Основы теории сигналов. ЛК 1

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go KHH3H£ xHlnaxcdatfoo ‘HHHatfaao ognu-xroiax axooHuAxoaoo — Bnnai\ido(|)H||
xoiaxAu oxoan aiadoxox ‘«aiaHHBtf» AHMwdax axcxax a ‘«BHtiKwdot[)HH» AHMwdax OHXBHOU axau xaAtrauo OHauaxMdaauadu ‘auaHJHO OHXKHOU axodxowooad xax ‘wax trodou ‘afxox
KM1KHOU M BMH3U3aO 3MtTigO V
iiiiniHMdo(|)Hii
AooHodou x axooHgooouo oxe — wauaHxno wooa oolnAoHdu ‘oaxonoao oolngo OHfQ f x H HWH.MOOhHuaadUHj ‘HWHxoohHxoAxa ‘HWHxoohHdxxoue — HwiaHead XAJOW iauaHJH'g
axxaago ojowaAuauooH xo HiaHHaxcadxo xAae KOxauuaK WOUBHJHO auoxooxxacfiaU OJOxoahHxoAxa KUU ‘UBHJHO HiaHHOHenaauax Koxanuau WOUBHJHO adoenaauax KUU ‘xauAtn OJ3 EH SHHSxcKduaH K3X3KU3K WOUBHJHO adxswxauoa KUU ‘dswnduag Hxxogadgo XH H HWXUBHJHO o iaxogad KUU iaHXKoo xoiaxogad iadogndu sag KHHSodxoodogHdu nxoaugo a adsHSxcHH BUU iaHxcaa xax axxogadgo XH H aouBHJHX KMdoax Awanou axKHOu owHUoxgoaH oxaoa 3toK3dx[

3MH3U3aa

npHMeHHMH B peaabHHM CHrHa^aM, xOTa u C HeKOTOpbIMu OrOBOpKaMu.
^aaMu o6pa6oTKu u aHaau3a curHaaoB o6waHO aBaaroTca:
• OnpegeaeHue uau o^HKa aucaoBbix napaMeTpOB curHaaoB ^Heprua, cpegHaa MO^HOCTb, cpegHee KBagpaTuaecKoe 3HaaeHue u np.);
• H3yaeHue u3MeHeHua napaMeTpOB curHaaoB BO BpeMeHu;
• Pa3ao*eHue curHaaoB Ha ^aeMeHTapHbIe cocraBaaro^ue gaa cpaBHeHua CBOHCTB pa3auaHwx curHaaoB;
• CpaBHeHue creneHu 6au3ocru, "noxo^ecru", "pogcrBeHHocru" pa3auaHwx curHaaoB, B TOM aucae c onpegeaeHHbiMu KoauaecrBeHHbiMu o^HKaMu.
MaTeMaTuaecKuH annapaT aHaau3a curHaaoB BecbMa o6mupeH, u mupoKO npuMeHaeTca Ha npaKTuKe BO Bcex 6e3 ucKaroaeHua o6aacrax HayKu u TexHuKu.

