ტელეკომუნიკაციის თეორია ა რობიტაშვილი, გ მურჯიკნელი, თ ვეკუა, გ რობიტაშვილი by georgianpolitics

a. robitaSvili, g. murjikneli, T. vekua, g. robitaSvili

telekomunikaciis Teoria

`teqnikuri universitetiâ&#x20AC;?@

saqarTvelos teqnikuri universiteti

a. robitaSvili, g. murjikneli, T. vekua, g. robitaSvili

telekomunikaciis Teoria

damtkicebulia stu-s saredaqcio-sagamomcemlo sabWos mier. 02.07.2009, oqmi #6

Tbilisi 2009

uak 621.391

wigni dawerilia `telekomunikaciis Teoriis~ programuli kursis Sesabamisad. masSi ganxilulia informaciis (Setyobinebebis) gadacemis Teoriuli safuZvlebi. mocemulia is ZiriTadi ganmartebebi,. romlebic exeba Setyobinebebs, signalebs da arxebs, warmodgenilia

determinirebuli

SemTxveviTi

procesebis maTematikuri aRweris sakiTxebi. Camoyalibebulia da gaanalizebulia Setyobinebebisa da signalebis gadacemis da miRebis procesis aucilebeli gardaqmnebi da ukugardaqmnebi. gankuTvnilia telekomunikaciis profilis studentebisaTvis,

magistrantebisaTvis,

doqtorantebi-

saTvis da dargis specialistebisaTvis. recenzenti profesori n. uRreliZe

© sagamomcemlo saxli ,,teqnikuri universiteti’’, 2009 ISBN 978-9941-14-663-3 http://www.gtu.ge/publishinghouse/ yvela ufleba daculia. am wignis arc erTi nawili (iqneba es teqsti, foto, ilustracia Tu sxva) aranairi formiT da saSualebiT (iqneba es eleqtronuli Tu meqanikuri), ar SeiZleba gamoyenebul iqnas gamomcemlis werilobiTi nebarTvis gareSe. saavtoro uflebebis darRveva isjeba kanoniT.

S i n a a r s i gv.

S e s a v a l i . . . . . . . . . . . . . . . . . . . . . . . . 6 Tavi I. signalebisa da xelSeSlebis maTematikuri aRwera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.1. determinirebuli signalebi . . . . . . . . . . 14 1.2. SemTxveviTi procesebi (signalebi) . . . . . 18 1.3. procesebis (signalebis) miaxloebiTi warmodgena . . . . . . . . . . . . . . . . . . . . . . 20 1.4. signalebis speqtruli warmodgena . . . . . 24 1.5. signalebis droiTi warmodgena. . . . . . . . 27 1.6. analizuri signali . . . . . . . . . . . . . . . . 33 1.7. Setyobinebebisa da signalebis fizikuri maxasiaTeblebi . . . . . . . . . . . . . . . . . . 34 1.8. SemTxveviTi signalebis (procesebis) maxasiaTeblebi . . . . . . . . . . . . . . . . . . 36 1.9. SemTxveviTi procesebis stacionaruloba42 1.10. stacionaruli SemTxveviTi procesebis korelaciis funqciis Tvisebebi . . . . . . 45 1.11. SemTxveviTi procesis speqtruli maxasiaTeblebi . . . . . . . . . . . . . . . . . . . 48 1.12. normaluri SemTxveviTi procesebi . . . . .51 1.13. viwrozoliani SemTxveviTi procesebi . . 52 1.14 SetyobinebaTa gadacemis procesis geometriuli warmodgena . . . . . . . . . . . 54 Tavi II. informaciis gadacemis Teoriis safuZvlebi . . . . . . . . . . . . . . . . . . . . . . 59 2.1. informaciis raodenobrivi zoma . . . . . . 59 2.2. diskretuli Setyobinebebis entropia . . 62

gv.

2.3. diskretuli arxis gadacemis siCqare da gamtarunarionoba . . . . . . . . . . . . . . . . . 67 2.4. Senonis ZiriTadi Teoremebi diskretuli arxebisaTvis . . . . . . . . . . . . . . . . . . . . 71 2.5. diskretul SetyobinebaTa wyaros mwarmoebluroba da siWarbe . . . . . . . . . 76 2.6. uwyveti Setyobinebebis entropia . . . . . . 78 2.7. uwyveti Setyobinebebis wyaros mwarmoebluroba da siWarbe . . . . . . . . . 82 Tavi III. telekomunikaciis arxebi . . . . . . . . . . . .86 3.1. telekomunikaciis arxebis klasifikacia da maxasiaTeblebi . . . . . . . . . . . . . . . . . . . 86 3.2. damaxinjebebi da xelSeSlebi telekomunikaciis arxebSi . . . . . . . . . . 90 3.3. diskretuli arxebis modelebi . . . . . . . 95 3.4. diskretul-uwyveti arxebis modelebi . . 98 3.5. uwyveti arxebis modelebi . . . . . . . . . . . 99 3.6. signalebis ZiriTadi gardasaxvebi telekomunikaciis arxebSi . . . . . . . . . . 102 Tavi IV. diskretuli Setyobinebebis gadacemis Teoria . . . . . . . . . . . . . . . . . . . . . . . . .109 4.1. uwyvet arxebSi diskretuli Setyobinebebis optimaluri miRebis amocana . . . . . . . . . 109 4.2. uwyvet arxebSi diskretuli Setyobinebebis optimaluri miRebis kriteriumebi . . . . . 112 4.3. telekomunikaciis diskretuli sistemebis potencialuri xelSeSlebisadmi mdgradoba fluqtuaciuri xelSeSlebis dros . . . . 118

gv.

Tavi 5. informaciids cifrul formaSi gadacemis safuZvlebi. . . . . . . . . . . . . . . . . . . . . . 124 5.1. informaciis cifruli gadacemis sqema . . 124 5.2. informaciis cifruli gadacemis boWkovan-optikuri sistemebi . . . . . . . . 127 5.3. informaciis cifruli gadacemis sistemebi ATMM teqnologiebis gamoyenebiT . . . . . 131 5.4. informaciis gadacemis cifruli sistemebi analoguri saabonento xazebisaTvis . . . 133 5.5. informaciis gadacemis cifruli radiosistemebi . . . . . . . . . . . . . . . . . . 134 5.6. analoguri signalebis cifruli kodirebis ZiriTadi meTodebi . . . . . . . . 142 5.6.1. impulsur-koduri modulaciis (ikm) meTodi . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.6.2. diferencialuri impulsur-koduri modulacia (dikm) . . . . . . . . . . . . . . . . . 148 5.6.3. adapturi diferencialuri impulsurkoduri modulaciis (adikm) meTodi . . . 149 5.7. cifruli signalebis multipeqsirebis ZiriTadi principebi . . . . . . . . . . . . . . . 152 5.8. cifruli signalebis saxazo kodirebis ZiriTadi principebi . . . . . . . . . . . . . . . 157 literatura . . . . . . . . . . . . . . . . . . . . . 166

S e s a v a l i telekomunikaciis teqnikuri

Teoria

saSualebebis

aris

gamoyenebiT

mecniereba manZilze

in-

formaciis gadacemis Sesaxeb. telekomunikacia (ing. Telecommunication) _ nebismieri gadacema da/an emisia da miReba signalebisa, romlebic warmoadgenen niSnebs, werilobiT dokumentebs,

gamosaxulebebs,

xmas

nebismieri

sxva

saxis informacias, sadeniani radio _ optikuri an sxva eleqtromagnituri sistemebis gamoyenebiT (Rec. 701, International Telecommunication Union Standartisation ITU â&#x20AC;&#x201C; T _ telekomunikaciis saerTaSoriso kavSiris standartizaciis seqtori-tsk-t, rekomendacia G.701). informacia movlenis

aris

erToblioba

materialuri

monacemebisa

sistemis

raime

mdgomareobis

Sesaxeb. informaciis warmodgenis formas ewodeba Setyobineba. sxva sityvebiT rom vTqvaT, Setyobineba aris gadacemisaTvis gankuTvnili informacia. imisaTvis, rom Setyobineba miewodos momxmarebels, aucilebelia gamoyenebul iqnas iseTi fizikuri procesebi, romlebsac gaaCniaT unari gavrceldes

gadamcemidan

mimRebamde

garkveuli

siCqariT.

droSi cvalebad fizikur sidides romelic asaxavs Sewyobinebas, ewodeba signali. signali Setyobinebis materialuri matarebelia. amasTan,

signali

SeiZleba

optikuri an hidroakustikuri. 6

iyos

eleqtruli,

wignSi ganxilulia mxolod eleqtruli telekomunikaciis sadac

(sadeniani,

Setyobinebebis

eleqtruli masala

radiokavSiris) gadacemisaTvis

signalebi.

magram,

samarTliania

aseve

hidroakustikuri

sistemebi, gamoiyeneba

wignSi

mocemuli

optikuri

signalebisaTvisac

SeiZleba

gamoyenebul iqnas maTi maTematikuri aRwerisaTvis. zogadad, Setyobinebis rogorc gamgzavni, aseve mimRebi SeiZleba iyos adamiani an sxvadasxva xelsawyoebi,

gadamcemi,

maregistrirebeli,

Semnaxveli

da informaciis momxmarebeli (gamomyenebeli). nax. S.1-ze mocemulia Setyobinebis gadamcemis struqturuli sqema misi wyarodan mimRebamde (erTi mimarTulebiT). gadamcemi mowyobiloba gardaqmnis sawyis Setyobinebas iseT signalad romlis warmodgenis forma mosaxerxebelia telekomunikaciis mocemuli xazisaTvis (fizikuri garemosaTvis). mimRebi mowyobiloba axorcielebs ukugardaqmnas. im saSualebaTa erTobliobas, romlebic gankuTvnilia Setyobinebis anu signalebis gadasacemad ewodeba

telekomunikaciis

arxi.

(farTo

gagebiT).

amasTan, saSualebebi _ es aris teqnikuri mowyobilobebi gadis

telekomunikaciis

signali.

sasargeblo

xazi,

romlebSic

signalebTan

erTad

telekomunikaciis arxSi rogorc wesi, warmoiSveba xelSeSlebi,

anu

xmaurebi

sxv.

rac

iwvevs

aRdgenili Setyobinebis araerTmniSvnelovnebas gadacemulTan. sxvadasxva saxis xelSeSlebi realur arxebSi zemoqmedeben

signalebze

misi

gavrcelebis

mTel

gzaze. magram analizis gamartivebis mizniT xelSeSlebis saerTo zemoqmedeba am procesze S.1-ze warmodgenilia jamuri eqvivalenturi blokiT `xelSeSlebis wyaro~. am wyarodan mimRebi mowyobilobis Sesasvlelze

miewodeba

SemTxveviTi

eleqtruli

aRgznebebi, romlebic sasargeblo signalTan urTierTqmedebisas amaxinjeben mas. xazis gamosasvlelidan signali miewodeba telekomunikaciis sistema telekomunikaciis arxi

2 a (t)

S(t)

Z(t)

5 aâ&#x20AC;˛ (t)

nax. S.1. erTi mimarTulebiT Setyobinebebis telekomunikaciis sistemebis ganzogadebuli struqturuli sqema 1 _ Setyobinebebis wyaro; 2 _ gadamcemi mowyobiloba; 3 _ telekomunikaciis xazi; 4 _ mimRebi mowyobiloba; 5 _ Setyobinebis mimRebi; 6 _ xelSeSlebis wyaro.

gadamcemi mowyobiloba

a (x,y,z,t)

b(t)

S(t)

Z(t)=S(t)+U(t)

u (t)

aâ&#x20AC;˛ (x,y,z,t)

mimRebi mowyobiloba

bâ&#x20AC;˛ (t)

nax. S.2. telekomunikaciis xazSi Setyobinebebis da signalebis gardaqmnis struqturuli sqema 1 _ Setyobinebis wyaro; 2 _ signalad gardamqmneli da koderi; 3 _ modulatori; 4 _ telekomunikaciis xazi; 5 _ demodulatori; 6 _ dekoderi da SetyobinebaSi gardamqmneli; 7_ Setyobinebis mimRebi; 8 _ xelSeSlebis wyaro

mimReb mowyobilobas. misi daniSnulebaa _ gardaqmnas miRebuli signalebi rac SeiZleba zustad im Setyobinebebad romlebic iqna gadacemuli, miuxedavad masze moqmedi sxvadasxva xelSeSlebisa. zogadad, Setyobinebebis wyaro, gadamcemi mowyobiloba, telekomunikaciis xazi (garemo) telekomunikaciis punqtebs Soris, mimRebi mowyobiloba da Setyobinebis mimRebi, qmnian telekomunikaciis sistemas. telekomunikaciis TeoriaSi sistema iTvleba

mocemulad, Tu masSi gansazRvrulia Setyobinebis signalad gardaqmnis meTodebi da signalis Setyobinebad ukuaRdgenis wesi miRebuli signalis Sesabamisad. Setyobinebebi, da rasakvirvelia maTi Sesabamisi signalebi, iyofian or did jgufad: uwyvetad da wyvetilad (diskretulad). uwyveti Setyobinebebis magaliTad SeiZleba CaiTvalos laparaki, musika, da sxv. im sistemebs, romlebic gankuTvnilni arian uwyveti Setyobinebebis gadacemisaTvis, ewodebaT uwyveti (analoguri). zemoaRniSnuli Setyobinebebis garda praqtikaSi xSirad gvxvdeba wyaroebi, romlebic xasiaTdebian mdgomareobis diskretuli mdgomareobiT. ase magaliTad, diskretuls warmoadgenen wyaroebi, romlebic Setyobinebebs gascemen teqstis, monacemebis da sxva msgavsi saxiT. ganvixiloT telekomunikaciis sistemaSi Setyobinebebis da signalebis gardaqmnis struqturuli sqema (nax. S.2.) gadamcemi mowyobilobis Sesasvlelis wyarodan miewodeba Setyobineba

a(t , x, y, z ), , romelic zogad

SemTxvevaSi (mag. telexedvis sistemaSi SeiZleba warmodgenil iqnas t drois funqciis da koordinatebis x, y, z funqciad. kerZo SemTxvevaSi Setyobineba a SeiZleba damokidebuli iyos x, y, z da t argumentebidan romelimeze an ramdenimeze. ase mag., fotodepeSebis gadacemisas Setyobineba damokidebulia gadasacemi gamosaxulebis wertilebis mxolod 10

or koordinatze. gadamcemi mowyobiloba gardaqmnis Setyobinebas a(t , x, y, z ) , saarxo signalad S (t ) , romelic warmoadgens mxolod drois funqcias. msgavsi gardaqmnebi, rogorc wesi, xorcieldeba ramdenime etapad, ris safuZvelze SeiZleba calke gamoiyos gadamcemi mowyobilobis ramdenime damoukidebeli funqcionaluri bloki. xSirad aseTi oria. blok 2-Si Setyobineba a (t) gardaiqmneba e.w. pirvelad signalad (e.w. dabalsixSiruli signali). blok 3-Si (modulatorSi) sabolood formirdeba saarxo signali S(t), romelsac gaaCnia unari gavrceldes telekomunikaciis xazSi da miewodos mimReb mowyobilobas. es gardaqmna, rogorc wesi, mdgomareobs mudmivi an cvladi gadamtanis garkveuli parametris SecvlaSi mamodulirebeli signalis b(t) Sesabamisad. xSirad saarxo signali formirdeba mravaljeradi mimdevrobiTi modulaciis safuZvelze. telekomunikaciis xazSi sawyisi

S (t ) signali

ganicdis damaxinjebebs U (t ) xmauris (xelSeSlebis) gamo.

Sedegad

miRebuli

signali

Z (t ) = S (t ) + U (t )

igivurad ar emTxveva S (t ) -s. mimRebi mowyobiloba, iseve rogorc gadamcemi mowyobiloba SeiZleba warmodgenil iqnas ramdenime blokiT (5, 6 nax. S.2-ze). demodulatori (bloki 5) miRebuli Z (t ) rxevebidan gamoyofs pirvelad signals bâ&#x20AC;˛(t ) , xolo dekoderi misi saSualebiT aRadgens Setyobinebas. zogadad, miRebuli Setyobineba

aâ&#x20AC;˛( x, y, z , t ). gansxvavdeba gadacemuli

a ( x, y, z , t ) -sgan

rac, rogorc zemoT aRiniSna gamowveulia gadacemis traqtSi arsebuli damaxinjebebiTa da xelSeSlebiT. rac ufro mcired gansxvavdeba miRebuli Setyobineba gadacemulisgan miT ufro metia gadacemis namdviloba (xarisxi). unda aRiniSnos, rom nax. S.2-ze mocemuli struqturuli sqema sworia telekomunikaciis uwyveti sistemebisaTvis. e.i. Tu Setyobineba a ( x, y, z , t ) , aris misi argumentebis uwyveti funqcia. am SemTxvevaSi saWiro araa blokebi 2 da 6. rasakvirvelia isini aucilebelia im SemTxvevaSi roca gvaqvs Sereuli sistemebi signalis diskretizaciiT. telekomunikaciis Teoriis kursis safuZvels warmoadgens telekomunikaciis statistikuri Teoria romelic intensiurad viTardeba. telekomunikaciis statistikuri Teoria anu informaciis Teoria xasiaTdeba fiziko-maTematikuri da inJinerul-teqnikuri meTodebis SerwymiT. aqedan gamomdinare, mas gaaCnia rigi damaxasiaTebeli Taviseburebani. erTis mxriv, igi warmoadgens calkeuli maTematikuri disciplinebis (albaTobis Teoria, SemTxveviTi funqciebis Teoria, analizuri geometria, simravleTa Teoria) Semdgom ganviTarebas da meores mxriv _ eleqtruli da lazeruli telekomunikaciis ganzogadebas. rogorc qvemoTaa naCvenebi albaTuri meTodebis telekomunikaciis TeoriaSi gamoyeneba saSualebas iZleva gadaWril iqnas mravali praqtikuli amocana, romlebic adre saerTod ar ismeboda, an ver wyde12

boda. es meTodebi Sesaswavli procesebis adeqvaturia. wigni gankuTvnilia telekomunikaciis profilis studentebisaTvis. magistrantebisaTvis, doqtorantebisaTvis da dargis specialistebisaTvis.

sakontrolo kiTxvebi 1. ra aris sagnis ` telekomunikaciis Teoria~ Seswavlis mizani? 2. telekomunikaciis TeoriaSi ra aris Setyobineba? 3. ra aris signali da ra gansxvavebaa signalsa da Setyobinebas Soris? 4. ras warmoadgens telekomunikaciis xazi, arxi da sistema? 5. ra gansxvavebaa diskretul da uwyvet Setyobinebebs Soris?

Tavi I. signalebisa da xelSeSlebis maTematikuri aRwera adamianTa garSemo mimdinare fizikuri da sxva procesebis mravalgvarobis miuxedavad (maTi droSi cvlilebis TvalsazrisiT) isini SeiZleba daiyos or did jgufad (klasad): determinirebuli (regularuli) da SemTxveviTi. pirvel jgufs miekuTvneba procesebi (signalebi), romelTa msvleloba droSi srulad aris gansazRvruli winaswar (apriorulad). SemTxveviTi ewodeba procesebs (signalebs) romelTa evolucia (cvlilebebi) aris mravali cvladis funqcia da damokidebulia ara marto droze, aramed SemTxveviT faqtorebzec. qvemoT ganxilulia determinirebuli da SemTxveviTi signalebis aRweris maTematikuri aspeqtebi.

1.1. determinirebuli signalebi vidre gadavidodeT determinirebuli signalebis dawvrilebiT aRweraze, unda aRinoSnos, rom bunebaSi ar SeiZleba arsebobdes `sufTa~ determinirebuli signalebi, radgan aseTi signalebi SeiZleba warmoiSvas mxolod izolirebul fizikur Tu sxva sistemebSi. amgvarad, realuri fizikuri Tu sxva procesi SeiZleba CaiTvalos determinirebulad da Sesabamisad aRiweros drois determinirebuli funqciiT mxolod miaxloebiT. aqedan gamomdinare, determinirebuli signalebi warmoadgenen realuri signalebis garkveul idealizacias.

miuxedavad zemoaRniSnulisa, determinirebuli signalebi dReisaTvis farTod gamoiyeneba sxvadasxva wrfivi, arawrfivi, parametruli da sxva wredebis gamosakvlevad da wredSi gardamavali procesebis, oTxpolusebis maxasiaTeblebis da a.S. Sesaswavlad. unda aRiniSnos, rom sxvadasxva formis determinirebuli signalebi farTod gamoiyenebian, rogorc gadamtanebi signalebis gardasaxvis, formirebisa da gadacemisaTvis. determinirebuli signalebi Tavis mxriv pirobiTad SeiZleba daiyos xuT qvejgufad. kerZod: diskretuli signalebi; kauziluri signalebi; perioduli signalebi; uwyveti signalebi, finituri signalebi. a) diskretuli signalebi. diskretuli signalebi ganisazRvrebian drois fiqsirebul momentebSi droiT RerZze ganlagebuli mniSvnelobebis Tvladi (sasruli) simravliT (nax. 1.1).

nax. 1.1.

b) kauzaluri signalebi kauzaluri ewodebaT signalebs romlebTac aqvT dasawyisi droSi (nax. 1.2). cxadia, yvela realuri signali kauzaluria, vinaidan warmoadgenen garkveuli fizikuri Tu sxva movlenebis Sedegs (ase mag., warmoiSobian generatoris gamosasvlelze drois garkveul momentSi kvebis miwodebis Semdeg). maTi analizisas mizanSewonilia maTi dasawyisi SevuTavsoT drois aTvlis nulovan dasawyiss

(t = 0) da

CavTvaloT, rom isini nulis tolia, rodesac t < 0. .

nax. 1.2.

g) perioduli signalebi periodulia signalebi, romelTa nebismieri mniSvnelobebi meordebian drois garkveuli t intervalis Semdeg (nax. 1.3). am intervals determinirebuli signalis periodi ewodeba. perioduli signalebisaTvis marTebulia piroba

x(t ) = x(t + mT ),

sa-

dac m mebismieri mTeli ricxvia. perioduli signalebis umartives, telekomunikaciaSi yvelaze gavrcelebul formas warmoadgenen harmoniuli signalebi 16

x(t ) = Ag sin(ωot + ϕo ); sadac AO , ωO , ϕO _ mudmivi sidideebia da Sesabamisad aris harmoniuli signalis amplituda, wriuli sixSire da faza.

nax. 1.3.

d) uwyveti signalebi uwyveti (analoguri) signalebi ewodebaT signalebs, romlebic gansazRvrulni arian drois nebismier wertilSi, e.i. ganisazRvrebian droiT RerZze ganlagebuli mniSvnelobebis araTvladi (usasrulo) simravliT (nax. 1.4)

nax. 1.4.

e) finituri signalebi finituri signalebi droSi lokalizebul signalebs ewodebaT, romlebic nulis toli arian drois garkveuli SezRuduli intervalis ta â&#x2030;¤ t â&#x2030;¤ tb gareT (nax. 1.5).

nax. 1.5.

1.2. SemTxveviTi procesebi (signalebi)

SemTxveviTi procesebi da Sesabamisad maTi aRmweri signalebi asaxaven fizikuri da sxva sistemebis iseT cvlilebebs droSi, romelTa winaswarmetyveleba SeuZlebelia. maTematikurad SemTxveviTi procesi (signali) aRiwereba drois SemTxveviTi

X (t ) funqciiT, rom-

lis erT-erT saxes x(t ) , miRebuls cdis an eqsperimentis Sedegad, ewodeba SemTxveviTi procesis realizacia. am SemTxvevaSi cdis an eqsperimentis cnebis qveS igulisxmeba SemTxveviTi procesis 18

wyaros erTjeradi CarTva garkveuli drois ganmavlobaSi da gamosasvleli rxevebis Cawera. magram, vinaidan realur pirobebSi wyaros parametrebi droze arian damokidebulni da icvlebian, amitom irRveva cdebis Catarebis pirobebis ucvleloba da aqedan gamomdinare cdis ganmeoradobis piroba. amitom zogadad, aucilebelia Catardes wyaroTa didi simravlis gamocda. swored, am gamocdaTa safuZvelze miRebuli Sedegebi warmoadgens realizaciaTa simravles (ansambls) (aRiniSneba

k (t )

(t )},

nax. 1.6). igi srulad asaxavs SemTxveviT process. SemTxveviTi procesis (signalis) X (t ) myisi mniSvnelobebis erTobliobas, romlebic aiRebian drois nebismier momentSi, ewodeba SemTxveviTi procesis kveTa. Tu X (t ) SemTxveviTi funqciaa argumentis fiqsirebuli t = ti

mniSvnelobisaTvis, igi

X (ti ) SemT-

xveviT sidides warmoadgens. es niSnavs imas, rom eqsperimentis ucvlelobis pirobebisas

X (t ) funq-

cia garkveuli albaTobiT Rebulobs sxvadasxva konkretul

formebs

x ( k ) (t ) , romlebsac SemTxvebviTi

procesis realizaciebi ewodeba. yovel k -uri x ( k ) (t ) realizacias

drois

fiqsirebul

momentSi

garkveuli X ( k ) (ti ) mniSvneloba gaaCnia da amgvarad SemTxveviTi procesis yoveli realizacia determinirebul funqcias warmoadgens. gansxvavebiT determinirebuli funqciebisagan (signalebisgan), romlebic savsebiT aRiwereba erTi 19

x(t ) realizaciiT, SemTxveviTi x(t ) procesi aRiwereba

realizaciaTa

ansambliT

(k )

(t )} ,

romelic

SeiZleba CaiTvalos savsebiT gansazRvrulad, Tu cnobilia realizaciaTa simravle da maTi gamoCenis albaTobebi.

(t)

nax. 1.6.

1.3. procesebis (signalebis) miaxloebiTi warmodgena procesebis aRmweri signalebi SeiZleba mocemul iqnan cxrilebis, grafikebis an Sesabamisi rTul analizuri gamosaxulebebis saxiT. ris gamoc saWiroa mivmarToT signalebis miaxloebiT warmodgenas. arsebobs signalebis miaxloebiTi warmodgenis ori meTodi interpolacia da aproqsimacia. orive SemTxvevaSi laparakia T muli

droiT intervalze moce-

f (t ) signalebis miaxloebiT da amave dros

garkveuli azriT saukeTeso aRweraze X (t ) funqciebiT, romlebic mravalwevrebs:

warmoadgenen

X (t ) = aoξ o (t ) + a1ξ1 (t ) + ... + aK ξ K (t ) = sadac a K funqciebi,

e.w.

ganzogadebul

∑a

K =O

ξ K (t ),

(1.1)

mudmivi koeficientebia, xolo ξ K (t )

romlebic

qmnian

mowesrigebul

{ξ K (t )}

sistemas. interpolaciisa da aproqsimaciis amocanebi SeiZleba mniSvnelovnad gamartivdes Tu ξ K (t ) sistema Sedgeba erTmaneTisagan damoukidebeli funqciebisagan. ZiriTadi gansxvaveba interpolaciasa da aproqsimacias Soris mdgomareobs SemdegSi. interpolaciis dros aucilebeli moTxovnaa

X (t ) funqciis

mniSvnelobebi emTxveodnen f (t ) signalis mniSvnelobebs garkveul wertilebSi, romlebsac interpolaciis kvanZebi ewodeba, xolo aproqsimaciis dros aseTi Tanxvedra ar aris aucilebeli. saWiroa mxolod

f (t )

X (t )

funqcia

funqciisagan

naklebad

gansxvavdebodes

ukanasknelis

gansazRvris

areze. zogjer aproqsimaciis amocanas funqciaTa moaxloebis amocanas uwodeben. aqve unda aRiniSnos, rom ar aris gamoricxuli funqciaTa aproqsimaciis dros X (t ) mniSvnelobebi daemTxves f (t ) − s.

f (t ) warmoiSoba

signalis

(1.1)

mwkrivis

ε (t ) = f (t ) − X (t )

warmodgenisas

cvlileba,

romlis

Sefa-sebis wesi gansazRvravs funqciaTa miaxloebiT wes-sa da kriteriums. zogierT SemTxvevaSi aucilebelia Secdomis maqsimaluri sidide iyos minimaluri

f (t ) -s gansa-

zRvris areze. aproqsimaciis aseT saxes (meTods), romlis drosac xdeba | ε (t ) | -s minimizacia, ewodeba Tanabari miaxloeba, an miaxloeba minimaqsimaluri kriteriumiT. zogadad, ufro farTo gavrceleba hpova miaxloebam saSualo kvadratuli cdomilebis minimizaciis kriteriumiT. saSualo kvadratuli cdomilebis Sefaseba xdeba Semdegi gamosaxulebiT: T

1 ε (t ) = ∫ [ f (t ) − X (t )]2 dt . TO 2

(1.2)

am kriteriumis farTo gamoyeneba gamowveulia imiT, rom igi iTvaliswinebs integralur efeqtscdomilebis dagrovebas signalis gansazRvris areze, rac damaxasiaTebelia telekomunikaciis teqnikuri sistemebis umravlesobisaTvis. rasakvirvelia, SeiZleba gamoyenebul iqnas miaxloebis sxva kriteriumebic, magram yvelaze farTo gamoyeneba hpova zemoaRniSnulma orma kriteriumma. signalis miaxloebiTi warmodgena sagrZnoblad martivdeba, Tu (1.1) mravalwevris agebisas gamoyenebul iqneba funqciaTa

{ξ K (t )}

sistema, romelTac

bazisuri sistemebi ewodebaT. (1.1)-Si Semavali a K koeficientebis gamosaTvlelad, gamosaxulebaSi Semavali orive mxare gavamrav-

ξ j (t ) -ze da movaxdinoT integrireba (O, T )

loT

sazRvrebSi. miviRebT: T

∫ X (t )ξ (t ) dt = ∑ a ∫ ξ j

k =1

vinaidan

(t )ξ j (t ).

