Digital Communications. Chapter 5 : Optimum Receivers for the Additive White Gaussian Noise Channel. Additive white Gaussian noise assumption

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1 igial Counicaion Chape 5 : Opiu Receive o he Aiive Whie Gauian oie Channel 5- Oveview Aiive whie Gauian noie aupion n whee n i a Gauian poce. l l l Queion: How o eign a eceive o copenae uch in o noie? 5. Siple analyi o eoyle oulae ignal an oulae ignal wih eoy 5. Pobabiliy o eo eceive ie l 5-

2 5. Opial eceive o ignal coupe by aiive whie Gauian noie Sye view anie ignal l Channel Receive ignal l l n l n l AWG : Aiive Whie Gauian oie Φ n W/Hz Opial eceive o ignal coupe by aiive whie Gauian noie Aupion Fo ipliciy aue ha he baeban ignal i ealvalue. n l l einiion o opialiy o eiae in oe o iniize he eo pobabiliy l 5-

3 5. Opial eceive o ignal coupe by aiive whie Gauian noie Receive Receive ignal l Signal eoulao eeco Oupu eciion oel o analyi Signal eoulao Vecoizaion l [... ] eeco : iniize he pobabiliy o eo in he above uncional bloc eiao [... ] ˆ Opial eceive o ignal coupe by aiive whie Gauian noie Realizaion o Signal eoulao Signal coelao ache ile 5-6

4 5.. Coelaion eoulao Coelaion eoulao { } ohonoal n n Receive ignal l o eeco Saple a Coelaion eoulao Vecoizaion o he eceive ignal Given he bai { } l whee l n n ' n n an n l n l. { } pan he channel ybol pace. oe ha { } ay no pan he noie pace an hence hee ay exi an exa e n l. hi e howeve i ielevan o he eciion eo o he anie ignal. Alenaive explanaion i ha n l i ielevan o eciion eo ince eciion i ae only bae on { }. 5-8

5 5.. Coelaion eoulao Given he bai whee { } ' l n nl n n n. Receive ignal l n n ' n l o eeco n Saple a Coelaion eoulao Saiical popey o noie {n n n }. hi ep explain why we equie Suppoe n l i a zeo-ean whie Gauian poce. ean : E[ n ] auocoelaion : E[ n n { } being ohonoal. n n E[ n ]. l ] E [ n n ] l δ δ. l Hence {n n n } ae zeo-ean uncoelae Gauian ano veco. 5-

6 5.. Coelaion eoulao oe I { n } ae no ohonoal {n n n } becoe zeo-ean epenen Gauian noie Coelaion eoulao Saiical popey o eceive ignal { }. { } uncoelae Gauian veco wih ean { }. n [ [ [ i ] [... ] uncoelae Gauian wih ean veco ] [... ] uncoelae Gauian wih ean veco ] [... ] uncoelae Gauian wih ean veco 5-

7 5.. Coelaion eoulao axiu-lielihoo eciion ae bae on Coelaion eoulao AP eciion ag ax P I equal pio hen j { [... ]} { } P j ag ax j P P j / o evey { } j [... ] P{ [... ] }. an AP eciion ae becoe L eciion ae. j L eciion ag ax P{ [... ] j}. j j Coelaion eoulao Exaple 5.-. Baeban PA ignal wih ecangula pule hape. channel ybol: l A g. whee A jπ [ A g e ] c A g co Re π an.... g c bai. ohewie a l l nl whee n i zeo - ean Gauian wih vaiance an l l an. l l / Coelaion eoulao becoe a pue inegao. 5-

8 5.. Coelaion eoulao l l P l l exp. π A g aa. g a o <. l L eciion ag ax P j ag in j l {[... ] } l j. j Coelaion eoulao In he ollowing lie we will aue he lowpa ignal ae alway eploye excep ohewie ae. Hence he lenghy ubcip l will be oie a he ex i. o ipliy he analyi we will alo aue he lowpa equivalen noie n l now enoe by n ollowing he peviou ea i a whie Gauian noie inea o a oe pacical banliie whie Gauian noie. 5-6

