A Maximum-Likelihood Based Feedback Carrier Synchronizer for Turbo-Coded Systems

Transcription

1 A Mxiu-Lielihoo Bse Feebc Crrier Synchronizer for Turbo-Coe Systes ele oels *, Heii Steen *, Mrc Moenecley *, Herwig Bruneel + Telecounictions n Infortion Processing Deprtent (DIGCOM *, SMACS + Group) Ghent University Gent, Belgiu {nnoels,hs,,hb}@telin.ugent.be Abstrct This contribution consiers the estition of tievrying crrier phse fro coe signls t low operting SR. ML estition theory is use to erive n itertive coe-ie feebc phse-trcer tht is well suite for ppliction in receivers with itertive MAP etection. Siultion results for turbo-coe syste inicte tht, in the presence of crrier frequency offsets n phse noise tht cnnot be hnle by feeforwr synchronizer, the propose synchronizer yiels only sll BER egrtion s copre to perfectly synchronize receiver. Crrier synchroniztion; phse loce loops; coes I. ITRODUCTIO The ipressive bit error rte (BER) perfornce of stteof-the-rt powerful coes iplicitly ssues coherent etection, i.e., the crrier phse ust be recovere ccurtely before t etection. Synchroniztion for encoe systes is yet very chllenging ts since the receiver usully opertes t extreely low signl-to-noise rtio (SR) vlues. Tritionl phse estition techniques y fil to cope with such high noise environent. Evlution of the Crer-Ro boun (CRB) for crrier phse estition in coe systes [1] hs shown tht synchronizers tht exploit the coe properties in the estition process (so-clle coe-ie synchronizers) re potentilly ore ccurte thn synchronizers tht o not exploit the coe properties (so-clle non-coe-ie synchronizers) when operting on coe signls. The evelopent of ccurte coe-ie phse estition lgoriths hs receive uch ttention in the recent technicl literture. In [1,2], we hve propose feeforwr (FF) crrier estition technique for coe systes tht exploits the nowlege bout the unerlying error-correcting coe. In this FF structure synchroniztion is crrie out itertively. At ech itertion, the current phse n frequency estites re use by the ecoer to prouce soft infortion tht is subsequently use by the synchronizer to upte the phse n frequency estite. We will refer to this schee s turbo -FF (T-FF) synchroniztion in the sequel. At the norl operting SR of coe systes, T-FF synchroniztion hs been shown to perfor very closely to fictitious t-ie (DA) synchronizer tht nows ll trnsitte sybols in vnce, provie tht sufficiently ccurte initil estites re vilble. Wheres the requireents on the initil phse estite re esily et by FF DA lgorith operting on short pilot sequence tht precees the ctul t, this is not the cse for the initil frequency estite. Convergence of the T- FF synchronizer requires tht the ifference between the frequency offset F n its initil estite is sll s copre to 1/LT, with T n L enoting the sybol intervl n the nuber of trnsitte sybols, respectively. For lrge L n low SR (i.e., the coon sitution for turbo coes), obtining such n ccurte initil frequency estite fro short pilot sequence is extreely hr becuse of the so-clle threshol-phenoenon [1]. Therefore, the initil frequency estite is set to zero, in which cse T-FF crrier synchroniztion is useful tool only for FT sll s copre to 1/L. In this contribution we evelop feebc (FB) phse estitor tht exploits the coe properties. The in vntge of FB lgoriths s copre to FF lgoriths is tht they inherently hve the bility to utoticlly trc slowly vrying crrier phse. In [3-6], FB phse estition hs been opte to cope with crrier phse which is tie-vrint ue to frequency offset n/or phse noise. In [3], the receiver is precee by n externl stnr FB loop, which runs once n effectively e-rottes the signl before it is processe by the stnr ecoer. In this siple scenrio, the estition process copletely ignores the unerlying coe structure. The coe-ie lgorith in [4] iplicitly ssues tht the phse vritions re sll over the fre urtion (LT), n its hoc erivtion is fr fro optiu. The tenttive ecisionie FB lgorith in [5] uses hr sybol ecisions extrcte fro one of the two seril conctente ecoers of turbo ecoer n therefore loses gret prt of the vilble infortion. The in rwbc of [6] is tht the stnr ecoer hs to be oifie consierbly in orer to ebe phse estition. In [7], lgoriths re erive fro fctor grph tht inclues both the coe constrints n the sttistics of Wiener phse noise oel. Unfortuntely, the ltter ethos re only pplicble if the crrier phse sttistics re vilble. Here, we present novel pproch bse on the itertive exchnge of infortion between n itionl phse trcing unit n conventionl ecoer, so-clle turbo -FB (T-FB) synchroniztion. The lgorith is erive fro the ML criterion n y be viewe s the FB counterprt of the T-FF estitor presente in [1,2]. When pplie to n itertive xiu posteriori (MAP) ecoer, it hs very low ipleenttion coplexity. Siultion results for turbocoe syste show tht the propose T-FB crrier This wor ws supporte by the InteruniversityAttrction Poles Progre P5/11 Belgin Science Policy