{ξ k (t )}

{

(1.2).

}

da ξ j (t )

sistemebi orTonormi-

rebulia, (1.2) gamosaxulebis marjvena mxaris yvela Sesakrebi iqneba nulis toli garda erTisa, romelic Seesabameba

k= j

SemTxvevas. ukanaskneli

Sesakrebi toli iqneba ak -si da amgvarad T

ak = ∫ X (t )ξ k (t ) dt .

(1.3)

ak koeficientebs, gansazRvruls (1.3) gamosaxulebis mixedviT, furies ganzogadebuli koeficientebi ewodeba, xolo (1.1) mwkrivs-furies ganzogadoebuli mkrivi. aqve unda aRiniSnos saSualo kvadratuli cdomileba, gansazRvruli (1.3) gamosaxulebis safuZvelze aRwevs minimums da udris

2 min

n 1 2 = ∫ X (t )dt − ∑ aK2 . TO K =1

2 min

(1.4)

gamosaxulebidan

rodesac

n →∞,

xolo

→ 0, miiReba e.w. parsevalis toloba

n 1 2 X ( t ) dt = aK2 . ∑ ∫ TO K =1

(1.5)

es gamosaxuleba erTis mxriv, amyarebs kavSirs signalis energias da am signalis ganzogadebuli 23

daSlis koeficientebs Soris da, meores mxriv warmoadgens orTonormirebul funqciaTa sistemis Sekrulobis pirobas. orTonormirebuli sistema Sekrulia, Tu misTvis marTebulia piroba (1.5). aRsaniSnavia, rom orTonormirebuli sistema Sekrulia, Tu misTvis marTebulia (1.5)-iT mocemuli piroba. aRsaniSnavia isic, rom Sekrulobis pirobidan gamomdinareobs sistemis sisrulis pirobac.

1.4. signalebis speqtruli warmodgena signalebis speqtrul analizs safuZvlad udevs maTi warmodgena elementaruli harmoniuli Semadgenlebis erTobliobiT. imisda mixedviT periodulia

araperioduli

signalebis

aRmweri

X (t )

funqcia, igi SeiZleba warmodgenil iqnes furies mwkrivis an integralis saxiT. Tu T

periodis mqone

X (t ) funqcia akmayofi-

lebs e.w. dirixles pirobebs, e.i, aris uwyveti Sekrul intervalze da am intervalze an ar gaaCnia, an aqvs sasruli raodenobebis eqstremumebi, igi furies trigonometriuli mwkrivis saxiT warmodgeba

X (t ) =

ao 2 ∞ + ∑ aK cosωk t + bk sin ωk t , T T k =1

(1,6)

sadac:

ωk = k 2π / T ,

(1.7)

T /2

ao =

∫ X (t )dt ,

(1.8)

−T / 2

T /2

∫ X (t ) cos ω t dt ,

ak =

(1.9)

−T / 2 T /2

bk =

∫ X (t ) sin ω t dt.

(1.10)

−T / 2

amgvarad, perioduili signali SeiZleba warmodgenil iqnes e.w. sixSiruli speqtris saxiT, anu perioduli Semadgenlebis (hermonikebis) jamiT, romelTa amplitudebia ak da bk . amasTan (1.6) mwkrivis wevrebi ZiriTadi

a1 cos ω1t

b1 sin ω1t

perioduli

erTad gansazRvraven

Semadgenlis,

e.w.

harmonokis, xolo danarCeni wevrebi

pirveli

ak cos ωk t

bK sin ω K t K -uri harmonikebis (k > 1) sidideebs. Sesabamisad

ω1 = 2π / T

ZiriTadi

pirveli

harmonikis,

ω K (k > 1) ki- K -uri harmonikis sixSireebia. zogadad iTvleba, rom signalebis energiis ZiriTadi nawili moTavsebulia nulovan sixSiresa da sixSiris im mniSvnelobas Soris, sadac speqtris momvlebi pirvelad xdeba nulis toli, anu sixSireTa diapazonSi

O ≤ω ≤

2π , T

amitom signalis speqtris zeda zRvrul sixSired miRebulia speqtris momvlebis pirveli nulis Sesabamisi sixSire, e.i.

ωz = 2πf z = 2π / τ aqedan ki

fz ⋅τ = 1 .

(1.11)

amgvarad, rac ufro moklea signali, miT ufro farToa misi sixSiruli speqtri da piriqiT. sazogadod, nebismieri formis signalisaTvis marTebulia ukuproporciuli damokidebuleba sprqtris zeda zRvrul sixSiresa da signalis xangZlivobas Soris, e.i. zogadad f z ⋅τ ≈ const . araperioduli signalebis SemTxvevaSi marTebulia daSveba imisa, rom igi periodulad SeiZleba warmovidginoT im SemTxvevaSi romlisTvisac T → ∞ . amasTan, sxvaoba mezobeli harmonikebis sixSireebs Soris miiswrafvis nulisaken, speqtri xdeba uwyveti, harmonikebis amplitudebi _ usasrulod mcire. zRvarze gadasvlis Sedegad rodesac T → ∞ , sixSireTa diskretuli mniSvnelobebis mimdevroba Seicvleba mimdinare ω sixSiriT, jami Seicvleba integraliT da amis Sedegad miviRebT

X (t ) =

1 2π

∞

∫ S ( jω )e

jwt

dω ,

(1.12)

−∞

sadac

S ( jω ) =

∞

∫ x(t )e

− jωt

dt ,

(1.13)

−∞

speqtruli simkvrivea. (1.12) gamosaxuleba aris furies integrali kompleqsur formaSi. (1.12)-is integralqveSa gamosaxuleba gamosaxavs calkeul usasrulod mcire Sesakrebs, e.i. harmoniul rxevas mcire amplitudiT dc: 26

e jωt

usasrulod

1 S ( jω ) ⋅ e jwt dω = dce jωt dt 2π aqedan simkvrives

vpoulobT

amplitudebis

speqtralur

dc , S ( jω ) = 2π dω rasac agreTve ewodeba araperioduli signalis kompleqsuri speqtri;

S ( jω ) -s absolutur mniSvne-

lobas ubralod speqtri ewodeba. (1.12) da (1.13) gamosaxulebebs, Sesabamisad furies uku da pirdapiri integraluri gardasaxvebi (ewodebaT). SemoklebiT isini aRiniSnebian F −1[ X (t )] da F [ X (t )].

1.5. signalebis droiTi warmodgena analoguri signalebis droiTi warmodgenis sferoSi, anu analoguri signalebis, droiTi diskre-tizaciis sferoSi klasikuri Sedegebi miRebul iqna g. naikvistis (1933) da v.a. kotelnikovis (1931) mi er. vinaidan, Cvenis azriT, v.a. kotelnikovis Teoremis mixedviT kargad Cans uwyveti signalebis droiTi dis-kretizaciiT gamowveuli damaxinjebebi, qvemoT moce-mulia aRniSnuli Teoremis mokle formulireba da misi dawvrilebiTi analizi. v.a. kotelnikovis mier damtkicebuli Teoremis Tanaxmad, drois uwyveti X (t ) funqcia, romelsac ar gaaCnia sixSiruli Semadgenlebi ω z sixSiris zeviT, 27

mTlianad ganisazRvreba X (kΔtd ) myisi mniSvnelobebiT (diskretebiT) wertilebSi, romlebic erTmaneTisagan

Δtd = π / ωz

daSorebulia

intervals

diskretizaciis

intervaliT. biji,

Δt d

xolo

f d = 1 / Δtd = ωz / π _ diskretizaciis sixSire ewodeba. am TeoremiT uwyveti

X (t )

funqcia SeiZleba

warmodgenil iqnes Semdegi mwkrivis saxiT:

X (t ) =

∞

∑ X (kΔt

k =−∞

) sin ωz (t − kΔtd ) / ωz (t − kΔtd )

(1.14)

v.a. kotelnikovis daSlaSi (1.1) bazisuri funqciebis rols asrulebs funqciebi:

ϕ k (t ) = sin ωz (t − kΔtd ) / ωz (t − kΔt z ) grafikulad ϕ k (t ) mocemuli

saxe

(1.15)

funqciebs aqvs nax. (1.7)-ze

(naxazze

Semotanilia

aRniSvna

t − kΔtd = τ ). ) ganvixiloT is ZiriTadi mizezebi, romlebic warmoadgens damaxinjebebis wyaroebs v.a. kotelnikovis Teoremis praqtikuli gamoyenebisas. damaxinjebebis pirveli klasi ganpirobubulia imiT, rom yvela realuri signali finituria. amitom, v.a. kotelnikovis Teoremis gamoyenebisas usasrulo mwkrivi icvleba sasruli mwkriviT, e.i. T xangZlivobis X (t ) signalebze iReba sasruli ricxvis n = T / Δtd + 1 ≈ f d ⋅ T = 2 f z ⋅ T diskretebi. cxadia, am SemTxvevaSi

X (t )

funqciis

sizuste

miT

ufro

naklebi

iqneba,

rac

ufro

naklebi

raodenobis

diskretebi monawileobs X (t ) funqciis aRdgenaSi;

nax. 1.7

_ damaxinjebebis meore klasi ganpirobebulia imiT, rom realur signalebs sasruli xangZlivobis gamo usasrulo sixSiruli speqtrebi gaaCnia, amitom sixSiruli speqtris SezRudvisas, zeda ω z zRvruli sixSiris gareT rCeba signalis energiis nawili; _ damaxinjebebis mesame klasi vlindeba demudulatorSi analogiuri signalis aRdgenisas da ganpirobebulia imiT, rom X (t ) funqciis aRdgenisaTvis diskretebis saSualebiT saWiroa ϕ k (t ) funqciis generacia, rac praqtikulad xorcieldeba dabali sixSireebis filtris gamoyenebiT. magram, vinaidan misi maxasiaTeblebi sakmaod gansxvavdeba idealurisagan, maxinjdeba ϕ k (t ) funqciis forma, rac Sedegad iwvevs X (t ) funqciis damatebiT damaxinjebebs. (nax. 1.8, 1-idealuri filtris maxasiaTebeli, realuri filtris maxasiaTebeli). 29

nax. 1.8.

ganvixiloT daqvantviT gamowveuli damaxinjebebis arsi. daqvantvisas adgili aqvs daqvantvis damaxinjebebs. tsk-t-s G.701 (03/93) rekomendaciaSi mocemuli ganmartebiT, daqvantvis damaxinjeba aris damaxinjeba, warmoqmnili diskretebis daqvantvis procesis Sedegad muSa diapazonis sazRvrebSi. daqvantvis damaxinjebebi, rogorcwesi, gamoisaxeba rogorc signalis saSualo simZlavris fardoba damaxinjebebis saSualo simZlavresTan. aRniSnuli fardoba SeiZleba Caiweros Semdegnairad:

{

} {

}

Ps / Pdq = f x 2 (t ) / E [ X (t ) − X * (t )]2 , sadac E{•} _ maTematikuri molodinia, X (t ) _ analoguri Sesasvleli signali,

X * (t ) _ dekodi-

rebuli analoguri signali. imisaTvis, rom ganisazRvros daqvantvis damaxinjebebis saSualo simZlavre, gaTvaliswinebul unda iqnes Semdegi: _ Secdomis [ X (t ) − X * (t )] amplituda SemosazRvrulia δ / 2

mniSvnelobiT, vinaidan dekodirebuli 30

gamosasvleli diskretebi ganlagebulia daqvantvis bijis SuaSi; _ diskretebis mniSvnelobebi Tanabari albaTobebiT SeiZleba moxvdes nebismier wertilSi daqvantvis bijis sazRvrebSi, e.i. diskretebis albaTobaTa simkvrive aris Tanabari da tolia 1 / δ ; aseve sainteresoa signalebis speqtraluri warmodgena. nax. 1.9. a,b da g-ze warmodgenilia diskretizebuli signalebis speqtrebi Semdegi SemTxvevebisaTvis: ωd = 2ω z ; ωd < 2ω z ; da ωd > 2ω z ; . _ signalis amplituda SemosazRvrulia koderis muSa diapazoniT, Tu davuSvebT, rom datvirTvis rezistoris winaRoba tolia 1 omis, maSin daqvantvis damaxinjebebis saSualo simZlavre tolia:

Pdq = δ 2 / 12 Tu Tanabari daqvantvisas davuSvebT, rom daqvantvis damaxinjebebi ararisdamokidebuli diskretebis mniSvnelobebze, maSin signali daqvantvis damaxinjebebi fardobisaTvis (decibelebSi gveqneba:

nax. 1.9. a

nax. 1.9. b

nax. 1.9. g

Ps / Pdq = 10 lg[ x 2 (δ 2 / 12] = 10,8 + 20 lg(v / δ ), sadac, v _ Sesasvleli signalis amplitudis saSualo kvadratuli mniSvnelobaa.

1.6. analizuri signali realuri procesebi (signalebi) zogadad SeiZleba aRweril iqnas drois namdvili funqciebiT. magram zogierT SemTxvevaSi mizanSewonilia maTi warmodgena kompleqsuri sibrtyis veqtorebis saxiT, rac saSualebas iZleva Semotanil iqnas `analizuri signalis~ cneba rogorc determinirebuli, aseve SemTxveviTi signalerbis analizis gamartivebis mizniT. rogorc cnobilia, drois namdvili

x(t ) fun-

qcia SeiZleba warmodgenil iqnes simboluri formiT

x (t ) = x(t ) + jx(t ) = u (t )e jϕ (t )

(1.16)

sadac x(t ) kompleqsuri signalis namdvili nawili Re[ x(t )] = x(t ) emTxveva sawyis funqcias, xolo warmosaxviTi nawili Im[ x(t )] = xˆ (t )

x(t ) funqciasTan

kvadraturaSia (daZrulia x(t ) -s mimarT 900–iT). (1.16) gamosaxulebaSi

Semaval

u (t )

ϕ (t )

funqciebs,

Sesabamisad x(t ) signalis momvlebi da faza ewodeba.

x(t ) signalis kompleqsuri saxiT warmodgenis safuZvelxze SesaZlebelia Semotanil iqnas `analizuri signalis~ cneba:

x(t ) signals ewodeba `analizuri~, Tu x(t ) da xˆ (t )

dakavSirebulia

erTmaneTTan

integralur gardasaxvaTa wyviliT:

e.w.

hilbertis

∞

xˆ (t ) = H [ x(t )] =

1 ⎛ x −τ ⎞ ⎜ ⎟dτ , π −∫∞⎝ t − τ ⎠

x(t ) = H −1[ xˆ (t )] = −

x(t )

xˆ (t )

(1.17)

∞

1 ⎛ xˆ (τ ) ⎞ ⎜ ⎟dτ . π −∫∞⎝ t − τ ⎠

funqciebs

ewodebaT

(1.18) urTierT-

SeuRlebuli funqciebi hilbertis mixedviT. aseT SemTxvevaSi signalis momvlebi da faza gamoisaxeba

U (t ) = [ x(t )]2 + xˆ (t ) 2 ,

ϕ (t ) = arctg

xˆ (t ) . x(t )

(1.19) (1.20)

e.i. x(t ) signalis momvlebisa da fazis gansazRvrisaTvis saWiroa hilbertis gardasaxvis safuZvelze gamoiTvalos x(t ) funqciis SeuRlebuli xˆ (t ) funqcia. (1.19) da (1.20) gamosaxulebebidan gamomdinareobs, rom Tu x(t ) = cos ωt , , maSin hilbertis mixedviT misi SeuRlebuli iqneba xˆ (t ) = sin ωt , funqcia.

1.7. Setyobinebebisa da signalebis fizikuri maxasiaTeblebi Setyobinebebis da signalebis Tvisebebis Sesaswavlad, maTi calkeuli realizaciis ganxilvis nacvlad, mizanSewonilia isini aRiweris ganzogadebuli fizikuri maxasiaTeblebiT, romlebic damaxasiaTebelia mocemuli saxis Setyobinebisa da signa-

lis simravlisaTvis. aseT fizikur maxasiaTeblebs miekuTvneba signalis: xangZlivoba _ Ts , speqtris sigane Fs da dinamikuri diapazoni D s . fizikuri maxasiaTeblebi Ts da Fs ganxiluli

Ds ganisazRv-

iyo zemoT. fizikuri maxasiaTebeli reba

signalis

saSualo

simZlavris

minimaluri

( Pmin ) da maqsimaluri ( Pmax ) sidideebiT da izomeba decibelebiT (db)

Ds = 10 lg( Pmax/ Pmin ).

(1.21)

praqtikaSi signalis saSualo simZlavris minimaluri mniSvneloba Pmin ganisazRvreba arxSi moqmed xmauris simZlavriT Pp -isa. Tu simZlavris maqsimaluri sididis Pmax -is nacvlad ganvixilavT signalis saSualo simZlavres Ps , miviRebT:

Ds = 10 lg( Ps / Pz ). Px / Px

fardobas

(1.22)

signal-xelSeSlis

fardoba

ewodeba. signalebs, romelTa xangZlivobis namravli speqtris siganeze, e.w. baza B s = TsFs sididiT axlosaa erTTan, ewodeba martivi anu elementaruli signalebi.

signalebs

romelTa

bazis

sidide

B s >> 1 ,

ewodeba xmaurisebri signalebi. signalebis, xelSeSlebisa da gadacemis arxebis erT-erT ganzogadebul maxasiaTebels warmoadgens maTi e.w. moculoba.

signalis fizikur Semdegi namravli

moculobaSi

Vs = Ts ⋅ Fs ⋅ Ds analogiurad arxis moculoba

igulisxmeba (1.23)

ganisazRvreba

telekomunikaciis

Va = Ta ⋅ Fa ⋅ Da

(1.24)

sadac Ta , Fa da Da Sesabamisad aris telekomunikaciis arxis gamoyenebis dro, misi gadacemis sixSiruli zoli da im doneebis dinamikuri diapazoni, romlebic telekomunikaciis arxSi gadaicema damaxinjebis gareSe. telekomunikaciis arxSi signalebis daumaxinjeblad gadacemisaTvis saWiroa dakmayofildes Semdegi utoloba

Vs ≤ Vs .

(1.25)

zogadad, signalebis fizikuri maxasiaTeblebis sidide mniSvnelovanwiladaa damokidebuli telekomunikaciis sistemis saxeze.

1.8. SemTxveviTi signalebis (procesebis) maxasiaTeblebi determinirebuli signalebisagan (procesebisagan) gansxvavebiT, romelTa msvleloba calsaxad aris gansazRvruli da romlebic aRiwerebian drois determinirebuli funqciiT, SemTxvevbiTi signalebi asaxaven fizikuri sistemebis iseT cvlilebebs

droSi, romelTa msvlelobis winaswarmetyveleba SeuZlebelia. rogorc aRiniSna, realuri signalebi, romelTac gadaaqvT Setyobineba SemTxveviT xasiaTs atareben. es gamowveulia imiT, rom bunebaSi arsebuli rTuli mizez-Sedegobrivi kavSirebis Sedegad realuri fizikuri Tu sxva procesebis msvleloba ganisazRvreba mravali sxvadasxva saxis faqtoriT, romelTa sruli gaTvaliswineba SeuZlebelia. amasTan unda aRiniSnos, rom am faqtorebis erToblivi zemoqmedeba emorCileba garkveul kanonzomierebebs, romelTa Seswavla SesaZlebelia albaTobis Tanamedrove Teoriis cnobili meTodebis saSualebiT. kerZod, SemTxveviTi procesebis Teoriis saSualebiT. a) SemTxveviTi procesebis (signalebis) tipebi imisda mixedviT, Tu ra saxis simravles (diskretuls Tu uwyvets) miekuTvneba argumentis (dro t) da SemTxveviTi procesis realizaciis X doneebi, SemTxveviTi procesebi (signalebi) pirobiTad SeiZleba daiyos Semdeg oTx tipad: 1. uwyveti SemTxveviTi procesi (signali): t da x Rebulobs nebismier mniSvnelobas namdvili RerZis monakveTze (an SeiZleba mTels RerZze). 2. diskretuli SemTxveviTi procesi (signali): t uwyvetia, xolo x sidideebi-diskretuli (Rebulobs mniSvnelobas erT SesaZlo mniSvnelobidan Î&#x201D;x bijiT). 3. uwyveti SemTxveviTi mimdevroba: t diskretulia (bijiT Î&#x201D;t ) xolo x -ma SeiZleba miiRos nebis37

mieri mniSvneloba ricxviTi RerZis monakveTze (an mTels RerZze). aseT procesebs xSirad ewodebaT procesebi (signalebi) diskretuli droiT. 4. diskretuli SemTxveviTi mimdevroba: t da x diskretulia, diskretuli SemTxveviTi mimdevroba xSirad gamoiyeneba SemTxveviTi procesebis (signalebis) aproqsimaciisaTvis da mniSvnelovnad amartivebs gamokvlevebs. im SemTxvevaSi rodesac ganawilebis funqciebi damokidebulia drois arCeul; momentze, Sesabamis SemTxveviT procesebs ewodeba arastacionaruli, vinaidan maTi mimdinareoba droSi aris araerTgvarovani. xolo Tu ganawilebis funqciebi akmayofilebs pirobas

F1 ( X , t ) = F1 ( X );Ď&#x2030;1 ( x, t ) = Ď&#x2030;1 ( x),

(1.26)

e.i. droze ar arian damokidebuli, maT stacionaruli SemTxveviTi procesebi ewodeba. qvemoT dawvrilebiTaa ganxiluli SemTxveviTi procesebis stacionalobis sakiTxebi. ganvixiloT SemTxveviTi procesebis raodenobrivi maxasiaTeblebi, vinaidan mxolod monacemi imis Sesaxeb, rom ori procesi X (t ) da Y (t ) SemTxveviTia ar iZleva maTi erTmaneTTan Sedarebis saSualebas.

X (t ) SemTxveviTi procesis ganxilvisas davafiqsiroT drois momenti t1 , . e.i. aviRoT kveTa X 1 = X (t1 ) , romelic rogorc yvela SemTxveviTi sidide emorCileba albaTobaTa ganawilebis ama Tu im kanons. aRvniSnoT F1 ( x, t1 ) -iT amorCeuli kveTis ganawilebis integraluri funqcia. es 38

niSnavs,

rom sadac P

F1 ( X , t1 ) = P( X 1 < x),

(1.27)

simboloTi aRniSnulia frCxilebSi miTiTebuli utolobis Sesrulebis albaToba. kveTas X 2 romelic aRebulia drois sxva momentSi t2 (t2 # t1 ), Seesabameba ganawilebis integraluri funqcia

F1 = ( x, t2 )

(indeqsi 1 miuTiTebs imaze,

rom ganawilebis funqcia aris erTganzomilebiani)

F1 ( X , t2 ) = P( X 2 < x), zogad

SemTxvevaSi

F1 = ( X , t1 )

(1.28) da

F1 = ( X , t 2 )

SeiZleba iyos gansxvavebuli sxvadasxva t1 da t2 saTvis.

amitom

X (t )

funqciis

daxasiaTebisaTvis

SeiZleba ganvixiloT ganawilebis funqciaTa ojaxi, romelic damokidebulia t parametrze:

F1 ( X , t ) = P[ X (t ) < x)],

(1.29)

Tu davafiqsirebT ori cvladis funqciaSi (1.29) meore arguments (dro t ), miviRebT drois Sesabamis momentSi X (t ) procesis kveTis ganawilebis integralur funqcias. Tu arsebobs (1.29) funqciis kerZo warmoebuli x cvladis mixedviT

∂F1 ( x, t ) , (1.30) ∂x maSin mas ewodeba X (t ) procesis ganawilebis

ω1 ( x, t ) =

erTganzomilebiani diferencialuri funqcia, anu albaTobaTa erTganzomilebiani simkvrive. x cvladis mixedviT sakmarisad mcire nazrdisas (1.29)-Si gveqneba

Δ[ F1 ( x, t )] = ω1 ( x, t )Δx, 39

(1.31)

zemoaRniSnulis garda SemTxveviTi procesebis kvlevisas didi mniSvneloba aqvs maT raodenobriv maxasiaTeblebs. (anu sxvadasxva gasaSualedebul mniSvnelobebs), vinaidan F1 ( x, t ) funqciebi warmoadgenen SemTxveviTi procesebis SedarebiT arasrul maxasiaTeblebs, da warmodgenas iZlevian procesebis Sesaxeb mxolod calkeul fiqsirebul momentebSi. cxadia, rom SemTxveviTi procesebis ufro sruli aRwerisaTvis gamoyenebuli unda iqnes ori, sami da zogadad n-ganzomilebiani ganawilebis integraluri da diferencialuri funqciebi, romlebsac aqvT Semdegi saxe:

Fn ( x1 , x2 ,..., x; t1t2 ...,tn ) = P{x(t1 ) < x1; x(t2 ) < x2 ..., x(t ) < xn }P , (1.32)

ωn ( x1 , x2 ,..., xn ; t1 , t2 ,...tn ) =

∂ n Fn ( x1 , x2 ,...xn ; t1 , t 2 .,..., tn ) ∂x1∂x2 ...∂xn

(1.33)

unda aRiniSnos, rom mravalganzomilebiani ganawilebis funqciebis gansazRvra warmoadgens rTul amocanas da misi gadawyveta umravles SemTxvevaSi SeuZlebelia. amitom mimarTaven ufro martiv, kerZod ricxobriv maxasiaTeblebs, e.w. momentur funqciebs, romlebic SemTxveviTi procesis droiTi maxasiaTebelia da ganisazRvreba Semdegnairad: ∞

mk = M [ X k (t )] = ∫ x ( k )ω1 ( x, t )dx. −∞

(1.33)-dan warmoadgenen

gamomdinare

{X (t )} k

momenturi

(1.34) funqciebi

funqciis maTematikur molo-

dins, e.i. X (t ) SemTxveviTi funqciis saSualo mniSv-

nelobas realizaciaTa erTobliobis anu ansamblis mixedviT.

M [ ] maTematikuri molodinis niSania. TanamamravlTa ricxvi iwodeba momentis rigad. amgvarad, ∞

m1 = M [ X (t )] = ∫ xω1 ( x, t )dx, −∞

(1.35)

aris pirveli rigis momenti; ∞

m2 = M [ X 2 (t )] = ∫ x 2ω1 ( x, t )dx, −∞

(1.36)

aris meore rigis momenti; ∞

B (t1 , t 2 ) = M [ X (t1 ) X (t 2 )] =

∫∫

−∞

∞

−∞

x1 x2ω2 ( x1 , x2 ; t1 , t 2 )dx1dx2 , (1.37)

aris meore rigis daZruli momenti, romelsac agreTve korelaciis (avtokorelaciis) funqcia ewodeba. xSirad gamoiyeneba meore rigis centraluri momenturi funqciebi e.w. dispersia, romelic gansazRvravs SemTxveviTi funqciis gadaxras saSualo funqciis mimarT ∞

δ 2 (t ) = m1{[ x(t ) − m1 (t )]2 } = ∫ [ x(t ) − m1 (t )]2 ω1 ( x, t )dx (1.38) −∞

am gamosaxulebidan miviRebT, rom meores mxriv

δ 2 (t ) = m2 (t ) − m12 (t ).

(1.39)

1.9. SemTxveviTi procesebis stacionaruloba rogorc zemoT aRiniSna, SemTxveviTi procesis stacionarulobas gansazRvravs maTi maxasiaTeblebis invariantuloba drois aTvlis momentis mimarT. amisada mixedviT arCeven: stacionalur SemTxveviT process viwro azriT; stacionalur SemTxveviT process farTo azriT, arastacionalur SemTxveviT process SemTxveviTi process ewodeba stacionaruli viwro azriT, Tu am process nebismieri rigis albaTobaTa simZlavreebi da momenturi funqciebi ar aris damokidebuli drois aTvlis momentze. aseTi procesebisaTvis nebismieri τ -Tvis marTebulia piroba

ω ( x, t ) = ω ( x, t + τ ).

