Optimum Demodulation. Lecture Notes 9: Intersymbol Interference

Transcription

1 d d Lcur os 9: Irsybol Irrc I his lcur w xai opiu dodulaio wh h rasid sigal is ilrd by h chal ad hr is addiiv whi Gaussia ois. h opiu dodulaor chooss h possibl rasid vcor ha would rsul i h rcivd vcor (i h absc o ois) o b as clos as possibl (i Euclida disac) o wha was rcivd. his w show ca b ipld by a ilr achd o h rcivd sigal or a giv daa sybol ollowd by a oliar procssig via h Virbi algorih. h ilr is sapld a h daa ra. W also aalyz h prorac o such a sys. h aalysis is vry siilar o ha o covoluioal cods. Bcaus h rcivd sigal is ilrd ad sapld, h oupu o h ilr cosiss o wo copos. O du o h rasid sigal ad o du o h ois. h oupu du o ois is, howvr, o whi. Howvr, i h x scio w show ha h oupu o h achd ilr ca b whid. Wih a whid achd ilr h opiu rcivr (Virbi algorih) bcos clar. Fially, i h las scio w show how o dsig a sys o liia irsybol irrc. Opiu Dodulaio Cosidr rasiig daa a ra disorig chal. W would li o id h opiu (iiu squc rror probabiliy) rcivr. Assu h odulaor is a ilr acig o a iii squc o ipulss (a ra wih ipuls rspos. h chal is characrizd by a ipuls rspos o g ad h rcivr is a ilr sapld a ra wih ipuls rspos h. Daa Ra odulaor s h rasid sigal is o h or s u hrough a chal wih badwidh W or hrough a g Chal z u r IX- IX- whr u is h daa sybol rasid durig h -h sigalig irval assud o b i h alphab A ad is h wavor usd or rasissio. W assu a rasissio o daa sybols (hi o as big vry larg). h oupu o h chal ilr is h whr z h g g τ s τ dτ τ u g u h g τ u τ τ dτ τ dτ τ dτ h rcivd sigal cosis o wo rs. O du o sigal ad o du o ois. r su u h whr Sic whr Λ s u u h is whi Gaussia ois h opiu rcivr copus or ach daa squc v v r s v r r s v v v y s v v h r h y sv r h d d v v x d v v v v h h h d h d IX-3 IX-4

2 d s x h h d Sic his is y did arlir h rcivd sigal should b ilrd ad sapld as show blow bor doig so procssig. hus h opiu dcisio rul is Chos v i Λ v ax u Sic h dcisio saisic dpds o h rcivd sigal oly hrough y i is clar ha y is a suici saisic o ipl opial rcivr. Cosidr a ilr h r oupu would b h u Λ. h i h rcivd squc is passd hrough his ilr h y s r h r r h d s d Daa Ra s g z r hr y odulaor Chal Dodulaor y h sapld oupu would b y r h d IX-5 IX-6 odl y η r h h d u h h d η + + y ow h origial coiuous i dcio probl ca b rplacd wih a discr i probl. η y u x η x x x x x u u u u u 3 u IX-7 IX-8

3 d v y v y v y v y v y Opiu Rcivr o ha η is Gaussia. E E Var η η ηη E h Howvr, η is o a i.i.d. squc. odl d h h h d s h s d h s E s dds h x (o ha v j Λ Λ j v v v y ). Assu x x v L (ii irsybol irrc) x v v v x x v x v x v L v v v j v v j i v v j L v x jx jx j i v v j jx j j jx j IX-9 IX- Virbi Algorih h dcisio rul ca b ipld i a Virbi algorih li srucur. Di h sa a i o b h las L daa sybols. (hs ar h oly sybols ha ac h oupu a i ). L h λ σ σ v Λ v σ hus w ca apply h Virbi algorih. λ y L v v x v σ σ v v y L v x x v L Γ σ b h lgh (opiizaio criria) o h shors (opiu) pah o sa σ a i. L Ω σ b h shors pah o sa σ a i. L ˆΓ b h lgh o h pah o sa σ a i ha gos hrough sa σ a i. h algorih wors as ollows., i idx, ˆΩ ˆΩ Γ ˆΓ, σ σ σ Γ σ σ σ v y L ˆΩ σ σ Γσ axσ ˆΓ σ σ Sorag: AL Γ Iiializaio Φ L A (Φ is h py s). x v σ λ σ σ σ argaxσ ˆΓ σ or ach σ σ. ˆΩ Rcursio σ σ σ σ A L ˆΩσ σ σ IX- IX-

