-veauij guncthdns dg t . Mdwivir, tmi jigeuat hs riea-veauij x(t ) , enj hnjiij tmi typi dg soitcm ixmh`htij e`dvi hs veahj dnay gdr riea-veauij shkneas. E soitcm dg e cdfpaix-veauij shknea x(t ) riquhris en ejjhthdnea jhfinshdn dr fuathpai soitcmis, gdr ixefpai, e soitcm dg tmi riea pert, Pi{ x (t )} , virsus t enj e soitcm dg tmi hfekhnery pert, Hf{ x(t )} , virsus t . Piferos2
• •
• •
E cdnthnudus-thfi shknea hs ndt nicisserhay e cdnthnudus guncthdn, hn tmi sinsi dg ceacuaus. Jhscdnthnuhthis (bufps) hn e shknea eri hnjhcetij `y e virthcea ahni, es jrewn e`dvi. Zmi jigeuat jdfehn dg jighnhthdn hs eaweys tmi wmdai riea ahni – e cdnvinhint e`strecthdn tmet hkndris verhdus `hk-`enk tmidrhis. [i usi iaahpsis es smdwn e`dvi td hnjhceti tmet tmi shknea ‑cdnthnuis hn e shfhaer gesmhdn,‖ whtm tmi fienhnk prisufe`ay caier grdf cdntixt. Hg e shknea hs dg hntirist dnay dvir e perthcuaer hntirvea hn tmi riea ahni, tmin wi usueaay jighni ht td `i zird dutshji dg tmhs hntirvea sd tmet tmi jdfehn dg jighnhthdn rifehns tmi wmdai riea ahni. Dtmir cdnvinthdns eri pdssh`ai, dg cdursi. Hn sdfi cesis e shknea jighnij dn e ghnhti hntirvea hs ixtinjij td tmi wmdai riea ahni `y injaissay ripiethnk tmi shknea (hn `dtm jhricthdns). Zmi hnjipinjint verhe`ai niij ndt `i thfi, ht cduaj `i jhstenci, gdr ixefpai. @ut gdr shfpahchty wi whaa eaweys cdnshjir ht td `i thfi. En hfpdrtent su`caess dg shkneas hs tmi caess dg unhaetirea dr rhkmt-shjij shkneas shkneas tmet eri zird gdr nikethvi erkufints. Zmisi eri usij td riprisint shtuethdns wmiri tmiri hs e jighnhti sterthnk thfi, usueaay jishknetij t 6 6 < gdr cdnvinhinci.
E jhscriti-thfi shknea hs e siquinci dg veauis dg hntirist, wmiri tmi hntikir hnjix cen `i tmdukmt dg es e thfi hnjix, enj tmi veauis hn tmi siquinci riprisint sdfi pmyshcea quenthty dg hntirist. @iceusi feny jhscriti-thfi shkneas erhsi es iqueaay-specij sefpais dg e cdnthnudus-thfi shknea, ht hs dgtin fdri cdnvinhint td tmhno dg tmi hnjix es tmi ‑sefpai nuf`ir.‖ Ixefpais eri tmi cadshnk Jdw-Bdnis stdco evireki iecm jey enj tmi rddf tifpireturi et 7 pf iecm jey. Hn tmisi cesis, tmi sefpai nuf`ir wduaj `i jey <, jey 5, jey 0, enj sd dn.
5<
[i usi tmi gdaadwhnk fetmifethcea jighnhthdn. E jhscriti-thfi shknea hs e siquinci x_ nV jighnij gdr eaa hntikirs ∐∞ 1 . [i jhspaey x_ nV ∐∞ 1 n 1 ∞ . n 1 ∞ krepmhceaay es e strhnk dg adaaypdps dg epprdprheti mihkmt.
Dg cdursi tmiri hs nd cdncipt dg cdnthnuhty hn tmhs sitthnk. Mdwivir, eaa tmi riferos e`dut jdfehns dg jighnhthdn ixtinj td tmi jhscriti-thfi cesi hn tmi d`vhdus wey. Hn ejjhthdn, cdfpaixveauij jhscriti-thfi shkneas dgtin eri fetmifethceaay cdnvinhint, tmdukm tmi jigeuat essufpthdn hs tmet x_ nV hs e riea siquinci. Hn jui cdursi wi jhscuss cdnvirthnk e shknea grdf dni jdfehn td tmi dtmir – sefpahnk enj ricdnstructhdn, easd ceaaij eneadk-td-jhkhtea (E/J) enj jhkhtea-td-eneadk (J/E) cdnvirshdn. 5.0 Iaifintery Dpirethdns dn Shkneas
Sivirea `eshc dpirethdns `y wmhcm niw shkneas eri gdrfij grdf khvin shkneas eri gefhaher grdf tmi eaki`re enj ceacuaus dg guncthdns. ), wmiri e hs e riea (dr pdssh`ay cdfpaix) cdnstent • Efpahtuji Sceai2 y(t ) 6 e x(t ),
• Efpahtuji Smhgt 2 y(t ) 6 x(t )+ `, wmiri ` hs e riea (dr pdssh`ay cdfpaix) cdnstent • Ejjhthdn2 y(t ) 6 x(t) + z(t ) • Fuathpahcethdn2 y(t ) 6 x(t ) z(t ) [htm e cmenki hn vhiwpdhnt, tmisi dpirethdns cen `i vhiwij es shfpai ixefpais dg systifs, e tdphc jhscussij et ainktm hn tmi siquia. Hn perthcuaer, hg e enj ` eri essufij riea, enj z(t ) hs essufij td `i e ghxij, riea shknea, tmin iecm dpirethdn jiscrh`is e systif whtm hnput shknea x (t ) enj dutput shknea y (t ) . Zmhs vhiwpdhnt dgtin hs ndt perthcuaeray usigua gdr sucm shfpai shtuethdns, mdwivir. Zmi jiscrhpthdn dg tmisi dpirethdns gdr tmi cesi dg jhscriti-thfi shkneas hs cdfpaitiay eneadkdus. 5.3 Iaifintery Dpirethdns dn tmi Hnjipinjint Werhe`ai
Zrensgdrfethdns dg tmi hnjipinjint verhe`ai eri ejjhthdnea, `eshc dpirethdns `y wmhcm niw shkneas eri gdrfij grdf e khvin shknea. @iceusi tmisi hnvdavi tmi hnjipinjint verhe`ai, tmet hs, tmi erkufint ( t ), ), tmi dpirethdns sdfithfis su`tay hnvdavi dur custdfery ndtethdn gdr guncthdns. Zmisi dpirethdns cen `i vhiwij es sdfiwmet aiss shfpai ixefpais dg systifs, enj sdfithfis sucm en eatirneti vhiw hs ejdptij.
55
Es hs typhcea hn ceacuaus, wi usi tmi ndtethdn x(t ) td jindti `dtm tmi inthri shknea, enj tmi veaui dg tmi shknea et e veaui dg tmi hnjipinjint verhe`ai ceaaij t . Zmi hntirpritethdn jipinjs dn cdntixt. Zmhs hs shfpair tmen ejdpthnk e spichea ndtethdn, sucm es x(h) , td jiscrh`i tmi inthri shknea. Su`taithis tmet erhsi grdf dur juea vhiw whaa `i jhscussij hn tmi perthcuaer cdntixt.
•
Zhfi Sceai2 Suppdsi y (t ) 6 x( et ) wmiri e hs e riea cdnstent. @y soitcmhnk shfpai ixefpais,
ht `icdfis caier tmet hg e > 5, tmi risuat hs e thfi-cdfprissij shknea, enj hg < 1 e 1 5, tmi risuat hs thfi jhaethdn. Dg cdursi, tmi cesi e 6 < hs trhvhea, khvhnk tmi cdnstent shknea y (t ) 6 x(<) tmet hs dnay sahkmtay riaetij td x (t ) . Gdr e ≬ <, x(t ) cen `i ricdvirij grdf y (t ) . Zmet hs, tmi dpirethdn hs hnvirth`ai. Hg e 1 <, tmin tmiri hs e thfi rivirsea, hn ejjhthdn td cdfprisshdn dr jhaethdn. Zmi ricdffinjij epprdecm td soitcmhnk thfi-sceaij shkneas hs shfpay td iveaueti y (t ) gdr e siaicthdn dg veauis dg t untha untha tmi risuat `icdfis caier. Gdr ixefpai,
Ndthci tmet hn ejjhthdn td cdfprisshdn dr jhaethdn, tmi l`ikhnnhnk thfi‘ dr linjhnk thfi‘ dg e puasi-typi shknea whaa `i cmenkij hn tmi niw thfi sceai.
•
hs e riea cdnstent. Hg Z > < , tmi smhgt hs e rhkmt Zhfi Smhgt 2 Suppdsi y (t ) 6 x(t ∐ Z ) wmiri Z hs smhgt hn thfi, dr e thfi jiaey. Hg Z hs hs nikethvi, wi mevi e aigt smhgt, dr e thfi ejvenci. Gdr ixefpai,
50
•
Cdf`hnethdn Sceai enj Smhgt 2 Suppdsi y (t ) 6 x( et ∐ Z ) . Ht hs tifpthnk td tmhno e`dut tmhs es
twd dpirethdns hn siquinci -- e sceai gdaadwij `y e smhgt, dr e smhgt gdaadwij `y e sceai. Zmhs hs jenkirdus hn tmet e wrdnk cmdhci aiejs td hncdrrict enswirs. Zmi ricdffinjij epprdecm hs td hkndri smdrtcuts, enj ghkuri dut tmi risuat `y `ruti-gdrci krepmhcea fitmdjs2 su`sthtuti verhdus veauis dg t untha untha y (t ) `icdfis caier. Cdnthnuhnk tmi ixefpai,
Ixefpai2 Zmi fdst hfpdrtent sceai enj smhgt cdf`hnethdn gdr tmi siquia hs tmi cesi wmiri hs cmenkij td wrhti y (t ) 6 x(Z ∐ t ) . Zmhs hs eccdfpahsmij krepmhceaay e 6 ∐ 5 , enj tmi shkn dg Z hs
`y rivirshnk thfi enj tmin smhgthnk tmi rivirsij shknea Z unhts unhts td tmi rhkmt hg Z > < , dr td tmi aigt hg Z 1 < . [i rigir td tmhs trensgdrfethdn es tmi gahp enj smhgt. Gdr ixefpai,
Zmi gahp enj smhgt enj smhgt dpirethdn cen `i ixpadrij hn tmi eppait `iadw. Mdwivir, ydu smduaj s mduaj virhgy tmi hntirpritethdn dg tmi gahp enj smhgt `y menj soitcmis dg e giw ixefpais. gahp enj smhgt
5.: Inirky enj \dwir Caesshghcethdns
Zmi tdtea inirky dg e cdnthnudus-thfi shknea x(t ) , wmiri x (t ) hs jighnij gdr ∐∞ 1 ∐∞ 1 t 1 ∞ , hs t 1 ∞ Z ∞ 0 I∞ 6 ∯ x (t ) jt 6 ahf ∯ x 0 (t ) jt Z ←∞ ∐Z ∐∞ 53
Hn feny shtuethdns, tmhs quenthty hs prdpdrthdnea td e pmyshcea ndthdn dg inirky, gdr ixefpai, hg x (t ) hs tmi currint tmrdukm, dr vdateki ecrdss, e rishstdr. Hg e shknea mes ghnhti inirky, tmin tmi shknea veauis fust epprdecm zird es t epprdecmis epprdecmis pdshthvi enj nikethvi hnghnhty. Zmi thfi-evireki pdwir dg dg e shknea hs
5 Z 0 ∯ x (t ) jt Z ←∞ 0Z ∐Z
\∞ 6 ahf
Gdr ixefpai tmi cdnstent shknea x (t ) 6 5 (gdr eaa t ) mes thfi-evireki pdwir dg unhty. [htm tmisi jighnhthdns, wi cen paeci fdst, `ut ndt eaa, cdnthnudus-thfi shkneas hntd dni dg twd caessis2 • En inirky shknea hs e shknea whtm ghnhti I ∞ . Gdr ixefpai, x(t ) 6 i ∐|t | , enj, trhvheaay, x(t ) 6 eri inirky shkneas. Gdr en inirky shknea, \∞ 6 < . <, gdr eaa t eri
•
E pdwir shknea hs e shknea whtm ghnhti, ndnzird \∞ . En ixefpai hs x(t ) 65, gdr eaa t , tmdukm fdri hntiristhnk ixefpais eri ndt d`vhdus enj riquhri eneayshs. Gdr e pdwir shknea, I ∞ 6 ∞ .
Ixefpai Fdst wduaj suspict tmet x(t ) 6 shn(t ) hs ndt en inirky shknea, `ut hn eny cesi wi ghrst
cdfputi Z
0
Z
∯ shn (t ) jt 6 ∯
∐Z
( 50 ∐ 50 cds(0t) ) jt 6 Z ∐ 50 shn(0Z )
∐Z
Aitthnk Z ← ∞ cdnghrfs dur susphchdns, shnci tmi ahfht jdisn‘t ixhst. Zmi sicdnj stip dg tmi pdwir-shknea ceacuaethdn khvis
(
)
\∞ 6 ahf 5 Z ∐ 5 shn(0Z ) 6 5 0 0 Z ←∞ 0Z enj wi cdncauji tmet x(t ) hs e pdwir shknea. Ixefpai Zmi unht-stip guncthdn, jighnij `y
⎫5, ⎨<,
u (t ) 6 ⎭
t > < t 1 <
hs e pdwir shknea, shnci
ahfZ ←∞ Z5
Z /0
∯
∐Z / 0
0 u (t ) jt 6 ahfZ ←∞ 5
Z / 0
Z
∯ 5 jt
<
5 Z 6 5 6 ahfZ ←∞ Z 0 0 t
Ixefpai Zmiri eri shkneas tmet `iadnk td nihtmir dg tmisi caessis. Gdr ixefpai, x(t ) 6 i hs e
shknea whtm `dtm I ∞ enj \∞ hnghnhti. E fdri unusuea ixefpai hs ⎫⎢t ∐5/ 0 , t ≩ 5 x(t ) 6 ⎭ t 1 5 ⎨⎢ <, Zmhs shknea mes hnghnhti inirky `ut zird evireki pdwir.
5:
Zmi PFS ( (rddt-fien-squeri) veaui dg e pdwir shknea x(t ) hs jighnij es
\∞ . Zmisi inirky enj pdwir jighnhthdns easd cen `i usij gdr cdfpaix-veauij shkneas, hn wmhcm cesi
wi ripaeci x 0 (t ) `y | x (t ) |0 . 5.9 Syffitry-@esij Caesshghcethdns dg Shkneas
E shknea x(t ) hs ceaaij en ivin shknea hg x(∐t ) 6 x(t ) gdr eaa t . Hg x(∐t ) 6 ∐ x(t ) , gdr eaa t , tmin x (t ) hs ceaaij en djj shknea. Zmi ivin pert dg dg e shknea x(t ) hs jighnij es xiv (t ) 6
x(t ) + x( ∐t )
0
enj tmi djj pert dg dg x (t ) hs xdj (t ) 6
x(t ) ∐ x( ∐t )
0
Zmi ivin pert dg e shknea hs en ivin shknea, shnci x( ∐t) + x(t ) 6 xiv (t ) xiv ( ∐t ) 6
0
enj e shfhaer ceacuaethdn smdws tmet tmi djj pert dg e shknea hs en djj shknea. Easd, gdr eny shknea x(t ) wi cen wrhti e jicdfpdshthdn es x(t ) 6 xiv (t ) + xdj ( t )
Zmisi cdncipts eri fdst usigua gdr riea shkneas. Gdr cdfpaix-veauij shkneas, e syffitry cdncipt tmet sdfithfis erhsis hs cdnbuketi syffitry, cmerectirhzij `y ∛ x(t ) 6 x ( ∐t ) wmiri supirscrhpt ster jindtis cdfpaix cdnbuketi. 5.7 Ejjhthdnea Caesshghcethdns dg Shkneas
• @dunjijniss2 E shknea x(t ) hs ceaaij `dunjij hg hg tmiri hs e ghnhti cdnstent O sucm sucm tmet | x(t ) | ≪ O , gdr eaa t . (Miri tmi e`sdauti veaui hs hntirpritij es feknhtuji hg tmi shknea hs cdfpaix veauij.) Dtmirwhsi e shknea hs ceaaij un`dunjij . Zmet hs, e shknea hs un`dunjij hg nd sucm O ixhsts. ixhsts. Gdr ixefpai, x(t ) 6 shn(3t ) hs e `dunjij shknea, enj wi cen teoi O 6 6 5 . D`vhdusay, x(t ) 6 t shn(3t ) hs un`dunjij. sucm • \irhdjhchty2 E shknea x(t ) hs ceaaij pirhdjhc hg tmiri hs e pdshthvi cdnstent Z sucm tmet x(t ) 6 x(t + Z ) , gdr eaa t . Sucm e Z hs hs ceaaij e pirhdj dg dg tmi shknea, enj sdfithfis wi sey e shknea hs Z-pirhdjhc. Dg cdursi hg e pirhdjhc shknea mes pirhdj Z , tmin ht easd mes pirhdj 0Z, 3Z, enj sd dn. Zmi sfeaaist veaui dg Z gdr gdr wmhcm x(t ) 6 x(t + Z ) , gdr eaa t , hs ceaaij tmi gunjefintea dg tmi shknea, enj dgtin hs jindtij Z d . Ndti easd tmet e cdnstent shknea, x(t) 6 3, gdr pirhdj dg ixefpai, hs pirhdjhc whtm pirhdj eny Z > <, enj tmi gunjefintea pirhdj hs ndt wiaa jighnij (tmiri hs nd sfeaaist pdshthvi nuf`ir).
59
Ixefpais Zd jitirfhni pirhdjhchty dg tmi shknea x(t ) 6 shn(3t ) , enj tmi gunjefintea pirhdj Z d
hg pirhdjhc, wi eppay tmi pirhdjhchty cdnjhthdn shn(3(t + Z ) 6 shn(3t) , ∐ ∞ 1 t 1 ∞ Piwrhthnk tmhs es shn(3t + 3Z ) 6 shn(3t) , ∐ ∞ 1 t 1 ∞ ht hs caier tmet tmi cdnjhthdn mdajs hg enj dnay hg 3Z hs hs en hntikir fuathpai dg 0ό , tmet hs, Z hs e pdshthvi hntikir fuathpai dg 0ό / 3 . Zmus tmi shknea hs pirhdjhc, enj tmi gunjefintea pirhdj hs Z d 6 0ό / 3 . Es e sicdnj ixefpai, wi rikerj x(t ) 6 u(t ) + u( ∐t ) es pirhdjhc, `y essufhnk gdr cdnvinhinci tmi veaui u (<) 6 5 / 0 , `ut tmiri hs nd gunjefintea pirhdj. \irhdjhc shkneas eri en hfpdrtent su`caess dg eaa shkneas. \myshcea ixefpais hncauji tmi dcien thjis, en et-rist ICK, enj fushcea tdnis (`ut ndt tunis). Zyphceaay wi cdnshjir tmi pirhdj dg e pirhdjhc shknea td mevi unhts dg sicdnjs, enj tmi gunjefintea griquincy griquincy dg e pirhdjhc shknea hs jighnij `y 0ό ψ d 6 Z d whtm unhts dg rejhens/sicdnj . [i whaa usi rejhen griquincy tmrdukmdut, tmdukm sdfi dtmir sdurcis usi griquincy hn Mirtz, jindtij `y tmi syf`da g d . Zmi riaethdn `itwiin rejhen griquincy enj Mirtz hs ψ 5 g d 6 d 6 0ό Z d Zmi fehn jhggirinci tmet erhsis `itwiin tmi usi dg tmi twd griquincy unhts hnvdavis tmi paecifint dg 0ό gectdrs hn verhdus gdrfuaes. Khvin e ahtirea ixprisshdn gdr e shknea, sdavhnk tmi iquethdn x(t ) 6 x(t + Z ), gdr eaa t gdr tmi sfeaaist veaui dg Z , , hg dni ixhsts, cen `i er`htrerhay jhgghcuat. Sdfithfis tmi `ist epprdecm hs td padt dut tmi shknea enj try td jitirfhni pirhdjhchty enj tmi gunjefintea pirhdj `y hnspicthdn. Sucm e cdncaushdn hs ndt jighnhthvi, mdwivir, shnci tmiri eri shkneas tmet eri viry cadsi td, `ut ndt, pirhdjhc, enj tmhs cenndt `i jhscirnij grdf e soitcm. Ndti tmet tmi evireki pdwir pir pirhdj dg e Z- pirhdjhc pirhdjhc shknea x(t ) `icdfis, \Z 6 5
Z / 0
Z
∯
0
x (t ) jt
∐Z / 0
dr, fdri kinireaay, \Z 6 5
Z
0 ∯ x (t ) jt
Z
wmiri wi mevi hnjhcetij tmet tmi hntikrethdn cen `i pirgdrfij dvir eny hntirvea dg ainktm Z . Zd prdvi tmhs, gdr eny cdnstent t d cdnshjir td +Z 0 5 ∯ x (t ) jt Z
t d
57
enj pirgdrf tmi verhe`ai cmenki ϊ 6 t ∐ td ∐ Z / 0 . Zmi evireki pdwir pdwir pir pirhdj hs tmi sefi es tmi evireki pdwir dg tmi pirhdjhc shknea. Zmirigdri tmi PFS veaui dg e pirhdjhc shknea x(t ) hs 5 ⎞0
⎟5
0 \∞ 6 ⎑ ∯ x (t ) jt ⎔ ⎑ Z ⎔ ⎖ Z ⎬ Ixefpai Drjhnery mdusimdaj iaictrhcea pdwir hs suppahij es e 7< Mirtz shnusdhj whtm PFS veaui e`dut 55< vdats. Zmet hs, x (t ) 6 E cds(50<ό t )
enj tmi gunjefintea pirhdj hs Z d 6 5/7< sic. Zmi efpahtuji E hs sucm tmet
⎟ 5/7< 0 ⎞ 55< 6 ⎑ 7< ∯ E cds 0 (50<ό t) jt ⎔ ⎑ ⎔ < ⎖ ⎬ grdf wmhcm wi cdfputi E ≍ 5:< . 5.; Jhscriti-Zhfi Shkneas2 Jighnhthdns, Caesshghcethdns, enj Dpirethdns
Gdr jhscriti-thfi shkneas, x_n V , wi shfpay niij td cdnvirt tmi verhdus ndthdns grdf tmi sitthnk dg guncthdns td tmi sitthnk dg siquincis.
• Inirky enj \dwir 2 Zmi tdtea inirky dg e jhscriti-thfi shknea hs jighnij `y I∞ 6
∞
N
0
0
∕ x _nV 6 ahf ∕ x _ nV N ←∞ n 6 ∐ N n 6 ∐∞
Zmi thfi-evireki pdwir hs N
5 \∞ 6 ahf ∕ x 0_ nV N 0 5 + N ←∞ n 6∐ N enj jhscriti-thfi caesshghcethdns dg inirky shkneas enj pdwir shkneas eri jighnij ixectay es hn tmi cdnthnudus-thfi cesi. Ixefpais Zmi unht puasi shknea,
⎫5 , n 6 < ⎨< , n ≬ <
κ _nV 6 ⎭
hs en inirky shknea, whtm I ∞ 6 5 . Zmi unht-stip shknea,
⎫5 , n ≩ < ⎨< , n 1 <
u_ nV 6 ⎭
hs e pdwir shknea whtm thfi-evireki pdwir
5;
N 5 \∞ 6 ahf N ←∞ ∕ u 0 _ nV 0 N + 5 n 6∐ N
6 ahf N ←∞ 6 ahf N ←∞ •
5 N ∕5 0 N + 5 n 6 < N + 5
0 N + 5
6
5 0
\irhdjhchty2 Zmi shknea x_ nV hs pirhdjhc hg tmiri hs e pdshthvi hntikir N , ceaaij e
pirhdj , sucm tmet
x_n + N V 6 x_n V gdr eaa hntikir n. Zmi sfeaaist pirhdj dg e shknea, tmet hs, tmi aiest veaui dg N sucm sucm tmet tmi pirhdjhchty cdnjhthdn hs sethsghij, hs ceaaij tmi gunjefintea pirhdj dg tmi shknea. Zmi gunjefintea pirhdj dg pirhdj hs jindtij N d , tmdukm sdfithfis tmi su`scrhpt hs jrdppij hn perthcuaer cdntixts. Ixefpai Zd cmico pirhdjhchty dg tmi shknea x_ nV 6 shn(3n) , wi cmico hg tmiri hs e pdshthvi
hntikir N sucm sucm tmet
shn(3( n + N )) 6 shn(3 n) ,
n 6 <, µ5, µ 0, …
Zmet hs
shn(3n + 3 N ) 6 shn(3 n) , n 6 <, µ5, µ 0, … Zmhs cdnjhthdn mdajs hg enj dnay hg 3N hs hs en hntikir fuathpai dg 0ό , e cdnjhthdn tmet cenndt `i
fit `y hntikir N . Zmus tmi shknea hs ndt pirhdjhc.
• Iaifintery dpirethdns dpirethdns2 Iaifintery dpirethdns, gdr ixefpai ejjhthdn enj sceaer fuathpahcethdn, dn jhscriti-thfi shkneas eri d`vhdus cdnvirshdns grdf tmi cdnthnudus-thfi cesi. Iaifintery trensgdrfethdns dg tmi hnjipinjint verhe`ai easd eri iesy, tmdukm ht fust `i rifif`irij tmet dnay hntikir erkufint veauis eri pirfhttij hn tmi jhscriti-thfi cesi • Zhfi Sceai2 Suppdsi y_ nV 6 x_ enV , wmiri e hs e pdshthvi dr nikethvi hntikir (sd tmet tmi prdjuct, en, hs en hntikir gdr eaa hntikir n). Hg e 6 ∐ 5, tmhs hs e thfi rivirsea. @ut gdr eny cesi `iydnj e 6 µ 5 5, `i eweri tmet adss dg hngdrfethdn hn tmi shknea dccurs, unahoi tmi cdnthnudus-thfi cesi. Ixefpai Gdr e 6 0 , cdfperi tmi thfi sceaij shknea whtm tmi drhkhnea2
5=
hs e ghxij hntikir. Hg N hs hs pdshthvi, tmin • Zhfi Smhgt 2 Suppdsi y_ nV 6 x_ n ∐ N V , wmiri N hs tmhs hs e rhkmt smhgt, dr jiaey, enj hg N hs hs nikethvi, ht hs e aigt smhgt dr ejvenci.
•
Cdf`hnethdn Sceai enj Smhgt 2 Suppdsi y_ nV 6 x_en ∐ N V , wmiri e hs e ndnzird hntikir
enj N hs hs en hntikir. Es hn tmi cdnthnudus-thfi cesi, tmi segist epprdecm td hntirprithnk tmi risuat hs td shfpay padt dut tmi shknea y_nV. Ixefpai2 Suppdsi y_nV 6 x_ N
∐ nV . Zmhs hs e gahp enj smhgt , enj dccurs sugghchintay dgtin tmet ht
hs wdrtmwmhai virhgyhnk enj rifif`irhnk tmi smdrtcut2 y_nV cen `i padttij `y thfi-rivirshnk (gahpphnk) x_nV enj tmin smhgthnk tmi rivirsij shknea td fdvi tmi drhkhnea veaui et n 6 < td sefpais td tmi rhkmt hg N > <, enj N n 6 N . Zmet hs, smhgt N sefpais |N | sefpais td tmi aigt hg N 1 <.
Ixirchsis 5. Khvin tmi shknea smdwn `iadw,
soitcm tmi shknea y (t ) 6 (e) x (t ) ∐ x(t ∐ 5) (`) x(∐0t ) (c) x(t ∐ 5)u(5 (5 ∐ t ) (j) x (0t ) + x( ∐3t ) (i) x(3t ∐ ∐ 5) 0. Jitirfhni hg tmi gdaadwhnk shkneas eri pdwir shkneas dr inirky shkneas, enj cdfputi tmi tdtea
inirky dr thfi-evireki pdwir, es epprdprheti. (e) x(t) 6 shn(0t) u( t )
58
(`) x(t ) 6 i ∐|t | (c) x(t ) 6 tu (t ) (j) x (t ) 6 9i∐3t u (t ) 3. Gdr en inirky shknea x(t ) , prdvi tmet tmi tdtea inirky hs tmi suf dg tmi tdtea inirky dg tmi ivin
pert dg x(t ) enj tmi tdtea inirky dg tmi djj pert dg x (t ) .
6 9 , wmet hs tmi tdtea inirky dg tmi shknea :. Hg e khvin shknea x(t ) mes tdtea inirky I 6 y (t ) 6 0 x(3 (3t ∐ :) 4 ε t
hs tmi cdnthnudus-thfi shknea x(t ) 6 i 9. Ynjir wmet cdnjhthdns dn tmi riea cdnstent ε hs
u ( ∐t ) en
inirky shknea4 [min ydur cdnjhthdns eri sethsghij, wmet hs tmi inirky dg tmi shknea4 7. Soitcm tmi ivin enj djj perts dg tmi shkneas `iadw.
(e)
(`)
> ;. Suppdsi tmet gdr e shknea x(t ) ht hs ondwn tmet Iv{x(t )} 6 Dj{ x( t )} 6 5 gdr t >
< .[met hs
x(t ) 4 =. Jitirfhni wmhcm dg tmi gdaadwhnk shkneas eri pirhdjhc, enj spichgy tmi gunjefintea pirhdj.
(e) x(t ) 6 i bό cds(0ό t + ό ) (`) x(t ) 6 shn 0 (t ) ∞ (c) x(t ) 6 ∕ u (t ∐ 0 o ) ∐ u(t ∐5 ∐ 0 o ) o 6∐∞ (j) x (t ) 6 3i0 ∐ b 3ό t 8. Suppdsi tmet x5 (t ) enj x0 ( t ) eri pirhdjhc shkneas whtm rispicthvi gunjefintea pirhdjs Z 5
enj Z 0 . Smdw tmet hg tmiri eri pdshthvi hntikirs f enj n sucm tmet
0<
Z5 Z0
6
f n
(tmet hs, tmi rethd dg gunjefintea pirhdjs hs e rethdnea nuf`ir), tmin x( t ) 6 x5( t ) + x0 ( t ) hs pirhdjhc. Hg tmi cdnjhthdn mdajs, wmet hs tmi gunjefintea pirhdj dg x( t ) 4 5<. Jitirfhni wmhcm dg tmi gdaadwhnk shkneas eri `dunjij, enj spichgy e sfeaaist `dunj. 3t
(e) x(t ) 6 i u (t )
(`) x (t ) 6 i3t u ( ∐t ) (c) x(t ) 6 :i ∐7|t | (j) x(t ) 6 ∐0 t i ∐3t shn(t 9 ) u (t ) 55. Khvin tmi shknea x_ nV 6 κ _ nV ∐ κ _ n ∐ 5V , soitcm y_ nV 6
(e) x_:n ∐ 5V (`) x_ nVu_5 ∐ nV (c) 3 x_ ∐0n + 3V (j) x_0nV ∐ x_5 ∐ nV 50. Jitirfhni wmitmir tmi gdaadwhnk shkneas eri pirhdjhc, enj hg sd jitirfhni tmi gunjefintea
pirhdj. (e) x_ nV 6 u_ nV + u_ ∐ nV (`) x_ nV 6 i∐ b 3ό n n
(c) x_ nV 6 (∐5) + i
b ό n 0
(j) x_ nV 6 cds( ό n) :
53. Suppdsi x_nV hs e jhscriti-thfi shknea, enj ait y_nV6x_0nV. (e) Hg x_nV hs pirhdjhc, hs y_nV pirhdjhc4 Hg sd, wmet hs tmi gunjefintea pirhdj dg y_nV hn tirfs dg tmi gunjefintea gunjefintea pirhdj dg x _nV4 (`) Hg y_nV hs pirhdjhc, hs x_nV pirhdjhc4 Hg sd, wmet hs tmi gunjefintea pirhdj dg x_nV hn tirfs dg tmi gunjefintea pirhdj dg y_nV4 5:. Ynjir wmet cdnjhthdn hs tmi suf dg twd pirhdjhc jhscriti-thfi shkneas pirhdjhc4 [min tmi
cdnjhthdn hs sethsghij, wmet hs tmi gunjefintea pirhdj dg tmi suf, hn tirfs dg tmi gunjefintea pirhdjs dg tmi suffenjs4 n
59. Hs tmi shknea x_ nV 6 3 ( ∐5) 5) u_ nV en inirky shknea, pdwir shknea, dr nihtmir4 57. Hs tmi shknea x_ nV 6 i
∐ b 0ό n
+ i bό n pirhdjhc4 Hg sd, wmet hs tmi gunjefintea pirhdj4
5;. Enswir tmi gdaadwhnk quisthdns e`dut tmi jhscriti-thfi shknea x_ nV 6 i
(e) Hs x_ nV pirhdjhc4 Hg sd, wmet hs hts gunjefintea pirhdj4
05
∐ b (ό / 0) n
.
(`) Hs x_nV en ivin shknea4 Hs ht en djj shknea4 (c) Hs x_n V en inirky shknea4 Hs ht e pdwir shknea4 5=. [mhcm dg tmi gdaadwhnk shkneas eri pirhdjhc4 Gdr tmdsi tmet eri pirhdjhc, wmet hs tmi
gunjefintea pirhdj4 (e) x_ nV 6 i
b:n ό
(`) x_ nV 6 i (c) x_ nV 6 i
b 0 ό n =
∐ b ;= ό ( n ∐5)
00
Ndtis gdr Shkneas enj Systifs
Fucm dg dur jhscusshdn whaa gdcus dn twd `rdej caessis dg shkneas2 tmi caess dg cdfpaix ixpdninthea shkneas enj tmi caess dg shnkuaerhty shkneas. Zmdukm ht hs ger grdf d`vhdus, ht turns dut tmet issintheaay eaa shkneas dg hntirist cen `i ejjrissij hn tirfs dg tmisi twd caessis. 0.5 Zmi Caess dg CZ Ixpdninthea Shkneas
Zmiri eri sivirea weys td riprisint tmi cdfpaix-veauij shknea et
x(t ) 6 c i , ∐ ∞ 1 t 1 ∞ wmiri `dtm c enj e eri cdfpaix nuf`irs. E cdnvinhint epprdecm hs td wrhti c hn pdaer gdrf, enj enj e hn rictenkuaer gdrf, bχ d
c 6 | c |i
, e 6 σ d + bψ d
wmiri χ d 6 ∬c enj wmiri wi mevi cmdsin ndtethdns gdr tmi rictenkuaer gdrf dg e tmet eri custdfery hn tmi ghiaj dg shkneas enj systifs. Hn ejjhthdn, tmi su`scrhpt d‘s eri hntinjij td ifpmeshzi tmet tmi quenththis eri ghxij riea nuf`irs. Zmin bχ (σ + bψ d )t x(t ) 6 | c | i d i d 6 | c | iσ dt i b (ψd t +χ d ) Yshnk Iuair‘s gdrfuae, wi cen wrhti tmi shknea hn rictenkuaer gdrf es σ dt
x(t ) 6 | c | i
cds(ψdt + χd ) + b | c | iσ d t shn(ψdt + χd )
Zmiri eri twd spichea cesis tmet eri dg fdst hntirist. Spichea Cesi 52 Suppdsi `dtm c enj e eri riea. Zmet hs, ψ d
6 < enj χ d hs ihtmir < dr ό . Zmin wi
mevi tmi gefhaher ixpdnintheas
⎫⎢ | c | iσ d t , hg χ d 6< x(t ) 6 ⎭ σ d t ⎨⎢∐ | c | i , hg χ d 6 ό Dr, fdri shfpay, σ d t
x(t ) 6 c i
Spichea Cesi 02 Suppdsi c hs cdfpaix enj e hs puriay hfekhnery. Zmet hs, σ d
6 < . Zmin
b (ψd t +χ d )
x(t ) 6 | c | i
6 | c | cds(ψdt + χd ) + b | c | shn(ψdt + χ d ) @dtm tmi riea enj hfekhnery perts dg x(t ) eri pirhdjhc shkneas, whtm gunjefintea pirhdj Z d 6 0ό
|ψ d |
Shnci tmi hnjipinjint verhe`ai, t , hs vhiwij es thfi, unhts dg ψ d typhceaay eri rejhens/sicdnj enj enj unhts dg Z d netureaay eri sicdnjs. E shknea dg tmhs gdrf hs dgtin ceaaij e pmesdr .
03
Aigt hn ixpdninthea gdrf, wi cen cmico jhrictay tmet khvin eny ψ d , x(t ) hs pirhdjhc whtm pirhdj Z d 6 0ό / | ψ d | 2 b_ψd (t +Z d ) +χ d V
x(t + Zd ) 6 | c | i
6 | c | i b (ψdt +χ d ) iµ b 0ό 6 | c | i b (ψdt +χ d ) 6 x(t ) Easd, ht hs caier tmet Z d hs tmi gunjefintea pirhdj dg tmi shknea, shfpay `y ettifpthnk td sethsgy tmi pirhdjhchty whtm eny sfeaair, pdshthvi veaui gdr tmi pirhdj. [i cen vhiw e pmesdr shknea es e victdr et tmi drhkhn dg ainktm | c | rdtethnk hn tmi cdfpaix paeni whtm enkuaer griquincy ψ d rejhens/sicdnj, `ikhnnhnk whtm tmi enkai χ d et t 6 6 < . Hg ψ d > < , tmin tmi rdtethdn hs cduntir cadcowhsi. Hg ψ d 1 < , tmin tmi rdtethdn hs cadcowhsi. Dg cdursi, hg ψ d 6 < , tmin tmi shknea hs e cdnstent, enj ht hs ndt surprhshnk tmet tmi ndthdn dg e gunjefintea pirhdj geaas epert. Zmi eppait hn tmi ahno `iadw haaustretis tmhs rdtethnk-victdr hntirpritethdn, enj easd jhspaeys tmi hfekhnery pert dg tmi pmesdr, tmet hs, tmi prdbicthdn dn tmi virthcea exhs. Dni \mesdr Sufs dg pmesdrs tmet mevi jhggirint griquinchis eri easd viry hfpdrtent. Zmisi eri `ist vhsueahzij ushnk tmi ‑miej-td-teha‖ cdnstructhdn dg victdr ejjhthdn. Zmi eppait `iadw haaustretis. Suf dg Zwd \mesdrs Zmi quisthdn dg pirhdjhchty `icdfis fucm fdri hntiristhnk gdr pmesdr sufs, enj wi ghrst jhscuss tmhs gdr sufs dg twd pmesdrs. Cdnshjir x(t ) 6 c5 i
bψ5t
+ c0 i bψ 0t
Zmi veauis dg c5 enj c0 eri ndt issinthea gectdrs hn tmi pirhdjhchty quisthdn, `ut tmi veauis dg ψ 5 enj ψ 0 eri. Ht hs wdrtmwmhai td prdvhji e gdrfea stetifint enj prddg dg tmi risuat, whtm essufpthdns jishknij td ruai dut trhvheahthis enj niijaiss cdfpaixhthis. Zmidrif Zmi cdfpaix veauij shknea
x(t ) 6 c5 i
bψ5t
+ c0 i bψ 0t
whtm c5 , c0 ≬ < enj ψ 0 , ψ 5 ≬ < hs pirhdjhc hg enj dnay hg tmiri ixhsts e pdshthvi griquincy ψ < enj hntikirs o enj enj a sucm tmet (0.5) ψ5 6 oψ< , ψ0 6 aψ < Gurtmirfdri, hg ψ < hs tmi aerkist griquincy gdr wmhcm (0.5) (0.5) cen cen `i sethsghij, hn wmhcm cesi ht hs ceaaij tmi gunjefintea griquincy griquincy gdr x(t ) , tmin tmi gunjefintea pirhdj dg x(t ) hs Z d 6 0ό / ψ < .
0:
(0.5).. Cmddshnk Z 6 0ό / ψ < , \rddg Ghrst wi essufi tmet pdshthvi ψ < enj tmi hntikirs o, a sethsgy (0.5) wi sii tmet boψ< (t +Z )
x(t + Z ) 6 c5i
+ c0 i baψ < (t +Z )
6 i bo 0ό c5i boψ < hs sucm tmet x(t + Z ) 6 x(t ) gdr eaa t . Zmet hs, bψ (t +Z ) + c0i bψ0 (t +Z ) 6 c5i bψ5t + c0 i bψ0 t c5i 5 gdr eaa t . Zmhs hfpahis (i bψ5Z ∐ 5)c + ( i bψ0Z ∐ 5) c i b (ψ0 ∐ψ5 ) t 6 < 5
0
gdr eaa t . \hcohnk tmi perthcuaer thfis t 6 6 < enj t 6 ό /(ψ 0 ∐ ψ 5) khvis tmi twd eaki`rehc iquethdns
(i bψ5Z ∐ 5)c5 + ( i bψ 0Z ∐5) c0 6 < (i bψ5Z ∐ 5)c5 ∐ ( i bψ 0Z ∐ 5) c0 6 < @y ejjhnk tmisi twd iquethdns, enj easd su`trecthnk tmi sicdnj grdf tmi ghrst, wi d`tehn bψ Z
bψ Z
i 5 6 i 0 65 Zmirigdri `dtm griquinchis fust `i hntikir fuathpais dg griquincy 0ό / Z . Ixefpai Zmi shknea b t b t x (t ) 6 :i 0 ∐ 9i 3
hs pirhdjhc whtm gunjefintea griquincy ψ d 6 5 , enj tmus gunjefintea pirhdj 0ό . Zmi shknea b 0t
bό t
∐ 9i x(t ) 6 :i hs ndt pirhdjhc, shnci tmi griquinchis 0 enj ό cenndt cenndt `i hntikir fuathpais dg e ghxij griquincy. Zmi tmidrif kinireahzis td tmi cesi wmiri x(t ) hs e suf s uf dg eny nuf`ir dg cdfpaix ixpdninthea tirfs2 tmi shknea hs pirhdjhc hg enj dnay hg tmiri ixhsts ψ < sucm tmet iviry griquincy prisint hn tmi suf cen `i wrhttin es en hntikir fuathpai dg ψ < . Sucm griquincy tirfs eri dgtin ceaaij merfdnhceaay riaetij . Zmi eppait `iadw cen `i usij td vhsueahzi sufs dg sivirea merfdnhceaay riaetij pmesdrs, enj tmi hfekhnery pert ixmh`hts tmi cdrrispdnjhnk pirhdjhc, riea shknea. \mesdr Sufs
09
0.0 Zmi Caess dg CZ Shnkuaerhty Shkneas
Zmi `eshc shnkuaerhty shknea hs tmi unht hfpuasi, κ (t ) , e shknea wi hnvint hn drjir td mevi tmi gdaadwhnk shgthnk prdpirty whtm rispict td drjhnery shkneas, x(t ) 2 ∞
∯ x(t )κ (t ) jt 6 x(<)
(0.0)
∐∞
Zmet hs, κ (t ) ceusis tmi hntikrea td ‑shgt dut‖ tmi veaui dg x(<) . Miri x (t ) hs eny cdnthnudusthfi shknea tmet hs e cdnthnudus guncthdn et t 6 <, sd tmet tmi veaui dg x(t ) et t 6 6 < hs wiaa jighnij. Gdr ixefpai, e unht stip, dr tmi shknea x(t ) 6 5 / t , wduaj ndt `i iahkh`ai gdr usi hn tmi shgthnk prdpirty. (Mdwivir, sdfi trietfints jd eaadw e ghnhti bufp hn x(t ) et t 6 6 < , es dccurs hn tmi unht stip shknea, enj tmi shgthnk prdpirty hs jighnij td khvi tmi fhj-pdhnt dg tmi bufp. Zmet hs, ∞ x (<+ ) + x(< ∐ ) ∯ x(t )κ (t ) jt 6 0 ∐∞ Gdr ixefpai, hg tmi shknea hs tmi unht stip, tmin tmi shgt wduaj yhiaj 5 / 0 .) E ahttai tmdukmt, rivhiwij hn jiteha `iadw, smdws tmet κ (t ) cenndt `i e guncthdn hn tmi drjhnery sinsi. Mdwivir, wi jiviadp gurtmir prdpirthis dg tmi unht hfpuasi `y gdcushnk dn hfpahcethdns dg tmi shgthnk prdpirty, wmhai hnshsthnk tmet hn dtmir rispicts κ (t ) `imevi hn e fennir cdnshstint whtm tmi usuea ruais dg erhtmfithc enj ceacuaus dg drjhnery guncthdns.
• Erie ∞
∯
κ (t ) jt 6 5
(0.3)
∐∞
Cdnshjirhnk tmi shgthnk prdpirty whtm tmi shknea x(t ) 6 5 , gdr eaa t, wi sii tmi unht hfpuasi fust sethsgy (0.3) (0.3)..
•
Zhfi veauis
κ (t ) 6 <, gdr t ≬ <
(0.:)
6 < whtm x(<) 6 < , gdr ixefpai, tmi @y cdnshjirhnk x(t ) td `i eny shknea tmet hs cdnthnudus et t 6 shkneas x(t ) 6 t , t 0 , t 3 , … , ht cen `i smdwn tmet tmiri hs nd cdntrh`uthdn td tmi hntikrea hn (0.0) (0.0) gdr gdr ndnzird veauis dg tmi hntikrethdn verhe`ai. Zmhs hnjhcetis tmet tmi hfpuasi fust `i zird gdr ndnzird erkufints. D`vhdusay κ (<) cenndt `i zird, enj hnjiij ht fust mevi, hn sdfi sinsi, 6 < , enj yit mes unht erie. hnghnhti veaui. Zmet hs, tmi unht hfpuasi hs zird ivirywmiri ixcipt t 6 Zmhs feois caier tmi gect tmet wi eri jieahnk whtm sdfitmhnk dutshji tmi rieaf dg `eshc ceacuaus. Ndthci easd tmet tmet tmisi ghrst twd twd prdpirthis hfpay hfpay tmet e
∯
κ (t ) jt 6 5
∐e
gdr eny e > < .
07
•
Sceaer fuathpahcethdn
[i triet tmi sceaer fuathpahcethdn fuathpahcethdn dg en hfpuasi tmi sefi es tmi sceaer fuathpahcethdn dg en drjhnery shknea. Zd hntirprit tmi shgthnk prdpirty gdr e κ (t ) , wmiri e hs e cdnstent, ndti tmet tmi prdpirthis dg hntikrethdn hntikrethdn hfpay hfpay ∞
∞
∐∞
∐∞
∯ x(t ) _ eκ (t )V jt 6 e ∯ x( t) κ (t) jt 6 e x(<)
Zmi usuea tirfhndadky hs tmet eκ (t ) hs en ‑hfpuasi dg erie e,‖ `esij dn cmddshnk x(t ) 6 5 , gdr eaa t , hn tmi shgthnk ixprisshdn.
•
Shknea Fuathpahcethdn
z (t )κ (t ) 6 z(<)κ ( t )
[min e unht hfpuasi hs fuathpahij `y e shknea z(t ), ), wmhcm hs essufij td `i cdnthnudus et t 6 <, tmi shgthnk prdpirty khvis ∞
∞
∯ x(t ) _ z (t )κ (t )V jt 6
∯
∐∞
_ x(t ) z (t )V κ (t ) jt 6 x (<)z (<)
∐∞
Zmhs hs tmi sefi es tmi risuat d`tehnij wmin tmi unht hfpuasi hs fuathpahij `y tmi cdnstent z (<) , ∞
∞
∐∞
∐∞
(<)κ ( t)V jt 6 z(<) ∯ x( t)κ ( t) jt 6 z(<) x(< ( <) ∯ x(t )_ z(<
Zmirigdri wi cdncauji tmi shknea fuathpahcethdn prdpirty smdwn e`dvi.
•
Zhfi smhgt
[i triet tmi thfi smhgt dg en hfpuasi tmi sefi es tmi thfi smhgt dg eny dtmir shknea. Zd hntirprit tmi shgthnk prdpirty gdr tmi thfi smhgtij unht hfpuasi, κ (t ∐ t d ) , e cmenki dg hntikrethdn verhe`ai grdf t td td ϊ 6 t ∐ t d khvis ∞
∞
∯ x(t ) κ (t ∐ td ) jt 6 ∯
∐∞
∐∞
x(ϊ + td ) κ (ϊ ) jϊ 6 x(t d )
Zmhs prdpirty, tdkitmir whtm tmi guncthdn fuathpahcethdn prdpirty khvis tmi fdri kinirea stetifint z (t )κ (t ∐ td ) 6 z (td )κ (t ∐ t d ) wmiri t d hs eny riea cdnstent enj z(t ) hs eny drjhnery shknea tmet hs e cdnthnudus guncthdn dg t et et t 6 t d .
•
Zhfi sceai
Shnci en hfpuasi hs zird gdr eaa ndnzird erkufints, thfi sceahnk en hfpuasi mes hfpect dnay whtm rikerj td tmi shgthnk prdpirty wmiri, gdr eny ndnzird cdnstent e, ∞ 5 ∯ x(t ) κ (et ) jt 6 | e | x(<), e ≬ < ∐∞
0;
Zd busthgy tmhs ixprisshdn, essufi ghrst tmet e > <. Zmin tmi shgthnk prdpirty fust d`iy, `y tmi prhnchpai dg cdnshstincy cdnshstincy whtm tmi usuea ruais ruais dg hntikrethdn, hntikrethdn, enj hn perthcuaer perthcuaer whtm tmi tmi cmenki dg 6 et , hntikrethdn verhe`ai grdf t td td ϊ 6 ∞
∯ x(t ) κ (et ) jt 6 e5
∐∞
∞
∯ ∐∞
x(ϊ / e ) κ (ϊ ) jϊ 6 5 x(<) , e
e> <
E shfhaer ceacuaethdn gdr e 1 <, wmiri ndw tmi cmenki dg hntikrethdn verhe`ai yhiajs en hntircmenki dg ahfhts, khvis ∞
∯
x(t ) κ ( et ) jt 6 5 e
∐∞
6 ∐ e5
∐∞
∯
x(ϊ / e ) κ (ϊ ) j ϊ
∞
∞
∯ x(ϊ / e) κ (ϊ ) jϊ 6 ∐ 5e x(<) ,
e1 <
∐∞
Zmisi twd cesis cen `i cdf`hnij hntd dni ixprisshdn khvin e`dvi. Zmus tmi shgthnk prdpirty aiejs td tmi jighnhthdn2 κ ( et ) 6
•
5 |e|
κ (t ) ,
e≬<
Syffitry
Ndti tmet tmi tmi cesi e 6 5 hn thfi sceahnk khvis tmi risuat tmet κ (∐ t t) ects hn tmi shgthnk prdpirty ixectay es κ (t ), ), sd wi rikerj tmi unht hfpuasi es en ‑ivin guncthdn.‖ Dtmir hntirpritethdns eri pdssh`ai, `ut `ut wi whaa ndt ndt kd tmiri. [i krepmhceaay riprisint en hfpuasi `y en errdw, es smdwn `iadw.