(1.40)

SemTxveviTi process ewodeba stacionaruli farTo azriT, Tu pirveli da meore rigis momentebi ar aris damokidebuli droze da korelaciis

B(t1, t2 ) = B(τ ) funqcia damokidebulia mxolod τ = t2 −t1 intervalze. Tu SemTxveviTi procesi ar akmayofilebs zemoT CamoTvlil pirobebs, igi arastacionaruli SemTxveviTi procesia. im stacionaruli procesebisaTvis, romlebic xasiaTdebian e.w. ergodikulobis TvisebiT (stacionaruli ergodikuli SemTxveviTi procesebi), gasaSualoeba realizaciaTa mTeli simravlis mixedviT, imave Sedegs iZleva, rasac gasaSualeba erTi realizaciisa drois mixedviT. amasTan, aucilebelia 42

gasaSualebis dro aRebul iqnes sakmaod didi. stacionaruli ergodikuri procesebisaTvis momenturi funqciebi ganisazRvreba Semdegi gamosaxulebiT T /2

mk = M [ x ( K ) (t )] = lim T −∞

1 x ( k ) (t )dt , T −T∫/ 2

(1.41)

da mudmiv sidides warmoadgens. ansamblis mixedviT gasaSualoebis operacia zogjer aRiniSneba swori xaziT gasaSualebuli sididis zemoT

mk (t ) = x ( k ) (t ) ,

(1.42)

xolo drois mixedviT gasaSualoebis operacia _ talRiseburi xaziT

mk = ~ x (k ) stacionaruli ergodikuri marTebulia toloba

(1.43) procesebisaTvis

x ( K ) (t ) = ~ x (k )

(1.44)

pirveli da meore rigis sawyisi momentebis gamosaxulebebi SeiZleba miREebul iqnes zemoT mocemuli zogadi (1.41) gamosaxulebidan. kerZod pirveli rigis momentisaTvis gveqneba

m1 = ~ x (t )

Lim 1 T / 2 x(t )dt. T − ∞ T −T∫/ 2

(1.45)

(1.45) gamosaxulebidan Cans, rom m1 warmoadgens SemTxveviTi procesis mudmiv Semdgens.

Lim 1 T / 2 2 2 ~ m2 = x (t ) = x (t )dt. T → ∞ T −T∫/ 2

(1.46)

Tu miviRebT, rom x(t ) asaxavs denis an Zabvis mniSvnelobas erTeulovan winaRobaze, maSin m2 warmoadgens SemTxveviTi procesis (signalis) saSualo simZlavres. stacionaruli ergodikuli procesebisaTvis dispersiis gamosaxuleba miiRebs Semdeg saxes

M 2 = δ = [ x(t ) − ~ x (t )]2 = Lim 2

T →∞

T /2

∫ [ x(t ) − x (t )] dt. ~

(1.47)

−T / 2

am SemTxvevaSi igi simZlavris saSualo mniSvnelobidan gadaxris proporciulia. (1.39) gamosaxulebis Tanaxmad am SemTxvevaSi

δ 2 = ~x 2 − [ ~x ]2 ,

(1.48)

e.i. aseT SemTxvevaSi dispersia warmoadgens sxvaobas saSualo simZlavres da SemTxveviTi procesis mudmivi Semdgenis simZlavres Soris. stacionaruli engodikuri procesebis SemTxvevaSi korelaciis (avtokorelaciis) funqcia Caiwereba Semdegnairad: T /2

1 B (τ ) = x(t ) x(t + τ ) Lim x(t ) ⋅ x(t + τ )dt. T −T∫/ 2 T →∞

(1.49)

Tu SemTxveviTi procesi Seicavs mudmiv Semdgens, maSin korelaciis funqcia K (τ ) Semdegnairad gamoisaxeba

K (τ ) = Lim T →∞

T /2

∫ [ X (t ) − x (t )][ x(t − τ ) − x(t − τ )]dt ~

(1.50)

−T / 2

korelasciis k (τ ) da B(τ ) funqciebi erTmaneTTan dakavSirebulia Semdegi TanafardobiT

K (τ ) = B(τ ) − [ ~ x (t )]2 .

(1.51)

im procesebisaTvis, romlebsac nulovani saSualo mniSvneloba aqvs, gveqneba

K (τ ) = B(τ )

(1.52)

1.10. stacionaruli SemTxveviTi procesebis korelaciis funqciis Tvisebebi ganvixiloT stacionaruli SemTxveviTi procesebis korelaciis funqciis ZiriTadi Tvisebebi. 1. avtokorelaciis (korelaciis) funqcia_ Tavisi argumentis τ -s klebadi funqciaa. igi miiswrafvis nulisaken τ -s usasrulo zrdisas, Tu SemTxveviT process nulovani saSualo mniSvneloba gaaCnia (ar gaaCnia mudmivi Semdgeni) (nax. 1.10a) winaaRmdeg SemTxvevaSi

B(∞)

miiswrafvis mudmivi

Semdgenis kvadratis mniSvnelobisaken (nax. 1.10b), e.i. mudmivi Semdgenis simZlavris mniSvnelobisaken. 2. avtokorelaciis funqcia τ argumentis luwi funqciaa:

B(τ ) = x(t ) x(t − τ ) = x(t ) ⋅ x(t + r ) = B(−τ ).

(1.53)

b) nax. 1.10

3. avtokorelaciis funqciis B(τ ) mniSvneloba

τ = 0 -is SemTxvevaSi ricxobrivad procesis saSualo simZlavris tolia B (o) = x 2 (t ) = m2 , amasTan B(0) > 0. 4. avtokorelaciis funqciis mniSvneloba τ ≠ 0 momentebisaTvis ar SeiZleba aRematebodes mis sawyis mniSvnelobas τ = 0 e.i.

| B(τ ) |≤ B(0) .

(1.54)

5. ori SemTxveviTi procesis statistikuri kavSiris Sefasebisas Semoitaneba urTierTkorelaciis funqciis cneba

Bxy (τ ) = M [ x(t )Y ( t +τ ) = Lim T →∞ maSin, rodesac

T /2

∫ x(t )y(t + τ )dt

(1.55)

−T / 2

x(t ) = y (t ) urTierTkorelaciis

funqcia gadadis avtokorelaciis funqciaSi. 6. ori SemTxveviTi procesis x(t ) da y (t ) jamis avtokorelaciis funqcia tolia 46

Bz (τ ) =[x(t) + y(t)]x(t +τ ) + y(t +τ )]= x(t)x(t +τ ) + y(t) y(t +τ ) + x(t)y(t +τ ) + + y (t ) ⋅ x(t + τ ) = Bxx (τ ) + Byy (τ ) + 2 Bxy (τ ). aq

Bx (τ ) da

By (τ )

Sesabamisad

x(t )

1.56) da

y (t )

procesebis avtokorelaciis funqciaa, xolo Bxy (τ ) _ urTierTkorelaciis funqcia. 7. avtokorelaciisa da urTierTkorelaciis funqciebi

damokidebulia

rogorc

x(t )

y (t )

SemTxveviTi procesebis Sepirispirebuli mniSvnelobebs Soris statistikur urTierTkavSirze, aseve am procesebis dispersiaze. statistikuri urTierTkavSiris zomad xSirad gamoiyeneba urTierTkorelaciis normirebuli funqcia

bxy (τ ) = Tu

x(t ) = y(t ),

(1.57)

Bxy (τ )

gamosaxulebaSi

miviRebT

(1.57)

Bxx (0) Byy (0) davuSvebT,

gamosaxulebas

rom

normirebuli

avtokorelaciis b(τ ) funqcias (korelaciis koeficienti). korelaciis koeficienti SeiZleba Seicvalos -1 dan +1 farglebSi, e.i.

− 1 ≤ bxy ≤ 1 8. stacionaruli SemTxveviTi procesebisaTvis SeiZleba naCvenebi iqnes τ -s iseTi τ 0 mniSvneloba, rom rodesac τ > τ o adgili ar hqondes statistikur urTierTkavSirs

SemTxveviTi

procesis

mniSvnelo-

bebs Soris, e.i. B(τ ) ≈ 0. τ o sidides ewodeba korelaciis intervali da igi Semdegnairad ganisazRvreba ∞

τo =

∫ | b(τ ) | dτ =

−∞

∞

1 | B(τ ) | dτ . B(O) −∫∞

(1.57)

geometriulad korelaciis intervali SeiZleba gansazRvrul iqnes rogorc im sworkuTxedis fuZis

b(o) = 1 -is tolia, xolo

sigrZe, romlis simaRle farTobi

farTobisa,

romelic

moTavsebulia

| b(τ ) | mrudsa da abcisTa RerZs Soris.

1.11. SemTxveviTi procesis speqtruli maxasiaTeblebi SemTxveviTi procesebis speqtrul maxasiaTeblad gamoiyeneba funqcia G (ω ) , romelsac energetikuli speqtri an simZlavris speqtraluri simkvrive ewodeba. stacionaruli SemTxveviTi procesis energetikuli speqtri G (ω ) da korelaciis funqcia

B(τ )

erTmaneTTan dakavSirebulia viner-xinCinis gardasaxvebis saSualebiT

1 G (ω ) = 2π B(τ ) =

∞

∫ B(τ )e

−∞

− jwτ

∞

dτ = 2 ∫ B(τ ) cos ωτ dτ . (1.59) o

∞

1 G (ω )e jωτ dω = ∫ G (ω ) cos ωτ dω. 2 −∫∞ 0

(1.60)

xSirad wriuli sixSiris ω − s nacvlad gamoiyeneba rxevebis sixSire hercebSi, maSin (1.59) da (1.60) gamosaxulebebi miiRebs Semdeg saxes: ∞

G1 ( f ) = 4 ∫ B(τ ) cos 2πfτ dτ .

(1.61)

da ∞

B (τ ) = ∫ G1 ( f ) cos 2πfτ df .

(1.62)

G1 ( f ) -s gaaCnia ganzomileba vati/hc. arastacionaruli procesebisaTvis sargebloben korelaciis saSualo funqciis B(τ , t ) da saSualo speqtris cnebiT, romlebic erTmaneTTan rebulia furies gardaqmnebis wyviliT:

dakavSi-

∞

B (τ ) = B(τ , t ) = ∫ G ( f , t ) cos 2πfτ df

(1.63)

da ∞

G1 ( f ) = G1 ( f , t ) = 4 ∫ B(τ , t ) cos 2πfτ dτ

(1.64)

unda aRiniSnos, rom determinirebuli procesebis (signalebis) analizisgan gansxvavebiT SemTxveviTi procesis simZlavris speqtruli simkvrive ar iZleva saSualebas aRsdges SemTxveviTi procesis romelime realizacia, radgan igi ar Seicavs monacemebs calkeuli speqtraluri Semadgenlebis Sesaxeb. ganvixiloT mokled energetikuli speqtris efeqturi sigane. im SemTxveviTi procesebis aRweri49

sas, romelTac aqvs araTanabari energetikuli speqtri da romlis intensivoba klebulobs sixSiris zrdisas, gamoiyeneba energetikuli speqtris eqvivalenturi anu efeqturi siganis cneba ∞

∫ G(ω )dω

Ωe =

(1.65)

Gmax (ω )

Gmax _ aris speqtraluri simkvrivis funqciis udidesi

mniSvneloba.

Ω e = 2πFe

SeiZleba

intervalTan

Semdegi

sidide,

dakavSirdes korelaciis gamosaxulebiT ∞

τo =

∫ | B(τ ) | dτ

−∞

B ( 0)

1 G1 (0) , 2 B ( 0)

(1,66)

∞

vinaidan,

G1 ( f ) = ∫ B(τ ) cos 2πfτ dτ = G1 (0)

B(0) = G1 (0)

Fe (procesis saSualo simZlavre). (ami-

tom, (1.65)-dan miviRebT).

τo =

1 2 Fe

(1.67)

(1.67) Tanafardoba warmoadgens amplitudebis speqtris siganesa da impulsis xangZlivobas Soris kavSiris ganzogadebas ( Feτ e = const )

1.12. normaluri SemTxveviTi procesi realur procesebi (signalebi da xelSeSlebi) umravles SemTxvevaSi SeiZleba warmodgenil iqnas normaluri anu hausis SemTxveviTi procesebis sa-

X k ganawilebulia norma-

xiT. SemTxveviTi sidide

lurad, Tu misi albaTobaTa simkvrive ganisazRvreba formuliT −

ω1 ( x) =

2πδ 2

x k2

δ2

(1.68)

sadac, δ 2 dispersiaa. normaluri SemTxveviTi procesis integraluri ganawilebis funqcia Semdegnairad gamoisaxeba ∞

2 1 F (uo ) = P(u < uo ) = e−u / 2du = 1/ 2[1 + φ(uo )]−∞ (1.69) ∫ 2π −∞

sadac x =

, xolo funacias

Φ (uo ) =

2 2π

∫e

−u 2 / 2

du ,

(1.70)

ewodeba albaTobis integrali, anu krampis funqcia da igi tabulirebul funqcias warmoadgens. stacionaruli normaluri procesis ergodikulobis piroba ganisazRvreba misi korelaciis funqciis gamoyenebiT. kerZod, ∞

∫ | B(τ ) | dτ < ∞ .

−∞

(1.71)

SemTxveviT process, romlis speqtruli simkvrive yvela sixSireze erTnairia, ewodeba `TeTri xmauri~. viner-xanCinis gardasaxvis safuZvelze advilad SeiZleba naCvenebi iqnas, rom `TeTri xmauris~ funqciiT

korelaciis

funqcia

B (τ ) = πGδ (τ )

gamoisaxeba

(1.72)

amgvarad, aseTi SemTxveviTi procesebis mniSvnelobebi ar aris korelirebuli, e.i. Sesabamisi korelaciis intervali τ o = 0

1.13. viwrozoliani SemTxveviTi procesebi realur SemTxveviT procesebs gaaCnia SezRuduli energetikuli speqtri, romlis eqvivalenturi sigane ( Ω e ) zeda ω z da qveda ωq sixSireebis mniSvnelobebiT ganisazRvreba: Ω e = ωz − ωq. amitom mizanSewonilia SemTxveviTi procesebi davyoT or jgufad _ viwroziolian da farTozolian SemTxveviT procesebad imisda mixedviT, Tu sixSireTa RerZis ra adgilas aris ganlagebuli Ω e. uwyveti (maT Soris Tanabari) energetikuli speqtris mqone SemTxveviT process ewodeba viwrozoliani, Tu misi energetikuli speqtri moTavsebulia ZiriTad viwro zolSi romelime fiqsirebuli

ωo sixSiris irgvliv. winaaRmdeg SemTxvevaSi process ewodeba farTozoliani.

SemTxveviTi procesis viwrozolianobis piroba analizurad SeiZleba Semdegnairad gamoisaxos:

(Ω e / ωo ) << 1. analizurad viwrozoliani procesi SeiZleba warmovadginoT Semdegi gamosaxulebis saxiT

x(t ) = u (t ) cos[ωot + ϕ (t )],

(1.73)

sadac u (t ) da ϕ (t ) procesis momvlebi da fazaa, xolo ωo _ saSualo sixSire. meores mxriv (1.72) gamosaxuleba SeiZleba Semdegnairad CavweroT

x(t ) = u1 (t ) cos ωot + u2 (t ) sin ωot , aq

u1 (t ) = u (t ) ⋅ cos ϕ (t )

e.w.

sinfazuri,

(1.74) xolo

u2 (t ) = u (t ) ⋅ sin ϕ (t ) _ kvadraturuli Semdgenebia. Tu viwrozoliani procesi emorCileba normalur ganawilebas misi Semdgenebi warmoadgenen hausis procesebs. sinfazuri da kvadratuli Semdgenebis damoukideblobis gamo maTi erToblivi albaTobaTa simkvrive toli iqneba erTganzomilebiani albaTobaTa simkvriveebis namravlisa. amasTan,

ω (u1 ) = 1 / 2π δ x e − (u

2 1

ω (u2 ) = 1 / 2π δ x e − (u

2 2

/ 2δ 2 )

(1.75)

(1.76)

1.14 SetyobinebaTa gadacemis procesis geometriuli warmodgena SezRuduli speqtrisa da sasruli xangZlivobis signalebi geometruilad SeiZleba warmodgenil iqnas n-ganzomilebiani sivrcis elementebis saxiT. gansxvaveba or romelime signals Soris ganisazRvreba manZiliT maT gamomxatvek veqtorebs Soris. imave sixSiruli zolis mqone xelSeSla SeiZleba warmodgenil iqnes n ganzomilebiani veqtoris sa-

xiT. signalis veqtorisagan ( S ) gansxvavebiT, xelSeSlis veqtors

G (ξ )

SeiZleba hqondes nebismieri

sidide da mimarTuleba, e.i. geometriulad igi xasiaTdeba SemTxveviTi veqtoriT romlis bolo ikavebs garkveul moculobas n - ganzomilebian sivrceSi, anu qmnis e.w. `Rrubels~ cvladi simkvriviT, romelic ganisazRvreba albaTobaTa simkvrivis funqciiT. xelSeSlebis signalze zemoqmedebisas, signalis veqtoris irgvliv warmoiSoba `Rrubeli~, romlis cvalebadi simkvrive gamoxatavs jamuri

G G G x = s + ξ veqtoris moxvedris albaTobas moculobis

mocemul

elementSi.

xelSeSlis

`Rrubels~

sferos forma efeqturi radiusiT: r = dac

–

signalis

xangZlivoba,

F-

2TFPx mis

aqvs samier

dakavebuli sixSireTa zoli, Px – xelSeSlis simZlavre.

nax.

1.11-ze

mocemulia

xelSeSlebis

signalze zemoqmedebis geometriuli aRweris gamartivebuli organzomilebiani modeli. ganvixiloT mokled Setyobinebis da signalebis is ZiriTadi gardasaxvebi, romlebsacadgili aqvs informaciis gadacemis sistemaSi.

nax. 1.11

gadamcemi a (t ) Setyobineba (veqtori a ) gardaiq-

mneba S (t ) signalad (veqtori S ). maTematikurad es procesi aRiwereba Semdegnairad

S (t ) = Φ g[a (t )],

(1.76)

sadac Φ g _ gadamcemis operatoria. geometriulad signalis formireba SeiZleba warmodgenil iqnes rogorc Setyobineba A sivrcis gardaqmna signalis S sivrced (aRiniSneba

A → ξ ).

1.12 naxazze warmodgenilia SetyobinebaTa, signalebis da xelSeSlebis organzomilebiani sivrcis modeli.

nax. 1.12

gadacemis arxSi signalze xelSeSlebis zemoqmedebisas warmoiSoba gaurkvevlobis are, romelSic

xdeba x = s + ξ

signali. signalisa da xelSeSlebis

urTierTqmedeba SeiZleba gamoisaxos arxis operatoriT

x(t ) = Φ a[ s (t ), ξ (t )];

(1.77)

sadac Φ a - arxis operatoria. romelic gardaqmnis gadacemuli signalis S sivrced.

nax.

12-ze

mocemuli

sivrces signalis x SemTxvevisaTvis

G S

veqtori gardaiqmneba x veqtorad. mimRebSi aRsdgeba

miRebuli

arxSi

x(t )

signalis

garkveulwilad

mixedviT

damaxinjebuli

Setyobineba _ a * (t ) e.i.

a* (t ) = Φ m[ x(t )],

(1.78)

sadac Φ m _ mimRebis operatoria. amgvarad, SetyobinebaTa gadacemis mTeli sistemis operatori SeiZleba Caiweros Semdegnairad. 56

a (t ) = Φ m{Φ a{Φ m[a (t )]}}

(1.79)

im SemTxvevaSi, Tu xelSeSlebs adgili ara aqvs

{

}

a * (t ) = a(t ) = Φ −g1 Φ g[a (t )] ,

(1.80)

sadac Φ −g1 _ gadamcemis operatoris Sebrunebuli operatoria.

sakontrolo kiTxvebi 1. rogor process ewodeba determinirebuli? 2. rogor process ewodeba SemTxveviTi? 3. miaxloebiT rogor SeiZleba warmovadginoT SemTxveviTi procesi? 4. rogor ganvsazRvroT SemTxveviTi procesis maTematikuri molodini, dispersia da korelaciis funqcia? 5. rogor SemTxveviT process ewodeba stacionaruli farTe gagebiT da romels viwro gagebiT? 6. ra Tvisebebi gaaCnia korelaciis funqciebs? 7. rogor stacionarul procesebs ewodeba ergodikuli? 8. rogor SemTxveviT procesebs ewodeba normaluri? 9. ra kavSiria energetikul speqtrsa da korelaciis funqcias Soris? 10. rogor formulirdeba naikvistis (kotelnikovis) Teorema?

11. rogoria furies mwkrivis zogadi saxe da rogor gamoiTvleba am mwkrivis koeficientebi? 12. ra igulisxmeba signalis doneTa dinamikur diapazonad da signalis moculobad? 13. rogoria furies integralis saxe? 14. ras ewodeba analizuri signali? 15. rogor SemTxveviT procesebs ewodeba viwrozoliani? 16. rogor SeiZleba warmovidginoT SetyobinebaTa gadacemis procesi geometriulad?

Tavi II. informaciis gadacemis Teoriis safuZvlebi rogorc winamdebare wignis SesavalSi aRiniSna telekomunikaciis sistema gankuTvnilia Setyobinebebis gadacemisaTvis. magram sistemis agebis wesi arCevisa da misi SefasebisaTvis aucilebelia am Tezisis Semdgomi dakonkreteba. rogorc cnobilia, telekomunikaciis sistemebSi Setyobineba ganicdis mravalricxovan gardaqmnebs, romlebic mkveTrad cvlis mis eleqtrul warmodgenas da fizikur maxasiaTeblebs. aqedan gamomdinare, SeiZleba gakeTdes daskvna imis Sesaxeb, rom telekomunikaciis sistemaSi gadacemis obieqts warmoadgens Setyobinebis ara eleqtruli warmodgena, aramed iseTi informacia masze, romelic saSualebas iZleva aRvadginoT Setyobinebebi aucilebeli sizustiT. swored es informacia unda darCes invariantuli yvela gardaqmnisas. amasTan gardasaqmneli procesebi (Setyobinebebi da signalebi) aris am informaciis matarebeli da aucilebelia misi raodenobrivi Sefaseba.

2.1. informaciis raodenobrivi zoma rogorc nebismieri fizikuri movlenis, ise informaciisaTvis Semotanil unda iqnes misi raodenobrivi zomis cneba.

cxadia, Tu romelime fizikuri movlenis an materialuri sistemis Sesaxeb Cven winaswar (apriorulad) gagvaCnia yvela cnoba, am movlenis Sesaxeb miRebuli Setyobineba ar Seicavs araviTar informacias, e.i. momxmareblisaTvis araviTar siaxles ar warmoadgens. magram im SemTxvevaSi, Tu adgili aqvs movlenas, romlis Sesaxeb araviTari winaswari (aprioruli) cnobebi ar gagvaCnia, an gvaqvs isini mcire raodenobiT, maSin movlenis Sesaxeb miRebuli Setyobineba momxmareblisaTvis moulodnelia da bevr siaxles Seicavs, e.i. informaciulad mdidaria. zogadad, SetyobinebaTa SemTxveviTi bunebis gamo misi momxmareblisaTvis yovelTvis arsebobs gaurkvevloba imis Sesaxeb, Tu SetyobinebaTa ansamblidan romeli konkretuli xi Setyobineba iqna gadacemuli. aqedan gamomdinare, im informaciis raodenobriv sazomad, romelsac Seicavs erTi Setyobineba gamoyenebul unda iqnes SetyobinebaTa mTeli ansamblidan am Setyobinebis amorCevis P ( xi ) albaTobis nebismieri funqcia. analizisaTvis sasurvelia SerCeul iqnas monotonurad klebadi 1 / P ( xi ) funqcia, vinaidan am SemTxvevaSi P ( xi ) -is Semcirebas Seesabameba momxmareblisaTvis xi Setyobinebis amorCevis moulodnelobis zrda. gamoTvlebisaTvis mizanSewonilia informaciis raodenoba gansazRvrul iqnas logariTmul erTeulebSi

I ( xi ) = log

1 = â&#x2C6;&#x2019; log P( xi ). P( xi ) 60

(2.1)

(2.1.) gamosaxulebiT gansazRvrul informaciis raodenobas Semdegi Tviseba gaaCnia: 1. vinaidan WeSmariti movlenis albaToba

P( xi ) = 1, amitom I ( xi ) = 0, e.i. informaciis raodenoba determinirebul SetyobinebebSi nulis tolia. 2. vinaidan SeuZlebeli movlenis albaToba

P ( xi ) = 0,

amitom am SemTxvevaSi

I ( xi ) = ∞. radgan

yvela realuri movlenis albaToba gansxvavdeba nulisagan, e.i. maTi albaToba akmayofilebs pirobas

0 < P( xi ) < 1, , amitom I ( xi ) yovelTvis dadebiTi da sasruli sididea. 3. informaciis logariTmul zomas gaaCnia e.w. aditiurobis Tviseba, rac imaSi mdgomareobs, rom informaciis raodenoba, romelsac Seicavs ramdenime damoukidebeli Setyobineba, calkeul SetyobinebaSi mocemuli informaciis raodenobis jamis tolia. es gamomdinareobs iqedan, rom n damoukidebeli Setyobinebebis erToblivi albaToba calkeuli Setyobinebebis albaTobebis namravlis tolia

P ( x1 , x2 ,..., xn ) = P( x1 ) ⋅ P ( x2 )...P( xn ).

(2.2)

informaciis raodenoba am SemTxvevaSi toli iqneba n

i −1

i =1

I ( x1 , x2 ...., xn ) = − log P( x1 , x2 ,..., xn ) = −∑ log P( xi ) = ∑ I ( xi ) logariTmis fuZe informaciis mosaxulebaSi (2.1) SeiZleba aRebul magram telekomunikaciaSi yvelaze leba hpova oris tolma fuZem.

(2.3)

raodenobis gaiqnes nebismieri, farTo gavrceam SemTxvevaSi

informaciis raodenoba izomeba orobiT erTeulebSi, anu bitebSi. diskretuli informaciis gadacemis sistemebSi, e.w. orobiT sistemebSi, Setyobinebebis gadacemisaTvis gamoiyeneba ori simbolo: `0~ da `1~.

2.2. diskretuli Setyobinebebis entropia zogadad informaciis raodenobrivi sazomi I(xi ) (2.1) saSualebas iZleva gamoiTvalos informaciis raodenoba im SemTxvevaSi, rodesac Setyobinebebis albaTobebi gansxvavdebian nulisagan. informaciis raodenoba I ( xi ) , romelic moTavsebulia x diskretul SetyobinebaTa wyaros calkeul elementarul xi SetyobinebaSi, axasiaTebs mxolod am konkretul Setyobinebas da ar iZleva warmodgenas informaciis im saSualo raodenobaze I (x) , romelsac gamoimuSavebs wyaro erTi nebismieri elementaruli Setyobinebis amorCevisas SetyobinebaTa mTeli ansamblidan. informaciis saSualo raodenoba, romelic axasiaTebs SetyobinebaTa wyarois mTlianad, informaciis Teorias erT-erTi ZiriTadi cnebaa. informaciis saSualo raodenoba, romelsac Seicavs erTi x Setyobineba, ganisazRvreba rogorc im informaciis raodenobis maTematikuri molodini, romelsac Seicavs SemTxveviTad (albaTobiT P ( xi ) ) arCeuli xi Setyobineba

i =1

H ( x) = I ( x) = ∑ P( xi ) I ( xi ) = −∑ P( xi ) log P( xi ) , (2.4) sadac H (x) _ diskretuli procesis gaurkvevloba anu entropia. es termini gadmoRebulia Termodinamikidan. informaciis TeoriaSi H (x) entropia agreTve axasiaTebs situaciis gaurkvevlobas Setyobinebis miRebamde, vinaidan winaswar ar aris cnobili Setyobinebebis ansamblidan romeli Setyobineba iqneba gadacemuli. amgvarad, rac ufro didia entropia, miT ufro met informacias Seicavs saSualod erTi Setyobineba. im SemTxvevaSi Tu yvela Setyobineba damoukidebeli da Tanabrad albaTuria e.i. rodesac

P ( x1 ) = P( x2 ) = ... = P( xm ) = P( x) = 1 / n. n -SetyobinebaTa sadac raodenobaa,

(2.5)

maSin entropia maqsimaluria da (2.4)-is Tanaxmad, tolia n 1 1 H ( x)... = H o ( x) = −∑ log 2 = log n n k =1 n

(2.6)

amgvarad, informaciis raodenoba SetyobinebaSi damokidebulia mxolod SetyobinebaTa n raodenobaze ansamblSi. rac ufro metia Setyobineba, miT ufro didia gaurkvevloba da miT ufro didi raodenobis informacias Seicavs gadacemuli Setyobineba . zogadad, ansamblis SetyobinebaTa araTanabroba amcirebs entropias mis maqsimalur mniSvnelobasTan SedarebiT. meores mxriv, SetyobinebaTa Soris statistikuri urTierTkavSiris gaTvaliswinebas miv-

yavarT entropiis Semdgomi Semcirebisaken, vinaidan am kavSiris gaTvaliswineba amcirebs Setyobinebebis amorCevis Tavisuflebas ansamblidan da amiT amcirebs yovel axlad arCeul Setyobinebaze mosul informaciis saSualo sidides. statistikuri kavSiri mosalodnel da winamdebare Setyobinebebs Soris SeiZleba gamosaxul iqnes an erToblivi albaTobiT P ( xi , x j ) , an pirobiTi albaTobiT P ( xi , x j ) . ukanaskneli warmoadgens xi Setyobinebis gadacemis albaTobas im pirobiT, rom ma-

xi Setyobineba (2.1) gamosa-

namde gadacemuli iyo

xulebis Tanaxmad, informaciis raodenoba, romelsac Seicavs

Setyobineba cnobili

x j Setyobi-

nebis SemTxvevaSi, toli iqneba

I ( xi / x j ) = − log P( xi / x j ).