4 x, x y y y y x x y y Exapl: L x x, v Λ v λ σ σ y v x v Exapl λ σ σ v y x v v vx v y x x v + + Cosidr a chal wih x lgh 5 ( ). y y 8 x. Cosidr h ollowig rcivd squc o y y x x Assu v. h h rllis is show blow. x x - - x x IX-3 IX-4 rllis rllis + y x y x 4 6 x - y - IX-5 IX-6

5 6 6 y x x y x x x y 4 x rllis rllis 6 x x x x y IX-7 IX y x x y x rllis 6 3 rllis 6 y x 8 y x x 6 IX-8 IX-

6 y x y rllis 6 x x 8 rllis 6 8 x x y x x 8 x x y 8 IX- IX-3 rllis y x rllis v h doubl lis rprs h pah chos by h Virbi dcodr. hus h Virbi dcodr would oupu h squc. y x x IX- IX-4

7 , x y v v v v Exapl: L, x, x, v Λ v λ σ σ v y x v λ σ σ v x v v v x v v x IX-5 IX-6 Error Probabiliy v v v v ric y y x x x x x y x x x x y x x x y x x x y y x x x x x y x x x x ow cosidr h prorac o h abov axiu lilihood squc dcor (LSD). W will valua h uio uppr boud o h rror probabiliy. o do his w d o dri h pairwis rror probabiliy bw wo squcs. his is h probabiliy ha squc v is dodulad giv squc u is rasid or a sys wih oly wo possibl rasid squcs v). L P (u vdo h codiioal rror probabiliy giv u rasid. h P u v u P Q Λ v s v s v s u u s u 4 As cd h pairwis rror probabiliy dpds oly o h squar Euclida disac bw h sigals s u ad s v. s v s u s v s u d Λ u IX-7 IX-8

8 L ε v u s v s u v u v u v 4 4 v u v u v u u h v u ε x 4 v u v u v u ε ε x d h h x h h ε ε x d d P u v 4 4 P ε x j ε x L j ε x ε ε ε ε L jx j jx j ε x ε hus h icral Euclida disac bw wo pahs or a giv i idx is dpds o h pas L rrors. h rror sa is did o b h las L rrors ε ε h all zro rror sa corrspods o h pas L sybols big corrcly dodulad. L A rror v is did o b a rror squc ha divrgs oc ro h all zro sa ad h rrgs lar. Sic a cssary codiio or a rror o a paricular yp (irs v rror or sybol rror) is ha a rror v occurs ha causs h dodulaor/dcodr o ollow a pah ha divrgs ad h a so lar i rrgs w ca calcula h rror probabiliy or a paricular od by couig h ubr o pahs (ad hir disac) ha divrg ad rrg. W ca us h sa diagra o dri h ubr o rror squcs wih a paricular disac.. IX-9 IX-3 L ε P E P b ε wh Haig wigh o (ubr o ozro rs) Firs v rror probabiliy P a i dcodr is o a corrc sa or h irs i Bi rror probabiliy P bi rror occurs or sybol h uio boud o h probabiliy o rror a i is P E u ε v ε P u v u u v whr h su is ovr all squcs ha divrg ro h all zro sa ad h rrg lar. Each o h u squcs ar qually lily. I ach posiio whr ε copos o h squcs u ad v ar drid. I ε h v ad u. Siilarly i ε h v ad u. h copos whr ε hr ar wo choics or u ad v (u v or u v ). Sic hr ar w H placs v P u P u v P u h whr ε hr ar wh P E suchsqucsuadv. Hc ε ε h bi rror probabiliy is boudd by For L P w P w h w H ε P b P wh w w w H ε wh P P ε x L ε x ε ε x εε x W calcula hs uio bouds by uraig h squcs ha divrg ro h all zro rror sa ad rrg (rror vs) ha corrspod o a giv Euclida disac bw wo daa squcs ad has a giv ubr o ozro rs (or is a giv lgh). o do his w draw a sa diagra (siilar o ha or covoluioal cods) ad labl ach pah wih IX-3 IX-3