2. KHACCMOM^Mfl CMrHAHOB

2.1 no pa3MepHOCTH
Pa3MepHOCTb curHaaa - ^T0 aucao He3aBucuMbix nepeMeHHbix, no KOTopwM
onpegeaaeTca ero 3HaaeHue.
Eo^bmuHCTBO curHaaoB, paccMarpuBaeMwx B reopuu, aBaaroTca OflHOMepHWMH u uMerar Bug s(x). HanpuMep, 3aBucuMOCTb Hanpa^eHua OT BpeMeHu, aMnauTygw OT aacroTbi T.g.
KpoMe Toro cy^ecrayeT TaK*e gocraToaHO 6oabmoH Kaacc gByMepHbix curHaaoB Buga s(x,y). KaaccuaecKuM npuMepoM gByMepHoro curHaaa aBaaeTca aro6oe naocKoe u3o6pa*eHue uau pacnpegeaeHue KaKOH-au6o BeauauHbi no reorpa^uaecKuM KOopguHaTaM.
TaK*e uHorga BcrpeaaroTca TpexMepHbie curHaaw, aa^e Bcero, Korga peab ugeT 06 onpegeaeHuu 3HaaeHua KaKOH-au6o BeauauHbi B pa3Hwx ToaKax npocTpaHcTBa, HanpuMep TpaeKTopuu gBu^eHua caMoaeTa uau KocMuaecKoro Kopa6aa.
Teopua gonycKaeT cy^ecrBOBaHue TaK*e MHoroMepHwx curHaaoB, a KpoMe Toro pacmupaeT MHorue MeTogbi o6pa6oTKu Ha ^T0T cayaaH. HO B npaKTuaecKOH geaTeabHocTu, TaKue cayaau BcrpeaaroTca pegKO.
2.2 no HenpepwBHOCTH
Hro6oM curHaa onpegeaeHHbie BO3MO*Hbie 3HaaeHua Ha onpegeaeHHOM npocTpaHcTBe 3HaaeHuH He3aBucuMOH nepeMeHHOH. KaK 3HaaeHua, TaK u He3aBucuMaa nepeMeHHaa Moryr 6wTb au6o HenpepwBHHMu, au6o gucKpeTHbiMu.
HenpepwBHocTb — CBOHCTBO, 3aKaroaaro^eeca B nocTeneHHOM, naaBHOM, 6e3 cKaaKOB u3MeHeHuu 3HaaeHuH KaKOH-au6o nepeMeHHOH, $yH^uu uau gpyroro MaTeMaTuaecKoro o6beKTa.
^HCKpeTHocTb — CBOHCTBO, npoTuBonocraBaaeMoe HenpepwBHocru, npepwBHocrb.
AHaaoroBWH (HenpepbiBHbirt) cnrHaa - curHaa, 3HaaeHua u He3aBucuMaa nepeMeHHaa KOToporo aBaaroTca HenpepwBHHMu MHO^ecraaMu BO3MO*HHX 3HaaeHuH.
^HCKpeTHHH cnrHaa - curHaa, He3aBucuMaa nepeMeHHaa KOToporo onpegeaeHa Ha
gucKpeTHOM MHO^ecTBe, a 3HaaeHua aBaaroTca HenpepwBHHMu.

curHaa - curHaa, 3HaaeHua KOToporo gucKpeTHw, a He3aBucuMaa nepeMeHHaa HenpepwBHa.
U,H^poBort curHaa — curHaa gaHHwx, y KoToporo Ka^gwfi u3 npegcraBaaro^ux napaMeTpoB onucwBaeTca ^yH^uefi gucKpeTHoro BpeMeHu u KoHeaHWM MHO^CTBOM BO3 MQSHHX 3HaaeHufi.

CooTBeTcTBeHHo, MO^HO onpegeauTb caegyro^ue npeo6pa3oBaHua curaaaoB: ^HCKpeTH3a^HH — npogecc npeo6pa3oBaHua aHaaoroBoro curHaaa B gucKpeTHwfi. KBaHTOBaHue — npeo6pa3oBaHue aHaaoroBoro curHaaa B KBaHToBaHHwfi.
Om«l)poBKa — npeo6pa3oBaHue aHaaoroBoro curHaaa B gu^poBofi.
BoccraHoBaeHue — npeo6pa3oBaHue curHaaa u3 gucKpeTHoro uau ^u$poBO^o B aHaaoroBwfi.

2.3 no BHgy MaTeMaTunecKofi Mogeau
nocKoabKy curHaaw nopo^garoTca $u3unecKuMu npo^ccaMu, a Te B CBOW onepegb MoryT uMeTb KaK caynafiHwfi, TaK u npegcKa3yeMwfi xapaKTep, curHaaw TaK*e MoryT 6wTb pa3Horo Tuna. CooTBeTcTBeHHo oTaunaroTca u Mogeau, co3gaBaeMwe gaa KoppeKTHoro aHaau3a curHaaa.
Ecau MaTeMaTunecKaa Mogeab no3BoaaeT TOHHO onpegeauTb 3HaneHue curHaaa B aro6ofi TonKe, TaKaa Mogeab u TaKofi curHaa Ha3WBaroTca geTepMHHupoBaHHbiMH. HanpuMep, npu aHaau3e Hanpa^eHua B ^aeKTpuHecKofi ceTu uMeeT cMwca ucnoab3oBaTb Mogeab
Buga s(t) = sin(rnt + p).
Ecau curHaa HOCUT caynafiHwfi xapaKTep, TO roBopaT o Haauauu caynafiHoro (cTOxacTunecKoro) curHaaa. MaTeMaTuaecKaa Mogeab B TaKoM cayaae 3agaeTca B Buge 3aKoHa pacnpegeaeHua BepoaTHocrefi, Koppeaa^uoHHofi $yH^uu, cneKTpaabHofi naoTHocru ^Hep^uu u gp.