(2.7)

informaciis saSualo raodenoba am SemTxvevaSi

I i ( xi / x j ) entropiiT, romelic

gamoisaxeba pirobiT

gamoiTvleba rogorc I ( xi / x j ) informaciis maTematikuri molodini n

H 2 ( x) = H ( xi / x j ) = −∑∑ P( xi , x j ) log P( xi / x j ).

(2.8)

i =1 j =1

analogiurad ganisazRvreba pirobiTi entropia, rodesac statistikuri kavSirebi vrceldeba n Setyobinebebze n

H n ( x) = H ( xi / x j , xk ,..., xn ) = −∑ ∑ ...∑ P ( xi , x j ,..., xn ) log P ( xi / x j , xk ,..., xn ). i =1 j −1

n=1

zemoaRniSnulidan

gamomdinareobs, rom zoga-

dad

H o ( x) ≥ H1 ( x) ≥ H 2 ( x) ≥ ... ≥ H n ( x). (2.9) im SemTxvevaSi, Tu adgili aqvs n Setyobinebebisagan Semdgar mimdevrobas, informaciis raodenoba toli iqneba

I n = nH (x).

(2.10)

aqedan gamomdinareobs, rom informaciis raodenoba SeiZleba gazrdil iqnes ara marto Setyobinebebis raodenobis gazrdiT, aramed SetyobinebaTa wyaros entropiis zrdis xarjze. zogierT SemTxvevaSi saWiro xdeba im informaciis raodenmobis gamoTvla, romelsac Seicavs romelime X (t ) SemTxveviTi procesi Y (t ) procesebis Sesaxeb. magaliTad, telekomunikaciis sistemis mimReb mxareze miRebuli signalebi gadacemuli signalebis Sesaxeb. informaciis saSualo raodenobas, romelsac

Y (t )

procesi

Seicavs

X (t )

procesis

Sesaxeb, ewodeba urTierTinformaciis saSualo raodenoba da aRiniSneba I ( x, y ). urTierTinformaciis saSualo raodenoba toli iqneba H (x) entropiisa mxolod maSin, rodesac Y (t ) procesis nebismieri xdeba

absoluturi

y k (t ) realizaciis aRwarmoeba sizustiT

ansambli calsaxad asaxavs

X (t )

rodesac

{y (t )}

process. magram

praqtikulad y k (t ) realizaciaze dakvirvebis Semdeg yovelTvis rCeba gaurkvevloba

x ( k ) (t ) realizaciis

Sesaxeb, rac gamowveulia gadacemis arxSi moqmedi xelSeSliT. procesis

gaurkvevloba

pirobiTi

H ( x / y)

ganisazRvreba entropiiT

X (t )

miRebuli

cnobili Y (t ) procesis dros. vinaidan aprioruli gaurkvevloba

X (t ) procesis Sesaxeb H (x) –is to-

lia, xolo I ( x, y ) gansazRvravs informaciis saSualo raodenobis Semcirebas Y (t ) -s miRebis Semdeg. amitom

I ( x, y ) = H ( x) − H ( x / y ).

(2.11)

Tu (2.11) gamosaxulebaSi SevitanT (2.4) da (2.8) gamosaxulebebs da mxedvelobaSi miviRebT, rom

P ( x / y ) = P( x, y ) / P ( y ), gveqneba n

I ( x, y ) = −∑∑ P( x, y ) log[ P( x, y ) / P( y )].

(2.12)

x =1 y −1

zemoaRniSnulidan

gamomdinareobs,

rom

X (t ) procesi ar aris damokidebuli Y (t ) -ze, e.i. P ( x / y ) = P( x), maSin pirobiTi entropia H(x/ y) = H(x) da amitom I ( x, y ) = 0, , xolo rodesac X (t ) da Y (t ) Soris arsebobs calsaxa urTierTdamokidebuleba, maSin H ( x / y ) = 0 da I ( x, y ) = H ( x) = I ( x). aRsaniSnavia, rom noba

I (x) -s ewodeba sakuTari informaciis raodeda

igi

asaxavs

informaciis

saSualo

raodenobas, romelsac Seicavs X (t ) procesi Tavis Tavis Sesaxeb. diskretuli SemTxveviTi procesis sakuTari informacia misi entropiis tolia.

2.3. diskretuli arxis gadacemis siCqare da gamtarunarionoba ganvixiloT telekomunikaciis arxi, romelSic gadaicema X SetyobinebaTa erToblioba, romelTa entropiaa H ( X ) . arxSi Setyobinebebi ganicdian H (N ) entropiis

mqone

Sedegadac

miRebis

xelSeSlebis adgilas

zemoqmedebas,

gveqneba

ris

Setyobi-

nebaTa erToblioba H (Y ) entropiiT. amgvarad, Tu gadacemuli informaciis raodenobaa H ( X ), , xolo miRebuli _ H (Y ) , maSin H ( X / Y ) aris informaciis is raodenoba, romelmac SeiZleba gansazRvros X -is mniSvneloba, rodesac cnobilia informaciis is raodenoba, romelsac Seicavs Y . amitom

H (X /Y )

SeiZleba miCneul iqnes rogorc informaciis is raodenoba, romelic ikargeba arxSi xelSeSlebis zemoqmedebisas da mas SeiZleba vuwodoT informaciis danakargi. amgvarad, Tu gadacemuli informaciis saerTo raodenobas

H ( X ) gamovaklebT xel-

SeSliT

informaciis

H ( X / Y ), ,

ganpirobebuli miviRebT

informaciis

danakargs raodenobas

I (Y , X ) , romelsac Seicaven miRebuli Y nebaTa erToblioba gadacemuli Sesaxeb, e.i.

Setyobi-

Setyobinebebis

I (Y , X ) = H ( X ) â&#x2C6;&#x2019; H ( X / Y ).

(2.13)

(2.13) gamosaxuleba warmoadgens informaciis im raodenobas, romelic saSualod gadaicema arxSi xelSeSlebis zemoqmedebisas.

unda aRiniSnos, rom rodesac xelSeSlebis done arxSi Zalian mcirea an xelSeSlebi saerTodar arsebobs, maSin pirobiTi entropia

H ( X / Y ) = 0 da

informacia, romelsac Y Seicavs X -is Sesaxeb, toli iqneba gadacemuli Setyobinebebis I (Y , X = H ( X ) entropiisa, xolo im SemTxvevaSi, rodesac xelSeSlebis done Zalian didia, Setyobinebebi X da Y xdebian statistikurad damoukidebelni, amitom

H ( X / Y = H ( X ) da I (Y , X ) = H ( X ) − H ( X ) = 0 , e.i. Y Setyobinebebi am pirobebSi ar Seicaven araviTar informacias X-is Sesaxeb. vinaidan

H ( X / Y ) = H ( X , Y ) − H (Y ),

(2.14)

amitom (2.13) SeiZleba Caiweros Semdegnairad

I (Y , X ) = H ( X ) + H (Y ) − H ( X , Y ).

(2.15)

Tu gaviTvaliswinebT, rom n

H ( X ) = −∑ P( xk ) log P( xk ) , k =1

H ( X , Y ) = −∑∑ P( xk , y j ) log B( xk , y j ), j =1 k −1 n

H (Y ) = −∑ P( y j ) log B( y j ) j =1

SeiZleba vaCvenoT, rom sabolood (2.15) miiRebs saxes n

I (Y , X ) = ∑∑ P( xk , y j ) log[ P( xk , y j ) : P( xk ) P( yi )]. (2.16) j =1 k =1

(2.16) gamosaxuleba saSualebas gvaZlevs gamovTvaloT informaciis raodenoba X -is Sesaxeb Y -Si albaTobebis P ( xk ), P ( y j ) da P ( xk , y j ) saSualebiT. vinaidan informaciis gadacema xdeba droSi, mizanSewonilia SemovitanoT informaciis gadacemis siCqaris cneba, rogorc informaciis im raodenobisa, romelic gadaicema saSualod drois erTeulSi. informaciis gadacemis siCqare ganisazRvreba Semdegnairad R = Lim T →∞

I ( X ⋅Y ) H (X ) − H (X /Y) = Lim , T T T →∞

biti/wm. (2.17)

informaciis gadacemis siCqaris maqsimalur mniSvnelobas garkveuli SezRudvebis mxedvelobaSi miRebiT ewodeba arxis gamtarunarianoba da igi Semdegnairad gamoisaxeba max/(X , Y ) max[H ( X ) − H ( X / Y )] = lim , T T T →∞ T →∞

C = max R = lim

biti/wm. (2.18)

im SemTxvevaSi, rodesac arxSi ara aqvs adgili xelSeSlebs, informaciis gadacemis siCqare da arxis gamtarunarianoba, (2.13) gaTvaliswinebiT, Semdeg saxes miiRebs:

R = lim T →∞

C = max R = lim T →∞

H (X ) , or. erT./wm (2.19) T

max H ( X ) , T

or. erT. /wm. (2.20)

Tu informaciis gadacema xdeba martivi orobiTi signalebiT (magaliTad, toli xangZlivobis denis impulsebis gagzavniT an ar gagzavniT arxSi), maSin arxis gatarebis zoli damokidebulia e.w. manipulaciis sixSireze, romelic aiReba periodis naxevris xangZlivobis mqone sworkuTxa impulsebis perioduli mimdevrobis (e.w. meandris) pirveli harmonikis toli, e.i. Fm = 1 / 2τ .

nax. 2.1.

am pirobebSi,

C = 2 Fm . maqsimaluri gadacemis

siCqaris aseT mniSvnelobas ewodeba naikvistis zRvari. im SemTxvevisaTvis, rodesac arxSi moqmedebs xelSeSla, SeiZleba naCvenebi iqnes, rom, magaliTad, orobiTi simetriuli damaxsovrebis armqone arxis gamtarunarianoba gamoisaxeba Semdegnairad

1⎡ 1 1 ⎤ ⎢1 − Po log − (1 − Po ) log ⎥, τ⎣ Po 1 − Po ⎦

(2.21)

sadac Po _ miRebis Secdomis sruli albaTobaa.

nax.

2.1-ze

mocemulia

C-s

damokidebuleba

Secdomis albaTobaze Po , saidanac Cans, rom, rodesac

Po = 0,5 gamtarunarianoba xdeba nulis toli,

vinaidan am SemTxvevaSi gadacemul da miRebul signalebs Soris ar arsebobs araviTari urTierTkavSiri.

2.4. Senonis ZiriTadi Teoremebi diskretuli arxebisaTvis diskretuli arxebisaTvis Senonis mier damtkicebuli iqna ori ZiriTadi Teorema, romelsac aqvs fundamentaluri mniSvneloba informaciis gadacemis TeoriaSi. pirveli Teorema exeba uxelSeSlo diskretul arxebs. aseTi saxis arxebisaTvis es Teorema Semdegnairad SeiZleba Camoyalibdes: im SemTxvevaSi, rodesac wyaros mwarmoebluroba Rw naklebia arxis gamtarunarianobaze C , ( Rw < C ) yovelTvis arsebobs kodirebis iseTi xerxi, romelic saSualebas iZleva arxSi gadacemul iqnes wyaros yvela Setyobineba. xolo Tu Rw > C , , aseTi gada-cemis ganxorcieleba SeuZlebelia. Teoremis bolo debulebis damtkiceba SeiZleba Semdegnairad: vTqvaT, rom wyaros mwarmoebluroba metia, vidre arxis gamtarunarianoba, e.i.

Rw > C , ,

maSin am SemTxvevaSi wyaros yvela Setyobinebis gadasacemad arxSi saWiro iqneba, rom informaciis

gadacemis siCqare ar iyos naklebi, vidre wyaros mwarmoebluroba, e.i. R â&#x2030;Ľ Rw, da Sesabamisad, R â&#x2030;Ľ C es ki SeuZlebelia, vinaidan ganmartebis Tanaxmad gamtarunarianoba tolia gadacemis siCqaris maqsimumisa ( C = max). . amgvarad, rodesac

Rw > C , wyaros

yvela Setyobinebis gadacema arxSi SeuZlebelia. Senonis zemoaRniSnuli Teorema safuZvlad udevs e.w. optimaluri kodirebis princips. optimaluri kodirebisas, informaciis wyaros Sesabamisi SeTanxmebiT arxTan, xorcieldeba informaciis gadacemis siCqaris miaxloeba arxis gamtarunarianobasTan. ganvixiloT e.w. Senonfanos optimaluri kodirebis procedura, romelic SemdegSi mdgomareobs. yvela SesaZlo Setyobineba ganlagdeba maTi albaTobebis klebadobis mixedviT. miRebuli ranJirebuli (mowesrigebuli) mimdevroba gaiyofa or jgufad, ise rom jgufebis jamuri albaToba daaxloebiT toli aRmoCndes. Semdeg zeda jgufs mieniWeba koduri simbolo ~0~, xolo qveda jgufs _ ~1~. Setyobinebebis miRebul qveda jgufs isev yofen or Tanabar albaTobis mqone jgufad da isev xdeba koduri simboloebis miniWeba zemoaRniSnuli wesiT. aseTi operacia grZeldeba manam, sanam saboloo jgufSi ardarCeba TiTo Setyobineba. kodirebis aseTi algoriTmi uzrunvelyofs koduri kombinaciebis saSualo sigrZis minimizacias, gadacemis siCqaris gazrdas da mis miaxloebas arxis gamtarunarianobasTan.

nax. 2.2.

Senon-fanis kodirebis procedura wyarosaTvis, romelic

gamoimuSavebs

aprioruli albaTobebiT

oTx

Setyobinebas

x1 ÷ x4

P ( x1 ) ÷ P ( x4 ), gamosaxulia

kodirebis grafiT (nax. 2.2), romelic gviCvenebs rogor xdeba ranJirebuli Setyobinebebis mimdevrobis dayofa jgufebad da romeli koduri simbolo mieniWeba jgufebs calkeul Setyobinebebs dayofis yovel bijze. rogorc vxedavT, Senon-fanos optimaluri kodirebisas yvelaze didi albaTobis mqone Setyobinebebs mieniWeba yvelaze mokle koduri kombinaciebi, mcire albaTobis mqone Setyobinebebs ki _ grZeli. ganvsazRvroT axla ris tolia gadacemis siCqare Senon-fanos kodirebisas. Tu (2.19) gamosaxulebaSi CavTvliT, rom H ( X ) = nH ( S ) da T = nτ sadac,

n _ Set-yobinebaTa raodenobaa, xolo τ Setyobinebis saSualo xangZlivoba, maSin

_ erTi

H (X ) nH ( S ) H ( S ) = lim = , T nτ τ T −∞ n−∞

R = lim

(2.22)

sadac, n

H ( S ) = −∑ P ( Si ) log P ( Si ),

(2.23)

i −1

i =1

τ = ∑τ i P( Si ) = τ ∑ ni P( Si )

(2.24)

(2.24) gamosaxulebaSi τ = ni τ o ;τ o _ koduri kombinaciebis erTi simbolos xangZlivobaa; n _ simboloebis raodenoba kodur kombinaciaSi. unda aRiniSnos, rom Senon-fanos kodebs axasiaTebs e.w. dauyvanlobis Tviseba, rac imaSi mdgomareobs, rom am kodebs ar sWirdebaT specialuri e.w. gamyofi simboloebi calkeuli koduri kombinaciebis dasawyisisa da daboloebis garCevisaTvis. es imiT aris gamowveuli, rom Senon-fanos kodebSi mokle koduri kombinaciebi arasdros ar emTxvevian ufro grZeli koduri kombinaciebis dasawyiss da piriqiT. Senonis meore ZiriTadi Teoirema exeba arxebs, romlebSic moqmedeben xelSeSlebi. es Teorema msgavsia zemoT moyvanili Teoremisa uxelSeSlo arxebisaTvis da mdgomareobs SemdegSi. Tu informaciis wyaros mwarmoebluroba naklebi an tolia arxis gamtarunarianobisa, maSin arsebobs kodirebis xerxi, romelic saSualebas iZleva wyaros yvela

Setyobinebis gadacemisa Secdomis mcire albaTobiT. Tu Rw > C , maSin aseTi gadacema SeuZlebelia. Cven aq ar moviyvanT am Teoremis damtkicebas. avRniSnavT mxolod, rom es Teorema ar aris konstruqciuli, e.i. ar iZleva optimaluri kodebis povnis inJinrul gzebs. magram Teoremas aqvs didi mniSvneloba, vinaidan igi safuZvlad udevs sul axal Sexedulebebs informaciis gadacemis SesaZleblobebze. ase, magaliTad, cxadia, rom Secdomis albaTobis SemcirebiT SeiZleba mkveTrad gaizardos informaciis gadacemis siCqare. amis miRweva SeiZleba, magaliTad, Setyobinebebis mravaljeradi gameoirebiT. Secdomebis albaTobis nulamde dayvanisaTvis intuiciurad gamodis, rom gadacemis siCqare unda iyos usasrulod didi. Senonis Teorema ki amtkicebs, rom Secdomebisa da xelSeSlebis arseboba arxSi TavisTavad ar qmnis dabrkolebas informaciis gadacemisaTvis didi sizustiT. igi mxolod zRudavs informaciis gadacemis siCqares. informaciis gadacemis saWiro didi sizuste da gadacemis siCqaris sasruli sidide ar gamoricxaven erTmaneTs. amaSi mdgomareobs am Teoremis didi mniSvneloba. amasTan, unda aRiniSnos rom, Teorema ar iZleva pasuxs kiTxvaze, Tu rogor unda ganxorcieldes Sesabamisi kodireba. am amocanis gadaWra rTulia da moiTxovs damatebiTi gamokvlevebis Catarebas.

2.5. diskretul SetyobinebaTa wyaros mwarmoebluroba da siWarbe SetyobinebaTa wyaros mwarmoebluroba ewodeba im sakuTari informaciis raodenobas I ′( x), , romelsac Seicavs x(t ) Setyobineba drois erTeulSi, misi ganzomilebaa ansambli

biti/wamSi.

diskretulia,

x(t )

Tu misi

yoveli

SetyobinebaTa Setyobinebis

I ( x) da wyaros gamomuSavdeba siCqariT v

sakuTari informaciis raodenoba mier Setyobinebebi Setyobineba wamSi, maSin

I ′( x) = vI ( x).

(2.25)

amgvarad, wyaros mwarmoebluroba mTlianad ganisazRvreba SetyobinebaTa statistikiT, romlis cvlilebiT SeiZleba miRweul iqnes wyaros maqsimaluri mwarmoebluroba. rogorc aRvniSneT, diskretuli Setyobinebebis SemTxvevaSi sakuTari informacia maTi entropiis tolia I ( x) = H ( x) da aseTi SetyobinebebisaTvis

I ′( x) = vH ( x).

(2.26)

meores mxriv, diskretuli Setyobinebebis entropia maqsimaluria, rodesac Setyobinebebi damoukidebelni da Tanabrad saalbaToni arian

H max ( x) = H ( x) = log n. amgvarad, diskretul SetyobinebaTa wyaros maqsimalurad SesaZlo mwarmoebluroba tolia

′ ( x) = v log n I max

(2.27)

Setyobinebebis araTanabaralbaToba da maT Soris statistikuri urTierTkavSirebis arseboba amcirebs, rogorc viciT, wyaros entropias da iwvevs Setyobinebebis informaciuli tevadobis Semcirebas. meore mxriv, Tu Setyobinebebs Soris arsebuli statistikuri kavSirebis xasiaTi cnobilia, maSin am Setyobinebis nawili SeiZleba saerTod ar gadaices, vinaidan maTi aRdgena mimReb mxares SesaZlebeli xdeba am cnobili kavSirebis safuZvelze. SetyobinebaTa ansamblSi Warbi Setyobinebebis arsebobisas izrdeba gadacemis xangZlivoba da mcirdeba telekomunikaciis sistemis efeqtianoba. SetyobinebaTa wyaros siWarbe raodenobrivad SeiZleba Sefasdes Semdegnairad

′ ( x) = 1 − H ( x) / H max ( x) = 1 − H ( x) / log n. (2.28) P = 1 − I ′( x) / I max siWarbis sidide icvleba 0 ≤ P ≤ 1 farglebSi.

k = H ( x) / H max ( x).

(2.29)

K koeficients ewodeba SekumSvis koeficienti da igi gviCvenebs, Tu ra sididemde SeiZleba SeikumSos Setyobinebebis ansambli siWarbis mospobisas. siWarbes, gamowveuls Setyobinebebis statistikis araTanabrobiT da urTierTkavSirebiT, statistikuri siWarbe ewodeba. aRsaniSnavia, rom SetyobinebaTa wyaros siWarbe yovelTvis ar aris am ukanasknelis uaryofiTi Tviseba. zogierT SemTxvevaSi statistikuri urTierTkavSirebi Setyobinebebs Soris saSualebas gvaZleven aRvadginoT calkeuli Setyobineba maTi damaxinjebis SemTxvevaSi, e.i. siWarbe SeiZleba gamoyenebul

iqnes SetyobinebaTa gadacemis sisworis gazrdisaTvis, an informaciis gadacemis siCqaris SemcirebisTvis.

2.6. uwyveti Setyobinebebis entropia vinaidan uwyveti Setyobinebebis romelime mniSvnelobis gamoCenis albaToba nulis tolia, amitom zogadad aseTi saxis Setyobinebebis entropia usasrulod didia. magram realur pirobebSi signalebs, jer erTi, SezRuduli speqtri gaaCniaT, rac saSualebas iZleva warmovadginoT isini anaTvlebis erTobliobiT da, meorec, xelSeSlebis gavlenis gamo am anaTvlebis garCevadi mniSvnelobebis raodenoba sasrulia. yovelive amis gamo SegviZlia SevcvaloT uwyveti signali diskretuli mniSvnelobebis erTobliobiT da ganvazogadoT uwyvet signalebze entropiis is gamosaxulebebi, romlebic miRebuli iyo zemoT diskretuli SetyobinebebisaTvis.

nax. 2.3

vTqvaT, mocemulia uwyveti Setyobinebis mniSvnelobebis albaTobaTa simkvrive (nax. 2.3). procesis xasiaTi mniSvnelovnad ar Seicvleba, Tu x uwyvet mniSvnelobas SevcvliT diskretuli

x1 , x2 ,..., xk ,..., xn

mniSvnelobebiT, romlebic erTmaneTisagan daSore-

Δx

bulia

intervaliT da romelTa albaTobebia

P = ( xk )Δx . cxadia, aseTi Secvla miT ufro zusti iqneba, rac ufro mcirea Δx . maSin n

k =1

R=1

H ( x) = −∑ω( xk )Δx log{ω( xk )Δx} = −∑ω( xR )Δx logω( xR ) − ∑ω( xR )Δx log Δx .

(2.30)

zRvarze gadasvlisas, rodesac Δx → 0, (2.30) gamosaxulebis pirveli Sesakrebi miiRebs Semdeg saxes ∞

− ∫ ω ( x) log ω ( x)dx.

(2.31)

−∞

xolo meore Sesakrebi toli iqneba- log Δx , vinaidan albaTobaTa simkvrivis Tvisebebis safuZvelze ∞

∫ ω ( x)dx = 1.

−∞

amgvarad miviRebT, rom ∞

H ( x) = − ∫ ω ( x) log ω ( x)dx − log Δx = h( x) − log Δx. (2.32) −∞

gamosaxulebas ∞

h( x) = − ∫ ω ( x) log ω ( x)dx.

(2.33)

−∞

ewodeba uwyveti Setyobinebebis diferencialuri entropia (vinaidan ganisazRvreba albaTobaTa diferencialuri kanoniT ω ( x) ). log Δx sidide damo-

kidebulia mxolod Δx intervalze da am ukanasknelis mudmivobis dros mudmiv sidides warmoadgens. SeiZleba naCvenebi iqnes, rom informaciis saSualo raodenoba, romelsac Seicavs uwyveti procesis

erTi

anaTvali

uwyveti

X (t )

y (t )

procesis

erTi anaTvlis Sesaxeb, diskretuli SemTxveviTi procesis analogiurad gamoisaxeba Semdegnairad

I ( x, y ) = h( x) − h( x / y ),

(2.34)

sadac, ∞ ∞

∞ ∞

−∞−∞

− ∞− ∞

h( x / y ) = − ∫ ∫ ω ( x, y ) log ω ( x / y )dxdy = − ∫ ∫ ω ( x, y ) log

ω (τ , y ) dxdy , ω ( y)

(2.35)

e.w. pirobiTi diferencialuri entropiaa. (2.33) da(2.35) gamosaxulebebis safuZvelze urTierTinformaciis saSualo raodenoba SeiZleba warmodgenil iqnes Semdegnairad ∞ ∞

I ( x, y ) =

ω ( x, y )

∫ ∫ ω ( x, y) log ω ( x)ω ( y) dxdy.

(2.36)

−∞−∞

unda aRiniSnos, rom diskretuli Setyobinebebis entropiisagan gansxvavebiT, diferencialuri entropia damokidebulia uwyveti Setyobinebebis ganzomelebaze. amitom, miuxedavad amisa, rom igi axasiaTebs Setyobinebebis wyaros gaurkvevlobis xarsixs, ar warmoadgens am ukanasknelis informaciul zomas. uwyveti Setyobinebebis informaciuli zoma ganisazRvreba mxolod diferencialuri entropiebis sxvaobiT (2.34). amgvarad, uwyveti Setyobinebebis rogorc upirobo (2.33), aseve pirobiTi (2.35) entropiebi ganisazRvrebian mxolod SetyobinebebaTa statistikiT. am

ukanasknelebis Tvisebebidan aRsaniSnavia Semdegi ori: a) Tu uwyveti Setyobinebebis dispersiebi SezRuduli sididea, maSin entropia maqsimaluria normaluri ganawilebis dros, e.i. rodesac

ω ( x) = 1 / σ 2π cxp[− x 2 / 2σ 2 ],

(2.37)

b) Tu uwyveti Setyobinebebis dispersia ar aris SezRuduli,

h(x)

maSin

entropia

maqsimaluria

Tanabari ganawilebis dros

ω ( x) = 1 /( xmax − xmin ), sadac ( xmax − xmin ) Setyobinebebis SesaZlo mniSvnelobebis intervalia. am SemTxvevaSi entropia

h( x) = log( xmax − xmin ).

(2.38)

aRsaniSnavia is garemoeba, rom sakuTari informaciis raodenoba I ( x) uwyveti SetyobinebebisaTvis ar aris toli am ukanasknelebis diferencialuri entropiisa. vinaidan uwyveti Setyobinebebis nebismieri

x(t ) realizaciis aRwarmoebis sizuste rea-

lur mowyobilobebSi SezRudulia, amitom aRwarmoebis

x * (t ) Sedegi SemTxveviT gansxvavdeba

x(t )

WeSmariti mniSvnelobebisagan. informaciis saSualo raodenoba, romelsac Seicavs sawyisi

x(t ) realizaciis damoukidebeli x anaTv-

lis Sesaxeb, (2.36) SeiZleba gamoisaxos ∞ ∞

I ( x, x*) =

x * (t ) anaTvali

safuZvelze

ω ( x, x*)

Semdegnairad

∫ ∫ ω ( x, x*) log ω ( x)ω ( x*) dxdx* = h( x) − h( x / x*).

