ASYNCHRONOUS DIRECT SEQUENCE SPREAD SPECTRUM

Transcription

1 Corso di Comuniczioni Mobii ASYCHOOUS DIECT SEQUECE SPEAD SPECTUM Prof Cro egzzoni

2 eferences Pichoz D L Schiing nd L B Misein Theory of Spred-Specrum Communicions A Tuori IEEE Trnscions on Communicions Vo COM- 3 o 5 Mggio 98 pp K Phvn AH Levesque Wireess Informion ewors Wiey: ew Yor AJ Vierbi CDMA: Principes of Spred Specrum Communicions : Addison Wesey: JG Prois Digi Communicions Terz Edizione McGrw-Hi: MB Pursey Performnce Evuion for Phse-Coded Spred-Specrum Muipe Access Communicions Pr I: Sysem Anysis IEEE Trns on Comm Vo 5 o 8 pp Agoso A Lm F Ozuur Performnce Bounds of DS/SSMA Communicions wih Compex Signure Sequences IEEE Trns on Comm vo 4 pp Oobre 99 7 D Srwe M B Pursey Correion Properies of Pseudorndom nd eed Sequences Proceedings of he IEEE Vo 68 o 5 pp Mggio 98 8 FM Ozuur S Tnrn AW Lm: Performnce of DS/SSMA Communicions wih MPSK Signing nd Compex Signure Sequences IEEE Trns on Comm Vo 43 o /3/4 Febbrio 995 pp7-33

3 Inroducion In he previous session TECICHE DI TASMISSIOE-DATI DIGITALI BASATE SUL COCETTO DI SPEAD SPECTUM Direc Sequence Spred Specrum sysem wih wo or more users using he sme bnd s usu in CDMA bu differen spreding codes hs been priy nyzed The users invoved in oher communicions re considered s inerference ced Cross Inerference whose power is reed o Process Gin By modifying nd choosing pricur spreding code heir effecs cn be reduced The previous insnces re min feures of Code Division Muipe Access which uses he srengh of Spred Specrum echniques o rnsmi over he sme bnd nd wih no empor imiion Asynchronous informion provided by sever users 3

4 Mui User DS-CDMA In Mui-user DS-CDMA ech rnsmier is idenified by is P sequence I is possibe o deec he informion rnsmied hrough receiver bsed on convenion mched fier The oher users differen by he rnsmiing one wi be considered s Mui User Inerference MUI genery non Gussin disribued The received sign fer he smping cn be considered s he conribuion of hree componens: y BPSK DE- MODULATO P cos π f θ P DE-SPEADE P Generor p s P P Z Tb η I Tb n g Firs Term is he x sign η is he AWG I is he MUI 4

5 Perfomnces AWG hp Usuy re sysems re composed by sever users so due o he cenr imi heorem he over inerference MUI cn be considered s Gussin disribued This hypohesis is refeced in BE compuion where is Gussin pproximion is considered Considering s firs cse very simpe siuion where - DS-SS users re Gussin heir power in he rnsmission bnd B is -P where P is he rnsmied power considered equ for users Is specr densiy is : I K P B The power of over noise MUI nd AWG is: B K P B K P TOT Wih previous d i is possibe o obin he Sign o oise io he receiver: S ou Eb Eb I K P B 5

6 Perfomnces AWG hp By using BPSK moduor he rnsmission bndwidh is B he BE is PE BPSK Q Sou wih Gussin hypohesis we hve: T c nd P E BPSK K Q 4 E b / x y Where Q x e dy is he Gussin Error Funcion π In singe user nd Gussin AWG scenrio he DS-CDMA hs he sme performnce of nrrow bnd BPSK moduion 6

7 BE Evuion - Gussin hp In he s wo sides pricur nd usuy wrong hypohesis hs been considered: he MUI is modeed s whie noise In re cse is specr densiy is OT f hus he Mui User Inerference cn no be considered s whie noise To crry ou deeper nysis he firs nd second order sisics of rndom vribes considered Gussin hve o be compued Being η nd I Gussin disribued he pdf of n g is Gussin wih zero men nd vrince given by: vr n g vr η vr I becuse I nd η re independen rndom vribes wih zero men η is he oupu of he receiver when n he AWG is he inpu: T η n p cos ω d whose vrince is T/4 c 7

8 Perfomnces - Gussin hp I s redy expined is he inerference genered by oher users I cn be defined s ou he receiver s: I P K Z P K T b p p d cos φ where φ is he phse dey nd is he ime dey for user The symbos hve he sme probbiiy nd he error probbiiy is : P e Pr n Pr Z b g P P T f n g x dx P Z b Pr Z n g Pr Z < Pr Z b < Pr Z b b where f n g x b is he gussin pdf of n g 8