CurHanbi

(fleTepMMHMpoBaHHbie

CnyMaPiHbie

(CnyMaPiHbie noMexn) (J~lcme3Hbie cnrHanbi)

riepMQflMMecKMe j

(J~apMQHHMecKHe) (Tlo/iMrapMOHMMecKMej (floMTM nepMQflMMecKMe] (AnepMQflMMecKHe) (CiauMQHapHbie) (HecTauHOHapHbie)

Puc. 2. KaaccmlmKamiH curHaaoB.

MHO^ecTBy nepHognaecKHx OTHOCAT rapMoHuaecKue u nonurapMoHuaecKue curHanw. ^na nepuoguaecKux curHanoB BbinonHaeTca o6^ee ycnoBue s(t) = s(t + kT), rge k = 1, 2, 3, ... - nro6oe ^noe aucno (u3 MHo^ecraa ^nbix aucen OT -ro go ro), T - nepuog, aBnaro^uaca KoHeaHbiM oTpe3KoM He3aBucuMoa nepeMeHHoa
r apMOHHnecKHe cnrHa^bi (cuHycouganbHbie), onucwBaroTca cnegyro^uMu $opMynaMu:
s(t) = A\sin (2rnfot+§) = A\sin (rnot+fy), s(t) = A*cos(wot+y),

rge A, fo, roo, ^, 9 - nocroaHHbie BenuauHbi, KoTopwe MoryT ucnonHaTb ponb uH^opMa^uoHHHx napaMeTpoB curHana: A - aMnnmyga curHana, fo - ^uKnuaecKaa aacroTa B rep^x, roo = 2nfo - yrnoBaa aacroTa B paguaHax, 9 u ^- HaaanbHbie we yrnw B paguaHax. nepuog ogHoro iaHua T = 1/fo = 2n/roo. npu 9 = ^-TC/2

cuHycHwe u KocuHycHbie $yH^uu onucwBaroT oguH u TOT *e curHan.
no^nrapMOHHHecKHe cnrHa^bi cocraBnaroT Hau6onee mupoKo pacnpocrpaHeHHyro rpynny nepuoguaecKux curHanoB u onucwBaroTca cyMMoa rapMoHuaecKux Kone6aHua:
N N
a a
s(t) = n_0 An sin (2rofnt+9n) = n_0 An sin (2roBnfpt+9n), Bn e I, (1.1.2) unu HenocpegcraeHHo $yH^uea s(t) = y(t ± kTp), k = 1,2,3,..., rge Tp - nepuog ogHoro nonHoro Kone6aHua curHana y(t), 3agaHHoro Ha ogHoM nepuoge. 3HaaeHue fp =1/Tp Ha3biBaroT ^yHgaMeHTanbHon aacroToa Kone6aHua.

nonurapMoHuaecKue curHanw npegcraBnaroT co6oa cyMMy onpegeneHHoa nocroaHHoa cocraBnaro^ea (fo=0) u npou3BonbHoro (B npegene - 6ecKoHeaHoro) aucna rapMoHuaecKux cocraBnaro^ux c npou3BonbHHMu 3HaaeHuaMu aMnnuTyg An u $a3 9n, c aacroTaMu, KpaTHHMu ^yHgaMeHTanbHoa aacToTe fp. ^pyruMu cnoBaMu, Ha nepuoge $yHgaMeHTanbHoa aacroTbi fp,

nepuogoB Bcex rapMOHUK, HTO u co3gaeT nepuoguHHocrb noBTopeHua curHana.