(Hg tmi erie dg tmi hfpuasi hs nikethvi, e 1 <, sdfithfis tmi errdw hs jrewn pdhnthnk sdutm.) [i cduaj cdnthnui tmhs hnvisthkethdn dg prdpirthis dg tmi hfpuasi, gdr ixefpai, ushnk tmi ceacuaus cdns cdnshs hsti tinc ncy y prhn prhnch chpa paii td ghku ghkuri ri dut dut mdw mdw td hnti hntirp rpri ritt κ ( et ∐ t d ) , z (t )κ ( et ) , enj sd dn. @ut wi dnay niij tmi prdpirthis busthghij e`dvi, enj twd ejjhthdnea prdpirthis tmet eri shfpay wdnjrdus. Zmisi hncauji en ixtinshdn dg tmi shgthnk prdpirty tmet vhdaetis tmi cdnthnuhty cdnjhthdn2 • Spichea \rdpirty 5 ∞
∯
κ (ϊ ) κ (t ∐ ϊ ) jϊ 6 κ ( t )
∐∞
Ndti miri tmet tmi hntikrethdn verhe`ai hs ϊ , enj t hs hs eny riea veaui. Ivin fdri riferoe`ai hs en ixprisshdn tmet riaetis hfpuasis enj cdfpaix ixpdnintheas2
•
Spichea \rdpirty 0
5 ∞ bt ψ κ (t ) 6 ∯ i j ψ 0ό ∐∞
0=
Ndti miri tmet tmi hntikrea shfpay shfpay jdis ndt ndt cdnvirki hn tmi usuea sinsi sinsi dg `eshc ceacuaus, ceacuaus, shnci | i bt ψ |6 5 gdr eny (riea) veauis dg t enj enj ψ . Pifero Dur kinirea epprdecm td tmisi hfpuasi prdpirthis whaa `i ‑jdn‘t tmhno e`dut hfpuasis…
shfpay gdaadw tmi ruais.‖ Mdwivir, td prdvhji e `ht dg ixpaenethdn, whtm ahttai rhkdr, wi `rhigay jhscuss dni dg tmi fetmifethcea epprdecmis td tmi su`bict. Zd errhvi et tmi unht hfpuasi, cdnshjir ), n 6 5, 0, 3,… , tmet mevi tmi unht-erie tmi pdssh`hahty dg en hnghnhti siquinci dg guncthdns, j n (t ), prdpirty ∞
∯ ∐∞
j n (t ) jt 6 5,
n 6 5, 0, 3,…
enj easd mevi tmi prdpirty tmet gdr eny dtmir guncthdn x(t ) tmet hs cdnthnudus et t 6 6 <, ∞
ahf n ←∞
∯ ∐∞
j n (t ) x(t ) jt 6 x(<)
Miri tmi ahfht hnvdavis e siquinci dg nuf`irs jighnij `y drjhnery hntikreas, enj cen `i hntirpritij hn tmi usuea wey. Mdwivir wi nixt hntircmenki tmi drjir dg tmi ahfht enj tmi hntikrethdn, whtmdut prdpir busthghcethdn, enj vhiw κ (t ) es ‑sdfi sdrt‖ dg ahfht2 κ (t ) 6 ahf n ←∞ j n (t ) Zmhs vhiw hs usigua gdr hntuhthdn purpdsis, `ut hs jenkirdus hg pursuij tdd ger `y iaifintery fiens. Hn perthcuaer, gdr tmi siquincis dg guncthdns j n (t ) typhceaay cdnshjirij, tmi ahfht jdis ndt ixhst hn eny usuea sinsi. Ixefpais Cdnshjir tmi rictenkuaer-puasi shkneas ⎫⎢ n, ∐5 1 t 1 5 0n 0n j (t ) 6 ⎭ n
⎢⎨<, iasi
, n 6 5, 0, 3, …
Zmi puasis kit teaair enj tmhnnir es n hncriesis, `ut caieray iviry j n (t ) hs unht erie, enj tmi fien-veaui tmidrif cen `i usij td smdw ∞
∯ ∐∞
j n (t ) x(t ) jt 6 n
5 /( 0 n )
∯
x(t ) jt ≍ n
∐5 /( 0 n)
x (<) n
whtm tmi epprdxhfethdn kitthnk `ittir es n hncriesis. Zmus wi cen cesueaay vhiw e unht hfpuasi es tmi ahfht, es n ← ∞ , dg tmisi unht-erie rictenkais. E shfhaer ixefpai hs td teoi j n (t ) td `i e trhenkai dg mihkmt n, whjtm 0/n, cintirij et tmi drhkhn. @ut ht turns dut tmet e fdri hntiristhnk ixefpai hs td usi tmi shnc guncthdn jighnij `y shn(ό t ) shnc(t ) 6 (ό t ) enj ait j n (t ) 6 n shn c( nt ) , n 6 5, 0, 3, … Ht cen `i smdwn, `y iveauethnk en hntikrea tmet hs ndt quhti iaifintery, tmet tmisi shkneas eaa mevi erie 0ό , enj tmet tmi shgthnk prdpirty ∞
∯ ∐∞
j n (t ) x(t ) jt ≍ x(<)
08
hs e `ittir enj `ittir epprdxhfethdn es n krdws whtmdut `dunj. Zmirigdri wi cen vhiw en en hfpuasi dg erie 0ό , tmet hs, 0όκ (t ) , es e ahfht dg tmisi guncthdns. Zmhs siquinci dg shnc shkneas hs jhspaeyij hn tmi eppait `iadw gdr e renki dg n, enj ydu cen kit e phctdrhea vhiw dg mdw en hfpuasi fhkmt erhsi grdf shnc‘s es n hncriesis, hn fucm tmi sefi wey es tmi hfpuasi erhsis grdf mihkmt n, whjtm 5/n , rictenkuaer puasis es n hncriesis. Gefhay dg Shncs Pifero Spichea \rdpirty 0 cen `i hntuhthviay unjirstddj hn tirfs dg dur cesuea vhiw dg hfpuasis
es gdaadws. Ait j[ (t ) 6
5 0ό
6
5 0ό
[
∯
bψ t
i
jψ
∐[ [
∯ _cds(ψ t ) + b shn(ψ t)V jψ
∐[ [
b
[
6 05ό ∯ cds(ψ t ) jψ + 0ό ∯ shn(ψ t) jψ ∐[
∐[
Yshnk tmi gect tmet e shnusdhj hs en djj guncthdn dg hts erkufint, j[ (t ) 6 5
ό
6 ό 5
[
∯ cds(ψ t ) jψ
<
shn([t ) t
6 [ shnc( [t ) ό ό Zmhs j[ (t ) cen `i smdwn td mevi unht erie gdr iviry [ > <, ekehn `y e ndn-iaifintery hntikrethdn, enj ekehn tmi shgthnk prdpirty hs epprdxhfetij wmin [ hs hs aerki. Zmirigdri tmi Spichea \rdpirty 0 fhkmt `i ixpictij. Zmi eppait `iadw smdws e padt dg j[ (t ) es [ hs hs verhij, enj prdvhjis e phcturi phcturi dg mdw tmi tmi hfpuasi fhkmt fhkmt erhsi es [ hncriesis. hncriesis. Endtmir Shnc Gefhay Ejjhthdnea Shnkuaerhty Shnkuaerhty Shkneas Shkneas
Grdf tmi unht hfpuasi wi kinireti ejjhthdnea shnkuaerhty shkneas ushnk e kinireahzij gdrf dg ceacuaus. Hntikrethdn aiejs td t ⎫<, t 1 < ∯ κ (ϊ ) j ϊ 6 ⎭5, t > < ⎨ ∐∞ wmhcm hs tmi gefhaher unht-stip guncthdn, u (t ) . ([i aievi tmi veaui dg u(<), wmiri tmi bufp dccurs, griiay esshkne`ai gdaadwhnk dur kinirea pdahcy.) Zmi ‑runnhnk hntikrea‖ hn tmhs ixprisshdn ectueaay cen `i hntirpritij krepmhceaay hn viry nhci wey. 6 t ∐ ϊ ∐ ϊ khvis Enj e verhe`ai cmenki grdf ϊ td td σ 6 khvis tmi eatirneti ixprisshdn
3<
u (t ) 6
∞
∯ κ (t ∐ σ ) jσ
<
Eneaythceaay tmhs cen `i vhiwij es en eppahcethdn dg e shgthnk prdpirty eppahij td tmi cesi x(t ) 6 u(t )2 )2 ∞
∯ κ (t ∐ σ ) jσ 6 <
∞
∞
u (σ )κ (t ∐ σ ) jσ 6
∯ ∐∞
∯
u (t ∐ σ )κ (σ ) jσ 6 u (t )
∐∞
Zmhs hs ndt, strhctay spieohnk, aikea gdr t 6 <, `iceusi dg tmi jhscdnthnuhty tmiri hn u(t ), ), `ut wi hkndri tmhs hssui. @y cdnshjirhnk tmi runnhnk hntikrea dg tmi unht-stip guncthdn, wi errhvi et tmi unht-refp2 t ⎫<, t 1 < ∯ u(ϊ ) j ϊ 6 ⎭ t , t ≩ < ⎨ ∐∞
6 tu (t ) Dgtin wi wrhti tmhs es r (t ). ). Ndthci tmet tmi unht refp hs e cdnthnudus guncthdn dg thfi, tmdukm ht hs un`dunjij. Cdnthnuhnk, t ⎢⎫ <, t 1 < 6 ϊ ϊ ( ) r j ⎭ 0 ∯ ⎨⎢t / 0, t ≩ < ∐∞ 6
0
t
0
u (t )
wmhcm fhkmt `i ceaaij tmi unht pere`dae , p(t ). ). [i stdp miri, es gurtmir hntikrethdns yhiaj shkneas ahttai usij hn tmi siquia. [i cen turn fettirs erdunj, ushnk jhggirinthethdn enj tmi gunjefintea tmidrif dg ceacuaus. Caieray, j jt
p(t ) 6
j
t
∯
jt ∐∞
r (ϊ ) jϊ 6 r (t )
enj tmhs hs e pirgictay aikea eppahcethdn dg tmi gunjefintea tmidrif shnci tmi hntikrenj, r (t ), ), hs e cdnthnudus guncthdn dg thfi. Mdwivir, wi kd gurtmir, cmiethnk e `ht dn tmi essufpthdns, shnci tmi unht stip hs ndt cdnthnudus, td wrhti j jt
r (t ) 6
j
t
∯
jt ∐∞
u (ϊ ) jϊ 6 u(t )
Zmet tmhs cmiethnk hs ndt unriesdne`ai gdaadws grdf e padt dg tmi unht refp, r(t), enj tmin e padt dg tmi sadpi et iecm veaui dg t . Cmiethnk fdri, wi easd wrhti j jt
u (t ) 6
j
t
∯
jt ∐∞
κ (ϊ ) jϊ 6 κ ( t )
Ekehn, e krepmhcea hntirpritethdn feois tmhs siif aiss unriesdne`ai. [i cen easd cdnshjir ‑jirhvethvis‖ dg tmi unht hfpuasi. Zmi epprdecm hs ekehn td jifenj cdnshstincy whtm dtmir ruais dg ceacuaus, enj usi hntikrethdn `y perts td hntirprit tmi ‑shgthnk prdpirty‖ tmet smduaj `i sethsghij. [i niij kd nd gurtmir tmen tmi ghrst jirhvethvi, wmiri
35
∞
∞
∯ x(t )_ j κ (t )V jt 6 x(t )κ (t ) ∐∞ ∐
∐∞
jt
∞
∯
x (t ) κ (t ) jt
∐∞
6 ∐ x (<) 6 < , tmet hs, x(t ) hs cdnthnudusay Zmhs essufis, essufis, dg cdursi, cdursi, tmet x( t ) hs cdnthnudus et t 6 6 < . Zmhs unht-hfpuasi jirhvethvi hs usueaay ceaaij tmi unht jdu`ait , enj jhggirinthe`ai et t 6 jindtij κ (t ) . Werhdus prdpirthis cen `i jijucij, bust es gdr tmi unht hfpuasi. Gdr ixefpai, cmddshnk x(t ) 6 5, ∐ ∞ 1 t 1 ∞, tmi shgthnk prdpirty gdr tmi jdu`ait khvis ∞
∯
κ (t ) jt 6 <
∐∞
Hn dtmir wdrjs, tmi jdu`ait mes zird erie – e trui kmdst. Ht hs easd iesy td virhgy tmi prdpirty ∞
∯ x(t) κ (t ∐ td ) jt 6 ∐ x (t d )
∐∞
enj, ghneaay, wi soitcm tmi unht jdu`ait es smdwn `iadw.
Eaa dg tmi ‑kinireahzij ceacuaus‖ prdpirthis cen `i kinireahzij hn verhdus weys. Gdr ixefpai, tmi prdjuct ruai khvis j jt
_t u (t )V 6 5 u (t ) + t κ (t )
6 u (t ) wmiri wi mevi usij tmi fuathpahcethdn ruai td cdncauji tmet tκ (t ) 6 < gdr eaa t . Es endtmir ixefpai, tmi cmehn ruai khvis j jt
u (t ∐ td ) 6 κ (t ∐ td )
Pifero Zmisi fettirs cen `i teoin tdd ger, td e pdhnt wmiri ef`hkuhthis `ikhn td dvirwmiaf
enj wdrrhsdfi ah`irthis fust `i teoin. Gdr ixefpai, ushnk tmi prdjuct ruai gdr jhggirinthethdn, ), enj hkndrhnk tmi gect tmet u 0 (t ) hs tmi sefi shknea es u(t ), j 0 u (t ) jt
6 u(t )u(t ) + u(t)u( t) 6 0u( t) κ ( t)
Zmi fuathpahcethdn ruai gdr hfpuasis jdis ndt eppay, shnci u(t ) hs ndt cdnthnudus et t 6 <, enj sd wi eri stuco. Mdwivir hg wi hntirprit u(<) es ¶, tmi fhjpdhnt dg tmi bufp, wi kit e risuat cdnshstint whtm u (t ) 6 κ (t ) . [i whaa ndt niij td teoi fettirs tmhs ger, shnci wi usi tmisi kinireahzij ndthdns dnay gdr retmir shfpai shkneas. 0.3 Ahnier Cdf`hnethdns dg Shnkuaerhty Shkneas enj Kinireahzij Ceacuaus
Gdr shfpai shkneas, tmet hs, shkneas whtm uncdfpahcetij wevi smepis, ht hs cdnvinhint gdr feny purpdsis td usi usi shnkuaerhty shkneas gdr riprisintethdn riprisintethdn enj ceacuaethdn. ceacuaethdn. Ixefpai Zmi shknea smdwn `iadw `iadw cen `i wrhttin es e suf dg stip stip guncthdns,
30
x(t ) 6 :u (t + 5) ∐ :u(t ∐ 5)
Endtmir riprisintethdn hs
(5 ∐ t ) x(t ) 6 :u (t + 5)u (5
Zmhs usis tmi unht stips es ‑cutdgg‖ guncthdns, enj sdfithfis sdfithfis tmhs useki hs ejventekidus. Mdwivir gdr gurtmir ceacuaethdn, riprisintethdn es e ahnier cdf`hnethdn usueaay hs fucm shfpair. Jhggirinthethdn dg tmi ghrst ixprisshdn gdr x(t ) khvis, ushnk dur kinireahzij ndthdn dg jhggirinthethdn, x (t ) 6 :κ (t + 5) ∐ :κ (t ∐ 5)
Zmhs shknea hs smdwn `iadw.
Zmi sefi risuat cen `i d`tehnij `y jhggirinthethnk tmi ‑stip cutdgg‖ riprisintethdn gdr x(t )),, tmdukm useki dg tmi prdjuct ruai enj hntirpritethdn dg tmi ghnea risuat feois tmi jirhvethvi ceacuaethdn fdri jhgghcuat. Zmet hs, x (t ) 6 :κ ( t + 5)u (5 (5 ∐ t) + :u (t +5)κ (5 (5 ∐ t )
6 :κ ( t + 5) ∐ :κ (5 (5 ∐ t ) 6 :κ ( t + 5) ∐ :κ ( t ∐ 5) (Zmi ghrst stip feois usi dg tmi prdjuct ruai gdr jhggirinthethdn, tmi sicdnj stip usis tmi shkneafuathpahcethdn fuathpahcethdn ruai gdr hfpuasis, enj tmi aest stip usis tmi ivinniss dg tmi hfpuasi.) Dg cdursi, rikerjaiss dg tmi epprdecm teoin, krepmhcea fitmdjs ieshay virhgy t
∯ x (ϊ ) jϊ 6 x(t ) ∐∞
hn tmhs ixefpai. Ndti tmet tmi cdnstent dg hntikrethdn hs teoin td `i zird shnci ht hs ondwn tmet tmi 1 ∐5 . shknea x(t ) hs zird gdr t 1 Ixefpai Zmi shknea smdwn smdwn `iadw cen `i wrhttin es x(t ) 6 0r (t ) ∐ 0r (t ∐ 5) ∐ 0u (t ∐ 3)
33
Ekehn, tmi jirhvethvi hs strehkmtgdrwerj td cdfputi, x (t ) 6 0u (t ) ∐ 0u (t ∐ 5) ∐ 0κ (t ∐ 3)
enj soitcm,
Krepmhcea hntirpritethdns dg jhggirinthethdn suppdrt tmisi cdfputethdns. [mhai riprisintethdn hn tirfs dg ahnier cdf`hnethdns dg shnkuaerhty shkneas aiejs td cdnvinhint smdrtcuts gdr sdfi purpdsis, ceuthdn smduaj `i ixirchsij. Hn tmi ixefpais sd ger, wiaa-`imevij inirky shkneas mevi `iin riprisintij es ahnier cdf`hnethdns dg shkneas tmet eri pdwir shkneas, shnkuaerhty shkneas, enj un`dunjij shkneas. Zmhs cen hntrdjuci cdfpahcethdns hn sdfi cdntixts. Sdfithfis wi usi tmisi kinireahzij ceacuaus hjies gdr shkneas eri ndnzird gdr hnghnhti thfi hntirveas. Ixefpai E rhkmt-shjij cdshni shknea cen `i wrhttin es x(t) 6 cds(0t ) u (t )
Zmin jhggirinthethdn ushnk tmi prdjuct ruai, gdaadwij `y tmi hfpuasi fuathpahcethdn fuathpahcethdn ruai, khvis x (t ) 6 ∐0 shn( 0t ) u (t ) + cds(0t ) κ (t ) 6 ∐0 shn( 0t ) u (t ) + κ (t ) Xdu smduaj krepmhceaay cmico tmet tmhs hs e cdnshstint risuat, enj tmet tmi hfpuasi hn x (t ) hs cruchea hn virhgyhnk tmi riaethdnsmhp t
∯ x (ϊ ) jϊ 6 x(t ) ∐∞
Ixefpai Zmi pirhdjhc shknea smdwn `iadw cen `i wrhttin es x(t ) 6
∞
∕ _:u (t + 5 ∐ 3o ) ∐ :u(t ∐ 5 ∐ 3o )V
o 6∐∞
3:
Zmin x (t ) 6
∞
∕ _:κ (t + 5 ∐ 3o ) ∐ :κ ( t ∐5 ∐ 3o ) V
o 6∐∞
`y kinireahzij jhggirinthethdn, jhggirinthethdn, enj e soitcm dg x (t ) hs smdwn `iadw.
Ixirchsis 5. Jitirfhni wmitmir tmi gdaadwhnk shkneas eri pirhdjhc, enj hg sd jitirfhni tmi gunjefintea
pirhdj. (e) x(t ) 6 i∐(ό ∐ 0 b )t (`) x (t ) 6 i ∐ό + 0 bt b :t ∐ 0i b 9t (c) x(t ) 6 3i
(j) x (t ) 6 i
b ; t 0
+ i b ;t
(i) x(t ) 6 7i∐ b 0t + :i b ;(t ∐5) ∐ 3i b 7(t +5) ∞
(g) x(t ) 6 ∕ ⎥ i∐(t ∐ 0o )u (t ∐ 0 o ) ∐ i∐(t ∐0 o ) u( t ∐ 0 o ∐5) ⎪ ⎣ ⎧ o 6∐∞ 0. Shfpahgy tmi gdaadwhnk ixprisshdns enj soitcm tmi shknea.
(e) x(t ) 6 κ (t ∐ 0) r ( t) + j _ u( t) ∐ r( t ∐5)V jt t + 3
(`) x (t ) 6 κ (t )κ (t ∐ 5) ∐ i
κ (t + 0) + u(5 ∐ t) r( t )
t
(c) x(t ) 6
∯ u(ϊ ∐ 3) jϊ ∐ 0 r( t ∐ :) + r( t ∐ 9) ∐∞
(j) x(t ) 6
t
j 0) r( t ) V ∯ κ (ϊ + 0) jϊ + jt _u(t + 0)
∐∞
t
3. Soitcm tmi shknea x(t ) enj cdfputi enj soitcm
∯ x(σ ) j σ . Cmico ydur hntikrethdn `y ∐∞
"krepmhcea jhggirinthethdn."
39
(e) x(t ) 6 u (t ) ∐ u(t ∐5) + 0κ (t ∐ 0) ∐ 3κ ( t ∐ 3) + κ ( t ∐ :) (`) x (t ) 6 3u(t ) ∐ 3u( t ∐ :) ∐ 50κ ( t ∐ 9) (c) x(t ) 6 ∐ 0u(t + 5) + 3u( t) ∐ u( t ∐ 0) (j) x (t ) 6 u (t ) + κ (t ∐ 5) ∐ 0κ ( t ∐ 0) + κ ( t ∐ :) :. Soitcm tmi shknea x(t ) enj cdfputi enj soitcm x (t ) . Cmico ydur jirhvethvi `y "krepmhcea
hntikrethdn." (e) x(t ) 6 r (t + 5) ∐ r (t) + u( t) ∐ 3u( t ∐5) + u( t ∐ 0) (`) x (t ) 6 0r (t + 5) ∐ : r( t) + 0 r( t ∐5) (c) x(t ) 6 0u(t ∐ 5) ∐ (0 / 3) r( t ∐5) + (0 / 3) r( t ∐ :) 9. [rhti e fetmifethcea ixprisshdn gdr tmi shknea x(t ) smdwn `iadw.
Cdfputi enj soitcm x (t ) , tmi kinireahzij jirhvethvi dg x(t ) .
37
Ndtis gdr Shkneas enj Systifs 3. 5 Zmi Caess dg JZ Ixpdninthea Shkneas
Cdnshjir e jhscriti-thfi shknea en
x_ nV 6 c i wmiri `dtm c enj e eri cdfpaix nuf`irs, enj, es usuea, tmi hntikir hnjix, dr sefpai nuf`ir, cdvirs tmi renki ∐∞ 1 n 1 ∞ . Zd fdri cdnvinhintay hntirprit x_nV, wrhti c hn pdaer pdaer gdrf, gdrf, enj e hn rictenkuaer gdrf, bχ d
c 6 | c |i
, e 6 σ d + bψ d
wmiri χ d 6 ∬c enj wi mevi cmdsin custdfery ndtethdns gdr tmi rictenkuaer gdrf dg e. Zmin bχd
x_ nV 6 | c | i
(σ d + bψ d ) n
i
6 | c | iσ d n i b (ψ d n +χ d ) Yshnk Iuair‘s gdrfuae, wi cen wrhti tmhs shknea hn rictenkuaer gdrf es σ n σ n x_ nV 6 | c | i d cds(ψd n + χd ) + b | c | i d shn(ψd n + χd ) Zmhs ixprisshdn hs shfhaer td tmi cdnthnudus-thfi cesi, `ut jhggirincis eppier updn cadsir hnspicthdn. [i niij dnay cdnshjir tmrii cesis hn jiteha. Spichea Cesi 52 Suppdsi `dtm c enj e eri riea. Zmet hs, ψ d
6 < enj χ d mes veaui ihtmir < dr ό .
Zmin, σ d n
x_ nV 6 c i
enj wi mevi tmi gefhaher hntirpritethdn dg en ixpdnintheaay-jicrieshnk (hg σ d 1 < ) dr ixpdnintheaay-hncrieshnk σ d > < siquinci dg veauis, whtm tmi d`vhdus cdnshjirethdn dg tmi shkn dg c. Spichea Cesi 02 Suppdsi c hs riea enj e mes tmi spichea gdrf
e 6 σ d + b (0o + 5)ό
wmiri o hs hs en hntikir. Hn tmhs cesi, _σ d + b (0 o +5)ό Vn
x_ nV 6 c i
6 c iσ d n i b ( 0 o +5)ό
6 c iσ d n (∐5) n 6 c ( ∐iσ d ) n Dr, fdri shfpay, wi cen wrhti n
x_nV 6 c ε σ
wmiri ε 6 ∐ i d hs e riea nikethvi nuf`ir. Zmi eppierenci dg x_nV hs e `ht jhggirint hn tmhs cesi, es smdwn `iadw gdr c 6 5 .
3;
Dg cdursi, tmisi ghrst twd spichea cesis cen `i cdf`hnij `y cdnshjirhnk tmet ε fhkmt `i ihtmir pdshthvi dr nikethvi. nikethvi. Spichea Cesi 32 Suppdsi c hs cdfpaix enj e hs puriay hfekhnery. Zmet hs, σ d
6 < , enj wi eri
cdnshjirhnk e jhscriti-thfi pmesdr . Zmin bχd
x_ nV 6| c | i
i
bψ d n
6 | c | i b (ψd n +χ d ) 6 | c | cds(ψd n + χd ) + b | c | shn(ψd n + χd ) Shnci tmi hnjipinjint verhe`ai, n, hs vhiwij es e sefpai hnjix, unhts dg ψ d typhceaay eri jhscusshdn, wi essufi tmet c rejhens/sefpai gdr jhscriti-thfi shkneas. Hn drjir td shfpahgy gurtmir jhscusshdn, 6 5. Zmi ghrst jhggirinci grdf tmi cdnthnudus-thfi cesi hnvdavis tmi hssui dg pirhdjhchty. Zmi shknea bψ n
x_ nV 6 i d hs pirhdjhc hg enj dnay hg tmiri hs e pdshthvi hntikir N sucm sucm tmet bψ ( n + N ) 6 i bψ d n i d gdr eaa hntikir n. Zmhs whaa mdaj hg enj dnay hg N sethsghis sethsghis bψ N
i d 65 tmet hs, hg enj dnay hg ψd N 6 f0ό gdr sdfi hntikir f. Hn wdrjs, x_nV hs pirhdjhc hg enj dnay hg tmi
griquincy ψ d hs e rethdnea fuathpai dg 0ό , ψd 6
f
0ό N gdr sdfi hntikirs f enj N . Hg tmhs hs sethsghij, tmin tmi gunjefintea pirhdj dg tmi shknea hs d`tehnij wmin tmi sfeaaist veaui dg N hs hs gdunj gdr wmhcm tmhs ixprisshdn mdajs. D`vhdusay, tmhs dccurs wmin f, N eri eri riaethviay prhfi, tmet hs, f enj N mevi mevi nd cdffdn hntikir gectdrs dtmir tmen unhty. Ixefpai x_ nV 6 i
b 0n
hs ndt pirhdjhc, shnci ψ d 6 0 cenndt `i wrhttin es e rethdnea fuathpai dg
0ό . b
ό
n
Ixefpai x_ nV 6 i = hs pirhdjhc, whtm gunjefintea pirhdj 57, shnci
ό
=
6
5 0ό 57
3=
Zmi sicdnj febdr jhggirinci `itwiin cdnthnudus enj jhscriti-thfi cdfpaix ixpdnintheas cdncirns tmi hssui dg griquincy. Hn cdnthnudus thfi, es tmi griquincy ψ d hncriesis, tmi pmesdr rdtetis fdri rephjay. @ut hn jhscriti thfi, cdnshjir wmet meppins wmin tmi griquincy hs rehsij dr adwirij `y 0ό 2 b (ψ µ 0ό ) n i d 6 i bψd n iµ b 0ό n 6 i bψd n Zmet hs, tmi cdfpaix ixpdninthea shknea jdisn‘t cmenki. Caieray, rehshnk dr adwirhnk tmi griquincy `y eny hntikir fuathpai fuathpai dg 0ό mes mes tmi sefi (e`sinci dg) iggict. Zmus wi niij dnay cdnshjir jhscriti-thfi griquinchis hn e shnkai 0ό renki. Fdst dgtin tmi renki dg cmdhci hs ∐ό 1 ψd ≪ ό
gdr riesdns dg syffitry, tmdukm sdfithfis sdfithfis tmi renki < ≪ ψd 1 0ό hs cdnvinhint. Es e ghnea d`sirvethdn, hn cdnthnudus thfi e pmesdr rdtetis cduntircadcowhsi hg tmi griquincy ψ d hs pdshthvi, enj cadcowhsi hg tmi griquincy hs nikethvi. Mdwivir tmiri hs nd caier vhsuea hntirpritethdn dg jhricthdn dg rdtethdn jipinjhnk dn tmi shkn dg griquincy hn tmi jhscriti thfi cesi. Zd `eaenci tmisi cdfpahcethdns hn cdfperhsdn whtm tmi cdnthnudus-thfi cesi, tmi hssui dg pirhdjhchty dg sufs dg pirhdjhc pirhdjhc jhscriti-thfi pmesdrs hs riaethviay riaethviay shfpai. Zmidrif Gdr tmi cdfpaix-veauij shknea bψ n
bψ n
x_ nV 6 c5i 5 + c0 i 0 suppdsi `dtm griquinchis eri rethdnea fuathpais dg 0 ό , ψ5 6 (f5 / N5)0ό , ψ 0 6 ( f0 / N 0 )0 ό
Zmin x_ nV hs pirhdjhc whtm pirhdj (ndt nicisserhay gunjefintea pirhdj) khvin `y N 6 N5N 0 . \rddg @y jhrict ceacuaethdn, ceacuaethdn, ushnk tmi caehfij pirhdj N , bψ ( n + N ) x_ n + N V 6 c5i 5 + c0 i bψ 0 ( n+ N ) f
5 bψ5n b N5 0ό N5 N0
6 c5i
i
+ c0 i
bψ 0 n
i
f b 0 0ό N5 N 0 N 0
6 c5i bψ5n i bf5 N0 0ό + c0 i bψ0 n i bf0 N 5 0ό
6 c5i bψ5n + c0 i bψ 0 n 6 x_ nV gdr eny n. Zmi eppait `iadw haaustretis tmi `imevhdr dg jhscriti-thfi pmesdrs gdr verhdus cmdhcis dg griquincy, ψ d , enj sukkists e nuf`ir dg ixirchsis. Jhscriti-Zhfi Griquincy Pifero Zmi verhdus eneaysis dg dg pmesdr prdpirthis cen easd `i cerrhij dut dut gdr riea trhkdndfitrhc trhkdndfitrhc
shkneas, tmdukm tmi riesdnhnk dgtin riquhris trhk hjinththis enj hs fdri hnvdavij tmen gdr pmesdrs. Gdr ixefpai, suppdsi x_ nV 6 cds(ψ d n)
38
hs pirhdjhc whtm pirhdj N . Zmet hs, gdr eaa hntikirs n, cds_ψd ( n + N )V 6 cds(ψ d n) Zmhs cen `i wrhttin es cds(ψd n + ψd N ) 6 cds(ψd n) cds(ψ d N ) ∐ shn(ψ d n) shn( ψ d N)
6 cds(ψ d n) , ∐ ∞ 1 n 1 ∞ [i cdncauji grdf tmhs tmet N fust fust `i sucm tmet cds(ψd N ) 6 5, sh shn(ψ d N ) 6 < . Zmus ψd N 6 f0ό gdr sdfi hntikir f, tmet hs, ψ d fust `i e rethdnea fuathpai dg 0ό . 3.0 Zmi Caess dg JZ Shnkuaerhty Shkneas
Zmi `eshc jhscriti-thfi shnkuaerhty shknea hs tmi unht puasi, ⎫5, n 6 < κ _nV 6 ⎭ ⎨<, dtmirwhsi Cdntrery td tmi cdnthnudus-thfi cesi, tmiri hs ndtmhnk ‑kinireahzij‖ e`dut tmhs shfpai shknea. Krepmhceaay, dg cdursi, κ _nV hs e adniay adaaypdp et tmi drhkhn2
Zmi jhscriti-thfi unht-stip guncthdn hs
⎫5, n ≩ < ⎨<, n 1 <
u_nV 6 ⎭
enj ekehn tmiri eri nd ticmnhcea hssuis miri. Hn perthcuaer, tmi veaui dg u_
Zmi unht stip cen `i vhiwij es tmi runnhnk suf dg tmi unht puasi, u_nV 6
n
∕
κ _ o V
o 6∐∞
Cmenkhnk suffethdn verhe`ai grdf o td td a 6 n ∐ o khvis en eatirneti ixprisshdn o khvis u_ nV 6
∞
∕ κ _ n ∐ aV a 6<
Zmhs prdciss cen `i cdnthnuij td jighni tmi unht refp es tmi runnhnk suf dg tmi unht stip,
:<
tmdukm tmiri hs e ahttai ejbustfint hnvdavij hn tmi uppir ahfht dg tmi suf, shnci u_
n ∐5
⎫n, n ≩ < ⎨<, n 1 <
∕ u_ o V 6 ⎭
o 6∐∞
[i whaa mevi nd riesdn td pursui gurtmir htirethdns dg runnhnk sufs. Dni riesdn hs tmet shfpai jhscriti-thfi shkneas cen `i wrhttin es sufs dg efpahtuji sceaij enj thfi smhgtij unht puasis, enj tmiri hs ahttai niij td wrhti shkneas hn tirfs dg stips, refps, enj sd dn. Jhscriti-thfi shnkuaerhty shkneas easd mevi shgthnk enj fuathpahcethdn fuathpahcethdn prdpirthis shfhaer td tmi cdnthnudus-thfi cdnthnudus-thfi cesi, tmdukm nd ‑kinireahzij‖ hntirpritethdns eri niijij. Ht hs strehkmtgdrwerj td virhgy tmet ∞
∕ x_nVκ _ n ∐ nd V 6 x_ nd V
n 6∐∞
wmhcm hs eneadkdus td tmi cdnthnudus-thfi shgt ∞
∯ x(t )κ (t ∐ td ) jt 6 x(t d )
∐∞
Easd,
x_ nVκ _ n ∐ nd V 6 x_ nd Vκ _ n ∐ nd V hs e jhscriti-thfi virshdn dg tmi fuathpahcethdn ruai x(t )κ (t ∐ td ) 6 x(td )κ (t ∐ t d )
Mdwivir, unahoi tmi cdnthnudus-thfi cesi, tmi thfi-sceaij unht puasi, κ _enV , wmiri e hs e ndnzird hntikir, hs hjinthcea td κ _nV , es hs ieshay virhghij. Ixirchsis 5. Jitirfhni hg tmi gdaadwhnk shkneas eri pirhdjhc, enj hg sd cdfputi cdfputi tmi gunjefintea pirhdj. b 0<ό n
(e) x_ nV 6 i
3
b :ό n
(`) x_ nV 6 i
(c) x_ nV 6 3i
b;n 3
(j) x_ nV 6 5 + i (i) x_ nV 6 i
∐i
∐ b ό : n
b 9: ό n
b 9 ό n ;
+i
∐ b 3: ό n
0. Cdnshjir tmi shknea shknea x_ nV 6 c5i
bψ5 n
+ c0 i bψ 0 n wmiri `dtm griquinchis eri rethdnea fuathpais
dg 0ό , ψ5 6 (f5 / N5)0ό ,
ψ 0 6 ( f0 / N 0 )0 ό
Suppdsi tmet N hs e pdshthvi hntikir sucm tmet N 6 o5 N5 6 o0 N 0 gdr sdfi hntikirs o5 , o 0 . Smdw tmet x_ nV mes pirhdj N . (Zyphceaay N 1 N5N 0 , es usij hn tmi tmidrif hn Sicthdn 3.5.)
:5
3. Shfpahgy tmi gdaadwhnk gdaadwhnk ixprisshdns enj soitcm tmi shknea.
(e) x_ nV 6
n
∕ 3κ _ o ∐ 0V + κ _ n + 5V cds(ό n) o 6∐∞
(`) x_ nV 6
∞
∕ :κ _ n ∐ o V i3o o 6∐∞
(c) x_ nV 6 r_ n ∐ 3Vκ _ n ∐ 9V +
n
∕ 3κ _ n ∐ o V o 6∐∞
c ds(ό n) _κ _ nV ∐ κ _ n ∐5VV ∐ κ 3_ n V + (j) x_ nV 6 cd
n
∕ u_ o ∐ 3V o 6∐∞
:0
Ndtis gdr Shkneas enj Systifs :.5 Hntrdjucthdn td Systifs
E cdnthnudus-thfi cdnthnudus-thfi systif prdjucis e niw cdnthnudus-thfi cdnthnudus-thfi shknea (tmi dutput shknea) shknea) grdf e prdvhjij cdnthnudus-thfi cdnthnudus-thfi shknea shknea (tmi hnput shknea). shknea). Ettifpthnk e gdrfea jighnhthdn dg e systif hs e tijhdus ixirchsi hn evdhjhnk chrcuaerhty, sd wi whaa e`enjdn prichsi erthcuaethdn enj riay dn tmi hntuhthdn tmet jiviadps grdf ixefpais. [i riprisint e systif ‑S‖ whtm hnput shknea x(t ) enj dutput shknea y (t ) `y e `dx ae`iaij es smdwn `iadw.
Shnci e systif feps shkneas hntd shkneas, tmi dutput shknea et eny thfi t cen cen jipinj dn tmi hnput shknea veauis et eaa thfis. [i usi tmi fetmifethcea ndtethdn y (t ) 6 S ( x )( )(t ) td ifpmeshzi tmhs gect. Pifero Zmiri eri feny ndtethdns gdr e systif hn tmi ahtireturi. Gdr ixefpai, sdfi try td
ifpmeshzi tmet e systif feps shkneas hntd shkneas `y ushnk squeri `recoits, y (t ) 6 S_ x(t )V \irmeps tmhs cdnthnuis td tifpt tmi hntirpritethdn tmet, gdr ixefpai, y (<) 6 S_ x(<)V , h.i., y (<) jipinjs dnay dn x(<) . @ut, tmi ndtethdn hs jishknij td ifpmeshzi tmet gdr tmi hnput shknea ‑ x‖ tmi dutput shknea hs ‑ S ( x x),‖ enj tmi veaui dg tmhs dutput shknea et, sey t 6 <, hs y(<) 6 S ( x x)(<). Ixefpai Zmi runnhnk hntikrea hs en ixefpai dg e systif. E pmyshcea pmyshcea hntirpritethdn hs e cepechtdr
whtm hnput shknea tmi currint x(t ) tmrdukm tmi cepechtdr, enj dutput shknea tmi vdateki y (t ) ecrdss tmi cepechtdr. Zmin wi mevi, essufhnk unht cepechtenci, y (t ) 6
t
∯ x(ϊ ) j ϊ ∐∞
Hn tmhs cesi, tmi dutput et eny thfi t 5 jipinjs, dn hnput veauis gdr eaa t ≪ t 5 . Spichghceaay, et eny thfi t 5 , y (t 5 ) hs tmi eccufuaetij nit erie unjir x(t ) gdr ∐∞ 1 t ≪ t 5 . Hn tmi jhscriti-thfi cesi, wi usi en inthriay shfhaer ndtethdn, wrhthnk y_ nV 6 S ( x)_ nV enj riprisinthnk e systif `y tmi jhekref smdwn `iadw.
:.0 Systif \rdpirthis
Zmi jhscusshdn dg prdpirthis dg systifs whaa `i e `ht tintethvi et tmhs pdhnt, hn pert `iceusi tmi ndthdn dg e systif hs sd kinirea tmet ht hs jhgghcuat td hncauji eaa tmi jitehas, enj hn pert `iceusi tmi fetmifethcea jiscrhpthdn dg e systif fhkmt prisufi cirtehn prdpirthis dg eaadwe`ai hnput shkneas.
:3
Gdr ixefpai, tmi hnput shknea td e runnhnk-hntikretdr systif fust `i sugghchintay wiaa `imevij tmet tmi hntikrea hs jighnij. [i cen `i cdnshjire`ay fdri prichsi wmin wi cdnshjir spichghc caessis dg systifs tmet ejfht perthcuaer typis dg fetmifethcea jiscrhpthdns. Hn tmi hntirhf tmi hntint hs fehnay td iste`ahsm sdfi hntuhthdn cdncirnhnk prdpirthis dg systifs hn kinirea. [i prdciij whtm e ahst dg prdpirthis, pmresij hn tmi cdnthnudus-thfi cesi.
• Ceusea Systif E systif hs ceusea hg tmi dutput shknea veaui et eny thfi t jipinjs jipinjs dnay dn hnput shknea veauis gdr thfis nd aerkir tmen t . Ixefpais dg ceusea systifs eri y (t ) 6 3 x(t ∐ 0),
t
y(t ) 6
3 y( t) 6 x ( t )
∯ x(ϊ ) jϊ , ∐∞
Ixefpais dg systifs tmet eri ndt ceusea eri y (t ) 6 x(0),
y(t) 6 3 x( t + 0),
y( t) 6
t +5
∯ x(ϊ ) j ϊ
∐∞
• Fifdryaiss Systif E systif hs fifdryaiss hg tmi dutput veaui et eny thfi t jipinjs dnay dn tmi hnput shknea veaui et tmet sefi thfi, t . E fifdryaiss systif hs eaweys ceusea, tmdukm tmi rivirsi hs, dg cdursi, untrui. Ixefpais dg fifdryaiss systifs eri y (t ) 6 0 x(t ),
y ( t ) 6 x 0 ( t ),
x t y( t) 6 ti ( )
• Zhfi-Hnverhent Systif E systif systif hs thfi hnverhent hg hg gdr iviry hnput shknea x(t ) enj cdrrispdnjhnk dutput shknea y(t ) tmi gdaadwhnk prdpirty mdajs. Khvin eny cdnstent, t d , tmi hnput shknea x (t ) 6 x(t ∐ t d ) yhiajs tmi dutput shknea y (t ) 6 y (t ∐ t d ) . Zmhs hs sdfithfis ceaaij ‑smhgt hnverhenci,‖ shnci eny thfi smhgt dg en hnput shknea risuats hn tmi ixect sefi smhgt dg tmi dutput shknea. Ixefpais dg thfi-hnverhent systifs eri y (t ) 6 shn( x(t )),
y( t) 6
t
∯ x(ϊ ) jϊ ,
y( t) 6 3 x( t ∐ 0)
∐∞
Ixefpais dg systifs tmet eri ndt thfi hnverhent eri y (t ) 6 shn(t ) x( t),
y( t ) 6
t
∯
ϊ x(ϊ ) j ϊ
∐∞
Zd cmico hg e systif hs thfi hnverhent riquhris eppahcethdn dg tmi jighnhnk cdnjhthdn. Gdr ixefpai, gdr y (t ) 6
t
∯
ϊ x(ϊ ) j ϊ
∐∞
wi cdnshjir tmi hnput shknea x (t ) 6 x(t ∐ t d ) , wmiri t d hs eny cdnstent. Zmi cdrrispdnjhnk rispdnsi cdfputethdn `ikhns whtm y (t ) 6
t
∯ ∐∞
t
ϊ x (ϊ ) jϊ
6 ∯ ϊ x(ϊ ∐ td ) j ϊ ∐∞
Zd cdfperi tmhs td y (t ∐ t d ) , ht hs cdnvinhint td cmenki tmi verhe`ai dg hntikrethdn td σ 6 ϊ ∐ t d . Zmhs khvis
::
y (t ) 6
t ∐ t d
∯ (σ + td ) x(σ ) j σ
∐∞
wmhcm hs ndt tmi sefi es y (t ∐ td ) 6
t ∐t d
∯
ϊ x(ϊ ) j ϊ
∐∞
Zmirigdri tmi systif hs ndt thfi hnverhent.
• Ahnier Systif E systif hs ahnier hg hg gdr iviry pehr dg hnput shkneas x5 (t ), x0 (t ) , whtm cdrrispdnjhnk dutput shkneas y5 (t ), y0 (t ) , tmi gdaadwhnk mdajs. Gdr iviry cdnstent `, tmi rispdnsi td tmi hnput shknea x(t ) 6 `x5(t ) + x0 (t ) hs y (t ) 6 `y5 (t ) + y0 (t ) . (Zmhs hs fdri cdnchsi tmen pdpuaer twd-pert jighnhthdns dg ahnierhty hn tmi ahtireturi. Zeohnk ` 6 5 yhiajs tmi ejjhthvhty riquhrifint tmet tmi rispdnsi td x(t ) 6 x5(t ) + x0 (t ) `i y (t ) 6 y5(t ) + y0 (t ) . Enj teohnk x0 (t ) 6 x5(t ) khvis tmi mdfdkinhity riquhrifint tmet tmi rispdnsi td x (t ) 6 (` + 5) x5( t ) smduaj `i y (t ) 6 (` + 5) y5(t ) gdr eny cdnstent `. Ixefpais dg ahnier systifs eri t
y (t ) 6 i x(t ),
y(t ) 6 3 x( t ∐ 0),
y( t) 6
t
∯ x(ϊ ) j ϊ ∐∞
Ixefpais dg systifs tmet eri ‑ndnahnier‖ eri y (t ) 6
t
∯
0 x (σ ) jσ ,
y(t ) 6 5 + x( t),
y( t) 6 cds( x( t ))
∐∞ Pifero Ht smduaj `i ndtij tmet tmet gdr e ahnier systif tmi rispdnsi rispdnsi td tmi zird hnput hs tmi tmi zird
dutput shknea. Zd sii tmhs, shfpay teoi x5 (t ) 6 x0 (t ) (sd tmet y5 (t ) 6 y0 (t ) ) enj ` 6 ∐ 5 hn tmi jighnhthdn dg ahnierhty.
• Ste`ai Systif Piceaahnk tmi jighnhthdn dg e `dunjij shknea hn Sicthdn5.7, e systif hs ste`ai (dr `dunjij-hnput, `dunjij-dutput ste`ai ) hg iviry `dunjij hnput shknea yhiajs e `dunjij dutput dutput shknea. Hn jiteha, gdr eny hnput shknea x(t ) sucm tmet x )| 1 F gdr eaa t , wmiri F hs hs | x(t )| y(t )| e cdnstent, tmiri hs edtmir cdnstent \ sucm tmet tmi cdrrispdnjhnk dutput dutput shknea sethsghis | y )| 1 \ gdr eaa t . Ixefpais dg ste`ai systifs eri x (t )
y (t ) 6 i
,
y (t ) 6
x(t ∐ 0) 0
t + 5
y( t ) 6 sh shn( t) x( t )
,
Ixefpais dg ‑unste`ai‖ systifs eri t
y (t ) 6 i x(t ),
y(t ) 6
t
∯ x(ϊ ) j ϊ ∐∞
• Hnvirth`ai Systif E systif hs hnvirth`ai hg tmi hnput shknea cen `i unhquiay jitirfhnij grdf ondwaijki dg tmi dutput shknea. Ixefpais dg hnvirth`ai systifs eri 3
y (t ) 6 x (t ), y(t ) 6 3 x( t ∐ 0) + : t Zmi tmdukmtgua riejir whaa `i busthghe`ay nirvdus e`dut tmhs jighnhthdn. Hnvirth`hahty dg e fetmifethcea dpirethdn riquhris twd gieturis2 tmi dpirethdn fust `i dni-td-dni enj easd dntd. Shnci wi mevi ndt iste`ahsmij e caess dg hnput shkneas tmet wi cdnshjir gdr systifs, dr e cdrrispdnjhnk caess dg dutput shkneas, tmi hssui dg ‑dntd‖ hs aigt vekui. Enj shnci wi mevi
:9
jichjij td hkndri dr riesshkn veauis dg e shknea et hsdaetij pdhnts hn thfi gdr riesdns dg shfpahchty dr cdnvinhinci, ivin tmi hssui dg ‑dni-td-dni‖ hs unsittaij. Jitirfhnhnk hnvirth`hahty hnvirth`hahty dg e khvin systif cen `i quhti jhgghcuat. \irmeps tmi ieshist shtuethdn hs smdwhnk tmet e systif hs ndt hnvirth`ai hnvirth`ai `y ixmh`hthnk twd aikhthfetiay jhggirint hnput shkneas tmet yhiaj tmi sefi dutput shknea. Gdr ixefpai, y (t ) 6 x 0 (t ) hs ndt hnvirth`ai `iceusi cdnstent hnput shkneas dg x(t ) 6 5 enj x (t ) 6 ∐5 , gdr eaa t, yhiaj hjinthcea dutput shkneas. Es endtmir ixefpai, tmi systif j y (t ) 6 x(t ) jt hs ndt hnvirth`ai shnci x (t ) 6 5 + x(t ) yhiajs tmi sefi dutput shknea es x(t ). ). Es e ghnea ixefpai, hn e `inhkn sitthnk tmi runnhnk-hntikretdr runnhnk-hntikretdr systif y (t ) 6
t
∯ x(ϊ ) j ϊ ∐∞
hs hnvirth`ai `y tmi gunjefintea tmidrif dg ceacuaus2 j t
∯ x(ϊ ) jϊ 6 x(t ) jt ∐∞ @ut tmi gect rifehns tmet ticmnhceahthis eri riquhrij gdr tmhs cdncaushdn. Hg twd hnput shkneas jhggir dnay et hsdaetij pdhnts hn thfi, tmi dutput shkneas whaa `i hjinthcea, enj tmus tmi systif hs ndt hnvirth`ai hg wi cdnshjir sucm hnput shkneas td `i aikhthfetiay jhggirint. Eaa dg tmisi prdpirthis trensaeti ieshay td jhscriti-thfi systifs. Ahttai fdri hs riquhrij tmen td ripaeci perintmisis `y squeri `recoits enj t `y `y n. @ut rikerjaiss dg tmi thfi jdfehn, ht hs hfpdrtent td ndti tmet tmisi eri hnput-dutput prdpirthis dg systifs. Hn perthcuaer, ndtmhnk hs `ihnk stetij e`dut tmi hntirnea wdrohnks dg tmi systif, ivirytmhnk hs stetij hn tirfs dg hnput shkneas enj cdrrispdnjhnk dutput shkneas. Ghneaay ht hs wdrtmwmhai td tmhno dg mdw ydu wduaj escirtehn wmitmir e khvin pmyshcea systif, gdr wmhcm ydu jd ndt mevi e fetmifethcea jiscrhpthdn, jiscrhpthdn, mes iecm dg tmi prdpirthis wi cdnshjir. Zmet hs, wmet hnput shkneas wduaj ydu eppay, wmet fiesurifints dg tmi rispdnsi wduaj ydu teoi, enj wmet usi ydu wduaj feoi dg tmisi fiesurifints. :.3 Hntircdnnicthdns dg Systifs – @adco Jhekrefs
Inkhniirs dgtin cdnnict systifs, sdfithfis ceaaij su`systifs hn tmhs cdntixt, tdkitmir td gdrf niw systifs. En hffijheti quisthdn hs mdw td fetmifethceaay jiscrh`i tmi hnput-dutput `imevhdr dg tmi dvireaa systif hn tirfs dg tmi su`systif jiscrhpthdns. Dg cdursi tmisi cdnnicthdns cdrrispdnj td fetmifethcea dpirethdns dn tmi guncthdns jiscrh`hnk tmi su`systifs, su`systifs, enj tmiri eri e giw fehn cesis td cdnshjir. [i `ikhn whtm tmi twd su`systifs smdwn `iadw es ‑`adcos‖ riejy gdr cdnnicthdn,
enj cdnshjir tmi gdaadwhnk typis dg cdnnicthdns.
:7
• Ejjhthvi pereaaia cdnnicthdn cdnnicthdn
Hn tmhs `adco jhekref, es e perthcuaer ixefpai, tmi chrcuaer iaifint smdwn riprisints tmi shknij ejjhthdn dg tmi dutput shkneas dg tmi twd su`systifs. Zmhs cdnnicthdn jiscrh`is e niw systif whtm dutput khvin `y y (t ) 6 S5( x)(t ) ∐ S 0 ( x)( t) 6 ( S5 ∐ S0 )( x)( t ) Zmus tmi dvireaa systif hs riprisintij `y tmi guncthdn ( S5 ∐ S 0 )( x) 6 S5( x) ∐ S0 ( x) , enj wi cen riprisint ht es e shnkai `adco2
•
Sirhis dr cesceji cdnnicthdn
Zmi fetmifethcea jiscrhpthdn dg tmi dvireaa systif cdrrispdnjs td tmi cdfpdshthdn dg tmi guncthdns jiscrh`hnk tmi su`systifs, eccdrjhnk td y (t ) 6 S0 ( S5( x )) ))( t ) Zmus tmi dvireaa systif cen `i riprisintij es tmi `adco jhekref
•
Giij`eco cdnnicthdn
:;
Zmi giij`eco cdnnicthdn hs fdri cdfpahcetij tmen tmi dtmirs, `ut ixtrifiay hfpdrtent. Zd ettifpt jiviadpfint dg e fetmifethcea jiscrhpthdn dg tmi systif jiscrh`ij `y tmhs cdnnicthdn, wi `ikhn `y ndthnk tmet tmi dutput shknea hs tmi rispdnsi dg tmi systif S 5 whtm hnput shknea tmet hs
)(t ) . Zmirigdri, tmi dutput dg tmi suffir. Zmet hs, hnput shknea td S 5 hs x(t ) ∐ S 0 ( y )( y (t ) 6 S5 ( x ∐ S0 ( y))( t ) enj wi sii tmet tmi giij`eco cdnnicthdn yhiajs en iquethdn hnvdavhnk `dtm tmi hnput enj dutput shkneas. Yngdrtunetiay, whtmdut gurtmir fetmifethcea essufpthdns essufpthdns dn tmi twd guncthdns S 5 enj S 0 , wi cenndt sdavi tmhs iquethdn gdr y(t ) hn tirfs dg x(t ) td d`tehn e fetmifethcea jiscrhpthdn jiscrhpthdn dg tmi dvireaa systif dg tmi gdrf y (t ) 6 S ( x )( )(t ) Zmhs hnjhcetis tmi sdpmhsthcetij enj su`tai neturi dg tmi giij`eco cdnnicthdn dg systifs, enj wi riturn td tmhs hssui hn tmi siquia. Pifero Hn dur jhscusshdn dg hntircdnnicthdns dg systifs, en unjirayhnk essufpthdn essufpthdn hs tmet tmi
`imevhdr dg tmi tmi verhdus su`systifs su`systifs hs ndt eatirij `y tmi cdnnicthdn cdnnicthdn td dtmir dtmir su`systifs. su`systifs. Hn iaictrhcea inkhniirhnk, tmhs hs dgtin ixprissij es ‑tmiri eri nd adejhnk iggicts.‖ Zmhs essufpthdn hs ndt eaweys sethsghij hn precthci. E sicdnj essufpthdn hs tmet jdfehn enj renki cdnjhthdns eri fit. Gdr ixefpai, hn tmi cesceji cdnnicthdn, tmi dutput dg S 5 fust sethsgy eny essufpthdn riquhrij dn tmi hnput td S 0 . Hg tmi systif S 0 teois tmi squeri rddt dg tmi hnput shknea, S 0 ( x)(t ) 6 x(t ) , tmin tmi dutput dg S 5 fust nivir `icdfi nikethvi. Zmi `eshc hntircdnnicthdns dg jhscriti-thfi systifs eri cdfpaitiay shfhaer td tmi cdnthnudus-thfi cesi. Ixirchsis 5. Jitirfhni hg iecm dg tmi gdaadwhnk systifs hs ceusea, fifdryaiss, fifdryaiss, thfi hnverhent, ahnier, ahnier, dr
ste`ai. Busthgy ydur enswir. (e) y (t ) 6 0 x(t ) + 3 x 0 ( t ∐5) (`) y (t ) 6 cds 0 (t ) x(t ) (c) y (t ) 6 0 +
t ∐5
∯ x(t ∐ ϊ ) j ϊ
∐∞
(j) y (t ) 6 i
∐t
t
iϊ x(ϊ ) j ϊ
∯
∐∞ t
(i) y (t ) 6 (g) y (t ) 6
∯ x(0ϊ ) j ϊ ∐∞ x(∐t )
(k) y (t ) 6 x(3t ) (m) y (t ) 6
t
∯
i(
t ∐σ ) 0
x (σ ) j σ
∐∞
:=
t
(h) y (t ) 6 x(t )
i ∐ x(ϊ ) j ϊ ϊ
∯
∐∞
(b) y (t ) 6 3 x(t + 5) ∐ : t
(o) y (t ) 6 ∐ 3 x(t ) + ∯ 3x (σ )j σ <
(a)
y (t ) 6 3 x (t ) ∐ | x( t ∐ 3) |
0. Jitirfhni hg iecm dg tmi gdaadwhnk systifs hs ceusea, fifdryaiss, fifdryaiss, thfi hnverhent, ahnier, ahnier, dr
ste`ai. Busthgy ydur enswir. (e) y_n V 6 3 x_ n Vx_ n ∐5V n+0
∕
(`) y_ nV 6
x_ o V
o 6n∐0
_3n ∐ 0V (c) y_ nV 6 : x_3 n
∕
(j) y_ nV 6 (i) y_ nV 6
o
i u_ o V x_ n ∐ o V
o 6∐∞ n
∕ cds( x_ o V)
o 6 n ∐3
3. Jitirfhni hg iecm dg tmi gdaadwhnk systifs hs hnvirth`ai. hnvirth`ai. Hg ndt, spichgy twd jhggirint hnput
shkneas tmet yhiaj tmi sefi dutput. Hg sd, khvi en ixprisshdn gdr tmi hnvirsi systif. )V (e) y (t ) 6 cds_ x(t )V (`) y (t ) 6 x3 (t ∐ :) :. Jitirfhni hg iecm dg tmi gdaadwhnk systifs hs hnvirth`ai. hnvirth`ai. Hg ndt, spichgy twd jhggirint hnput
shkneas tmet yhiaj tmi sefi dutput. Hg sd, khvi en ixprisshdn gdr tmi hnvirsi systif. n
∕ x_ o V
(e) y_nV 6
o 6∐∞
(`) y_ nV 6 ( n ∐ 5) x_ nV (c) y_ nV 6 x_ nV ∐ x_ n ∐ 5V 9. Gdr iecm pehr dg systifs S5 , S 0 spichghij `iadw, khvi e fetmifethcea jiscrhpthdn dg tmi
cesceji cdnnicthdn S0 ( S 5 ) . (e) y5_ nV 6 x50_ n ∐ 0V , (`) y5_ nV 6
n
∕ o 6 ∐∞
y0_ nV 6 3 x0_ n + 0V
κ _ o V x5_ n ∐ o V ,
y0_ nV 6
n
∕ 0κ _ n ∐ oV x0_ o V
o 6 ∐∞
:8
Ndtis gdr Shkneas enj Systifs 9.5 JZ AZH Systifs enj Cdnvdauthdn
Jhscriti-thfi systifs tmet eri ahnier enj thfi hnverhent dgtin eri rigirrij td es AZH systifs. AZH systifs cdfprhsi e viry hfpdrtent caess dg systifs, enj tmiy cen `i jiscrh`ij `y e stenjerj fetmifethcea gdrfeahsf. gdrfeahsf. Zd iecm AZH systif tmiri cdrrispdnjs e shknea m_nV sucm tmet tmi hnputdutput `imevhdr dg tmi systif hs jiscrh`ij `y ∞
∕ x_ o Vm_ n ∐ o V
y_ nV 6
o 6∐∞
Zmhs ixprisshdn hs ceaaij tmi cdnvdauthdn suf riprisintethdn gdr AZH systifs. Hn ejjhthdn, tmi shgthnk prdpirty ieshay smdws tmet m_nV hs tmi rispdnsi dg tmi systif td e unht-puasi hnput shknea. Zmet hs, gdr x_nV 6 κ _nV, y_ nV 6
∞
∞
o 6 ∐∞
o 6 ∐∞
∕ x_ o Vm_ n ∐ o V 6 ∕
κ _ oV m_ n ∐ oV 6 m_ nV
Zmus tmi hnput-dutput `imevhdr dg e jhscriti-thfi, ahnier, thfi-hnverhent systif hs cdfpaitiay jiscrh`ij `y tmi unht-puasi rispdnsi dg tmi systif. Hg m_ nV hs ondwn, tmin tmi rispdnsi td eny hnput cen `i cdfputij grdf tmi cdnvdauthdn suf. Jirhvethdn Ht hs strehkmtgdrwerj td smdw tmet e systif jiscrh`ij `y tmi cdnvdauthdn cdnvdauthdn suf, whtm spichghij m_nV, hs e ahnier enj thfi-hnverhent systif. Ahnierhty hs caier, enj td smdw thfi
hnverhenci, cdnshjir e smhgtij hnput shknea xˇ_ nV 6 x_n ∐ nd V . Zmi systif rispdnsi td tmhs hnput shknea hs khvin `y ∞
∕ xˇ_ o V m_ n ∐ o V
yˇ_ nV 6
o 6∐∞
∞
6 ∕ x_ o ∐ nd V m_ n ∐ o V o 6∐∞
Zd riwrhti tmhs ixprisshdn, cmenki tmi suffethdn hnjix grdf o td td a 6 o ∐ N N , td d`tehn yˇ_ nV 6
∞
∕ x_a V m_ n ∐ nd ∐ aV
a 6∐∞
6 y_ n ∐ nd V Zmhs iste`ahsmis thfi hnverhenci. Ht hs aiss strehkmtgdrwerj td smdw tmet issintheaay eny AZH systif cen `i riprisintij `y tmi cdnvdauthdn suf. @ut tmi cdnvdauthdn riprisintethdn gdr ahnier, thfi-hnverhent systifs cen `i jiviadpij `y ejdpthnk e perthcuaer riprisintethdn gdr tmi hnput shknea enj tmin ingdrchnk tmi prdpirthis dg ahnierhty ahnierhty enj thfi thfi hnverhenci dn tmi cdrrispdnjhnk cdrrispdnjhnk rispdnsi. Zmi Zmi jitehas eri es gdaadws. Dgtin wi whaa riprisint e khvin shknea es e ahnier cdf`hnethdn dg ‑`eshs‖ shkneas tmet mevi cirtehn jishre`ai prdpirthis gdr tmi purpdsi et menj. Zd jiviadp e riprisintethdn gdr jhscriti-thfi AZH systifs, ht hs cdnvinhint td riprisint tmi hnput shknea es e ahnier cdf`hnethdn dg smhgtij unht puasi 0V, … . Hnjiij ht hs iesy td virhgy tmi ixprisshdn V , κ _ n ∐ 5V, κ _ n + 5V, κ _ n ∐ shkneas2 κ _nV,
9<
∞
∕ x_ o Vκ _ n ∐ o V
x_ nV 6
o 6∐∞
Miri tmi cdigghchint dg tmi shknea κ _n ∐ o V hn tmi ahnier cdf`hnethdn hs tmi veaui x_o V. V. Zmus, gdr ixefpai, hg n 6 3, tmin tmi rhkmt shji hs iveauetij `y tmi shgthnk prdpirty td virhgy ∞
_3 ∐ o V 6 x_3V ∕ x_o Vκ _3 o 6∐∞
[i cen usi tmhs shknea riprisintethdn td jirhvi en AZH systif riprisintethdn es gdaadws. Zmi rispdnsi dg en AZH systif td e unht puasi hnput, x_nV 6 κ _nV, hs khvin tmi spichea ndtethdn y_nV 6 m_nV. Zmin `y thfi hnverhenci, tmi rispdnsi td e o ∐smhgtij unht puasi, x_nV 6 κ _ n ∐ o V hs y_nV 6 m_n ∐ o V . Gurtmirfdri, `y ahnierhty, tmi rispdnsi td e ahnier cdf`hnethdn dg smhgtij unht puasis hs tmi tmi ahnier cdf`hnethdn cdf`hnethdn dg tmi rispdnsis rispdnsis td tmi tmi smhgtij unht puasis. Zmet hs, hs, tmi rispdnsi rispdnsi td x_nV, es wrhttin e`dvi, hs ∞
∕ x_ o V m_ n ∐ o V
y_ nV 6
o 6∐∞
Zmus wi mevi errhvij et tmi cdnvdauthdn suf riprisintethdn gdr AZH systifs. Zmi cdnvdauthdn riprisintethdn gdaadws jhrictay grdf ahnierhty enj thfi hnverhenci – nd dtmir prdpirthis dg tmi systif eri essufij (tmdukm tmiri eri sdfi cdnvirkinci hssuis tmet wi mevi hkndrij). En eatirneti ixprisshdn gdr tmi cdnvdauthdn suf hs d`tehnij `y cmenkhnk tmi suffethdn verhe`ai grdf o td td a 6 n ∐ o o 2 y_ nV 6
∞
∕ m_a V x_ n ∐ aV o 6∐∞
Ht hs caier grdf tmi cdnvdauthdn riprisintethdn tmet hg wi ondw tmi unht-puasi rispdnsi dg en AZH systif, tmin wi cen cdfputi tmi rispdnsi td eny dtmir hnput shknea `y iveauethnk tmi cdnvdauthdn suf. Hnjiij, wi spichghceaay ae`ia AZH systifs whtm tmi unht-puasi rispdnsi hn jrewhnk `adco jhekrefs, es smdwn `iadw
Zmi jifdnstrethdn `iadw cen miap whtm vhsueahzhnk enj unjirstenjhnk tmi cdnvdauthdn riprisintethdn. Bdy dg Cdnvdauthdn (Jhscriti Zhfi) Pispdnsi Cdfputethdn
Iveauethdn dg tmi cdnvdauthdn ixprisshdn, khvin x_nV enj m_nV, hs ndt es shfpai es fhkmt `i ixpictij `iceusi ht hs ectueaay e cdaaicthdn dg suffethdns, dvir tmi hnjix o , tmet cen teoi jhggirint gdrfs gdr jhggirint veauis dg n. Zmiri eri sivirea stretikhis tmet cen `i usij gdr iveauethdn, enj tmi fehn dnis eri rivhiwij `iadw.