−∞−∞

(2.39)

(2.39) gamosaxulebidan gamomdinareobs, rom informaciis

raodenoba

I ( x, x*), , garda imisa, rom

damokidebulia x(t ) procesis statistikaze, damokidebulia agreTve x * aRwarmoebis xerxze, rac ganisazRvreba pirobiTi albaTobebis ω ( x / x*)

simkvri-

viT. vinaidan ω ( x, x*) = ω ( x*)ω ( x / x*). , amitom sakuTrivi informaciis I (x) raodenoba, romelsac Seicavs procesis erTi damoukidebeli anaTvali, SeiZleba gansazRvrul iqnes rogorc orobiTi erTeulebis is minimaluri raodenoba, romelic saWiroa am anaTvlis moTxovnili sizustiT aRwarmoebisaTvis, e.i. rogorc I ( x, x*) gamosaxulebis minimumi, aRebuli yvela ω ( x / x*) -is mixedviT (aRwarmoebis yvela SesaZlo xerxis mixedviT), romlis drosac aRwarmoebis cdomileba ar aRemateba dasaSveb ε sidides I ( x) = H 2 ( x ) = min I ( x, x*) = h( x) − max h( x / x*). ω ( x1 x*) ω ( x1 x*)

(2.40)

2.7. uwyveti Setyobinebebis wyaros mwarmoebluroba da siWarbe imisda mixedviT, Tu ra xasiaTisaa dro (diskretuli Tu uwyveti), uwyveti Setyobinebebis wyaroebi SeiZleba daiyos or jgufad _ diskretuli drois uwyveti Setyobinebebisa da uwyveti drois uwyveti Setyobinebebis wyaroebad.

diskretuli drois uwyveti Setyobinebebis wyaroebis mwarmoeblurobasa da siWarbis gansazRvrisaTvis, (2.13) gamosaxulebis Tanaxmad, saWiroa gamoTvlil iqnes sakuTrivi informaciis is raodenoba, romelsac Seicavs wyarosgamosavali procesis uwyveti anaTvali, e.i. unda ganisazRvros anaTvlis entropia. im kerZo SemTxvevaSi, rodesac wyaros gamosavali procesi normaluria, ε entropiis maqsimaluri mniSvneloba

H max ( x) = log(σ x / σ ρ ). aseT SemTxvevaSi, cnobili v, σ 2

(2.41) da σ p sidi-

deebis dros, diskretuli drois uwyveti Setyobinebebis wyaros mwarmoebluroba da siWarbe Semdegnairad gamoisaxeba:

′ ( x) = υ log(σ k / σ p ), I max

(2.42)

P = 1 − H t ( x) / log(σ x / σ p ).

(2.43)

uwyveti drois Setyobinebebis wyaros SemTxvevaSi mwarmoebluroba da siWarbe Semdegnairad gamoiTvleba. Tu Setyobinebebis sixSirul speqtrs SevzRudavT zevidan

f z sixSiriT, maSin, kotelnikovis

Teoremis Tanaxmad, aseTi tipis Setyobinebebi SeiZleba warmovidginoT

{x1}

anaTvlebis mimdevrobis sa-

xiT, romlebsac wyaro gamoimuSavebs υ = 2 f z

siCqa-

riT im SemTxvevaSi, rodesac anaTvlebi damoukidebulia. (2.13) da (2.28) Tanaxmad

I ′( x) = υ /( x) = υH ε ( x).

(2.44)

uwyveti drois uwyveti mwarmoebluroba, mocemuli

zedazRvruli

sixSiris,

SetyobinebaTa

σx2 = Ps saSualo simZlavrisa da cdomilebaTa σ p2 = Ps simZlavris

dros

iqneba

maqsimaluri,

rodesac

wyaros mier 2 f z siCqariT gamomuSavebuli anaTvlebi damoukidebeli da normalurad ganawilebulia, e.i. rodesac Setyobinebebs eqneba `TeTri xmauris~ maxasiaTeblebi. aseT SemTxvevaSi, (3.30)-is Tanaxmad

P 1 ′ ( x) = 2 f z log x = f z D s; I max 5 Px

(2.45)

aq D s _ SetyobinebaTa dinamikuri diapazonia. aseTi Tanaxmad,

wyaros

siWarbe,

(2.43)

P = 1 − I ′( x) /( f g D s ) .

gamosaxulebis (2.46)

rogorc (1.22) da (2.45) gamosaxulebebis Sedareba gviCvenebs, SetyobinebaTa V s moculoba romelime Ts drois ganmavlobaSi tolia am informaciis maqsimaluri raodenobisa, romelsac Seicaven es Setyobinebebi.

sakontrolo kiTxvebi 1. ras ewodeba informaciis raodenobrivi zoma? 2. ras ewodeba diskretuli Setyobinebebis entropia? 3. ras ewodeba diskretuli arxis gadacemis siCqare da gamtarunarianoba? 4. raSi mdgomareobs Senonis ZiriTadi Teoremebi diskretuli arxebisaTvis? 5. ras ewodeba diskretul SetyobinebaTa wyaros mwarmoebluroba da siWarbe? 6. ras ewodeba uwyveti Setyobinebebis entropia? 7. ras ewodeba uwyveti Setyobinebebis wyaros mwarmoebluroba da siWarbe?

Tavi III. telekomunikaciis arxebi 3.1. telekomunikaciis arxebis klasifikacia da maxasiaTeblebi rogorc wignis SesavalSi aRvniSneT, telekomunikaciis arxi warmoadgens im teqnikur saSualebaTa da fizikuri aris erTobliobas, romelTa saSualebiT xdeba signalebis gavrceleba gadamcemidan mimRebisaken. zogadad, telekomunikaciis arxs SeiZleba hqondes ramdenime Sesavali da gamosavali da axorcielebdes signalebis ormxriv gadacemas. magram simartivisaTvis Cven SemdegSi ganvixilavT mxolod erTi Sesavlisa da erTi gamosavlis mqone arxebs, romlebic uzrunvelyofen signalebis calmxriv gavrcelebas. aseTi tipis arxebi eleqtrulad SeiZleba warmodgenil iqnes oTxpolusas saxiT. telekomunikaciis arxis Semadgenel nawils _ fizikur ares, romelSic xdeba signalebis gavrceleba gadamcemidan mimRebisaken _ e.w. telekomunikaciis xazebs SeiZleba sxvadasxva buneba hqondes. telekomunikaciis Tanamedrove sistemebSi farTod gamoiyeneba sadeniani telekomunikaciis (sahaero da sakabelo), radio â&#x20AC;&#x201C; da radiosareleo (maT Soris meteoruli, kosmosuri, ionosferuli, tropisferuli), optikuri da a.S. xazebi. arxis Sedgeni-lobaSi SeiZleba Sediodes ramdenime telekomunika-ciis xazi, magram umravles SemTxvevaSi erTi da igive xazi emsaxureba ramdenime arxs.

telekomunikaciis xazis garda arxi Seicavs agreTve mTel rig teqnikur saSualebebs, romlebic ganlagebulni arian arxis saSualedo da damaboloebel punqtebSi. saSualdo punqtebSi ganlagebul teqnikur mowyobilobebs miekuTvneba sxvadasxva saxis maZliereblebi, regeneratorebi, koreqtorebi da a.S. rac Seexeba damaboloebeli punqtebis mowyobilobebs, isini, imisda mixedviT, Tu gardaqmnaTa ra TvalsazrisiT ganixileba arxSi moqmedi signalebi, SeiZleba miekuTvnon rogorc sakuTriv arxs, aseve gadamcems an mimRebs. telekomunikaciis arxebis klasifikacias SeiZleba safuZvlad daedos sxvadasxva pirobebi. telekomunikaciis sistemebis daniSnulebis mixedviT, romlebsac emsaxureba telekomunikaciis arxebi, es ukanasknelebi SeiZleba daiyos: satelefono, satelevizio, satelegrafo, radiosamauwyeblo fiWuri mobiluri da a.S. arxebad. imisda mixedviT, Tu ra gziT vrceldeba signalebi gadamcemidan mimRebisaken _ Tavisufal sivrceSi Tu mimmarTveli xazebis gaswvriv _ gamoiyofa radio da sadeniani telekomunikaciis arxebi. xSirad arxebis klasifikacia xdeba gamoyenebuli sixSiruli diapazonis mixedviT. ase, magaliTad, koaqsialur kabelebSi sixSiruli diapazoni aRwevs ramdenime aTas kilohercs, radio telekomunikaciis Tanamedrove sistemebSi ki izrdeba ramdenime aseul aTas megahercamde da a.S. yvelaze farTo gavrceleba pova telekomunikaciis arxebis dayofam im signalebis xasiaTis mixed-

viT, romelTa gadacemasac isini axorcielebs am niSnis mixedviT asxvaveben: uwyvet arxebs, romelTa Sesasvlelsa da gamosavalze signalebi icvleba uwyvetad SesaZlo mniSvnelobebis garkveul diapazonSi; diskretul arxebs, romelTa Sesavali da gamosavali signalebi diskretulia mniSvnelobebis mixedviT; diskretul-uwyvet arxebs, romelTa Sesavali signalebi diskretulia da gamosavali _ uwyveti an piriqiT. telekomunikaciis arxebis maxasiaTeblebis saxe ZiriTadad damokidebulia imaze, Tu ra tipis signalebis (diskretuli Tu uwyveti) gadacemisTvisaa isini gankuTvnili. garda amisa, maxasiaTeblebi damokidebulia agreTve telekomunikaciis im sistemebis daniSnulebaze, romelTa SedgenlobaSi Sedis gansaxilveli arxebi. uwyveti arxebis SemTxvevaSi maT maxasiaTeblebs SeiZleba miekuTvnos gadasacemi signalebis saSualo da pikuri simZlavreebi, gadasacemi signalebis sixSiruli zoli, arxebis amplituduri maxasiaTeblebi da a.S. ase, magaliTad, standartuli satelefono arxis ZiriTadi maxasiaTeblebia: narCeni milevadoba, romelic axasiaTebs signalis doneebis sxvaobas arxis Sesavalsa da gamosavalze; arxis milevadobis sixSiruli maxasiaTebeli, romelic warmoadgens narCeni milevadobis damokidebulebas sixSireze; gadasacemi sixSireebis efeqturi zoli, romelic Semofarglulia sixSireTa im mniSvnelovnebiT, sadac narCeni milevadoba garkveuli sididiT aRemateba arxis saSualo sixSiris Sesabamis narCen

milevadobas; arxis amplituduri maxasiaTebeli, romelic warmoadgens arxis narCeni milevadobis damokidebulebas Semavali signalis doneze. zogierT SemTxvevaSi mxedvelobaSi miiReba agreTve arxis faza _ sixSiruli maxasiaTebeli, romelic warmoadgens arxis Semaval da gamomaval signalebs Soris fazuri Zvris damokidebulebas sixSireze. im SemTxvevaSi, Tu saqme gvaqvs diskretul arxebTan, maxasiaTeblebis gansazRvrisas mxedvelobaSi unda iyos miRebuli im signalebis wyaroebis Taviseburebani, romlebsac emsaxureba aseTi tipis arxebi. diskretuli arxebis ZiriTadi maxasiaTeblebs SeiZleba miekuTvnon: gadasacemi Setyobinebebis moculoba, magaliTad, gadasacemi signalebis raodenoba; drois erTeulSi gadasacemi Setyobinebebis raodenoba, anu gadacemis siCqare, romelic izomeba bodebSi; informaciis gacemis periodi, anu dro, romlis ganmavlobaSi informaciis wyaro miawodebs gadacemis traqts morig Setyobinebas; arxSi informaciis dayovnebis dro; SetyobinebaTa elementebis gadacemis siswore, romelic axasiaTebs arxis xelSeSlebisadmi mdgradobas; arxis saimedooba da sxv. cxadia, yovel konkretul SemTxvevaSi, zemoT CamoTvlil maxasiaTeblebs daemateba aseve is maxasiaTeblebi, romlebic saWiroa arxis SedgenilobaSi myofi xazebis aRwerisaTvis. rac Seexeba diskretul-uwyvet arxebs, maTi maxasiaTeblebis Camoyalibebisas gamoiyeneba ro-

gorc uwyveti, aseve diskretuli CamoTvlili maxasiaTeblebi.

arxebis

zemoT

3.2. damaxinjebebi da xelSeSlebi telekomunikaciis arxebSi telekomunikaciis realur arxebSi elementaruli signalebi gavrcelebisas ganicdian cvlilebebs, ris Sedegadac miRebuli signalebi gansxvavdebian gadacemulisagan. gansxvavebas SeiZleba hqondes rogorc determinebuli, aseve SemTxveviTi xasiaTi. determinirebuli cvlilebebidan yvelaze saxifaToa signalis formis cvalebadoba, vinaidan sxva saxis cvlilebebis (signalis gaZliereba an Sesusteba, misi droiTi dayovneba da a.S.) koreqtireba ar aris rTuli, signalis formis cvlileba realur arxebSi gamowveulia am ukanasknelis amplituduri da sixSiruli maxasiaTeblebis saxiT, xolo radioarxebis SemTxvevaSi eleqtromagnituri talRebis gavrcelebis mravalsxivobiT. analizis gamartivebis mizniT arxi SeiZleba warmodgenil iqnes rogorc mimdevrobiT CarTuli wrfivi da arawrfivi uinercio oTxpolebi, romlebic gansazRvraven signalis Sesabamisad wrfiv da arawrfiv damaxinjebebs. wrfivi ewodeba damaxinjebebs, romlebic warmoiSobian mudmivi parametrebis mqone inerciul xazur oTxpolesbSi da romlebic ganpirobebuli arian am oTxpolusebSi Semavali reaqtiuli elemen-

tebiT. aseTi oTxpolusebis ampltudur-sixSiruli maxasiaTeblebis araTanabaroba da faza-sixSiruli maxasiaTeblebis arawrfivoba iwvevs signalis formis damaxinjebebs, vinaidan am SemTxvevaSi irRveva Tanafardoba signalis harmoniuli Semdgenelebis amplitudebsa da fazebs Soris. sixSiruli da fazuri damaxinjebebis Tavidan acilebisaTvis, rogorc cnobilia, saWiroa arxis gadacemis zolSi oTxpulsas gadacemis koeficientis moduli iyos mudmivi,

e.i.

| K (ω ) |= K o ,

xolo

faza

icvlebodes

wrfivad (Φ (ω ) = ωτ ). im SemTxvevaSi, Tu wrfiv oTxpulsas cvladi parametrebi aqvs (maaliTad, im radioarxis SemTxvevaSi, romelsac axasiaTebs talRebis mravalsxivuri gavrceleba), saWiroa signalis harmoniuli

Semdgenlebis

fazuri

Zvra

ω=0

sixSireze 2π -s jeradi iyos, e.i. Φ (0) = R 2π . winaaRmdeg SemTxvevaSi warmoiSoba e.w. kvadraturuli damaxinjebebi. aseTi saxis damaxinjebebis magaliTis moyvanilia 3.1. a da b naxazebze nax. 31 a-ze mTliani xaziT gamosaxulia signali, miRebuli ori harmoniuli SemadgenliT (wyvetili xazebi); b-ze naCvenebia, Tu sawyisi signalis rogoc damaxinjebebs iwvevs garemoeba, rodesac harmoniuli Semadgenlobis sawyisi faza 2π -s jeradi ar aris. arawrfivi ewodeba damaxinjebebs, romlebic warmoiSoba uinercio arawrfivi mudmivi parametrebis mqone oTxpolusebSi da ganpirobebulia arawrfivi elementebis arsebobiT. telekomunikaciebis arxis arawrfivoba zRudavs signalis dauma-

xinjebeli mniSvnelobis maqsimalurad SesaZlo sidides, anu zRudavs gadasacemi signalis dinamikur diapazons.

nax. 3.1.

arxis apmlituduri maxasiaTebls arawrfivobis gamo, mis Sesasvlelze rTuli formis signalis miwodebisas, arxis gamosaval signalSi SeiZleba warmoiSvas iseTi Semdgenebi, romlebsac adgili ar hqonda Sesaval signalSi. kerZod, gamosaval signalSi SeiZleba warmoiSvas Semdegi tipis kombinaciuri sixSireebi:

Rf1 ± lf 2 ; mf1 ± nf 2 ; pf 2 ± qf 2 ;... sadac

R, l , m, n, p, q

mTeli

ricxvebia,

(3.1.) xolo

f1 , f 2 , f 3 ,... Semavali signalis harmoniuli Semdgenebi. aseTi tipis damaxinjebebis Sefaeba xdeba e.w. intermodulaciuri damaxinjebebis koeficientis saSualebiT. aRsaniSnavia is garemoebac, rom, Tu rawrfivi oTxpolusa Sedis romelime arxis SedgenilobaSi, maSin masSi gamavali signalis arawrfivobis produqtebs eqneba iseTi sixSireebi, romlebic SeiZleba moxvdnen mezobeli arxis gada-

cemis zolSi da amgvarad gaxdnen am arxisaTvis damatebiTi damaxinjebis wyaro. ganvixiloT SemTxveviTi xasiaTis cvlilebebi, romlebsac ganicdian arxSi gamavali signalebi. SemTxveviTi xasiaTis cvlilebebs xelSeSlebi ewodeba. xelSeSlebis zemoqmedebis Sedegad arxis gamosavali signali SeiZleba warmodgenil iqnes Semdegnairad

x * (t ) = x(t )両 m(t ) + 両 e(t ),

(3.2)

sadac 両 m(t ) da 両 e(t ) xelSeSlis e.w. multiplikaciuri da aditiuri Semdgenebia. zogadad, xelSeSlebis fizikur wyaroebs sxvadasxvanairi buneba aqvT da isi SeiZleba moTavsdnen rogorc arxSi (e.w. Sinagani xelSeSlebi) aseve mis gareTac (garegani xelSeSlebi). aditiuri xelSeSlebi maTi speqtruli da droiTi maxasiaTeblebis mixedviT SeiZleba daiyos sam jgufad: xelSeSlebi, ganawilebuli sixSiris da drois mixedviT (e.w. fluqtuaciuri xelSeSlebi), Seyursuli sixSiris mixedviT (e.w. harmoniuli xelSeSlebi) da Seyursuli droSi (e.w. impulsuri xelSeSlebi). fluaqtuaciur xelSeSlebSi igulisxmeba droSi uwyveti normaluri ganawilebis da nulovani saSualo mniSvnelobis mqone SemTxeviTi procesi (xSirad agreTve stacionaruli da ergodokuli); romlis eneretikuli speqtri arxis gatarebis zolSi SeiZleba CaiTvalos Tanabrad. fluqtuaciuri xelSeSlebi telekomunikaciis arxebSi ZiriTadad

ganpirobebulia mowyobilobaTa Sinagani xmaurebiT (magaliTad, Tburi xmaurebiT), mzis da varskvlavebis radiogamosxivebiT, ucxo radiosadgurebis jamuri signaliT (Tu cxadia, maTi raodenoba imdenad didia, rom adgili aqvs jamuri signalis statistikuri maxasiaTeblebis normalizaciis movlenas) da a.S. harmoniul xelSeSlebSi igulsxmeba iseTi aditiuri xelSeSlebi, romelTa energetikuli speqtri moTavsebulia sixSireTa viwro zolSi, romelic gadasacemi signalis sixSiruli zolis toli an masze naklebia. aseTi saxis xelSeSlebi damaxasiaTebelia radioarxebisaTvis da ganpirobebulia ucxo radiosadgurebis signalebiT. xSirad harmoniuli xelSeSla warmoiSoba sxvadasxva ara mxolod radioarxebSi, aramed Soreuli telekomunikaciis sadenian arxebSic. impulsur xelSeSlebSi igulisxmeba iseTi aditiuri xelSeSlebi, romlebic warmoadgenen SemTxeviTi impulsebis mimdevrobebs. sadeniani telekomunikaciis arxebSi impulsuri xelSeSlebi ganpirobebulia ZiriTadad sakomutacio xelsawyoebis xmauriT. sxvadasxa saxis aRZruli emZ-iT da a.S. radioarxebSi ki-impulsuri xelSeSlebi ZiriTadad atmosferuli da sawarmooo warmoSobisaa. impulsur xelSeSlebs gaaCniaT sakmaod farTo energetiuli speqtri, romlis sidide mkveTrad mcirdeba nulovani da ramdenime aTeuli megahercis sixSireebis ubnebze.

rac Seeeba multiplikaciur xelSeSlebs, isini ganpirobebulia arxis gadacemis koeficientis SemTxveviTi cvalebadobiT. amis ZiriTad mizezs warmoadgenen arxis maZlierebeli mowyobilobebis gaZlierebis koeficientebis cvalebadoba mkvebavi Zabvebis meryeobis gamo, radioalRebis gavrcelebis milevadoba da a.S. multiplikaciuri xelSeSlebi maTi realizaciebis droSi cvalebadobis siCqaris mixedviT arxSi gadacemuli signalis cvlilebebis siCqaresTan SedarebiT SeiZleba daiyos or gjufad â&#x20AC;&#x201C; nel da Cqar multiplikaciur xelSeSlebad.

3.3. diskretuli arxebis modelebi realur arxebSi zemoT aRniSnuli damaxinjebebi da xelSeSlebi moqmedebs erToblivad, ris Sedegadac arxebSi mimdinare procesebi rTuleba. aseT pirobebSi signalebis gardasaxvis maTematikuri aRwera, umravles SemTxevaSi, gdauWreli amocanaa da amitom analizis gamartivebis mizniT mimarTaven arxebis idealizebul maTematikur modelebs. ganvixiloT mokled diskretuli arxebis maTematikuri modelebi da maTi aRwera. diskretuli arxis aRwerisaTvis saWiroa vicodeT: mis Sesasvleze moqmedi koduri xi (i = 1, 2, 3,..., n) simboloebis

alfabeti

simboloebis

P ( xi )

albaTobebi; gamosavali xj ( j = 1, 2,..., n) simboloebis alfabeti;

drois

erTeulSi

gatarebuli

koduri

simboloebis n

P ( x j / xi )

raodenoba da gadasvlis

albaTobebi, e.i. albaTobebi imisa, rom arxis gamosavalze gamoCndeba xi simbolo mis Sesasvlelze x j simbolos miwodebisas. im movlenebis erToblivi albaToba, romelic mdgomareobs arxis Sesasvlelze xi simbolos miwodebisas

mis

gamosavalze

simbolos

miRebaSi,

Semdegnairad gamoisaxeba

P ( xi , x j ) = P( xi ) P( x j / xi ) = P( x j ) P( xi / x j ),

(3.3)

sadac P ( x j ) − x j simbolos miRebis upirobo albaTobaa, romelic Semdegnairad gamoiTvleba. n

P ( xi ) = ∑ p ( xi ) P( x j / x j ).

(3.4)

i =1

rac Seexeba imis alaTobas, rom arxis gamosavalze gamoCndeba x j koduri simbolo xi simbolos gadacemisas, e.w. aposteriul albaTobas, igi gamoiTvleba baiesis formulis safuZvelze

⎡ n ⎤ P ( xi / x j ) = P( xi ) P( x j / xi ) ⎢∑ P( xi ) P( x j / xi )⎥. ⎣ i=1 ⎦

(3.5.)

zemoTaRniSnuli albaTobebi zogadad damokidebulia imaze, Tu romeli simbolo iyo gadacemuli an miRebuli adre. im SemTxvevaSi, roca gadasvlis

P ( x j / xi ) albaTobebi yoveli i , j wyvilisaTvis ar icvleba droSi da damokidebuli ar aris imaze, romeli simbolo iyo gadacemuli an miRebuli adre, diskretul arxs ewodeba umaxsovro da erTgva-

rovani an stacionaruli), xolo Tu es albaTobebi damokidebulia droze, maSin arxs ewodeba araerTgvarovani, anu arastacionaruli. rodesac zemoaRniSnuli albaTobebi damokidebulia imaze, Tu romeli signali iyo gadacemuli da miRebuli adre, Sesabamis arxs ewodeba maxsovrobis mqone. aRsaniSnavia, rom reluri arxebis Tvisebebis gaTvaliswinebiT diskretuli arxebis yvelaze ufro srulyofil models warmoadgens araerTgvarovani damaxsovrebis mqone arxi. magram ufro xSirad mimarTaven erTgvarovani umaxsovro diskretuli arxebis models, rogorc ufro martivs. Tu erTgvarovan arxSi Semavali koduri simboloebis alfabeti alfabeti

{x } j

{xi }

da gamosavali simboloebis

erTnairia, e.i. n = n′ da yoveli

wyvilisaTvis albaToba

j≠i

P( x j / xi ) = P, xolo, rode-

sac i = j , albaToba P ( x j / xi ) = Q = 1 − (n − 1) P, Sesabamis arxs ewodeba simetriuli. aRniSnul arxebs, romelTa Semavali da gamomavali simboloebis alfabetebi ar emTxveva, e.i. arxebs, romelTa gamomavali alfabeti Seicavs zedmet simbolos Semaval alfabetebTa SedarebiT

n = n + 1 , ewodeba arxebi `waSliT~. damatebiTi simbolos gamoCena gviCvenebs, rom mimRebi mowyobiloba ver Rebulobs calsaxa gadawyvetilebas miRebuli simbolos Sesaxeb. am damatebiT simbolos ewodeba `kiTxvis simbolo~.

miuxedavad imisa, rom aseT arxebSi koduri kombinaciebis nawili SeiZleba waiSalos (damaxinjdes), kodis da misi damuSavebis xerxebis arCeviT SeiZleba mkveTrad gaizardos xelSeSlebisadmi mdgradoba.

3.4. diskretul-uwyveti arxebis modelebi rogorc aRvniSneT, diskretul uyveti arxebis Sesavalze moqmedeben diskretuli simboloebi (magaliTad, bi ), arxebis gamosavali signali uwyvetia da SeiZleba aRiweros uwyveti η (t ) funqciiT. diskretul-uwyveti vicodeT:

arxis

sruli

Semavali

aRwerisaTvis

diskretuli

alfabeti ( i = 1, 2,..., n)

saWiroa

simboloebis

Sesabamisi aprioruli

P (bi )

alaTobebiT; arxis Sesavalze drois erTeulSi miwodebuli koduri simboloebis saSualo n raodenoba da gadasvlis albaTobebi,

rom

ω (η / bi ) albaTobebi. e.i. imis arxis

gamosavalze

gamoCndeba

η (t ) uwyveti signalis elementi arxis Sesavalze bi diskretuli simbolos moqmedebisas. albaToba imisa, rom uwyveti η (t ) signalis miRebuli elementi Seesabameba gadacemul bi diskretul simbolos, gamoiTvleba baiesis formuliT

P(bi / η ) = P(bi )ω (η / bi ) / ω (η ).

(3.6)

sadac ω (η ) warmoadgens η (t ) signalis albaTobaTa simkvrives.

im SemTxvevaSi, rodesac η (t ) uwyveti signalis elementebis da simboloebis nebismieri erTobliobis Sesabamis albaTobaTa simkvrive ar aris damokidebuli droze da imaze, Tu romel elementebs da simboloebs hqondaT adgili adre, Sesabamis diskretul-uwyvet arxs ewodeba damaxsovrebis mqone da stacionaruli Tuki albaTobaTa ω (η / bi ) simkvrive damokidebulia droze, arxs ewodeba arastacionaruli (araerTgvarovani). xolo im SemTxvevaSi, rodesac ω (η / bi ) damokidebulia winamdebare simboloebis da elementebis mniSvnelobaze, disketuluwyvet arxs ewodeba damaxsovrebis mqone arxi. zogadad, yvela realuri arxi aris arastacionaruli da damaxsovrebis mqone. magram, rogorc gamocdileba gviCvenebs, realuri diskretul-uwyveti arxebis analizisaTvis sruliad sakmarisia SemovisazRvroT stacionaruli damaxsovrebis armqone arxebis modelebiT.