9 Perfomnces - Gussin hp From he previous formu he error probbiiy becomes: P eg Q S ou Q P T vr n g Q P T T vr I 4 where he S for he considered user he receiver is: S ou / Ε E Ψ vr I vr η S Ψ P is he mui-user inerference I normized wih respec o is he specr densiy of AWG S Ε is he sign o noise rio in he rnsmier 9

10 MUI Vrince The vrince of I vri or he men squre vue of E Ψ hs o be compued o obin he fin formu of P e I is sufficien he men squre vue becuse E Ψ K {[ b ρ b ρ ]cos } E Ψ E φ b φ Ψ where ρ p p d nd o T ρ p p d oe: ime dey nd phse dey re uniformy disribued vribes in [T nd [p nd he rnsmied symbos hve he sme probbiiy

11 Exmpe In he figures n exmpe of synchronous rnsmission wih dey is presened b eference User b -T T b b -T T Inereference User

12 MUI Vrince The previous quniies cn be defined considering he -periodic crosscorreion beween P sequence of reference user nd P sequence of user K The inegrs of side cn be compued s: j p j p p j p j j χ { } c c T T χ χ χ ρ { } c c T T χ χ χ ρ for such s c c T T

13 3 MUI Vrince { } { } cos K E E E φ ρ ρ φ Ψ Using he previous vues he vrince of normized MUI hs been reduced o: { } cos cos π φ φ φ π φ d E where nd { } { } ρ ρ ρ ρ d T E T This inegr cn be divided in summion of inegrs in he inerv [ ] c T c T where { } { } T T d T E c c ρ ρ ρ ρ

14 4 MUI Vrince { } ρ ρ By subsiuing he inegr wih he summion of inegrs nd wih he vues obined in side he vrince becomes: [ ] Ψ K v b b f E 3 6 b χ χ where b χ χ f x y z w x y z w xy zw v $

15 MUI Vrince - Concusion The s formu ow us o concude: The higher he process gin he ower he MUI vrince This mens h by incresing he SS bndwidh he power of he Mui-User inerference wi be reduced A fundmen prmeer is he cross-correion funcion mong P sequences Wih ow correion he MUI wi be reduced nd he inerference cn hve we effecs In he foowing secion hese specs wi be nyzed in deis 5

16 Opim eceiver: Asynchronous Trnsmission We ssume h rnsmied sign is corruped by AWG in he chnne; received sign cn be so expressed s: r s n where s is rnsmied sign nd n is noise wih specr densiy Opim receiver is for definiion receiver which seec bi sequence: { n n K } b Which is he mos probbe given received sign r observed during empor period TT ie: { } b n b n P rg mx b n r T T 6

17 Opim eceiver: Asynchronous Trnsmission Two consecuive symbos from ech user inerfere wih desired sign { } eceiver nows energies of signs nd heir rnsmission deys { } Opim receiver evues he foowing ieihood funcion: Λ b K T T r E T T K r E E b i EE b i b j i j i d K K b i c i T T it c T T r c d it d it c jt d Where b represens he d sequences received from K users 7

18 Opim eceiver: Asynchronous Firs inegr: Trnsmission doesn depend on K so cn be ignored in mximizion whie he second inegr: i T r i r c it d i it T T r d represens correor o mched fier oupus for K-h user in ech sign inerv Third inegr cn be esiy decomposed in erms regrding cross-correion: T T c it T T it c it c c it jt jt d d 8

19 Opim eceiver: Asynchronous Trnsmission Indeed cn be wrien: ρ ρ ρ for for > Λb cn be expressed s correion mesure one for ech K idenifier sequences which invoves he oupus: { i K i } r of K correors or mched fiers 9

20 Opim eceiver: Asynchronous Trnsmission { } By using vecori noion cn be shown h K oupus r i of correors or mched fiers cn be expressed in form: r b n where [ ] r r r r r i [ ri r i r i] K b b [ ] b b [ E b i E b i E b i ] b i K K n [ ] n n n [ ] n i n i ni nki

21 Opim eceiver: Asynchronous Trnsmission m is KxK mrix which eemens re: d mt c c m

22 Opim eceiver: Asynchronous Trnsmission Gussin noise vecor ni is zero men nd is uocorreion mrix is: Vecor r consiues se of sisics which re sufficien for esimion of { b i } rnsmied bis E [ ] n n j j The mximum ieihood deecor hs o ccue K correion mesures o seec he K sequences of engh which correspond o he bes correion mesures The compuion od of his pproch is oo high for re ime usge