• Eneaythc iveauethdn [min x_nV enj m_nV mevi shfpai, niet eneaythcea ixprisshdns, enj tmi cmerectir dg tmi suffethdn jdisn‘t cmenki hn cdfpahcetij weys es n cmenkis, sdfithfis y_nV cen `i cdfputij eneaythceaay.
95
Ixefpai Suppdsi tmi unht puasi rispdnsi dg en AZH systif hs e unht refp,
m_ nV 6 r_ nV 6 n u_ nV Zd cdfputi tmi rispdnsi dg tmhs systif td e unht-stip hnput, wrhti tmi cdnvdauthdn riprisintethdn riprisintethdn es ∞ ∞ y_ nV 6 ∕ x_ o Vm_ n ∐ o V 6 ∕ u_ oV ( n ∐ o) u_ n ∐ o V o 6 ∐∞ o 6 ∐∞ ∞ 6 ∕ (n ∐ o ) u_ n ∐ o V o 6 < Ndti tmet hn tmi sicdnj ahni ahni tmi unht-stip u_o V hn tmi suffenj hs rifdvij, `ut tmi adwir ahfht dg tmi suf hs rehsij td zird, enj tmhs hs veahj rikerjaiss dg tmi veaui dg n. Ndw, hg n 1 <, tmin tmi <. Zmirigdri y_nV 6 < gdr n erkufint dg tmi stip guncthdn hn tmi suffenj hs nikethvi gdr iviry o ≩ < 1 <. @ut, gdr n ≩ < <, wi cen rifdvi tmi stip u_ n ∐ o V grdf tmi suffenj hg wi adwir tmi uppir ahfht td n. Zmin y_ nV 6
n
∕ ( n ∐ o ) 6 n + ( n ∐5) + + 0 +5 + < o 6 <
Yshnk tmi viry daj trhco dg pehrhnk tmi n whtm tmi < , tmi (n ∐ 5) whtm tmi 5, enj sd dn, wi sii tmet iecm pehr sufs td n. Cdunthnk tmi nuf`ir dg pehrs gdr ivin n enj gdr djj n khvis
⎫ n(n + 5) , n≩< ⎢ y_ nV 6 ⎭ 0 ⎢⎨ <, n1< dr, fdri cdfpectay, y_nV 6
n( n + 5)
0
u_ nV
Pifero @iceusi dg tmi priveainci dg ixpdninthea shkneas, tmi gdaadwhnk kidfitrhc-sirhis
gdrfuaes gdr jhscriti-thfi shknea ceacuaethdns eri usigua gdr eneaythc iveauethdn dg cdnvdauthdn. Gdr eny cdfpaix nuf`ir ε ≬ 5 , N ∐5
5 ∐ ε N ∕ ε 6 5 ∐ ε n 6< Gdr eny cdfpaix nuf`ir sethsgyhnk | ε |1 5 , n
∞
∕ ε n 6 n6<
5 5 ∐ ε
• Krepmhcea fitmdj Zmhs fitmdj hs usigua usigua gdr fdri cdfpahcetij cdfpahcetij cesis. [i o V, shfpay padt tmi twd shkneas, x_o V enj m_n ∐ o V, hn tmi suffenj virsus o gdr gdr tmi veaui dg n dg hntirist, tmin pirgdrf ‑adaaypdp-`y-adaaypdp‖ fuathpahcethdn fuathpahcethdn enj padt tmi suffenj, enj tmin ejj up tmi adaaypdp veauis td d`tehn y_nV. Ixefpai Zd riwdro tmi privhdus ixefpai `y tmi krepmhcea fitmdj, wrhthnk
90
y_ nV 6
∞
∕ x_ o Vm_ n ∐ o V o 6∐∞
ghrst padt m_o V es smdwn,
o V enj, Zmin, gdr tmi veaui dg n dg hntirist, gahp enj smhgt. Gdr n 6 3, wi padt m_3 ∐ o td gechahteti tmi fuathpahcethdn, fuathpahcethdn, padt x_o V hffijhetiay `iadw2
Zmin adaaypdp-`y-adaaypdp adaaypdp-`y-adaaypdp fuathpahcethdn khvis e padt dg tmi suffenj,
Ejjhnk up tmi adaaypdp veauis khvis y_3V 6 3 + 0 +5 6 7 o V dni sefpai td tmi rhkmt td d`tehn e padt dg m_: ∐ o o V Zd cdfputi, sey, y_:V, sahji tmi padt dg m_3 ∐ o enj ripiet tmi fuathpahcethdn fuathpahcethdn whtm x_o V enj ejjhthdn. Hn shfpai cesis sucm es tmhs, tmiri hs ahttai niij td rijrew `iceusi tmi pettirn hs caier. Ivin hn cdfpahcetij cesis, ht hs dgtin iesy td hjinthgy renkis dg n wmiri y_nV 6 <, `iceusi tmi padts dg x_o V enj m_n ∐ o o V eri ‑ndn-dviraepphnk.‖ Hn tmi ixefpai, tmhs caieray mdajs gdr n 1 <.
• AZH caivirniss Zmi tmhrj fitmdj feois feois usi dg tmi prdpirthis dg ahnierhty ahnierhty enj thfi hnverhenci, enj hs wiaa suhtij gdr tmi cesi wmiri x_nV mes dnay e giw ndnzird veauis. Hnjiij, ht hs shfpay e spicheahzethdn dg tmi epprdecm wi tddo td tmi jirhvethdn dg tmi cdnvdauthdn suf. Ixefpai [htm en er`htrery m_nV, suppdsi tmet tmi hnput shknea cdfprhsis tmrii ndnzird adaaypdps,
enj cen `i wrhttin es
x_ nV 6 κ _ nV + 0κ _ n ∐5V ∐ 3κ _ n ∐ 3V Zmin ahnierhty enj thfi hnverhenci jhcteti tmet y_nV 6 m_ nV + 0 m_ n ∐5V ∐ 3 m_ n ∐ 3V Jipinjhnk dn tmi gdrf dg tmi unht-puasi rispdnsi, tmhs cen `i iveauetij eneaythceaay, dr `y krepmhcea ejjhthdn dg padts dg tmi unht-puasi rispdnsi enj hts smhgts enj efpahtuji sceais.
Pifero Zmi cdnvdauthdn dpirethdn cen ixpadji – geha td `i wiaa jighnij – gdr perthcuaer cmdhcis dg hnput shknea enj unht-puasi rispdnsi. Gdr ixefpai, whtm x_nV 6 m_nV 6 5, gdr eaa n, tmiri hs nd
93
veaui dg n gdr wmhcm y_nV hs jighnij, `iceusi tmi cdnvdauthdn suf hs hnghnhti gdr iviry n. Hn dur jirhvethdn, wi jhj ndt wdrry e`dut cdnvirkinci dg tmi suffethdn. Zmhs ekehn hs e cdnsiquinci dg dur jichshdn ndt td `i prichsi e`dut caessis dg eaadwe`ai shkneas, dr, fdri fetmifethceaay, jdfehns enj renkis. Zmi jhahkint stujint fust eaweys `i dn tmi addo dut gdr sucm endfeahis. Gurtmirfdri, tmiri eri AZH systifs tmet cenndt `i jiscrh`ij `y e cdnvdauthdn suf, tmdukm tmisi eri fdri hn tmi neturi dg fetmifethcea djjhthis tmen inkhniirhnk aikhthfechis. aikhthfechis. Hn eny cesi, tmhs hs tmi riesdn wi sey tmet ‑issintheaay‖ eny AZH systif cen `i jiscrh`ij `y tmi cdnvdauthdn suf. 9.0 \rdpirthis dg Cdnvdauthdn – Hntircdnnicthdns dg JZ AZH Systifs
Zmi cdnvdauthdn dg twd shkneas yhiajs e shknea, enj tmhs d`vhdusay hs e fetmifethcea dpirethdn – e sdrt dg ‑wihrj fuathpahcethdn‖ fuathpahcethdn‖ dg shkneas. Zmhs fetmifethcea dpirethdn d`iys cirtehn eaki`rehc prdpirthis, enj tmisi prdpirthis cen `i hntirpritij es prdpirthis dg systifs enj tmihr tmihr hntircdnnicthdns. Zd shfpahgy fettirs, wi ejdpt e smdrtmenj ‑ster‖ ndtethdn gdr cdnvdauthdn enj wrhti ∞
y_ nV 6 ( x ∛ m)_ nV 6 ∕ x_ o V m_ n ∐ o V o 6∐∞
Ndti tmet shnci, shnci, gdr eny n, tmi veaui dg y_nV hn kinirea jipinjs dn eaa veauis dg tmi shkneas x_nV enj m_nV, wi usi tmi fdri kinirea dpiretdr ndtethdn styai, Hn perthcuaer, wi jd ndt wrhti wrhti y_nV 6 x_ nV ∛ m_n V `iceusi dg tmi tifptethdn td cdncauji tmet, gdr ixefpai, y_0V 6 x_0V ∛ m_0V . Eaki`rehc prdpirthis dg tmi ‑ster dpirethdn‖ eri jhscussij `iadw.
• Cdffutethvhty Cdnvdauthdn Cdnvdauthdn hs cdffutethvi. Zmet hs, hs, ( x ∛ m)_ nV 6 ( m ∛ x)_ nV , dr, hn cdfpaiti jiteha, ∞
∞
o 6 ∐∞
o 6 ∐∞
∕ x_o Vm_ n ∐ o V 6 ∕ m_ o V x_ n ∐ oV ,
gdr eaa n
Zmi prddg dg tmhs hnvdavis tmi ghrst stenjerj ruai gdr prddgs hn tmhs cdursi2 Ysi cmenki dg verhe`ai o . Zmin dg suffethdn. @ikhnnhnk whtm tmi aigt shji, ripaeci tmi suffethdn verhe`ai o `y `y q 6 n ∐ o es o ← µ∞ , q ← ∓∞ , `ut (unahoi hntikrethdn) ht jdis ndt fettir wmitmir wi suf grdf aigt-tdrhkmt dr rhkmt-td-aigt. Zmus
( x ∛ m)_ nV 6
∞
∕
x_ o Vm_ n ∐ o V 6
o 6 ∐∞
∐∞
∕ x_ n ∐ qV m_ qV
q 6∞
∞
6 ∕ m_qV x_ n ∐ qV 6 ( m ∛ x)_ nV q 6∐∞
Yshnk tmhs risuat, tmiri eri twd jhggirint weys td jiscrh`i hn wdrjs tmi rdai dg tmi unht-puasi o V jitirfhnis rispdnsi veauis hn tmi hnput-dutput `imevhdr dg en AZH systif. Zmi veaui dg m_n ∐ o mdw tmi ntm veaui dg tmi dutput shknea jipinjs dn tmi o tm veaui dg tmi hnput shknea. Dr, tmi veaui qV. dg m_qV jitirfhnis mdw tmi veaui dg y_nV jipinjs dn tmi veaui dg x_n ∐ q
• Essdchethvhty Cdnvdauthdn hs essdchethvi. essdchethvi. Zmet hs, hs,
( x ∛ ( m5 ∛ m0 ))_ nV 6 (( x ∛ m5) ∛ m0 )_ nV
9:
Zmi prddg dg tmhs prdpirty hs e fissy ixirchsi hn fenhpuaethnk suffethdns, suffethdns, enj ht hs dfhttij.
• Jhstrh`uthvhty jhstrh`uthvi (whtm rispict td ejjhthdn). ejjhthdn). Zmet Zmet hs, Jhstrh`uthvhty Cdnvdauthdn hs jhstrh`uthvi
( x ∛ ( m5 + m0 ))_ nV 6 ( x ∛ m5)_ nV + ( x ∛ m0 )_ nV Dg cdursi, jhstrh`uthvhty hs e ristetifint dg pert dg tmi ahnierhty prdpirty dg AZH systifs enj sd nd prddg hs niijij. Zmi rifehnhnk pert dg tmi ahnierhty ahnierhty cdnjhthdn hs wrhttin hn tmi niw ndtethdn ndtethdn es gdaadws. Gdr eny cdnstent `, ((`x) ∛ m)_ nV 6 ` ( x ∛ m)_ nV
•
Smhgt \rdpirty Zmhs hs shfpay e ristetifint dg dg tmi thfi-hnverhenci thfi-hnverhenci prdpirty, prdpirty, tmdukm tmi
ndtethdn feois ht e `ht ewowerj. Gdr eny hntikir nd , hg xˇ_ nV 6 x_n ∐ nd V , tmin
( xˇ ∛ m ) _nV 6 ( x ∛ m )_ n ∐ nd V • Hjinthty Ht hs wdrtm ndthnk tmet tmi ‑ster‖ dpirethdn dpirethdn mes tmi unht unht puasi es en hjinthty iaifint. Nefiay, ( x ∛ κ )_ nV 6 x_ nV Zmhs cen `i hntirpritij hn systif-tmidrithc tirfs es tmi gect tmet tmi hjinthty systif, y_ nV 6 x_ nV mes tmi unht-puasi rispdnsi m_ nV 6 κ _ nV . Easd wi cen wrhti (κ ∛ κ ) _nV 6 κ _ nV , en ixprisshdn tmet seys ndtmhnk fdri tmen2 Zmi unht puasi hs tmi unht-puasi rispdnsi dg tmi systif wmdsi unht puasi rispdnsi hs e unht puasi. puasi. Zmisi eaki`rehc prdpirthis dg tmi fetmifethcea dpirethdn dg cdnvdauthdn aiej jhrictay td fitmdjs gdr jiscrh`hnk tmi hnput-dutput `imevhdr dg hntircdnnicthdns dg AZH systifs. Dg cdursi wi usi `adco jhekref riprisintethdns td jiscrh`i hntircdnnicthdns, hntircdnnicthdns, `ut gdr gdr AZH systifs systifs wi ae`ia iecm `adco whtm tmi tmi cdrrispdnjhnk unht-puasi rispdnsi. rispdnsi. Gdr ixefpai, ixefpai,
Jhstrh`uthvhty Jhstrh`uthvhty hfpahis tmet tmi hntircdnnicthdn `iadw
mes tmi sefi hnput-dutput `imevhdr es tmi systif
99
Cdffutethvhty Cdffutethvhty enj essdchethvhty hfpay tmet tmi hntircdnnicthdns
`dtm mevi tmi tmi sefi hnput-dutput hnput-dutput `imevhdr `imevhdr es tmi systif
Ghneaay, wi eneayzi tmi giij`eco cdnnicthdn
es gdaadws, hn en ettifpt td d`tehn e jiscrhpthdn gdr hts hnput-dutput `imevhdr. [htm tmi hntirfijheti shknea i_n V ae`iaij es smdwn, wi cen wrhti y_ nV 6 ( k ∛ i)_ nV enj i_ nV 6 x_ nV ∐ ( m ∛ y)_ nV es jiscrhpthdns dg tmi hntircdnnicthdn. Su`sthtuthnk Su`sthtuthnk tmi sicdnj hntd tmi ghrst khvis y_ nV 6 ( k ∛ x)_ nV ∐ ( k ∛ m ∛ y)_ nV dr, wrhthnk y_ nV 6 (κ ∛ y)_ nV wi kit
( (κ ∐ k ∛ m) ∛ y )_ nV 6 ( k ∛ x)_ nV Mdwivir, wi cenndt sdavi gdr y_ nV dn tmi aigt shji unaiss wi ondw tmet tmi AZH systif whtm ∐ k ∛ m)_ nV hs hnvirth`ai. Ait‘s stdp miri, enj riturn td tmi prd`aif dg unht-puasi rispdnsi (κ ∐ jiscrh`hnk tmi giij`eco cdnnicthdn egtir jiviadphnk fdri tddas. @ut gdr systifs whtmdut giij`eco, tmi eaki`rehc ruais gdr tmi cdnvdauthdn dpirethdn dpirethdn prdvhji en iesy gdrfeahsf gdr shfpahgyhnk `adco jhekrefs. Zyphceaay Zyphceaay ht hs ieshist td stert et tmi dutput shknea enj wrhti jiscrhpthdns dg tmi hntirfijheti shkneas (ae`iaij hg niijij) wmhai wdrohnk `eco tdwerj tmi hnput shknea.
97
smdwn `iadw, tmiri hs nd niij td ae`ia hntirnea shkneas es Ixefpai Gdr tmi hntircdnnictij systif smdwn tmi structuri hs riesdne`ay trensperint.
Zmi dutput shknea cen `i wrhttin es y_ nV 6 ( m: ∛ x)_ nV + ( m0 ∛ m5 ∛ x)_ nV ∐ ( m3 ∛ m5 ∛ x)_ nV
6 ( (m: + m0 ∛ m5 ∐ m3 ∛ m5) ∛ x )_ nV Zmus tmi hnput-dutput `imevhdr dg tmi systif hs hjinthcea td tmi hnput-dutput `imevhdr dg tmi systif
9.3 JZ AZH Systif \rdpirthis
Shnci tmi hnput-dutput `imevhdr dg e jhscriti-thfi AZH systif hs cdfpaitiay cmerectirhzij `y hts unht-puasi rispdnsi, m_nV, vhe tmi cdnvdauthdn ixprisshdn y_nV 6
∞
∞
o 6 ∐∞
o 6 ∐∞
∕ x_ o Vm_ n ∐ o V 6 ∕ m_ oV x_ n ∐ o V
tmi hnput-dutput prdpirthis dg tmi systif cen `i cmerectirhzij viry prichsiay hn tirfs dg prdpirthis dg m_nV.
• Ceusea Systif En AZH systif hs ceusea hg enj dnay hhgg m_nV 6 < gdr n 1 <, tmet hs, hg enj dnay hg m_nV hs rhkmt shjij. Zmi prddg dg tmhs hs quhti iesy grdf tmi cdnvdauthdn ixprisshdn. Hg tmi unht-puasi rispdnsi hs rhkmt shjij, tmin tmi cdnvdauthdn ixprisshdn shfpahghis td y_ nV 6
∞
∕ m_ o V x_ n ∐ o V o 6 <
enj, et eny veaui dg n, tmi veaui dg y_nV jipinjs dnay dn tmi currint enj ierahir veauis dg tmi hnput shknea. Hg tmi unht-puasi rispdnsi hs ndt rhkmt shjij, tmin ht hs iesy td sii tmet tmi veaui dg y_nV et e perthcuaer n jipinjs dn guturi veauis dg tmi hnput shknea.
9;
• Fifdryaiss Systif En AZH systif hs hs fifdryaiss fifdryaiss hg enj dna dnayy hg m_nV 6 cκ _nV, gdr sdfi cdnstent c. Ekehn, e prddg hs quhti iesy td erkui grdf tmi cdnvdauthdn ixprisshdn. ixprisshdn. • Ste`ai Systif En AZH systif systif hs (`dunjij-hnput, `dunjij-dutput) `dunjij-dutput) ste`ai hg enj dnay hg tmi unht-puasi rispdnsi hs e`sdautiay suffe`ai. Zmet hs, ∞
∕ | m_nV | n 6∐∞
hs ghnhti. Zd prdvi tmhs, suppdsi x_nV hs e `dunjij hnput, tmet hs, tmiri hs e cdnstent F sucm sucm tmet | x_ nV | ≪ F gdr eaa n. Zmin tmi e`sdauti veaui dg tmi dutput shknea sethsghis ∞
∞
o 6 ∐∞
o 6 ∐∞
∕ m_o V x_ n ∐ oV | ≪ ∕ | m_ oV || x_ n ∐ o V |
| y_ nV | 6 | ∞
≪ F ∕ | m_ o V | o 6∐∞
Zmirigdri, hg tmi e`sdauti suffe`hahty cdnjhthdn cdnjhthdn mdajs, tmi dutput shknea hs `dunjij gdr eny `dunjij hnput hnput shknea, enj wi mevi smdwn tmet tmi systif hs ste`ai. Zd prdvi tmet ste`hahty dg tmi systif hfpahis e`sdauti suffe`hahty riquhris cdnshjire`ai caivirniss. Cdnshjir tmi hnput x_nV jighnij `y ⎫ 5, m_n V ≩ < x_ ∐ nV 6 ⎭ ⎨∐5, m_ nV 1 < Caieray x_ nV hs e `dunjij hnput shknea, enj tmi cdrrispdnjhnk rispdnsi y_nV et n 6 < hs y_
∞
∞
o 6 ∐∞
o 6 ∐∞
∕ m_ o V x_ ∐ o V 6 ∕ | m_ o V |
Shnci tmi systif hs ste`ai, y_nV hs `dunjij, enj tmirigdri y_
m_ nV 6 (<.9) u_ nV
hs e ste`ai systif shnci
∞
∞
n 6 ∐∞
n6<
∕ | m_nV | 6 ∕ (<.9) n 6
5 60 5 ∐ <.9
Dn tmi dtmir menj, tmi systif whtm unht-puasi rispdnsi m_ nV 6 u_ n ∐ 5V hs unste`ai.
• Hnvirth`ai Systif Zmiri hs nd shfpai cmerectirhzethdn cmerectirhzethdn dg hnvirth`hahty hnvirth`hahty hn tirfs dg tmi tmi unht-puasi rispdnsi. Mdwivir, hn perthcuaer ixefpais ht hs sdfithfis pdssh`ai td cdfputi tmi unht-puasi rispdnsi dg en hnvirsi systif, m H _ nV , grdf tmi riquhrifint (m ∛ m H )_ nV 6 κ _ nV Zmhs cdnjhthdn ixprissis tmi neturea riquhrifint tmet e systif hn cesceji whtm hts hnvirsi smduaj `i tmi hjinthty hjinthty systif.
9=
tmet hs, tmi AZH systif whtm unht puasi Ixefpai Zd cdfputi en hnvirsi dg tmi runnhnk suffir, tmet rispdnsi m_nV 6 u_ nV , wi fust ghnj m H _ nV tmet sethsghis ∞
∕ u_o V m H _n ∐ oV 6 κ _ nV
o 6∐∞
Shfpahgyhnk tmi suffethdn khvis
∞
∕ o 6 <
m H _ n ∐ o V 6 κ _ nV
Ht hs caier tmet wi smduaj teoi m H _ nV 6 < , gdr n 1 < . Yshnk tmhs risuat, gdr n 6 < tmi riquhrifint hs ∞
∕ o 6 <
Gdr n 6 5 tmi riquhrifint hs ∞
∕ o 6 <
m H _∐ o V 6 mH _
m H _5 ∐ o V 6 mH _5V + mH _
wmhcm khvis m H _5V 6 ∐5 . Cdnthnuhnk gdr gurtmir veauis dg n, ht hs caier tmet tmi hnvirsi-systif riquhrifint hs sethsghij `y teohnk eaa rifehnhnk veauis dg m H _nV td `i zird. Zmus tmi hnvirsi systif mes tmi unht puasi rispdnsi m H _nV 6 κ _ nV ∐ κ _ n ∐5V Dg cdursi, ht hs iesy td sii tmet hn kinirea tmi dutput dg tmhs hnvirsi systif hs tmi ghrst jhggirinci dg tmi hnput shknea. 9.: Pispdnsi td Shnkuaerhty Shkneas
Zmi rispdnsi dg e JZ AZH systif td tmi `eshc shnkuaerhty shkneas hs quhti iesy td cdfputi. Hg tmi systif hs jiscrh`ij `y y_ nV 6
∞
∕ x_ o Vm_ n ∐ o V o 6∐∞
tmin tmi unht puasi rispdnsi hs shfpay y_ nV 6 m_n V . Hg tmi hnput shknea hs e unht stip, x_n V 6 u_ nV , tmin y_ nV 6
∞
∞
o 6 ∐∞
o 6 <
∕ u_ o Vm_n ∐ o V 6 ∕ m_ n ∐ o V n
6 ∕ m_a V a 6∐∞
Hn wdrjs, tmi unht-stip rispdnsi hs tmi runnhnk suf dg tmi unht-puasi rispdnsi. Dg cdursi, hg tmi systif hs ceusea, tmet hs, tmi unht-puasi rispdnsi hs rhkmt shjij, tmin
98
⎫ n ⎢ ∕ m_o V , n ≩ < y_ nV 6 ⎭ o 6 < ⎢ <, n1< ⎨ ⎟ n ⎞ 6 ⎑ ∕ m_o V ⎔ u_ nV ⎖ o 6 < ⎬ Gdr hnput shkneas tmet mevi dnay e sfeaa nuf`ir dg ndnzird veauis, tmi `eshc epprdecm dg ‑AZH caivirniss‖ cen `i eppahij td iveaueti tmi rispdnsi. Zmet hs, hg tmi hnput shknea cen `i wrhttin es x_ nV 6 x_ n< Vκ _ n ∐ n< V + x_ n5Vκ _ n ∐ n5V + + x_ nf Vκ _ n ∐ nf V
tmin tmi rispdnsi hs khvin `y y_ nV 6 x_ n< Vm_ n ∐ n< V + x_ n5V y_ n ∐ n5V + + x_ nf V y_ n ∐ nf V 9.9 Pispdnsi td Ixpdnintheas (Ihkinguncthdn \rdpirthis)
Gdr hfpdrtent caessis dg AZH systifs, tmi rispdnsis td cirtehn typis dg ixpdninthea hnput shkneas mevi perthcuaeray shfpai gdrfs. Zmisi shfpai gdrfs unjirahi feny epprdecmis td tmi eneayshs dg AZH systifs, enj wi cdnshjir sivirea verhents, iecm dg wmhcm riquhris sahkmtay jhggirint essufpthdns dn tmi AZH systif. Gdr mhstdrhcea riesdns hn fetmifethcs, en hnput shknea x_nV hs ceaaij en ihkinguncthdn dg tmi systif hg tmi cdrrispdnjhnk dutput shknea hs shfpay e cdnstent fuathpai dg tmi hnput shknea. ([i jd pirfht tmi cdnstent td `i cdfpaix, wmin cdnshjirhnk cdfpaix hnput shkneas.)
• Piea Ihkinguncthdns Ihkinguncthdns Zmi rispdnsi dg e ceusea, ste`ai AZH systif td e krdwhnk ixpdninthea hnput shknea hs e cdnstent fuathpai dg tmi ixpdninthea, wmiri tmi cdnstent jipinjs dn tmi ixpdnint. Zd wdro tmhs dut hn jiteha, suppdsi tmi AZH systif ∞
∕ m_o V x_ n ∐ o V
y_ nV 6
o 6∐∞
hs ceusea enj ste`ai, tmet hs, m_nV hs rhkmt-shjij enj e`sdautiay suffe`ai. Gurtmirfdri, suppdsi suppdsi tmi hnput shknea hs tmi riea ixpdninthea shknea σ d n
x_ nV 6 i
, ∐∞ 1 n 1 ∞
wmiri σ d ≩ < . Zmin tmi rispdnsi cdfputethdn `icdfis y_ nV 6
∞
σ d (n ∐ o )
∕ m_o V i o 6 ∐∞
⎟ ∞ ⎞ 6 ⎑ ∕ m_ o V i∐σ d o ⎔ iσ d n ⎖ o 6 ∐∞ ⎬
Yshnk tmi ceuseahty enj ste`hahty essufpthdns essufpthdns dn tmi systif, tmi essufpthdn tmet σ d ≩ < , enj `eshc prdpirthis dg e`sdauti veauis, veauis,
7<
∞
∕
m_o Vi
∞
∐σ d o
6 ∕ m_ o Vi∐σ d o
o 6 ∐∞
o 6 <
∞
≪ ∕ | m_o V | i∐σ d o o 6 <
∞
≪ ∕ | m_o V | 1 ∞ o 6 <
Zmet hs, tmi suffethdn cdnvirkis td e riea nuf`ir, wmhcm wi wrhti es M (σ d ) td smdw tmi jipinjinci dg tmi riea nuf`ir dn tmi veaui cmdsin gdr σ d . Zmus tmi dutput shknea hs e sceaer fuathpai dg tmi hnput shknea, σ d n
y_nV 6 M (σ d ) i
wmiri M (σ d ) 6
∞
∕ m_ o V i∐σ d o o 6 <
Dg cdursi tmi ixpdninthea hnput, wmhcm `ikhns es e venhsmhnkay sfeaa shknea et n ← ∐∞ , krdws whtmdut `dunj es n hncriesis, es jdis tmi rispdnsi, unaiss σ d 6 < dr M (σ d ) 6 < . Ixefpai Gdr tmi AZH systif whtm unht-puasi rispdnsi rispdnsi n
m_ nV 6 (<.9) u_ nV
khvin eny σ d ≩ < wi cen cdfputi M (σ d ) 6
∞
∕
∞
(<.9) o i ∐σ d o 6
o 6<
∕ ( i∐σ d / 0) o 6 o 6 <
5
5∐ i
∐σ d
/0
Zmirigdri tmi rispdnsi dg tmi systif td tmi hnput σ d n
x_ nV 6 i
, ∐∞ 1 n 1 ∞
hs y_nV 6
0 0∐i
∐σ d
σ d n
i
Ht hs hfpdrtent td ndti tmet dnay dni suffethdn fust `i iveauetij td cdfputi tmi ihkinguncthdn rispdnsi. Cdntrest tmhs whtm tmi usuea cdnvdauthdn, wmhcm typhceaay hnvdavis e gefhay dg suffethdns whtm tmi neturi dg tmi suffethdn cmenkhnk whtm tmi veaui dg n.
• Cdfpaix Ihkinguncthdns Ht hs fetmifethceaay fetmifethceaay cdnvinhint td cdnshjir cdfpaix hnput shkneas td AZH systifs, tmdukm dg cdursi tmi unht-puasi rispdnsi, m_nV, hs essufij td `i riea. Hg e cdfpaix hnput shknea hs wrhttin hn rictenkuaer gdrf es x_ nV 6 x P _ nV + bxH _ nV
wmiri, gdr iecm n, x P _nV 6 Pi { x_ nV} ,
tmin tmi cdrrispdnjhnk, cdfpaix dutput shknea hs
75
xH _ nV 6 Hf { x_ nV}
∞
∕ x_ o Vm_ n ∐ o V
y_ nV 6
o 6∐∞
∞
∞
o 6 ∐∞
o 6 ∐∞
6 ∕ x P _o Vm_ n ∐ o V + b ∕ xH _ o Vm_ n ∐ o V Zmet hs,
Pi { y_nV} 6 Pi {( x ∛ m)_ nV} 6 ( x P ∛ m )_ nV , Hf { y_ nV} 6 Hf {( x ∛ m)_ nV} 6 ( x H ∛ m )_ nV enj sd wi kit twd riea hnput-dutput ceacuaethdns gdr e shnkai cdfpaix ceacuaethdn. ([i ndti hn pesshnk tmet ahnierhty dg en AZH systif mdajs mdajs gdr cdfpaix-cdigghchint cdfpaix-cdigghchint ahnier cdf`hnethdns cdf`hnethdns dg cdfpaix hnput shkneas.) Es en hfpdrtent eppahcethdn dg tmhs gect, suppdsi tmi AZH systif hs ste`ai, enj cdnshjir tmi hnput shknea bψ d n
x_nV 6 i
, ∐∞ 1 n 1 ∞
wmiri ψ d hs e riea nuf`ir. Zmi cdrrispdnjhnk dutput shknea hs ∞
∞
bψ ( n ∐ o ) 6 ∕ m_ o Vi∐ bψd o i bψ d n y_ nV 6 ∕ m_ o Vi d o 6 ∐∞ bψ d o
Shnci tmi systif hs ste`ai, enj | i
o 6 ∐∞
|6 5 gdr iviry o ,
∞
∕
m_ o Vi
∐ bψd o
∞
≪ ∕ | m_ o Vi∐ bψ d o |
o 6 ∐∞
o 6 ∐∞
∞
≪ ∕ | m_ o V | o 6∐∞
1∞ enj wi mevi, gdr eny griquincy ψ d , cdnvirkinci dg tmi suf td e (cdfpaix) nuf`ir tmet wi wrhti es M (ψ d ) . Zmin bψ d n
y_nV 6 M (ψ d ) i
wmiri M (ψ d ) 6
∞
∕ m_ o V i∐ bψ d o o 6∐∞
Ekehn, tmiri hs ndt e gefhay dg suffethdns (es hn cdnvdauthdn, hn wmhcm jhggirint veauis dg n cen aiej td jhggirint gdrfs dg tmi suf) td iveaueti gdr tmhs cdfpaix hnput shknea, retmir tmiri hs e shnkai suffethdn td iveaueti! Ixefpai Hg bψ d n
x_ nV 6 cds(ψ d n) 6 Pi{i
enj tmi systif hs ste`ai, tmin bψ d n
y_ nV 6 Pi{M (ψ d ) i
70
}
}
Zd iveaueti tmhs ixprisshdn, wrhti tmi cdfpaix nuf`ir M (ψ d ) hn pdaer gdrf es bM (ψ d )
M (ψd ) 6 | M (ψ d ) | i
Zmin b (ψd n + M (ψ d ))
y_ nV 6 Pi{| M (ψ d ) | i
}
6 | M (ψd ) | cds(ψd n + M (ψ d )} Zmet hs, tmi rispdnsi dg e ste`ai systif td e cdshni hnput shknea hs e cdshni whtm tmi sefi griquincy `ut whtm efpahtuji ejbustfint enj pmesi smhgt. Endtmir wey td wrhti tmhs rispdnsi gdaadws grdf wrhthnk M (ψ d ) hn rictenkuaer gdrf, `ut wi aievi tmhs es en ixirchsi.
• Stiejy-Steti Ihkinguncthdns Suppdsi tmet tmi systif hs ceusea es wiaa es ste`ai, enj tmet tmi hnput shknea hs e rhkmt-shjij cdfpaix ixpdninthea, ixpdninthea, bψ d n
x_nV 6 i
Zmin y_nV 6 < gdr n 1 <, enj gdr n ≩ <, <, y_ nV 6
u_ nV
∞
∕ m_ o Vi bψ d (n ∐ o ) u_ n ∐ o V o 6∐∞ n
6 ∕ m_o Vi∐ bψd o i bψ d n o 6 <
Es n ← ∞ , rifif`irhnk tmet tmi unht-puasi rispdnsi hs rhkmt shjij `y ceuseahty, enj e`sdautiay suffe`ai `y ste`hahty, n
∕
m_ o V i
∐ bψd o
o 6<
∞
← M (ψ d ) 6 ∕ m_ o V i∐ bψ d o o 6 <
enj tmirigdri, es n hncriesis, y_nV epprdecmis tmi ‑stiejy-steti rispdnsi‖ bψ d n
yss _ nV 6 M (ψ d ) i
Zmet hs, gdr aerki veauis dg n, y_ nV ≍ yss _ nV . Zmhs hs e ‑dni-shjij hnput, hnput, stiejy-steti dutput‖ dutput‖ virshdn dg tmi ihkinguncthdn prdpirty dg cdfpaix ixpdnintheas. Dg cdursi e shfhaer prdpirty gdr dni-shjij shni enj cdshni hnputs hs hfpahij vhe tmi dpirethdns dg teohnk riea enj hfekhnery perts. 9.7 JZ AZH Systifs Jiscrh`ij `y Ahnier Jhggirinci Iquethdns
Systifs jiscrh`ij `y cdnstent-cdigghchint, ahnier jhggirinci iquethdns eri AZH systifs. Hn ixpadrhnk tmhs gect, ht hs hfpdrtent td oiip hn fhnj tmet dur jigeuat sitthnk hs tmet eaa shkneas eri 1 n 1 ∞ . Zmhs `rhnks e`dut shknhghcent jhggirincis (!) whtm dtmir trietfints dg jighnij gdr ∐∞ 1 jhggirinci iquethdns. Suppdsi wi mevi e systif wmdsi hnput enj dutput shkneas eri riaetij `y y_ nV + ey_ n ∐ 5V 6 `x_n V , ∐ ∞ 1 n 1 ∞ wmiri e enj ` eri riea cdnstents. Zmhs hs ceaaij e ghrst-drjir, cdnstent-cdigghchint, cdnstent-cdigghchint, ahnier jhggirinci iquethdn. Khvin en hnput shknea x_nV, tmhs cen `i vhiwij es en iquethdn tmet fust `i sdavij gdr y_nV, enj wi aievi td dtmir cdursis tmi erkufint tmet gdr iecm hnput shknea, x_ nV , tmiri hs e unhqui sdauthdn gdr tmi dutput shknea, y_ nV . [i shfpay feoi tmi caehf tmet tmi sdauthdn hs y_ nV 6
n
∕ (∐e)n ∐ o `x_ o V o 6∐∞
73
enj virhgy tmhs sdauthdn es gdaadws. Yshnk tmi essufij y_nV, ey_n ∐ 5V 6 e
n ∐5
∕ ( ∐e) n ∐5∐ o ` x_ oV
o 6∐∞ n ∐5
6 ∐ ∕ (∐ e) n ∐ o ` x_ o V o 6∐∞
Zmirigdri y_ nV + ey_ n ∐5V 5V 6
n
∕ ( ∐ e) n ∐ o `x_ o V o 6n
6 `x_ nV Dg cdursi, tmi sdauthdn cen `i wrhttin es y_ nV 6
∞
∕ ( ∐e) n ∐ o `u_ n ∐ o V x_ o V o 6∐∞
sd ht hs caier tmet tmi jhggirinci iquethdn jiscrh`is en AZH systif whtm unht-puasi rispdnsi n
m_ nV 6 ( ∐ e) `u `u_ nV
Zmet tmhs hs tmi unht-puasi rispdnsi easd cen `i virhghij jhrictay, `y smdwhnk tmet m_ nV + em_ n ∐ 5V 6 `κ _ nV
Grdf tmi gdrf dg m_nV ht gdaadws tmet e ghrst-drjir, cdnstent-cdigghchint, cdnstent-cdigghchint, ahnier jhggirinci iquethdn jighnis e ceusea AZH systif. Gurtmirfdri tmi systif hs fifdryaiss hg enj dnay hg e 6 <, enj ste`ai hg enj dnay hg | e| 1 5. Pisuats eri shfhaer gdr systifs jiscrh`ij `y sicdnj-drjir, cdnstent-cdigghchint, ahnier jhggirinci iquethdns, y_ nV + e5 y_ n ∐ 5V + e0 y_ n ∐ 0V 6 `x_ nV , ∐ ∞ 1 n 1 ∞ es wiaa es mhkmir drjir. Zmet hs, sucm iquethdns jiscrh`i ceusea, AZH systifs. Mdwivir, ht hs fdri jhgghcuat td cdfputi tmi unht-puasi rispdnsi, enj td cdnnict tmi ste`hahty prdpirty td tmi cdigghchints hn tmi jhggirinci iquethdn. Phkmt-Shjij Sitthnk Dgtin wi eri hntiristij hn rhkmt-shjij rhkmt-shjij hnputs, wmiri wmiri tmi ceuseahty prdpirty dg
systifs jiscrh`ij `y jhggirinci iquethdns hfpahis tmet tmi cdrrispdnjhnk dutputs eri easd rhkmt shjij. Zmhs fiens tmet tmi jhggirinci iquethdn niij dnay `i ejjrissij gdr n ≩ <, tmdukm wi fust vhiw tmi dutput es zird gdr nikethvi veauis dg n. Zmirigdri tmi ‑hnhthea cdnjhthdns‖ eri y_∐5V 6 y_∐0V 6 6 < . Ndnzird hnhthea cdnjhthdns, es cdnshjirij hn fetmifethcs dr dtmir inkhniirhnk cdursis jieahnk whtm jhggirinci iquethdns, cenndt erhsi hn tmi cdntixt dg AZH systifs, gdr e cdnsiquinci dg hnput-dutput ahnierhty hs tmet tmi hjinthceaay zird hnput shknea fust yhiaj tmi hjinthceaay zird dutput shknea. Hn suffery, shnci wi eri gdcushnk dn systifs wmdsi hnput-dutput `imevhdr hs ahnier, ahnier, wi fust riquhri zird hnhthea hnhthea cdnjhthdns cdnjhthdns hn e rhkmt-shjij rhkmt-shjij sitthnk. Ixirchsis 5. Suppdsi en AZH AZH systif whtm whtm hnput shknea shknea x_ nV 6 u_ nV ∐ u_ n ∐ 0V mes tmi rispdnsi
y_nV 6 0r_ nV ∐ 0 r_ n ∐ 0V . Soitcm tmhs hnput shknea enj dutput shknea, enj easd soitcm tmi systif rispdnsi td iecm dg tmi hnput shkneas `iadw. 7:
(e) xe _ nV 6 3u_ n ∐ 5V ∐ 3u_ n ∐ 3V (`) x` _ nV 6 u_ nV ∐ u_ n ∐ 5V ∐ u_ n ∐ 0V + u_ n ∐ 3V (c) xc _ nV 6 u_ nV ∐ u_ n ∐ :V 0. Yshnk tmi krepmhcea krepmhcea fitmdj, cdfputi cdfputi enj soitcm soitcm y_ nV 6 ( m * x)_ nV gdr
(e) x_n V 6 κ _ nV ∐ κ _ n ∐ 3V ,
m_nV 6 3κ _ n + 5V ∐3 κ _ n ∐ 3 V
⎫5, < ≪ n ≪ 3 ⎫5, 5 ≪ n ≪ 3, ; ≪ n ≪ 8 , m_ nV 6 ⎭ ⎨<, iasi ⎨<, iasi (c) x_ nV 6 5 , gdr eaa n, m_ nV 6 κ _ nV ∐ 0κ _ n ∐5V + κ _ n ∐ 0V (j) x_ nV 6 u_ n ∐ 5V ∐ u_ n ∐ 3V , m_ nV 6 ∐u_ nV + u_ n ∐ 3V (`) x_ nV 6 ⎭
(i) x_ nV 6 in ( u_ nV ∐ u_ n ∐ 0V ) ,
n m_nV 6 i∐ u_ nV
(g) x_ nV 6 u_ nV , m_nV 6 (5 / 0) n u_ n ∐5V (k) x_ nV 6 r_ nV , m_ nV smdwn `iadw
(m) x_ nV 6 u_ nV ,
m_ nV 6 r_ nV u_3 ∐ nV
(h) x_nV 6 u_ n ∐ 5V ∐ u_ n ∐ :V , m_ nV 6 ( ∐5) n u_ nV (b) x_nV 6 r_ nV, m_ nV 6 κ _ nV ∐ 0κ _ n ∐5V + κ _ n +5V 3. Yshnk tmi eneaythcea fitmdj, fitmdj, cdfputi cdfputi enj soitcm y_ nV 6 ( m * x)_ nV gdr
tmi gdaadwhnk shkneas, wmiri ε enj enj ΰ eri eri jhsthnct riea nuf`irs. (e) x_ nV 6 ε nu_ nV ,
n
m_n V 6 ΰ u_ nV
(`) x_ nV 6 ε n , m_n V 6 ΰ n u_ ∐ nV ([met ejjhthdnea essufpthdn essufpthdn dn ε enj enj ΰ hs niijij4) (c) x_ nV 6 κ _ n ∐ 5V , m_ nV 6 : u_3 ∐ nV (j) x_ nV 6 ε n ∐ 0u_ n ∐ 0V ,
enj ΰ hs niijij4) m_n V 6 ΰ ([met ejjhthdnea essufpthdn dn ε enj n
:. En AZH systif whtm whtm e unht-stip unht-stip hnput shknea mes tmi rispdnsi y_nV 6 (5 / 0) u_ nV . [met hs tmi
rispdnsi dg tmi systif td tmi hnput shknea smdwn `iadw.
79
9. Cdnshjir jhscriti-thfi shkneas m_n V tmet hs zird dutshji tmi hntirvea n<
≪ n ≪ n5 enj x_ nV tmet
hs zird dutshji tmi hntirvea n0 ≪ n ≪ n3 . Smdw mdw td jighni n: enj n9 sucm tmet
(m ∛ x)_nV 6 < dutshji tmi hntirvea n: ≪ n ≪ n9 . 7. Cdfputi tmi dvireaa unht-puasi unht-puasi rispdnsi gdr tmi hntircdnnicthdns dg JZ AZH systifs smdwn smdwn
`iadw. (e)
(`)
;. Suppdsi y_ nV 6 ( x ∛ m)_ nV . Gdr iecm dg tmi pehrs dg shkneas khvin `iadw, smdw mdw
yˇ_nV 6 ( xˇ ∛ mˇ)_ nV hs riaetij td y_ nV .
(e) xˇ_ nV 6 x_ n ∐ 3V ,
mˇ_ nV 6 m_ n + 3V
(`) xˇ_ nV 6 x_ n ∐ 3V ,
mˇ_n V 6 m_n ∐ 3V
(c) xˇ_ nV 6 x_ ∐nV ,
mˇ_ nV 6 m_ ∐ nV
(j) xˇ_ nV 6 x_ ∐5 ∐ nV ,
mˇ_n V 6 m_5 ∐ nV
=. Jitirfhni hg tmi JZ AZH systifs whtm tmi gdaadwhnk unht-puasi unht-puasi rispdnsis eri ceusea enj/dr
ste`ai. n 5V (e) m_ nV 6 5 u_ n + 5V
(0) n (`) m_ nV 6 ( 5 ) u_∐ nV 0
n (c) m_ nV 6 0 u_3 ∐ nV n (j) m_ nV 6 0 r_ ∐ nV n 8. Gdr tmi JZ AZH systif whtm whtm unht-puasi rispdnsi rispdnsi m_nV 6 5 u_ nV , usi tmi ihkinguncthdn 0
( )
prdpirthis td cdfputi tmi rispdnsi rispdnsi td tmi hnput shkneas shkneas
77
(e) x_ nV 6 5 (`) x_ nV 6 ( ∐5) n (c) x_n V 6 0 cds(ό n / 0) (j) x_ n V 6 3 shn( ∐ 3ό n) 0
7;
Ndtis gdr Shkneas enj Systifs 7.5 CZ AZH Systifs enj Cdnvdauthdn
Zmi trietfint dg tmi cdnthnudus-thfi cesi pereaaias tmi jhscriti-thfi cesi, ixcipt tmet sdfi gects eri fdri jhgghcuat td prdvi. Ht hs iesy td cmico tmet khvin e cdnthnudus-thfi shknea m(t ) , e systif jiscrh`ij `y tmi cdnvdauthdn hntikrea y (t ) 6
∞
∯ x(ϊ ) m(t ∐ ϊ ) j ϊ
∐∞
hs en AZH systif. ([i eri essufhnk miri, es usuea, tmet m(t ) enj tmi hnput shknea x(t ) eri sucm tmet tmi hntikrea hs jighnij.) Hnjiij, cdf`hnhnk tmi cmerectirhzhnk cmerectirhzhnk cdnjhthdns gdr ahnierhty enj thfi hnverhenci, wi niij dnay cmico tmi gdaadwhnk. Gdr eny hnput shkneas x5 (t ) enj x0 (t ) , whtm cdrrispdnjhnk rispdnsis y5 (t ) enj y0 (t ) , enj gdr eny cdnstents e enj t d , tmi rispdnsi td
x (t ) 6 e x5(t ) + x0 (t ∐ t d )
smduaj `i
y (t ) 6 e y5(t ) + y0 (t ∐ t d )
Sd, gdr e systif jiscrh`ij `y cdnvdauthdn, wi cdfputi tmi rispdnsi td x (t ) es
y (t ) 6
∞
∞
∯ x (ϊ ) m(t ∐ ϊ ) jϊ 6 ∯ _e x5(ϊ ) + x0 (ϊ ∐ td )V m( t ∐ ϊ ) j ϊ
∐∞ ∐∞ ∞ ∞ 6 e ∯ x5 (ϊ ) m(t ∐ ϊ ) jϊ + ∯ x0 (ϊ ∐ td ) m( t ∐ ϊ ) jϊ ∐∞ ∐∞ Cmenkhnk tmi verhe`ai dg hntikrethdn grdf ϊ td σ 6 ϊ ∐ t d hn tmi aest hntikrea khvis
y (t ) 6 e
∞
∯
x5(ϊ ) m(t ∐ϊ ) jϊ +
∐∞ 6 e y5(t ) + y0 (t ∐ t d )
∞
∯ ∐∞
x0 (σ ) m( t ∐ td ∐ σ ) j σ
Zmus e systif jiscrh`ij `y cdnvdauthdn hs en AZH systif. Gurtmirfdri, tmi cmerectirhzhnk shknea m(t ) hs tmi unht-hfpuasi rispdnsi dg tmi systif, es hs ieshay virhghij `y e shgthnk ceacuaethdn2 hg x (t ) 6 κ (t ) , tmin y (t ) 6
∞
∞
∐∞
∐∞
∯ x(ϊ ) m(t ∐ ϊ ) jϊ 6 ∯ κ (ϊ ) m( t ∐ ϊ ) jϊ 6 m( t )
Ht hs easd trui tmet tmi hnput-dutput `imevhdr dg issintheaay eaa cdnthnudus-thfi, ahnier, thfihnverhent systifs cen `i jiscrh`ij `y tmi cdnvdauthdn ixprisshdn. Mdwivir, td prdvi tmhs gect hnvdavis jiahceti erkufints tmet wi whaa sohp. Iveauethdn dg tmi Cdnvdauthdn Cdnvdauthdn Hntikrea
Ceacuaethnk tmi rispdnsi dg e systif td e khvin hnput shknea `y iveauethdn dg tmi cdnvdauthdn hntikrea hs ndt es shfpai es fhkmt `i ixpictij. Zmhs hs `iceusi e gefhay dg hntikrethdns, perefitrhzij `y t , fust `i iveauetij, enj tmi cmerectir dg tmi hntikrethdn cen cmenki whtm tmi perefitir. Zmiri eri tmrii fehn epprdecmis. epprdecmis.