9 Error Sa Diagra L b + D x x D x l l whr x is h icral Euclida disac squard (dividd by 4) i goig ro o sa o aohr ad l is i h rror pah is ozro ad is zro i h rror is zro. (his rduda us o l will b laid wh w dri h bi rror probabiliy). D x D x Dx x x a d D x c - D x x IX-33 IX-34 rasr Fucio h rasr ucio is calculad by solvig h ollowig quaios or d d c b b Dx xb c Dx Dx xc Dx x c Dxa x b Dxa Addig h las wo quaios ad solvig or b cad subsiuig h rsul io h irs quaio yilds d a D D x D x x D x x Dx x Dx x Dx a. pahs wih wo rrors ad Euclida disac squard o 4x 4x ad so o. P E P b D D x D D x x x x Dx x x D x x x Dx x D For larg SR his is h sa as o ISI! Jus as wih covoluioal cods his Uio-Bhaacharyya boud ca b iprovd by usig h xac rror probabiliy or h irs w rs ad h uppr boudig h rror probabiliy or highr ordr rs wih h Bhaacharyya boud. hus hr ar wo pahs wih rror ad Euclida disac squard o 4x. hr ar wo IX-35 IX-36

10 Ergy oic ha h rgy o h sigal a h oupu o h chal is E Ex s d h uio boud ca b ighly boudd by P b J w j 5 j Q Uio Boud E b j 5 D j 5 w D D x E x E x u h u h l E x u u l h d l u l h l h l d whr E x dos caio wih rspc o h daa bi. Bcaus h daa ar assud idpd, idically disribud wih E x u u l i ladis ohrwis h rgy o h sigal is E u h d IX-37 IX-38 Uio Boud Siulaio Uio Bhaacharrya Boud h x x d P b AWG Hard Dcisio Prorac 8 h rgy pr bi is E be x E b / (db) Figur 6: Prorac o Opial Rcivr (x x ) IX-39 IX-4

11 apl Cod x:=.; x:=.; wih(lialg); a:=**dˆ(x-*x); b:=**dˆ(x+*x); c:=**dˆx; :=arix(3,3,[[-, -, ],[-a, -b, ],[-b, -a, ]]); d:=[, **Dˆx, **Dˆx]; xx:=lisolv(,d); xx:=xx[3]; xx3:=di(xx, ); xx4:=val(xx3, =); xx5:=val(xx4, =.5); xx6:=sipliy(xx5); xx7:=sris(xx6, D=, 5); P b D w D 85 Bouds (x x 75D 5D 5 5D 4 5 5D3 35D 7 5 5D D D D7 5 j w j D j D 4375D6 D ) 875D4 5D 5 5D D D D D D D5 6465D D D D D D IX-4 IX-4 Sigal Dsig or Filrd Chals Bcaus h coplxiy o h Virbi algorih grows as h A L whr is h alphab siz ad L is h ory o h chal i is dsirabl o dsig a sys wih zro irsybol irrc. So cosidr rasiig daa a ra badwidh W. A wha ra is his possibl wihou craig irsybol irrc? Assu h odulaor is a ilr acig o a iii squc o ipulss (a ra ipuls rspos. h chal is characrizd by a ipuls rspos o g ad h rcivr is a ilr sapld a ra wih ipuls rspos h. A hrough a chal wih wih Daa s() g Ra z r h odulaor Chal Dodulaor y y h rasid sigal is o h or s u IX-43 IX-44