3.5. uwyveti arxebis modelebi realur uwyvet arxebSi signalebis, xelSeSlebisa da damaxinjebebis urTierTqmedeba rTuli xasiaTisaa da amitom aseT arxebSi signalebis gardaqmnebis maTematikuri aRweris amocana metad rTuli da zogadad aqamde gadauWrelia. am arxebSi signalebis gadacemia da gardaqmnis sakiTxebis gamokvlevisas sargebloben ramdenime arsebiTad idealizebuli modelebiT. ganvixiloT zogierTi maTgani.

idealur arxSi adgili ara aqvs xelSeSlebs. gamosavali da Sesavali signalebi erTmaneTTan dakavSirebulia determinirebulad. aseTi arxebis maTematikuri aRwerisaTvis sakmarisia vicodeT kavSiri Sesaval da gamosaval signalebs Soris da is SezRudvebi, romlebsac unda akmayofilebdnen Semavali signalebi da arxi. kerZod, SezRudvebi Seexeba arxis gatarebis sixSirul zols da gadacemebi signalis pikuri an saSualo simZlavreebis dasaSveb sididebs. unda aRiniSnos, rom realuri arxebi sakmaod Sorsaa aseTi saxis modelebisagan. hausis arxi uwyveti arxebis yvelaze gavrcelebuli modelia. masSi moqmedebs mxolod aditiuri xelSeSla, romelic warmoadgens hausis process nulovani mudmivi SemdgeniT, romlis sixSiruli speqtri emTxveva an gadafaravs arxis gatarebis zols. hausis arxi savsebiT ganisazRvreba gatarebis sixSiruli zoliT da xelSeSlebis simZlavris speqtruli simkvriviT (an korelaciis funqciiT). igi warmoadgens arxebis sakmaod karg models. im SemTxvevaSi, Tu miRebis adgilas signalis faza ar aris cnobili an SemTxveviT icvleba, arxis modelad SeiZleba gamoyenebul iqnes hausis arxi signalis gaurkveveli faziT. aseT SemTxvevaSi xmauris speqtruli simkvrivis garda cnobili unda iyos signalis fazis cvlilebis statistika. miyuCebis mqone arxi. umravles relur radioarxebSi moqmedebs multipliaciuri xelSeSlebi. aseT arxebSi signalis gavrceleba gadamcemidan mimRebi-

100

saken xdeba ramdenime gziT, romelTa sigrZe da Sesabamisad, signalis gavrcelebis dro icvleba, SemTxeviT. amis gamo SemTxveviT icvleba agreTve am gzebis gadacemis koeficientebi da amitom Sesasvlelze jamuri signalis amplituda da faza SemTxveviTia, e.i. adgili aqvs miyuCebis movlenas. miyuCebis mqone arxis aRwerisaTvis saWiroa vicodeT arxis sixSiruli da fazuri maxasiaTeblebis cvlilebis statistika. imisda mixedviT, Tu rogoria korelacia signalis sxvadasxva Semdgenebs Soris, ganasxvaveben miyuCebis or saxes: saerTos da seleqciurs. miyuCebas ewodeba saerTo, Tu signalis Semdgenebis fluqtuaciebis korelacia imdenad mniSvnelovania, rom sixSiruli da fazuri maxasiaTeblebis fluqtuaciebi SeiZleba CaiTvalos erTnairad signalis yvela SemdgenisaTvis. im SemTxevaSi, rodesac signalis calkeuli Semdgenebi fluqtuireben erTmaneTisagan damokideblad, adgili aqvs e.w. seleqciur miyuCebas. amgvarad, zemoT ganxiluli uwyveti arxebis yvela modelSi Semavali signalebi SeiZleba warmovidginoT drois SemTxveviTi funqciebiT. aseT SemTxvevaSi arxis aRwerisaTvis saWiroa vicodeT am signalebis statistikuri maxasiaTeblebi, zogad SemTxvevaSi albaTobaTa simkvriveebi. uwyvet arxs ewodeba stacionaruli da damaxsovrebis armqone, Tu arxis ganmsazRvreli statistikuri maxasiaTeblebi damokidebuli ar aris droze, xolo gadasvlebis albaTobebi damokidebuli ar

101

aris gamosavali signalis winamdebare gadacemul an miRebul elementebze.

signalis

3.6. signalebis ZiriTadi gardasaxvebi telekomunikaciis arxebSi telekomunikaciis sistemebis analizis Catarebisas mniSvnelovani amocanaa is wrfivi da arawrfivi gardasaxvis gamokvleva, romelsac ganicdis signalebi, rogorc SemTxveviTi procesebi, telekomunikaciis arxebSi gavlisas. zogad SemTxvevaSi, signalis wrfiv wredebSi gavlis amocana mdgomareobs gamosavali signalis ganawilebis kanonis gansazRvraSi wredis mocemuli sidideebisa da Semavali signalis cnobili ganawilebis kanonis mixedviT. aseTi amocanis amoxsna dakavSirebulia did siZneleebTan da amitom igi daiyvaneba gamosavali procesis ricxobrivi maxasiaTeblis, kerZod, maTematikuri molodinisa da korelaciis (an energetikuli speqtris) gansazRvris amocanaze. vTqvaT, mocemuli gvaqvs wrfivi sistema kompleqsuri gaacemis K ( jω ) = K o e jϕ (ω ) funqciiT, romlis Sesasvlels miewodeba cnobili m1x maTematikuri molodinis, Gx (ω ) energetikuli speqtris mqone stacionaruli SemTxeviTi x(t ) signali. saWiroa ganisazRvros gamosavali

y (t ) signalis

102

m1 y

maTematikuri

molodini da G y (ω ) eneretikuli speqtri (an korelaciis funqcia B y (τ )). vinaidan stacionaruli SemTxveviTi procesis maTematikuri molodini warmoadgens am procesis mudmiv Semdgens, amitom gamosavali procesis maTematikuri molodini ganisazRvreba Semdegnairad

m1 y = K 0 m1x .

(3.6)

Tu cnobilia wrfivi wredis impulsuri gardamavali g (t ) maxasiaTebeli, gamosavali SemTxveviTi procesis m1 y maTematikuri molodini iqneba ∞

m1v = m1x ∫ g (t )dt.

(3.7)

diuamelis integralis Tanaxmad, ∞

y (t ) = ∫ g (τ ) x(t − τ )dτ ,

(3.8)

maSin ∞

m1 y = M [ y (t )] = ∫ g (τ ) M [ x(t − τ )]dτ .

(3.9)

vinaidan x(t ) stacionaluri SemTxveviTi procesia, amitom

M [ x(t − τ )] = M [ x(t )] = m1x

(3.10)

mudmivi sididea da, amgvarad, ∞

m1 y = m1x ∫ g (τ )dτ . o

103

(3.11)

(3.11) gamosaxulebiT SeiZleba damtkicdes (3.6) gamosaxulebis marTebulobac. rogorc cnobilia, kompleqsuri

K ( jω )

gadacemis

funqcia

dakavSi-

rebulia g (t ) -Tan furies gardasaxvis saSalebiT ∞

K ( jω ) = ∫ g (t ) exp(− jωt )dt.

(3.12)

Tu (3.12) gamosaxulebaSi SevitanT ω = 0 , miviRebT ∞

K (0) = ∫ g (t )dt.

(3.13)

(3.13) da (3.11) gamosaxulebebis gaTvaliswinebiT miviRebT (3.6) gamosaxulebas. wrfivi wredis gamosavali SemTxveviTi signalis korelaciis funqcia diuamelis (3.8) gamoyenebis safuZvelze Semdegnairad ganisazRvreba ∞

∞

B y (τ ) = y (t1 ) y (t 2 ) = ∫ g (τ 1 ) x(t1 − τ )dτ 1 ∫ g (τ 2 ) x(t 2 − τ 2 )dτ 2 = ∞∞

= ∫ ∫ g (τ 1 ) g (τ 2 ) x(t1 − τ 1 ) x(t2 − τ 2 )dτ 1dτ 2 = o o

∞∞

= ∫ ∫ g (τ 1 ) g (τ 2 ) Bx(τ + τ 1 − τ 2 )dτ 1dτ 2 , o o

sadac τ − t 2 − t1.

104

(3.14)

amgvarad, gamosavali procesis korelaciis funqcia damokidebulia mxolod τ droiT intervalze. aqedan gamomdinare, wrfiv wredze stacionaruli SemTxveviTi signalis gamosavali signalic stacionarulia. ganvixiloT gamosavali SemTxveviTi signalis energetikuli speqtri. viner-xinCinis pirdapiri gardasaxvisa da (3.14) gamosaxulebis safuZvelze gveqneba

G y (ω ) =

∞

∫ B (τ )e

jωτ

−∞

e − jωτ dτ =

⎤ ⎡∞ ∞ dτ = ∫ ⎢ ∫ ∫ g (τ 1 ) g (τ 2 ) B2 (τ + τ 1 − τ 2 ) dτ 1dτ 2 ⎥ −∞ ⎣ −∞−∞ ⎦ ∞

∞

−∞

jωτ − jω (τ +τ −τ ) ∫ BK (τ + τ 1 − τ 2 )e 1 2 dτ ∫ g (τ 1 )e 1 dτ 1 .

∞

⋅ ∫ g (τ 2 )e jωτ 2 dτ 2 = Gx (ω ) K (− jω ) K ( jω ) =| K ( jω ) | ⋅Gx (ω ).

(3.15)

−∞

SemTxveviTi procesis ganawilebis kanoni wrfivi wredis gamosavalze zogadad gansxvavdeba mis Sesavalze miwodebuli SemTxveviTi signalis ganawilebis kanonisagan. magram im SemTxvevaSi, Tu wrfivi wredis Sesasvlels miewodeba hausis procesi, gamosavali procesic emorCileba normalur kanons, icvleba mxolod procesis zogierTi maxasiaTebeli (dispersia, korelaciis funqcia da a.S.) unda aRiniSnos Semdegi garemoebac. viwrozoliani wrfivi sistemis SemTxvevaSi, romlis gatarebis zoli gacilebiT ufro viwroa, vidre Semavali signalis speqtris sigane, adgili aqvs e.w. normalizaciis movlenas, rac imaSi mdgomareobs, rom,

105

miuxedavad imisa, Tu ra ganawilebis kanoni aqvs Semaval process, gamosavali procesis ganawilebis kanoni axlosaa normalurTan. ganvixiloT SemTxveviTi signalebis arawrfivi gardasaxvebi. iseve, rogorc wrfivi gardasaxvebis SemTxvevaSi, amocana mdgomareobs gamosavali procesis statistikuri maxasiaTeblebis gansazRvraSi mocemuli Semavali procesis statistikuri maxasiaTeblebis mixedviT. SemTxveviTi signalis cxadia, rom arawrfivi gardasaxvebis amocana wrfivi gardasaxvebis amocanaze gacilebiT ufro rTulia. amitom Cven ganvixiloT yvelaze martivi amocana _ SemTxveviTi signalis gavlis amocana arawrfiv uinercio sistemaSi. sadac gamosavali y (t ) procesi calsaxad aris damokidebuli Semaval x(t ) procesTan. vTqvaT

cnobilia

arawrfivi

gardasaxvis

y = f (x) funqcia, maSin x = φ ( y ) ukufunqcia agreTve calsaxad gansazRvruli funqcia iqneba, albaToba imisa,

rom

Semavali

procesis

mniSvneloba moTavsebulia

romelime

intervsalSi, toli

unda iyos ( x da y Soris calsaxa damokidebulebis gamo) imis albaTobisa, rom gamosavali procesis η mniSvneloba moTavsebulia Sesabamis y, y + dy intervalSi

P ( x < ξ < x + dx) = P( y < η < y + dy ),

ω2 ( x)dx = ω y ( y )dy.

106

(3.16) (3.17)

aqedan gamomdinareobs, rom gamosavali procesis ganawileba Semavali procesis ganawilebaze damokidebulia Semdegnairad:

ω y ( y ) = ω x ( x) | dx / dy | .

(3.18)

amasTan, aRebuli unda iqnes warmoebulis absoluturi sidide, vinaidan ganawilebis funqcia yovelTvis dadebiTia. arawrfivi gardaqmnebisas gamosavali procesis ricxobrivi maxasiaTeblis gansazRvra xdeba Semdegnairad. gamosavali procesis maTematikuri molodini. ∞

M [ y (t )] = m y =

∫ yω

( y )dy.

(3.19)

−∞

(3.17) gamosaxulebis gamoyenebiT miviRebT, rom ∞

m2 y =

∫ f ( x) ω

( x)dx.

(3.20)

−∞

analogiurad gamosavali procesis korelaciis funqcia Semavali stacionaruli SemTxveviTi signalis dros iqneba

B y (τ ) =

∞ ∞

∫ ∫ f ( x ) f ( x )ω ( x , x )dx dx . 1

(3.21)

−∞−∞

rac Seexeba gamosavali procesis energetikul speqtrs, igi SeiZleba napovni iyos viner-xinCinis gardasaxvebis safuZvelze.

107

sakontrolo kiTxvebi 1. ra principebiT xorcieldeba telekomunikaciis arxebis klasifikacia? 2. rogori damaxinjebebi da xelSeSlebia telekomunikaciis arxebSi? 3. gaaanalizeT diskretuli arxebis modelebi 4. gaaanalizeT diskretul-uwyveti arxebis modelebi 5. gaaanalizeT uwyveti arxebis modelebi 6. signalebis ra ZiriTadi gardasaxvebia telekomunikaciis arxebSi?

108

Tavi IV. diskretuli Setyobinebebis gadacemis Teoria 4.1. uwyvet arxebSi diskretuli Setyobinebebis optimaluri miRebis amocana zogadad,

Setyobinebebis

gadacemis

sistemaSi

Setyobinebebis gadacemis xerxi winaswar aris cnobili da ZiriTad amocanaswarmoadgens Setyobinebebis miRebis xelSeSlisadmi yvelaze mdgradi xerxis povna. xelSeSlebisadmi mdgradobis cneba Semdegnairad ganisazRvreba: Setyobinebis gadacemis sistemis xelSeSlebisadmi mdgradoba ewodeba sistemis unarianobas gaarCios (aRmoaCinos, aRadginos) signalebi mocemuli sizustiT. maqsimalurad miRwevad xelSeSlebisadmi mdgradobas ewodeba potencialuri. ama Tu im mimRebis potencialuri da realuri xelSeSlebisadmi mdgradobis

urTierT

Sedareba

saSualebas

gvaZlevs

SevafasoT mocemuli mowyobilobis SesaZleblobebi da davsaxoTgzebi misi Semdgomi srulyofisa. vTqvaT, gvaqvs sxvadasxva xi (t ) signalebi, romlebidanac dakvirvebis intervalze gadaicema mxolod erTi. zogad SemTxvevaSi mimRebs Sesasvlelze ewodeba

SemTxveviTi

signalis

realizacia

y (t ), ,

romelic warmoadgens sasargeblo signalis xi (t ) da

Ď&#x2030; (t ) SemTxveviTi xelSeSlis aditiur narevs

109

y (t ) = xi (t ) + Ď&#x2030; (t ).

(4.1)

amasTan, gadacemuli signalisa da xelSeSlis sxvadasxva variantebs SeiZleba Seesabamebodes jamuri

y (t )

amgvarad,

signalis

y (t )

erTi

igive

realizacia.

signalis miRebis Semdeg arsebobs

gaurkvevloba imis Sesaxeb, Tu romeli xi (t ) signali iyo

gadacemuli.

cnobilia

signalis

xelSeSlis statistikuri Tvisebebi, SeiZleba aigos iseTi mimRebi mowyobiloba, romelic am Tvisebis garkveuli analizis safuZvelze miiRebs gadawyvetilebas. gadacemuli signalis Sesaxeb gadawyvetileba miiReba

garkveuli

wesis

safuZvelze,

romelic

ganisazRvreba mocemuli kriteriumiT. mimRebs, romelic arCeuli kriteriumis safuZvelze saukeTesod aRadgens (aRmoaCens, gaarCevs) gadacemul Setyobinebas, ewodeba optimaluri, xolo misi

xelSeSlebisadmi

mdgradoba

iqneba

maqsima-

luri. sanam SevudgebodeT diskretuli Setyobinebebis optimaluri miRebis amocanis garkvevas, ramdenime sityva im aditiuri xelSeSlebis Sesaxeb, romlebic arxeboben arxSi da moqmedeben mimRebis Sesasvlelze. rogorc zemoT aRiniSna, aditiuri xelSeSlebis

mravalferovneba

SeiZleba

pirobiTad

daiyos

Semdeg sam ZiriTad jgufad: funqcionaluri, harmo-

110

niuli,

anu

Seyursuli

sixSiris

mixedviT

impulsuri, anu Seyursuli drois mixedviT. fluqtuaciur xelSeSlaSi igulisxmeba nulovani saSualo mniSvnelobis normaluri ganawilebis mqone droSi uwyveti SemTxveviTi procesi, romlis energetikuli speqtri Tanabrad aris ganawilebuli mTel sixSirul diapazonSi. unda aRiniSnos, rom fluqtuaciuri xelSeSlis Tavidan acileba praqtikulad

SeuZlebelia,

vinaidan

isini

ZiriTad

Setyobinebis sakuTari xmaurebiT arian gamowveulni. SeiZleba mxolod maTi nawilobriv Sesusteba SetyobinebaTa gadacemis sistemis Sesabamisi agebiT. harmoniul xelSeSlaSi igulisxmeba iseTi aditiuri

xelSeSla,

Seyursulia

sixSireTa

SeiZleba

iyos

naklebi,

romelic

harmoniuli xorcieldeba

romlis

energetikuli

viwro

sixSireTa ukavia

xelSeSlebis mimRebi

zolSi, zolis

mowyobilobis

romelic toli

sasargeblo Sesusteba

speqtri an

signals. ZiriTadad

arCevadobis

gazrdiT. rac Seexeba impulsuri xelSeSlebs, es ukanasknelebi warmoadgenen SemTxveviTi impulsebis mimdevrobas, romelTa xangZlivoba signalis elementze naklebia. aseTi saxis xelSeSlebTan brZolis yvelaze efeqtir xerxs warmoadgens, magaliTad, signalis amplituduri SezRudva an mimRebi mowyobilobis

Sesasvlelis

myisieri

xelSeSlis moqmedebis dros. 111

Caketva

impulsuri

amgvarad, droisa da sixSiris mixedviT Seyursuli aditiuri xelSeSlebis mniSvnelovnad Semcireba SeiZleba moxdes iseTi mimRebi mowyobilobis agebiT, romlis drosac mcirdeba iseTi saxis xelSeSlebis

gadamwyveti

moxvedris

albaToba.

xelSeSlebs,

maTi

mowyobilobis rac

Seexeba

Semcireba

Sesasvlelze fluqtuaciur

moiTxovs

mTeli

informaciis gadacemis sistemis optimizacias garkveuli azrioT daswored es problema ganixileba qvemoT.

4.2. uwyvet arxebSi diskretuli Setyobinebebis optimaluri miRebis kriteriumebi ganvixiloT dawvrilebiT, uwyvet arxebSi diskretuli, Setyobinebebis optimaluri miRebis Semdegi ori kriteriumi: Secdomis minimaluri albaTobis kriteriumi da minimizaciis riskis kriteriumi. Secdomis

minimaluri

telekomunikaciis

realur

albaTobis arxebSi,

kriteriumi.

rogorc

wesi

yovelTvis adgili aqvs xelSeSlebs. amitom Setyobinebis signalebis uSecdomod aRdgena SeuZlebelia, radganac

xelSeSlebis

gadacemul

miRebul

SemTxveviTi signalebs

bunebis Soris

gamo

Sesaba-

misoba araerTmniSvnelovania. im SemTxvevaSi, rodesac gadaicema Setyobineba

n -uri simboloebisagan Sedgenuili

ai (0,1,2,..., n â&#x2C6;&#x2019; 1), , mimRebma mowyobilobam 112

miRebuli rxevebis S * (t ) = S (t ) + 両 (t ) analizis (diskretuli Setyobinebebis wyaros, signalebis S

arxis Tvisebebis gaTvaliswinebiT) safuZvelze, unda miiRos

gadawyvetileba

Sesaxeb.

mimRebis

gadacemuli

moqmedeba

simboloebis

dros

SeiZleba

warmodgenil iqnas rogorc miRebuli S * (t ) signalis sivrcis dayofa arakveTad qvesimravleebad ( n simboloebis

ricxvis

tol)

unda

moaxdinos

miRebuli S * signalis gaigiveba im ak

simbolos-

Tan, romlis areSi igi moxvda. amgvarad, mimRebis arsebiT

moqmedebas

warmoadgens

gadawyvetileba

Setyobinebebis gadacemuli simbolos Sesaxeb. amis gamo,

zogjer

mimRebi

gadamwyvet

mowyobilobad

iwodeba. rasakvirvelia, realur pirobebSi mimRebSi xorcieldeba mis Sesasvlelze miwodebuli signalis

rigi

gardaqmnebi

(filtracia,

gaZliereba,

demodulacia da sxv.) magram winamdebare ganxilvis SemTxvevaSi isini araarsebiTi mniSvnelobisaa.

S sivrcis dayofa S *k qvesivrceebad SesaZlebelia sxvadasxva wesebis gamoyenebiT. is dayofa, romelic Seesabameba optimizaciis romelime kriteriums iwodeba optimalurad, xolo mimRebi romelic muSaobs

kriteriumis

Sesabamisad

optimalur

mimRebad. yovel kriteriums Seesabameba wesi, romliTac mimRebi iRebs gadawyvetilebas; es wesi gansazRvravs optimaluri mimRebis funqcionalur sqemas. 113

avRniSnoT S * signalis S k* qvesimravleSi moxvedris albaTobaTa ai simbolos gadacemisas P(Sk* | ai )-

P( Si* | ai )

Ti. maSin cxadia, miRebis

albaTobaa,

xolo

simbolos sworad

1− P(Si* | ai ) = ∑P(Sk* | ai )

k ≠i

misi SecdomiT miRebisa. Setyobinebebis simbolos SecdomiT miRebis sruli (saSualo) albaToba (an Secdomis albaToba) tolia n −1

Pl = 1 − ∑ P(ai ) P( Si* | ai ).

(4.2)

i =0

optimaluri mimRebis agebassafuZvlad SeiZleba miRebul iqnas

P -is minimumis kroteriumi, magram

aseTi

yovelTvis

midgoma

ZiriTadi mcdari

naklia

is,

rom

gadawyvetilebebis

zogadad

sxvadasxvaa

aris

igi

swori.

misi

iTvaliswinebs

mniSvnelobas,

romelic

Setyobinebebis

sxvadasxva

telekomunikaciis

sistemebSi

simbolosaTvis.

e.i.

optimalurobis

cneba

mWidrodaa

dakavSirebuli

Secdomis mniSvnelobasTan. minimaluri riskis kriteriumi. optimalurobis yvelaze zogad kriteriums warmoadgens minimaluri riskis kriteriumi. misi arsi mdgomareobs SemdegSi: gadacemuli simboloebis yvela wyvils (gadacemuli simbolo ak ; miRebuli simbolo ai , i ≠ K ) eniWebaT romeliRac

ricxviTi

danakargebi

ewodebaT.

koeficientebi rac 114

ufrTo

L(ak , ai )

maT

arasasurveli

Secdoma, mas miT ufro meti danakargebi miewereba. (danakargebis Sefaseba damoukidebuli amocanaa da winamdebare wignSi igi ar ganixileba). Tu gaviTvaliswinebT zemoaRniSnuls, maSin optimalurobis moTxovnas

safuZvlad

SeiZleba

daedos

saSualo

danakargebis minimumi, anu riskis minimumi. n −1

r = ∑ P(ai ) L(ak , ai ) P( S k* | ai )

(4.3)

zogjer minimaluri riskis kriteriumi iwodeba baiesis

kriteriumad.

gamoyeneba

moiTxovs

telekomunikaciis

unda didi

arxis

aRiniSnos raodenobis * k

P ( S | ai ),

rom

misi

monacemebs

Setyobinebis

wyaros P(ai ) da danakargebis L(ak , ai ) Sesaxeb rac praqtikaSi yovelTvis ar gvaqvs. amitom unda ganvixiloT optimalurobis kriteriumi, romlebic ama Tu im sasruliT Seesabameba im apriorul monacemebs romlebic gamomdinareobs beisis kriteriumidan. upirveles

yovlisa

ganvixiloT

situacia,

rodesac nebismieri Secdomiani gadasvlebi ai → ak Tanabrad arasasurvelia, e.i. L(ak , ai ) yvela wyvilisaTvis k , i (k ≠ i ) warmoadgens raRac erTnair sidides. aseTive mdgomareoba warmoiqmneba maSinac rodesac danakargebis dasabuTebuli arCeva SeuZlebelia. Tu miviRebT rom L(ak , ai ) = L , gveqneba

115

n −1 n −1

n −1

i =o k =o

i =o

r − L∑∑ P(ai ) P( S k* | ai ) = L∑ P(ai )[1 − P( Si* | ai )].

(4.4)

riski minimaluria, rodesac Secdomis sruli albaToba minimaluria, anu rodesac swori miRebis albaToba maqsimaluria. mimRebis moqmedeba efuZneba simboloebis apostorioruli ganawilebis analizs, romelic

Tavis

mxriv

ganisazRvreba

beisis

formuliT:

P (ai | S * ) = P ( S * ) −1 P(ai ) P( S * | ai ). Tu yoveli

(4.5)

ai simbolos gadacemisas regist-

rirdeba is, romlisTvisac maqsimaluria apostoriuruli albaToba. mimRebi, romelic axdens gadacemuli simbolos arCevas apostorioruli albaTobis maqsimumis mixedviT iwodeba kotelnikovis optimalur mimRebad. informaciis gadacemis Tanamedrove sistemebs waeyenebaT gazrdili moTxovnebi gadacemis sisworis TvalsazrisiT da Secdomebi Setyobinebis nebismier simboloSi

amcirebs

mis

Rirebulebas

ise

mniSv-

nelovnad, rom mizanSewonilia Secdomebi CaiTvalos erTnairad

gadacemuli

miRebuli

simbolos

yvela SesaZlo wyvilisaTvis. Tu mimRebSi araa cnobili aprioruli albaTobebi P (ai ) , magram danakargebi L(ak , ai ) gansazRvrulia, maSin minimaluri riskis kriteriumis gamoyeneba ar SeiZleba. gadawyvetilebis miRebis wesisaT-

116

vis SeiZleba gamoyenebul iqnes mxolod pirobiTi riski

ri =

n −1

∑ L( a , a ) P ( S k

k =0 , k ≠ i

* k

| ai ).

(4.6)

S k* –s dayofis Sereuli xerxisas pirobiTi riski aris mxolod ai -s funqcia. maSin optimaluri wesi iqneba

imdagvari,

riskis

romelsac

maqsimaluri

kriteriums

ewodeba

Seesabameba

mniSvnelobis minimaqsuri

pirobiTi

minimumi.

aseT

kriteriumi.

kriteriumis gamoyenebis SemTxvevaSi garantirebulia is,

rom

saSualod

danakargebi

gadaaWarbebs

maqsimalur mniSvnelobas. aqve unda ganvixiloT SemTxveva, rodesac ar aris

monacemebi

simboloebis

albaTobaTa

ganawi-

lebis da danakargebis Sesaxeb. maSin gadawyvetilebebis SeiZleba

miRebis

optimaluri

gamoviyenoT

wesis

mxolod

SerCevisaTvis

pirobiTi

albaTo-

bebis P( S * | ai ) codna. Tu mas ganvixilavT, rogorc

ai -is

funqcias,

SemTxvevaSi

albaTobebis

ganawileba iwodeba simarTlis msgavs funqciad.

117

4.3. telekomunikaciis diskretuli sistemebis potencialuri xelSeSlebisadmi mdgradoba fluqtuaciuri xelSeSlebis dros potencialuri

xelSeSlebisadmi

mdgradobis

arsi. signalebis optimaluri miRebis problemisadmi zemoT (4.2) mocemuli midgoma aris sakmaod zogadi da SeiZleba gamoyenebul iqnas signalebis, xelSeSlebisa

arxebis

farTe

klasisaTvis.

yvelaze

ufro advilad optimaluri miRebis amocana wydeba mudmivi parametrebis mqone arxisaTvis, romelSic moqmedebs TeTri xmauris tipis aditiuri normaluri fluqtuaciuri

xelSeSla.

dros

warmoqmnili

amocana pirvelad gadaWril iqna v.a. kotelnikovis mier

1946

wels.

cialuri

mis

mier

damuSavebuli

xelSeSlebisadmi

saSualebas

iZlevba

mdgradobis

ganisazRvros

potenTeoria

optimaluri

mimRebis xelSeSlebisadmi mdgradoba da dadgindes misi funqcionaluri sqema im SemTxvevisaTvis, roca gadasacemi signalebis forma mimReb mxares zustadaa cnobili, xolo yvela Secdomis mniSvneloba erTnairia. am xelSeSlebisadmi mdgradobas potencialuri

ewodeba,

ukeTesad

iqnes

radganac

miRebuli

igi

arcerTi

SeiZleba

sxva

mimRebis

mier. nebismieri realuri mimRebisaTvis xelSeSlebisadmi mdgradoba SeiZleba ganisazRvros gaangariSebis

eqsperimentaluri

gziT.

misi

Sedareba

potencialuri xelSeSlebisadmi mdgradobasTan sa118

Sualebas iZleva dadgindes, Tu ramdenad srulyofilia mocemuli mimRebi da saWiroa Tu ara misi gaumjobeseba. optimaluri

mimRebis

agebis

wesi.

mTlianad

cnobili signalis SemTxveva gulisxmobs, rom ai -is gardaqmna Si (t ) signalad aris erTmniSvnelovani da misi yvela parametri signalis gadacemis dasawyisisa da dasasrulis dros T aris

cnobili

mimReb

intervalis CaTvliT

mxareze.

(tolia

signalis

xangZlivobis). magram, praqtikaSi es piroba yovel-

ai -sa da

Tvis ar sruldeba.

Si (t ) -s Sesabamisoba

SeiZleba ar iyos erTmniSvnelovani: erT

sim-

bolos SeiZleba Seesabamebodes Si signalis mTeli erToblioba. signali

maSin

Seicavs

gadacemuli erT

(da

miRebuli)

ramdenime

ucnob

(SemTxveviT) parametrebs. signalis parametrebi aseve SeiZleba Seicvalos signalis gadacemisas, cvladi parametrebis mqone telekomunikaciis arxSi. aseTi signalebi iwodeba signalebad cvladi parametrebiT. cvladi

parametrebis

analizs,

magram

gaTvaliswineba

amasTan

erTad

arTulebs

cxadia,

rom

signalebis parametrebis umniSvnelo variaciebisas, xelSeSlebisadmi

mdgradoba

mcired

gansxvavdeba

potencialurisagan, roca parametrebi iTvleba mudmivad.

mosazreba

garkveulwilad

amarTlebs

signalis parametrebis mudmivobis daSvebas.