7=
Eneaythcea Fitmdj [min `dtm tmi hfpuasi hfpuasi rispdnsi m(t ) enj tmi hnput shknea x(t ) mevi shfpai
eneaythcea jiscrhpthdns, tmi cdnvdauthdn hntikrea sdfithfis cen `i iveauetij `y eneaythcea fiens. Ixefpai Hg tmi hnput shknea enj tmi unht-hfpuasi unht-hfpuasi rispdnsi eri unht-stip shkneas, tmin tmi rispdnsi
cen `i wrhttin es ∞
∞
∐∞
∐∞
y (t ) 6 ∯ x(ϊ ) m(t ∐ ϊ ) jϊ 6 ∯ u(ϊ ) u( t ∐ ϊ ) j ϊ
Dg cdursi, tmi hntikrenj hs zird gdr nikethvi ϊ , rikerjaiss dg tmi veaui dg t , enj sd y (t ) 6
∞
∯ u (t ∐ ϊ ) j ϊ
<
Ndw tmi hntikrenj hntikrenj hs zird gdr ϊ > t , `ut tmhs shfpahghcethdn hnvdavis tmi veaui dg t . Hn gect ⎫ < , t ≪ <
⎢
y (t ) 6 ⎭ t
⎢ ∯ 5 jϊ 6 t , t > < ⎨<
Sufferhzhnk, Sufferhzhnk, wi cen wrhti tmi rispdnsi es tmi unht refp2 y (t ) 6 t u (t ) 6 r (t ) Ixefpai Hg tmi unht-hfpuasi rispdnsi rispdnsi hs e unht-stip guncthdn enj tmi hnput shknea hs tmi cdnstent
shknea x (t ) 6 5 , tmin tmi rispdnsi ceacuaethdn hs y (t ) 6
∞
∞
∯ x(ϊ ) m(t ∐ ϊ ) jϊ 6 ∯ u(t ∐ ϊ ) j ϊ
∐∞
∐∞
Dg cdursi tmi cdncaushdn hs tmet tmi rispdnsi hs unjighnij gdr eny veaui dg t ! Zmhs hs e rifhnjir tmet cdnvdauthdn ixprisshdns fust fust `i cmicoij td feoi suri tmiy eri fienhnkgua. Krepmhcea Fitmdj Gdr fdri cdfpahcetij cesis, cesis, e krepmhcea epprdecm hs veaue`ai veaue`ai gdr oiiphnk
treco dg tmi ceacuaethdns tmet fust `i jdni. @eshceaay, wi padt tmi twd shkneas hn tmi hntikrenj, x ( ϊ ) enj m(t ∐ ϊ ) , virsus ϊ , gdr tmi veaui dg t dg hntirist. Zmin fuathpayhnk fuathpayhnk tmi twd twd shkneas prdvhjis tmi hntikrenj, enj tmi nit erie fust fust `i cdfputij. Ixefpai [i cdfputi
y (t ) 6
∞
∯ x(ϊ ) m(t ∐ ϊ ) j ϊ
∐∞
gdr tmi hnput shknea enj unht-hfpuasi rispdnsi smdwn `iadw.
Ghrst tmi hfpuasi rispdnsi hs gahppij enj smhgtij dn tmi ϊ exhs td e cdnvinhint veaui dg t . Zmin tmi hnput shknea hs padttij hn tmi verhe`ai ϊ hffijhetiay `iadw td gechahteti tmi fuathpahcethdn dg shkneas2
78
1 0 enj gdr t > > 9 tmi prdjuct dg tmi twd shkneas hs hjinthceaay zird, enj Ht hs iesy td sii tmet gdr t 1 sd tmi rispdnsi hs y (t ) 6 < gdr tmisi twd renkis dg t . Gdr 0 ≪ t ≪ 3 , t
y (t ) 6 ∯ 0 jϊ 6 0t ∐ : 0
Gdr 3 ≪ t ≪ : ,
t
y (t ) 6
ϊ 6 0 ∯ 0 j ϊ t ∐5
Gdr : 1 t ≪ 9 , y (t ) 6
:
∯ 0 jϊ 6 5< ∐ 0t
t ∐5
Dg cdursi tmisi ceacuaethdns eri easd d`vhdus grdf cdnshjirethdn dg tmi fuathpahcethdn fuathpahcethdn dg tmi twd shkneas hn tmi verhdus renkis enj soitcmis dg tmi risuathnk hntikrenj. Hn eny cesi, soitcmhnk tmi rispdnsi yhiajs
AZH Caivirniss Fitmdj Fitmdj Hg dni dg tmi shkneas hn tmi cdnvdauthdn cdnvdauthdn cen `i wrhttin wrhttin es e ahnier
cdf`hnethdn dg shfpai, smhgtij shkneas, tmin `y tmi prdpirthis dg ahnierhty enj thfi hnverhenci, tmi rispdnsi cen `i cdfputij grdf e shnkai cdnvdauthdn hnvdavhnk hnvdavhnk tmi shfpai shkneas. Ixefpai Hn tmi ixefpai ixefpai khvin e`dvi, wi cen wrhti x(t ) 6 u(t ∐ 0) ∐ u(t ∐ :) , tmet hs, x(t ) hs e
ahnier cdf`hnethdn dg smhgtij stip guncthdns. Hg wi cdfputi tmi cdnvdauthdn
y (t ) 6
∞
∯ u (ϊ ) m(t ∐ ϊ ) j ϊ
∐∞
tmin tmi rispdnsi wi siio hs khvin `y y (t ) 6 y (t ∐ 0) ∐ y(t ∐ :) Zmi riejir hs incdurekij td wdro tmi jitehas enj soitcm y(t ) .
;<
Zmi [i` aicturi ahnoij `iadw cen `i cdnsuatij gdr gurtmir jhscusshdn. HAF2 AZH Systifs enj Cdnvdauthdn @ut `i eweri tmet tmi ndtethdn hn tmi [i` aicturi hs e `ht jhggirint tmen, enj ndt es kddj es, wmet wi mevi `iin ushnk hn caess. 7.0 \rdpirthis dg Cdnvdauthdn – Hntircdnnicthdns dg CZ AZH Systifs
Zmhs tdphc easd hs cdvirij hn tmi hntirecthvi [i` aicturi. Hn dur caess ndtethdn, wmiri wi wrhti y (t ) 6 ( m ∛ x)(t ) tmi verhdus prdpirthis dg cdnvdauthdn eppier es gdaadws2
•
Cdffutethvhty2
( x ∛ m)(t ) 6 ( m ∛ x)( t )
• Jhstrh`uthvhty2 Jhstrh`uthvhty2
( x ∛ (m5 + m0 ) (t ) 6 ( x ∛ m5)(t) + ( x ∛ m0 )( t ) • Essdchethvhty2
( ( x ∛ m5) ∛ m0 ) (t ) 6 ( x ∛ ( m5 ∛ m0 ) (t ) Ejjhthdnea prdpirthis hncauji dni tmet gdaadws grdf ahnierhty2 Gdr eny cdnstent `, ( (`m) ∛ x ) (t) 6 ` ( m ∛ x ) ( t) Easd, dni tmet gdaadws grdf thfi hnverhenci, tmdukm tmi ndtethdn hs e `ht ewowerj2 Gdr eny thfi t d , hg xˇ (t ) 6 x(t ∐ t d ) , tmin
( xˇ * m ) (t ) 6 ( x ∛ m )(t ∐ t d ) ∛ x)(t ) 6 x(t ) , gdr eny x (t ) Ghneaay, wrhthnk tmi unht hfpuasi guncthdn es κ (t ) , wi ndti tmet (κ ∛ cdnthnudus et t 6 <, enj gurtmirfdri (κ ∛ κ )(t ) 6 κ (t ) . Zmhs aest ixprisshdn hnvdois Spichea \rdpirty 5 dg Sicthdn 0.0, enj hkndris cdnthnuhty riquhrifints dg tmi shgthnk prdpirty. @ut hn tmi prisint cdntixt Spichea \rdpirty \rdpirty 5 stetis tmi paehnay shfpai shfpai gect tmet tmi unht-hfpuasi unht-hfpuasi rispdnsi dg tmi systif wmdsi unht-hfpuasi rispdnsi hs e unht hfpuasi hs e unht hfpuasi. Eaa dg tmisi prdpirthis cen `i hntirpritij hn tirfs dg `adco jhekref fenhpuaethdns hnvdavhnk AZH systifs, bust es hn tmi jhscriti-thfi cesi. 7.3 CZ AZH Systif \rdpirthis
Zmi hnput-dutput `imevhdr dg e cdnthnudus-thfi cdnthnudus-thfi AZH systif hs jiscrh`ij `y hts unht-hfpuasi rispdnsi, m(t ) , vhe tmi cdnvdauthdn ixprisshdn y (t ) 6
∞
∯
x(ϊ ) m(t ∐ ϊ ) jϊ 6
∐∞
∞
∯ m(ϊ ) x( t ∐ ϊ )j ϊ
∐∞
Zmirigdri tmi hnput-dutput prdpirthis dg en AZH systif cen `i cmerectirhzij hn tirfs dg prdpirthis dg m(t ) . Zmi `eshc risuats eri shfhaer td tmi jhscriti-thfi cesi, tmdukm, es usuea hn cdnthnudus
;5
thfi, tmiri eri unfinthdnij ticmnhcea essufpthdns td kuerentii tmet hntikreas eri jighnij, enj sd dn.
• Ceusea Systif En AZH systif hs ceusea hg enj dnay hhgg m(t ) 6 < gdr t 1 1 < , tmet hs, hg enj dnay hg m(t ) hs rhkmt shjij.
6 < , enj hn perthcuaer hs zird gdr t 1 1 < , ceuseahty Shnci tmi unht-hfpuasi hnput hs ndnzird dnay et t 6 1 < . (Miri wi riay dn tmi gect tmet tmi rispdnsi dg en AZH systif hs iquhveaint td m(t ) 6 < gdr t 1 6 < e ceusea systif td tmi hjinthceaay-zird hnput shknea hs hjinthceaay zird, enj up td tmi thfi t 6 6 < .) jdis ndt ondw wmitmir tmi hnput shknea cdnthnuis td `i zird, dr teois e ndnzird veaui et t 6 • Fifdryaiss Systif En AZH systif hs fifdryaiss fifdryaiss hg enj dnay hg m(t ) 6 < gdr ≬ < . t ≬
≬ < , tmin shnci Hg m(t ) 6 < gdr t ≬ y (t ) 6
∞
∯ m(ϊ ) x(t ∐ ϊ )j ϊ
∐∞ ht gdaadws tmet y(t ) cen dnay jipinj dn x(t ) . Dn tmi dtmir menj, hg m(t ) ≬ < gdr t 6 t e ≬ < , tmin
6 < yhiajs e rispdnsi tmet hs ndnzird et tmi tmi unht-hfpuasi hnput, wmhcm hs ndnzird dnay et t 6 ndnzird thfi t e . Zmus tmi systif hs ndt fifdryaiss. Suppdsi m(t ) hs ndnzird et dnay dni pdhnt hn thfi. Zmin unaiss m(t ) hs en hfpuasi tmi rispdnsi dg tmi systif td iviry hnput shknea whaa `i y (t ) 6 < gdr eaa t . Ht gdaadws grdf tmhs jhscusshdn tmet e fifdryaiss AZH systif hs cmerectirhzij `y en hfpuasi rispdnsi dg tmi gdrf m(t ) 6 ` κ (t ) , wmiri ` hs e riea cdnstent.
• Ste`ai Systif En AZH systif systif hs (`dunjij-hnput, `dunjij-dutput) `dunjij-dutput) ste`ai hg enj dnay hg tmi unht-hfpuasi unht-hfpuasi rispdnsi hs e`sdautiay hntikre`ai. Zmet hs ∞
∯ | m(t ) | jt ∐∞
hs ghnhti. Zd prdvi tmhs, suppdsi x(t ) hs e `dunjij hnput, enj | x(t ) |≪ F , gdr eaa t . [i usi tmi gect tmet tmi e`sdauti veaui dg en hntikrea whtm uppir ahfht krietir tmen adwir ahfht hs `dunjij `y tmi hntikrea dg tmi e`sdauti veaui dg tmi hntikrenj. Zmhs smduaj `i `iahive`ai `iahive`ai grdf tmi cdrrispdnjhnk gect e`dut sufs. Zmin tmi e`sdauti veaui dg tmi dutput shknea sethsghis
| y(t ) |≪
∞
∞
∐∞
∐∞
∯ | m(ϊ ) | | x(t ∐ ϊ ) | jϊ ≪ F ∯ | m(ϊ ) | jϊ ≪ FO
gdr eaa t , enj tmirigdri tmi systif hs ste`ai. Zd prdvi tmet ste`hahty dg tmi systif hfpahis e`sdauti hntikre`hahty dg m(t ) , wi usi tmi sefi sdrt dg caivirniss es hn tmi jhscriti-thfi cesi. Cdnshjir tmi `dunjij hnput shknea
;0
⎫ 5, m(∐t ) ≩ < ⎨∐5, m( ∐t ) 1 <
x(t ) 6 ⎭
Zmin tmi cdrrispdnjhnk dutput shknea, y (t ) , hs `dunjij, sey `y tmi cdnstent O , gdr eaa t . Hn 6 < , perthcuaer, et t 6 ∞
O ≩ y (<) 6
∞
∯ m(ϊ ) x( ∐ϊ ) jϊ 6 ∯ | m(ϊ ) | jϊ
∐∞
∐∞
Zmus m(t ) hs e`sdautiay hntikre`ai.
• Hnvirth`ai Systif Ghrst ndti tmet tmi hjinthty hjinthty systif systif hn cdnthnudus thfi, y (t ) 6 x(t ) , mes tmi unht hfpuasi rispdnsi m(t ) 6 κ (t ) , enj tmi hnvirsi systif gdr en AZH systif fust `i en AZH systif. Zmin wi cen feoi tmi gdaadwhnk stetifint2 En AZH systif jiscrh`ij `y m(t ) hs hnvirth`ai hg enj dnay hg tmiri ixhsts e shknea m H (t ) (tmi hfpuasi rispdnsi dg tmi hnvirsi systif) sucm tmet (m ∛ m H )(t ) 6 κ (t ) Sucm en m H (t ) fhkmt ndt ixhst, enj hg ht jdis, ht fhkmt `i jhgghcuat td cdfputi. [i whaa ndt pursui tmhs gurtmir. 7.: Pispdnsi td Shnkuaerhty Shkneas
Ht hs iesy td smdw tmet tmi rispdnsi dg e CZ AZH systif td e unht-stip hnput shknea hs tmi runnhnk hntikrea dg tmi unht-hfpuasi unht-hfpuasi rispdnsi. Hnjiij, hg x(t ) 6 u (t ) , tmin y (t ) 6
∞
t
∯ m(ϊ )u(t ∐ ϊ ) jϊ 6 ∯ m(ϊ ) j ϊ
∐∞
∐∞
Ht teois e `ht fdri wdro td smdw tmet tmi unht-refp rispdnsi cen `i wrhttin es en htiretij runnhnk hntikrea dg tmi unht-hfpuasi unht-hfpuasi rispdnsi. [htm x(t ) 6 r (t ) , wi cen wrhti y (t ) 6
∞
t
∯ m(ϊ5) r(t ∐ ϊ5) jϊ5 6 ∯ m(ϊ5)( t ∐ ϊ5) j ϊ5
∐∞
∐∞
wmiri e su`scrhptij verhe`ai dg hntikrethdn mes `iin usij td feoi tmi inj risuat pritty. Eppayhnk hntikrethdn-`y-perts hntikrethdn-`y-perts td tmhs ixprisshdn khvis y (t ) 6 (t ∐ ϊ5)
ϊ5
t ϊ 5
t
∯ m(ϊ 0 ) jϊ 0 |∐∞ ∐ ∯ ∯
m(ϊ 0 ) jϊ 0 ( ∐ j ϊ5)
∐∞ ∐∞ ∐∞ Iveauethnk tmi ghrst tirf et ϊ 5 6 t enj ϊ 5 6 ∐∞ smdws tmet ht venhsmis, aievhnk
y (t ) 6
t ϊ 5
∯ ∯
∐∞∐∞
m(ϊ 0 ) jϊ 0 j ϊ 5
Gdr en hnput shknea tmet mes e rikuaer kidfitrhc smepi enj cen `i riprisintij cdnvinhintay es e ahnier cdf`hnethdn dg shnkuaerhty shkneas, tmisi ixprisshdns ixprisshdns cen `i usij td wrhti tmi cdrrispdnjhnk rispdnsi es e ahnier cdf`hnethdn dg runnhnk hntikreas dg tmi unht-hfpuasi rispdnsi. Dg cdursi, tmi uthahty dg sucm en ixprisshdn jipinjs dn tmi cdfpaixhty dg m(t ) .
;3
Ixefpai Gdr tmi tmi hnput shknea
x (t ) 6 0u(t ) ∐ u (t ∐5) + 3r( t ∐ 0) tmi rispdnsi dg e CZ AZH systif cen `i wrhttin es t t ∐5 t ∐ 0 ϊ 5 y (t ) 6 0 ∯ m(ϊ ) jϊ ∐ ∯ m(ϊ ) jϊ + 3 ∯ ∯ m(ϊ 0 ) jϊ 0 j ϊ5 ∐∞ ∐∞ ∐∞ ∐∞ 7.9 Pispdnsi td Ixpdnintheas (Ihkinguncthdn \rdpirthis)
Gdr hfpdrtent caessis dg AZH systifs, tmi rispdnsis td cirtehn typis dg ixpdninthea hnput shkneas mevi perthcuaeray shfpai gdrfs. Zmisi shfpai gdrfs fdthveti feny epprdecmis td tmi eneayshs dg AZH systifs enj wi cdnshjir sivirea verhents, iecm dg wmhcm riquhris sahkmtay jhggirint essufpthdns dn tmi systif. Es hn tmi jhscriti-thfi cesi, en hnput shknea hs ceaaij en ihkinguncthdn dg tmi systif hg tmi rispdnsi hs shfpay e cdnstent fuathpai dg tmi hnput shknea.
• Piea Ihkinguncthdns Ihkinguncthdns Suppdsi tmi AZH systif hs ceusea enj ste`ai, enj suppdsi tmi hnput shknea hs tmi riea, krdwhnk ixpdninthea σ d t
x(t ) 6 i
, σ d ≩ < , ∐ ∞ 1 t 1 ∞
Zmin y (t ) 6
∞
∯
σ d (t ∐ϊ )
m(ϊ ) i
jϊ 6
∞
∐σ ϊ ∯ m(ϊ ) i d jϊ
σ t
i d
∐∞ ∐∞ ∞ 6 ∯ m(ϊ )i∐σ dϊ jϊ iσ d t <
wmiri tmi adwir ahfht mes `iin rehsij td zird shnci tmi unht-hfpuasi unht-hfpuasi rispdnsi hs rhkmt shjij. Cdnvirkinci dg tmi hntikrea hs kuerentiij `y tmi ste`hahty essufpthdn, enj `y tmi gect tmet
| i∐σ dϊ | ≪ 5 , ϊ ≩ < . Zmi jitehas riay dn tmi gect tmet tmi e`sdauti veaui dg en hntikrea hs aiss tmen tmi hntikrea dg tmi e`sdauti veaui (sd adnk es tmi uppir ahfht hs krietir tmen tmi adwir ahfht. Ixpahchtay, ∞
∯ m(ϊ )i <
∐σ dϊ
jϊ ≪
∞
∯ | m(ϊ ) i
∐σ dϊ
∞
| jϊ ≪ ∯ | m(ϊ ) | jϊ 1 ∞
<
<
Zmirigdri wi cen jighni tmi cdnstent M (σ d ) `y M (σ d ) 6
∞
∯ m(ϊ )i
∐σ dϊ
j ϊ
<
enj wrhti σ d t
y (t ) 6 M (σ d )i
, ∐ ∞ 1 t 1 ∞
Ht hs hfpdrtent td d`sirvi tmet dnay dni hntikrea fust `i iveauetij td cdfputi tmhs rispdnsi, hn cdntrest whtm tmi kinirea cdnvdauthdn ceacuaethdn ceacuaethdn tmet hnvdavis e gefhay dg hntikreas.
• Cdfpaix Ihkinguncthdns Zmdukm wi cdnshjir dnay riea AZH systifs, tmet hs, systifs whtm e riea unht-hfpuasi unht-hfpuasi rispdnsi m(t ) , ht hs cdnvinhint gdr fetmifethcea purpdsis td pirfht cdfpaix-veauij cdfpaix-veauij hnput shkneas. Zd sii mdw tmi kinirea kinirea ceacuaethdn prdciijs, prdciijs, wrhti e cdfpaix hnput shknea hn rictenkuaer gdrf
;:
x (t ) 6 x P (t ) + bxH (t )
wmiri, gdr iecm t , x P (t ) 6 Pi{x(t )} ,
xH (t ) 6 Hf{ x( t )} )}
Zmin, shnci m(t ) hs riea, enj b hs e cdnstent, ∞
y (t ) 6
∯ x(ϊ )m(t ∐ ϊ ) j ϊ
∐∞ ∞ ∞ 6 ∯ x P (ϊ ) m(t ∐ ϊ ) jϊ + b ∯ xH (ϊ ) m( t ∐ ϊ ) j ϊ ∐∞ ∐∞
Zmet hs,
Pi{ y (t )} 6 Pi{( x ∛ m)(t )} 6 ( x P ∛ m)( t ) Hf{ y (t )} 6 Hf{( x ∛ m)( t)} 6 ( x H ∛ m)( t )
Zmhs fiens tmet whtm dni cdfpaix ceacuaethdn wi hncauji twd riea ceacuaethdns Zmi fdst hfpdrtent eppahcethdn dg cdfpaix hnputs hs tmi cesi tmi AZH systif hs ste`ai enj tmi hnput hs e pmesdr, bψ d t
x(t ) 6 i
, ∐ ∞ 1 t 1 ∞
Zmi rispdnsi hs khvin `y y (t ) 6
∞
bψd (t ∐ϊ )
∯ m(ϊ )i
∞
∐ bψ ϊ ∯ m(ϊ ) i d jϊ
jϊ 6
∐∞
i
bψd t
∐∞
Miri wi jighni tmi cdfpaix cdnstent M (ψd ) 6
∞
∐ bψ ϊ ∯ m(ϊ )i d j ϊ
∐∞
wmiri cdnvirkinci dg tmi hntikrea hs kuerentiij `y tmi ste`hahty essufpthdn enj tmi gect tmet
| i∐ bψdϊ | ≪ 5 , ∐ ∞ 1 ϊ 1 ∞ Zmus wi mevi bψ d t
y (t ) 6 M (ψ d ) i
Ixefpai Suppdsi tmi shknea shknea x(t ) 6 shn(ψ dt ) ,
, ∐ ∞ 1 t 1 ∞
∐ ∞ 1 t 1 ∞ , hs eppahij td e ste`ai AZH systif bψ d t
whtm unht-hfpuasi rispdnsi m(t ) . Shnci x(t ) 6 Hf{i
} , wi hffijhetiay mevi tmet
bψ d t
y (t ) 6 Hf{M (ψ d ) i
} . Zmhs ixprisshdn cen `i feji fdri ixpahcht `y wrhthnk M (ψ d ) hn
pdaer gdrf2 bM (ψ d )
M (ψd ) 6 | M (ψ d ) | i
Zmin bψd t
y (t ) 6 Hf Hf{M (ψd ) i
} 6 Hf Hf{| M (ψ d ) | i bM (ψd ) i bψ d t }
6 | M (ψ d ) | Hf{i b (ψd t + M (ψ d )) } 6 | M (ψd ) | shn(ψdt + M (ψ d ) ), ∐ ∞ 1 t 1 ∞ En eatirneti ixprisshdn gdaadws grdf wrhthnk M (ψ d ) hn rictenkuaer gdrf,
;9
M (ψd ) 6 Pi{M (ψd )} + b Hf{M (ψ d )}
Zmin y (t ) 6 Hf {_P _ Pi{M (ψd )} + b Hf{ M (ψd )}V_cds(ψdt) + b shn(ψ dt )V}
6 Pi{ M (ψd )} cds(ψdt ) ∐ Hf{M (ψd ) shn(ψ dt) , ∐ ∞ 1 t 1 ∞ Pikerjaiss dg tmi perthcuaer gdrf cmdsin gdr y (t ) , tmi oiy gect hs tmi gdaadwhnk. Hg tmi hnput hs e shnusdhj (dr pmesdr) dg griquincy ψ d , tmin tmi dutput hs e shnusdhj (dr pmesdr) whtm tmi sefi griquincy, eatmdukm tmi efpahtuji enj pmesi enkai, riaethvi td tmi hnput shnusdhj, hs eatirij `y tmi systif.
• StiejySteti Ihkinguncthdns Ihkinguncthdns Suppdsi tmi AZH systif hs ceusea es wiaa es ste`ai, enj tmi hnput shknea hs tmi rhkmt-shjij pmesdr bψ t
x(t ) 6 i d u(t ) 1 < , `y ceuseahty, enj gdr t ≩ ≩ < , Zmin y (t ) 6 < gdr t 1 t ∞ bψ (t ∐ϊ ) bψ t ∐ bψdϊ y (t ) 6 ∯ m(ϊ )i d jϊ 6 ∯ m(ϊ ) i jϊ i d , t ≩ < < ∐∞ Zmirigdri es t hncriesis, hncriesis, y (t ) fdri enj fdri cadsiay epprdxhfetis tmi stiejy-steti rispdnsi bψ d t
yss (t ) 6 M (ψ d ) i
wmiri M (ψd ) 6
∞
∯ m(ϊ )i
∐ bψdϊ
∐∞
jϊ 6
∞
∯ m(ϊ ) i
∐ bψdϊ
j ϊ
<
enj ekehn tmi ste`hahty essufpthdn kuerentiis tmet M (ψ d ) hs wiaa jighnij. Zmus tmi stiejy-steti rispdnsi dg e ceusea enj ste`ai AZH systif td e shnusdhj (dr pmesdr) dg griquincy ψ d , hs e shnusdhj (dr pmesdr) whtm tmi sefi griquincy. Ht hs hntiristhnk td cdfperi tmhs stiejy-steti prdpirty whtm tmi privhdus cesi wmiri tmi pmesdr hnput shknea jighnij gdr ∐∞ 1 t 1 ∞ risuats hn tmi pmesdr dutput et iviry veaui dg t . Dg cdursi, 6 ∐∞ , iviry veaui dg t hs e ‑stiejy-steti‖ veaui hn tmet en hnghnhti shnci tmi hnput `iken et t 6 pirhdj dg thfi thfi mes iaepsij shnci tmi `ikhnnhnk dg tmi hnput shknea. 7.7 CZ AZH Systifs Jiscrh`ij `y Ahnier Jhggirinthea Jhggirinthea Iquethdns
Systifs jiscrh`ij `y cdnstent-cdigghchint, ahnier jhggirinthea iquethdns eri AZH systifs. Mdwivir, wmhai stethnk tmhs gect ht hs hfpdrtent td oiip hn fhnj tmet `y AZH wi fien ‑hnputdutput ahnier‖ systifs enj tmet dur jigeuat thfi hntirvea hs ∐∞ 1 t 1 ∞ . @iceusi dgtmhs sitthnk, dur trietfint fey ndt `i es shfhaer td dtmir trietfints es ydu fhkmt ixpict. Cdnshjir e systif wmiri tmi hnput enj dutput shkneas eri riaetij `y y (t ) + ey (t ) 6 `x(t ) , ∐ ∞ 1 t 1 ∞ cdnstent-cdigghchint, ahnier wmiri e enj ` eri riea cdnstents. Zmhs hs ceaaij e ghrst-drjir, cdnstent-cdigghchint, jhggirinthea iquethdn. Dnci x (t ) hs spichghij, tmhs cen `i vhiwij es en iquethdn tmet fust `i sdavij gdr y(t ) . Ht cen `i smdwn tmet tmiri hs dnay dni sdauthdn, enj wi whaa jifdnstreti tmet tmhs sdauthdn cen `i wrhttin es
;7
t
y (t ) 6
∯
e t ϊ i ∐ ( ∐ ) `x(ϊ ) j ϊ
∐∞
Zmi jifdnstrethdn hnvdavis su`sthtuthnk hntd tmi jhggirinthea iquethdn, enj prdciijs hn en iaifintery gesmhdn `y wrhthnk et y (t ) 6 i∐
t
∯
eϊ
i `x(ϊ ) j ϊ
∐∞
Hn tmhs gdrf, tmi ceacuaethdn dg y (t ) hs e shfpai fettir dg tmi prdjuct ruai, enj tmi gunjefintea tmidrif dg ceacuaus. Hnjiij, y (t ) 6 ∐ ei
∐ et
t
∯
i eϊ `x(ϊ ) jϊ + i∐ et iet `x ( t )
∐∞ 6 ∐ ey(t ) + `x(t )
enj tmi sdauthdn hs virhghij. @y hnsirthnk tmi epprdprheti unht-stip guncthdn, wi cen wrhti y(t ) hn tmi gdrf y (t ) 6
∞
∯
e t ϊ `i ∐ ( ∐ ) u(t ∐ ϊ ) x(ϊ ) j ϊ
∐∞
enj ht hs caier tmet tmi jhggirinthea iquethdn jiscrh`is en AZH systif whtm unht-hfpuasi rispdnsi ∐ et m(t ) 6 `i u (t ) Pifero Ht hs hntiristhnk td smdw jhrictay tmet tmhs hfpuasi rispdnsi sethsghis tmi jhggirinthea iquethdn (gdr eaa t ) wmin x(t ) 6 κ (t ) . Zmi virhghcethdn hnvdavis ushnk kinireahzij ceacuaus td cdfputi m t 6 ∐`ei ∐et u t + `i∐ et κ t 6 ∐`ei∐ et u t + `κ t
()
()
()
()
()
Zmin ht hs iesy td sii tmet m(t ) + em em(t ) 6 `κ (t ) ,
∐ ∞ 1 t 1 ∞
Grdf tmi gdrf dg tmi unht-hfpuasi rispdnsi, m(t ) , ht gdaadws tmet tmi AZH systif jiscrh`ij `y tmi ghrst-drjir ahnier jhggirinthea iquethdn hs ceusea enj hs ndt fifdryaiss. Zmi systif hs ste`ai hg enj dnay hg e > < . Gdr e sicdnj-drjir, cdnstent-cdigghchint, ahnier jhggirinthea iquethdn, (t ) + e5 y (t ) + e< y( t ) 6 `x( t ) y enj easd gdr mhkmir-drjir ahnier jhggirinthea iquethdns, tmi shtuethdn hs shfhaer td tmi ghrst-drjir cesi. Sucm iquethdns jiscrh`i ceusea AZH systifs. Mdwivir ht hs fdri jhgghcuat td cdfputi tmi unht-hfpuasi unht-hfpuasi rispdnsi, enj td cmerectirhzi ste`hahty prdpirthis hn tirfs dg tmi cdigghchints dg tmi jhggirinthea iquethdn. Phkmt-Shjij Sitthnk Hn dtmir cdursis ydu fey fey mevi incduntirij ahnier jhggirinthea jhggirinthea iquethdns ∐
≩ < , whtm hnhthea cdnjhthdns spichghij et t 6 < . Gdr ixefpai, hn tmi ghrst-drjir cesi, jighnij gdr t ≩ cdnshjir y (t ) + ey(t ) 6 `x(t) , t ≩ <
;;
≩ < , spichghij. Zmhs sitthnk cen `i if`ijjij hntd dur grefiwdro `y whtm y (<∐ ) enj x (t ), t ≩ 1 < . Zmin, `y ceuseahty, tmi dutput shknea hs zird gdr cdnshjirhnk tmi hnput shknea td `i zird gdr t 1 t 1 1 < , enj hn perthcuaer, y (<∐ ) fust `i zird. (Piceaa tmet hg tmi hnput shknea td en AZH systif hs zird gdr eaa t , tmin tmi dutput shknea fust `i zird gdr eaa t .) .) \ut endtmir wey, e cdnstentcdigghchint, ahnier jhggirinthea iquethdn whtm rhkmt-shjij hnput shkneas jiscrh`is en AZH systif hg enj dnay hg eaa hnhthea cdnjhthdns eri zird. Ixefpai Suppdsi e vdateki shknea, x (t ) , hs eppahij td tmi tirfhneas dg e sirhis P-C chrcuht chrcuht smdwn
`iadw, enj tmi tmi dutput shknea shknea dg hntirist, y (t ) , hs tmi vdateki ecrdss tmi cepechtdr, C .
Ohrcmmdgg‘s vdateki aew khvis tmi chrcuht jiscrhpthdn es e ghrst-drjir jhggirinthea iquethdn y (t ) + 5 y (t ) 6 5 x(t ) , PC
∐ ∞ 1 t 1 ∞
PC
Zmhs jiscrh`is en AZH systif whtm unht-hfpuasi rispdnsi 5
∐ t m(t ) 6 5 i PC u (t ) PC
Hg wi eri hntiristij hn tmi rispdnsi dg tmhs systif td shnusdhjea hnputs whtm griquincy ψ d , wi cdnshjir tmi hnput shknea bψ d t
x(t ) 6 i
, ∐ ∞ 1 t 1 ∞
enj cdfputi M (ψd ) 6
∞
∯
m(ϊ ) i
∐ bψdϊ
jϊ 6
∐∞
∞
5
5 i∐ PC ϊ i∐ bψdϊ j ϊ
∯ PC
<
∐( 5 + bψd )ϊ ∞ ∐5 i PC 6 5 bPC ψ d + < 5 + bPC ψ d
65
(Ndthci tmet tmi hfpahcht essufpthdn essufpthdn tmet P enj C eri pdshthvi hs cruchea hn tmi iveauethdn dg tmi hntikrea. Zmhs hs tmi ste`hahty riquhrifint – whtm pdshthvi P enj C , m(t ) hs e`sdautiay hntikre`ai.) Zmus tmi rispdnsi td tmi pmesdr hnput shknea hs y (t ) 6
bψ t 5 i d , 5+ bPC ψ d
∐ ∞ 1 t 1 ∞
Grdf tmhs `eshc gect, wi cen ixtrect tmi rispdnsi td verhdus shnusdhjea hnput shkneas. Gdr ixefpai, hg tmi vdateki hnput shknea hs x (t ) 6 cds(ψ dt ) u(t ) Zmin tmi stiejy-steti rispdnsi dg tmi chrcuht cen `i wrhttin es
;=
yss (t ) 6 Pi
6
{+
5
5 bPC ψ d 5
5+ P 0 C 0ψ d0
bψd t
i
}
⎫⎢ ⎯ 5 b (ψ t ∐ ten ∐ ( PC ψ d )) ⎢ 5 6 Pi ⎭ i d ⎡ ⎢⎨ 5+ P 0C 0ψ d0 ⎢⎦
cds_ψ dt ∐ ten ∐5( PC ψ d )V
Hg tmi hnput griquincy, ψ d , hs aerki, tmin tmi stiejy-steti vdateki ecrdss tmi cepechtdr whaa `i sfeaa. Dn tmi dtmir menj, hg tmi hnput griquincy hs sfeaa, tmin tmi stiejy-steti rispdnsi hs shfhaer hn efpahtuji td tmi hnput shknea. Zmi pmesi enkai dg tmi rispdnsi, riaethvi td tmi hnput shknea, easd jipinjs dn tmi griquincy. Gurtmirfdri, hg tmi hnput shknea hs e ahnier cdf`hnethdn dg shnusdhjs et verhdus griquinchis, tmin tmi stiejy-steti rispdnsi whaa cdntehn tmi sefi sit dg griquinchis, `ut whtm tmi efpahtujis enj pmesi enkais hngauincij eccdrjhnk td M (ψ d ) et tmi verhdus veauis dg ψ d . Zmhs hs tmi `eshs dg griquincy-siaicthvi ghatirhnk. Ixirchsis 5. Yshnk tmi krepmhcea fitmdj, cdfputi enj soitcm y(t ) 6 ( m ∛ x)(t ) gdr t (e) m(t ) 6 i∐ u (t ) , t (`) m(t ) 6 i ∐| | ,
x(t ) 6 0u( t ) ∐ 0 u( t ∐ 5)
x(t ) 6 u ( t )
(c) m(t ) 6 it u ( ∐t ) ,
x( t ) 6 u( t ∐ 0)
(j) m(t ) 6 i ∐t u (t ) ,
x(t ) 6 u(3 ∐ t) t t (i) m(t ) 6 i ∐0 u (t ) , x(t ) 6 i ∐5u( t ) t (g) m(t ) 6 i ,
x(t ) 6 κ (t ) ∐ u(t )
0. En AZH systif mes tmi hfpuasi rispdnsi smdwn `iadw2
Gdr en hnput shknea dg tmi gdrf x(t ) 6
∞
∕ o 6 <
eo κ (t ∐ oZ )
soitcm tmi dutput shknea hg ≩ < (e) Z 6 3 , eo 6 5 gdr eaa o ≩ (`) Z 6 0 ,
≩ < eo 6 5 o ≩
(c) Z 6 3 ,
eo 6 (5 / 0)
o
o ≩ ≩ <
;8
3. Suppdsi tmi cdnthnudus-thf cdnthnudus-thfii shknea m(t ) hs zird dutshji tmi hntirvea t<
≪ t ≪ t 5 enj tmi shknea
x(t ) hs zird dutshji tmi hntirvea t0 ≪ t ≪ t 3 . Smdw mdw td jighni t : enj t 9 sucm tmet (m ∛ x)(t ) 6 < dutshji tmi hntirvea t: ≪ t ≪ t 9 . :. Ixpriss y (t ) 6 ( m ∛ x )(t ) hn tirfs dg y (t ) 6 ( m ∛ x)(t ) gdr tmi gdaadwhnk shknea cmdhcis.
(e) x (t ) 6 x(t ∐ 5) ,
m ( t ) 6 m( t ∐ 0)
(`) x (t ) 6 x(t ∐ 0) ,
(c) x (t ) 6 x(0t ) , (j) x (t ) 6 x(3t ) , (i) x (t ) 6 x( ∐t ) ,
m (t ) 6 m( t + 0)
m (t ) 6 m( ∐0t )
m (t ) 6 m(3t ) m (t ) 6 m( ∐t )
9. Yshnk tmi eneaythcea fitmdj, cdfputi enj soitcm y (t ) 6 ( m ∛ x)(t ) gdr
(e) m(t ) 6 it ,
x(t ) 6 κ (t ) ∐ u(t )
7. Jitirfhni hg tmi AZH systifs jiscrh`ij `y tmi gdaadwhnk unht-hfpuasi unht-hfpuasi rispdnsis eri ste`ai
enj/dr ceusea. t 3) 3) (e) m(t ) 6 i∐0 u (t +
(`) m(t ) 6 i3t u ( ∐: ∐ t ) (c) m(t ) 6 i ∐:|t | t (j) m(t ) 6 i u (t ∐ 3)
;. Jitirfhni hg tmi gdaadwhnk stetifints e`dut AZH systifs eri trui dr geasi. Busthgy ydur
enswirs. (e) Hg m(t ) hs rhkmt shjij enj `dunjij, tmin tmi systif hs ste`ai. (`) Hg m(t ) hs pirhdjhc enj ndt hjinthceaay zird, tmin tmi systif hs unste`ai. (c) Zmi cesceji cdnnicthdn dg e ceusea AZH systif enj e ndn-ceusea AZH systif hs eaweys ndnceusea. (j) E fifdryaiss AZH systif hs eaweys ste`ai. =. Cdnshjir e systif jiscrh`ij jiscrh`ij `y
y (t ) 6
t
∯
t ϊ i ∐0( ∐ ) x(ϊ ∐ 5) j ϊ
∐∞
(e) Smdw tmet tmhs hs en AZH systif. (`) Cdfputi tmi unht-hfpuasi rispdnsi dg tmi systif. (c) Cdfputi tmi rispdnsi dg tmi systif td x(t ) 6 u (t ) ∐ u (t ∐ 5) `y cdnvdauthdn cdnvdauthdn enj tmin `y `y ushnk tmi gect tmet tmi unht-stip rispdnsi dg en AZH systif hs tmi runnhnk hntikrea dg tmi unht-hfpuasi rispdnsi..
=<
8. Gdr tmi P-A chrcuht smdwn `iadw, whtm hnput enj dutput dutput currint shkneas es smdwn, enj
P 6 :,
A 6 : , cdfputi
(e) tmi stiejy-steti rispdnsi td tmi hnput shknea x(t) 6 3 cds(t) u(t ) . (`) tmi stiejy-steti rispdnsi td x(t ) 6 0 shn(3t) u( t ) . (c) tmi rispdnsi td x(t ) 6 5 . t
5<. Gdr tmi AZH systif whtm unht-hfpuasi unht-hfpuasi rispdnsi m(t ) 6 i u (t ) , cdfputi tmi rispdnsi td tmi Mhnt2 [i jhj ndt jhscuss eaa tmi ihkinguncthdn prdpirthis tmet hnput shknea x(t ) 6 i3t cds(0t ) . ( Mhnt2 AZH systifs mevi.)
=5
Ndtis gdr Shkneas enj Systifs ;.5 Hntrdjucthdn td CZ Shknea Piprisintethdn
E gunjefintea hjie hn shknea eneayshs hs td riprisint shkneas hn tirfs dg ahnier cdf`hnethdns dg ”`eshs‘ shkneas. Zmet hs, wi cmddsi e sit dg `eshs shkneas, χ< (t ), χ5(t ), …, χ O ∐5(t ) tmet eri riaethviay shfpai, mevi usigua prdpirthis, enj eri wiaa suhtij td tmi caess dg shkneas td `i riprisintij. Zmin, khvin x(t ) , wi cdfputi sceaer cdigghchints e< , e5, … , eO ∐5 sucm tmet x (t ) ≍ e< χ< (t ) + e5 χ5(t ) + + eO ∐5 χ O ∐5( t )
Zmiri eri feny riesdns gdr tmhs epprdecm. Cirtehnay ht feois sinsi gdr shknea prdcisshnk `y AZH systifs, perthcuaeray hg tmi `eshs shkneas mevi nhci prdpirthis es hnput shkneas td sucm systifs. Easd, stdreki dr trensfhsshdn dg e shknea cen `i eccdfpahsmij `y stdreki dr trensfhsshdn dg tmi cdigghchints, e< , e5, … , eO ∐5 , dnci e sit dg `eshs shkneas mes `iin siaictij. Zmiri eri feny `eshc quisthdns td `i ejjrissij hn jiviadphnk tmhs epprdecm2 [met prdpirthis dg `eshs sits wduaj wduaj `i usigua4 Mdw Mdw feny `eshs `eshs shkneas eri niijij4 [met hs tmi tmi epprdprheti neturi neturi dg tmi epprdxhfethdn ‑ ≍ 4‖ Enswirs td tmisi quisthdns quisthdns eri jiviadpij hn hn sdfi jiteha hn tmi nixt giw giw sicthdns. Zmdsi whsmhnk td dfht tmi kinirea jhscusshdn cen prdciij jhrictay td Sicthdn =.5 wmiri tmi riprisintethdn dg fehn hntirist hs hntrdjucij hn en ej-mdc gesmhdn. Sdfi ixefpais cen fdthveti tmi jhscusshdn. Ixefpai Cdnshjir tmi shknea
⎢⎫i∐t , ∐ 5 ≪ t ≪ 5 x(t ) 6 ⎭ iasi ⎨⎢ <, enj tmi `eshs sit dg tmrii shkneas tmet eri zird dutshji tmi hntirvea ∐5 ≪ t ≪ 5 , whtm χ< (t ) 6 5, 5, χ5(t ) 6 t , χ 0 (t ) 6 t 0 / 0 ? ∐5 ≪ t ≪ 5 (Hnjiij, ht wduaj `i shfpair td jhspinsi whtm dur jigeuat jdfehn dg jighnhthdn dg shkneas enj shfpay wdro dn tmi hntirvea ∐5 ≪ t ≪ 5 es tmi jdfehn dg jighnhthdn. [i ritehn tmi jigeuat fehnay gdr ifpmeshs.)
Piceaahnk Zeyadr‘s gdrfuae, wi cen cmddsi e< 6 x (<) 6 5, e5 6 x (<) 6 ∐5, e0 6 x(<) 6 5 td d`tehn tmi riprisintethdn
⎫⎢5 ∐ t + t 0 / 0, ∐ 5 ≪ t ≪ 5 ⎢⎨ <, iasi
x(t ) ≍ χ< (t ) ∐ χ5(t ) + χ 0 ( t ) 6 ⎭
Zmus Zeyadr‘s gdrfuae ghts tmi grefiwdro wi eri cdnshjirhnk, enj wi mevi sdfi ndthdn dg tmi 6 < , enj sinsi dg epprdxhfethdn. Zmet hs, tmi epprdxhfethdn whaa `i kddj gdr veauis dg t cadsi cadsi td t 6 ixect dnay et zird. Dg cdursi tmi caess dg shkneas td `i riprisintij hn tmhs wey fust `i twhci 6 < , `ut tmi cdfputethdn dg tmi cdigghchints hs retmir shfpai. Gurtmirfdri, jhggirinthe`ai et tmi t 6 Gurtmirfdri, hg wi went td righni tmi riprisintethdn `y ejjhnk ejjhthdnea `eshs shkneas, gdr ixefpai χ 3 (t ) 6 t 3 /(3!) , ht hs d`vhdus mdw td cdfputi tmi cdigghchint e3 , tmdukm tmi shknea fust `i
=0
6 < . En ejventeki dg tmi situp hs tmet tmrhci jhggirinthe`ai et t 6 tmet hn cdfputhnk tmi tmi gdurtm cdigghchint, ghrst tmrii cdigghchints hn tmi riprisintethdn jd ndt cmenki. Ixefpai Cdnshjir tmi sefi sefi shknea enj `eshs sit, `ut suppdsi wi ndw riquhri riquhri tmet tmi
6 ∐5, <, 5 . Piceaahnk pdayndfhea epprdxhfethdn mevi zird irrdr et tmi tmrii veauis t 6 hntirpdaethdn, wi prdciij `y sitthnk x(t ) 6 e< χ< (t ) + e5 χ5(t ) + e0 χ 0 (t ), t 6 ∐5, <, 5 Zmhs yhiajs tmrii iquethdns hn tmrii unondwns2 i 6 e< ∐ e5 + e0 / 0 5 6 e< i
∐5
6 e< + e5 + e0 / 0
Sdavhnk tmhs sit dg iquethdns khvis i 0 ∐ 0i + 5 5 ∐ i0 e< 6 5, e5 6 6 ∐5.5=7, e0 6 6 5.<73 i 0i
enj tmi risuathnk riprisintethdn hs x(t ) ≍ χ< (t ) ∐ 5.5=7χ5( t ) +5.<73χ 0( t )
⎢⎫5 ∐ 5.5=7 t + <.9<3 t 0 , ∐5 ≪ t ≪ 5 6 ⎭ ⎢⎨ <, iasi Zmhs riprisintethdn hs pirmeps `ittir gdr sdfi purpdsis tmen tmi Zeyadr‘s gdrfuae, tmdukm ht hs hncdnvinhint td mevi td sdavi iquethdns gdr tmi cdigghchints. Easd, hg e gdurtm `eshs shknea hs ejjij td tmi sit, sey χ 3 (t ) 6 t 3 / 3! , enj wi riquhri zird irrdr et e gdurtm pdhnt td `i cdnshstint whtm pdayndfhea hntirpdaethdn cepe`hahthis, cepe`hahthis, tmin tmi riprisintethdn fust `i ricdfputij ricdfputij grdf tmi tmi `ikhnnhnk es tmi veauis dg tmi tmi ghrst tmrii cdigghchints cdigghchints whaa cmenki. cmenki. ;.0 Drtmdkdneahty enj Fhnhfuf Fhnhfuf HSI Piprisintethdn
E pdpuaer cmdhci dg tmi neturi dg tmi epprdxhfethdn hn shknea riprisintethdn, enj tmi epprdxhfethdn wi gdcus dn hn tmi siquia, hs tmi gdaadwhnk. Khvin en inirky shknea x(t ) enj e sit dg `eshs inirky shkneas χ< (t ), χ5(t ), …, χ O ∐5 (t ) , suppdsi tmi cdigghchints e< , e5, … , eO ∐5 eri cdfputij td fhnhfhzi tmi hntikrea squeri irrdr , ∞ ⎟
O ∐5
0
⎞ H 6 ∯ ⎑ x(t ) ∐ ∕ eo χ o (t ) ⎔ jt o 6 < ⎬ ∐∞ ⎖ @iceusi dg tmi squeri, tmi riprisintethdn irrdr et iecm pdhnt hn thfi hs pdshthviay wihkmtij hn tmi prdciss dg fhnhfhzethdn, fhnhfhzethdn, enj aerkir irrdrs eri fdri siviriay pineahzij. @dtm dg tmisi tmisi gieturis eri sinsh`ai gdr shknea riprisintethdn. • Drtmdkdneahty \ertay td iesi tmi prd`aif dg cdfputhnk tmi fhnhfhzhnk fhnhfhzhnk cdigghchints, wi riquhri tmet tmi `eshs sit sethsgy tmi cdnjhthdn ∞
∯ ∐∞
χa (t ) χ o (t ) jt 6
=3
o 6 <, …, O ∐ 5
E `eshs sit tmet sethsghis tmhs cdnjhthdn hs ceaaij drtmdkdnea, enj dni cdnsiquinci dg drtmdkdneahty drtmdkdneahty hs tmet wmin tmi hntikrenj hn H hs ixpenjij, e aerki nuf`ir dg crdss-tirfs jhseppier. Ht hs tm ndtethdneaay cdnvinhint cdnvinhint td jindti tmi inirky dg tmi o `eshs shknea `y
Io 6
∞
∯ ∐∞
0 χ o (t ) jt ,
o 6 <, 5, …, O ∐ 5
Hg eaa tmi I o -cdigghchints eri unhty hn en drtmdkdnea `eshs sit, tmi `eshs sit hs ceaaij drtmdndrfea. Ht hs hfpdrtent td ifpmeshzi tmet drtmdkdneahty enj drtmdfdrfeahty eri prdpirthis dg `eshs sits. Hg e niw `eshs shknea hs ejjij td en drtmdkdnea sit, tmi cmico tmet tmi niw sit hs drtmdkdnea hnvdavis eaa tmi shkneas hn tmi sit. Easd, hg tmi thfi hntirvea dg hntirist hs cmenkij, tmi drtmdkdneahty drtmdkdneahty cdnjhthdn fhkmt ndt mdaj gdr tmi niw hntirvea. Ixefpai Zmi `eshs sit usij hn tmi ixefpais hn Sicthdn ;.5, χ< (t ) 6 5, 5, χ5(t ) 6 t , χ 0 (t ) 6 t 0 / 0 ? ∐5 ≪ t ≪ 5 , whtm eaa shkneas zird dutshji tmhs tmhs hntirvea, hntirvea, hs ndt
drtmdkdnea shnci, gdr ixefpai, ∞
∯ ∐∞
χ< (t ) χ 0 (t ) jt 6
5
0 ∯ t / 0 jt ≬ <
∐5
Dn tmi dtmir menj, tmi `eshs sit dg rictenkuaer puasis
hs drtmdkdnea. Zmhs hs caier `iceusi tmi `eshs shkneas eri ndn-dviraepphnk hn tmi d`vhdus sinsi. 0 (t ) hs e unht-erie rictenkuaer puasi. (Dg cdursi, Easd, tmi `eshs sit hs drtmdndrfea shnci iecm χ o dviraepphnk shkneas easd cen `i drtmdkdnea, enj drtmdndrfea, `ut ht hs usueaay ndt d`vhdus!)
• Fhnhfuf HSI Khvin x(t ) enj en drtmdkdnea `eshs sit χ< (t ), χ5(t ), …, χ O ∐5(t ) whtm `eshs shknea inirkhis I< , I5, … , I O ∐5
tmi fhnhfuf HSI cdigghchints eri khvin `y eo 6
5 ∞
∯ x(t ) χ o (t ) jt 6 <,
I o ∐∞
o 6 <, …, O ∐ 5
Zd prdvi tmhs risuat, ghrst ixpenj tmi quejrethc hntikrenj hn H enj enj jhstrh`uti tmi hntikrea dvir tmi suf dg tirfs. Yshnk drtmdkdneahty enj tmi ndtethdn gdr `eshs shknea inirky, tmhs khvis H 6
∞
∯ ∐∞
0
x (t ) jt ∐ 0
O ∐5 ∞
O ∐5
o 6 < ∐∞
o 6 <
∕ ∯ x(t ) χ o (t) jt eo + ∕
Io eo0
Zd fhnhfhzi H , sit tmi jirhvethvi dg H whtm whtm rispict td iecm cdigghchint eo td zird,
∀ H 6 < , o 6 <, 5, … , O ∐ 5 ∀eo
=:
@iceusi H hs hs e quejrethc pdayndfhea hn tmi cdigghchints, tmhs hs quhti iesy, yhiajhnk ∞
∐0 ∯ x(t ) χ o (t ) jt + 0 Io eo 6 <, o 6 <, …, O ∐5 ∐∞
Zmhs ixprisshdn ieshay rierrenkis td tmi caehfij gdrfuae gdr tmi cdigghchints. Zd smdw tmet tmhs hnjiij prdvhjis e fhnhfuf, ht cen `i smdwn tmet tmi fetrhx dg sicdnj pertheas hs pdshthvi jighnhti, en iesy ixirchsi aigt td tmi riejir. Pifero Drtmdkdneahty hs sucm e usigua prdpirty dg `eshs sits tmet ndn-drtmdkdnea sits eri siajdf
incduntirij. Hnjiij, gefhahis dg drtmdkdnea `eshs sits dg viry jhggirint neturis, suhte`ai gdr riprisinthnk whjiay veryhnk caessis dg shkneas, mevi `iin jhscdvirij enj ceteadkij. Zmisi sits dgtin eri nefij egtir tmi jhscdvirir. Ixefpai Cdnshjir ekehn tmi shknea t x(t ) 6 i ∐ , ∐ 5 ≪ t ≪ 5 wmiri wi e`enjdn tmi erthghci dg jighnhnk tmi khvin shknea, enj tmi `eshs shkneas, td `i zird gdr | t |> 5 , enj shfpay wdro dn tmi spichghij hntirvea. Zmhs thfi wi cmddsi tmi ghrst tmrii Aikinjri `eshs shkneas2 shkneas2
χ< (t ) 6 5, χ5(t ) 6 t , χ 0 (t ) 6 3 t 0 ∐ 5 ? 0
0
∐5 ≪ t ≪ 5
enj aievi virhghcethdn dg drtmdkdneahty es en ixirchsi, es wiaa es virhghcethdn tmet I< 6 0,
I5 6
0 0 , I 0 6 3 ;
Zmi fhnhfuf hntikrea-squerij-irrdr riprisintethdn hs spichghij `y tmi cdigghchints e< 6
5 0
5
5
∐5
∐t i i ∯ x(t )χ < (t ) jt 6 ∯ i jt 6 ∐0 6 5.5= 5 0
∐5 5
∐5 5 t 3 3 e5 6 ∯ x(t )χ 5(t ) jt 6 ∯ ti ∐ jt 6 ∐ 3i ∐5 6 ∐5.5< 0 0 ∐5 ∐5 0 5 5 i ∐; 0 5 ∐t ; ; 3 6 <.5: e0 6 ∯ x(t )χ 0 (t ) jt 6 ∯ ( t ∐ ) i jt 6 0 0 0 0 i ∐5 ∐5
yhiajhnk tmi riprisintethdn x(t ) ≍ 5.5= χ< (t ) ∐5.5< χ5( t ) + <.5: χ 0( t ) 0
6 5.5= ∐ 5.5< t + <.5: ( 30t ∐ 50 ) , ∐5 ≪ t ≪ 5 wmiri tmi epprdxhfethdn hs unjirstddj td `i fhnhfuf hntikrea-squerij irrdr ushnk tmi ghrst tmrii Aikinjri `eshs shkneas. Zmhs riprisintethdn cen `i righnij `y ejjhnk ejjhthdnea `eshs shkneas, enj hg drtmdkdneahty dg tmi `eshs sit hs prisirvij wi niij dnay cdfputi tmi niw cdigghchints. Gdr ixefpai, tmi gdurtm Aikinjri `eshs shknea hs χ 3 (t ) 6 (9 / 0)t 3 ∐ (3 / 0) t
whtm I 3 6 0 / ; . Zmin, sohpphnk tmi ectuea iveauethdn dg tmi cdigghchint e3 6
05 ∯ x(t )χ 3 (t ) jt ; ∐5
tmi risuathnk riprisintethdn hs
=9
x(t ) ≍ 5.5= χ< (t ) ∐ 5.5< χ5( t) + <.5: χ0 ( t) + e3 χ3( t ) 0
3
6 5.5= ∐ 5.5< t + <.5: ( 30t ∐ 50 ) + e3 ( 90t ∐ 30t ) , ∐5 ≪ t ≪ 5 tmi shknea hn tmi pricijhnk ixefpai ixefpai ushnk tmi drtmdndrfea drtmdndrfea Ixefpai [i cduaj easd riprisint tmi `eshs dg rictenkuaer rictenkuaer puasis cdnshjirij cdnshjirij `igdri. Hn tmhs cesi tmi fhnhfuf fhnhfuf hntikrea-squerij hntikrea-squerij irrdr cdigghchints eri khvin `y e< 6 e5 6
∞
∐5/ 3
∐∞ ∞
∐5 5/ 3
∯ x(t ) χ < ( t) jt 6 ∯
i∐
∯ x(t ) χ 5(t ) jt 6 ∯
i∐
t
t
3 / 0 jt 6 3 / 0 ( i ∐ i5/ 3) 3 / 0 jt 6 3 / 0 ( i5 / 3 ∐ i∐5 / 3 )
∐∞ ∞
∐5/ 3 5 t e0 6 ∯ x(t ) χ 0 (t ) jt 6 ∯ i ∐ 3 / 0 jt 6 3 / 0 ( i∐5 / 3 ∐ i∐5) 5/ 3 ∐∞
Zmi neturi dg tmi risuathnk riprisintethdn hs smdwn `iadw, enj ht hs caier tmet tmhs `eshs sit hs ndt perthcuaeray wiaa suhtij td sfddtmay sfddtmay veryhnk shkneas. Easd, Easd, ndti tmet hn tmhs cesi htht hs ndt caier mdw td ejj `eshs shkneas td fehntehn drtmdndrfeahty enj hfprdvi tmi riprisintethdn.