2.3.2. HenepnoflHHecKHe cnrHanw
K HenepnoflHHecKHM curHanaM OTHOCHT noHTu nepuoguHecKue u anepuoguHecKue curaanbi.
noHTH nepnoflHHecKHe cnrHanw 6nu3Ku no CBoen $opMe K nonurapMoHuHecKuM. OHU TaK*e npegcraBnaroT CO6OH cyMMy gByx u 6onee rapMoHuHecKux curHanoB (B npegene -go 6ecKoHeHHocru), HO He c KpaTHbiMu, a c npou3BonbHbiMu HacToTaMu, oTHomeHua KOTOPHX (XOTH 6H gByx HacToT MuHuMyM) He OTHOCHTCH K pa^uoHanbHbIM HucnaM, BcnegcTBue Hero ^yHgaMeHTanbHHH nepuog cyMMapHbix Kone6aHun 6ecKoHeHHo BenuK.

TaK, HanpuMep, cyMMa gByx rapMoHuK c HacToTaMu 2fo u 3.5fo gaeT nepuoguHecKun curHan (2/3.5 - pa^uoHanbHoe Hucno) c ^yHgaMeHTanbHon HacroTOH 0.5fo, Ha ogHoM nepuoge KOTOPOH 6ygyr yKnagbiBaTbca 4 nepuoga nepBon rapMoHuKu u 7 nepuogoB BTopon. Ho ecnu
3HaHeHue HacroTbi BTopon rapMoHuKu 3aMeHuTb 3HaHeHueM v 12 fo, TO curHan nepengeT B
pa3pag HenepuoguHecKux, nocKonbKy oTHomeHue 2/ v 12 He OTHOCUTCH K Hucny
pa^uoHanbHHx Hucen. KaK npaBuno, noHTu nepuoguHecKue curHanH nopo^garoTca ^u3uHecKuMu npo^ccaMu, He cBH3aHHHMu Me*gy CO6OH. MaTeMaTuHecKoe oTo6pa*eHue curHanoB To*gecrBeHHo nonurapMoHuHecKuM curHanaM (cyMMa rapMoHuK), a HacroTHbiH cneKTp TaK*e gucKpeTeH (puc. 6).

AnepnogHHecKHe cnrHanw cocraBnaroT ocHoBHyro rpynny HenepuoguHecKux curHanoB u 3agaroTca npou3BonbHHMu ^yH^uaMu BpeMeHu. Ha puc. 6 noKa3aH npuMep

exp(-b*t),
rge a u b - KOHcraHTbi, B gaHHOM caynae a = 0.15, b = 0.17.

K anepuogunecKuM curHaaaM OTHocaTca TaK*e HMnyabCHue cnrHaabi, KOTopwe B paguoTexHUKe u B OTpacaax, mupoKO ee ucnoab3yro^ux, nacro paccMaTpuBaroT B Buge OTgeabHoro Kaacca curHaaoB. HMnyabcw npegcraBaaroT CO6OH curHaaw onpegeaeHHOH u gocraTonHO npocroM $opMbi, cy^ecrayro^ue B npegeaax KOHenHbix BpeMeHHbix uHTepBaaoB. CurHaa, npuBegeHHHH Ha puc. 7, OTHocurca K nucay uMnyabcHbix.