23 Opim eceiver: Aernive Approch Considering mximizion of Λb ie probem of forwrd dynmic progrmming cn be possibe by using Vierbi gorihm fer mched fiers bench Vierbi gorihm Ech rnsmied symbo is overpped wih no more hn K- symbos b i b i- b K i- b i b K i When he gorihm uses finie decision dey sufficien number of ses he performnces degrdion becomes negigibe 3

24 Opim eceiver: Aernive Approch The previous considerion poins ou h here is no singur mehod o Λb decompose Some versions of Vierbi gorihm for mui-user deecion proposed in he se of he r re chrcerized by K ses nd compuion compexiy O4 K /K which is si very high This ind of pproch is so used for very ie number of users K< When number of users is very high sub-opim receivers re considered 4

25 Sub-opim eceivers: Convenion eceiver The convenion receiver for singe user is demoduor which: Correes received sign wih user s sequence Connec mched fier oupu o deecor which impemens decision rue Convenion receiver for singe user suppose h he over noise chnne noise nd inerference is whie Gussin 5

26 Sub-opim eceivers: Convenion eceiver The convenion receiver is more vunerbe o MUI becuse is impossibe o design orhogon sequences for ech coupe of users for ny ime offse The souion cn be he use of sequences wih good correion properies o conin MUI God Ksmi The siuion is criic when oher users rnsmi signs wih more power hn considered sign ner-fr probem Prcic souions require power conro mehod by using sepre chnne moniored by users The souion cn be mui-user deecors 6

27 7 Sub-opim eceiver: De-correing Deecor n b r The correor oupu is: Lieihood funcion is: Where b r b r b K Λ

28 Sub-opim eceiver: De-correing Deecor I cn be proved h he vecor b which mximize mximum ieihood funcion is: b r This ML esimion of b is obined rnsforming mched fiers bench oupus Bu r b n see side 7 b b n So b is n unbised esimion of b The inerference is so eimined 8

29 Sub-opim eceiver: De-correing Deecor The souion is obined by serching iner rnsformion: b Where mrix A is compued o minimize he men squre error MSE Ar J b [ ] b b b b [ ] b Ar b Ar E E I cn be proved h he opim vue A o minimize Jb in synchronous cse is: A I b I r 9

30 Sub-opim eceiver: Minimum Men Squre Error Deecor The oupu of deecor is: b sgn b When is ow compred o oher digon eemens in minimum MSE souion pproxime ML souion of de-correing receiver When noise eve is high wih respec o sign eve in digon eemens in mrix A pproxime idenic mrix under sce fcor So when S is ow deecor subsniy ignore MUI becuse chnne noise is dominn Minimum MSE deecor provides bised esimion of b hen here is residu MUI 3

31 Sub-opim eceiver: Minimum Men Squre Error Deecor To obin b iner sysem is o be compued: I b r An efficien soving mehod is he squre fcorizion* of mrix: I Wih his mehod 3K muipicions re required o deec K bis Compuion od is 3K muipicions per bi nd i is independen from boc engh nd increse inery wih K * Prois ppendix D 3

32 3 Concusions: BE Evuion For n synchronous DS/CDMA sysem BE expression cn be wrien priy repored in side 4[5] s: [ ] K b b f E K v Ψ / 3 / vr vr Ψ Ε S K S E I ou S η I eds o: K E 3 Ψ If sochsic P sequences re considered: / 3 S K Q P E This formuion is wrong for few users wheres cn be used for rge number of users I is usefu for simpe evuion of DS/CDMA sysem performnces

33 Concusions: BE Evuion From P E expression cn be derived n evuion of CDMA sysem cpciy in erms of number simuneous users served wih cerin Quiy of Service QoS For high vues of x: Q x exp x π x Considering dmissibe P E -3 sufficien for voc ppicions Q 3 3 K 3 3 E b Considering he righ side of equion s upper bound: K < 3 3 E b 33

34 Concusions: BE Evuion For high vues of sign-o-noise rio n pproximion is possibe: K 3 A simpe guidnce bou DS/CDMA sysem o esime sysem cpciy is h more hn /3 synchronous users cn be served where is he process gin wih probbiiy error ower hn -3 34

35 Exmpe: numeric resus 7 3 K 6 K 6 K 4 K 4 K K Ε Ε 7 K K 6 K 4 BE Gussin evuion for DS/CDMA sysems BPSK moduion God sequences K number of users 35

36 Commens BE Gussin evuion is ony n pproximion of re BE For S < db Gussin noise is predominn nd BE is brey infuenced by new users For very high S MUI is predominn nd he higher he number of users he ower re performnces if process gin is ow Incresing S over cerin hreshod BE sures: his is he boe-nec given by MUI presence To increse performnces higher process gin is needed; his fc invoves n expnsion of rnsmission bnd he equ bi-re 36