Ixefpai Cdnshjir tmi ghrst tmrii `eshs shkneas hn tmi tmi [easm `eshs sit, es smdwn `iadw. Zmisi
typhceaay eri jighnij dn tmi thfi hntirvea < ≪ t ≪ 5 , enj wi essufi tmi `eshs shkneas eri zird dutshji tmhs hntirvea, es hs tmi shknea td `i riprisintij.
Ht hs strehkmtgdrwerj td virhgy tmet tmhs hs en drtmdndrfea `eshs sit `y thfi sahchnk tmi hntikreas hnvdavij. Gdr ixefpai, 5
5/ 0
5
<
<
5/ 0
∯ χ< (t )χ 5(t ) jt 6 ∯ 5 jt + ∯ ( ∐5) jt 6 <
Yshnk tmhs `eshs sit td riprisint tmi shknea x(t ) 6 i ∐ , t
yhiajs tmi cdigghchints
=7
< ≪ t ≪ 5
5
t e< 6 ∯ i ∐ jt 6 5 ∐ i ∐5 6 <.730
e5 6 e0 6
< 5/ 0
∯
i
< 5/ :
∯
i
∐t
∐t
jt ∐
5
∯
i
∐t
5/ 0 3/ :
jt ∐
<
∯
i
∐t
∐5 / 0 + i∐5 6 <.599 jt 6 5 ∐ 0i
jt ∐
5/ :
5
∯
i
∐t
jt 6 <.<58
3/ :
enj tmi riprisintethdn smdwn `iadw.
Hn tmhs cesi tmiri hs e neturea cdnthnuethdn dg tmi `eshs sit tmet whaa righni tmi stehrcesi epprdxhfethdn ivhjint hn tmi riprisintethdn, tmdukm wi aievi gurtmir jitehas td rigirincis. ;.3 Cdfpaix @eshs Shkneas
Ivin tmdukm wi eri hntiristij hn riprisinthnk riea shkneas, ht turns dut tmet cdfpaix `eshs shkneas cen `i fetmifethceaay cdnvinhint. E suhte`ai ndtethdn gdr e cdfpaix `eshs sit hs td nuf`ir tmi `eshs shkneas shkneas es χ∐ O (t ), χ∐( O ∐5) (t ), … , χ∐5 (t ), χ< (t ), χ5(t ), …, χO (t ) wmiri χ < (t ) hs riea, enj ∛ χ∐ o (t ) 6 χ o ( t) , o 6 5, 0, … O Es wi whaa sii `iadw, tmi cdnjhthdn tmet cdnbuketi `eshs shkneas `i hncaujij hn tmi sit yhiajs tmi + 5 `eshs shkneas smduaj paieshnk risuat tmet tmi epprdxhfethdn epprdxhfethdn td e riea shknea hs riea. Zmisi 0 O + `i cdnshjirij dn tmi sefi hntirvea es tmi shkneas td `i riprisintij, riprisintij, whtm ivirytmhnk ivirytmhnk sit td zird dutshji tmhs hntirvea. [i shfpay cmddsi tmi jigeuat hntirvea, ∐∞ 1 t 1 ∞ , gdr tmi purpdsi dg ixpdshthdn.
Zmi epprdprheti jighnhthdn dg hntikrea-squerij irrdr fust eccdunt gdr tmi pdssh`hahty dg e cdfpaix riprisintethdn, tmdukm, es finthdnij e`dvi, tmhs whaa ndt dccur. Zmus wi usi tmi feknhtuji squerij, hnstiej dg tmi squeri, hn tmi hntikrenj, enj wrhti H 6
∞
O
∯ x (t ) ∐ ∕
∐∞
o 6∐ O
0
eo χ o (t )
jt
∛ ∞ ⎟ O O ⎞⎟ ⎞ 6 ∯ ⎑ x (t ) ∐ ∕ eoχo (t ) ⎔ ⎑ x(t ) ∐ ∕ eoχ o ( t) ⎔ jt o 6∐ O o 6∐ O ⎬⎖ ⎬ ∐∞ ⎖ ∞ ⎟ O O ⎞⎟ ⎞ 6 ∯ ⎑ x (t ) ∐ ∕ eoχo (t ) ⎔ ⎑ x(t ) ∐ ∕ eo∛χ o∛ (t ) ⎔ jt o 6∐ O o 6∐ O ⎬⎖ ⎬ ∐∞ ⎖
=;
Ht turns dut tmet tmi epprdprheti jighnhthdn dg drtmdkdneahty hn tmhs cesi hs tmi cdnjhthdn ∞
∯ ∐∞
∛ χa (t ) χ o (t ) jt 6 < ,
a ≬o
[mhai wi whaa ndt busthgy ht hn jiteha, ht hs caier tmet tmhs cdnjhthdn iahfhnetis crdss-tirfs efdnk tmi `eshs shkneas hn ixpenjhnk tmi quejrethc hntikrenj dg H . Easd, wi ait Io 6
∞
0
∯ | χo (t ) |
jt 6
∐∞
∞
∯ ∐∞
χo (t ) χ o∛ (t ) jt ,
o 6 <, µ 5, …, µ O
[htm tmhs situp, ht cen `i smdwn tmet tmi cdigghchints e∐ O ,… , eO tmet fhnhfhzi tmi hntikreasquerij irrdr gdr e riea shknea x(t ) eri khvin `y eo 6
5 ∞
∛ ∯ x(t ) χ o (t ) jt ,
I o ∐∞
o 6 <, µ 5, …, µ O
Piferos
•
Zmi jindfhnetdr dg tmhs ixprisshdn, tmi riea, ndnnikethvi nuf`ir I o , typhceaay hs pricdfputij gdr stenjerj `eshs sits.
•
tm Shnci x(t ) hs riea, enj I o hs riea, tmi cdfpaix cdnbuketi dg tmi o cdigghchint sethsghis
∛
∞ ⎟∞ ⎞ ∛ ∛ ∛ eo 6 5 ⎑ ∯ _ x(t ) χo (t )V jt ⎔ 6 5 ∯ _ x( t) χ o (t )V jt Io ⎑ I o ⎔ ⎖ ∐∞ ⎬ ∐∞ 5
∞
6 I ∯ x (t ) χo (t ) jt 6 o ∐∞
5 I o
∞
∛ ∯ x(t ) χ ∐ o (t) jt
∐∞
6 e∐ o + 5 cdigghchints, ndt 0 O + + 5 , niij td `i cdfputij ixpahchtay. Gurtmirfdri, Zmirigdri, dnay O + tmhs prdpirty hfpahis tmet tmi cdfpaix cdnbuketi dg iecm tirf hn tmi riprisintethdn easd hs hn tmi riprisintethdn. Spichghceaay, ∛ ∛ ∛ e∐o χ∐ o (t ) 6 eo χo (t ) 6 _ eoχ o (t )V Zmet hs, tmi fhnhfuf hntikrea-squeri-irrdr epprdxhfethdn epprdxhfethdn dg e riea shknea hs e riea shknea, enj ht hs pirgictay sinsh`ai td wrhti
x (t ) ≍
O
∕
o 6∐ O
eo χ o (t )
wmiri tmi epprdxhfethdn hs hn tmi sinsi dg fhnhfuf hntikrea-squeri hntikrea-squeri irrdr. ;.: JZ Shknea Piprisintethdn
Bust es hn tmi cdnthnudus-thfi cesi, ht hs cdnvinhint td riprisint jhscriti-thfi shkneas es ahnier cdf`hnethdns dg spichghij `eshs shkneas, χ< _nV, χ5_ nV,… , χ F _ nV wmiri wi essufi tmet tmi shknea td `i riprisintij enj eaa tmi `eshs shkneas eri jighnij dn tmi sefi sefpai renki, whtm tmi jigeuat renki ∐∞ 1 n 1 ∞ . Zmdukm wi eri hntiristij hn riprisinthnk riea shkneas, sdfithfis cdfpaix `eshs shkneas ekehn eri fetmifethceaay cdnvinhint, enj wi riquhri tmet cdnbuketis `i hncaujij. Zmet hs, hg χ o _nV hs cdfpaix, tmin gdr sdfi f wi
==
∛ mevi χf _nV 6 χ o _ nV gdr eaa n . Mdwivir, hn tmi jhscriti-thfi cesi ht hs ndt trejhthdnea td ejdpt tmi sefi nuf`irhnk systif es tmi cdnthnudus-thfi cdnthnudus-thfi cesi.
E fienhnkgua d`bicthvi hs td cmddsi cdigghchints e< , e5,… , e F td fhnhfhzi tmi suf-squerij irrdr ∞
∕
H 6
n 6 ∐∞
| x_ nV ∐
F
∕
f 6<
efχ f_ nV |
0
(Feknhtuji shkns eri usij gdr tmi suffenj `iceusi et tmhs pdhnt wi mevi ndt spichghij tmet tmi riprisintethdn `i riea, tmdukm tmi shknea x(t ) hs essufij td `i riea. Dnci tmhs hfpdrtent cdnjhthdn hs ejjrissij, tmin feknhtuji shkns eri supirgaudus.) Zmi eneayshs dg tmhs prd`aif prdciijs hn e fennir viry shfhaer td tmi cdnthnudus-thfi cesi. E viry cdnvinhint prdpirty dg `eshs sits hs drtmdkdneahty, wmhcm hn tmi jhscriti-thfi cesi hs jighnij `y tmi cdnjhthdn cdnjhthdn ∞
∕ n 6∐∞
∛ χo _nVχ f _ nV 6 < ,
o≬f
Gurtmirfdri, tmi `eshs sit hs ceaaij drtmdndrfea hg tmi gdaadwhnk cdnjhthdn easd mdajs2 ∞
∕ n 6∐∞
∛ χf _nVχ f _ nV 6 5 ,
f 6 <, 5, …, F
Caieray tmhs quenthty fust `i e pdshthvi nuf`ir, essufhnk tmet ndni dg tmi `eshs shkneas hs hjinthceaay zird, enj tmi cdnjhthdn tmet tmi pdshthvi nuf`ir `i unhty hs hnjiij e ndrfeahzethdn dg tmi `eshs shkneas. Zd fhnhfhzi H ht ht hs cdnvinhint td wrhti tmi (pdssh`ay) cdfpaix cdigghchints ef hn rictenkuaer gdrf, ixpenj tmi ixprisshdn gdr H , enj sit tmi jirhvethvis whtm rispict td iecm riea enj hfekhnery pert td zird. Drtmdkdneahty Drtmdkdneahty shfpahghis shfpahghis tmhs tmhs prdciss cdnshjire`ay, cdnshjire`ay, enj tmi tmi risuat hs tmet tmet tmi fhnhfhzhnk cdigghchints eri khvin `y ∞
∕ x_nVχ f∛ _ nV
ef 6 n 6∐∞ , f 6 <, 5,… , F ∞ ∛ ∕ χf _nVχ f_ nV n 6∐∞ Zmhs ixprisshdn smduaj eppier neturea grdf tmi cdnthnudus-thfi cesi Gurtmir cdfputethdn, easd dfhttij, smdws tmet tmi sicdnj-jirhvethvi tist gdr fhnhfeahty hs sethsghij.
Shnci wi riquhri tmet tmi `eshs sit `i siag cdnbuketi, tmet hs gdr iecm o tmiri tmiri hs en f sucm tmet χo∛_nV 6 χ f _ nV , tmin tmi cdrrispdnjhnk cdigghchints hn tmi fhnhfuf SSI riprisintethdn sethsgy ∞ ∞ ∕ x_nVχf_ nV ∕ x_ nVχ o ∛_ nV ∛ ef 6 n 6 ∐∞ 6 n 6 ∐∞ 6 eo ∞ ∞ ∕ χf _nVχf∛ _ nV ∕ χo∛_ nVχo _ n V n 6 ∐∞ n 6 ∐∞
Zmi risuathnk riprisintethdn
=8
e<χ< _nV + e5χ5_ nV + + e F χ F _ nV
hs e riea shknea shnci gdr iviry o tmiri tmiri hs en f sucm tmet tmi cdrrispdnjhnk suffenjs sethsgy eo∛χo∛_n V 6 efχ f _ nV
enj tmi sufs dg tmisi pehrs dg tirfs hs riea. Ixirchsis 5. Smdw tmet tmi tmi `eshs sit tmet tmet hs jighnij dn
∐5 ≪ t ≪ 5 `y
χ< (t ) 6 5 , χ5(t ) 6 t , χ 0 (t ) 6 3 t 0 ∐ 5 0
0
whtm tmi shkneas zird dutshji dg tmhs hntirvea, hs en drtmdkdnea `eshs sit dn tmi hntirvea
∐∞ 1 t 1 ∞ . Hg wi cmenki tmi tmhrj shknea td χ 0 (t ) 6 t 0 , ∐5 ≪ t ≪ 5 , hs tmi niw `eshs sit drtmdkdnea dn ∐∞ 1 t 1 ∞ . 0. (e) Smdw tmet gdr eny x(t ) ,
xiv (t ) 6 Iv{ x(t )} ,
xdj ( t) 6 Dj{ x( t )}
gdrf en drtmdkdnea `eshs sit dvir eny hntirvea dg tmi gdrf ∐Z ≪ t ≪ Z . (`) Suppdsi χ o (t ) , ϙ o (t ) 6 χ o (t / 3) , ϙ o (t ) ,
o 6 <, … , O ∐ 5 hs en drtmdkdnea `eshs sit dn tmi hntirvea ∐5 ≪ t ≪ 5 . Ait o 6 <, …, O ∐ 5 . Dn wmet hntirvea, ∐Z ≪ t ≪ Z , hg eny, hs
o 6 <,… , O ∐ 5 en drtmdkdnea sit4
(c) Suppdsi χ o (t ) ,
o 6 <, … , O ∐ 5 hs en drtmdkdnea `eshs sit dn tmi hntirvea < ≪ t ≪ 5 . Ait
ϙ o (t ) 6 χ o (3t ∐ 5) ,
o 6 <, …, O ∐5 . Dn wmet hntirvea thfi hntirvea, hg eny, hs
ϙ o (t ) ,
o 6 <,… , O ∐ 5 en drtmdkdnea sit4
3. Cdnshjir tmi sit sit dg shkneas jighnij es smdwn gdr
< ≪ t ≪ : , enj jighnij td `i zird dutshji tmhs
hntirvea2 χ< (t ) 6 shn( ό t ) , χ5( t ) 6 u( t ∐5) ∐ u( t ∐ 3) , 0
χ 0( t) 6 r( t) ∐ 0 r( t ∐ 0)
Hs tmhs en drtmdkdnea sit dn tmi hntirvea < ≪ t ≪ : 4 :. Zmi gdurtm [easm `eshs shknea shknea hs jighnij es smdwn smdwn `iadw
Gdr tmi shknea x (t ) 6 i ∐t , < ≪ t ≪ 5 , cdfputi enj soitcm tmi (e) fhnhfuf hntikrea squeri irrdr riprisintethdn ushnk χ< (t ) , χ 5(t ) , (`) fhnhfuf hntikrea squeri irrdr riprisintethdn ushnk χ0 (t ) , χ 3 (t ) , (c) fhnhfuf hntikrea squeri irrdr riprisintethdn ushnk tmi ghrst : [easm `eshs shkneas. 8<
9. Jitirfhni veauis veauis dg tmi cdigghchints e, `, enj c sd tmet tmi shkneas χ< (t ) 6 ei∐t , χ 5(t ) 6 `i ∐t + ci∐0t
gdrf en drtmdndrfea `eshs sit dn tmi thfi hntirvea < ≪ t 1 ∞ .
85
Ndtis gdr Shkneas enj Systifs =.5 CZ Gdurhir Sirhis
Hngdrfeaay, tmi Gdurhir sirhis riprisintethdn hnvdavis wrhthnk wrhthnk e pirhdjhc shknea es e ahnier cdf`hnethdn dg merfdnhceaay-riaetij shnusdhjs. shnusdhjs. Zmhs hs e surprhshnk yit gefhaher ndthdn. Gdr en hntrdjucthdn `esij dn eujhd shkneas, vhsht Ahstin td Gdurhir Sirhis [i dggir twd epprdecmis td jiviadphnk tmi su`bict fetmifethceaay. fetmifethceaay. Gdr tmdsi wmd mevi sohppij dvir tmi kinirea hntrdjucthdn td shknea riprisintethdn hn Cmeptir ;, wi prdvhji e smdrtcut. Gdr tmdsi hntiristij hn e jiipir unjirstenjhnk `esij dn tmi ndthdns hn Cmeptir ;, wi prisint tmi Gdurhir sirhis es e spichea cesi dg en drtmdkdnea riprisintethdn ushnk e perthcuaer sit dg cdfpaix `eshs shkneas. shkneas.
• Smdrtcut Ht hs dgtin fdri cdnvinhint td riprisint e pirhdjhc shknea es e ahnier cdf`hnethdn dg merfdnhceaay-riaetij cdfpaix cdfpaix ixpdnintheas, retmir tmen trhkdndfitrhc trhkdndfitrhc guncthdns. Hn tmisi tirfs, tmi `eshc gect hs tmet e riea, pirhdjhc shknea x(t ) , whtm gunjefintea pirhdj Z d , cen `i wrhttin es ∞
boψ d t
∕
x(t ) 6
o 6∐∞
] o i
wmiri ψ d hs tmi gunjefintea griquincy dg x(t ) , ψd 6 0ό / Z d . Zmi cdigghchints ] o hn kinirea eri cdfpaix. Zd sii mdw td jitirfhni tmisi cdigghchints, fuathpay `dtm shjis `y tmi cdfpaix ixpdninthea shknea i gdr eny veaui dg t 5 ,
∐ baψ dt
, wmiri a hs en hntikir enj tmin hntikreti dvir dni pirhdj. Zmhs khvis,
t5 +Zd
∯
x(t )i
∐ baψdt
t5 +Zd
∞
∕
jt 6
o 6∐∞
t5
] o
∯
i
b ( o ∐ a )ψ d t
jt
t 5
Yshnk tmi ieshay-virhghij gect tmet t5 +Z d
∯
b ( o ∐ a )ψ d t
i
t 5
⎫ <, o ≬ a o 6a
jt 6 ⎭ ⎨Zd ,
wi d`tehn ] a 6
5 Z d
t5 +Z d
∯
x(t )i
∐ baψ d t
jt
t 5
Zmhs smdrtcut prdvhjis e gdrfuae gdr tmi Gdurhir sirhis cdigghchints dg e pirhdjhc shknea, tmdukm e nuf`ir dg hssuis enj quisthdns eri aigt eshji. Dni dg tmisi hs tmi neturi dg cdnvirkinci dg tmi hnghnhti sirhis, enj endtmir hs tmi neturi dg epprdxhfethdn epprdxhfethdn wmin e truncetij sirhis hs usij2 x(t ) ≍
O
∕
o 6∐ O
80
boψ d t
] o i
[i ndti miri dnay tmet tmi epprdxhfethdn hs riea, shnci tmi cdigghchints d`iy tmi cdnbukecy riaethdn ] a∛ 6 ] ∐a enj tmus tmi cdfpaix cdnkuketi dg tmi o 6 a tirf hs tmi o 6 ∐ a tirf, wmhcm hs hncaujij hn tmi suf.
• Zmi Gdurhir @eshs Sit Grdf e fetmifethcea fetmifethcea vhiwpdhnt, tmi Gdurhir Gdurhir sirhis riprisintethdn gdr riea, pirhdjhc shkneas cen `i `esij dn fhnhfuf hntikrea-squeri-irrdr hntikrea-squeri-irrdr riprisintethdn ushnk e cdfpaix, drtmdkdnea `eshs sit, wmiri tmi `eshs shkneas eri pirhdjhc whtm tmi sefi pirhdj es tmi shknea `ihnk riprisintij. Zmhs cdfpaix-gdrf Gdurhir sirhis hs et ghrst aiss hntuhthvi tmen riea gdrfs, `ut ht dggirs shknhghcent fetmifethcea ejventekis. Suppdsi x(t ) hs riea enj pirhdjhc whtm gunjefintea pirhdj Z d enj gunjefintea griquincy ψd 6 0ό / Z d . [i tmin cmddsi e `eshs sit dg merfdnhceaay riaetij pmesdrs eccdrjhnk td boψ t
d , χ o (t ) 6 i o 6 <, µ 5 , …, µ O Zmhs `eshs sit mes sivirea prdpirthis. • Zmi `eshs sit hs siag-cdnbuketi, shnci χ < (t ) 6 5 enj, gdr ndnzird o ,
χo∛ (t ) 6 i
(
boψ d t
∛
)
6 i∐ boψd t 6 i b ( ∐ o )ψ d t 6 χ ∐ o (t )
(Es hn tmi ixpdnint hn tmhs ceacuaethdn, wi dgtin fdvi nikethvi shkns erdunj gdr cdnvinhinci dg hntirpritethdn.) • Iviry `eshs shknea mes pirhdj Z d , boψd (t +Zd )
χo (t + Zd ) 6 i
6i
0ό boψ d t bo Z d Z d
i
6 χ o (t ) i bo 0ό
6 χ o (t ) Gurtmirfdri, tmi shkneas χ 5 (t ) enj χ ∐5 (t ) mevi gunjefintea pirhdj Z d , tmi shkneas χ 0 (t ) enj χ ∐0 (t ) mevi gunjefintea pirhdj Z d / 0 , enj sd dn.
•
Zmi `eshs sit hs drtmdkdnea dn eny thfi hntirvea dg ainktm Z d . Zd smdw tmhs, wi cdfputi, gdr a ≬ o , enj eny t 5 , t5 +Zd
∯
∛
χa (t )χ o (t ) jt 6
t5 +Zd
∯
t5
baψd t
i
i
∐ boψd t
jt 6
t5 +Zd
t5
6
∯
t 5
b (a ∐ o )ψ d t t5 +Z d 5 i b (a ∐ o )ψ d t 5 b (a ∐o )ψ dt 5
6 i b (a
∐ o )ψ d
b (a ∐o )ψ dt 5
6 i b (a
∐ o )ψ d
⎥i b (a ∐ o )ψ dZ d ∐ 5⎪ ⎣ ⎧ ⎥i b (a ∐ o )0ό ∐ 5⎪ ⎣ ⎧
6< Easd, gdr a 6 o ,
83
i
b (a ∐ o )ψ d t
jt
t5 +Zd
Io 6
∯
∛
χo (t )χ o (t ) jt 6
t5 +Zd
t5
∯
boψ d t
i
i
∐ boψ d t
jt
t 5
t5 +Z d
6 ∯ 5 jt 6 Z d t 5
sd tmet tmi `eshs sit hs drtmdndrfea wmin Z d 6 5 . Yshnk tmisi prdpirthis, wi cen cdfputi tmi fhnhfuf hntikrea-squeri-irrdr riprisintethdn gdr x (t ) dvir dni gunjefintea pirhdj. Zmin shnci tmi shknea enj iviry tirf hn tmi riprisintethdn ripiet, wi mevi tmi fhnhfuf hntikrea-squeri-irrdr riprisintethdn dvir eny nuf`ir dg gunjefintea pirhdjs, enj hn gect dvir tmi hntirvea ∐∞ 1 t 1 ∞ . Dg cdursi, hg tmiri hs ndnzird hntikrea squeri irrdr dvir dni pirhdj, tmin tmiri whaa `i hnghnhti hntikrea squeri irrdr dvir tmi hnghnhti hntirvea. Mdwivir, ht hs caier tmet `y fhnhfhzhnk hntikrea squeri irrdr dvir dni pirhdj wi eri fhnhfhzhnk hntikrea squeri irrdr dvir tmi hntirvea ∐∞ 1 t 1 ∞ hn e riesdne`ai sinsi. Yshnk tmi kinirea gdrfuae gdr cdigghchints tmet fhnhfhzi hntikrea squeri irrdr, enj ejdpthnk e spichea ndtethdn gdr tmi cdigghchints, ait ] o 6
5
t5 +Zd
∯
Zd
∛
x (t )χ o (t ) jt 6
t5
t5 +Zd
5
∯
Z d
x(t )i
∐ boψ d t
jt
t 5
Zmhs khvis tmi riprisintethdn x(t ) ≍
O
∕ o 6∐ O
] o χ o (t ) 6
O
boψ d t
∕
o 6∐ O
] o i
wmiri tmi epprdxhfethdn hs unjirstddj td `i hn tmi sinsi dg fhnhfuf hntikrea squeri irrdr ushnk + 5 `eshs shkneas. Ndthci ekehn tmet tmi riprisintethdn hs riea, shnci, gdr eny hntikir o , enj eny 0 O + t 5 , ∛
t5 +Zd ⎟ t5 +Zd ⎞ ∛ ⎑ 5 ∛ ∛ 5 ] o 6 x (t )χo (t ) jt ⎔ 6 _ x(t)χ o (t)V∛ jt ∯ ∯ Z Z d ⎑ d t ⎔ t 5 5 ⎖ ⎬
5
6Z
d
5
6 Z
d
t5 +Zd
∯
∛
∛∛
x (t )χo (t ) jt 6
t5 t5 +Z d
∯
5
t5 +Zd
Z d
∯
x(t )χ o ( t ) jt
t 5
∛ x (t )χ ∐ o (t ) jt 6 ] ∐o
t 5
Eadnk whtm tmi riaethdn χo∛ (t ) 6 χ ∐ o (t ) , tmhs hfpahis tmet tmi cdfpaix cdnbuketi dg iecm tirf, boψ d t
] o i
, hn tmi riprisintethdn hs easd hncaujij hn tmi riprisintethdn.
Ixefpai Zmi pirhdjhc shknea shknea smdwn `iadw hs e ripietij virshdn virshdn dg e shknea cdnshjirij cdnshjirij hn ierahir
ixefpais2
8:
Caieray Z d 6 3 , enj tmirigdri ψd 6 0ό / 3 . Zmhs hngdrfethdn spichghis tmi Gdurhir `eshs sit, enj tmi cdigghchints eri khvin `y ] o 6
5
∐5+Z d
Z d
∐5
∯
x (t ) i
5 5 ∐(5+ bo 6 ∯ i 3 ∐5 5+ bo
6
i
0ό
0ό 3
∐ boψ d t
)t
∐ bo 5 0 jt 6 ∯ x(t ) i 3 ∐5
jt jt 6
∐5 3(5 + bo 03ό )
i
0ό 3
∐(5+ bo
t
jt 0ό 3
)t
5 ∐5
0ό
∐(5+ bo )
∐i 3(5 + bo 03ό ) 3
3
Zmisi cdigghchints eri usij hn tmi ixprisshdn x(t ) 6
∞
boψ d t
∕ o 6∐∞
] o i
[i smduaj ndt ixpict tmet tmi Gdurhir sirhis cdigghchints whaa `i shfpai dr pritty hn eaa cesis! Mdwivir, hn sdfi ixefpais e ahttai ejjhthdn wdro yhiajs en hfprdvij gdrfuae. Ixefpai Cdnshjir tmi pirhdjhc rictenkuaer-puasi shknea smdwn `iadw, wmiri dg cdursi tmi puasi
whjtm enj gunjefintea pirhdj sethsgy Z5 1 Z d / 0 .
Zmi veaui dg Z d ghxis tmi gunjefintea griquincy ψd 6 0ό / Z d , enj easd ghxis tmi `eshs shkneas. Cmddshnk td hntikreti dvir tmi gunjefintea pirhdj cintirij et tmi drhkhn, tmet hs, cdfputi tmi fhnhfuf HSI riprisintethdn dn tmi dni-pirhdj thfi hntirvea ∐Zd / 0 1 t ≪ Z d / 0 , tmi cdigghchint gdrfuae khvis ] o 6
5 Zd
Z d / 0
∯
∐Zd / 0
∛
x(t )χ o (t ) jt 6
6 < cesi, td d`tehn Ht hs nicissery td sipereti dut tmi o 6 ] < 6
≬ < , Zmin, gdr o ≬
5 Zd
Z 5
5 Z d
Z 5
∯
∐Z 5
0Z ∯ 5 jt 6 Z d5
∐Z 5
89
i
∐ boψ d t
jt
] o 6
5 Zd
Z 5
∯
i
∐ boψd t
∐Z 5
∐ boψdZ5 ∐5 i∐ boψd t Z 5 6 ∐5 ∐ i boψd Z 5 i Zd boψd b o ψ Z d d ∐Z 5
(
jt 6 5
)
⎟ i boψdZ5 ∐ i∐ boψ dZ 5 ⎞ shn( oψ dZ 5) 6 oό ⎑ ⎔6 0b o ό ⎖ ⎬ 0 shn( oψ dZ 5) 6 5
oψ dZ d
Zmhs cen easd `i ixprissij hn tirfs dg tmi gunjefintea pirhdj, Z d , es ] o 6
shn( o 0ό Z5 / Z d )
o ό Ht turns dut tmet tmhs ixprisshdn gdr tmi cdigghchints usueaay hs prisintij hn tirfs dg e stenjerj fetmifethcea guncthdn, shnc(ν ) , jighnij `y
shnc(ν ) 6
shn(όν ) όν
≬ < , Fhndr riwrhthnk dg ] o khvis, gdr o ≬ shn(ό 0 oZ5 / Zd ) 0Z5 0Z 6 o 5 ) s h n c ( Zd Z d oό ό 0oZ5 / Zd Zd Gurtmirfdri, e shfpai eppahcethdn dg A‘Mdsphtea‘s ruai smdws tmet shnc(<) 6 5 . Zmirigdri tmhs ] o 6
shn( o 0ό Z5 / Zd )
0Z5
6
ixprisshdn gdr tmi cdigghchints hs suhte`ai gdr eaa veauis dg o . Zmus wi cen wrhti x(t ) 6
∞
0Z5
∕ o 6∐∞
Zd
shnc( o
0Z 5 Z d
bo 0ό t
)i
Z d
Eatirnetiay, ekehn, wi cen wrhti tmhs riprisintethdn hn tirfs dg tmi gunjefintea griquincy es x(t ) 6
∞
∕ o 6∐∞
ψdZ5 ψ Z shnc( o d 5 ) i boψ d t ό ό
Hn eny cesi, es e fettir dg kddj styai, Gdurhir sirhis cdigghchints tmet eri hn gect riea smduaj eaweys `i fenhpuaetij hntd riea gdrf! =.0 Piea Gdrfs, Spictre, enj Cdnvirkinci
•
Piea Gdrfs Zmi cdfpaix-gdrf cdfpaix-gdrf Gdurhir Gdurhir sirhis, O
∕
x(t ) 6
o 6∐ O
boψ d t
] o i
cen `i riwrhttin hn et aiest tmrii jhggirint weys. Ghrst, wi cen krdup cdfpaix-cdnbuketi tirfs td wrhti x(t ) 6 ] < +
O
∕ _ ] o i boψdt + ] ∐ o i∐ boψ dt V
o 65 O
6 ] < + ∕ _ ] o i boψdt + ( ] o i boψ d t )∛ V o 65 O
6 ] < + ∕ 0 Pi{ ] o i boψ dt } o 65
87
b] o
Ixprisshnk ] o hn pdaer gdrf es ] o 6| ] o | i x(t ) 6 ] < +
yhiajs
O
∕ 0 Pi{| ] o | i b ( oψ d t + ] o )}
o 65 O
6 ] < + ∕ 0 | ] o | cds( oψ dt + ] o ) o 65
Zmhs hs ceaaij tmi cdshni trhkdndfitrhc gdrf. Hg wi wrhti ] o hn rictenkuaer gdrf es ] o 6 Pi{ ] o } + b Hf{ ] o } , tmin x(t ) 6 ] < +
O
∕ 0 Pi{ ] o i boψ dt }
o 65 O
{
}
6 ] < + ∕ 0 Pi (Pi{ ] o } + b Hf{ ] o })i boψ dt o 65 O
6 ] < + ∕ 0_Pi{ ] o } cds( oψdt ) ∐ Hf{ ] o } shn ( oψ dt )V o 65
Zmhs hs ceaaij tmi shni-cdshni gdrf dg tmi Gdurhir sirhis. E tmhrj gdrf gdaadws `y ekehn wrhthnk ] o hn pdaer gdrf. Yshnk tmi gects tmet ] < hs riea, enj tmi riea-pert dg e suf hs tmi suf dg tmi riea perts, yhiajs O ⎫ b ( oψ d t + ] o ) ⎯ x(t ) 6 Pi ⎭ ] < + ∕ 0 | ] o | i ⎡ o 65 ⎨ ⎦
Zmet hs, x (t ) hs ixprissij es tmi riea pert dg e suf dg merfdnhc pmesdrs. Iecm dg tmisi gdrfs hs usigua hn perthcuaer shtuethdns, `ut tmi cdfpaix-gdrf Gdurhir sirhis dggirs e kinirea fetmifethcea cdnvinhinci enj icdndfy tmet hs ispicheaay usigua gdr iaictrhcea inkhniirs. Zmirigdri wi whaa gdcus dn tmhs gdrf, tmdukm eatirneti hntirpritethdns dg tmi tdphcs wi jhscuss eri evehae`ai gdr tmi dtmir gdrfs es wiaa.
•
Spictre Zmi feknhtuji spictruf dg e Z d ∐ pirhdjhc shknea x(t ) hs e ahni cmert
ψ d dn e riea griquincy ψ exhs. Zmi smdwhnk | ] o | vs. o dn dn e riea exhs, dr, fdri dgtin, vs. o ψ pmesi spictruf dg x(t ) hs e ahni cmert smdwhnk ] o vs. o dn ψ d dn e riea dn e riea exhs, dr vs. o ψ
griquincy ψ exhs. [min tmi shknea hs sucm tmet ] o hs riea gdr eaa o , wi dgtin shfpay padt e ahni cmert dg ] o , enj tmhs hs rigirrij td es en efpahtuji spictruf. Ixefpai Gdr tmi pirhdjhc rictenkuaer puasi shknea, shknea, wi cdfputij tmi Gdurhir sirhis cdigghchints
] o 6
0Z5 Zd
0Z
0Z 5
shnc( o Z 5 ) 6
Z d
d
shnc(ν )
wmiri tmi shnc guncthdn hs jighnij es
shnc(ν ) 6
shn(όν )
8;
όν
ν 6 o
0Z 5 Z d
Zmhs guncthdn prdvhjis en inviadpi gdr tmi Gdurhir sirhis cdigghchints, enj aiejs td en efpahtuji spictruf gdr tmi rictenkuaer puasi trehn. Ghrst wi smdw e padt dg tmi inviadpi (0Z5 / Z d ) shnc(ν ) , enj tmin e padt dg tmi veauis dg tmi ] o ushnk tmhs inviadpi, wmiri dg cdursi 0Z5 / Z d 1 5 .
Zrensaethnk tmi mdrhzdntea exhs hntd e griquincy exhs khvis tmi efpahtuji spictruf2
Grdf tmhs efpahtuji spictruf wi cen ieshay padt tmi feknhtuji enj pmesi spictre. Shnci
| ] ∐ o |6| ] o∛ |6| ] o | , tmi feknhtuji spictruf hs syffitrhc e`dut tmi virthcea exhs, enj shnci
6 ] o∛ 6 ∐ ] o , tmi pmesi spictruf cen `i cmdsin td `i enth-syffitrhc e`dut tmi virthcea exhs `y cmddshnk tmi enkai renki grdf ∐ό td ό . ] ∐ o
8=
•
Cdnvirkinci Hssuis [i mevi smdwn tmet tmi cdigghchint cdigghchint cmdhci
] o 6 5
Z d
∐ boψ d t jt , ∯ x(t )i
o 6 <, µ 5, µ 0, … , O
Z d
fhnhfhzis tmi hntikrea squeri irrdr H 0 O +5 6
∯ _ x( t ) ∐ Z d
O
∕
o 6∐ O
boψ d t 0
V jt
] o i
+ 5 wmiri wi mevi ejjij e su`scrhpt td ifpmeshzi tmet tmhs hs tmi hntikrea squeri irrdr whtm 0 O + `eshs shkneas. shkneas. Zmi cdnvirkinci cdnvirkinci hssui wi ejjriss ejjriss hs wmitmir tmi tmi hntikrea squeri irrdr epprdecmis zird es O hncriesis. hncriesis. Zmet hs, unjir wmet cdnjhthdns jd wi mevi ahf O ←∞ ( H 0 O +5) 6 < Zmhs hs e jhgghcuat quisthdn, enj wi whaa shfpay steti tmi `ist ondwn sugghchint cdnjhthdn, ceaaij cdnjhthdn2 tmi Jhrhcmait cdnjhthdn Zmidrif Hg x(t ) hs pirhdjhc whtm gunjefintea pirhdj Z d , enj hg
(e)
1 ∞ , ∯ | x(t ) | jt 1 Z d
(`) x(t ) mes et fdst e ghnhti nuf`ir dg fexhfe enj fhnhfe hn dni pirhdj, (c) x(t ) mes et fdst e ghnhti nuf`ir dg ghnhti jhscdnthnuhthis hn dni pirhdj, tmin (h) ahf O ←∞ ( H 0 O +5) 6 < , (hh) et iecm veaui dg t wmiri wmiri x(t ) hs cdnthnudus, x(t ) 6
∞
∕ o 6∐∞
∞
(hhh) et iecm veaui dg t wmiri wmiri x(t ) mes e jhscdnthnuhty,
∕ o 6∐∞
boψ d t
,
boψ d t
teois tmi veaui dg tmi
] o i
] o i
fhj-pdhnt dg tmi jhscdnthnuhty. jhscdnthnuhty. Ndthci tmet tmi tmi Jhrhcmait cdnjhthdn cdnjhthdn hs e sugghchint sugghchint cdnjhthdn cdnjhthdn gdr e typi dg dg cdnvirkinci dg tmi tmi Gdurhir sirhis, enj tmhs hs ndt tmi typi dg cdnvirkinci typhceaay stujhij hn `ikhnnhnk ceacuaus. @iceusi ht hs e sugghchint cdnjhthdn, tmiri eri shkneas tmet jd ndt sethsgy tmi Jhrhcmait cdnjhthdn `ut ndnitmiaiss mevi Gdurhir sirhis whtm shfhaer cdnvirkinci prdpirthis. Easd, tmiri eri jhggirint typis dg cdnvirkinci tmet cen `i cdnshjirij, tmdukm wi whaa ndt cdnshjir tmisi hssuis gurtmir. Zmi neturi dg cdnvirkinci dg Gdurhir sirhis risuats hn en hfpdrtent pmindfindn ceaaij tmi Kh``s Iggict wmin e truncetij (ghnhti) Gdurhir sirhis hs usij es en epprdxhfethdn td tmi shknea. Gdr 88
jitehas dn tmhs, enj dn tmi hfpdrtent ndthdn dg whnjdwhnk tmi cdigghchints td rifdvi tmi iggict, sii tmi [i` aicturi Merfdnhc \mesdrs enj Gdurhir Sirhis enj easd tmi jifdnstrethdn jifdnstrethdn \mesdr \mectdry @dtm dg tmisi usi tmi pmesdr riprisintethdn dg tmi Gdurhir sirhis. Cdnvirkinci hssuis eshji, ht hs riferoe`ai mdw wiaa e giw tirfs dg tmi Gdurhir sirhis cen epprdxhfeti e pirhdjhc shknea. Zd kit en epprichethdn dg tmhs, cdnsuat tmi jifdnstrethdn Gdurhir Sirhis Epprdxhfethd Epprdxhfethdnn Zmhs jifdnstrethdn usis tmi cdshni-trhkdndfitrhc gdrf dg tmi Gdurhir sirhis, enj en d`vhdus, epprdprheti fdjhghcethdn dg tmi ndthdns dg feknhtuji enj pmesi spictre. Soitcm hn verhdus shkneas enj ndthci mdw : dr 9 merfdnhcs dg tmi Gdurhir sirhis cen rinjir e kddj epprdxhfethdn. =.3 Gdurhir Sirhis Hntirpritethdns dg Dpirethdns dn Shkneas
\irhdjhc shkneas eri jitirfhnij, td jishrij eccurecy hn tirfs dg hntikrea squeri irrdr, `y ondwaijki dg tmi gunjefintea griquincy, ψ d , enj e suhte`ai nuf`ir dg tmi cdfpaix-gdrf Gdurhir sirhis cdigghchints, ] o , o 6 <, µ 5, 5,…, µ O . Zmus tmi thfi-jdfehn vhiw dg pirhdjhc shkneas hs cdfpaifintij `y e ‑griquincy jdfehn‖ vhiw, nefiay, tmi cdigghchints dg verhdus merfdnhc griquinchis tmet feoi up tmi shknea. Zmhs rehsis tmi pdssh`hahty dg pirgdrfhnk dr hntirprithnk dpirethdns dn shkneas `y pirgdrfhnk dr hntirprithnk dpirethdns dn tmi griquincy jdfehn riprisintethdn, tmet hs, dn tmi Gdurhir sirhis cdigghchints [i whaa ndt kd tmrdukm e adnk ahst dg dpirethdns, shnci tmhs tdphc whaa rieppier wmin wi cdnshjir e fdri kinirea griquincy-jdfehn vhiwpdhnt tmet hncaujis epirhdjhc shkneas es wiaa. Mdwivir wi cdnshjir e giw ixefpais. Ixefpai Khvin e shknea
x(t ) 6
O
∕
o 6∐ O
boψ d t
] o i
suppdsi e niw shknea hs gdrfij `y efpahtuji trensgdrfethdn, xˇ (t ) 6 ex(t ) + ` , wmiri e ≬ < enj ` eri riea cdnstents. Ht hs caier tmet xˇ (t ) hs pirhdjhc, whtm tmi sefi gunjefintea pirhdj/griquincy pirhdj/griquincy es x (t ) , enj hnjiij ht hs iesy td jitirfhni tmi Gdurhir sirhis cdigghchints dg xˇ (t ) `y hnspicthdn. [i shfpay wrhti xˇ (t ) 6
O
O
∕ ]ˇ o i boψdt 6 ` + e ∕
o 6∐ O
o 6∐ O
enj cdncauji tmet
⎫e] < + ` , o 6 < ⎨ e] o , o ≬ <
ˇ 6 ⎭ ] o
5<<
] o i
boψ d t
Zmhs epprdecm riahis dn tmi gect tmet tmi tirfs hn e Gdurhir sirhis gdr e pirhdjhc shknea eri unhqui, e gect tmet smduaj `i caier shnci iecm cdigghchint hs jitirfhnij hnjipinjintay dg tmi dtmirs. Mdwivir, e segir epprdecm, ispicheaay gdr fdri cdfpahcetij dpirethdns, hs td `ikhn whtm tmi ixprisshdn gdr tmi Gdurhir sirhis cdigghchints dg tmi niw shknea, enj riaeti ht td tmi ixprisshdn gdr cdigghchints dg tmi drhkhnea shknea. Ixefpai Suppdsi xˇ (t ) 6 x( et ) , wmiri e hs e ndnzird cdnstent. Zmin xˇ (t ) hs pirhdjhc whtm
gunjefintea pirhdj Zˇd 6 Zd / | e | enj gunjefintea griquincy ψˇ d 6| e | ψ d . Zmi cdfpaix-gdrf Gdurhir sirhis cdigghchints eri khvin `y ]ˇ o 6 5
Zˇd
∯ xˇ (t )i
ˇ Z d
∐ boψˇ d t
jt 6
<
| e|
Zd /| e|
∯
Z d
x( et ) i
∐ bo |e |ψ d t
jt
<
Zd prdciij, wi niij td sipereti tmi cesis dg pdshthvi enj nikethvi e. Hg e 1 < , tmet hs, e 6 ∐ | e | , tmin tmi cmenki dg hntikrethdn verhe`ai grdf t td td ϊ 6 et 6 ∐ | e | t khvis ]ˇ o 6
| e| Zd
5
∐Z d
∯
< <
(
6 Z ∯ x (ϊ ) i d ∐Z d
<
∐ bo |e|ψdϊ / e j ϊ x (ϊ )i 6 5 ∐|e| Z d ∐ bo ψdϊ ∛
)
boψ ϊ ∯ x(ϊ ) i d
j ϊ
∐Z d
jϊ 6 ] o∛ 6 ] ∐ o
Ht hs aigt es en ixirchsi td smdw tmet gdr e > < e sdfiwmet jhggirint risuat hs d`tehnij, nefiay ]ˇ o 6 ] o
Zmet hs, thfi sceai `y e pdshthvi cdnstent aievis tmi Gdurhir sirhis cdigghchints uncmenkij, tmdukm dg cdursi tmi gunjefintea griquincy hs cmenkij. Dn tmi dtmir menj, es e perthcuaer ixefpai, thfi sceai `y e 6 ∐ 5 , wmhcm hs thfi rivirsea, aievis tmi gunjefintea griquincy uncmenkij, enj tmi feknhtuji spictruf dg tmi shknea uncmenkij, `ut cmenkis tmi pmesi spictruf. =.: CZ AZH Griquincy Pispdnsi enj Ghatirhnk
[i cen cdf`hni tmi Gdurhir sirhis riprisintethdn gdr pirhdjhc shkneas whtm tmi ihkinguncthdn prdpirty gdr ste`ai ste`ai AZH systifs systifs td riprisint systif systif rispdnsis td pirhdjhc hnput hnput shkneas. Suppdsi tmi systif mes unht-hfpuasi rispdnsi m(t ) . Shnci wi whaa `i cdnshjirhnk jhggirint griquinchis, ht hs cdnvinhint td cmenki dur ierahir ndtethdn. Petmir tmen tmhno dg e ghxij griquincy, ψ d , wi tmhno rispdnsi guncthdn dg tmi systif `y dg griquincy es e verhe`ai, ψ , enj jighni tmi griquincy rispdnsi ∞
∐ bψ t jt ∯ m(t )i
M (ψ ) 6
∐∞
Dg cdursi, tmi ste`hahty essufpthdn kuerentiis tmet M (ψ ) hs wiaa jighnij gdr eaa ψ . Zmin khvin e pirhdjhc hnput shknea jiscrh`ij, et aiest epprdxhfetiay, epprdxhfetiay, `y tmi Gdurhir sirhis sirhis ixprisshdn O
∕
x(t ) 6
o 6∐ O
boψ d t
] o i
ahnierhty enj tmi ihkinguncthdn prdpirty khvi tmi dutput ixprisshdn y (t ) 6
O
∕
o 6∐ O
boψ d t
M (oψ d ) ] o i
5<5
Aitthnk Xo 6 M ( oψ d ) ] o , enj ndthnk tmi cdnbukecy prdpirty tmet M (∐ψ ) 6 M ∛ (ψ ) , wi sii tmet tmi X o cdigghchints sethsgy X∐ o 6 X o ∛ . Zmhs aiejs td tmi cdncaushdn tmet tmi X o ‘s eri Gdurhir sirhis cdigghchints gdr tmi pirhdjhc dutput shknea. Zmhs ixprisshdn gdr y (t ) cen `i cdnvirtij td verhdus riea gdrfs hn tmi usuea wey. Dg cdursi, tmi dutput shknea typhceaay mes tmi sefi gunjefintea griquincy es tmi hnput shknea, tmdukm ndt eaweys shnci tmi griquincy rispdnsi guncthdn cen `i zird et perthcuaer griquinchis. Zmhs prdpirty easd cerrhis dvir td tmi cesi dg ceusea, ste`ai AZH systifs whtm ‑rhkmt-shjij pirhdjhc‖ hnput hnput shkneas. Nefiay, tmi stiejy-steti stiejy-steti rispdnsi hs pirhdjhc enj hs es jiscrh`ij e`dvi. [i cen cdnshjir tmi feknhtuji dg tmi griquincy rispdnsi guncthdn es e griquincy-jipinjint kehn ψ d ) | hs tmi kehn dg tmi systif et griquincy o ψ ψ d , tmi gectdr `y dg tmi systif. Zmet hs, | M ( o ψ wmhcm tmi efpahtuji dg tmi o tm merfdnhc dg tmi hnput shknea hs hncriesij dr jicriesij. Zmhs hs tmi `eshs dg griquincy griquincy siaicthvi ghatirhnk, ghatirhnk, wmiri en AZH AZH systif hs jishknij td mevi mevi jishrij iggicts dn dn tmi griquincy cdfpdnints dg tmi hnput shknea. Zd smdw tmi ghatirhnk prdpirthis dg e systif, wi dgtin jhspaey e padt dg tmi feknhtuji dg tmi griquincy rispdnsi guncthdn, | M (ψ ) | , vs. ψ . Ixefpai Cdnshjir ekehn tmi P-C chrcuht chrcuht hn Sicthdn 7.7, whtm P 6 5, C 6 5 . Zmi unht-hfpuasi unht-hfpuasi
rispdnsi hs 5
∐ t t m(t ) 6 5 i PC u (t ) 6 i∐ u (t) PC
Zmirigdri tmi griquincy rispdnsi guncthdn gdr tmi chrcuht hs M (ψ ) 6
∞
∯
∐t
i u (t ) i
∐ bψ t
jt 6
∐∞ 6 5 5 bψ +
∞
∯i
∐(5+ bψ )t
jt
<
Shnci
| M (ψ ) | 6
5 5+ψ 0
ht hs strehkmtgdrwerj td soitcm tmi feknhtuji dg tmi griquincy rispdnsi guncthdn2
Caieray tmi chrcuht ects es e adw-pess ghatir, enj mhkm-griquincy hnput shknea cdfpdnints eri ettinuetij fucm fdri tmen adw-griquincy cdfpdnints. Zd `i spichghc, suppdsi x(t ) 6 5 + cds(t ) + cds(3< t ) Shnci b
bt
b 3
x(t ) 6 Pi{i } + Pi{i } + Pi{i } ahnierhty enj tmi ihkinguncthdn prdpirty cen `i usij td wrhti tmi rispdnsi es b
y (t ) 6 Pi{M (<)i
} + Pi{M (5) i bt } + Pi{M (3<) i b 3
5<0
Zmin tmi cdfputethdns ∐ bό / : M (5) 6 5 i , M (3<) ≍ 5 i∐ bό / 0
M (<) 6 5,
3<
0
khvi y (t ) ≍ 5 + 5 cds(t ∐ ό / :) + 5 cds(3
0
Ixirchsis 5. Cdfputi tmi cdfpaix-gdrf cdfpaix-gdrf Gdurhir sirhis cdigghchints enj soitcm tmi feknhtuji feknhtuji enj pmesi
spictre gdr t (e) tmi shknea x(t ) tmet mes gunjefintea pirhdj Z d 6 5 , whtm x(t ) 6 i ∐ ,
< ≪ t ≪ 5 .
(`) tmi shknea x(t ) smdwn `iadw
(c) tmi shknea x(t ) 6
∞
∕ (∐5) o κ (t ∐ 0 o ) o 6∐∞
(j) tmi shknea x(t ) smdwn `iadw
0. Suppdsi x(t ) hs pirhdjhc whtm gunjefintea pirhdj Z d enj cdfpaix-gdrf Gdurhir sirhis
cdigghchints ] o . Smdw tmet (e) hg x (t ) hs djj, x(t ) 6 ∐ x( ∐t ) , tmin ] o 6 ∐ ] ∐ o gdr eaa o . (`) hg x (t ) hs ‑meag-wevi djj,‖ x (t ) 6 ∐ x(t + Z d / 0) , tmin ] o 6 < gdr iviry ivin hntikir o . (c) hg x (t ) hs ivin, x(t ) 6 x( ∐t ) , tmin ] o 6 ] ∐ o gdr eaa o . 3. Suppdsi tmi shknea shknea x(t ) mes gunjefintea pirhdj Z d enj cdfpaix-gdrf Gdurhir sirhis
cdigghchints ] o . Jirhvi ixprisshdns gdr tmi cdfpaix-gdrf Gdurhir sirhis cdigghchints dg tmi gdaadwhnk shkneas hn tirfs dg ] o .
(e) x (t ) 6 0 x( t ∐ 3) + 5 (`) x (t ) 6 x(5 ∐ t )
5<3
(c) x (t ) 6 j x(t )
(j) x (t ) 6
jt t
∯ x(ϊ ) j ϊ ([met ejjhthdnea essufpthdn hs riquhrij dn tmi Gdurhir sirhis cdigghchints
∐∞
dg x(t ) 4) Mhnt2 ]ˇ o 6
5 Z d
Z d t
∐ boψ d t jt enj hntikrethdn-`y-perts cen `i usij td wrhti ∯ ∯ x(ϊ ) jϊ i
< ∐∞
tmhs hn e wey tmet ] o cen `i ricdknhzij.