9 ache-ile eoulao Linea ile eoulaion eplacing he coelao ache-ile eoulao ieence beween coelao an ile. i.e. hen apling a pope ie : ile : coelao h h Coelaion-ype eoulaion i a pecial cae o ile-ype eoulaion!!. apling a : hen ile. Le h h. o <

10 5.. ache-ile eoulao ieence beween coelao an ile coelao : ile : h.. hen apling a pope ie i.e Ye coelaion eoulao equie ha { } be paiwie ohogonal ache-ile eoulao A coelaion-ype eoulao i a ile-ype eoulao wih each ile eine by h. A ile-ype eoulao i a coelaion-ype eoulao eine ove he bai {h }. oe ha {h } ay no o a bai. A ile-ype eoulao wih ile h. i calle he ache-ile eoulao o he ignal. 5-

11 5-5.. ache-ile eoulao Exaple o ache ile 5- < < < < ohewie A A A A A ohewie h y ] [ ] [ ache-ile eoulao 3 A / In he above exaple he auocoelaion uncion o ile oupu pea a. A A A A

12 ache-ile eoulao Popeie o ache ile ache ile axiize SR oupu ignal-o-noie aio une AWG. Poo. [ ] n n n h h h h n n E h y E y SR n h n y h y y E y SR / ] [ ii. oe ha. an whee ] [ eine i 5-. whee he uppe boun i achieve by eing / Hence iv. wih equaliy hol i inequaliy By Cauchy - Schwaz iii C h h h SR C h h h 5.. ache-ile eoulao

13 ache-ile eoulao Fequency-oain inepeaion o he ache ile Φ Φ Φ S S S e Y y SR S H H e S e S S H S Y e S e S e e e H C h n j n n j j j j j j j * i eal. i π π π π π π π π Paeval elaion ache-ile eoulao Exaple. Biohogonal ignal wih bai he ache ile o he above ignal ae:

14 5.. ache-ile eoulao Channel ybol A A A 3 A / / / / [ A [ A [ A [ A / ] / ] / ] / ] SR une AWG SR / A ache-ile eoulao oie-ee oupu o epecively ue o h an h. Sapling oupu wih noie. [ A [ n [ A [ n / / n n n n A / A ] i i anie; n] i i anie; ] i 3 i anie; n ] i i anie. / 5-8

15 5..3 he opiu eeco P{ } i uually calle he lielihoo uncion. he opiu eeco o eoyle oulae ignal Opial iniizaion o pobabiliy o eo. AP axiu a poeio pobabiliy cieion ag ax P AP axiu-lielihoo cieion ag ax L ag ax { } P{ } P P P { } axiu-lielihoo cieion AP cieion i equal pio he opiu eeco AP i opial in pobabiliy o eo. Poo. Le R { : AP ag ax P i }. { R } ae pai - wie ijoin wih unio ie beae. Pcoec Pcoec. R R P P which oely belong o i P Pcoec P P he poo i coplee by obeving ha any e - aignen o R o ohe e R ' ' will no inceae 5-3

16 he opiu eeco axiu-lielihoo cieion une AWG uncoelae Gauian wih ean veco ]... [ ]... [ uncoelae Gauian wih ean veco ]... [ ]... [ uncoelae Gauian wih ean veco ]... [ ]... [ P / exp π 5-3 { } { } ag in ag in log ag ax log P ag ax P ag ax Eucliean L π 5..3 he opiu eeco

17 5..3 he opiu eeco Exaple o ignal pace iaga o L eciion ae o one in o ignal aignen he opiu eeco Eoneou eciion egion o he 5h ignal

18 5..3 he opiu eeco Exaple o alenaive ignal pace aignen he opiu eeco Eoneou eciion egion o he 5h ignal

19 5..3 he opiu eeco Obevaion hee ae wo aco ha eeine he eo pobabiliy.. he Eucliean iance aong ignal veco. Geneally peaing he lage he Eucliean iance aong ignal veco he alle he eo pobabiliy.. he poiion o he ignal veco. he oe wo exepliie ignal pace iaga have he ae pai-wie Eucliean iance aong ignal veco. iniizing he Eucliean iance axiulielihoo L cieion only i he noie i AWG. I he noie i no AWG coul we eine a pope iance uncion o ha he iniizaion o he iance L cieion? he opiu eeco iniu iance cieion eine a eic eic ag in eic. iniu iance cieion axiu lielihoo cieion une AWG i Eucliean eic i coniee. ag in L v Alo ag in < > ag ax v < > 5-38