119

rogorc wina paragrafSi aRiniSna, aseT SemTxvevaSi optimalurma mimRebma unda miiRos gadawyvetileba gadacemul ai simboloze (anu Si (t ) signalze) albaTobaTa aposterialuri ganawilebis analizis safuZvelze (4.5) Secdomiani gadawyvetilebebis minimaluri

SesaZlo

ricxvi

miiReba

maSin,

roca

gadacemulad iTvleba simbolo, romelsac Seesabameba udidesi aposterioruli albaToba, e.i. kotelnikovis iTvleba

mimRebis

gadawyvetilebis

gadacemulad

wesi

simbolo

ak ,

Semdegia: romlis-

Tvisac sruldeba piroba

P (ak | S * ) > P(ai | S *),

(4.6)

yvela i ≠ k -saTvis. (4.5) gamosaxulebaSi S * rxevis miRebis albaToba ganisazRvreba sruli albaTobis formuliT n−1

P ( S *) = ∑ P(ai ) P( S * | Si ),

(4.7)

i =0

da S * (t ) miRebis Semdeg aris raRac cnobili sidide. romelic erTnairia yvela ai -saTvis. amitom, gadawyvetilebis nairad:

wesi

gadacemulad

SeiZleba

Caiweros

iTvleba

Semdeg-

simbolo

ak ,

romlisTvisac

P (ak ) P ( S * | S k ) > P(ai ) P( S * | Si ); yvela i ≠ k -saTvis.

120

(4.8)

S * rxevis pirobiTi albaToba, maSin rodesac gadaicema signali Si aditiuri xelSeSlis zemoqmedebisas, tolia imis albaTobisa, rom dakvirvebis intervalSi xelSeSlam miiRo mniSvneloba

ξ (t ) = S * (t ) − Si (t ).

(4.8)

xelSeSla, aris uwyveti procesi, amitom albaTobis nacvlad unda ganvixiloT albaTobis simkvrive. yvela Gξ (vt/hc) sixSireze simZlavris Tanabari speqtraluri albaTobaTa simkvrivis mqone normaluri fluqtuaciuri xelSeSlis albaTobaTa simkvrive gamoiTvleba formuliT T

− ∫ ξ 1 G1 o e ωn (ξ1 , ξ 2 ,...ξ n = n ( 2π δ )

(4.10)

–is

gamoyenebiT

(4.8)

( t ) dt

(4.10)

SeiZleba Caiweros

Semdegnairad: T

∫

−Gξ−1 ( S *− Sk ) 2 dt

P ( ak ) e

> P(ai )e

∫

−Gξ−1 ( S *− Si ) 2 dt o

(4.11)

yvela i ≠ k -saTvis. zogjer mosaxerxebelia ganxilul iqnas simboloebis

ara

aposterioruli

albaTobebi,

aramed

maTi naturaluri logariTmi, romelic argumentis monotonuri funqciaa. maSin optimaluri mimRebis gadawyvetilebis wesi iqneba

121

* 2 ∫ ( S * − Sk )dt − Gξ ln P(ak ) < ∫ (S − Si ) dt − Gξ ln P(ak ), o

(4.12)

yvela i ≠ k -saTvis. amgvarad, imisaTvis rom miRweul iqnes gadacemis udidesi miRwevadi siswore mimRebma gadacemulad unda amoirCios is simbolo

ak , romlis-

Tvisac

∫ (S * −S

)dt − Gξ ln P(ak ) -s

gaaCnia

umciresi

mniSvneloba. Tu

yvela

simbolo

TanabaralbaTuria

maSin

optimaluri mimRebis piroba Caiwereba Semdegnairad: T

2 * 2 ∫ (S * −S k ) dt < ∫ (S − Si ) dt , o

(4.13)

yvela i ≠ k -saTvis. yvelaze didi albaTobiT gadacemuli signalia is, romelic yvelaze mcired gansxvavdeba sagan (saSualo kvadratuli azriT).

122

S * (t ) -

sakontrolo kiTxvebi 1. raSi mdgomareobs uwyvet arxebSi diskretuli Setyobinebebis optimaluri miRebis amocana? 2. romlebia uwyvet arxebSi diskretuli Setyobinebebis miRebis kriteriumebi? 3. ra aris telekomunikaciis diskretuli sistemebis potencialuri xelSeSlebisadmi mdgradobis arsi fluqtuaciuri xelSeSlebis dros?

123

Tavi 5. informaciids cifrul formaSi gadacemis safuZvlebi 5.1. informaciis cifruli gadacemis sqema gadacemis cifruli sistema (gcs) Sedgeba sami ZiriTadi

kvanZisagan:

cifruli

signalis

formi-

rebis mowyobiloba, mimRebi mowyobiloba da saxazo traqtis mowyobiloba (nax. 5.1).

nax. 5.1. gadacemis cifruvli sistemis zogadi struqturuli sqema

cifruli signalis formirebis mowyobilobis ZiriTad kvanZs warmoadgens cifruli multipleqsori, zogadad, telekomunikaciis saerTSoriso kavSiris standartizaciis seqtoris ITU-T-s (Interna-tional Telecomunication Union) G. 701 (03/93) mocemuli

ganmartebiT,

rekomendaciaSi

cifruli

multipleqsori

aris aparatura, romelic droiTi jgufwarmoqmnis meSveobiT

aerTianebs

ramdenime

cifrul

signals

erT Sedgenil cifrul signalad. Tanamedrove cifrul sistemebSi gamoyenebul cifrul multipleq124

sorebs SeuZlia gaaerTianos sxvadasxva saxis cifruli signalebi: satelefono, satelevizio, monacemTa gadacemis da a.S. amasTan im SemTxvevaSi, roca signali

analoguria,

mas

winaswar

gardaqmnian

cifrul formaSi analogur cifruli gardamsaxis (acg) saSualebiT. mimReb mowyobilobaSi xdeba ukuoperacia, romelsac asrulebs cifruli demultipleqsori. tskt-s zemoaRniSnul rekomendaciaSi mocemuli ganmartebiT cifruli demultipleqsori aris aparatura Sedgenili cifruli signalis Semadgenel cifrul signalebad dayofisaTvis. cifruli demultipleqsirebis Semdeg, saWiroebis SemTxvevaSi, xdeba sawyisi analoguri signalebis aRdgena cifrul-analoguri gardamsaxis (cag) saSualebiT. cifruli nakadebis gadacema SeiZleba ganxorcieldes sxvadasxva tipis saxazo traqtiT: sakabelo (simetriuli, koaqsialuri, boWkovan-optikuri), radiosareleo

Tanamgzavruli.

miuxedavad

traqtebis specifikuri Taviseburebebisa, maTi ageba warmoebs erTi struqturuli sqemiT (nax. 5.2). cifruli nakadis xazSi gadacemisas warmoqmnili damaxinjebebis Sesamcireblad saxazo traqtis gadamcem damaboloebel mowyobilobaSi dayenebulia kodis

gardamqmneli

(kggad),

romelic

cifruli

multipleqsoris gamosasvlel kods gardaqmnis e.w. saxazo

kodad.

kodis

gardamqmneli

dayenebulia

agreTve mimReb damaboloebel mowyobilobaSi (kgmim), 125

oRond aq igi asrulebs Sebrulebul operacias _ saxazo kods gardaqmnis sawyis kodad. radiosareleo, optikur

Tanamgzavrul

sistemebSi

modulireba

warmoebs

saxazo

kodiT,

gadamtani ris

boWkovansixSiris

Semdeg

xazSi

gadaicema radioimpulsTa mimdevroba.

nax. 5.2. saxazo traqtis gamartivebuli struqturuli sqema

cifruli signalebis damaxinjebebi,gamowveuli xelSeSlebiTa da danakargebiT xazSi, nawilobriv swordeba regeneratorebSi, romlebic SeiZleba iyos aramosmaxure

(ar)

momsaxure

(mr).

tsk-t-s

G.701(03/93) mocemuli ganmartebiT, zogadad, regeneratori aris maZlierebeli, romelic axdens cifruli signalebis regenerirebas. aqve unda aRiniSnos, rom ar-Si warmoebs xazSi gadacemuli

cifruli

koreqcia, amasTan, droiTi

gaZliereba

aRdgeba misi amplituduri

Tanafardobebi.

regenerirebisa,

signalis mr-Si,

SesaZlebelia

garda

signalis

jgufuri

cifruli

nakadis raRac nawilis gamoyofac.

126

kodebis

gardaqmnisa

saxazo

signalebis

regenerirebis saboloo mizans warmoadgens Secdomebis albaTobisa da impulsebis fizikuri fluqtuaciebis albaTobisa da impulsebis fizikuri fluqtuaciebis Semcireba. es ukanaskneli gansakuTrebiT saSiSia

farTozoliani

signalebisaTvis,

amitom

aseTi signalebis gadacemisas damaboloebel mowyobilobebSi xSirad iyeneben fizikuri fluqtuaciebis CamxSobebs.

5.2. informaciis cifruli gadacemis boWkovan-optikuri sistemebi rogorc cnobilia, optikuri gadacemis dros xdeba

cifruli

eleqtruli

signaliT

optikuri

gadamtanis modulireba, ris Semdeg modulirebuli sinaTlis signali gadaicema optikuri kabeliT. boWkovan-optikuri kavSiris sistemis struqturuli sqema moyvanilia nax. 5.3-ze. nax 5.1-ze moyvanil sqemasTan SedarebiT igi Seicavs damatebiT kvanZebs _ optikur gadamcems (og) da optikur mimRebs (om). garda amisa, saxazo traqti regeneratorebTan (r)

erTad

Seicavs

optikur

maZliereblebs

(m).

gadacemis ares warmoadgens optikuri kabeli ok.ogSi xdeba eleqtruli signalis gardaqmna optikur signalad (eog), om-Si ki ukuoperacia (oeg).

127

nax. 5.3. boWkovan-optikuri kavSiris sistemebis zogadi struqturuli sqema

eog-is ZiriTad elements warmoadgens naxevargamtaruli

lazeri

oeg-is

sxivis

sikaSkaSis

Suqgamomsxivebeli

fotodiodi.

eog-Si

modulireba

xdeba

diodi,

sinaTlis

gadacemis

kodis

gardamqmnelis (kggad) gamosavali signaliT, romlis Sesasvlelze multipleqsoridan miewodeba sawyisi cifruli signali. eog-is gamosasvlelze dayenebulia gadamcemi SemaTanxmebeli mowyobiloba (Smgad), romlis

daniSnulebaa

gadamcemi

mowyobilobis

gamosavali signalis parametrebis SeTanxmeba ok-is maxasiaTeblebTan. gadis ra optikur kabels, optikuri signali mimRebi SemaTanxmebeli mowyobilobis (SmmimR) saSualebiT miewodeba oeg-s da gardaiqmneba masSi eleqtrul signalad. amis Semdeg mimRebis kodis gardamqmnelSi

(kgmimR)

saxazo

128

signali

gardaiqmneba

sawyis cifrul mimdevrobad, demultipleqsirdeba da miewodeba momxmarebels. amgvarad,

gadamcem

mxares

multipleqsoris

Sesasvlelebidan eog-mde, agreTve mimReb mxares oegdan demultipleqsoris gamosasvlelamde moqmedebs eleqtruli signali, eog-dan oeg-mde ki ok-Si gadis optikuri signali. rogorc og-is, ise om-is elementebi dReisaTvis mzaddeba kompaqturi mowyobilobis saxiT, romelsac ewodeba qvantur-eleqtronuli moduli. aseTi moduli Seicavs eog-s da Smgad-s an oeg-s da SmmimR-s. konstruqciulad igi asanTis kolofis zomisaa da erTi mxridan mierTebulia kg-Tan, meore mxridan ki ok-Tan. saxazo warmoadgens

traqtis

erT-erT

regeneratori.

ZiriTad

dReisaTvis

elements jer

ver

xerxdeba optikuri signalis uSualod regenerireba, amitom optikur regeneratorSi Tavdapirvelad xdeba xazSi

gadacemuli

optikuri

signalis

gardaqmna

eleqtrulad, am eleqtruli signalis regenerireba, Semdeg ki regenerirebuli signali kvlav gardaiqmneba optikur signalad. aqedan gamomdinare, optikuri signalis regeneratori, eleqtronuli signalis

regenerirebis

sqemis

garda,

Seicavs

Sesasvlelze da eog-s gamosasvlelze (nax. 1.4).

129

oeg-s

nax. 5.4. optikuri regeneratori.

optikur Soris

sistemebSi

gacilebiT

manZili

didia

regeneratorebs

eleqtrul

sistemebTan

SedarebiT. Tu eleqtrul sistemebSi igi ramdenime erTeuli km-is rigisaa, optikurSi igi dReisaTvis ramdenime aTeul km-s aRwevs. Tanamedrove optikur sistemebSi

regeneratorebTan

regeneratorebis

magierac

erTad

ki)

sul

(zogjer ufro

ki did

gamoyenebas poulobs optikuri signalebis maZliereblebi

(m).

aseTi

maZliereblis

dasamzadeblad

sakmarisia 10 sm sigrZis boWko, romlis gulara legirebulia

erbiumiT.

maZliereblebi

uzrunvel-

yofs saregeneracio seqciis sigrZis zrdas 110-160 km-mde lazeruli gamomsxiveblis 1550 nm talRis sigrZisaTvis,

rac

praqtikulad

2-jer

amcirebs

regeneratorebis saWiro raodenobas. aseTi maZliereblis upiratesobas warmoadgens is, rom isini ar saWiroebs signalebis optikur-eleqtrul da eleqtro-optikur gardamsaxebs gaZlierebis procesSi. bolo wlebamde iTvleboda, rom dupleqsuri optikuri

kavSiris ganxorcielebis sWirdeba ori

optikuri boWko, romelTagan TiToeuli gamoyene-

130

buli iqneboda informaciis gadasacemad erTi mimarTulebiT.

dReisaTvis

ukve

inergeba

sixSiruli

gancalkevebis iseTi optikuri sistemebi, romlebic ormxrivi kavSiri xorcieldeba erTi da igive boWkoTi sxvadasxva talRis sigrZeebze (mag., gadacema warmoebs

λ = 1,3

mkm,

xolo

miReba

− λ = 1,55 mkm

sigrZis talRebze).

5.3. informaciis cifruli gadacemis sistemebi ATM teqnologiebis gamoyenebiT ATM teqqnologiebis bazaze agebuli sistemebi ZiriTadad gamoiyeneba cifrul qselebSi momsaxurebis integraciiT (B-ISDN Broadand Integrated Service Digital Network). msgavsi qselebis arqiteqtura efuZneba calkeuli fenebis koncefcias, romlebic uzrunvelyofs sami jgufis funqcias: momxmareblis, marTvisa da administraciuli marTvis funqciis gamoyofas (nax. 5.5). igi dawvrilebiT ganxiluli iqneba qvemoT. farTozoliani ISDN (nax. 5.6) (am naxazze BNT1Broadand Network Termination1 _ farTozoliani qseluri daboloeba; B-LT-Broadand Termination _ farTozolovani saxazo daboloeba) fenebad iyofa gamoyenebuli protokolebis emyareba

Sesabamisad

winamorbeds-viwrozolian

igi

ISDN-s,

samsaxurebs uzrunvelyofs ori qvesistema: 131

ZiriTadad sadac

_ gamanawilebeli qseli abonantsa da SeRwevis adgilobriv

sakomutacio

mowyobilobas

Soris,

romelSic gadaicema ujredebis uwyveti nakadi an sinqronuli cifruli kadrebi;

nax. 5.5. farTozoliani ISDN-is funqcionaluri sqemebi

_ magistraluri satransporto qseli, romelic aerTianebs

adgilobrivi

mowyobilobebs

sinqronuli

saSualebiT.

132

SeRwevis

sakomutacio

cifruli

gadacemis

nax. 5.6. farTozoliani ISDN

qselis arqiteqtura

5.4. informaciis gadacemis cifruli sistemebi analoguri saabonento xazebisaTvis ganvixiloT

analoguri

saabonento

xazebis

cifrul formaSi damkvrivebis sistemebis muSaobis principi.

nax.

5.7-ze

mocemulia

sistemis

zogadi

struqturuli sqema.

nax. 5.7. analoguri saabonento xazebis cifrul formaSi damkvrivebis sistema

133

rogorc sistemebSi genili

struqturuli

gamoiyeneba

satelefono

sqemidan

cifrul

Cans,

formaSi

laparakebis

droiTi

msgavs warmodmulti-

pleqsirebis meTodi. analoguri signali avtomaturi satelefono

sadguris

(ats)

saabonento

komp-

leqtis gamosasvlelidan miewodeba sasadguro terminals

(naxevarkompleqts)

COT (Central Office

Terminal), sadac gardaiqmneba cifrul formaSi cifruli kodirebis romelime meTodis gamoyenebiT: impulsur-koduri modulacia (ikm); delta-modulacia (dm); adapturi delta-modulacia (adm); adapturi deferencialuri impulsur-koduri modulacia (adikm) (es meTodebi dawvrilebiT ganxilulia qvemoT). amis

Semdgom

n-cifruli

nakadi

erTiandeba,

ganicdis saxazo kodirebas da cifrul formaSi gadaecema

saabonento

raodenobis

xaziT

(k<n,

k=1,2...), amis gamo maT cifruli saabonento xazebi ewodeba. daSorebul terminalSi _ saabonento naxevar kompleqtSi

(Rovote

Terminal)

xorcieldeba

zemoaRniSnulis ukuoperacia.

5.5. informaciis gadacemis cifruli radiosistemebi gadacemis radiosistema ewodeba teqnikur saSualebaTa da saxazo traqtis erTobliobas, romlebic uzrunvelyofs gadacemis tipuri arxebisa da 134

jgufuri

traqtebis

formirebasa

telekomuni-

kaciis signalebis gadacemas radiotalRebis saSualebiT Ria sivrceSi. aseve, rogorc mravalarxiani sakabelo sistemebi, radiosistemebic arsebobs analoguri da cifruli. radiosistemis orive es nairsaxeoba igeba erTi da igive struqturuli sqemiT, romelic naCvenebia nax. 1.8-ze. ganvixiloT am sqemis calkeuli kvanZebis agebisa da moqmedebis principebi.

nax. 5.8. radiosistemebis struqturuli sqema

radiosistemebis

sawyis

bolo

kvanZebs

Sesabamisad warmoadgens multipleqsori da demultipleqsori, romelTac igive daniSnuleba aqvs, rac sakabelo sistemebSi. radiosistemebis sadgurebi, romlebic awarmoebs

signalebis

Setanas,

gamotanas

tranzits,

rogorc wesi, teritoriulad daSorebulia qseluri sadgurebisa

komutciis

kvanZebidan,

amitom

radiosistemebis SemadgenlobaSi Sedis gamtariani SemaerTebeli xazebic. gadacemis damaboloebel mowyobilobaSi xdeba iseTi

saxazo

signalis 135

formireba,

romliTac

modulirdeba maRalsixSiruli rxevebi. saxazo signali Sedgeba sainformacio da sasamsaxuro signalebisgan.

mimRebis

damaboloebel

mowyobilobaSi

warmoebs maRalsixSiruli radiosignalis demodulireba da sainformacio da sasamsaxuro signalebis gamoyofa. radiolulis daniSnulebaa modulirebuli radiosignalebis gadacema garkveul manZilze radiotalRebis saSualebiT. radiolula Seicavs radiolulis lobas

gadamcem da

mowyobilobas,

radiotalRebis

mimReb

gavrcelebis

mowyobitraqts.

gadamcemi da mimRebi mowyobilobebi erTad Sedis damaboloebeli radiosadguris SemadgenlobaSi. radiolulas ewodeba martivi, Tu igi Seicavs mxolod or damaboloebel sadgurs da radiotalRebis gavrcelebis erT traqts da Sedgenili, roca mis SemadgenlobaSi ori damaboloebeli radiosadguris garda Sedis erTi an ramdenime saretranslacio sadguri, romelic uzrunvelfgyofs radiosignalebis miRebas, gardaqmnas, gaZlierebasa da xelaxla gadacemas. nax. 5.9-ze mocemulia ormxrivi radiosistemis lulis struqturuli sqema. lulis gadamcemi damaboloebeli

mowyobilobidan

radiolulis

Sesasv-

lelze miewodeba maRalsixSiruli radisignali, romelic

modulirebulia

saxazo

signaliT.

radio-

gadamcemSi gad 1 radiosignalis simZlavre izrdeba nominalur

mniSvnelobamde, 136

misi

sixSire

gar-

daqimneba speqtris gadasatanad saWiro sixSireTa mocemul diapazonSi. fideruli traqtiT gadasacemi radiosignalebi miwodeba antena 1-s, romelic uzrunvelyofs radiotalRebis gasxivebas Ria sivrceSi saWiro

mimarTulebiT.

rogorc

wesi,

Tanamedrove

ormxriv radiosistemebSi sawinaaRlmdego mimarTulebis radiosignalebis gadacemisa da mirebisaTvis gamoiyeneba saerTo saanteno-sadifero traqti.

nax. 5.9. ormxrivi radiosistemis lulis struqturuli sqema

mimReb mxareze (damaboloebeli radiosadguri _ 2) eleqtromagnituri talRebi miiReba antena 2-iT, ris

Semdeg

sadifero

traqti

2-iT

miewodeba

radiomimRebs mimR2, sadac xorcieldeba miRebuli radiosignalebis

sixSiruli

seleqcia,

sixSiris

ukugardasaxva da saWiro gaZliereba. radiolulis gamosasvlelidan miRebuli radiosignali miewodeba 137

lulis damaboloebel mowyobiloba 2-s. analogiurad gadaecema radiosignalebi sawinaaRmdego mimarTulebiT damaboloebel radiosadgur 2-dan radiosadgur 1-saken. saretranslacio tipis:

sadgurebi

telekomunikaciis

arsebobs

gadasacemi

ori

signalebis

gamoyofisa da axlebis Setanis gareSe, gamoyofiT da axlebis SetaniT. pirvel SemTxvevaSi Sedgenili radiolula warmoadgens ramdenime martivi radiolulis mimdevrobiT SeerTebas, xolo mowyobiloba _

radiolulis

mowyobilobis

kompleqtis

mim-

devrobiT SeerTebas. meore tipis retranslatorebis mowyobilobis lulis

SemadgenlobaSi

damaboloebeli

damatebiT

mowyobiloba,

Sedis

romelic

Seicavs modulatorsa da demodulators. rogorc

aRiniSna,

radiosistemebis

agebis

zemoT moyvanili principebi marTebulia, zogadad, rogorc analoguri, ise cifruli sistemebisaTvis. ZiriTadi gansxvaveba maT Soris gamoixateba saxazo signalis formaSi da, aqedan gamomdinare, damaboloebeli radiosadgurebis calkeuli kvanZebi agebis principebSi. cifrul radiosistemebSi analogurisgan gansxvavebiT gamoiyeneba iseTive multipleqsorebi, rogorc cifrul sakabelo sistemebSi. cifrul radiosistemebSi gamoyenebuli signalis

forma

gansazRvravs

lulis

damaboloebeli

mowyobilobebis agebis principebs. saxeldobr, aq adgili aqvs ori tipis modulirebas: pirvelads da 138

meorads. pirveladi modulirebis dros yalibdeba cifruli

saxazo

maRalsixSiruli rebas

dros

meorad

moduli-

manipulirebas

mamodulirebeli

ricxvisagan

damokidebulebiT.

doneTa

manipuflireba

meoradis

radiosignali.

uwodeben

signalis

signali,

SeiZleba

iyos

ordoniani

mravaldoniani. cifrul

radiosistemebSi

saxazo

signalis

maformirebels warmoadgens modulatoris koderi, xolo ukugardamsaxs _ demodulatoris dekoderi. ukugardasaxvis Sedegad aRdgeba sawyisi orobiTi signali, manamde ki demodulatorSi xdeba radioarxidan

miRebuli

signalis

deteqtireba

(maRal-

sixSiruli rxevis moSoreba) da regenerireba. am ukanaskneli operaciis Sedegad deteqtoris gamosasvleli damaxinjebuli signali gardaiqmneba signalad, romelsac gaaCnia gadacemis modulirebuli signalis struqtura. modulatorisa da demudulatoris erTobliobas

ewodeba

modemi.

misi

ganzogadebuli

sqema

naCvenebia nax. 5.10-ze. cxadia,

rom

modemi

warmoadgens

ara

marto

cifruli radiosistemebis damaboloebeli mowyobilobebis, aramed Semtan-gamomtani retranslatorebis Semadgenel kvanZs. Tanamedrove cifrul radiosistemebSi ZiriTadad gamoiyeneba amplituduri (am), sixSiruli (sm) da fazuri (fm) manipulireba. 139

nax. 5.10. modemis ganzogadebuli sqema

am-is

dros

radiosignalis

samodulirebel

parametrs warmoadgens misi amplituda, sm-is dros _ sixSire, fm-is dros ki adgili aqvs radiosignalis fazis cvlilebas 1800-iT saxazo signalSi ~1~ → ~0~ da ~0~ → ~1~ gadasvlisas. fm-is erT-erT nairsaxeobas warmoadgens fardobiTi fazuri manipulireba _ ffm. fm-sagan

gansxvavebiT

fazis

cvlileba

1800-iT

xdeba yovelTvis, roca saxazo signalis sataqto intervalze

Cndeba

simbolo

~1~

(e.i.

~0~ → ~1~

~1~ → ~1~ gadasvlebis dros). unda

aRiniSnos,

rom

dReisaTvis

ZiriTad

mravalarxian radiosistemebs warmoadgens pirdapiri xedvis

radiosareleo

sistemebi

gadacemis

Tanamgzavruli sistemebi. maTi umravlesoba muSaobs decimetrul da santimetrul diapazonSi. swored

140

gadacemis aseTi maRali sixSireebis gamoyeneba ganapirobebs Sedgenili radiolulebis gamoyenebas, radgan aRniSnuli diapazonebis radiotalRebis gavrceleba xdeba mxolod pirdapiri xedvis farglebSi. pirdapiri xedvis miwispira radiosareleo sistemebSi manZili or mezobel retranslators Soris ramdenime

aTeuli

km-ia.

manZili

sagrZnoblad

metia Sereul troposferul radiosareleo sistemebSi, magram maTSi izrdeba radiosignalebis damaxinjebebi,

rac

iwvevs

mZlavri

gazrdili

mgrZnobiarobis

gadacemebisa

mimRebebis

gamoyenebis

aucileblobas. aseT pirobebSi ZiriTadad gamoiyeneba didi sigrZis radiosistemebis agebis ori meTodi:

didi

raodenobis

miwispira

retransla-

torebis gamoyeneba (radiosareleo xazebSi) da

retranslatoris dayeneba xelovnur Tanamgzavrze orive miwispira damaboloebeli sadguris radioxedvis areSi (Tanamgzavrul radiosistemebSi). miwispira radiosareleo

sistemebi

igeba

retranslatorebis

mwkriviT, romelTa raodenoba 100-s aRemateba, xolo manZili maT Soris 50-70 km-is rigisaa. troposferul radiosareleo sistemebSi manZili retranslatorebs Soris SeiZleba Seadgendes 150 รท 700 km-s, xolo erTi Tanamgzavris gamoyenebisas, romelic ganlagebulia geocentrul an gawelil elifsur orbitaze, miiRweva radiokavSiris siSore 15000 km. kavSiris siSoris gazrda am SemTxvevaSi SeiZleba kidev erTi Tanamgzavris gamoyenebiT. 141

5.6. analoguri signalebis cifruli kodirebis ZiriTadi meTodebi analoguri signalebis cifrul formaSi gardaqmnis sferoSi intensiuri gamokvlevebi, gamomdinare

rogorc

miRebuli

misi

Sedegebis

mimdinareobs

saintereso praqtikuli

ukanaskneli

bunebidan,

aseve

mniSvnelobidan,

ramdenime

aTwleulis

ganmavlobaSi. telekomunikaciis dargSi cifruli signalebis gamoyenebis sfero pirobiTad SeiZleba da iyos xuT nawilad: a) gadamuSaveba; b) gadacema; g) komutacia; d) Senaxva; e) maTi kombinaciebi. aRniSnuli miznebisaTvis cifruli signalebis gamoyenebis

mizanSewonilobis

gansazRvrisaTvis

erT-erT ZiriTad maxasiaTebels warmoadgens analoguri signalebis cifrul formaSi gardaqmnis (cifruli kodirebis) meTodi. ganvixilod mokled cifruli kodirebis dReisaTvis gavrcelebuli meTodebi.