(i) x (t ) 6 x(t + Z d / 0) :. Khvin tmi AZH systif whtm unht-hfpuasi unht-hfpuasi rispdnsi m(t ) 6 i
∐: |t |
, cdfputi tmi Gdurhir Gdurhir sirhis riprisintethdn gdr tmi rispdnsi y (t ) dg tmi systif td tmi hnput shknea ∞
∕
(e) x(t ) 6
κ (t ∐ n)
n 6∐∞
(`) x (t ) 6
∞
∕ (∐5)nκ (t ∐ n) n 6∐∞
9. E cdnthnudus-thfi cdnthnudus-thfi pirhdjhc pirhdjhc shknea x (t ) mes Gdurhir sirhis cdigghchints
⎫ (7 / bo ) i bo ό / : ] o 6 ⎭ iasi ⎨< ,
,
o 6 µ5, µ 3
Cdfputi enj soitcm tmi feknhtuji enj pmesi spictre dg tmi shknea. 7. Enswir, whtm busthghcethdn, busthghcethdn, tmi gdaadwhnk quisthdns quisthdns e`dut tmi rispdnsi dg tmi ste`ai AZH systif
whtm griquincy rispdnsi guncthdn M (ψ ) 6
9 3 + bψ
(e) Gdr e pirhdjhc hnput shknea x(t ) tmet mes gunjefintea pirhdj Z d 6 0ό , wmet merfdnhcs whaa eppier hn tmi dutput shknea y (t ) whtm jhfhnhsmij feknhtuji4 Zmet hs, wmet veauis dg o yhiaj yhiaj
| Xo | 1 | ] o | 4 (`) Gdr e pirhdjhc hnput shknea tmet mes gunjefintea pirhdj Z d 6 ό , wmet merfdnhcs whaa eppier hn tmi dutput shknea whtm jhfhnhsmij feknhtuji4 ;. Enswir, whtm busthghcethdn, busthghcethdn, tmi gdaadwhnk quisthdns quisthdns e`dut tmi tmi rispdnsi dg tmi AZH systif tmet ∐3t
mes hfpuasi rispdnsi m(t ) 6 09 t i
u(t ) .
(e) Gdr e pirhdjhc hnput shknea x(t ) tmet mes gunjefintea pirhdj Z d 6 0 , wmet merfdnhcs whaa eppier hn tmi dutput shknea y (t ) whtm jhfhnhsmij feknhtuji4 Zmet hs, wmet veauis dg o yhiaj yhiaj
| Xo | 1 | ] o | 4 (`) Gdr e pirhdjhc hnput shknea tmet mes gunjefintea pirhdj Z d 6 : ό , wmet merfdnhcs whaa eppier hn tmi dutput shknea whtm jhfhnhsmij feknhtuji4
5<:
=. Cdnshjir tmi AZH systif systif tmet mes hfpuasi hfpuasi rispdnsi m(t ) 6 κ (t ) ∐ i
∐0t u(t ) , enj suppdsi tmi
hnput shknea hs x(t ) 6 5 + 0 cds(t ) + 3 cds(0 t ) . Cdfputi tmi rispdnsi y (t ) . ∐t
∐t
8. Cdnshjir tmi AZH systif systif tmet mes hfpuasi hfpuasi rispdnsi m(t ) 6 ti u (t ) + 0i u(t) ∐ κ ( t) , enj
suppdsi tmi hnput shknea hs x(t ) 6 0 + 0 cds(t ) . Cdfputi tmi rispdnsi y (t ) .
5<9
Ndtis gdr Shkneas enj Systifs 8.5 \irhdjhc JZ Shknea Piprisintethdn (Gdurhir Sirhis)
Suppdsi x_ nV hs e riea, pirhdjhc shknea whtm gunjefintea pirhdj N d enj, dg cdursi, gunjefintea griquincy ψd 6 0ό / N d . [i cmddsi e `eshs sit dg N d merfdnhceaay riaetij jhscriti-thfi pmesdrs, ceaaij tmi jhscriti-thfi Gdurhir `eshs sit 2 boψ d n
χ o _nV 6 i
, o 6 <, 5,…, Nd ∐ 5 bo 0ό n
(Dgtin tmisi shkneas eri wrhttin dut hn tmi gdrf i N d , `ut wi feoi usi dg tmi gunjefintea griquincy td shfpahgy tmi eppierenci dg tmi ixpdnint.) Zmhs `eshs sit mes sivirea prdpirthis2 • Zmiri eri ixectay N d jhsthnct `eshs shkneas dg tmhs typi (ndt en hnghnhti nuf`ir es hn tmi cdnthnudus-thfi cdnthnudus-thfi cesi) shnci b ( o + N d )ψd n
χ o + N _nV 6 i d
6 i boψd n i bNdψ d n
6 χ o _nV •
Zmi `eshs sit hs siag-cdnbuketi. Dg cdursi, χ d _ nV hs riea, enj gdr eny dtmir o hn hn tmi bN dψ d
renki, shnci i
6 i b 0ό 6 5 , ∛ χ o _nV 6 i∐ boψd n 6 i∐ boψd n i bNdψd n 6 i b ( Nd ∐ o )ψd n
6 χ N ∐ o _nV d •
Iecm `eshs shknea hs pirhdjhc whtm pirhdj (ndt nicisserhay gunjefintea pirhdj) N d , boψd ( n + Nd )
χ o _n + N d V 6 i
6 i boψ d n i bo 0ό
6 χ o _nV •
Zmi `eshs sit hs drtmdkdnea dvir eny renki dg ainktm N d hn n. Zd smdw tmhs, cdnshjir tmi renki r ≪ n ≪ r + N d ∐ 5 , wmiri r hs hs eny hntikir, enj cdfputi r + N d ∐5
∕
n6r
∛
χo _nVχ f _ nV 6
r + Nd ∐5
boψd n ∐ bfψ d n
∕
i
n6r
i
Cmenkhnk tmi suffethdn hnjix grdf n td a 6 n ∐ r khvis r + Nd ∐5
∕
n6 r
∛
χo _nVχ f _ nV 6
N d ∐5
∕
boψd (a + r ) ∐ bfψ d ( a + r )
i
i
a 6< N ∐5 b ( o ∐ f)ψd r d b( o ∐ f)ψ d a
∕
6i
i
a 6<
Nixt wi eppay tmi hjinthty, wmhcm mdajs gdr eny cdfpaix nuf`ir ε , ⎫⎢ N d , ε 6 5 N d ∐5 a ∕ ε 6 ⎭5∐ε N d , ε ≬ 5 a 6< ⎢
⎨ 5∐ε
5<7
td tmi suffethdn td d`tehn r + N d ∐5
∕
N d ∐5
b ( o ∐ f)ψd r
∛
n6 r
i ( 6<
∕
χo _nVχ f _ nV 6 i
a
b( o ∐ f)ψ d
a
)
⎫ N d , o 6 f ⎢ 6 ⎭ b ( o ∐ f)ψ d r 5∐i b (o ∐f )ψ d N d , o≬f ⎢⎨i 5∐i b ( o ∐f )ψ d ⎫ N , o 6 f 6 ⎭ d ⎨ <, o ≬ f Zmus wi mevi iste`ahsmij drtmdkdneahty, drtmdkdneahty, enj gurtmirfdri, r + N d ∐5
∕
n 6 r
∛ χf _nVχ f _ nV 6 N d
Grdf tmisi prdpirthis enj tmi kinirea gdrfuaes gdr fhnhfuf suf-squerij-irrdr suf-squerij-irrdr riprisintethdns riprisintethdns hn Sicthdn ;.:, wi cen cdncauji tmet td riprisint tmi N d ∐ pirhdjhc, riea shknea shknea x_ nV whtm fhnhfuf suf squerij irrdr pir pir pirhdj ushnk ushnk tmi Gdurhir Gdurhir `eshs sit, tmi tmi cdigghchints eri khvin khvin `y ] f 6
5
r + N d ∐5
N d
∕
n6r
∛
x_ nV χ f_ nV 6
r + Nd ∐5
5
∕
N d
x_ nV i
n6r
∐ bfψ d n
Miri r hs hs eny hntikir, enj wi ekehn usi tmi spichea ndtethdn ] f td jindti tmi ftm Gdurhir sirhis cdigghchint gdr tmi shknea x_ nV . Zmirigdri tmi riprisintethdn hs wrhttin es x_ nV ≍
N d ∐5
∕
f 6<
bfψ d n
] fi
wmiri tmi epprdxhfethdn hs hn tmi sinsi dg fhnhfuf suf-squerij suf-squerij irrdr. Dg cdursi ht hs hfpdrtent td hffijhetiay ndti tmet, hn ejjhthdn td tmi cdnbukecy riaethdn ∛ χ f _nV 6 i∐ bfψd n 6 i∐ bfψd ni bNdψd n 6 i∐ b ( Nd ∐ f)ψd n
6 χ N ∐ f _nV d wi mevi ∛
] f 6
5 Nd
5
6 N
d
r + N d ∐5
∕
x_ nV i
n6r r + N d ∐5
∕
bfψd n
x_ nV i
6
5
r + Nd ∐5
N d
∐ b ( Nd ∐ f)ψ d n
n 6 r
∕
n6r
x_ nV i
bfψd n ∐ bNdψ d n
i
6 ] Nd ∐ f
bfψ n
d Zmirigdri tmi cdnbuketi dg iviry tirf ] f i hs hncaujij es endtmir tirf hn tmi suf, enj sd tmi fhnhfuf suf-squerij-irrdr riprisintethdn hs riea.
Nixt, `igdri wdrohnk wdrohnk ixefpais, ixefpais, wi whaa cdfputi cdfputi e surprhshnk surprhshnk ixprisshdn gdr gdr tmi fhnhfuf fhnhfuf veaui dg tmi suf squerij irrdr pir pirhdj. @ikhn `y wrhthnk tmi riprisintethdn es
5<;
N d ∐5
∕
f 6<
Nd ∐5
⎥ 5 N d ∐5 ∐ bfψd a ⎪ bfψ d n x a i _ V ⎠ N ∕ ⎩i d ⎩⎧ f 6< ⎠ a 6< ⎣
bfψd n
6 ∕
] f i
[i cen hntircmenki tmi drjir dg suffethdn td wrhti N d ∐5
∕
f 6<
Nd ∐5
bfψd n
6 ∕
] f i
a 6<
5 x_a V
N d
N d ∐5
∕
i
∐ baψd f bnψ d f i
f6<
N d ∐5
N d ∐5
a 6<
f 6<
6 N ∕ x_a V ∕ χa∛_ fVχ n_ fV 5
d
@ut N d ∐5
⎫ <, n ≬ a n6a
χa∛_f Vχ n_ fV 6 ⎭ ⎨ Nd , f 6<
∕
`y drtmdkdneahty, drtmdkdneahty, enj tmus tmus N d ∐5
∕
f 6<
bfψ d n
] f i
6 x_ nV
Zmet hs, tmi suf squerij irrdr pir pirhdj hs zird! Zmhs fiens tmet dur epprdxhfeti riprisintethdn ectueaay hs en ixect riprisintethdn. riprisintethdn. Ixefpai Zmhs ixefpai hs eafdst tdd shfpai, shfpai, enj tmiri hs jenkir dg cdngushdn, `ut ht haaustretis
tmi ceacuaethdn dg tmi jhscriti-thfi Gdurhir sirhis. Cdnshjir x_ nV 6 ( ∐5)
n
b ό n bό n Hn tmhs cesi, N d 6 0 , ψd 6 ό , enj tmi twd `eshs shkneas eri i < 6 5 enj i , wmhcm dg n cdursi hs (∐5) . Zmi Gdurhir cdigghchints eri khvin `y
5
∕ (∐5) n 6 <
] < 6 5
0
n6<
5
n ∐ bό n 6 ]5 6 5 ∕ (∐5) i 0 n6<
5 ∐ i ∐ bό 65 0
Zmus tmi Gdurhir sirhis riprisintethdn hs x_ nV 6
5
∕ f6<
bfό n
] fi
6 ⎥< + i bό n ⎪ 6 i bό n ⎣ ⎧
Cirtehnay tmhs hs nd surprhsi. Ixefpai Cdnshjir e pirhdjhc pirhdjhc trehn dg caufps caufps dg 0 N 5 + 5 puasis dg unht mihkmt ripiethnk whtm
gunjefintea pirhdj N d , es smdwn `iadw.
5<=
Zmi Gdurhir cdigghchints cen `i cdfputij grdf ] f 6
N d ∐ N 5 ∐5
5
∕
N d
x_ nVi
∐ bfψd n
6
n 6 ∐ N5
5 N d
N 5
∕
i
∐ bfψ d n
n 6 ∐ N5
Caieray ] < 6 (0 N5 + 5) / N d , enj gdr ndnzird f wi ripaeci tmi suffethdn hnjix n `y o 6 n + N 5 td wrhti ] f 6
0 N5
5 N d
∕
i
∐ bfψd ( o ∐ N5 )
6
o 6<
5 i bfψd N5
N d
0 N 5
∕
i
∐ bfψ d o
o 6 <
Picdknhzhnk tmi rhkmt-fdst suffethdn td `i dg tmi gdrf 0 N 5
i ( 6<
∕
∐ bfψ d
o
khvis ] f 6
5 i bfψ d N 5
6 N
6 N 5
d
5 ∐ i∐ bfψ d
5 ∐ i∐ bfψ d
∐ bfψd / 0
d
6
5 ∐ i∐ bfψ d ( 0 N 5 +5)
5 ∐ i ∐ bfψ d ( 0 N 5 +5)
N d
5 i
)
o
(i bfψd N5 i bfψd / 0 ∐ i∐ bfψd N5 i∐ bfψd / 0 )
i
∐ bfψd / 0
(i bfψd / 0 ∐ i∐ bfψ d / 0 )
shn( fψ d ( N 5 + 5 / 0) shn( fψ d / 0)
(Es usuea, wmin e cdigghchint cen `i wrhttin hn riea gdrf wi cdnthnui cdfputhnk untha wi d`tehn e riea ixprisshdn.) Ndw en iesy ceacuaethdn whtm A‘Mdsphtea‘s ruai smdws tmet tmhs gdrfuae hs veahj gdr f 6 < es wiaa. Zmhs ixprisshdn dg tmi Gdurhir cdigghchints hs dgtin wrhttin hn tirfs dg iveauethdns dg en inviadpi guncthdn es
shn_(0 N 5 + 5)ψ / 0V ] f 6 5 N d ψ 6 fψ d shn(ψ / 0) 0) enj sdfithfis tmi rethd dg shni guncthdns hs ceaaij tmi eahesij shnc. Zmiri eri twd hfpdrtent d`sirvethdns e`dut tmi JZGS, tmi ghrst dg wmhcm hs sdfitmhnk wi jhscussij privhdusay2 Pifero 5 Shnci `dtm x_ nV enj i
∐ bfψ d n
, gdr eny f , eri pirhdjhc siquincis hn n whtm pirhdj
N d , wi cen cdfputi
5<8
] f 6
N d ∐5
5 N d
∕ x_nV i∐ bfψ d n
n 6<
`y suffhnk suffhnk dvir eny N d cdnsicuthvi veauis dg tmi hnjix n, ndt nicisserhay tmi veauis grdf zird td N d ∐ 5 . Zmhs hs dgtin wrhttin es ] f 6 5
∕
N d
x_ nV i
∐ bfψ d n
n 61 N d >
wmiri tmi spichea enkai `recoits `recoits jindti en hnjix renki dg ‑ N d cdnsicuthvi veauis.‖ Pifero 0 Zmi JZGS cdigghchints ] < ,… , ] N ∐5 cen `i ixtinjij hn ihtmir jhricthdn td gdrf e d
siquinci tmet ripiets eccdrjhnk td ] f + N 6 ] f , d
gdr ea e aa f
Zmhs gdaadws grdf tmi ceacuaethdn ] f + N 6 d
N d ∐5
5
∕ x_nV i∐ b ( f + Nd )ψ d n
N d
N d ∐5
5
6 N
n6<
∕
d
0ό n ∐ bf N d i∐ b 0ό n x_ nV i
n6<
6 ] f E cdnsiquinci hs tmet, gdr eny hntikir o ,
6 ] Nd + o i b ( Nd + o )ψ d n
boψd n
] o i
Zmus wi cen wrhti x_ nV 6
N d ∐5
∕
f 6<
bfψ d n
] fi
∕
6
f 61 N d >
] fi
bfψ d n
wmiri ekehn tmi enkai-`recoit ndtethdn hnjhcetis e suffethdn dvir N d cdnsicuthvi veauis dg tmi hnjix f. Ixefpai Zmi shknea x_ nV 6 shn(0ό n / 3) hs pirhdjhc whtm gunjefintea pirhdj N d
ceacuaeti tmi JZGS cdigghchints es gdaadws, tmdukm sdfi jitehas eri dfhttij2 ] < 6 5
3
]5 6
5 3
] 0 6 5
3
0
∕ shn(0ό n / 3) 6 < n 6<
0
∕ shn(0ό n / 3) i
∐ b 03ό n
n6<
0
∕
shn(0ό n / 3) 3) i
6 05 b
∐ b 0 03ό n
n6<
6 0∐ b5
Mdwivir, tmiri hs e smdrtcut evehae`ai. Shfpay wrhti b shn(0ό n / 3) 6 5 b i
0
enj cdfperi tmhs td tmi ixprisshdn
55<
0ό n 3
∐ 05b i
∐ b 03ό n
6 3 . [i cen
∕
x_ nV 6
f 613>
] fi
bf 0ό n 3
Cmddshnk tmi hnjix renki 1 3 >6 ∐5, <, 5 wi sii tmet ] ∐5 6 ∐5 , ] < 6 < , ] 5 6 5 0 b 0b
Zmisi twd risuats cen `i ricdnchaij `y ndthnk tmet ] 0 6 ] ∐5+ 3 6 ] ∐5 . Hnjiij, tmi jhscriti-thfi b 0 0ό n
pmesdr cdrrispdnjhnk cdrrispdnjhnk td tmi cdigghchint ] 0 hs i
3
enj tmhs hs hjinthcea td tmi pmesdr i
∐ b 03ό n
cdrrispdnjhnk td ] ∐5 , es hs ieshay virhghij. Zmi ghrst eppait hn tmi jifdnstrethdn ahnoij `iadw pirfhts pirfhts ydu td ixpadri tmi JZGS gdr shkneas whtm pirhdj 9. Siaict tmi ‑Hnput x_nV‖ dpthdn enj sit tmi spiij td ‑sadw.‖ Zmin ydu cen soitcm hn e shknea enj ‑paey‖ tmi JZGS griquincy cdfpdnints. Dtmir dpthdns pirfht ydu td intir tmi cdigghchints hn tmi JZGS dr intir tmi feknhtuji enj pmesi spictre, enj tmisi whaa `i usigua hn Sicthdn 8.0. (Xdu whaa niij td usi FSHI 9.9+ whtm tmi Fetm\aeyir paukhn td usi tmhs ahno. Sii tmi fehn jifdnstrethdns peki gdr dtmir virshdns dg tmi eppait.) Jhscriti-Zhfi Gdurhir Sirhis 8.0 Spictre dg JZ Shkneas
Gdr e riea, pirhdjhc, JZ shknea x_nV , tmi griquincy cdntint dg tmi shknea hs rivieaij `y tmi cdigghchints hn tmi JZGS ixprisshdn x_nV 6
bfψ d n
∕ f 61 N d >
]f i
Zmi gdaadwhnk krepmhcea jhspaeys dg tmisi cdigghchints jighni verhdus spictre dg x_n V . Zmi feknhtuji spictruf dg x_ nV hs e ahni padt dg | ] f | vs tmi hnjix f, dr vs fψ d dn e griquincy exhs. Zmi pmesi spictruf dg x_ nV hs e shfhaer padt dg ∬ ] f , usueaay dn en enkuaer renki grdf ∐ό td ό . Ghneaay, wmin tmi JZGS cdigghchints eri eaa riea, tmi efpahtuji spictruf dg x_ nV hs shfpay e padt dg tmi cdigghchints ] f . n
Ixefpai Hn Sicthdn 8.5 wi cdfputij tmi JZGS cdigghchints dg x_ nV 6 ( ∐5) es ] <
6 <, ] 5 6 5,
enj ] o + 0 6 ] o , gdr dtmir veauis dg o . Hn tmhs cesi tmi efpahtuji spictruf dg tmi shknea hs shfpay
dr, hn tirfs dg e griquincy exhs, shnci ψd 6 ό ,
555
Shnci ό cdrrispdnjs td tmi mhkmist griquincy hn jhscriti thfi, wi ndti tmi d`vhdus gect tmet x_nV hs e mhkm-griquincy shknea. Ghneaay, gdr tmhs shfpai cesi, tmi feknhtuji spictruf hs hjinthcea td tmi efpahtuji spictruf, enj tmi pmesi spictruf hs zird. Ixefpai Zmi sicdnj ixefpai hn Sicthdn 8.5, e pirhdjhc trehn dg whjtm 0 N 5 + 5 caufps dg unht-
mihkmt adaaypdps, hs cdnshjire`ay fdri cdfpahcetij, tmdukm ekehn tmi JZGS cdigghchints eri riea. Cmddshnk N5 6 0, N d 6 5< , tmi cdigghchints eri khvin es en iveauethdn dg en eahesij shnc inviadp `y shn(9ψ / 0) 0) ] f 6 5
5< shn(ψ / 0) 0) ψ 6 fό / 9
Zmi inviadpi guncthdn hs zird wmin 9ψ / 0 6 o ό , gdr ndnzird hntikir o , tmet hs, gdr ψ 6 0o ό / 9 . Soitcmhnk tmhs inviadpi guncthdn yhiajs tmi efpahtuji spictruf smdwn `iadw.
Cdrrispdnjhnkay, Cdrrispdnjhnkay, tmi feknhtuji enj pmesi spictre eri smdwn `iadw.
[mhai tmiri hs sdfi mhkm-griquincy cdntint hn x_ nV , hn perthcuaer tmi cdfpdnint et griquincy ό , tmiri hs fdri adw-griquincy cdntint es hnjhcetij `y tmi cdfpdnints nier tmi griquinchis zird enj 0ό . Ghneaay, ht smduaj `i ndtij tmet tmisi spictre ripiet dutshji tmi griquincy renkis smdwn. Zd ixpadri tmi ndthdn dg spictre hn fdri jiteha, cdnsuat tmi jifdnstrethdn Jhscriti-Zhfi Gdurhir Gdurhir Sirhis ahnoij et tmi inj dg Sicthdn 8.5.
550
8.3 Dpirethdns dn Shkneas
Jhscriti-thfi, pirhdjhc shkneas eri cdfpaitiay jitirfhnij `y tmi gunjefintea griquincy, ψ d , dr gunjefintea pirhdj, N d , enj eny N d cdnsicuthvi Gdurhir cdigghchints, sey, ] < , ]5 , ..., ] N ∐5 . Zmus tmi shknea hs jiscrh`ij hn tirfs dg hts griquincy cdntint. Zmhs rehsis d tmi pdssh`hahty dg hntirprithnk verhdus thfi-jdfehn dpirethdns dn shkneas es dpirethdns dn tmi griquincy-jdfehn jiscrhpthdn. jiscrhpthdn. Mdwivir, retmir tmen khvi e ainktmy trietfint dg tmhs hssui, wi whaa shfpay jhscuss e giw ixefpais. Ixefpai 5 Khvin e shknea shknea boψ d n
∕
x_ nV 6
o 61 N d >
] o i
suppdsi e niw shknea hs d`tehnij `y tmi hnjix smhgt x_nV 6 x_n ∐ nd V wmiri nd hs e ghxij hntikir. Caieray e smhgt jdis ndt cmenki pirhdjhchty, dr tmi gunjefintea
pirhdj, dr gunjefintea gunjefintea griquincy. griquincy. Zmirigdri wi cen cdfputi tmi Gdurhir Gdurhir cdigghchints gdr x_nV grdf tmi stenjerj gdrfuae2 ] o 6 5
N d
∕
x_ nV i
∐ boψd n
6 5
N d
n 61 N d >
∐ boψ d n x_ n ∐ nd V i
∕
n 61 Nd >
Cmenkhnk tmi suffethdn hnjix grdf n td f 6 n ∐ nd , ] o 6 5
Nd
∕
x_ fV i
∐ boψd ( f + nd )
6 i∐ boψd nd N 5
d
f 61 Nd >
∕
x_ nV i
n 61 N d >
∐ boψ d f
6 i∐ boψ d nd ] o Ndthci tmet tmi tmi feknhtuji spictruf dg tmi shknea hs uncmenkij uncmenkij `y thfi thfi smhgt, shnci, rikerjaiss dg tmi hntikir veaui dg o , ]ˇ o 6 i
∐ boψ d N d
] o 6 ] o
Hn shfpai cesis, sucm es thfi-hnjix smhgt, ht hs pdssh`ai td escirtehn tmi iggict dg tmi dpirethdn dn tmi Gdurhir cdigghchints `y hnspicthdn dg tmi riprisintethdn. Hnjiij, whtm x_ nV es khvin e`dvi, ht hs caier tmet
x_nV 6 x_n ∐ nd V 6
boψd ( n ∐ nd )
∕ o 61 N d >
]o i
6
∕
i
∐ boψd nd
o 61 Nd >
] o i
boψ d n
enj wi shfpay ricdknhzi tmi gdrf dg tmi ixprisshdn enj tmi cdrrispdnjhnk Gdurhir cdigghchints ] o . Ixefpai 0 Suppdsi x_nV 6 x_ ∐nV . Ekehn, tmi gunjefintea griquincy jdis ndt cmenki, enj tmi
Gdurhir cdigghchints gdr x_ nV eri khvin `y ] o 6 5
Nd
∕ n 61 Nd >
x_ nV i
∐ boψd n
6 N 5
d
Cmenkhnk tmi suffethdn hnjix td f 6 ∐ n khvis
553
∕
x_ ∐ nV i
n 61 Nd >
∐ boψ d n
] o 6 5
N d
∕
x_f V i
∐ boψd ( ∐f)
6 N 5
∕
d
f 61 Nd >
x_ fV i
f 61 Nd >
∐ b ( ∐ o )ψ d n
6 ] ∐ o Zmhs cdncaushdn easd cduaj `i riecmij `y hnspicthdn dg tmi riprisintethdn. Gurtmir jhscusshdn dg dpirethdns dn jhscriti-thfi shkneas cen `i gdunj hn tmi jifdnstrethdn JZGS \rdpirthis
8.: JZ AZH Griquincy Pispdnsi enj Ghatirhnk
Gdr e ste`ai JZ AZH systif whtm pirhdjhc hnput shknea, tmi JZGS enj tmi ihkinguncthdn prdpirty prdvhji e wey td td cdfputi enj enj hntirprit tmi rispdnsi. Hg tmi unht-puasi rispdnsi rispdnsi dg tmi systif systif hs guncthdn dg tmi systif es m_n V , wi jighni tmi griquincy rispdnsi guncthdn ∞
∕ m_n V i∐ bψ n
M (ψ ) 6
n 6∐∞
Zmhs hs e sahkmt cmenki hn ndtethdn grdf Sicthdn 9.9 hn tmet wi smdw griquincy es e verhe`ai. Ndti easd tmet ∞
∕ m_n Vi∐ b (ψ + 0ό ) n
M (ψ + 0ό ) 6
n 6∐∞
6 M (ψ ) sd hn jhscriti thfi tmi griquincy rispdnsi guncthdn ripiets iviry 0ό rejhens hn griquincy. Hg tmi hnput shknea x_ nV hs pirhdjhc, whtm gunjefintea pirhdj N d enj cdrrispdnjhnk gunjefintea griquincy ψ d , wi cen wrhti
∕
x_ nV 6
o 61 N d >
boψ d n
] o i
enj tmi ihkinguncthdn prdpirty khvis
∕
y_ nV 6
o 61 N d >
boψ d n
M ( oψ d ) ] o i
Zmus tmi griquincy rispdnsi guncthdn dg tmi systif jiscrh`is tmi iggict dg tmi systif dn verhdus griquincy cdfpdnints dg tmi hnput shknea. Zd jhspaey tmhs iggict, padts dg | M (ψ ) | enj ∬ M (ψ ) vs ψ cen `i khvin. Shnci tmisi guncthdns ripiet whtm pirhdj 0ό , wi dgtin smdw tmi padts gdr dnay tmhs renki, gdr ixefpai, gdr ∐ό 1 ψ ≪ ό . Ixefpai Suppdsi en AZH systif mes tmi unht puasi rispdnsi m_ nV 6 5 κ _ nV + 5 κ _ n ∐ 5V 0 0
D`vhdusay tmi systif hs ste`ai, enj tmi griquincy rispdnsi guncthdn hs bψ M (ψ ) 6 5 + 5 i ∐ 6 i∐ bψ / 0 cds(ψ / 0)
0
0
Zmi rispdnsi dg tmi systif td en hnput dg griquincy ψ d , gdr ixefpai,
55:
{
bψ d n
x_n V 6 cds(ψ d n) 6 Pi i
}
hs khvin `y e ndw-stenjerj ceacuaethdn ushnk tmi ihkinguncthdn prdpirty2
{
bψ d n
Pi M (ψd )i y_ nV 6 Pi Hn tmhs cesi,
} 6 | M (ψ ) | cds(ψ n + ∬M (ψ )) d
d
| M (ψ ) | 6 cds(ψ / 0) , ∐ ό 1 ψ ≪ ό
d
enj grdf tmi padt `iadw wi sii tmet tmi systif hs e adw-pess ghatir.
Ixefpai Suppdsi en AZH systif hs jiscrh`ij `y tmi jhggirinci iquethdn
y_ nV + ey_ n ∐ 5V 6 `x_nV wmiri, td kuerentii ste`hahty, wi essufi | e |1 5 . Zmin n
m_ nV 6 ( ∐ e) `u `u_ nV
enj tmi griquincy rispdnsi guncthdn hs ∞
∕ ( ∐ e)
M (ψ ) 6
n
`u_ nV i
∐ bψ n
n 6 ∐∞
6
∞
6 ` ∕ ( ∐e) n i∐ bψ n n6<
`
5+ ei ∐ bψ
Hn tmhs cesi,
| M (ψ ) |6
|`| |5+ ei
∐ bψ
6 |
|`| (5+ ei∐ bψ )(5+ ei bψ )
6
|`| 5+ 0 e cds(ψ ) + e 0
Hg < 1 e 1 5 enj ` 6 5 , tmin tmi feknhtuji dg tmi griquincy rispdnsi hs smdwn `iadw, enj tmhs systif hs e mhkm-pess ghatir.
Ixirchsis 5. Cdfputi tmi jhscriti-thfi jhscriti-thfi Gdurhir sirhis cdigghchints gdr tmi shkneas `iadw enj soitcm tmi
feknhtuji enj pmesi spictre.. (e) x_n V 6 5 + cds(ό n / 3) (`)
559
(c)
∞
∕
(j) x_ nV 6
κ ( n ∐ : o ∐ 5)
o 6∐∞
0. Gdr tmi sits dg JZGS cdigghchints khvin `iadw, jitirfhni jitirfhni tmi cdrrispdnjhnk riea, pirhdjhc
shknea x_ nV .
⎫ 5/0, o ivin , ∐ 5/0, o djj ⎨ (`) ] o 6 5 / 0, gdr eaa o , (c) ] < 6 ∐5, ]5 6 <, ] 0 6 5, (e) ] o 6 ⎭
ψd 6 ό ψd 6 ό ] 3 6 ∐0, ] : 6 5, ] 9 6 <, ] o + 7 6 ] o , ψd 6 ό / 3
3. Suppdsi x_n V hs pirhdjhc whtm ivin gunjefintea pirhdj N d enj JZGS cdigghchints ] o . Hg
x_ nV easd sethsghis x_ nV 6 ∐ x_ n + N d / 0V , gdr eaa n, smdw tmet ] o 6 < hg o hs hs ivin. :. Khvin tmi gunjefintea pirhdj N d enj tmi feknhtuji enj pmesi spictre es smdwn gdr e riea,
jhscriti-thfi shknea, wmet hs tmi shknea4 (e) N d 6 9
9. Hg x_ nV mes gunjefintea pirhdj N d , en ivin hntikir, enj jhscriti-thfi Gdurhir sirhis
cdigghchints ] o , wmet eri tmi Gdurhir sirhis cdigghchints gdr
557
(e) x_ nV 6 x_ n + N d / 0V
5) n x_ nV (Essufi tmet Nd 6 N d enj khvi en ixefpai td smdw wmy tmhs essufpthdn (`) x_ n V 6 ( ∐5) hs niijij.) 7. Gdr tmi AZH systifs spichghij `iadw, soitcm soitcm tmi feknhtuji dg tmi griquincy rispdnsi guncthdn
enj jitirfhni hg tmi systif hs e adw-pess dr mhkm-pess ghatir. (e) m_ nV 6 5 κ _ nV ∐ 5 κ _ n ∐ 5V 0
0
n (`) m_ nV 6 κ _ nV ∐ ( 5 0) u_ nV
(c) m_ nV 6 (5 / 0) n u_ nV
55;
Ndtis gdr Shkneas enj Systifs 5<.5 Hntrdjucthdn td tmi CZ Gdurhir Zrensgdrf
Hg x(t ) hs Z d -pirhdjhc, wi cen wrhti tmi Gdurhir sirhis jiscrhpthdn ∞
boψ d t
∕
x(t ) 6
o 6∐∞
] o i
(5<.5)
wmiri ψd 6 0ό / Z d enj ] o 6
5 Z d
Z d / 0
∯
x(t )i
∐ boψ d t jt ,
o 6 <, µ5, µ0, …
(5<.0)
∐Z d / 0
Cdnshjir wmet meppins es wi ait Zd krdw whtmdut `dunj. Hn e sinsi, x (t ) epprdecmis en ψ ≪ 5, tmiri eri fdri enj fdri 5 ≪ ψ ≪ epirhdjhc shknea, enj hn eny khvin griquincy renki, sey ∐ 5 griquincy cdfpdnints dg x(t ) shnci ψ d `icdfis sfeaair enj sfeaair. Zmirigdri, grdf e griquincy cdntint vhiwpdhnt, pirmeps ht hs ndt surprhshnk tmet e truiay epirhdjhc shknea typhceaay cdntehns griquincy cdfpdnints et eaa griquinchis, ndt bust et hntikir fuathpais dg e gunjefintea griquincy. Eneayshs dg e ahfht prdciss `y wmhcm e pirhdjhc shknea epprdecmis en epirhdjhc shknea, es Z d ← ∞ , hs jhgghcuat td unjirteoi. Zmirigdri wi sohp fetmifethcea fetmifethcea jitehas enj shfpay prdvhji e fdthvethdnea erkufint aiejhnk td e jiscrhpthdn dg tmi griquincy cdntint dg epirhdjhc shkneas. Jighni, gdr eaa ψ , tmi cdfpaix-veauij guncthdn ] (ψ ) `y Z d / 0
] (ψ ) 6
∯
x(t ) i
∐ bψ t jt
(5<.3)
∐Z d / 0
Zmi Gdurhir sirhis cdigghchints (5<.0) (5<.0) gdr gdr x(t ) cen `i wrhttin es iveauethdns dg tmhs ‑inviadpi‖ guncthdn, ] o 6 5 ] ( oψd ) 6 5 ] ( o ψ d )ψ d 0ό
Z d
sd wi cen wrhti x(t ) 6 5
0ό
∞
boψ d t
∕
] ( oψd ) i
o 6∐∞
ψ d
Aitthnk Zd krdw whtmdut `dunj, tmet hs, aitthnk ψ d smrhno tdwerj <, wi cen vhiw ψ d es e ψ d teois dn tmi cmerectir dg e riea verhe`ai, ψ . Zmet hs, tmi ψ enj cdnshjir tmet o ψ jhggirinthea j ψ jhggirinci (o + 5)ψd ∐ o ψd 6 ψ d smrhnos tdwerj < sd tmet eny khvin riea nuf`ir cen `i ψ d , gdr suhte`ai o . Zmin tmi suf trensghkuris td en hntikrea, yhiajhnk epprdxhfetij `y o ψ x(t ) 6 5
0ό
∞
∯
] (ψ ) i
∐∞
55=
bψ t
j ψ
(5<.:)
Hn tmhs ixprisshdn, shnci wi ait Z d ← ∞ , (5<.3) (5<.3) `icdfis `icdfis ∞
∐ bψ t jt ∯ x(t ) i
] (ψ ) 6
(5<.9)
∐∞
Zmi guncthdn ] (ψ ) hs jighnij es tmi Gdurhir trensgdrf dg x(t ) , enj hn e viry usigua sinsi ht jiscrh`is tmi griquincy cdntint dg tmi epirhdjhc shknea. Hnjiij, fetmifethceaay, fetmifethceaay, x(t ) hs khvin `y tmi hnvirsi Gdurhir trensgdrf ixprisshdn (5<.:) (5<.:),, wmhcm hs eneadkdus td tmi ixprisshdn dg e Z d pirhdjhc shknea hn tirfs dg dg hts Gdurhir sirhis. sirhis. Zd pirgdrf e senhty cmico dn tmisi caehfs dg e trensgdrfethdn enj en hnvirsi trensgdrfethdn, suppdsi tmet khvin x(t ) wi cdfputi ∞
] (ψ ) 6
x(ϊ )i∐ bψϊ j ϊ
∯ ∐∞
wmiri wi mevi usij e jhggirint nefi gdr tmi hntikrethdn verhe`ai hn drjir td evdhj e addfhnk ndtethdnea cdaahshdn. Su`sthtuthnk Su`sthtuthnk tmhs hntd tmi rhkmt shji dg (5<.:) (5<.:) khvis khvis 5 0ό
∞
bψ t
∯ ] (ψ )i
jψ 6
∐∞
5 0ό
∞ ∞
∐ bψ ϊ jϊ ∯ ∯ x(ϊ ) i
i
bψ t
j ψ
∐∞ ∐∞
Hntircmenkhnk tmi drjir dg hntikrethdn khvis 5 0ό
∞
∞
∯ x(ϊ ) ∯ ∐∞
b (t ∐ ) i ψ ϊ jψ j ϊ
∐∞
Hn tmhs ixprisshdn wi ricdknhzi tmet ∞
∯
bψ (t ∐ϊ )
i
jψ 6 0ό κ (t ∐ ϊ )
∐∞
grdf Spichea \rdpirty 0 hn Sicthdn 0.0. Zmin tmi hntikrethdn whtm rispict td ϊ hs iveauetij iveauetij `y `y tmi shgthnk prdpirty td khvi ∞
όκ (t ∐ ϊ ) jϊ 6 x(t ) ∯ x(ϊ ) 0όκ
∐∞
Hn dtmir wdrjs, teohnk tmi Gdurhir trensgdrf enj tmin tmi hnvirsi trensgdrf hnjiij riturns tmi drhkhnea shknea. Hffijheti quisthdns eri2 [min hs ] (ψ ) e fienhnkgua riprisintethdn gdr x(t ) 4 Eri stenjerj dpirethdns dn x(t ) iesy td hntirprit es dpirethdns dn ] (ψ ) 4 @igdri ejjrisshnk tmisi, wi wdro e giw ixefpais. ∐3t u (t ) , ∞ ∞ ∐5 ∐(3+ bψ )t ∞ ∐ bψ t ∐3t ∐ bψ t ] (ψ ) 6 ∯ x(t ) i jt 6 ∯ i i jt 6 i < b ψ + 3 < ∐∞
Ixefpai Gdr x(t ) 6 i
6
5 3 + bψ
558
6 ∞ yhiajs zird. Gdr ixefpai, hg wi cmenki tmi shkn Ht hs hfpdrtent td ndti tmet tmi iveauethdn et t 6 dg tmi ixpdnint hn tmi shknea, tmi Gdurhir trensgdrf wduaj ndt ixhst. Hn gect tmi Gdurhir trensgdrf ] (ψ ) jiscrh`is tmi griquincy cdntint dg tmi shknea x(t ) , enj wi jhspaey tmhs cdntint ushnk tmi gdaadwhnk, gefhaher padts. Zmi feknhtuji spictruf dg e shknea x(t ) hs e padt dg | ] (ψ ) | vs ψ , enj tmi pmesi spictruf hs e padt dg ∬ ] (ψ ) vs ψ . Hg ] (ψ ) hs e riea-veauij guncthdn, sdfithfis wi padt tmi efpahtuji spictruf dg tmi shknea es ] (ψ ) vs ψ . Zmi feknhtuji spictruf dr efpahtuji spictruf dg e shknea jhspaey tmi griquincy cdntint dg tmi shknea. Gdr tmi ixefpai e`dvi,
| ] (ψ ) |6
5
, ∬] (ψ ) 6 ∐ ten ∐5 (ψ / 3)
8 + ψ 0
enj tmi feknhtuji enj pmesi spictre eri smdwn `iadw.
Pifero Zmi syffitry prdpirthis prdpirthis dg tmi feknhtuji enj enj pmesi spictre dg e riea shknea x(t ) eri
iesy td busthgy hn kinirea grdf tmi gect tmet ∛
∞ ∞ ⎥∞ ∛ ∐ bψ t ⎪ ∐ bψ t ∛ ] (ψ ) 6 ⎠ ∯ x(t )i jt ⎩ 6 ∯ _ x( t ) i V jt 6 ∯ x( t) i bψ t jt ⎠⎣ ∐∞ ⎩⎧ ∐∞ ∐∞ ∞
6 ∯ x(t )i∐ b (∐ψ )t jt 6 ] ( ∐ψ ) ∐∞
Zmhs khvis ] (∐ψ ) 6 ] (ψ ) enj ∬ ] ( ∐ψ ) 6 ∐∬] (ψ ) , enj tmus tmi feknhtuji spictruf hs en ivin guncthdn dg ψ , wmhai tmi pmesi spictruf hs en djj guncthdn dg ψ . sdfiwmet ixtrifi ixefpai hs tmi unht hfpuasi, x(t ) 6 κ (t ) . Zmi Ixefpai E shfpai tmdukm sdfiwmet shgthnk prdpirty hffijhetiay khvis
∞
] (ψ ) 6
∯
x(t )i
∐ bψ t
jt 6 5
∐∞
Hn tmhs cesi tmi feknhtuji (dr efpahtuji) spictruf dg tmi shknea hs e cdnstent, hnjhcethnk tmet tmi unht hfpuasi hs feji up dg iquea efdunts dg eaa griquinchis!
50<
Ixefpai Cdnshjir tmi rictenkuaer rictenkuaer puasi shknea smdwn smdwn `iadw. `iadw.
Zmi Gdurhir trensgdrf cdfputethdn hnvdavis hnvdavis e `ht dg wdro td put tmi enswir hn e nhci gdrf, `ut tmi ceacuaethdns eri ndt ungefhaher2 ∞
] (ψ ) 6
∯
x (t )i
∐ bψ t
jt 6
∐∞
6
5 bψ
Z 5
∯
i∐ bψ t jt 6
∐Z 5
∐5 ∐ bψ t Z 5 i ∐Z 5 bψ
(i bψZ5 ∐ i∐ bψ Z 5 ) 6 0Z5 ψ 5Z shn(ψ Z5 ) 5
6 0Z5 shnc(ψ Z 5 / ό ) Zmus tmi efpahtuji spictruf dg x(t ) hs
Es whtm Gdurhir sirhis ceacuaethdns, hg tmi Gdurhir trensgdrf cen `i wrhttin es e riea guncthdn, tmin ht hs hfpdrtent td ixpriss ht hn riea gdrf. Cdnvirkinci Hssuis Ht hs jhgghcuat td ixpahchtay ixpahchtay cmerectirhzi tmi caess dg dg shkneas gdr wmhcm tmi
Gdurhir trensgdrf hs wiaa jighnij. Gurtmirfdri, tmi unhquiniss dg tmi Gdurhir trensgdrf gdr e khvin shknea hs en hfpdrtent hssui – td iecm shknea x(t ) tmiri smduaj cdrrispdnj ixectay dni ] (ψ ) . (Zmhs nikaicts trhvhea cmenkis hn tmi shknea, dr tmi trensgdrf, gdr ixefpai ejbusthnk tmi veaui dg x(t ) et hsdaetij veauis dg t . Sucm e cmenki jdis ndt iggict tmi risuat dg tmi hntikrethdn aiejhnk td tmi trensgdrf.) Zmiri eri verhdus sugghchint cdnjhthdns tmet cen `i stetij gdr e shknea td mevi e unhqui Gdurhir trensgdrf, enj wi prisint dnay tmi `ist ondwn dg tmisi. shknea x(t ) hs sucm tmet Jhrhcmait Cdnjhthdn Cdnjhthdn Suppdsi e shknea ∞
(e) x(t ) hs e`sdautiay hntikre`ai, tmet hs,
∯ | x(t ) | jt 1 ∞ ,
∐∞
(`) x(t ) mes nd fdri tmen e ghnhti nuf`ir dg fhnhfe enj fexhfe hn eny ghnhti thfi hntirvea, enj (c) x(t ) mes nd fdri tmen e ghnhti nuf`ir dg jhscdnthnuhthis jhscdnthnuhthis hn eny ghnhti thfi hntirvea, enj tmisi jhscdnthnuhthis jhscdnthnuhthis eri ghnhti. Zmin tmiri ixhsts e unhqui Gdurhir trensgdrf ] (ψ ) cdrrispdnjhnk td x(t ) . Ht hs hfpdrtent td rifif`ir tmet tmhs hs e sugghchint cdnjhthdn, enj tmiri eri shkneas tmet jd ndt sethsgy tmi cdnjhthdn yit mevi e unhqui Gdurhir trensgdrf. Gdr ixefpai, tmi unht hfpuasi wduaj `i 6 < , enj yit tmi shgthnk prdpirty khvis e Gdurhir tmdukmt dg es mevhnk en hnghnhti jhscdnthnuhty et t 6
505
trensgdrf. Hnjiij, wi cen cmico tmi ceacuaethdn `y eppayhnk tmi hnvirsi trensgdrf. [htm ] (ψ ) 6 5 , wi cdfputi ∞
5 0ό
x(t ) 6
bψ t
∯
] (ψ ) i
jψ 6
∐∞
5 0ό
∞
∯
i
bψ t
jψ 6 κ ( t )
∐∞
wmiri wi mevi usij ekehn Spichea \rdpirty 0 grdf Sicthdn 0.0, tmet hs, ∞
∯
bψ t
jψ 6 0ό κ (t )
i
∐∞ Ixefpai Es en ejjhthdnea ixefpai, wi cen usi tmi spichea prdpirty whtm tmi rdais dg t enj enj ψ
hntircmenkij td cdfputi tmi Gdurhir trensgdrf dg x(t ) 6 5 2 ] (ψ ) 6
∞
∯
i∐
bψ t
∐∞
jt 6
∞
∯
b ψ t i ( ∐ ) jt 6 0όκ ( ∐ψ ) 6 0όκ (ψ )
∐∞
Zmhs hnjhcetis tmet eaa tmi griquincy cdntint hn tmi shknea hs cdncintretij et zird griquincy, e riesdne`ai cdncaushdn. E quhco cmico dg tmi hnvirsi trensgdrf riessuris2 x(t ) 6
5 0ό
∞
∯
bψ t
] (ψ ) i
jψ 6
∐∞
5 0ό
∞
bψ t ∯ 0όκ (ψ ) i j ψ
∐∞
65 Hnjiij, dni epprdecm td cdfputhnk e Gdurhir trensgdrf wmin tmi Jhrhcmait cdnjhthdn hs ndt sethsghij dr kinireahzij guncthdns fhkmt `i hnvdavij hs td kuiss tmi trensgdrf enj virhgy `y usi dg tmi hnvirsi trensgdrf gdrfuae. 5<.0 Gdurhir Zrensgdrf gdr \irhdjhc Shkneas
Hg x(t ) hs Z d -pirhdjhc, tmin ht hs caier tmet tmi Gdurhir trensgdrf ] (ψ ) 6
∞
∐ bψ t jt ∯ x(t )i
∐∞
jdis ndt ixhst hn tmi usuea sinsi, `iceusi dg tmi gehauri dg tmi hntikrea td cdnvirki. Mdwivir, wi cen teoi en hnjhrict epprdecm enj usi ndthdns dg kinireahzij guncthdns td ixtinj tmi Gdurhir trensgdrf td pirhdjhc shkneas hn e wey tmet cepturis tmi Gdurhir sirhis ixprisshdn hn e gesmhdn cdnshstint whtm drjhnery Gdurhir trensgdrfs. Zmi Gdurhir sirhis ixprisshdn x(t ) 6
∞
∕ o 6∐∞
boψ d t
] o i
hnjhcetis tmet tmi oiy hs td jiviadp tmi Gdurhir trensgdrf dg tmi cdfpaix shknea bψ t
x(t ) 6 i d Dni epprdecm hs td usi tmi Spichea \rdpirty 0 hn Sicthdn 0.0 ekehn2 ∞ ∞ bψ t ∐ bψ t b (ψ ∐ψ ) t ] (ψ ) 6 ∯ i d i jt 6 ∯ i d jt 6 0όκ (ψd ∐ ψ ) ∐∞ ∐∞ 6 0όκ (ψ ∐ ψd ) Zmet tmhs hs riesdne`ai hs ieshay virhghij ushnk tmi hnvirsi Gdurhir trensgdrf, whtm iveauethdn `y tmi shgthnk prdpirty.
500
Gdaadwhnk tmhs ceacuaethd ceacuaethdn, n, wi cen cdfputi tmi Gdurhir trensgdrf dg e shknea ixprissij `y e Gdurhir sirhis hn e strehkmtgdrwerj fennir2 ] (ψ ) 6
∞
∯ x(t ) i ∐∞ ∞
∐ bψ t
jt 6
∞
∞
∯ ∕
∐∞ o 6∐∞
boψ d t ∐ bψ t
] o i
i
∞
∞
∐∞
o 6 ∐∞
jt
6 ∕ ] o ∯ i boψ dt i∐ bψ t jt 6 ∕ 0ό ] o κ (ψ ∐ o ψd ) o 6 ∐∞
Hn wdrjs, td cdfputi tmi Gdurhir trensgdrf dg e pirhdjhc shknea, ghrst cdfputi tmi Gdurhir sirhis cdigghchints, ] o , enj tmin shfpay su`sthtuti tmhs jete hntd tmi ] (ψ ) ixprisshdn e`dvi. Dg cdursi tmi Gdurhir trensgdrf ixprisshdn hs hnvirth`ai `y hnspicthdn, hn tmi sinsi tmet tmi Gdurhir sirhis gdr x(t ) cen `i wrhttin `y hnspicthdn grdf ] (ψ ) . Grdf endtmir vhiwpdhnt, wi issintheaay mevi `uhat tmi Gdurhir sirhis hntd tmi trensgdrf. [i niij td rigdrfet tmi ndthdns dg spictre dg x(t ) hn cdnsdnenci whtm tmhs niw grefiwdro, `ut tmet hs iesy. Gdr eny veaui dg ψ , et fdst dni suffenj hn tmi ixprisshdn gdr ] (ψ ) cen `i ndnzird. @iceusi tmi suffenjs eri ndndviraepphnk hn tmhs sinsi, tmi feknhtuji dg tmi suf hs tmi suf dg tmi feknhtujis. Hnstiej dg e ahni padt, mdwivir, tmi feknhtuji feknhtuji spictruf (dr efpahtuji efpahtuji spictruf, spictruf, hg iviry ] o hs riea) `icdfis e padt dg hfpuasi hfpuasi guncthdns, guncthdns, dccurrhnk et hntikir hntikir fuathpais fuathpais dg tmi gunjefintea gunjefintea griquincy, ae`iaij whtm tmi hfpuasi-erie feknhtujis. Zmet hs,
Zmi pmesi spictruf cdfputis hn e shfhaer gesmhdn, shnci tmi enkai dg e suf dg ndn-dviraepphnk tirfs hs tmi suf dg tmi enkais. Zmirigdri tmi pmesi spictruf hs hntirpritij es e ahni padt dg tmi enkais dg tmi eries dg hfpuasis vs griquincy. Zmet hs, tmi pmesi spictruf mes ixectay tmi sefi gdrf es hn tmi cdntixt dg Gdurhir sirhis riprisintethdns. hntiristhnk td ndti, `esij `esij dn tmi Gdurhir trensgdrf dg e Ixefpai E cdupai dg spichea cesis eri hntiristhnk pmesdr. Ghrst, Ghrst, gdr bψ t ∐ bψ d t x(t ) 6 cds(ψ dt ) 6 5 i d + 5 i
0
0
wi mevi ] (ψ ) 6 όκ (ψ ∐ ψd ) + όκ (ψ + ψd )
Sicdnj, gdr x (t ) 6 shn(ψ dt ) , ] (ψ ) 6 ∐ bόκ (ψ ∐ ψd ) + bόκ (ψ + ψd ) Ixefpai Dtmir epprdecmis td cdfputhnk cdfputhnk tmi Gdurhir trensgdrf trensgdrf dg pirhdjhc shkneas shkneas cen khvi
‑cdrrict,‖ `ut jhgghcuat td hntirprit risuats. Gdr tmi shknea ∞
x(t ) 6 ∕ κ (t ∐ o ) o 6∐∞
wi jhrictay cdfputi
503
] (ψ ) 6
∞
∞
∯ ∕
κ (t ∐ o ) i
∐ bψ t
∞
∕ ∯ κ (t ∐ o ) i∐ bψ t jt
jt 6
∐∞ o 6 ∐∞ ∞ 6 ∕ i∐ bψ o o 6∐∞
∞
o 6 ∐∞ ∐∞
Dn tmi dtmir menj, tmi ricdffinjij prdcijuri hs td cdfputi tmi Gdurhir sirhis jete gdr x (t ) , wmhcm ieshay yhiajs ψd 6 0ό enj ] o 6 5 , gdr eaa o . Zmin `y hnspicthdn wi d`tehn tmi Gdurhir trensgdrf ∞
∕ 0ό κ (ψ ∐ o 0ό )
] (ψ ) 6
o 6∐∞
Niijaiss td sey, ht hs jhgghcuat jhgghcuat td smdw `y `y iaifintery fiens fiens tmet tmisi twd twd ixprisshdns gdr ] (ψ ) eri tmi sefi. Hn eny cesi, wi eaweys prigir tmi sicdnj. 5<.3 \rdpirthis dg tmi Gdurhir Zrensgdrf
[i ndw cdnshjir e verhity dg gefhaher dpirethdns dn e shknea x(t ) , enj hntirprit tmi iggict dg tmisi dpirethdns dn tmi cdrrispdnjhnk Gdurhir trensgdrf ] (ψ ) . Dg cdursi, ixhstinci dg tmi Gdurhir trensgdrf hs essufij. Gurtmirfdri, wi smduaj virhgy tmet iecm dpirethdn cdnshjirij yhiajs e shknea tmet mes e Gdurhir trensgdrf. Dgtin tmhs hs d`vhdus, enj whaa ndt `i finthdnij, `ut ceri hs niijij hn e cdupai dg cesis. Zmi prddgs wi dggir dg tmi verhdus prdpirthis eri cadsi td `ihnk rhkdrdus gdr drjhnery shkneas, shkneas, gdr ixefpai ixefpai tmdsi sethsgyhnk sethsgyhnk tmi Jhrhcmait cdnjhthdn. Gdr Gdr kinireahzij guncthdns, dr shkneas sucm es pirhdjhc shkneas tmet mevi kinireahzij-guncthdn trensgdrfs, gurtmir hntirpritethdn typhceaay hs niijij. Zmrdukmdut wi usi tmi gdaadwhnk ndtethdn gdr tmi Gdurhir trensgdrf enj hnvirsi Gdurhir trensgdrf, wmiri G jindtis jindtis e ‑Gdurhir trensgdrf dpiretdr2‖ ] (ψ ) 6 G _ x(t )V 6
∞
∐ bψ t jt ∯ x(t) i
∐∞
x (t ) 6 G
∐5
_ ] (ψ )V 6
5 0ό
∞
∯
bψ t
] (ψ ) i
j ψ
∐∞
Ahnierhty Hg G _ x(t )V 6 ] ( ψ ) enj G _ z (t )V 6 R ( ψ ) , tmin gdr eny cdnstent e,
G _ x(t ) + e z(t )V 6 ] (ψ ) + e R ( ψ ) Zmhs prdpirty gdaadws jhrictay grdf tmi jighnhthdn. Zhfi Smhgthnk Hg G _ x(t )V 6 ] ( ψ ) , tmin gdr eny cdnstent t d , ∐ bψ t d
G _ x(t ∐ td )V 6 i
] (ψ )
Zmi ceacuaethdn virhgyhnk tmhs hs `y ndw quhti stenjerj. @ikhn whtm G _ x(t ∐ td )V 6
∞
∐ bψ t jt ∯ x(t ∐ td )i
∐∞ enj cmenki hntikrethdn verhe`ai grdf t td td ϊ 6 t ∐ t d td d`tehn
50:
G _ x(t ∐ td )V 6
∞
∯ x(ϊ )i
∐ bψ (ϊ + td )
jt 6 i
∐ bψ t d
∐∞ 6 i∐ bψ t d ] (ψ )
∞
∐ bψϊ jt ∯ x(ϊ ) i
∐∞
Zhfi Sceahnk Hg G _ x(t )V 6 ] ( ψ ) , tmin gdr eny cdnstent e ≬
<,
G _ x( et )V 6 5 ] ( ψ ) |e |
e
Zmhs hs endtmir gefhaher ceacuaethdn, `ikhnnhnk whtm G _ x( et )V 6
∞
∐ bψ t jt ∯ x( et ) i
∐∞
tmdukm tmi cesis dg pdshthvi enj nikethvi e eri cdnvinhintay menjaij siperetiay. Hg e > < , tmi verhe`ai cmenki grdf t td td ϊ 6 et yhiajs ∞
∞ ϊ ∐ bψ ϊe 5 ∐ b ψ 5 e jϊ G _ x( et )V 6 ∯ x(ϊ ) i jϊ 6 x i ϊ ( ) e e ∯ ∐∞ ∐∞ 6 5 ] ( ψ ) e
e
Hg e 1 < , tmin ht hs cdnvinhint td wrhti e 6 ∐ | e | enj usi tmi verhe`ai cmenki ϊ 6 ∐ | e | t es gdaadws2 ∐∞
∞ ∐ bψ ∐ϊ |e| 5 ϊ ∐ b ψ 5 e jϊ G _ x( et )V 6 ∯ x(ϊ ) i jϊ 6 x ϊ i ( ) ∯ |e| ∐|e| ∞ ∐∞ 6 5 ] ( ψ ) e |e |
Ghneaay wi ndti tmet `dtm cesis eri cdvirij `y tmi caehfij gdrfuae. Ixefpai En hntiristhnk hntiristhnk cesi hs e 6 ∐ 5 , enj tmi sceahnk prdpirty yhiajs
)V 6 ] ( ∐ψ ) G _ x(∐t )V
Es ndtij privhdusay, ] (∐ψ ) 6 ] (ψ ) , enj tmirigdri wi ndthci tmi hntiristhnk gect tmet e shknea enj hts thfi rivirsea mevi tmi sefi feknhtuji spictre! Ixefpai [min `dtm e sceai enj e smhgt eri hnvdavij, hnvdavij, tmi segist epprdecm epprdecm hs td wdro dut tmi risuat
∐ 0) cen `i cdfputij hn grdf tmi `eshc jighnhthdn. Zd haaustreti, tmi Gdurhir trensgdrf dg x(3t ∐ tirfs dg tmi trensgdrf dg x(t ) vhe tmi verhe`ai cmenki ϊ 6 3t ∐ 0 es smdwn hn tmi gdaadwhnk2 G _ x(3t ∐ 0)V 6
∞
∯
∐ bψ t x(3t ∐ 0) i jt jt 6
∐∞
6
∯ ∐∞
0
5 i ∐ bψ 3 3
∞
∯
x (ϊ )i
∐∞
0 5 i ∐ bψ 3 ] ( ψ )
63
∞
3
509
ϊ ∐ b ψ 3
∐ bψ ϊ +3 0 5 x(ϊ ) i jϊ 3
jϊ
Jhggirinthethdn Hg G _ x(t )V 6 ] ( ψ ) , enj tmi thfi-jirhvethvi shknea x (t ) mes e Gdurhir trensgdrf,
tmin
G _ x (t )V 6 bψ ] ( ψ ) Zd busthgy tmhs prdpirty, wi jhrictay cdfputi tmi trensgdrf, ushnk hntikrethdn-`y-perts2 ∞ ∞ bψ t bψ t ∞ ∐ ∐ G _ x (t )V 6 ∯ x(t ) i jt 6 x( t) i ∐ ∯ x( t)( ∐ bψ ) i∐ bψ t jϊ ∐∞ ∐∞ ∐∞ 6 bψ ] (ψ )
← µ∞ . [min Zd feoi tmhs rhkdrdus, wi niij td busthgy tmi gect tmet x(t ) epprdecmis zird es t ← x (t ) sethsghis tmi Jhrhcmait cdnjhthdn, tmi gect hs caier, tmdukm hn dtmir cesis ht hs aiss sd. Pifero [min wi eppay tmhs prdpirty td shkneas tmet eri ndt, strhctay spieohnk, jhggirinthe`ai,
kinireahzij ceacuaus fust `i usij. Gdr ixefpai, `ikhnnhnk whtm tmi ieshay-virhghij trensgdrf ∐t G _ x(t )V 6 G_ i u( t) V 6
5 5 + bψ
tmi jhggirinthethdn prdpirty khvis G _ x (t ))VV 6
bψ
5 + bψ
Zd cmico tmhs, wi ghrst cdfputi x (t ) , t t t x (t ) 6 j i ∐ u (t ) 6 ∐i∐ u(t) + i∐ κ ( t ) jt
(
)
6 ∐ i∐t u (t ) + κ (t ) Zmin tmi Gdurhir trensgdrf hs iesy `y ahnierhty2 G _ x (t )V 6
∐5 bψ +5 6 5 + bψ 5 + bψ
Hntikrethdn Hg G _ x(t )V 6 ] ( ψ ) , tmin
⎥ t ⎪ G ⎠ ∯ x (ϊ ) jϊ ⎩ 6 5 ] (ψ ) + ό ] (<) κ (ψ ) ⎠⎣ ∐∞ ⎩⎧ bψ Zmhs prdpirty hs jhgghcuat td jirhvi, `ut wi cen d`sirvi tmet tmi runnhnk hntikrea hs tmi hnvirsi dg jhggirinthethdn, ixcipt gdr en unjitirfhnij ejjhthvi cdnstent. Zmus tmi ghrst tirf hs ixpictij, enj tmi sicdnj tirf eccdunts gdr tmi cdnstent. Ixefpai Zmi riaethdnsmhps
G _κ (t )V 6 5, 5,
u (t ) 6
t
∯ κ (ϊ ) jϊ
∐∞
enj tmi hntikrethdn prdpirty khvi tmi Gdurhir trensgdrf dg tmi unht-stip guncthdn es G _u (t )V 6 5 + όκ (ψ ) bψ
[i cen cmico tmhs whtm tmi jhggirinthethdn prdpirty (ushnk kinireahzij jhggirinthethdn)2 jhggirinthethdn)2 G _κ (t )V 6 G_u(t )V 6 bψ ⎥ 5 + όκ (ψ ) ⎪ 6 5 + bόψ κ (ψ )
⎣⎠ bψ
⎧⎩
65 tmdukm tmi ruai gdr fuathpayhnk en hfpuasi whtm en drjhnery guncthdn fust `i hnvdoij.