12 τ h oupu o h rcivd ilr is h y u x η I ordr ha hr b o irsybol irrc w rquir ha L x x h τ h h τ h d τ d hus by h saplig hor whr I W h x x φ W x W x W si πw πw si πw πw φ W W W W W si x π π h H H whr h. h x is h covoluio o h ad x X H wih. I h (absolu) badwidh o h chal is W h X has badwidh W. ha is h X W x. hus also For o irsybol irrc w rquir ha x x si π π or. L x h IX-45 IX-46 Puls Shap.5 which iplis ha x() hus X i 6 x 5 Spcru H hus W pulss pr scod ca yild zro irsybol irrc. I is asy o s ha by sigalig asr ha ra W w ca o guara ha hr is o irsybol irrc. X() rqucy x 4 Figur 7: yquis Puls Shap ad Spcru IX-47 IX-48

13 .5 Daa Wavor 3.5 x() i Figur 8: yquis Wavor Figur 9: yquis Ey Diagra IX-49 IX yquis Pulss 3 Probls wih his puls shap ar: I is hard o gra A sligh iig rror rsuls i iii sris dcayig as or irsybol irrc. Soluios Sigal slowr Allow irsybol irrc i a corolld ashio Figur : yquis Ey Diagra IX-5 IX-5

14 W, W π π π Irsybol-Irrc Fr Puls Shaps Cosidr slowr sigalig irs. Cosidr. (his iplis aliasig a h rcivr. Sic W wcadivid h irval W io sgs o lgh L W bh ubr o such sgs. Wh h sigal is sapld hs sgs g ovd o h origi ad caus aliasig. x W W X jπ d X X X W jπ d jπ jπ d d. L h x X q X X q Clarly w ca hav zro irsybol irrc i Exapls () X q jπu d IX-53 IX-54 X q x si π cosαπ 4α oic ha h puls dcays as 3 isad o or h yquis puls. h parar α is calld h rollo acor. H si α α α α () h si α π 4α cos 4α π α X si π α α α α IX-55 IX-56

15 Puls Shap.5 Daa Wavor.5 x() i 8 x 5 Spcru x() 6.5 X() rqucy x i Figur : Raisd Cosi Puls Shap ad Spcru (α 5) Figur : Raisd Cosi Wavor (α 5) IX-57 IX-58 x() 3 Raisd Cosi Ey Diagra i Ral(x()) Iag(x()) i Daa Wavor Daa Wavor i Figur 3: Raisd Cosi Ey Diagra α 5 Figur 4: Raisd Cosi Wavor (α 5) IX-59 IX-6

16 Figur 5: Raisd Cosi Wavor Ey Diagra (α 5) Figur 6: Raisd Cosi Wavor (α 5) IX-6 IX-6 Ral(x()) Iag(x()) Daa Wavor i Daa Wavor i Figur 7: Raisd Cosi Wavor π 4 QPSK (α ) Figur 8: Raisd Cosi Wavor Ey Diagra π 4 QPSK (α ) IX-63 IX-64

17 .5 Daa Wavor Ral(x()) Iag(x()) i Daa Wavor i Figur 9: Raisd Cosi Wavor π 4 QPSK (α ) Figur : Raisd Cosi Cosllaio π 4 QPSK (α 5) IX-65 IX Figur : Raisd Cosi Wavor Ey Diagra π 4 QPSK (α 5) Figur : Raisd Cosi Wavor π 4 QPSK (α 5) IX-67 IX-68