20 5..3 he opiu eeco Realizaion o he opiu AWG eceive v ag ax < > he e Pojecion o eceive ignal ono each channel ybol. he n e Copenaion o channel ybol wih unequal powe uch a PA. v ag ax < > E ag ax he opiu eeco Realizaion o Opial eeco oe ha he on-en pa i no neceaily a coelaion-ype eoulaion ince channel ybol ay no o a bai. he eic i nae he coelaion eic. 5-

21 5..3 he opiu eeco einology: Coelaion eic eic iance uncion I i nae he coelaion eic becaue i i a eaue o he coelaion beween he eceive veco an he -h ignal. v v C < >. Une AWG axiu iance cieion wih coelaion eic L cieion. einology: P eic v v v v P p p p p. axiu iance cieion wih P eic AP cieion he opiu eeco Exaple. Binay PA wih E. P P p. Poble: eeine he opiu AP eeco une AWG wih wo-ie powe pecu eniy /. Soluion. ± vaiance E n whee n i zeo - ean Gauian iibue wih /. i Gauian iibue wih ean ± E an vaiance /. 5-

22 he opiu eeco { } { } { } { } ohewie p p ohewie e p pe e p pe P P P P P P AP ln i ag ax ag ax ag ax ag ax ag ax / / / / E E E E E 5- Obevaion. he hehol epen no only on he pio bu alo on he noie powe. Wih equal pio he noie powe becoe ielevan he opiu eeco ohewie p p AP ln i E

23 5.. he axiu-lielihoo equence eeco Opial eeco o ignal wih eoy no channel-wih-eoy o noie-wih-eoy i.e ill he noie i AWG I i iplicily aue ha he oe o he ignal eoy i nown. axiu-lielihoo equence eeco axiu a poeioi pobabiliy bae on a equence o eceive ignal veco he axiu-lielihoo equence eeco axiu-lielihoo equence eeco Exaple uy : RZI A A Fo he peviou icuion ± A n whee n i zeo - ean Guaian iibue wih vaiance / an i he inex o ie. 5-6

24 5.. he axiu-lielihoo equence eeco PF o a equence o eoulaion oupu P exp / π have eoy o i i avanageou o eec he oiginal ignal bae on a equence o oupu. I L ule i eploye he eulan eeco i calle he axiu-lielihoo equence eeco he axiu-lielihoo equence eeco L equence eeco o RZI ignal L... ag ax ag ax ag in We heeoe nee o each o all poible cobinaion o { A A}... { A A}... { A A} P... π / which coni o... exp Eucliean iance poibiliie. 5-8

25 5.. he axiu-lielihoo equence eeco L equence eeco o uli-ienional ignal wih eoy L... ag ax ag ax ag in P... π... exp whee S. We heeoe nee o each o all poible cobinaion o... which coni o poibiliie.... S... S... S j / j j j j j Eucliean iance he coplexiy o eaching he opial oluion becoe a buen he axiu-lielihoo equence eeco Viebi eoulaion Algoih A equenial elli each algoih ha peo L equence eecion anoing a each ove veco poin ino a equenial each ove a veco elli equenial bea he veco ino coponen an peo he each bae on each coponen in equence o he veco Alo i i a ecoing algoih o convoluional coe. A A 5-5

26 5.. he axiu-lielihoo equence eeco he nube o equence in he elli each ay be euce by uing he Viebi algoih. L... ag in i A A i... { } i i I oaion : I { AA}i he channel ybol which ha eoy. i I I {} i he igial inpu which oe no have eoy he axiu-lielihoo equence eeco he ignal eoy oe o RZI ignal i L. he cuen channel ybol only epen on he peviou channel ybol. Aue he iniial ae i S. hen he elli will each i egula o ae he ecepion o he i wo ignal. 5-5