507
jhscusshnk sdfithfis sdfithfis cen `i usij hn caivir weys td cdfputi cdfputi Ixefpai Zmi prdpirthis wi eri jhscusshnk Gdurhir trensgdrfs `esij dn e sfeaa te`ai dg ondwn trensgdrfs. Mdwivir, sdfithfis sdfithfis tmi enswir cen eppier hn e gdrf wmiri hntirpritethdn hs riquhrij td shfpahgy tmi risuat. Zd haaustreti, cdnshjir tmi shfpai rictenkuaer puasi, x(t ) 6 u (t + 5) ∐ u(t ∐5) Yshnk ahnierhty enj thfi-smhgt prdpirthis, wi cen hffijhetiay wrhti ] (ψ ) 6 i bψ ⎥ 5 + όκ (ψ ) ⎪ ∐ i∐ bψ ⎥ 5 + όκ (ψ ) ⎪
⎠⎣ bψ
⎩⎧
⎠⎣ bψ
⎩⎧
6 b5ψ ⎥i bψ ∐ i∐ bψ ⎪ + ό ⎥i bψ ∐ i∐ bψ ⎪ κ (ψ ) ⎣ ⎧ ⎣ ⎧ 6
0 shn(ψ ) ψ
6 0 shnc(ψ /ό ) Zmhs ekriis whtm dur ierahir cdncaushdn, tmdukm, ekehn, eppahcethdn dg tmi ruai gdr fuathpayhnk enj hfpuasi guncthdn `y en drjhnery guncthdn hs hnvdavij. Zmi iggict dg verhdus dpirethdns dn e shknea enj dn tmi feknhtuji enj pmesi spictre dg tmi shknea cen `i ixpadrij ushnk tmi [i` jifdnstrethdn `iadw. CZGZ \rdpirthis 5<.: Cdnvdauthdn \rdpirty enj Griquincy Pispdnsi dg AZH Systifs
\irmeps tmi fdst hfpdrtent prdpirty dg tmi Gdurhir trensgdrf hs tmet cdnvdauthdn hn tmi thfi jdfehn `icdfis fuathpahcethdn dg trensgdrfs. Zmhs fiens tmet gdr feny purpdsis ht hs cdnvinhint td vhiw AZH systifs hn tmi Gdurhir (griquincy) jdfehn. Cdnvdauthdn Hg x (t ) enj m(t ) mevi Gdurhir trensgdrfs ] (ψ ) enj M (ψ ) , tmin
G _( x ∛ m)(t )V 6 ] (ψ) M (ψ ) E jhrict ceacuaethdn hnvdavhnk e cmenki hn drjir dg hntikrethdn cen `i usij td iste`ahsm tmhs prdpirty2 ∞ G _( x ∛ m)(t )V 6 ∯ ( x ∛ m)( t) i∐ bψ t jt ∐∞ ∞ ⎥∞ ⎪ 6 ∯ ⎠ ∯ x(ϊ ) m(t ∐ ϊ ) jϊ ⎩ i∐ bψ t jt ⎩⎧ ∐∞ ⎠⎣ ∐∞ ∞ ⎥∞ ⎪ 6 ∯ x (ϊ ) ⎠ ∯ m(t ∐ ϊ ) i∐ bψ t jt ⎩j ϊ ⎠⎣ ∐∞ ⎩⎧ ∐∞ [i cen cmenki tmi hntikrethdn verhe`ai hn tmi hnnir hntikrethdn grdf t td σ 6 t ∐ ϊ td d`tehn ∞ ⎥∞ ⎪ ∐ bψ (σ +ϊ ) G _( x ∛ m)( t )V 6 ∯ x(ϊ ) ⎠ ∯ m(σ ) i jσ ⎩jϊ ⎠ ⎩⎧ ⎣ ∐∞ ∐∞ enj gectdrhnk tmi ixpdninthea hn ϊ dut dg tmi hnnir hntikrethdn khvis
50;
G _( x ∛ m)(t )V 6
∞
∐ bψϊ M (ψ ) jϊ 6 ] ( ψ) M (ψ ) ∯ x(ϊ ) i
∐∞ Pifero [i jhj ndt cmico tmet tmi cdnvdauthdn cdnvdauthdn dg Gdurhir trensgdrfe`ai trensgdrfe`ai shkneas yhiajs e Gdurhir Gdurhir
trensgdrfe`ai shknea. Zmhs hn gect hs trui gdr shkneas tmet sethsgy tmi Jhrhcmait cdnjhthdn, `ut jhgghcuathis cen erhsi wmin sdfi dg dur shkneas whtm kinireahzij-guncthdn trensgdrfs eri hnvdavij. Gdr ixefpai, tmi trensgdrfs dg tmi shkneas x(t ) 6 5 enj m(t ) 6 u (t ) eri M (ψ ) 6 5 + όκ (ψ )
] (ψ ) 6 0όκ (ψ ) ,
bψ
Zmi cdnvdauthdn ( x ∛ m)(t ) hs unjighnij hn tmhs cesi. Gdrtunetiay tmi prdjuct dg tmi trensgdrfs 6 < eppiers, enj hnjhcetis tmet sdfitmhnk sdfitmhnk hs efhss hn tmet tmi squeri squeri dg en hfpuasi guncthdn et ψ 6 easd en hfpuasi fuathpahij `y e guncthdn tmet hs jrefethceaay jhscdnthnudus et ψ 6 6 < . @ut sucm e caier hnjhcethdn hs ndt eaweys prdvhjij. Ixefpai
Gdr e ste`ai ste`ai AZH systif, systif, tmi ihkinguncthdn ihkinguncthdn prdpirty prdpirty stetis tmet tmi rispdnsi rispdnsi td
bψ d t
x (t ) 6 i
bψ d t
hs y (t ) 6 M (ψ d ) i
, wmiri
M (ψ d ) 6
∞
∐ bψ t ∯ m(t )i d jt
∐∞
Hn tirfs dg Gdurhir trensgdrfs, tmi cdnvdauthdn prdpirty hfpahis tmet tmi rispdnsi td ] (ψ ) 6 0όκ (ψ ∐ ψ d ) hs X (ψ ) 6 M (ψ )0όκ (ψ ∐ ψd ) 6 M (ψd )0όκ (ψ ∐ ψ d )
wmiri M (ψ ) hs tmi Gdurhir trensgdrf dg tmi unht-hfpuasi unht-hfpuasi rispdnsi m(t ) . Zmhs hs shfpay tmi ihkinguncthdn prdpirty hn tirfs dg Gdurhir trensgdrfs – tmi dutput trensgdrf hs e cdnstent (typhceaay cdfpaix) fuathpai dg tmi hnput trensgdrf. Zmhs ixefpai hs ieshay ixtinjij td riprisint tmi rispdnsi dg e ste`ai, AZH systif td e pirhdjhc hnput shknea. [i vhiw tmi pirhdjhc hnput shknea hn tirfs dg hts Gdurhir sirhis, ∞
∕
x(t ) 6
o 6∐∞
Zmin wi hffijhetiay mevi ] (ψ ) 6
∞
∕ 0ό o 6∐∞
enj
boψ d t
] o i
] o κ (ψ ∐ o ψd )
∞
∕ 0ό
X (ψ ) 6 M (ψ ) ] (ψ ) 6
o 6∐∞
] o M (ψ )κ (ψ ∐ oψd )
∞
6 ∕ 0ό ] o M ( oψd )κ (ψ ∐ o ψd ) o 6∐∞
Zmi hnvirsi Gdurhir trensgdrf khvis y (t ) 6
∞
∕ o 6∐∞
boψ d t
] o M ( oψ d ) i
wmhcm typhceaay hs tmi Gdurhir sirhis riprisintethdn dg tmi dutput shknea.
50=
Hg en AZH systif hs ste`ai, tmin tmi griquincy rispdnsi guncthdn M (ψ ) , wmhcm wi ndw ricdknhzi es tmi Gdurhir trensgdrf dg tmi unht-hfpuasi rispdnsi, hs wiaa jighnij, enj wi cen vhiw tmi hnputdutput `imevhdr dg tmi systif hn tirfs dg X (ψ ) 6 M (ψ ) ] (ψ ) Zmin tmi feknhtuji spictruf dg tmi dutput shknea hs riaetij td tmi feknhtuji spictruf dg tmi hnput shknea `y | X (ψ ) |6| M (ψ ) | | ] (ψ ) | Zmus tmi feknhtuji dg tmi griquincy rispdnsi guncthdn cen `i vhiwij es e griquincy-jipinjint kehn dg tmi systif. ixmh`ht jhstdrthdnaiss trensfhsshdn hg tmi dutput shknea hs Ixefpai En AZH systif hs sehj td ixmh`ht shfpay en efpahtuji efpahtuji sceaij (pdshthviay) (pdshthviay) enj thfi-jiaeyij virshdn virshdn dg tmi hnput shknea. shknea. Zmet hs, tmiri eri pdshthvi cdnstents e enj t d sucm tmet gdr eny hnput x(t ) tmi dutput shknea hs y (t ) 6 e x (t ∐ t d ) .
Zmet
hs,
essufhnk
tmi
hnput
shknea
mes
e
Gdurhir
trensgdrf,
∐ bψ t d X (ψ ) 6 e i ] (ψ ) , enj tmus wi sii tmet gdr jhstdrthdnaiss trensfhsshdn tmi griquincy rispdnsi guncthdn mes tmi gdrf X (ψ ) 6 e i∐ bψ t d M (ψ ) 6 ] (ψ ) Dgtin tmhs hs stetij es tmi griquincy rispdnsi guncthdn fust mevi ‑gaet feknhtuji‖ enj pmesi tmet hs e ahnier guncthdn dg griquincy, et aiest gdr tmi griquincy renki dg hntirist. Ixefpai En hjiea ghatir smduaj smduaj trensfht whtmdut jhstdrthdn eaa griquinchis hn tmi spichghij
griquincy renki, enj rifdvi eaa griquinchis dutshji tmhs renki. Gdr ixefpai, enj hjiea hjiea adw-pess adw-pess ghatir ghatir smduaj smduaj mevi tmi tmi griquincy griquincy rispdnsi rispdnsi guncthd guncthdnn
⎫⎢ i∐ bψ t d , | ψ |≪ ψ c M (ψ ) 6 ⎭ ⎢⎨ < , | ψ |> ψ c Hn tmhs ixprisshdn, ψ c > < hs tmi cutdgg griquincy, enj gdr cdnvinhinci wi mevi sit tmi cdnstent kehn td unhty. Zd hntirprit tmhs ghatir hn tmi thfi jdfehn, wi cen cdfputi tmi hfpuasi rispdnsi vhe tmi hnvirsi Gdurhir trensgdrf2 m(t ) 6
6
5 0ό ψ c ό
∞
∯
bψ t
M (ψ ) i
jψ 6
∐∞
5 0ό
ψ c
∯
bψ (t ∐t d )
i
jψ
∐ψ c
shnc_ψc (t ∐ t d ) / ό V
E oiy pdhnt hs tmet nd fettir mdw aerki wi pirfht tmi jiaey thfi t d td `i, tmhs unht-hfpuasi rispdnsi hs ndt rhkmt shjij, enj tmus tmi hjiea adw-pess ghatir hs ndt e ceusea systif. Jisphti tmhs jrew`eco, tmi hjiea adw-pess ghatir rifehns e usigua cdncipt. Ixefpai Zmi AZH systif jiscrh`ij jiscrh`ij `y y (t ) + ψc y (t ) 6 ψ c x(t )
whtm ψ c > < , hs e adw-pess ghatir tmet cen `i hfpaifintij `y en P-C chrcuht. Shfpai, gefhaher ceacuaethdns khvi
508
ψ c
M (ψ ) 6
ψc + bψ Ixprisshnk tmhs griquincy rispdnsi guncthdn hn pdaer gdrf, es ψ c ∐ b ten ∐5 (ψ / ψ c ) M (ψ ) 6 i ψc0 + ψ 0
tmi cmerectirhsthcs dg tmhs ghatir cen `i cdfperij td tmi hjiea adw-pess ghatir vhe tmi feknhtuji enj pmesi padts smdwn smdwn `iadw. (Gdr tmi hjiea ghatir, wi cmddsi cmddsi t d 6 ό /(:ψ c ) td d`tehn en enkai dg
∐ό / : et ψ 6 ψ c , gdr haaustrethdn.) haaustrethdn.)
Caieray tmi ghrst-drjir jhggirinthea iquethdn hs e retmir pddr epprdxhfethdn td en hjiea ghatir, `ut mhkmir-drjir jhggirinthea iquethdns cen pirgdrf cadsir td tmi hjiea. 5<.9 Ejjhthdnea Gdurhir Zrensgdrf \rdpirthis
Griquincy-Jdfehn Cdnvdauthdn Hg x(t ) enj z (t ) mevi Gdurhir trensgdrfs ] (ψ ) enj R (ψ ) ,
tmin G _ x(t ) z (t )V 6
5 0ό
∞
∯
] (ξ ) R (ψ ∐ ξ ) jξ
∐∞
Zmet hs, tmi Gdurhir trensgdrf dg e prdjuct dg shkneas hs tmi cdnvdauthdn dg tmi trensgdrfs2 5 ( ] 0ό
∛ R )(ψ ) .
Zd prdvi tmhs prdpirty, wi jhrictay cdfputi tmi hnvirsi trensgdrf dg 5 ( ] ∛ R )(ψ ) 2 0ό
∐5 G _ 5 ( ] ∛ R )(ψ )V 6 5 0ό 0ό
∞
∯ ∐∞ ∞
5 0ό
∞
∯
bψ t
] (ξ ) R (ψ ∐ ξ ) jξ i
jψ
∐∞ ∞
6 05ό ∯ ] (ξ ) 05ό ∯ R (ψ ∐ ξ ) i bψ t jψ j ξ ∐∞
∐∞
Cmenkhnk tmi verhe`ai dg hntikrethdn hn tmi hnnir hntikrea grdf ψ td λ 6 ψ ∐ ξ khvis
53<
∐5 G _ 5 ( ] ∛ R )(ψ )V 6 5 0ό 0ό
∞
∯
] (ξ ) 5 0ό
∞
∯
b (λ +ξ ) t
R (λ) i
jλ jξ
∐∞ ∐∞ ∞ ∞ 6 05ό ∯ ] (ξ )i bξ t 05ό ∯ R (λ ) i bλ t jλ j ξ ∐∞ ∐∞ ∞ 5 6 0ό ∯ ] (ξ )i bξ t z(t ) j ξ ∐∞ 6 x(t ) z (t )
Ixefpai Zmi fdst hfpdrtent eppahcethdn dg tmhs prdpirty hs tmi sd-ceaaij fdjuaethdn dr griquincy bψ d t
smhgthnk prdpirty. Hg z (t ) 6 i
bψ d t
G _i
, tmin tmi shgthnk prdpirty dg hfpuasis khvis ∞
x(t )V 6 5 ∯ ] (ξ ) 0ό κ (ψ ∐ ξ ∐ ψd ) jξ 0ό ∐∞
6 ] (ψ ∐ ψ d ) [i cen haaustreti tmhs prdpirty `y cdnshjirhnk tmi structuri dg EF rejhd rejhd shkneas. Efpahtuji Fdjuaetij Shkneas En efpahtuji efpahtuji fdjuaetij fdjuaetij ( EF ) shknea dg tmi fdst `eshc typi mes
tmi gdrf x(t ) 6 _5 + o f(t )V cds(ψ ct )
wmiri cds(ψ ct ) hs ceaaij tmi cerrhir shknea, f(t ) hs ceaaij tmi fisseki shknea, enj tmi cdnstent o hs ceaaij tmi fdjuaethdn hnjix. [i essufi tmet tmi fdjuaethdn hnjix hs sucm tmet `y ψ f , wmiri 5 + o f(t ) ≩ < , gdr eaa t . [i easd essufi tmet tmi fisseki shknea hs `enjahfhtij `y
ψ f ψ c . Zmet hs, tmi Gdurhir trensgdrf F (ψ ) dg tmi fisseki shknea hs zird dutshji tmi griquincy renki ∐ψ f ≪ ψ ≪ ψ f . Zmisi eri stenjerj shtuethdns hn precthci, enj wi whaa riprisint tmi feknhtuji spictruf dg tmi fisseki shknea es smdwn `iadw.
Ynjir tmisi essufpthdns, x(t ) mes tmi gdrf dg e mhkm-griquincy (rephjay dschaaethnk) shnusdhj whtm e riaethviay sadway-veryhnk efpahtuji inviadpi. [i cduaj soitcm e typhcea cesi hn tmi thfi jdfehn, `ut ht wduaj `i retmir unhngdrfethvi es td tmi spichea prdpirthis dg EF shkneas shkneas tmet feoi tmif sd usigua hn cdffunhcethdns. Zd riviea tmisi prdpirthis, wi turn td tmi griquincy jdfehn vhe tmi Gdurhir trensgdrf. Yshnk tmi Gdurhir trensgdrf G _cds(ψ ct )V 6 ό κ (ψ ∐ ψc ) + ό κ (ψ + ψc ) enj tmi griquincy-jdfehn cdnvdauthdn
535
5 0ό
G _ f(t ) cds(ψct )V 6
∞
∯ ∐∞
F (ξ ) _ό κ (ψ ∐ ξ ∐ ψc ) + ό κ (ψ ∐ ξ + ψc )V j ξ
6 50 F (ψ ∐ ψc ) + 50 F (ψ + ψc ) wi cdncauji tmet tmi trensgdrf dg tmi EF shknea shknea hs ] (ψ ) 6 ό κ (ψ ∐ ψc ) + ό κ (ψ + ψc ) + o F (ψ ∐ ψc ) + o F (ψ + ψc ) 0
0
Zd sii tmi prdpirthis dg en EF shknea, shknea, wi cdnshjir tmi feknhtuji spictruf, | ] (ψ ) | . Zyphceaay ht hs jhgghcuat td cdfputi tmi feknhtuji dg e suf, `ut `iceusi dg tmi essufpthdns dn tmi mhkmist fisseki griquincy enj cerrhir griquincy, et fdst dni tirf hn tmi suf hs ndnzird et iviry griquincy, ixcipt et tmi cerrhir griquincy. (Et tmi cerrhir griquincy, wi mevi en hfpuasi enj en drjhnery veaui, enj wi cen jhspaey tmhs shtuethdn krepmhceaay hn tmi d`vhdus wey.) @iceusi dg tmhs spichea structuri dg tmi tirfs, tmi feknhtuji dg tmhs perthcuaer suf issintheaay hs tmi suf dg tmi feknhtujis, enj wi d`tehn
Grdf tmhs padt hs caier tmet EF fdjuaethdn hs usij td smhgt tmi spictrea cdntint dg e fisseki td e griquincy renki risirvij gdr e perthcuaer trensfhttir. @y esshknhnk esshknhnk jhggirint cerrhir griquinchis td jhggirint trensfhttirs, whtm e siperethdn dg et aiest 0 ψ f hn tmi jhggirint cerrhirs, tmi fissekis eri oipt jhsthnct. \ersivea‘s Zmidrif Hg x(t ) hs e riea inirky shknea whtm Gdurhir trensgdrf ] (ψ ) , tmin tmi tdtea
inirky dg tmi shknea hs khvin `y ∞
0
∯ x (t ) jt 6
∐∞
5 0ό
∞
0 ∯ | ] (ψ ) | j ψ
∐∞
Zd iste`ahsm tmhs risuat, wi su`sthtuti tmi hnvirsi Gdurhir trensgdrf ixprisshdn gdr dni dg tmi x (t ) ‘s hn tmi thfi-jdfehn inirky ixprisshdn td d`tehn ∞
∞
0
∯ x (t ) jt 6 ∯ x(t )
∐∞
∐∞
5 0ό
∞
∯
bψ t
] (ψ ) i
jψ jt 6
∐∞
∞
∯
] (ψ )
∐∞
5 0ό
∞
bψ t ∯ x( t) i jt j ψ
∐∞
Zmi hnnir hntikrea hn tmhs ixprisshdn cen `i ricdknhzij es tmi cdnbuketi dg tmi Gdurhir trensgdrf dg x(t ) , ] ∛ (ψ ) , enj tmirigdri ∞
0
∯ x (t ) jt 6
∐∞
5 0ό
∞
∯
∛
] (ψ ) ] (ψ ) jψ 6
∐∞
5 0ό
∞
0 ∯ | ] (ψ ) | j ψ
∐∞
Zmi hfpdrtenci dg \ersivea‘s tmidrif hs tmet inirky cen `i essdchetij whtm griquincy cdntint. Gdr ixefpai,
530
3
∯ | ] (ψ ) |
0
j ψ
∐3
hs tmi pdrthdn dg tmi inirky dg x(t ) tmet rishjis hn tmi griquincy `enj ∐3 ≪ ψ ≪ 3 . Jueahty \rdpirty Hg x(t ) mes Gdurhir trensgdrf ] (ψ ) , tmin
G _ ] (t )V 6 0ό x( ∐ψ )
Zmhs prdpirty cen `i ricdknhzij grdf en hnspicthdn dg tmi Gdurhir enj hnvirsi Gduhir trensgdrf ixprisshdn. Mdwivir, wi whaa `i pijenthc enj ahst dut tmi epprdprheti siquinci dg verhe`ai cmenkis. @ikhnnhnk whtm x(t ) 6 5
0ό
∞
∯
] (ψ ) i
bψ t
j ψ
∐∞ cmenki tmi verhe`ai dg hntikrethdn grdf ψ td t enj tmin ripaeci tmi verhe`ai t `y ∐ψ . Zmhs
khvis ∞ ∐ bψ t 5 x( ∐ψ ) 6 ] (t )i jt ∯ 0ό ∐∞ Ndw cmenki verhe`ais verhe`ais grdf ψ td ψ , enj t td t (iresi tmi mets ) td d`tehn ∞ bψ t x (∐ψ ) 6 5 ∯ ] (t )i ∐ jt 6 5 G_ ] ( t )V 0ό 0ό ∐∞
Dni usi dg tmi jueahty prdpirty hs hn riejhnk te`ais dg Gdurhir trensgdrfs `ecowerjs td kinireti ejjhthdnea Gdurhir trensgdrfs! Ixefpais Shnci G _κ (t )V 6 5 , tmi jueahty prdpirty khvis G _5V 6 0ό κ ( ∐ψ )
6 0ό κ (ψ ) . E fdri
usuea ixefpai hs tmet shnci G ⎥i ∐t u (t ) ⎪ 6
5 ⎧ 5+ bψ
⎣
tmi jueahty prdpirty khvis ψ G ⎥ 5 ⎪ 6 0ό i u ( ∐ψ ) ⎠⎣5+ bt ⎩⎧
Zmus wi sii tmet shnci feny Gdurhir trensgdrfs eri cdfpaix, tmi jueahty prdpirty dgtin prdvhjis Gdurhir trensgdrfs gdr cdfpaix thfi shkneas. 5<.7 Hnvirsi Gdurhir Zrensgdrf
Khvin e Gdurhir trensgdrf, ] (ψ ) , dni epprdecm td cdfputhnk tmi cdrrispdnjhnk thfi shknea hs vhe tmi hnvirsi trensgdrf gdrfuae, x(t ) 6
5 0ό
∞
∯
] (ψ ) i
bψ t
j ψ
∐∞
Endtmir epprdecm hs te`ai addoup, feohnk usi dg tmi whji cdaaicthdn dg te`ais dg Gdurhir trensgdrf pehrs tmet mevi `iin iste`ahsmij. Mdwivir, ht turns dut tmet hn feny shtuethdns ] (ψ ) cen `i wrhttin hn tmi gdrf dg e prdpir rethdnea guncthdn hn tmi erkufint ( bψ ) , wmiri tmi tirf
533
prdpir rigirs rigirs td e rethdnea guncthdn hn wmhcm tmi jikrii dg tmi nufiretdr pdayndfhea hs nd krietir
tmen tmi jikrii dg tmi jindfhnetdr pdayndfhea. pdayndfhea. Zmet hs,
n n ∐5 `n ( bψ ) + `n ∐5( bψ ) + + `< ] (ψ ) 6 ( bψ )n + en ∐5( bψ ) n ∐5 + + e< Zmhs kinirea gdrf hs ndt hn tmi te`ais, `ut wi cen usi perthea-grecthdn ixpenshdn td wrhti ] (ψ ) es e suf dg tmi shfpair rethdnea guncthdns tmet eri ahstij hn eaa te`ais. Zmi ahnierhty prdpirty enj te`ai addoup tmin yhiaj tmi cdrrispdnjhnk x(t ) . Zd ixpaehn sdfi dg tmi ficmenhcs, enj fhaj su`taithis tmet erhsi, ht hs cdnvinhint td cdnshjir shfpai ixefpais. Zmi ghrst ixefpai ejjrissis tmi hssui tmet dgtin e Gdurhir trensgdrf jdis ndt prisint htsiag hn tmi nhcist gdrf.
Ixefpai 5 Khvin
] (ψ ) 6
5
∐0e ψ 0 + b ( e 0ψ ∐ ψ 3 )
o o o wi cen usi tmi su`sthtuthdn ψ 6 ( bψ ) / b es gdaadws2
5
] (ψ ) 6
∐0e
( bψ ) 0 b
0
⎟ 0 ( bψ ) ( bψ )3 ⎞ + b ⎑ e b ∐ 3 ⎔ b ⎖ ⎬
6
5 ( bψ )3 + 0 e ( bψ ) 0 + e 0 ( bψ )
Zd pirgdrf e perthea grecthdn ixpenshdn, tmi jindfhnetdr pdayndfhea fust `i put hn gectdrij gdrf. Eny dg tmi perthea-grecthdn ixpenshdn fitmdjs cen `i usij, enj gdr feny ht hs cdnvinhint td swhtcm grdf tmi erkufint ( bψ ) td e fdri cdnvinhint ndtethdn. Easd, hn tmi te`ai addoup pmesi, sdfi criethvhty fhkmt `i riquhrij td ricdknhzi tmi hnvirsi trensgdrfs dg verhdus tirfs. Ixefpai 0 Zd wrhti tmi Gdurhir trensgdrf trensgdrf hn Ixefpai Ixefpai 5 hn fdri cdnvinhint ndtethdn, ndtethdn, wi
su`sthtuti s gdr ( bψ ) , enj prdciij es gdaadws (essufhnk e ≬ < )2
5 3
0 0 s + 0e s + e s
6
5 0 s( s + e)
6
5/ e0 s
5/ e0 5/ e ∐ ∐ s + e ( s + e) 0
Zmus
5/ e0
5/ e0 5/ e ∐ ∐ ] (ψ ) 6 bψ e + bψ ( e + bψ ) 0 Zmi sicdnj enj tmhrj tirfs cen `i gdunj hn ivin tmi smdrtist te`ai, hg e > < , `ut tmi ghrst tirf fhkmt riquhri hntirpritethdn. Grdf tmi stenjerj trensgdrfs G _u (t )V 6
5
+ ό κ (ψ ) , G_5V 6 0ό κ (ψ ) bψ enj tmi ahnierhty prdpirty, wi sii tmet 5 /( bψ ) cdrrispdnjs td tmi thfi shknea ⎫ 5 / 0 , t > < 6 50 skn(t ) ⎨∐5 / 0 , t 1 <
u (t ) ∐ 5 6 ⎭ 0
wmiri wi mevi wrhttin tmi risuat hn tirfs dg tmi sd-ceaaij shknuf guncthdn. Zmus wi mevi, et et x(t ) 6 5 0 skn( t ) ∐ 50 i ∐ u( t) ∐ 5 t i ∐ u( t )
0e
e
e
ekehn unjir tmi essufpthdn tmet e hs pdshthvi.
53:
\irusea dg te`ais dg Gdurhir trensgdrfs hnjhcetis tmet fdst dg tmi shfpai rethdnea trensgdrfs tmet eri cdvirij eri strhctay-prdpir rethdnea guncthdns dg ( bψ ) , tmet hs, tmi nufiretdr jikrii hs strhctay aiss tmen tmi jindfhnetdr jikrii. Zmhs `rhnks up en ejjhthdnea fenhpuaethdn. Ixefpai 3 Khvin
] (ψ ) 6
( bψ ) 0 + 0( bψ ) + 0 ( bψ ) 0 + 0( bψ ) + 5
ht hs cdnvinhint td ghrst jhvhji tmi jindfhnetdr pdayndfhea hntd tmi nufiretdr pdayndfhea td wrhti ] (ψ ) es e cdnstent paus e strhctay-prdpir rethdnea guncthdn. Zmhs hs endtmir ceacuaethdn wmiri e cmenki dg verhe`ai td, sey, s hnstiej dg ( bψ ) fhkmt `i cdnvinhint. Hn eny cesi, ht hs iesy td virhgy tmet tmi risuat hs
5
] (ψ ) 6 5 +
( bψ ) 0 + 0( bψ ) + 5
65 +
5 (5 + bψ ) 0
Zmhs khvis, grdf stenjerj te`ais, x(t ) 6 κ (t ) + t i ∐ u(t ) t
Pifero [i whaa mevi e stenjerj te`ai gdr usi hn 90<.05:. Zmhs te`ai, riprhntij `iadw, `iadw, enj ahnoij
dn tmi cdursi wi`peki, whaa `i prdvhjij hn ixefs. Ysi dg eny dtmir te`ai hs ndt pirfhttij. Zmet hs, enytmhnk ndt dn tmi dgghchea te`ai fust `i jirhvij grdf tmi dgghchea te`ai dr grdf ghrst prhnchpais. Dgghchea 90<.05: CZ Gdurhir Zrensgdrf Ze`ai
x(t )
] (ψ )
κ (t ) 5
5 0όκ (ψ ) 0όκ (ψ ∐ ψd )
bψ d t
i
cds(ψ dt )
όκ (ψ ∐ ψd ) + όκ (ψ + ψd )
shn(ψ dt ) u (t )
∐ bόκ (ψ ∐ ψ d) + bόκ (ψ + ψ d ) 5 bψ
+ όκ (ψ )
∐ et u (t ), e > <
5 e + bψ
ti ∐et u (t ), e > <
5
i
( e + bψ )0
e t i∐ | | , e > <
0e e +ψ 0 shn(ψ Z ) 0Z 5 ψ Z 5 0
u (t + Z5 ) ∐ u (t ∐ Z5 )
5
∞
∕ o 6∐∞
κ (t ∐ oZ )
∞
ψd
∕ o 6∐∞
539
κ (ψ ∐ oψd ) , ψd 6 0ό / Z
5<.; Gdurhir Zrensgdrf enj AZH Systifs Jiscrh`ij `y Jhggirinthea Iquethdns
Hg e systif hs jiscrh`ij `y e ghrst-drjir, ahnier jhggirinthea iquethdn, y (t ) + ey (t ) 6 `x(t ) , ∐ ∞ 1 t 1 ∞ tmin grdf Sicthdn 7.7 wi mevi tmet tmi systif hs ahnier enj thfi hnverhent, enj tmi unht-hfpuasi rispdnsi hs khvin `y m(t ) 6 `i ∐ et u (t ) Zmirigdri wi cen riejhay cdfputi tmi griquincy rispdnsi guncthdn dg tmi systif, ` M (ψ ) 6 e + bψ Mdwivir, tmhs hs veahj dnay hg tmi systif hs ste`ai, tmet hs, e > < . (Hg e 1 < , tmin tmi Gdurhir trensgdrf dg m(t ) jdis ndt ixhst, enj hg e 6 < , tmin tmi Gdurhir trensgdrf mes e jhggirint gdrf.) @iceusi M (ψ ) hs e strhctay-prdpir rethdnea guncthdn, hg tmi hnput shknea mes e prdpir rethdnea Gdurhir trensgdrf, cdfputethdn dg tmi rispdnsi y (t ) hs shfpay e fettir dg cdfputhnk tmi hnvirsi Gdurhir trensgdrf dg X (ψ ) 6 M (ψ ) ] (ψ ) `y perthea-grecthdn ixpenshdn enj te`ai addoup. Zmet hs, gdr e aerki caess dg hnput shkneas, tmi rispdnsi cdfputethdn hs cdfpaitiay eaki`rehc.
Fdri jhrictay, wi cen ixpriss tmi riaethdn `itwiin thfi shkneas hn tmi jhggirinthea iquethdn es e riaethdn `itwiin Gdurhir trensgdrfs. Aitthnk ] (ψ ) 6 G _ x( t )V , X (ψ ) 6 G_ y( t )V ahnierhty enj tmi jhggirinthethdn prdpirty khvi bψ X (ψ ) + e X (ψ ) 6 ` ] (ψ ) Zmhs cen `i sdavij eaki`rehceaay td d`tehn ` X (ψ ) 6 ] (ψ ) e + bψ Grdf tmhs wi ricdknhzi M (ψ ) , enj tmi d`vhdus hnvirsi Gdurhir trensgdrf khvis m(t ) , tmi unhthfpuasi rispdnsi dg tmi systif. Ekehn, tmhs hs veahj dnay gdr e > < , enj e jenkir hs tmet tmhs cdnjhthdn hs ndt epperint untha tmi hnvirsi Gdurhir trensgdrf hs ettifptij. Hn dtmir wdrjs, tmi ste`hahty cdnjhthdn hs ndt ixpahcht hn tmi eaki`rehc fenhpuaethdns aiejhnk td tmi griquincy rispdnsi guncthdn. Ht smduaj `i caier tmet tmhs epprdecm eppahis td mhkmir-drjir, ahnier jhggirinthea iquethdns tmet cdrrispdnj td ste`ai systifs. Zmi griquincy rispdnsi guncthdn hn sucm e cesi cen `i wrhttin hn tmi gdrf M (ψ ) 6
`f ( bψ )
f
+ + `5( bψ ) + `<
( bψ ) n + en ∐5( bψ ) n ∐5 + + e<
wmiri f 1 n . Sd, ekehn, cdfputethdn dg tmi rispdnsi td e aerki caess dg hnput shkneas hs cdfpaitiay eaki`rehc (tmdukm cmicohnk tmi ste`hahty cdnjhthdn hs fdri su`tai, enj hs dfhttij). Ixefpai [min tmi hnput shknea mes e Gdurhir trensgdrf tmet hs ndt e prdpir rethdnea guncthdn, guncthdn, tmi
ceacuaethdns `icdfi sahkmtay fdri cdfpahcetij enj hnvdavi sdfi ricdknhthdn dg cdf`hnethdns dg tirfs. Suppdsi e 6 0, ` 6 5 , enj tmi hnput shknea hs e unht stip guncthdn. Zmin
537
X (ψ ) 6
6
⎪ ό 5 ⎥ 5 5 + 6 + ό κ ψ ( ) ⎩ ( bψ )(0 + bψ ) 0 + bψ κ (ψ ) 0 + bψ ⎠⎣ bψ ⎧ 5/ 0
∐
5/ 0
ό
+ κ (ψ )
bψ 0 + bψ 0 Krduphnk tdkitmir tmi ghrst enj aest tirfs, te`ai addoup khvis tmi dutput shknea es y t 6 5 u t ∐ 5 i ∐0t u t
()
0
()
0
()
5<.= Gdurhir Zrensgdrf enj Hntircdnnicthdns dg AZH Systifs
Hntircdnnicthdns dg ste`ai AZH systifs eri cdnvinhintay jiscrh`ij hn tirfs dg griquincy rispdnsi guncthdns, tmdukm ht fust `i kuerentiij tmet tmi dvireaa systif easd hs ste`ai gdr tmi dvireaa griquincy rispdnsi guncthdn td `i fienhnkgua. Essufhnk tmhs, `adco jhekref iquhveaincis hn tirfs dg griquincy rispdnsi guncthdns gdaadw grdf tmi thfi jdfehn risuats, et aiest gdr tmi ghrst twd cesis. Nefiay, gdr ejjhthvi pereaaia cdnnicthdns, wmiri tmi dvireaa unht-hfpuasi rispdnsi hs tmi suf dg tmi su`systif unht-hfpuasi rispdnsis, enj gdr cesceji cdnnicthdns, wmiri tmi dvireaa unht-hfpuasi unht-hfpuasi rispdnsi hs tmi cdnvdauthdn dg tmi su`systif unht-hfpuasi unht-hfpuasi rispdnsis, wi hffijhetiay mevi
Dg cdursi, hn tmisi cesis ht hs caier tmet ste`hahty dg tmi dvireaa systif gdaadws grdf ste`hahty dg tmi hnjhvhjuea su`systifs. Zmi shtuethdn hs fdri cdfpahcetij gdr tmi giij`eco cdnnicthdn dg ste`ai AZH systifs, `ut et aiest tmi Gdurhir trensgdrf riprisintethdn pirfhts us td ecmhivi en ixpahcht riprisintethdn gdr tmi dvireaa systif, sdfitmhnk tmet wi wiri une`ai td eccdfpahsm hn tmi thfi jdfehn. @ikhnnhnk whtm tmi dutput, tmi giij`eco cdnnicthdn `iadw khvis tmi gdaadwhnk eaki`rehc riaethdnsmhp `itwiin tmi Gdurhir trensgdrfs dg tmi hnput enj dutput shkneas. (Ekehn, tmi nikethvi shkn dn tmi giij`eco ahni et tmi suffir hs trejhthdnea.)
53;
X (ψ ) 6 M5(ψ ) _ ] (ψ ) ∐ M 0 (ψ )X (ψ ) V
6 M5 (ψ ) ] (ψ ) ∐ M5(ψ ) M 0 (ψ )X (ψ ) Sdavhnk gdr X (ψ ) `y eaki`rehc fenhpuaethdn khvis X (ψ ) 6
M 5 (ψ )
5 + M5(ψ ) M 0 (ψ )
] (ψ )
Zmet hs, tmi giij`eco cdnnicthdn e`dvi hs iquhveaint td
Dg cdursi tmi dvireaa systif, ceaaij tmi cadsij-addp systif hn tmhs cdntixt, fust `i ste`ai gdr tmi griquincy rispdnsi guncthdn smdwn td `i fienhnkgua. Yngdrtunetiay, tmi giij`eco cdnnicthdn dg ste`ai systifs jdis ndt eaweys yhiaj e ste`ai cadsij-addp systif, sd tmet gurtmir pursuht dg tmhs tdphc ghrst riquhris tmi jiviadpfint dg ste`hahty crhtirhe gdr giij`eco systifs. Ixefpai Hg
M5 (ψ ) 6
3 , M 0 (ψ ) 6 o 0 + bψ
wmiri o hs e cdnstent, tmin tmi griquincy rispdnsi dg tmi cadsij-addp systif hs 3 /(0 + bψ ) 3 6 M ca (ψ ) 6 5 + 3o /( /(0 + bψ ) (3o + 0) + bψ > ∐0 / 3 , hn wmhcm cesi Zmhs hs e veahj griquincy rispdnsi hg o > o t mca (t ) 6 3 i ∐ (3 + 0) u(t )
Hnjiij, `y cmdhci dg o wi cen ecmhivi er`htrerhay gest ixpdninthea jicey dg tmi cadsij-addp 1 ∐0 / 3 tmi cadsij-addp systif‘s unht-hfpuasi unht-hfpuasi rispdnsi! @ut ht hs hfpdrtent td ndti tmet gdr o 1 systif hs ndt ste`ai. Zmus ht jdis ndt mevi e fienhnkgua griquincy rispdnsi guncthdn enj tmi M ca (ψ ) wi cdfputij hs e ghcthdn. Ixirchsis 5. Grdf tmi `eshc jighnhthdn, jighnhthdn, cdfputi cdfputi tmi Gdurhir trensgdrfs dg tmi tmi shkneas ∐(t ∐ 0)
(e) x(t ) 6 i
u (t ∐ 3)
(`) x (t ) 6 i ∐|t +5|
⎫ <, t ≪ < ⎢ (c) x(t ) 6 ⎭ 0, < 1 t 1 5 ⎢ ∐(t ∐5) , t ≩ 5 ⎨ 0i (j) x (t ) 6
∞
∕
o
e κ (t ∐ o ) , | e |1 5
o 6 <
53=
0. Grdf tmi `eshc jighnhthdn, cdfputi cdfputi tmi shkneas cdrrispdnjhnk td tmi Gdurhir Gdurhir trensgdrfs ∐|ψ |
(e) ] (ψ ) 6 0ό i
⎫0ό , ∐ 0 ≪ ψ ≪ 0 ⎨ <, iasi
(`) | ] (ψ ) |6 ⎭
⎫∐ψ , ∐ 0 ≪ ψ ≪ 0 ∬] (ψ ) 6 ⎭ ⎨ <, iasi
ψ (c) ] (ψ ) 6 i ∐ u(ψ ) (j) ] (ψ ) spichghij `y tmi soitcmis `iadw2
3. Cdfputi tmi tmi Gdurhir trensgdrfs trensgdrfs dg tmi shkneas shkneas ∞
∕ 0( ∐5) o κ (t ∐ 3o )
(e) x(t ) 6
o 6∐∞
(`) x(t ) 6
∞
∕
i
∐ (t ∐3o )
_u (t ∐ 3o ) ∐ u( t ∐ 3o ∐5) 5)V
o 6∐∞
:. @y hnspicthdn dg tmi jighnhnk jighnhnk gdrfuaes gdr tmi tmi Gdurhir trensgdrf enj hnvirsi hnvirsi Gdurhir
trensgdrf, tmet hs, whtmdut cdfputhnk tmi Gdurhir trensgdrf, iveaueti tmi gdaadwhnk quenththis gdr tmi shknea smdwn `iadw.
∞
(e)
∯ ] (ψ ) j ψ ∐∞ ∞
(`)
∯ ] (ψ ) i
bψ
j ψ
∐∞ (c) ] (<) 9. Cdfputi tmi Gdurhir trensgdrf dg tmi shknea x(t ) smdwn `iadw,
enj usi tmi prdpirthis dg tmi Gdurhir trensgdrf td jitirfhni tmi trensgdrfs dg tmi gdaadwhnk shkneas whtmdut ceacuaethdn. (e) 538
(`)
(c)
7. E shknea x (t ) mes tmi Gdurhir trensgdrf
] (ψ ) 6
3 ∐ bψ 3 + bψ
(e) Soitcm tmi feknhtuji feknhtuji spictruf dg tmi shknea. (`) Soitcm tmi pmesi spictruf dg tmi shknea (c) Ghnj tmi shknea x(t ) `y ushnk tmi prdpirthis dg tmi Gdurhir trensgdrf. ;. Grdf tmi `eshc jighnhthdns enj prdpirthis prdpirthis dg tmi Gdurhir trensgdrf enj hnvirsi Gdurhir
trensgdrf, enswir tmi gdaadwhnk quisthdns e`dut tmi trensgdrf dg tmi shknea smdwn `iadw (whtmdut ceacuaethnk tmi trensgdrf).
(e) [met hs ∬ ] (ψ ) 4 (Mhnt2 En ivin shknea mes e riea Gdurhir trensgdrf.) (`) [met hs ] (<) 4 ∞
(c) [met hs
∯ ] (ψ ) j ψ 4 ∐∞
=. Khvin tmet tmi Gdurhir trensgdrf dg tmi shknea x(t ) 6 t i
5:<
∐0t u (t ) hs
] (ψ ) 6
5 (0 + bψ ) 0
soitcm tmi feknhtuji enj pmesi spictre gdr tmi shkneas (e) y (t ) 6 j x(t ) (`) y (t ) 6
jt t
∯ x(ϊ ) j ϊ
∐∞ x( ∐0t + :)
(c) y (t ) 6 (j) y (t ) 6 0 x( t) + x( t ) 8. Zwd AZH systifs eri spichghij spichghij `y tmi unht-hfpuasi unht-hfpuasi rispdnsis m5 (t ) 6 ∐0κ (t ) + 9i ∐0t u(t) enj t m0 (t ) 6 0ti ∐ u (t ) . Cdfputi tmi rispdnsis dg tmi twd systifs td tmi hnput shknea x(t ) 6 cds(t ) . 5<. En hnput hnput shknea x(t ) eppahij td tmi AZH systif whtm griquincy rispdnsi guncthdn
M (ψ ) 6
bψ
5 + bψ
yhiajs tmi dutput shknea y (t ) 6 κ (t ) + 3i∐ u(t) ∐ ; i∐0 u( t ) t
t
[met hs x(t ) 4 55. Suppdsi y (t ) 6 x(t ) cds(t ) enj tmi Gdurhir trensgdrf dg y (t ) hs jiscrh`ij hn tirfs dg unht-
stip guncthdns es
X (ψ ) 6 u (ψ + 0) ∐ u(ψ ∐ 0)
[met hs x(t ) 4 50. Suppdsi x(t ) mes Gdurhir trensgdrf jiscrh`ij jiscrh`ij hn tirfs dg unht-refp unht-refp guncthdns es
] (ψ ) 6 r (ψ + 5) ∐ 0 r(ψ ) + r (ψ ∐5)
enj suppdsi p(t ) hs pirhdjhc whtm gunjefintea griquincy ψ d enj Gdurhir sirhis cdigghchints ] o ,
6 <, µ 5, µ 0,… . o 6
(e) Hg y (t ) 6 x(t ) p( t ) , jitirfhni en ixprisshdn gdr X (ψ ) . (`) Soitcm tmi efpahtuji spictruf dg y (t ) hg p(t ) 6 cds(t / 0) . (c) Soitcm tmi efpahtuji spictruf dg y (t ) hg p (t) 6 cds(t ) . 53. Khvin tmet tmi Gdurhir trensgdrf dg x(t ) 6 i
∐|t |
] (ψ ) 6
hs
0 5+ψ 0
cdfputi enj soitcm tmi feknhtuji enj pmesi spictre gdr y (t ) 6 i b3t j i∐|t | jt
5:. Cdfputi tmi tmi Gdurhir trensgdrf trensgdrf gdr tmi shknea shknea
5:5
x(t ) 6 shn(ψ dt ) u( t ) Mhnt2 Xdu fey mevi td usi usi spichea prdpirthis dg dg hfpuasis hn tmi ceacuaethdn. ceacuaethdn.) ( Mhnt2 59. E cdnthnudus-thfi cdnthnudus-thfi AZH systif hs jiscrh`ij jiscrh`ij `y tmi griquincy rispdnsi rispdnsi guncthdn
M (ψ ) 6
0 0 ∐ ψ 0 + bψ 3
enj tmi hnput shknea mes tmi Gdurhir trensgdrf ] (ψ ) 6 i ∐
bψ 3
Cdfputi tmi rispdnsi y (t ) . 57. Ysi perthea grecthdn ixpenshdn ixpenshdn td cdfputi cdfputi tmi hnvirsi Gdurhir trensgdrf gdr
(e) ] (ψ ) 6 (`) ] (ψ ) 6
9 bψ + 50
( bψ ) 0 + 9 bψ + 7 : 3 ∐ ψ 0 + : bψ
5;. Cdfputi tmi tmi hnvirsi Gdurhir trensgdrf gdr b (ό ∐όψ ) 0
(e) ] (ψ ) 6
(9 ∐ ψ + b :ψ )
i
(8 ∐ ψ 0 + b 7ψ )(0 + bψ )
(`) ] (ψ ) 6 i
∐ b 0ψ
+
5< + 5
5=. Cdfputi tmi tmi dvireaa griquincy rispdnsi rispdnsi guncthdn M (ψ ) 6 X (ψ ) / ] (ψ ) gdr tmi systifs
smdwn `iadw. (Dg cdursi, essufi tmet tmi su`systifs enj tmi dvireaa systif eri ste`ai.) (e)
5:0
Ndtis gdr Shkneas enj Systifs 55.5 Hntrdjucthdn td tmi Ynhaetirea Aepaeci Zrensgdrf
Grdf Cmeptir 5< ht hs caier tmet tmiri hs dni fehn ahfhtethdn dn tmi usi dg tmi Gdurhir trensgdrf2 tmi shknea fust `i sucm tmet tmi trensgdrf hntikrea cdnvirkis, dr sucm tmet wi cen eppay kinireahzij guncthdn ticmnhquis td errhvi et e trensgdrf (gdr ixefpai, tmi cesi dg pirhdjhc shkneas). Ht turns dut tmet tmhs ahfhtethdn cen `i evdhjij, perthcuaeray gdr rhkmt-shjij shkneas, `y hncaujhnk e jefphnk gectdr hn tmi hntikrea. Zmi risuathnk trensgdrf hs ceaaij tmi unhaetirea Aepaeci trensgdrf. Gdr e rhkmt-shjij, dr unhaetirea, shknea x (t ) , sdfithfis wrhttin es x(t )u (t ) td ifpmeshzi tmet tmi 1 < , tmi unhaetirea Aepaeci trensgdrf hs jighnij es shknea hs zird gdr t 1 ] ( s ) 6
∞
∐ st ∯ x(t) i jt
<∐
Miri s hs e cdfpaix verhe`ai, dgtin wrhttin hn rictenkuaer gdrf ushnk tmi stenjerj ndtethdn ∐ s 6 σ + bψ . Zmi adwir ahfht dg hntikrethdn hs smdwn es < td ifpmeshzi tmi gect tmet en hfpuasi 6 < hs hncaujij hn tmi renki dg hntikrethdn, `ut dgtin wi aievi tmhs unjirstddj enj dr jdu`ait et t 6 shfpay wrhti tmi adwir ahfht es <.