18 Ral(x()) Daa Wavor i Daa Wavor 3 Iag(x()) i Figur 3: Raisd Cosi Wavor π 4 QPSK (α 35) Figur 4: Raisd Cosi Wavor Ey Diagra π 4 QPSK (α 35) IX-69 IX avoidig irsybol irrc has h sa prorac as BPSK. h dirc is ha his odulaio sch rquirs absolu badwidh o W α. Howvr, sic his dos o rsul i a cosa vlop sigal or applicaios rquirig cosa vlop rasissio his odulaio sch is o accpabl Figur 5: Raisd Cosi Wavor π 4 QPSK (α 35) Bcaus w do o hav ay irsybol irrc h prorac o his hod or IX-7 IX-7

19 d Corolld Irsybol Irrc: Parial-Rspos h scod hod is o allow so irsybol irrc. his irsybol irrc is allowd i a corolld ashio. W sill sigal a ra Wbudoorquir zro irsybol irrc. x si x π π his is calld Duobiary rasissio (also calld parial rspos class I). Exapl (): X x jπ W W cos π W W W Exapl (): X x x jπ W W W W his is calld odiid Duobiary rasissio (also calld Parial Rspos Class IV). X j W si π W W W his is usd i agic rcordig (wih axiu lilihood dcodig). For hs syss wih corolld irsybol irrc w sill d a way o dc h daa. O hod is dcisio dbac. h ohr hod is prcodig. IX-73 IX-74 Orogoal Frqucy Divisio uliplxig (OFD) Orhogoal rqucy divisio uliplxig (OFD) is a alraiv hod o liiaig irsybol irrc. b L b do h pac o codd daa. his pac is srial-o-paralll covrd io sras o daa ach o lgh L bp. h squcs ar b whr bbp b b b b b b b b b b b b L b bl L Equivally h daa sras ca b wri as a squc o L colu vcors o lgh copos. b b b h daa vcor b is augd wih zros o gra h daa colu vcor C whr C b b b b whr h ladig ad railig zros ar usd o or a guardbad aroud h sigal as will b dscribd subsquly. h vcor C has lgh d. h vcor C is h ipu o a IFF which producs a squc o i sapls o lgh d. hs i doai sapls corrspod o sapls i i wih a saplig ra o IFF wh C is h ipu. ow or purposs o iigaig irsybol irrc prpd a ubr o hs sapls. c c d ν ν c d cd c d. L c do h oupu o h h lgh o c is d νwhrνrprss h lgh o h chal ory (icludig ay ilrig i h rasir ad rcivr). Sic h sapl ra is s corrspodig o c is s ν d. h daa ra is h d ν codd sybols/sc. For d ν h daa ra is approxialy squc o sapls ar h prix isrio or a pac h is giv by d c c 3 c c hs sapls h rprs h sigal a a sapl ra o d l dl l cl ν c s d h duraio d s sybols pr scod. h. hus L IX-75 IX-76

20 4 I h rqucy doai his sigal is S h dsird rasid sigal is s l l c l sic cos π H whr H is a rqucy hoppig par giv by H hp p h Hr hp dos h ubr o hops pr pac, h dos h duraio o a dwll (hop) irval ad p or, is giv by φ l l φl h ad zro lswhr. h ioraio barig phas, pr pac is dod by sp. h dsird RF sigal is obaid by ixig a basbad sigal up o a carrir rqucy via s b s Rs b jπ c Rcosjπ c h cssary basbad rasid sigal is s b l φ l s b sp p l I s b s si jπ c jπ l jbl I ordr or h sigal o hav orhogoal copos i is cssary ha ay pair o rqucis usd ad l saisy l or so igr. W will assu l l sybol irval. I h sigal s b is sapld a spacigs s s s. cosidr h sigal rasid durig a sigl h h sapls ar φ l sp π l bl whr l is h l-h carrir ad b l sybol irval which as valus π π ps s is h daa sybol or h l h carrir durig h -h 3. h ubr o OFD sybols s b l l b l p b l p s jπ l jπ l s IX-77 IX-78 Rcivr A h rcivr h rcivd sigal is r b h opial rcivr dos a coplx corrlaio o dri h daa. sb l b l IFF b jπl Z l s b s b jπ l jπ l d d whr h IFF is h ivrs Fourir rasor. Suppos ha h saplig ra a h rcivr is δ. h Z l L L s L jπ l L L s L L L L b jπ jπl L L jπl L IX-79 IX-8