27 5.. he axiu-lielihoo equence eeco Explaining he Viebi Algoih o S a. hee ae wo pah eneing each noe a. pah I I o noe S a enoe by S. S S / / / / / pah I I o noe S a enoe by S. S S / / / 5-53 / 5.. he axiu-lielihoo equence eeco Eucliean iance o each pah Eucliean iance o pah eneing noe S A A Eucliean iance o pah eneing noe S A A Viebi algoih. ica aong he above wo pah he one wih lage Eucliean iance. he eaining pah i calle uvivo a. ow you houl ene a lea oughly he ey o he Viebi algoih. S / / / / S / 5-5

28 5.. he axiu-lielihoo equence eeco Eucliean iance o each pah A A A A Viebi algoih. ica aong he above wo pah he one wih lage Eucliean iance. he eaining pah i calle uvivo a. We heeoe have wo uvivo pah ae obeving. S / / / S / he axiu-lielihoo equence eeco Suppoe he wo uvivo pah ae an S / / / S hen hee ae wo poible pah eneing S a 3 i.e. an. S / / / / / S

29 5.. he axiu-lielihoo equence eeco Eucliean iance o each pah A A 3 3 Viebi algoih. ica aong he above wo pah he one wih lage Eucliean iance. he eaining pah i calle uvivo a 3. / / / S / / S he axiu-lielihoo equence eeco Eucliean iance o each pah A A 3 3 Viebi algoih. ica aong he above wo pah he one wih lage Eucliean iance. he eaining pah i calle uvivo a 3. S / / / / S /

30 5.. he axiu-lielihoo equence eeco Viebi algoih Copue wo eic o he wo ignal pah eneing a noe a each age o he elli each Reove he one wih lage Eucliean iance he uvivo pah o each noe i hen exene o he nex ae. he eliinaion o one o he wo pah i one wihou copoiing he opialiy o he elli each becaue any exenion o he pah wih he lage iance will alway have a lage eic han he uvivo ha i exene along he ae pah he axiu-lielihoo equence eeco he nube o pah eache euce by a aco o wo a each age c. he nex lie. he Viebi algoih oe no euce he copuaional coplexiy ill eic copuaion ae equie. L... ag in i A A i... { } i Wha he Viebi algoih iniize i he nube o elli pah eache in peoing L equence eecion. 5-6

31 5.. he axiu-lielihoo equence eeco S S / / / / / / / / / / / / / / uvivo pah an / / / / / / / / / / / / / / hee oe Pah ae eove he axiu-lielihoo equence eeco Apply he Viebi algoih o elay oulaion eneing pah o each noe L eoy oe uvivo pah a each age 5-6

32 5.. he axiu-lielihoo equence eeco In he peviou icuion we only icue how o eove he pah? bu i no ouch he iue o how o ae eciion?. einiion o eciion elay o Viebi ecoing elay ean ha he anie bi coeponing o channel ybol a ie inan i houl be eiae ae he ecepion o i.... i he opial eciion i ˆ L L... ag in... { A A} i i i Iˆ L I Viebi pah eove ae elay ˆ L eiae o i i he axiu-lielihoo equence eeco Une he peie ha he Viebi algoih yiel he L eciion wha i he axiu eciion elay poibly encounee?... i Viebi pah eove ae elay L eiao ˆ i he opial eciion i ˆ L L... ag in... { A A} i i i Iˆ L I 5-6

33 5.. he axiu-lielihoo equence eeco Le boow an exaple o Exaple.3- o igial an Analog Counicaion by J.. Gibon. A coe wih L Aue he eceive coewo i i Viebi pah eove L eiao ˆ ae elay ˆ? i L eiao iniu Haing iance aociae wih each pah A ie inan one ill oe no now wha he i wo anie bi ae. hee ae wo poibiliie o ie peio ; hence eciion elay >.I eciion wee ae now he eciion elay. 5-66

34 iniu Haing iance aociae wih each pah 3 Hence we ge 3 an copue he accuulae eic o each pah i Viebi pah eove ae elay L eiao ˆ i 3 L eiao ˆ? 3 A ie inan 3 one ill oe no now wha he i wo anie bi ae. Sill hee ae wo poibiliie o ie peio ; hence eciion elay >.I eciion wee ae now he eciion elay. 5-68