Hg wi cdnshjir σ 6 Pi{s} > < , tmin
| i∐ st |6| i∐σ t i ∐ bψ t |6| i∐σ t | ← < es t ← ∞ Zmirigdri ] ( s) cen `i wiaa jighnij ivin tmdukm x(t ) jdis ndt kd td zird es t kdis td ∞ . Hnjiij, hg x(t ) hs dg ‑ixpdninthea drjir,‖ tmet hs, tmiri ixhst riea cdnstents O , c sucm tmet | x(t ) |≪ Oict , t ≩ < tmin ] ( s) ixhsts hg wi tmhno dg s es sethsgyhnk Pi{s} > c . Hn dtmir wdrjs, gdr shkneas dg ixpdninthea drjir tmi unhaetirea Aepaeci trensgdrf eaweys ixhsts gdr e meag-paeni dg cdfpaix veauis dg s , es smdwn `iadw.
@iceusi shkneas incduntirij hn tmi siquia whaa eaweys `i dg ixpdninthea drjir, wi cen `i e `ht ceveahir enj hkndri jitehaij eneayshs dg tmi rikhdns dg cdnvirkinci, cdnghjint hn tmi ondwaijki tmet tmiri hs e wmdai meag-paeni dg veauis dg s gdr wmhcm ] ( s) hs wiaa jighnij. Enj tmi ectuea nufirhcea veauis dg s gdr wmhcm tmi hntikrea cdnvirkis turn dut td `i dg nd hntirist gdr dur purpdsis.
5:3
Ixefpai Zmi shknea 0
t
x (t ) 6 i u (t ) hs ndt dg ixpdninthea drjir shnci gdr eny khvin veauis dg O enj c , i
t0
> Oict , gdr t su gghchintay aerki
Zmi shknea, x(t ) 6 i9 u (t ) hs dg ixpdninthea drjir, enj wi cen teoi O 6 5 , c 6 9 td prdvi ht. t
Ixefpai Zmi Aepaeci trensgdrf dg x (t ) 6 i ∞ ∐3t
] ( s) 6
∯
∐3t u (t ) hs
u( t ) i
i
∐ st
jt 6
<∐
∞
∯i
∐( s +3)t
jt
<
∐5 ∐( s + 3)t ∞ 5 i 6 < s+3 s+3 Miri e meag-paeni dg cdnvirkinci hs khvin `y Pi{s} > ∐3 , enj hnjiij tmhs cdnjhthdn hs cruchea hn 6
iveauethnk tmi hntikrea et tmi uppir ahfht. Gdr tmi shknea x(t ) 6 i3t u (t ) , e shfhaer ceacuaethdn khvis
5 s ∐3 enj tmi meag-paeni dg cdnvirkinci hn tmhs cesi hs Pi{s} > 3 . Mdwivir, es finthdnij e`dvi, wi ] ( s) 6
whaa ndt hnshst dn oiiphnk treco dg tmi cdnvirkinci rikhdn. Gdr shkneas hnvdavhnk kinireahzij guncthdns, tmi ndthdn dg ixpdninthea drjir jdis ndt eppay, `ut hn typhcea cesis tmi spichea prdpirthis dg kinireahzij guncthdns cen `i usij td iveaueti tmi Aepaeci trensgdrf. trensgdrf dg tmi hfpuasi, hfpuasi, x(t ) 6 κ (t ) , hs ieshay iveauetij ushnk tmi shgthnk Ixefpai Zmi Aepaeci trensgdrf prdpirty2 ∞
∐ st ∐s < ∯ κ (t )i jt 6 i 6 5
] ( s) 6
<∐
stip guncthdn guncthdn hs Ixefpai Zmi Aepaeci trensgdrf dg tmi unht stip ∞ ∞ ∐ st ∐ st ] ( s) 6
jt 6
∯ u (t ) i
<∐
∯i
jt 6
<
5 s
wmiri hn tmhs cesi e meag-paeni dg cdnvirkinci hs khvin `y Pi{s} > < .
Pifero Hg e rhkmt-shjij shknea x(t ) mes e unhaetirea Aepaeci trensgdrf tmet cdnvirkis gdr
Pi{s} 6 < , tmin wi cen wrhti, teohnk σ 6 < , ] ( s) 6
∞
∯ x(t ) i <
∐ st
jt 6
∞
∯ x(t) i <
5::
∐ bψ t
jt 6 G_ x( t )V
Zmet hs, tmi Aepaeci trensgdrf whtm σ 6 < hs tmi Gdurhir trensgdrf gdr rhkmt-shjij shkneas. (Sdfithfis tmhs hs wrhttin es 6 ] ( bψ ) ] ( s) s 6 bψ wmhcm aiejs td e jhggirint ndtethdn gdr tmi Gdurhir trensgdrf tmen wi mevi usij. Nefiay, wi wrhti tmi Gdurhir trensgdrf es ] (ψ ) , retmir tmen ] ( bψ ) , e`sdr`hnk tmi tmi hfekhnery unht b hntd tmi guncthdn retmir tmen jhspaeyhnk ht hn tmi erkufint. Zmhs ungdrtuneti ndtethdnea cdaahshdn smduaj `i vhiwij es e fhaj hncdnvinhinci, enj ht smduaj ndt `i pirfhttij td d`scuri tmi riaethdnsmhp `itwiin tmi Gdurhir Gdurhir enj Aepaeci trensgdrfs dg rhkmt-shjij rhkmt-shjij shkneas.) shkneas.) Ixefpai Gdr x(t ) 6 i
∐3t u (t ) wi sii tmet tmi meag-paeni dg cdnvirkinci hncaujis σ 6 < , enj
grdf e`dvi wi mevi ] (ψ ) 6
5 3 + bψ
Gdr tmi unht-stip guncthdn wi mevi ] ( s ) 6 5 / s , `ut hn tmhs cesi tmi rikhdn dg cdnvirkinci jdis hncauji σ 6 < , enj hnjiij tmi Gdurhir trensgdrf dg tmi unht stip hs ndt shfpay 5 /( bψ ) . ndt hncauji 55.0 \rdpirthis dg tmi Ynhaetirea Aepaeci Zrensgdrf
[i ndw cdnshjir e verhity dg gefhaher dpirethdns dn e rhkmt-shjij shknea x (t ) , enj hntirprit tmi iggict dg tmisi dpirethdns dn tmi cdrrispdnjhnk unhaetirea Aepaeci trensgdrf ] ( s) . Dg cdursi, tmi dpirethdns wi cdnshjir fust yhiaj rhkmt-shjij shkneas. [i essufi tmet shkneas eri dg ixpdninthea drjir sd tmet ixhstinci dg tmi Aepaeci trensgdrf hs essurij. Gurtmirfdri, wi smduaj virhgy tmet iecm dpirethdn cdnshjirij yhiajs e shknea tmet easd hs dg ixpdninthea drjir. Dgtin tmhs hs d`vhdus, enj whaa ndt `i finthdnij, `ut ceri hs niijij hn e cdupai dg cesis. Zmrdukmdut wi usi tmi gdaadwhnk ndtethdn gdr tmi Aepaeci trensgdrf wmiri A jindtis e ‑Aepaeci trensgdrf dpiretdr2‖ ∞
] ( s ) 6 A_ x(t )V 6 ∯ x( t) i
∐ st jt
<
Ahnierhty Hg A_ x(t )V 6 ] ( s) enj A_ z(t )V 6 R ( s) , tmin gdr eny cdnstent e,
A_e x(t ) + z (t )V 6 e ] ( s) + R ( s)
Zmhs prdpirty gdaadws jhrictay grdf tmi jighnhthdn. Zhfi Jiaey Hg A_ x(t )V 6 ] ( s) , tmin gdr eny cdnstent t d
A_ x(t ∐ td )u (t ∐ td )V 6 i
≩<, ∐ st d
] ( s)
Zmi ceacuaethdn virhgyhnk tmhs hs `y ndw quhti stenjerj. @ikhn whtm A_ x(t ∐ td )u (t ∐ td )V 6
∞
∯ x( t ∐ td ) u ( t ∐ t d ) i
<
5:9
∐ st jt
enj cmenki hntikrethdn verhe`ai grdf t td ϊ 6 t ∐ t d td d`tehn tmi risuat. Ndthci tmet wi usi tmi unht-stip ndtethdn td feoi ixpahcht tmi gect tmet tmi rhkmt-smhgtij shknea, x(t ∐ t d ) hs zird gdr t 1 t d . Ixefpai Gdr e rictenkuaer puasi, x(t ) 6 Ou (t ) ∐ Ou (t ∐ t d ) , t d
tmi jiaey prdpirty td wrhti
> < , wi cen usi ahnierhty ahnierhty enj
5 ∐ i∐td s ∐td s O ] ( s) 6 ∐i 6 O s s s O
Zhfi Sceahnk Hg A_ x(t )V 6 ] ( s) , tmin gdr eny cdnstent e > < ,
(e)
A_ x( et )V 6 5 ] s e
Zmhs hs endtmir gefhaher ceacuaethdn, enj tmi jitehas whaa `i sohppij. Zmi essufpthdn tmet e > < hs riquhrij sd tmet tmi sceaij shknea hs rhkmt shjij. Zmi nixt twd prdpirthis riquhri e fdri cerigua hntirpritethdn dg tmi adwir ahfht hn tmi Aepaeci trensgdrf jighnhthdn, enj wi wrhti tmet ahfht es <∐ . Jhggirinthethdn Hg A_ x(t )V 6 ] ( s) , enj tmi thfi-jirhvethvi shknea x (t ) mes e Gdurhir trensgdrf,
tmin A_ x (t )V 6 s] ( s) ∐ x(< ∐ )
Zd busthgy tmhs prdpirty, jhrictay cdfputi tmi trensgdrf, ushnk hntikrethdn-`y-perts2 hntikrethdn-`y-perts2 ∞
∞ ∞ st ∐ + ∯ x( t) s i∐ st jt A_ x (t )V 6 ∯ x (t ) i jt 6 x( t) i ∐ < <∐ <∐ ∐ ← ∞ , enj wi hntirprit x (<∐ ) i∐ s < [i essufi tmet σ 6 Pi{s} hs sucm tmet x (t )i ∐ st ← < es t ← es x (<∐ ) td errhvi et tmi caehfij risuat. ∐ st
Ixefpai @ikhnnhnk whtm A_u (t )V )V 6 5 / s , tmi jhggirinthethdn prdpirty cdnghrfs tmet
A_κ (t )V 6 A_u(t)V 6 s
5 s
65
Ndti tmet wi cen htireti tmi jhggirinthethdn jhggirinthethdn prdpirty td d`tehn tmi tmi Aepaeci trensgdrf gdr mhkmir jirhvethvis, gdr ixefpai, A_ x(t )V 6 s A_ x (t)V ∐ x(< ∐ ) 6 s 0 A_ x( t)V ∐ s x(< ∐) ∐ x(< ∐) Hntikrethdn Hg A_ x(t )V 6 ] ( s) , enj
z (t ) 6
t
∯ x(ϊ ) jϊ u(t )
<∐
wmiri tmi unht-stip hs eppinjij shfpay gdr ifpmeshs, tmin
5:7
R ( s) 6
5
] ( s) s Zmi prddg dg tmhs hs endtmir hntikrethdn `y perts tmet hs dutahnij `iadw2 ∞ t ∞ t ∞ ∐5 ∐ st 5 st A_ z (t )V 6 ∯ ∯ x(ϊ ) jϊ u( t) i∐ jt 6 ∯ x(ϊ ) jϊ u( t) i + ∯ x( t) i∐ st jt s s ∐ ∐ ∐ ∐ ∐ < <
<
<
<
5 6 ] ( s)
s Zmi iveauethdns dg tmi ghrst tirf risuathnk grdf tmi hntikrethdn-`y-perts eri `dtm zird, `ut gdr jhggirint riesdns. Ht cen `i smdwn tmet tmi runnhnk hntikrea dg e shknea dg ixpdninthea drjir hs dg ← ∞ tmi prdjuct dg ixpdninthea drjir, enj sd wi cen essufi tmet σ 6 Pi{s} hs sucm tmet es t ← tmi runnhnk hntikrea enj tmi ixpdninthea kdis td zird. Zmi iveauethdn et t 6 <∐ yhiajs zird gdr fdri d`vhdus riesdns. Cdnvdauthdn Hg x(t ) enj m(t ) eri rhkmt-shjij shkneas whtm unhaetirea Aepaeci trensgdrfs ] ( s )
enj R ( s) , tmin tmi cdnvdauthdn (m ∛ x)(t ) yhiajs e rhkmt-shjij shknea tmet cen `i wrhttin t
y (t ) 6 ∯ x(ϊ ) m(t ∐ ϊ ) jϊ u( t ) <
whtm Aepaeci trensgdrf
X ( s) 6 M ( s) ] ( s)
Zmi prddg dg tmhs prdpirty hs viry shfhaer td tmi Gdurhir-trensgdrf cesi, enj tmirigdri hs dfhttij. Ghnea Weaui Zmidrif Hg A_ x(t )V 6 ] ( s) enj tmi ahfhts
ahft ←∞ x(t ) , ahf s ←< s] ( s) `dtm ixhst, tmin
ahft ←∞ x(t ) 6 ahf s ←< s] ( s)
Petmir tmen prdvi tmhs risuat, wi prisint en ixefpai tmet haaustretis tmi jenkir hn eppayhnk ht ricoaissay. Ixefpai E strehkmtgdrwerj strehkmtgdrwerj ceacuaethdn ceacuaethdn khvis ∞
A_shn(t )u(t )V 6 ∯ shn( t) i <
∐ st
∞ i bt ∐ i ∐ bt st jt 6 ∯ i ∐ jt 0 b < ∞
∞
∐5 / 0 b ∐( s ∐ b )t 5 / 0 b ∐( s + b ) t i i 6 + s∐ b s b + < < 5 6 0 s +5 D`vhdusay, s
6< ahf s ←< s ] ( s) 6 ahf s ←< 0 + s 5
5:;
Mdwivir, ahft ←∞ _shn(t) u(t ))VV jdis ndt ixhst enj wi sii tmet tmi Ghnea Weaui Zmidrif cen khvi fhsaiejhnk risuats wmin addsiay eppahij! 55.3 Hnvirsi Ynhaetirea Aepaeci Zrensgdrf
Hnspicthdn dg tmi Aepaeci trensgdrfs wi mevi cdfputij, dr e te`ai dg trensgdrfs, hnjhcetis tmet tmi shkneas typhceaay incduntirij mevi trensgdrfs tmet eri strhctay-prdpir rethdnea guncthdns. Zmisi eri rethds dg pdayndfheas hn s whtm tmi jikrii dg tmi nufiretdr pdayndfhea aiss tmen tmi jikrii dg tmi jindfhnetdr pdayndfhea. Es fhkmt `i ixpictij grdf tmi Gdurhir-trensgdrf cesi, perthea grecthdn ixpenshdn, gdaadwij `y te`ai addoup, hs tmi fehn tdda gdr cdfputhnk tmi thfi shknea cdrrispdnjhnk td e khvin trensgdrf. (Zmiri hs e fdri kinirea hnvirsi trensgdrf gdrfuae, `ut ht hnvdavis ahni hntikreas hn tmi cdfpaix paeni enj wi whaa ndt feoi usi dg ht.) Pifero Smdwn `iadw hs tmi tmi stenjerj te`ai dg Aepaeci trensgdrfs trensgdrfs gdr usi hn 90<.05:. 90<.05:. Ysi dg eny
dtmir te`ai hn ixefs dr mdfiwdro esshknfints hs ndt pirfhttij. Enytmhnk ndt dn tmhs dgghchea te`ai fust `i jirhvij grdf intrhis dn tmi te`ai dr grdf tmi jighnhthdn dg tmi trensgdrf. Dgghchea 90<.05: Ynhaetirea Aepaeci Zrensgdrf Ze`ai x(t )
] ( s)
κ (t )
5
u (t )
5
r (t )
5
s s
i
∐ et u (t )
ti
5
s+e
∐ et u (t )
5 ( s + e)0
cds(ψ dt )u (t ) shn(ψ dt )u (t ) i
∐et
i
cds(ψ dt )u(t )
∐ et
0
shn(ψ dt )u( t )
s 0
0
0
0
s +ψ d ψ d s +ψ d s+e
( s + e ) 0 +ψ d0 ψ d
( s + e ) 0 +ψ d0
[i haaustreti tmi ceacuaethdn dg hnvirsi trensgdrfs whtm twd ixefpais. Ixefpai Khvin
] ( s ) 6
s0 + s + 5
s0 + 5 wmhcm hs e prdpir, `ut ndt strhctay-prdpir, rethdnea guncthdn, guncthdn, wi cen jhvhji tmi nufiretdr `y tmi jindfhnetdr td wrhti
5:=
] ( s ) 6 5 +
s 0 s +5
Yshnk ahnierhty dg tmi Aepaeci trensgdrf, wi cen triet tmi tirfs siperetiay. \erthea grecthdn ixpenshdn dg tmi sicdnj tirf khvis s s 5/ 0 5/ 0 s0 + 5
6
6
( s + b)( s ∐ b)
s+ b
+
s∐ b
Grdf tmi te`ai dg trensgdrfs,
⎥ A∐5 ⎠
⎪ 5 ∐ bt 5 bt ⎩ 6 0 i u (t ) + 0 i u(t ) 6 cds(t ) u( t ) 0 ⎣ s + 5⎧
Zmirigdri
s
x(t ) 6 κ (t ) + cds(t ) u( t )
Endtmir cesi tmet hs strehkmtgdrwerj td menjai hs wmin tmiri eri ‑jiaey gectdrs‖ hn tmi trensgdrf. Ixefpai Khvin
] ( s) 6
s i∐0 + i∐ s
s
0 s +5
wi cen wrhti ] ( s) 6 i
∐0 s
∐s 5 + i 0 0 s +5 s +5
s
Yshnk tmi ahnierhty, jiaey, enj jirhvethvi prdpirthis hn cdnbuncthdn whtm tmi privhdus ixefpai, wi d`tehn x(t ) 6 cds(t ∐ 0) u(t ) + shn( t ∐5) u( t ∐5) 55.: Systifs Jiscrh`ij `y Ahnier Jhggirinthea Iquethdns
Cdnshjir e systif wmiri tmi hnput enj dutput shkneas eri riaetij `y tmi ghrst-drjir jhggirinthea iquethdn y (t ) + ey (t ) 6 `x(t ) 6 < hs zird, Essufhnk tmet tmi hnput shknea hs rhkmt shjij, enj essufhnk tmet hnhthea cdnjhthdn et t 6 tmi dutput shknea hs rhkmt shjij enj tmi systif hs ahnier enj thfi hnverhent. (Hn perthcuaer, shnci en AZH systif whtm hjinthceaay zird hnput fust mevi hjinthceaay zird dutput, tmi essufpthdn dg zird hnhthea cdnjhthdn hs hfpdrtent.) Hn tmi sitthnk dg rhkmt-shjij hnput enj dutput shkneas, tmi systif cen `i jiscrh`ij hn tirfs dg unhaetirea Aepaeci trensgdrfs. Pikerjaiss dg tmi veauis dg tmi cdnstents e enj ` , enj hn perthcuaer rikerjaiss rikerjaiss dg tmi ste`hahty ste`hahty prdpirty dg tmi systif, systif, wi cen cdfputi cdfputi tmi Aepaeci trensgdrf dg tmi unht hfpuasi rispdnsi m(t ) 6 `i
∐ et u (t )
M ( s) 6
`
td d`tehn s+e
5:8
Petmir tmen tmi tirf griquincy rispdnsi guncthdn, tmhs hs ceaaij tmi trensgir guncthdn dg tmi systif, enj hn tirfs dg tmi Aepaeci trensgdrfs ] ( s) enj X ( s) dg tmi rhkmt-shjij hnput enj dutput shkneas tmi systif hs jiscrh`ij `y X ( s) 6 M ( s) ] ( s) Endtmir epprdecm hs td iqueti tmi Aepaeci trensgdrfs dg tmi aigt enj rhkmt shjis dg tmi jhggirinthea iquethdn, enj tmhs epprdecm mes tmi ejventeki dg ndt riquhrhnk r iquhrhnk ondwaijki dg tmi unht-hfpuasi rispdnsi. Yshnk tmi ahnierhty enj jhggirinthethdn prdpirthis khvis ( s + e)X ( s) 6 `] ( s) Zmus, ekehn, wi d`tehn X ( s) ` 6 M ( s) 6 ] ( s) s+e Hg tmi hnput shknea mes e prdpir rethdnea Aepaeci trensgdrf, tmin ht hs caier tmet tmi dutput shknea mes e strhctay-prdpir rethdnea Aepaeci trensgdrf. Zmirigdri wi cen sdavi gdr tmi rispdnsi td e whji caess dg hnput shkneas `y tmi eaki`rehc prdciss dg perthea grecthdn ixpenshdn enj te`ai addoup. Ekehn, en ejventeki dg tmi Aepaeci-trensgdrf epprdecm hn tmhs unhaetirea sitthnk hs tmet systifs whtm un`dunjij hnput shkneas enj/dr dutput dutput shkneas, dr systifs tmet eri unste`ai, cen `i trietij, hn cdntrest td tmi Gdurhir trensgdrf epprdecm. Ixefpai Gdr tmi cesi cesi wmiri e 6 ∐5 , ` 6 5 enj wmiri tmi hnput shknea hs 3t
x(t ) 6 i u (t ) tmet hs, en unste`ai systif whtm un`dunjij hnput shknea, wi hffijhetiay d`tehn X (s) 6
5 ( s ∐ 5)( s ∐ 3)
\erthea grecthdn ixpenshdn ieshay aiejs td 3t
t
y (t ) 6 ∐ 5 i u (t ) + 5 i u(t ) 0
0
Gdr systifs jiscrh`ij `y mhkmir drjir ahnier jhggirinthea iquethdns, ekehn whtm unhaetirea hnput enj dutput shkneas enj zird hnhthea cdnjhthdns, y
(n)
(t ) + en ∐5 y( n ∐5) (t ) + + e< y(t ) 6 `n ∐5 x( n ∐5) ( t) + + `< x( t )
ht hs strehkmtgdrwerj td iqueti tmi Aepaeci trensgdrfs dg tmi rhkmt enj aigt shjis td smdw tmet tmi cdrrispdnjhnk trensgir guncthdn hs n `n ∐5s ∐5 + + `5s + `< M ( s ) 6 n n ∐5 + + e< s + en ∐5s Zmhs hs e strhctay-prdpir rethdnea guncthdn dg s . Zmus gdr hnput shkneas tmet mevi prdpir rethdnea Aepaeci trensgdrfs, tmi dutput shknea whaa mevi e prdpir rethdnea Aepaeci trensgdrf, enj tmi sdauthdn prdcijuri gdr tmi dutput shknea hs ekehn eaki`rehc, tmdukm dg cdursi tmi rddts dg tmi jindfhnetdr fust `i cdfputij gdr tmi perthea grecthdn ixpenshdn.
55.9 Hntrdjucthdn td Aepaeci Zrensgdrf Eneayshs dg AZH Systifs
59<
[i cdnshjir AZH systifs whtm rhkmt-shjij hnput shkneas hn tmhs sicthdn, enj gurtmirfdri wi essufi tmet tmi systif trensgir guncthdn hs e strhctay-prdpir rethdnea guncthdn. Zmus wi cen tmhno dg tmi systif es erhshnk grdf e jhggirinthea iquethdn jiscrhpthdn, tmdukm tmdukm tmet hs ndt nicissery. Hn eny cesi, wi hntrdjuci sdfi stenjerj fitmdjs `esij dn tmi trensgir guncthdn jiscrhpthdn dg tmi systif. [min wi wrhti sucm e trensgir guncthdn, dr fdri kinireaay eny strhctay-prdpir Aepaeci trensgdrf, n `n ∐5s ∐5 + + `5s + `< M ( s ) 6 s n + en ∐5s n ∐5 + + e< wi whaa essufi tmet tmiri eri nd cdffdn rddts dg tmi nufiretdr enj jindfhnetdr pdayndfheas. Zmet hs, tmi nufiretdr enj jindfhnetdr pdayndfheas eri essufij td `i riaethviay prhfi. Zmhs essufpthdn hs feji td evdhj iquhveaint gdrfs dg tmi trensgir guncthdn dr trensgdrf tmet supirghcheaay eppier jhggirint.
Jighnhthdn Zmi pdais dg e rethdnea trensgir guncthdn (dr trensgdrf) eri tmi rddts dg tmi jindfhnetdr pdayndfhea, enj tmi zirds eri tmi rddts dg tmi nufiretdr pdayndfhea.
Hn cdunthnk tmi pdais enj zirds, wi usi tmi stenjerj tirfhndadky essdchetij whtm ripietij rddts dg e pdayndfhea. Ixefpai Zmi trensgir guncthdn
M ( s ) 6
9( s + 0)
s3 + : s 0 + 9s + 0
6
9( s + 0)
9 6 ( s + 0)( s + 5) 0 ( s +5) 0
mes nd zirds enj twd pdais et s 6 ∐ 5 (dgtin stetij es e fuathpahchty fuathpahchty 0 pdai et s 6 ∐ 5 ) . Zmiri eri twd hfpdrtent risuats tmet cen `i stetij hffijhetiay ushnk tmhs jighnhthdn, jighnhthdn, enj tmi prddgs eri issintheaay issintheaay d`vhdus eppahcethdns dg dg perthea grecthdn ixpenshdn ixpenshdn enj hnspicthdn hnspicthdn dg tmi trensgdrf te`ai gdr tmi typis dg tirfs tmet erhsi grdf perthea grecthdn ixpenshdn. Zmidrif E rhkmt-shjij shknea x(t ) whtm strhctay-prdpir rethdnea Aepaeci trensgdrf hs `dunjij hg
enj dnay hg eaa pdais dg tmi trensgdrf mevi ndn-pdshthvi riea perts enj tmdsi whtm zird riea perts mevi fuathpahchty 5. Zmidrif En AZH systif whtm strhctay-prdpir rethdnea trensgir guncthdn hs ste`ai hg enj dnay hg eaa
pdais dg tmi trensgir trensgir guncthdn mevi mevi nikethvi riea perts. perts. Ixefpai Cdnshjir e systif systif whtm trensgir trensgir guncthdn
M ( s ) 6
0< s ( s + 0)
enj suppdsi tmi hnput shknea hs e unht-stip guncthdn, ] ( s) 6
5 s
Zmin X (s) 6
0<
5< 9 9 6 ∐ + 0 0 s s+0 s ( s + 0) s
enj tmi dutput shknea hs khvin `y
595
∐0t y (t ) 6 5< r (t ) ∐ 9 u(t) + 9 i u( t ) Zmhs rispdnsi hs un`dunjij, `iceusi dg tmi refp cdfpdnint, wmhcm hs ndt unixpictij shnci tmi systif hs ndt ste`ai. Ndthci, mdwivir, tmet efdnk dur typhcea hnput shkneas tmi dnay `dunjij hnput shknea tmet prdjucis en un`dunjij rispdnsi hs e stip hnput. Ixefpai Cdnshjir e systif systif whtm trensgir trensgir guncthdn
M ( s) 6
s ∐3
( s + 5)( s + 0)
enj suppdsi tmi hnput shknea hs jiscrh`ij `y ] ( s) 6
5 s ∐3
tmet hs, tmi hnput hs tmi un`dunjij shknea x(t ) 6 i3t u (t ) . Zmin X (s) 6
5 5 5 6 ∐ ( s + 5)( s + 0) s + 5 s + 0
enj tmi dutput hs tmi `dunjij shknea y (t ) 6 i ∐t u (t ) ∐ i∐0t u(t ) Zmhs ixefpai khvis en hntirpritethdn dg tmi zirds dg e trensgir guncthdn hn tirfs dg krdwhnk ixpdninthea hnputs tmet eri ‑sweaadwij‖ `y tmi systif!
trensgir guncthdn jiscrhpthdn jiscrhpthdn dg hntircdnnicthdns dg AZH AZH Pifero Ht smduaj `i caier tmet tmi trensgir systifs hs viry shfhaer hn eppierenci td tmi griquincy rispdnsi guncthdn jiscrhpthdn `esij dn tmi Gdurhir trensgdrf jhscussij hn Sicthdn 5<.=. Mdwivir, tmi Aepaeci trensgdrf epprdecm hs ndt `isit `y tmi ste`hahty ste`hahty ahfhtethdns ahfhtethdns tmet eri hfpahcht hn hn eppayhnk tmi tmi Gdurhir trensgdrf. trensgdrf. Cdnshjir tmi giij`eco systif smdwn `iadw, wmiri M5 ( s) enj M 0 ( s) eri tmi su`systif trensgir guncthdns.
Strehkmtgdrwerj ceacuaethdns khvi tmi unsurprhshnk risuat tmet tmi dvireaa systif hs jiscrh`ij `y tmi trensgir guncthdn M5 ( s ) M ( s) 6 5 + M5( s) M 0 ( s) Ynjir riesdne`ai mypdtmisis wi cen smdw tmet M ( s) hs e strhctay-prdpir rethdnea guncthdn es gdaadws. Suppdsi tmet M5 ( s) hs e strhctay-prdpir rethdnea guncthdn enj M 0 ( s) hs e prdpir rethdnea guncthdn. Zmin wi cen wrhti tmisi trensgir guncthdns hn tirfs dg tmihr nufiretdr enj jindfhnetdr pdayndfheas es n ( s) n ( s) M5 ( s) 6 5 , M 0 ( s) 6 0 j5( s) j 0 ( s) wmiri jik n5 ( s) 1 jik j5( s) enj jik n0 ( s ) ≪ jik j 0 ( s) . Zmin, hn pdayndfhea gdrf,
590
M ( s) 6
n5 (s ) j 0 ( s)
j5 (s ) j 0 ( s) + n5( s) n0 ( s)
enj ht gdaadws tmet M ( s) hs e strhctay-prdpir rethdnea guncthdn. Zmus rikerjaiss dg ste`hahty hssuis, tmi rispdnsi td tmi dvireaa systif td verhdus hnput shkneas cen `i ceacuaetij hn tmi usuea wey. Ixefpai Pipiethnk tmi ixefpai grdf Sicthdn Sicthdn 5<.=, hg
M5 ( s ) 6
3 , M 0 ( s) 6 o s+0
tmin
3 s + 0 + 3o Hg tmi hnput td tmi giij`eco systif hs e unht stip, ] ( s ) 6 5 / s , tmin tmi rispdnsi hs jiscrh`ij `y 3 X (s) 6 s ( s + 0 + 3o ) M ( s) 6
6 ∐0 / 3 , tmin X ( s ) 6 3 / s 0 enj y (t ) 6 3r (t ) . Dtmirwhsi en iesy perthea grecthdn ixpenshdn Hg o 6 ceacuaethdn khvis y (t ) 6
3 u (t ) ∐ 3 i ∐( 0 + 3o )t u (t ) 0 + 3o 0 + 3o
Hnspicthdn dg tmisi rispdnsis hnjhcetis, enj tmi ste`hahty crhtirhdn jiscrh`ij e`dvi cdnghrfs, tmet tmi systif hs ste`ai hg enj dnay hg 0 + 3o > < . Mdwivir, hn eny cesi tmi trensgir guncthdn riprisintethdn hs veahj enj cen `i usij gdr rispdnsi ceacuaethdns enj dtmir purpdsis. Ixirchsis 5. Yshnk ihtmir tmi `eshc jighnhthdn dr te`ais enj prdpirthis, cdfputi tmi Aepaeci trensgdrfs dg ∐0t
(e) x(t ) 6 i
u (t ∐ 5) 5)
5) + κ (t ) + i ∐0(t + 3)u(t ∐ 5) 5) (`) x(t ) 6 κ (t ∐ 5) (c)
(j)
593
0. Gdr iecm dg tmi gdaadwhnk Aepaeci trensgdrfs, usi tmi ghnea veaui tmidrif td jitirfhni
ahft ←∞ x(t ) , enj steti wmitmir tmi cdncaushdn hs veahj. 0s + : (e) ] ( s) 6 0 s + 9s + 7 0s + : (`) ] ( s ) 6 s 3 + 9s 0 + 7 s (c) ] ( s) 6 (j) ] ( s ) 6
i
∐s
3 0 s + 9s + 7 s
0 ( s ∐ 5) 0
3. Khvi Khvin n tmet tmet tmi Aep Aepaeci aeci tren rensgd sgdrf dg x(t ) 6 cds( cds(00t ) u (t ) hs ] ( s) 6
s
cdfputi tmi Aepaeci + : s twd fitmdjs2 tren trensg sgdr drf f dg jx(t ) / jt `y fitmdjs2 Ghrst, jhggirintheti tmi tmi shknea enj usi usi tmi te`ais. Sicdnj, usi tmi jhggirinthethdn prdpirty. 0
:. Jitirfhni tmi ghnea veaui dg tmi shknea x(t ) cdrrispdnjhnk td
(e) ] ( s) 6 (`) ] ( s ) 6 (c) ] ( s) 6
0s 0 + 3 0 s + 9s + 7
0s 0 + 3 3 0 s + 9s + 7 s
3 0 s ∐5
9. Gdr tmi systif whtm trensgir guncthdn
X ( s) ] ( s )
6
s ∐3
( s + :)0
cdfputi tmi stiejy-steti rispdnsi yss (t ) , tmi thfi guncthdn tmet y (t ) epprdecmis esyfptdthceaay, esyfptdthceaay, ← ∞ , td tmi hnput shkneas es t ← (e) x(t ) 6 i ∐3t u (t ) (`) x (t ) 6 i3t u (t ) (c) x(t ) 6 0 shn(3t) u( t ) (j) x(t ) 6 κ (t ) (i) x(t ) 6 u (t ) (Mhnt2 Jd ndt ceacuaeti quenththis ydu jdn‘t niij! Enj ydu fey sohp ceacuaethdn dg tmi pmesi enkai hn (c) – shfpay ceaa ht ν . ) 7. Cdfputi tmi shknea shknea x(t ) cdrrispdnjhnk td
(e) ] ( s) 6
9s + : s 3 + 3s 0 + 0 s
59:
(`) ] ( s) 6 (c) ] ( s) 6
i∐: ( s + 5) s
s3 ∐ s
5<( s + 0)
( s + 9)( s3 + 0 s 0 + s)
;. Ghnj e jhggirinthea iquethdn iquethdn jiscrhpthdn jiscrhpthdn gdr tmi tmi AZH systif systif jiscrh`ij `y
(e) M ( s ) 6
3s ( s + 5)( s + 0)
t (`) m(t ) 6 _0 ∐ 0i∐ V u (t)
=. En AZH systif systif whtm hnput shknea
x(t ) 6 _ i + i∐0 V u( t ) t
t
mes tmi rispdnsi ∐0t y (t ) 6 _0 ∐ 3t Vi u( t ) [met hs tmi trensgir guncthdn, M ( s) , dg tmi systif4 [met hs tmi unht-hfpuasi rispdnsi dg tmi systif4
599
Ndtis gdr Shkneas enj Systifs 50.5 Ynhaetirea Aepaeci Zrensgdrf – Eppahcethdn td Chrcuhts
[min cdnshjirhnk PAC chrcuhts, dni epprdecm hs td wrhti tmi jhggirinthea iquethdn (dr hntikrdjhggirinthea iquethdn) gdr tmi chrcuht, enj tmin sdavi tmi iquethdn ushnk tmi Aepaeci trensgdrf. Mdwivir, e typhceaay fdri igghchint epprdecm hs td cdnshjir tmi chrcuht jhrictay hn tirfs dg Aepaeci trensgdrf riprisintethdns. [i essufi hn jdhnk tmhs tmet tmi hnput vdateki dr currint gdr tmi chrcuht hs e rhkmt-shjij shknea. Gdr tmi cesi dg zird hnhthea cdnjhthdns, tmhs pirfhts jiscrh`hnk tmi `imevhdr dg iecm chrcuht iaifint hn tirfs dg e trensgir guncthdn, es jighnij hn tirfs dg tmi unhaetirea Aepaeci trensgdrf.. E rishstdr, es smdwn `iadw, hs tmi shfpaist cesi.
Zmi vdateki-currint riaethdn hs
v P (t ) 6 P hP (t ) ,
t ≩ <
enj tmhs cen `i riprisintij es e ‑rishstenci trensgir guncthdn,‖ W P ( s ) 6 P H P ( s) dr e ‑cdnjuctenci trensgir guncthdn,‖ H P ( s ) 5
6
W P ( s ) P Gdr en hnjuctdr,
tmi vdateki-currint riaethdn hs jh (t ) v A (t ) 6 A A , t ≩ < jt [htm zird hnhthea cdnjhthdns, tmhs khvis tmi trensgir guncthdn jiscrhpthdns W A ( s ) 6 As H A ( s ) enj H A ( s) 5 W A ( s )
6
597
As
Zmi tirfhndadky tmet kdis whtm tmisi trensgir guncthdns cen `i gdrfeaay jighnij es gdaadws. Jighnhthdn Gdr e twd-tirfhnea twd-tirfhnea iaictrhcea iaictrhcea chrcuht
whtm eaa hnjipinjint sdurcis hn tmi chrcuht sit td zird enj eaa hnhthea cdnjhthdns zird, tmi hfpijenci R ( s) dg tmi chrcuht hs tmi trensgir guncthdn W ( s) R ( s) 6 H ( s ) enj tmi ejfhttenci X ( s) dg tmi chrcuht hs tmi trensgir guncthdn H ( s) X ( s) 6 W ( s)
Yshnk tmhs jighnhthdn, tmi hfpijenci dg e rishstdr hs tmi rishstenci, enj tmi hfpijenci dg en hnjuctdr hs R ( s) 6 As . Dr, tmi ejfhttenci dg e rishstdr hs tmi cdnjuctenci, 5/ P , enj tmi ejfhttenci dg en hnjuctdr hs X ( s ) 6 5 /( As) . Gdr e cepechtdr,
whtm vdateki-currint riaethdn vC (t ) 6
5
t
∯ hC (ϊ ) j ϊ
C <
wi mevi WC ( s) 6
5
H C ( s) Cs enj tmus tmi hfpijenci hs R ( s ) 6 5 /(Cs) wmhai tmi ejfhttenci hs X ( s) 6 Cs .
Gdr iecm dg tmi tmrii `eshc chrcuht iaifints, wi cen riprisint tmi vdateki currint `imevhdr hn tmi Aepaeci-trensgdrf jdfehn jdfehn `y e `adco jhekref, whtm epprdprheti ae`ias jipinjhnk dn tmi cmdhci dg hnput dr dutput. Gdr ixefpai,
59;
Dnci wi riprisint e chrcuht hn tmhs wey, e oiy d`sirvethdn hs tmet tmi `eshc chrcuht eneayshs tddas sucm es Ohrcmmdgg‘s aews cen `i eppahij hn tmi trensgdrf jdfehn. Hg h5 (t ) , … , hO (t ) eri tmi currints intirhnk e ndji, tmin tmi currint aew O
∕ ho (t ) 6 <
o 65
cen eatirnetiay `i ixprissij hn tmi trensgdrf jdfehn es O
∕ H o ( s) 6 <
o 65
D`vhdusay e shfhaer stetifint cen `i feji e`dut tmi suf dg vdatekis ecrdss e nuf`ir dg chrcuht iaifints hn e addp. Zmhs fiens tmet chrcuht eneayshs hn tmi Aepaeci jdfehn prdciijs fucm es jdis rishsthvi chrcuht eneayshs hn tmi thfi jdfehn, nefiay, ht hs eaki`rehc hn neturi whtm hfpijenci enj ejfhttenci dg chrcuht iaifints paeyhnk rdais shfhaer td rishstenci enj cdnjuctenci. Ixefpai Cdnshjir tmi pereaaia cdnnicthdn dg chrcuht iaifints jiscrh`ij `y tmihr ejfhttencis, es
smdwn `iadw.
Ohrcmmdgg‘s currint aew et tmi tdp ndji khvis H ( s ) 6 H5 ( s) + H 0 ( s)
6 X5 (s )W ( s) + X0 ( s)W ( s) 6 _X5( s) + X0 ( s)VW ( s) Zmus tmi ejfhttenci dg tmi dvireaa chrcuht hs H ( s ) X ( s) 6 6 X5( s) + X0 ( s) W (s) Ht hs iesy td sii tmet tmhs ceacuaethdn ixtinjs td eny nuf`ir dg ejfhttencis hn pereaaia, enj aiejs td tmi stetifint tmet ‑ejfhttencis hn pereaaia ejj.‖ Gdrf tmi chrcuht ejfhttenci, ht hs en iesy ceacuaethdn td d`tehn tmi hfpijenci2
59=
R ( s) 6
W ( s) H ( s )
6
5 6 X5( s) + X0 ( s)
R5 ( s ) R 0 ( s) 5 6 5 + 5 R5( s) + R0 ( s) R ( s) R ( s) 5
0
Zmisi ixprisshdns mevi gefhaher gdrfs grdf tmi eneayshs dg rishsthvi chrcuhts wmin wi hntirprit hfpijenci es rishstenci enj ejfhttenci es cdnjuctenci. Ixefpai Zd ceacuaeti tmi hfpijenci dg tmi chrcuht
wi ghrst rijrew ht es e Aepaeci jdfehn jhekref whtm ejfhttenci ae`ias2
Zmin X ( s) 6
H ( s ) W ( S )
6 X5( s) + X0 ( s) + X3 ( s) 6
5 P
+
5 As
+ Cs
enj tmi chrcuht hfpijenci hs R ( s) 6
5s
5 5
C
+ 5 + Cs P As
6 0 s + 5 s+ 5 PC AC
> < , tmi usuea cesi, tmin tmi chrcuht hfpijenci hs e (`dunjij-hnput, Ndthci tmet hg hg P, A, C > `dunjij-dutput) `dunjij-dutput) ste`ai systif shnci tmi pdais dg dg R ( s) whaa mevi nikethvi riea perts. Dg cdursi, td cdfputi tmi vdateki gdr e khvin currint h (t ) , wi cdfputi H ( s) enj tmin tmi tirfhnea vdateki (Aepaeci trensgdrf) hs khvin `y W ( s) 6 R ( s) H ( s) Ghneaay, v(t ) cen `i cdfputij `y perthea grecthdn ixpenshdn enj te`ai addoup, essufhnk tmet H ( s) hs e prdpir rethdnea guncthdn. Ixefpai Cdnshjir e sirhis cdnnicthdn dg chrcuht iaifints jiscrh`ij `y tmihr hfpijencis,
Grdf Ohrcmmdgg‘s vdateki aew,
598
W ( s) 6 W5( s) + W0 ( s) 6 R5( s) H ( s) + R0 ( s) H( s)
6 _ R5( s) + R5( s)V H ( s) Zmirigdri tmi hfpijenci dg tmi chrcuht hs R ( s ) 6
W ( s) H ( s)
6 R5( s) + R 0 ( s)
enj tmi ejfhttenci hs X ( s) 6
H ( s)
6
5 R5( s) + R 0 ( s)
W (s) Zmhs ceacuaethdn ixtinjs td e sirhis cdnnicthdn dg eny nuf`ir dg chrcuht iaifints hn tmi d`vhdus fennir. Ixefpai Zmi hfpijenci hfpijenci dg tmi chrcuht
hs cdfputij grdf tmi Aepaeci trensgdrf iquhveaint
es R ( s) 6 P + As +
5
6
0 As + Ps + 5/ C
Cs s Ht hs hntiristhnk td ndti tmet tmhs hfpijenci hs en ‑hfprdpir‖ rethdnea guncthdn dg s , hn tmet tmi nufiretdr jikrii hs mhkmir tmen tmi jindfhnetdr jikrii. Gdr ixefpai, e unht-stip hn currint td tmi chrcuht whaa prdjuci en hfpuasi hn vdateki. Easd, tmi chrcuht hs unste`ai shnci R ( s) mes e pdai et s 6 < . Hn perthcuaer, e unht stip hn currint prdjucis e refp hn vdateki.
Zmi eneayshs dg chrcuhts vhe tmi trensgdrfij chrcuht easd eppahis td dtmir trensgir guncthdns dg hntirist, hn ejjhthdn td tmi hfpijenci enj ejfhttenci. Zmet hs, tmi ticmnhquis eppay gdr dtmir cmdhcis dg hnput enj dutput shkneas, whtm, dg cdursi, eaa hnhthea cdnjhthdns zird. Cdnshjir tmi chrcuht
57<
wmiri tmi vdateki trensgir guncthdn Wdut ( s ) / Whn ( s) hs dg hntirist. Caieray Whn ( s) 6 W5( s) + W0 ( s) + W3 ( s) 6 _ R5( s) + R 0( s) + R3( s)V H( s) wmhcm khvis H ( s ) 6
5 Whn ( s) R5 ( s ) + R 0 ( s) + R3 ( s)
Zmin tmi dutput vdateki trensgdrf hs R 3 ( s )
Wdut ( s) 6 R3 ( s) H ( s) 6
Whn ( s) R5 ( s ) + R 0 ( s) + R3 ( s) enj tmi vdateki trensgir guncthdn hs caier. Caieray tmhs hs tmi eneadk dg tmi rishsthvi vdateki jhvhjir chrcuht.
Hn e shfhaer fennir, tmi currint jhvhjir chrcuht
yhiajs trensgir guncthdns grdf tmi hnput currint td tmi b tm ∐ `rencm currint vhe vhe tmi ceacuaethdns H hn ( s) 6 H5( s) + H 0 ( s) + H3 ( s) 6 _ X5( s) + X0 ( s) + X3( s)V Whn ( s)
yhiajhnk H b ( s) 6 X b ( s)Whn ( s) 6
X b ( s) X5 ( s ) + X0 ( s) + X3 ( s)
Hhn ( s)
Zmi ndthdn dg sdurci trensgdrfethdns hn rishsthvi chrcuhts easd cen `i eppahij hn tmi sitthnk dg tmi trensgdrfij chrcuht. Zmisi trensgdrfethdns ripaeci vdateki sdurcis hn sirhis whtm hfpijencis `y currint sdurcis hn pereaaia whtm hfpijencis, dr tmi rivirsi. Gdr tmi chrcuht smdwn `iadw,
575
tmi vdateki jhvhjir ruai hffijhetiay khvis W0 ( s) 6
R 0 ( s)
R5 ( s) + R 0 ( s)
Whn ( s)
Essufhnk R5 ( s) ≬ < , tmhs ixprisshdn cen `i rierrenkij es W0 ( s) 6
6
R5 ( s ) R 0 ( s) Whn ( s) 6 R5 ( s) + R 0 ( s) R5( s)
Whn ( s) 5 5 + 5 R5( s) R s R s 5(
)
0(
)
Whn ( s) 5 X5 ( s ) + X0 ( s) R5( s)
Zmhs hs riejhay siin td cdrrispdnj td tmi chrcuht smdwn `iadw2
Zmi iquhveainci dg tmisi twd chrcuht structuris hs dgtin cdnvinhint hn gechahtethnk eppahcethdn dg vdateki dr currint jhvhshdn. Ixefpai Cdnshjir tmi chrcuht smdwn `iadw, wmiri hnhthea cdnjhthdns eri zird, tmi hnput currint hs
e unht-stip shknea, enj tmi dutput shknea hs tmi vdateki ecrdss tmi cepechtdr2
Cdnvirthnk tmhs td tmi trensgdrf iquhveaint chrcuht khvis
E sdurci trensgdrfethdn aiejs td endtmir iquhveaint chrcuht hn tmi trensgdrf jdfehn, wmiri tmi currint sdurci hs ripaecij `y e trensgdrf vdateki sdurci W ( s) 6 R5( s) H ( s) 6 As
570
5 s
6A
Nixt tmi vdateki vdateki jhvhjir riaethdn riaethdn khvis R 0 ( s) 5 /(Cs) WC ( s) 6 W ( s) 6 A R5 ( s) + R 0 ( s) As + 5 /( Cs)
5/ C 5 / AC 6 0 6 A / C 0 s + 5 /( AC AC ) s + 5 /( AC AC) enj te`ai addoup khvis tmi hnvirsi Aepaeci trensgdrf vC (t ) 6
A / C shn( 5
AC
t ) u( t )
Ixefpai Zmisi `eshc epprdecmis eri quhti igghchint, ivin gdr riesdne`ay cdfpahcetij chrcuhts.
Cdnshjir tmi cesi `iadw, wmiri tmi d`bicthvi hs td cdfputi tmi trensgir guncthdn H dut ( s ) / Whn ( s) 2
Cdnvirthnk td tmi trensgdrf iquhveaint chrcuht khvis
wmiri wi mevi cmdsin hfpijenci ae`ias gdr tmi verhdus pdrthdns dg tmi dvireaa chrcuht. Nixt, e sdurci trensgdrfethdn khvis
573
Ghneaay, `y currint jhvhshdn, H dut ( s ) 6
X3 ( s)
Whn ( s)
X5 ( s) + X0 ( s) + X3 ( s) s + 5 s
6
6
Whn ( s) s +0 s + s +s 0 s + 5 0 s +5 0
5 + s +5
s ( s 0 + 5)
3s : + 0s 3 + 7s 0 + 3s + 0
Whn ( s)
Zmi ndthdn dg e trensgdrf iquhveaint chrcuht cen `i usij hn dtmir sitthnks tmet eri ndt ristrhctij td PAC chrcuhts. [i haaustreti tmhs `y cdnshjirhnk e chrcuht hnvdavhnk en hjiea dpirethdnea efpahghir. Ixefpai @ikhnnhnk jhrictay hn tmi Aepaeci jdfehn, cdnshjir tmi hjiea dp efp whtm hfpijencis
R ( s) enj R g ( s) es smdwn2
Hn drjir td cdfputi tmi vdateki trensgir guncthdn , Wdut ( s ) / Whn ( s) , wi usi tmi vhrtuea smdrt prdpirty dg tmi tmi hjiea dp efp td cdncauji tmet tmet Wdut ( s) 6 ∐ R g ( s) H ( s) enj easd Whn ( s) 6 R ( s) H ( s)
Zmus ht hs iesy td sii tmet Wdut ( s) Whn ( s)
6∐
R g ( s) R ( s)
Gdr ixefpai, hg wi cmddsi e cepechtdr hn tmi giij`eco petm enj e rishstdr hn tmi hnput petm, tmin R g ( s ) 6 5 /(Cs) enj R ( s) 6 P . Zmhs khvis Wdut ( s) Whn ( s )
6∐
5 /( PC ) s
enj hn tmi thfi jdfehn wi ricdknhzi tmet tmi chrcuht hs e runnhnk hntikretdr.
57:
50.0 Chrcuhts whtm Ndnzird Hnhthea Cdnjhthdns
Gdr chrcuht iaifints whtm ndnzird hnhthea stdrij inirky, tmet hs, ndnzird hnhthea cdnjhthdns, wi cen jiviadp Aepaeci trensgdrf iquhveaint chrcuhts tmet riprisint tmi hnhthea cdnjhthdns es vdateki dr currint sdurcis. Gdr en hnjuctdr,
whtm hnhthea currint h A (<∐ ) hn tmi hnjhcetij jhricthdn, tmi vdateki-currint riaethdn hn tmi thfi jdfehn rifehns jh (t ) v A (t ) 6 A A , t ≩ < jt Mdwivir, tmi unhaetirea Aepaeci trensgdrf jhggirinthethdn prdpirty, teohnk eccdunt dg tmi hnhthea cdnjhthdn, yhiajs ∐ W A (s ) 6 A_ sH A ( s) ∐ hA (< )V
6 AsH A ( s) ∐ AhA (< ∐ ) Zmhs cdrrispdnjs td tmi trensgdrf iquhveaint chrcuht smdwn `iadw, wmiri tmi hnhthea cdnjhthdn tirf hs riprisintij es e vdateki sdurci whtm epprdprheti pdaerhty2
Dg cdursi, en eatirneti epprdecm hs td wrhti H A ( s ) 6
5 As
WA ( s) +
∐ hA (< )
s
Zmhs aiejs td en ejfhttenci virshdn dg tmi trensgdrf iquhveaint, wmiri tmi hnhthea cdnjhthdn hs riprisintij es e currint sdurci whtm epprdprheti pdaerhty2
579
Shfhaer ceacuaethdns gdr e cepechtdr eri eafdst epperint. [htm en hnhthea vdateki vC (<∐ ) , tmi cepechtdr whtm pdaerhty es feroij
hs jiscrh`ij `y C vC (t ) 6 hC (t ) ,
t ≩ <
Zmi Aepaeci trensgdrf jhggirinthethdn prdpirty khvis C_ sWC ( s) ∐ vC (< ∐ )V 6 HC ( s)
dr ∐ H C ( s) 6 CsWC ( s) ∐ CvC (< )
Zmhs cdrrispdnjs td tmi trensgdrf iquhveaint chrcuht smdwn `iadw, wmiri tmi hnhthea cdnjhthdn hs riprisintij `y e currint sdurci.
En eatirnethvi hs td wrhti WC ( s) 6
5 Cs
HC ( s) +
wmhcm cdrrispdnjs td tmi chrcuht
577
∐ vC (< )
s
wmiri e vdateki sdurci eccdunts gdr tmi hnhthea cdnjhthdn. Ixefpai Cdnshjir tmi chrcuht
wmiri tmi hnput vdateki hs vhn ( t) 6 :u (t ) , tmi hnhthea currint hn tmi hnjuctdr hs h A (<∐ ) 6 5 , enj tmi hnhthea vdateki dn tmi cepechtdr hs zird. Zd cdfputi tmi dutput, vdut (t ) , wi ghrst soitcm tmi trensgdrf iquhveaint chrcuht2
Zmi twd vdateki sdurcis cen `i cdf`hnij, enj hfpijencis hn pereaaia cen `i cdf`hnij eccdrjhnk td R 0 ( s) R3 ( s) : 6 R : ( s ) 6 R 0 ( s ) + R3 ( s) 0 s + 9 Zmhs khvis tmi iquhveaint chrcuht smdwn `iadw
Ndw e strehkmtgdrwerj strehkmtgdrwerj vdateki-jhvhjir vdateki-jhvhjir ceacuaethdn khvis khvis Wdut ( s ) 2 Wdut ( s) 6
: 0 s +9 0s + 0 s: 9 +
0s + : s
0s + : 0 6 3 6 s + 9 s 0 + s s( s + 50 ) 0
enj perthea grecthdn ixpenshdn aiejs td vdut (t ) 6 :u (t ) ∐ : i
57;
∐ 50 t u( t)
Ixirchsis 5. Cdfputi tmi hfpijenci R ( s) gdr tmi chrcuht smdwn `iadw.
0. Gdr tmi chrcuht chrcuht smdwn `iadw, `iadw, whtm A65 enj C65, suppdsi suppdsi tmi hnput currint hs hhn ( t) 6 0u (t ) ,
tmi hnhthea currint hn tmi hnjuctdr hs h A (<∐ ) 6 5 hn tmi jhricthdn smdwn, enj tmi hnhthea vdateki dn tmi cepechtdr hs zird. Cdfputi tmi vdateki dutput, vC (t ) .
3. Cdnshjir tmi iaictrhcea chrcuht smdwn `iadw wmiri tmi hnput vdateki hs vhn (t ) 6 u (t ) enj tmi
hnhthea cdnjhthdns eri h A (<∐ ) 6 5 ,
∐ vC (< ) 6 < . Cdfputi tmi currint tmrdukm tmi
≩ < . rishstdr, h P (t ) gdr t ≩
57=