142

5.6.1. impulsur-koduri modulaciis (ikm) meTodi ikm principebi SemuSavebul iqna 1937 wels rivzis mier (safrangeTis patenti #852183, 1938 w; didi britaneTis patenti #535860, 1939 w; aSS-is patenti #227070, 1942 w.). tsk-t-s G. 701 (03/93) rekomendaciaSi mocemuli ganmartebiTi, kodireba aris procesi, romlis drosac signali ganicdis diskretizacias, TiToeuli diskreti iqvanteba sxva diskretebisagan damoukideblad

kodirebis

gziT

gardaqimneba

cifrul

signalad. wrfivi ikm-is modemis gamartivebul struqturul sqemas aqvs nax. 5.11-ze naCvenebi saxe. ganvmartoT ikm procesis Catarebuli TiToeuli gardasaxva. tsk-t-s G. 701 (03/93) rekomendaciaSi mocemuli ganmartebiT, diskretizacia aris procesi, romlis drosac miiReba signalis diskretebi, rogorc wesi, drois Tanabari SualedebiT. diskreti aris drois arCeul

momentSi

signalis

warmomdgeni

miRebuli am signalis monakveTisagan.

143

sidide,

nax. 5.11. wrfivi ikm-is modemis struqturuli sqema a) modulatori

b) demudulatori

daqvantva aris procesi, romlis drosac sidideebis uwyveti diapazoni iyofa momijnav intervalebis rigad da nebismieri sidide mocemuli intervalis sazRvrebSi warmodgeba intervalisaTvis erTaderTi winaswar gansazRvruli sididiT (nax. 5.12-ze mocemulia

terminebis

ilustracia,

romlebic

ekuTvnis daqvantvas). kodireba aris koduri kombinaciebis generacia daqvantuli sidideebis warmodgenisaTvis.

144

nax. 5.12. daqvantvisas gamoyenebuli terminebis ilustracia

wrfivi

ikm-is

dros

Sesasvleli

analoguri

signali zoluri filtriT izRudeba sixSiris mixedviT f z ≤ f ≤ f z , ris Sedegadac formirdeba signali X(t), romelic Semdgom diskretizatorSi ganicdis diskretizacias diskretizebuli

2f z

X z ( t)

sixSiriT. damqvantvelSi

signali iqvanteba. amasTan,

wrfivi ikm-is dros doneebs Soris intervali (δ ) mudmivia.

daqvantuli

X dq ( t)

signali

miewodeba

koders, romlis gamosasvlelze miiReba ikm signali

145

m(t). koderSi TiToeuli diskreti gardaiqmneba n â&#x20AC;&#x201C; Tanriga kodur sityvad (kodireba). Tu n â&#x20AC;&#x201C; Tanrigidan TiToeuli Rebulobs L sxvadasxva mniSvnelobas, maSin N=Ln. Tu

L=2, maSin m(t) signali aris

orobiTi da Sesabamisad N=2n. ikm signali gadacemis traqtis gavlis Semdeg miewodeba demudulatoris Sesasvlels. Tu davuSvebT, rom gadacemis traqtSi gadacemuli simboloebis dakargvas adgili ara aqvs, maSin regeneratoris gamosasvlelze miiReba signali m(t). dekoderSi m(t) cifruli signalidan formirdeba maaproqsimirebeli Zabva q*(t). es ukanaskneli, rogorc wesi, warmoadgens amplitudurad modulirebuli impulsebis mimdevrobas (amplitudur-impulsuri modulacia

aim).

aRdgenis

saboloo

etapze

demodu-

latoris analogur nawilSi aim impulsebis mimdevrobidan aRmdgeni filtris saSualebiT gamoiyofa mamodulirebeli gamosasvleli gansxvaveba

signali. uwyveti

amgvarad,

signali

X(t) signalisagan

formirdeba

X*(t),

romelic

diskretizaciis

daqvantvis procesiT gamowveuli damaxinjebebiT. nax. 5.13-ze mocemulia im gardasaxvdebis magaliTebi, romelTac adgili aqvs wrfivi impulsurkoduri modulaciis dros modulatorSi, roca L=2, N=8.

146

nax. 5.13. orobiTi nakadis formireba 3-bitiani ikm-is dros

147

5.6.2. diferencialuri impulsur-koduri modulacia (dikm) dikm-is principi efuZneba analoguri signalebis diskretebs Soris korelaciur kavSirs. tsk-t-s G. 701 (03/93) rekomendaciaSi mocemuli ganmartebiT diferencialuri impulsur-koduri modulacia aris procesi, romlis drosac signali diskretizdeba, sxvaoba am signalis TiToeul diskretsa da mis savaraudo

mniSvnelobas

Soris

iqvanteba

kodirebis gziT gardaiqmneba cifrul signalad. dikm-is gamoyeneba analoguri signalebis cifruli kodirebisaTvis zogadad saSualebas iZleva Semcirdes is informaciuli siWarbe, romelic gaaCnia Sesasvlel analogur signals da Sedegad Semcirdes gadasacemi cifruli informaciis siCqare. dikm-is

modemis

gamartivebul

struqturul

sqemas aqvs nax. 5.14-ze moyvanili saxe. dikm-is modemSi Semavali winaswarmetyveli aris mowyobiloba, romelic gamoimuSavebs diskretizebuli signalis savaraudo mniSvnelobas, miRebuls amave

signalis

winamavali

diskretebiT

diskretebis daqvantuli sidideebiT (tsk-t-s rek. G. 701 (03/93). diferencialuri impulsur-koduri modulaciis nairsaxeobas warmoadgens delta-modulacia (dm), romlis drosac mxolod erTi bitiT deteqtirdeba da kodirdeba

sxvaoba TiToeul diskretsa

148

da mis winaswarmetyveleb mniSvnelobas Soris (tskt-is rek. G.701 (03/93). dikm-is gamoyenebisas gamosasvleli cifruli nakadis Semdgomi Semcireba SesaZlebelia daqvantvisa da winaswarmetyvelebis procesebis adaptaciis meTodebis

gamoyenebiT,

rasac

mivyavarT

adaptur

diferencialur impulsur-kodur modulamde (adikm) (adaptur delta-modulaciamde-adm).

nax. 5.14. dikm modemis struqturuli sqema a) modulatori, b) modulacia

5.6.3. adapturi diferencialuri impulsur-koduri modulaciis (adikm) meTodi adikm dReisaTvis farTod gamoiyeneba analoguri

signalebis,

kerZod

cifruli kodirebisaTvis.

149

bgeriTi

signalebis

tsk-t-s G. 701 (03/93) rekomendaciaSi mocemuli ganmartebiT: _ adikm-is algoriTmebi warmoadgens SekumSvis algoriTmebs, romlebic SesaZleblobas iZleva Semcirdes bitebis gadacemis siCqare adapturi winaswarmetyvelebis da adapturi daqvantvis gamoyenebis meSveobiT; _ adapturi winaswarmetyveli aris winaswarmetyveli, romlis SefasebiTi funqcia icvleba diskretizebuli signalis speqtruli maxasiaTeblebis Sesabamisad drois xanmokle SualedebSi; _ adapturi daqvantva aris daqvantva, romlis drosac zogierTi parametri Secvladia daqvantuli signalis statistikuri maxasiaTeblebis Sesabamisad drois xanmokle SualedebSi. adapturi daqvantvisa da adapturi winaswarmetyvelebis meTodebis (an orives erTdroulad) gamoyeneba saSualebas iZleva mniSvnelovnad Semcirdes gamosasvleli cifruli nakadis gadacemis siCqare mimRebSi aRdgenili analoguri signalebis damaxinjebebis mniSvnelovani gazrdis gareSe. aRniSnuli bgeriTi

meTodebis

signalebis

gamoyeneba,

kodirebisaTvis,

magaliTad, saSualebas

iZleva es signalebi gadaices 32,16 da 8 kbiti/wm siCqariT. amJamad, bgeriTi signalebis cifruli kodirebisaTvis adikm-is meTodiT, damuSavebulia tsk-t-s rekomendaciebi G.721-726. 150

nax. 5.15-ze magaliTis saxiT naCvenebia bgeriTi signalebis adikm modemis gamartivebuli struqturuli sqema gadacemis siCqareebisaTvis _ 32 kbiti/wmSi da 16 kbiti/wm-Si. adikm modemis gadamcem mxareze Sesasvleli ikm signali

S(k) gadacemis

siCqariT

kbiti/wm-Si,

romelSic modulirebulia A(Îź ) kanoniT, gardamqmnelSi

gardaiqmneba

aramodulirebul

wrfiv

ikm

signalad S1(k). gamomklebi mowyobilobis gamosasvlelze miRebuli sxvaobiT signali d(k) warmoadgens Sesasvleli S1(k) signalisa da winaswarmetyvelebi Se(k)

signalis

sxvaobas.

adaptur

damqvantelSi

miiReba sxvaobiTi signalis sididis oTxTanriga an orTanriga orobiTi kodi, romelic gadaicema demodulatorSi.

mainvensirebeli

adapturi

damqvantve-

lis gamosavlelze miiReba 16 an 4 doned daqvantuli sxvaobiTi signalis invensirebuli mniSvnelobebi. amis Sedegad mis gamosasvleleze miiReba S1(k) signalis aRdgenili mniSvnelobebi. adaptur winaswarmetyvels miwodeba rogorc sxvaobiTi, aseve aRdgenili mniSvnelobebi, riTac ikvreba ukukavSiris maryuJi. adikm

demudulatorSi

Semavali

sinqronuli

kodirebis dayenebis mowyobiloba aucilebelia im damaxinjebebis

dagrovebis

Tavidan

asacileblad,

romelic SeiZleba warmoiqmnas mimdevrobiTi (adikmikm-adikm) kodirebisas.

151

nax. 5.15. adikm modemis gamartivebuli struqturuli sqema

5.7. cifruli signalebis multipeqsirebis ZiriTadi principebi tsk-t-is

R.140(1988)

rekomendaciaSi

mocemuli

ganmartebebiT: _

multipleqsireba

aris

procesi

ramdenime

calkeuli daqvemdebarebuli arxebis damoukidebeli signalebis gaerTianebisa da saerTo arxiT imave mimarTulebiT gadacemisaTvis;

152

_ demultipleqsireba aris procesi, romelic gamoiyeneba multipleqsuri signalebis aRdgenisaTvis

signalebis

sxvadasxva

individualur

arxebSi ukuganawilebisaTvis. multipleqsirebis operacias asrulebs multipleqsori, romelsac gaaCnia ramdenime Sesasvleli da erTi gamosasvleli. SesasvlelTa mTliani ricxvidan garkveul raodenobas ewodeba sainformacio, danarCenebs ki _ mmarTveli signalebi. telekomunikaciis cifrul sistemebSi cifruli multipleqsirebis operacia xorcieldeba komutatoris saSualebiT. es ukanaskneli mimdevrobiT aerTebs

TiToeul

arxs

drois

gansazRvruli

intervaliT (mas agreTve uwodeben `taim-slots~ anu `komutaciis intervals~), romelic saWiroa signalis diskretis

(an

romelime

fiqsirebuli

nawilis)

gasagzavnad mocemul arxSi. tsk-t-is G. 701.Q.9 rekomendaciebSi mocemuli ganmartebiT: taim-sloti (TS-Time Slot) aris nebismieri perioduli droiTi intervali, romelic SeiZleba calsaxad iqnes amocnobili da gansazRvruli. ase

formirebuli

diskretebis

nakadi

sxvadasxva

Sesasvleli arxidan miewodeba kavSiris arxs. mis mimReb mxares cifruli demultipleqsori analoguri komutatorisa da Semdeg qveda sixSiris filtris saSualebiT gamoyofs garkveul diskretebs da anawilebs maT Sesabamis arxebSi. aucilebelia, rom gadamcemi

mimRebi

mxareebis 153

komutatorebi

muSaobdes sinqnorulad, e.i. iyos sinqnorizebuli. tsk-t-is, rek. G. 701 (03/93)-is Tanaxmad sinqronizacia aris

procesi

signalebis

Sesabamisi

aRniSnuli

momentebis Sewyobisa maTi sinqnorulobis uzrunvelyofisaTvis. ikm

satelefono

qselebSi

komutatori

unda

`brunavdes~ periodiT, romelic Td diskretizebis periodis tolia. maSin arxis komutirebis intervali ^Tk =Td /n=125/n (mkwm), sadac n aris multipleqsoris Semavali arxebis ricxvi. komutatorebis sinqnorizebisaTvis gamoyenebuli unda iqnas specialuri sinqrosignali (magaliTad, garkveuli sigrZis ~11...11~ tipis mimdevroba). igi SeiZleba

gadacemuli

iqnas

marTvis

gareSe,

gamoyenebul iqnas Sigasaarxo sinqnorizeba. am ukanasknelis dros sinqronizebis procesi daiyvaneba an damatdebiTi, e.w. `gamaTanabrebeli~ bitis an bitebis jgufis CarTvaze m anaTvalis Semdeg, an anTvlebis nakadSi

ufro

organizebaze,

rTuli

ganmeorebadi

romelic

Seicavs

struqturis anaTvals

gansazrvruli sigrZis k vels, an gamaTanabrebel bitebs. aseT struqturas ewodeba kadri anu freimi. ramdenime freimi SeiZleba gaerTiandes ufro zogad struqturaSi, freimis

romelsac

gameorebis

periodi

ewodeba aris

multifremi. dro,

romelic

sWirdeba komutirebis erT mTlian cikls bitebis

154

gamaTanabrebeli jgufis Camatebis drois gaTvaliswinebiT. unda aRiniSnos is garemoeba, rom telekomunikaciaSi kompiuteruli teqnikis intensiur danergvamde terminebis _ `freimi~ (`kadri~) `multifreimi~, nacvlad gamoiyeneboda, Sesabamisad, terminebi `cikli~,

`zecikli~.

qvemoT

moyvanil

masalebSi

terminebi gamoiyeneba tsk-t-is G-701 rekomendaciaSi mocemuli Semdegi ganmartebebis mniSvnelobiT: _ freimi (cikli) â&#x20AC;&#x201C; aris cikluri erToblioba mimdevrobiTi

sataqto

intervalebisa,

SesaZlebelia

ganisazrvros

sataqto

romelSic intervalis

fardobiTi mdebareoba; _

multifreimi

erToblioba

(zecikli)

mimdevrobiTi

aris

ciklebisa,

cikluri romelSic

SesaZlebelia ganisazRvros TiToeulis fardobiTi mdebareoba. multipleqsoris komutators SeuZlia mimdevrobiT amoirCios arxebidan bitebis nebismieri mimdevroba.

monacvleoba.

process

ewodeba

ganasxvaveben

interlivingi,

interlivingis

anu

Semdeg

ZiriTad saxeobebs: _ bit-interlivingi, anu bitebis monacvleoba _ gamosasvlelze mimdevrobiT komutirdeba TiTo biti yoveli arxidan; _ bait-interlivingi, anu baitebis monacvleoba _ gamosasvlelze mimdevrobiT komutirdeba TiTo baiti yoveli arxidan; 155

blok-interlivingi,

anu

blokebis

monacv-

leoba _ gamosasvlelze morigeobiT komutirdeba TiTo

bloki

(romelic

SeiZleba

iyos

ramdenime

baitis sigrZis, an warmoadgendes sxva standartuli formatis mTel jerad vels) yoveli arxidan. telekomunikaciis cifrul sistemebSi cifruli multipleqsori aformirebs n Sesasvleli cifruli

mimdevrobisagan

erT

gamosasvlels,

Sedge-

nils n erTsaxela blokebis (biti, baiti, ramdenime baiti) Semcveli ganmeorebadi jgufebisagan, romlebic formirdeba `taim-slotis~ ganmavlobaSi. multipleqsorma am dros Teoriulad unda uzrunvelyos monacemTa gadacemis nxv rigis siCqare (igulisxmeba, rom igi yvela arxisaTvis erTnairia). Tu Sesavali signalis saxiT gamoiyeneba ZiriTadi

cifruli

siCqariT

arxis kbiti/wm,

DSO-is maSin

signali n:1

gadacemis

tipis

erTi

multipleqsoris saSualebiT SeiZleba Teoriulad vaformiroT nX64 kbiti/wm siCqaris nakadebi. Tu am multipleqsors CavTvliT pirvel rgolad m:1, L:1, k:1... tipis meore, mesame da a.S. doneebis ramdenime multipleqsoris kaskaduri SeerTebis sqemaSi, maSin SeiZleba gavaformoT gadacemis cifrul siCqareTa sxvadasxva ierarqiuli nakadebi.

156

5.8. cifruli signalebis saxazo kodirebis ZiriTadi principebi tsk-t-s-G.701

rekomendaciis

Tanaxmad

saxazo

kodi (saxazo signali) aris kodi, SerCeuli arxis maxasiaTeblebTan Sesabamisobis pirobidan gamomdinare, romelic gansazRvravs Sesabamisobas gadacemisaTvis gankuTvnili simboloebis erTobliobisa da signalis elementebis Sesabamis mimdevrobas Soris, romelic gadacema am arxiT. multipleqsirebis Sedegad miRebuli cifruli mimdevrobisaTvis damaxasiaTebelia, rom gadasacemi Setyobinebebis yovel simbolos `1~-s Seesabameba T xangrZlivobis xangrZlivobis

impulsi, pauza

(nax.

xolo

`0~-s

5.16.).

aseTi

igive

principiT

agebul kods ewodeba `kodi nulisken dabrunebis gareSe~ (inglisurad `Non Return to Zero~. am sityvebis sawyisi asoebis mixedviT mas mieca dasaxeleba NRZ).

nax. 5.16.NRZ kodi

157

aRniSnuli kodi martivia, energetikulad SedarebiT maRalefeqturi, magram mas gaaCnia cnobili naklovanebebi. amis gamo telekomunikaciis Tanamedrove sistemebSi gamoiyeneba aseve sxva efeqturi saxazo kodebi, kerZod, AMI (Alternate Mark Inovarsion-kodi impulsebis monacvleobiT, warmoadgens orpolarul samdonian kods),

HDB2

orpolaruli warmoadgens

(High-Density kodi

Bipolar

meore

orpolarul

code

rigis

order

simWidrovis,

samdonian

kods),

HDB3

(High-High-Density Bipolar code of order orpolaruli kodi mesame rigis simWidrovis, warmoadgens orpolarul samdonian kods), bipolaruli kodebi sami, eqvsi, rva nolisagan Semdgari blokebis specialuri koduri B3ZS, B6ZS, B8ZS (Bipolar

kombinaciebis CanacvlebiT:

with 3,6,8 Zero Sybstitution) da sxva. es kodebi kargadaa cnobili

maTi

aRwera

mocemulia

mraval

literaturaSi, maT Soris qarTul enazec, amitom isini qvemoT dawvri-lebiT ar ganixileba. ukanasknel periodSi saxazo traqtSi salaparako

signalebis

sxvadasxva bamisad

gadacemisaTvis

Tanamedrove saxazo

magaliTad,

SemuSavebulia

teqnologiebi

kodirebis

HDSL teqnologiaSi

axali

Sesa-

meTodebic.

(Hugh-bit-rate Digital

Subscriber loop â&#x20AC;&#x201C; maRalsiCqariani cifruli saabonento xazi) upiratesad gamoiyeneba saxazo kodirebis ori meTodi â&#x20AC;&#x201C; 2B1Q (2 Binary-orobiTi, 1Quartenary-oTxeuli)

158

aq da qvemoT pirveli cifri gviCvenebs simboloebis ricxvs kodirebis Sesasvlel orobiT jgufSi, aso B – sawyisi informaciis warmodgenisaTvis gamoyenebul aTvlis

sistemas,

ricxvs

kodis

Semdegi

jgufSi,

cifri

simboloebis

Q – oTxeuli)

CAP

(Carrierless Amplitude and phase Modulation – amplitudur – fazuri modulacia gadamtanis gadacemis gareSe). 2B1Q kodi warmoadgens modulirebul signals oTxi doniT, e.i. drois yovel momentSi gadaicema 2 biti informacia (4 koduri mdgomareoba). saxazo signalis

speqtri

simetriuli

sakmaod

maRa-

lsixSirulia. speqtri Seicavs agreTve dabalsixSirul da mudmiv Semdgenebs. ganvixiloT sxvadasxva faqtorebis gavlena 2B1Q kodis gadacemaze. qalaqis

pirobebSi

iqmneba

didi

raodenobis

dabalsixSiruli gavlenebi (metro, tramvai, eleqtroSeduReba, impulsuri xelSeSlebi da sxva). 2B1Q teqnologiaSi gamoyenebuli integraluri sqemebis kompleqtebi uzrunvelyofen sakmaod efeqtur koreqcias xelSeSlebisagan dasacavad da uzrunvelyofen gadacemis damakmayofilebel xarisxs. miuxedavad aRniSnulisa, 2B1Q kodireba mainc mgrZnobiarea xelSeSlebisadmi, radgan signals gaaCnia mudmivi Semdgeni. 2B1Q signalis speqtrSi sixSireTa didi gafantva iwvevs jgufuri drois SeyovnebasTan dakavSirebuli

problemebis

gadawyvetis

159

aucileblobas,

rac signalis damuSavebis algoriTms mniSvnelovnad arTulebs. 2B1Q kodis Semdgenebs.

speqtri

energiis

Seicavs

maqsimumi

maRalsixSirul

gadaicema

pirvel

`foTolSi~, romlis sigane xazSi siCqaris proporciulia. signalis mileva kabelSi izrdeba gadacemis manZilis

zrdiT,

amitom

moTxovnili

gadacemis

manZilis Sesabamisad gamoiyeneba saxazo signalis samidan erT-erTi siCqare: (784 kbiti/wm, 1168 kbiti/wm an 2320 kbiti/wm). 2B1Q teqnologia 2 mbiti/wm nakadis gadasacemad iTvalisiwnebs spilenZis kabelis erTi, ori an sami wyvilis gamoyenebas. TiToeuli wyvili gadascems nakadis nawils (nax. 5.17) zemoT moyvanili siCqareebiT. udidesi manZili miiRweva sami wyvilis gamoyenebiT 4 km 0,4 mm ZarRviT, umciresi erTi wyvilis

gamoyenebiT

gavrcelebulia 2B1Q

km-mde)

yvelaze

metad

kodirebis sistema, romelic

muSaobs ori wyviliT (3 km-de 0,4 mm ZarRviT).

nax. 5.17. 2B1Q teqnologia

160

gadacemaze

did

gavlenas

axdens

radio-

sixSiruli interferencia. radiogadamcemebi grZel da saSualo talRebis diapazonSi, mZlavri radiosareleo xazebis muSaoba iwvevs gavlenas sakabelo xazebze da xels uSlian 2B1Q kodis gadacemas im SemTxvevaSi, Tu gaaCniaT speqtrebis Tanxvedrili ubnebi. es faqtori gansakuTrebiT negatiurad moqmedebs HDSL aparaturis gamoyenebisas studiebisa da radiogadamcemi

centrebis

urTierT

dasakavSireb-

lad. 2B1Q teqnologiis gamoyeneba efeqturia mcire sigrZis (3 km-de) abonentis xazebisaTvis (aSS da dasavleT evropa). CAP teqnologiaSi gamoyenebulia modulaciis Tanamedrove

meTodebi

mikroeleqtronika.

CAP

signalis modulaciuri diagrama gvagonebs satelefono arxebis modemebis signalis diagramas. gadamtani sixSire modulirdeba amplitudiT an faziT da qmnis

kodur

amasTan,

xazSi

gadascems energias, aRdgeba

sivrces

gadacemis

win

informacias, `moiWreba~ mimRebis

mdgomareobiT.

gamtani,

magram

udides

xolo

Semdeg

mikroprocesoriT.

diagramis

nalis

yovel

Sesabamisad,

momentSi

romelic

Seicavs

signalidan,

modulaciuri drois

128

64-poziciuri CAP-64 sig-

gadascems

bit

informacias, e.i. 16-jer mets, vidre 2B1Q. CAP-128 modulacias

gaaCnia

128-poziciuri

modulaciuri

diagrama da Sesabamisad erT taqtSi gadaicema 7

161

biti. saxazo signalis informaciulobis amaRlebis Sedegs warmoadggens signalis sixSiris speqtris siganis

mniSvnelovani

vicilebT

speqtris

Semcireba,

riTac

diapazonebs,

Tavidan

romlebzec

warmoiqmneba didi xelSeSlebi da damaxinjebebi. 5.18. nax-ze wsarmodgenilia CAP signalis speqtri.

nax. 5.18. CAP teqnologia

nax. 5.19 HDB3, 2B1Q da CAP signalTa speqtrebi

162

Sedarebis

mizniT

HDB3 (ikm 30), 2B1Q

5.19

nax-ze

da CAP

warmodgenilia

signalTa speqtrebi,

romlidanac Cans CAP modulaciis upiratesobebi: 1. HDSL aparaturis muSaobis maqsimaluri manZili. mileva kabelSi signalis sixSiris proporciulia.

amitom

CAP signali,

romlis

SemdgenTa

speqtri 260 khc-s ar aRemateba, vrceldeba ufro did manZilze vidre 2B1Q an HDBB kodis SemTxvevaSi. 2. maRali xelSeSlebisadmi mdgradoba da aramgrZnobiaroba jgufuri drois Seyovnebis mimarT. vinaidan CAP teqnologia ar Seicavs maRal sixSirul (260 khc-ze zeviT) da dabalsixSirul (40 khcze qveviT) Semdgenlebs, misi mgrZnobiereba xelSeSlebis gavlenisgan dabalia. CAP teqnologiis speqtris mcire siganis (200 khc) gamo ar igrZnoba agreTve jgufuri dros Seyovnebis gavlena. 3. minimaluri gavlena mezobel wyvilebze. CAP ar iwvevs urTierTgavlenas da xelSeSlebs analogur satelefono signalis speqtrSi, radgan 4 khc-ze qveviT mas ar gaaCnia Semdgenebi. 4. SeTavsebadoba mezobel wyvilebze momuSave SemWidroebis aparaturasTan. saabonento da maerTebeli xazebis analoguri SemWidroebis aparaturasTan. saabonento da maerTebeli xazebis analoguri SemWidrovebis aparaturis umravlesoba muSaobs 1 mhc-mde sixSireTa speqtrSi. CAP modulaciis sistemebs sixSirul arxebze gavlena SeuZliaT moax-

163

dinon mxolod 4-260 khc diapazonSi. aqedan SeiZleba davaskvnaT, rom HDSL aparaturas CAP modulaciiT SesaZlebelia imuSaos. rac Seexeba 2B1Q teqnologias, mas yovel sixSirul arxze SeuZlia gavlena iqonios da, rogorc wesi, ar gamoiyeneba analogur sistemasTan erT kabelSi samuSaod.

164

sakontrolo kiTxvebi 1. ra Taviseburebebi aqvs impulsur-koduri modulaciis meTods? 2. ra Taviseburebebi aqvs

informaciis cifruli

gadacemis boWkovan-optikuri sistemebs? 3. ra Taviseburebebi aqvs informaciis cifruli gadacemis sistemebs ATM teqnologiebis gamoyenebiT? 4. ra Taviseburebebi aqvs informaciis gadacemis cifrul sistemebs analoguri saabonento xazebisaTvis? 5. ra Taviseburebebi aqvs informaciis gadacemis cifrul radiosistemebs? 6. ra Taviseburebebi aqvs impulsur-koduri modulaciis (ikm) meTods? 7. ra Taviseburebebi aqvs diferencialuri impulsur-kodur modulaciis (dikm) meTods? 8. ra Taviseburebebi aqvs adapturi diferencialuri impulsur-koduri modulaciis (adikm)

meTods?

9. raSi mdgomareobs cifruli signalebis multipleqsirebis ZiriTadi principebi? 10 raSi mdgomareobs cifruli signalebis saxazo kodirebis Taviseburebani?

165

literatura 1. n. xaratiSvili. signadlebis gadacemis Teoria. Tb., ganaTleba, 1984, 208 gv. 2. Зюко А.Г., Кловский Р.Д., Назаров М.В., Ринк Л.М. – Теория передачи сигналов. М.:Связь, 1980, 376с. 3. Назаров М.В., Кувшинов Б.И., Попов О.В. Теория передачи сигналов. М.:Связь, 1970, 368 с. 4. n. adeiSvili, a. robitaSvili, v. abulaZe, g. murjikneli, T. vekua. mokle cnobari telekomunikaciis

Tanamedrove

teqnologiebSi.

Tb.,

`cotne~,

2005, 120gv. 5. g. murjikneli, a. robitaSvili, T. vekua, j. xunwaria, p. boCikaSvili, v. abulaZe. telekomunikaciis Tanamedrove cifruli teqnologiebi. Tb., i/m `momavlidan~ 2006, 308gv.

166

ibeWdeba avtorTa mier warmodgenili saxiT

gadaeca

warmoebas

03.07.2009.

xelmowerilia

dasabeWdad

08.07.2009. qaRaldis zoma 60X84 1/16. pirobiTi nabeWdi Tabaxi 10. tiraJi 100 egz.

sagamomcemlo saxli `teqnikuri universiteti~, Tbilisi, kostavas 77

gamomcemloba `momavlidan~, q. Tbilisi, 26 maisis moedani #1