35 3 Hence we ge an copue he accuulae eic o each pah i Viebi pah eove ae elay L eiao ˆ i 3 L eiao ˆ 3? 3 A ie inan one ill oe no now wha he i wo anie bi ae. hee ae wo poibiliie o ie peio ; hence eciion elay > 3.I eciion wee ae now he eciion elay

36 3 5 Hence we ge 5 an copue he accuulae eic o each pah i Viebi pah eove ae elay L eiao ˆ i 5 3 L eiao ˆ? ie inan : Sae Coewo Receive : 3 A ie inan 5 one ill oe no now wha he i wo anie bi ae. hee ae wo poibiliie o ie peio ; hence eciion elay >.I eciion wee ae now he eciion elay. 5-7

37 ie inan : Sae Coewo Receive : Hence we ge 6 an copue he accuulae eic o each pah i Viebi pah eove ae elay L eiao ˆ i L eiao ˆ 5? ie inan : Sae Coewo Receive : A ie inan 6 one ill oe no now wha he i wo anie bi ae. hee ae wo poibiliie o ie peio ; hence eciion elay > 5.I eciion wee ae now he eciion elay

38 ie inan : Sae Coewo Receive : Hence we ge 7 an copue he accuulae eic o each pah i Viebi pah eove L eiao ˆ ae elay 6 L eiao ˆ? i A ie inan 7 one ill oe no now wha he i wo anie bi ae. hee ae wo poibiliie o ie peio ; hence eciion elay > 6.I eciion wee ae now he eciion elay

39 Hence we ge 8 an copue he accuulae eic o each pah i Viebi pah eove ae elay L eiao ˆ i L eiao ˆ 7! A ie inan 8 one inally now wha he i wo anie bi ae which i. Hence he eciion elay

40 5.. he axiu-lielihoo equence eeco Opial Viebi L-eciion ae : o wai unil hee i only one poibiliy o he anie ybol. he eciion elay coul be a lage a he anie coewo lengh. Subopial Viebi L-eciion ae : Se a lii o he eciion elay. he peviou i 5L anie ybol P o all uvivo pah ae ienical.... i Viebi pah eove ae 5L elay ˆ i 5 L L eiae o i5l he axiu-lielihoo equence eeco Subopial Viebi algoih I hee ae oe han one uvivo pah which eul in oe han one poible ecoing eul eain o ie peio i5l ju elec he one wih alle eic an oceully eove he ohe. Fo exaple RZI coe wih L. Suppoe ha wo uvivo pah ae b b b3 b b5 b6 o ae S an b b b3 b b5 b6 o ae S. I b b elec one o b b whoe eic beween b b b3 b b5 b6 an b b b3 b b5 b6 i alle ay he oe. hen he wo new uvivo pah becoe b b b3 b b5 b6 o ae S an b b b3 b b5 b6 o ae S. Since he conibuion o accuulae eic beoe bi i he ae o boh pah we can op i wihou changing he ollowing eciion eul. I.e. he nex eciion will bae on b b b b b b o ae S an b b b b b b o ae S

41 5.. he axiu-lielihoo equence eeco he ubopial Viebi algoih ay ill yiel a ba eciion bu wih a vey all pobabiliy; A a a he eciion elay i concene wih pacical conain how abou o iniize he pobabiliy o eciion eo bae on a ixe elay. Appaenly hi will eul in a alle eo pobabiliy han he ubopial Viebi algoih. he exaple in he peviou lie i acually a ybolby-ybol eeco wih ybol lengh 6 inoaion bi A ybol-by-ybol eeco o ignal wih eoy Aben an Fichan 97 algoih Opial in he ene o iniizing he ybol eo o a given elay ˆ ag ax { 3... } P... whee i he eciion elay wih L. ˆ ag ax ag ax ag ax { 3... } { 3... } { 3... } P P P P P P uncion o... L I i he channel ybol which ha eoy whee I {... }i he bloc igial inpu which o no have eoy n uncion o... L I n 5-8

42 A ybol-by-ybol eeco o ignal wih eoy Eiaion oula }... }... }... }... }... }... } ax ag ax ag... ax ag ˆ q P P P P { { { { { { { L L whee P P P q A ybol-by-ybol eeco o ignal wih eoy }... { }... { }... { }... { }... { }... { }... { ax ag ax ag... ax ag ˆ q P P P P L L