2.4. fflyMbi H noMexH
OTgeabHyro KaTeropuro curHaaoB cocraBaaroT myMbi u noMexn - curHaaw, ucKa^aro^ue uHTepecyro^uM curHaa. CTporo roBopa, OHU He aBaaroTca curHaaaMu B ucxogHOM onpegeaeHuu, T.K. He HecyT HuKaKOH noae3HOH uH$opMa^uu. Ho B TO *e BpeMa, nacro ux Ha3WBaroT curHaaaMu B TOM cMwcae, nro OHU uMeroT 3aBucuMocTb OT TOH *e He3aBucuMOH nepeMeHHOH, nro u OCHOBHOH curHaa, u TaK*e nopo^garoTca $u3unecKuMu npo^ccaMu. ^o^TOMy, nro6w He co3gaBaTb nyraHu^i, nacro roBopaT O «noae3HOM cnrHaae» u «myMax H noMexax».
OgHaKO cy^ecTByroT caynau, Korga OHU MeHaroTca MecTaMu u noMexu cTaHOBaTca noae3HbiM curHaaoM. HanpuMep, npu onpegeaeHuu xapaKTepucruK myMa gaa ^$$eKTUBHOH 6opb6w c HUM.
noMexaMu Ha3WBaroT cTopoHHue (r.e. BHemHue no OTHomeHuro K cucreMe) BO3geMcrBua, ucKa^aro^ue noae3HWH curHaa. KaaccunecKuMu npuMepaMu noMex aBaaroTca ceTeBaa noMexa nacroTOH 50 ^ u ^aeKTpoMa^HUTHaa HaBogKa (noMexa), aBaaro^aaca pe3yabTaTOM BO3geMcTBua pa3aunHwx ^aeKTpoMa^HUTHwx Koae6aHuH Ha cucreMy.
nog myMOM 3anacTyro noHuMaroT ucKa^eHua, nopo^gaeMwe caMOH cucreMOH u Hoca^ue caynaHHHH xapaKTep. HanpuMep, cy^ecrayeT TenaoBOH myM pe3ucTopoB u ^auKKep-myM TparoucropoB. fflyMw, Bcrpenaro^ueca B peaabHOH *U3HU, nacTO uMeroT cxo^ue xapaKTepucTuKu, ^o^TOMy MO*HO BwgeauTb pag TunoBwx myMOB.
Cy^ecTByeT u TaKoe noHaTue, KaK ^erawe myMw. ^Tu myMw Ha3BaHw pa3aunHWMu ^eTaMu cneKTpa no aHaaoruu co cBeTOM BuguMoro cneKTpa.
«EeabiM» myM - caynaHHWH curHaa, 3HaneHua KOToporo onpegeaaroTca HopMaabHWM (raycoBcKuM) pacnpegeaeHueM. npHMep: TenaoBOH (g^OHcoBcKuM) myM

u aepHbiH myMbi.
3. nAPAMETPbl CMrHAHOB
^^HTe^bHOCTb curHaaa T onpegeaaeT uHTepBaa BpeMeHu, B TeaeHue KOToporo curHaa cy^ecTByeT (oTauaeH OT Hyaa);
MaKcuMaabHoe H MHHHMa^bHoe 3HaneHHH noKa3brnaroT guana3oH u3MeHeHua 3HaaeHuM
curHaaa.
^HHaMHnecKHH guana3oH — aorapu^M OTHomeHua HauGoabmero BO3MO*Horo 3HaaeHua KaKOH-au6o BeauauHbi K HauMeHbmeMy Bo3Mo*HoMy:
D = 10 IgPjnax/Prmn
UupuHa cneKTpa curHaaa — noaoca aacTOT, B npegeaax KOTOPOH cocpegoToaeHa ocHoBHaa ^Hep^ua curHaaa;
OTHomeHHe curHaa/myM paBHo oTHomeHuro MO^HOCTU noae3Horo curHaaa K MO^HOCTU myMa;
HacTOTa gncKpeTH3a^in gucKpeTHoro curHaaa — aacroTa caegoBaHua oTcaeToB.
nepuog flHCKpeTH3a^HH — paccToaHue Me*gy gByMa cocegHuMu oTcaeTaMu.
TeKy^ee cpegHee 3HaaeHue 3a onpegeaeHHoe BpeMn:
(l/T) 6,' *T s(t) dt.
nocTOHHHaa cocTaBanro^an:

(l/T) 6oT s(') dt.
CpegHee BunpHMaeHHoe 3HaaeHue:
(l/T) 6oT |s(t)| dt.
CpegHee KBagpaTHHHoe 3HaaeHue:
.1 6 oT x(t)2 dt '' T .

3.1. ^Hep^eTHHecKHe napaMeTpw
B Teopuu curHaaoB mupoKo ucnoab3yroTca ^Hep^eTuaecKue xapaKTepucTuKu. nycTb curHaa npegcraBaaeT u3MeHeHue Hanpa^eHua U(t) uau ToKa i(t). Torga Ha pe3ucrope c conpoTuBaeHueM R BbigeaaeTca MrHoBeHHaa (reKy^aa) Mo^HocTb
p(t) = ^P- = Ri2(t)
A