21 L bl L L b l b b jπ jπ l h calculaio o Z l ca b do via h FF. Z l L jπl L L L L L L s l FF s L jπl whr FF is h as Fourir rasor. h irs valus ro h FF ar h dsird dcisio variabls. L L Cosidr a chal wih irsybol irrc. h i doai daa sigal is c l b l jπl h cyclic prix guard irval orcs c irsybol irrc) is whr h daa via z r c l l l ν h c ν. h rcivd sigal (wih ν is h chal ipuls rspos. h rcivr procsss h rcivd r jπ ν c ν h c ν h u h c u jπ jπ jπ u IX-8 IX-8 whr ν ν ν h h h Hb H jπ jπ jπ u b l l u b ν h l jπ b l jπlu jπ l u jπu Dsig Exapl As a possibl dsig xapl cosidr a sys whr h carrirs ar spacd 5Hz apar ad hr ar d4carrirs wih a guard bad o chals o ach sid. I his cas 64. Igorig h prix isrio w obai a sigal as show i Figur 6. h sigal is upsapld by isrig zros bw h sapls o h origial daa sigal ad ilrig. I Figur 7 w show h cas whr h ovrsaplig is by a acor o 8. h rsulig daa sigal is h ilrd (usig a idal bric wall ilr) ad rsuls i h dsir spcru as s i Figur 8. h daa ra possibl ca b calculad as ollows. Firs, h daa is codd. ypically h cod ra, r,is o h ordr o /4-/. Assu QPSK is usd o ach o h d carrirs ad ach sybol has duraio scods. h daa ra is h R d r ν h badwidh ha is usd dpds o h ilrig do. I hory h badwidh could b as sall as Hz. IX-83 IX-84

22 Spcru Spcru o Ovrsapld Sigal.5 i doai sigal (ral ad iagiary) *log( S() ).5 Spcru o Sigal 3 4 x Phas o Sigal 4 x x 5 Figur 6: Oupu o IFF x 6 Figur 7: Oupu o IFF IX-85 IX-86.5 Badliid vrsio o Ovrsapld Sigal rasir Bloc Diagra x 4 b i FEC b S/P C Zro Pad IFF c Badliid vrsio o Ovrsapld Sigal *log( X() ) x 6 Figur 8: Oupu o IFF Cyclic Prix I DSP S 3C67 c Ovrsapl Filr IQ od D/A Aalog Dvics 9857 (DDS/IQ od) IX-87 IX-88

23 Ey Diagras 5 Sys Parars h DSP gras a pac o codd bis as ollows. A pac o ioraio bis o lgh 48 is usd o as h ipu o a ra /3 cod (covoluioal, urbo or LDPC) producig 644 codd bis. Dpdig o h ulipah o h chal h codword is pucurd o wr bis icrasig h civ ra o h cod. Cosidr h cas o o ulipah iiially. h h codd bis ar pariiod io blocs o lgh 64 which is usd o dri 3 coplx (rqucy doai) sapls. h rqucy doai sapls ar paddd wih zros o ihr sid (6 o ihr sid i his xapl). h rsulig 64 coplx sapls ar usd as ipu o a IFF which producs 64 coplx sapls. For ulipah chals a cyclic prix is addd which allows us o dal wih h irsybol irrc x Figur 9: Ey Diagra x Figur 3: Ey Diagra IX-9 Ey Diagra IX-89 IX-9