2 replce synchronizer cn cope with phse noise n frequency offsets tht the T-FF crrier synchronizer fro [1,2] cnnot hnle. II. A ML-BASED PHASE TRACKIG SYCHROIZER FOR CODED TRASMISSIO r PED x α β/(z-1) 2π ˆ ( F i ) + T A. Uni-irectionl T-FB lgorith Consier liner oultion, n chnnel chrcterize by itive white Gussin noise (AWG) w, phse noise Θ n frequency offset F. Assuing perfect tiing infortion t the receiver, the tche filter output sples t the correct ecision instnts t=t re given by r = exp(j ) + w, =,..., L-1 (1) where is the -th trnsitte sybol, =Θ +2πFT, n T enotes the sybol intervl. The t sybols in the sequence { } re obtine fro the encoing of sequence of infortion bits n proper pping on signl constelltion. Pilot sybols y lso be inserte in { }. Fro siilr resoning s in [1], the erivtive of the loglielihoo function, relte to (1), is given by ( [ ] j * p ; = IA ( r, ) r e ln r (2) where, is pilot sybol A = M ( r, ) 1 = α α (3) Pr[ r, ], otherwise = n (α, α 1,...,α Μ 1 ) enotes the set of M constelltion points. The posteriori en A (, r ) cn be consiere s soft ecision (SD) regring, bse upon the receive vector r n the phse vector = (,..., L 1). In the cse of coe trnsission, the posterior probbilities Pr[ = α r, ] cn be efficiently copute by ens of MAP soft ecoing lgorith [8]. In FB phse estitor erive fro the ML criterion, the output signl x of the phse error etector (PED) t instnt T is given by (2), with by the phse estite ˆ ˆ. However copliction rises, s the SD A fro (3) epens not only on ˆ but rther on the entire phse estite vector ˆ = (ˆ,..., ˆ L 1 ), which is not vilble t instnt T. This ifficulty cn be circuvente by consiering itertive FB synchroniztion. During the i-th itertion, the PED output t instnt T is given by x ˆ ) I[ * i ( i j = r e ] (4) ) with (, ˆ ( i 1 = A r ) fro (3) n (i = 1, 2,..., I). ote tht uring the i-th itertion the SDs i re copute fro the ( 1) phse estite vector ˆ i obtine uring the (i-1)-th itertion. Figure 1. Bloc schee of iscrete-tie type-ii phse loce loop The PED is incorporte in type-ii phse-loce loop (PLL) [9]. The vntge of type-ii loop over type-i loop is the zero stey-stte phse error in the presence of frequency offset. The stte vribles of the type-ii loop uring the i-th itertion re the phse estite ˆ i n the frequency estite ˆ ( i ) F : ˆ ( i ) ˆ ˆ πFˆ T + αx = (5) 2πF ˆ + 1T 2πFˆ T + βx where α n β re loop preters tht control the loop equivlent noise bnwith B L n ping fctor ζ. At ech itertion, the PLL ust be initilize with proper vlues of ˆ i n F ˆ i. Irrespective of the itertion inex i, we te ˆ i F = n ˆ i = ˆ, where ˆ is the DA phse estite resulting fro short pilot sequence tht precees the ctul t sybols. Running the PLL uring the i-th itertion using (4,5) yiels the phse estite vector ˆ i (). The phse estite vector ˆ neee to strt the itertions is obtine by running the PLL with the following SDs: E M s j C 1 ˆ() 2 () = exp r e α α (6) = where E s = E[ 2 ], = E[ w 2 ] n C is norliztion constnt. The SD in (6) is obtine by isregring the coe structure, in which cse the SD () epens only on r n ˆ (). The i of iterting with respect to i is to iprove the relibility of the SDs i, which in turn iproves the qulity of the phse n frequency estites. A perfornce liit of the synchronizer is obtine by ssuing i =, which correspons to DA FB opertion (ll sybols priori nown). When pplie to turbo receiver with itertive MAP etection/ecoing, the propose phse estition /copenstion schee yiels very low itionl coplexity when the synchronizer itertions re erge with the ecoer itertions [1,2]. I.e., fter ech synchronizer itertion one ecoer itertion is perfore without resetting extrinsic probbilities. This pproch will be opte in the siultions. 1/(z-1) u

3 B. Bi-irectionl itertive FB lgorith The rwbc of the uni-irectionl T-FB crrier synchronizer escribe in section II.A is tht in ech itertion the presence of nonzero frequency offset F gives rise to n cquisition trnsient, uring which the phse error ssues lrge vlues. The urtion of this trnsient is in the orer of 1/B L, with B L enoting the loop bnwith. This proble cn be solve by pplying bi-irectionl T- FB synchroniztion, where uring o n even itertions, the upting is perfore in the forwr (fro the first sybol to the lst) n bcwr (fro the lst sybol to the first) irection, respectively. The initil phse n frequency estites for given itertion equl the lst phse estite n inus the lst frequency estite fro the previous itertion. In this wy, n cquisition trnsient occurs only in the first itertion of the itertive FB synchronizer. In orer to voi tht the SDs { (1) =,..., L-1} copute uring itertion i=1 re ffecte by lrge phse trnsients, we perfor forwr n bcwr recursion of the non-coe-ie PLL using (6) before strting the itertive process. C. Cycle slipping When the signl constelltion is invrint uner rottion over n ngle p, n the coe structure is isregre (e.g. uring non-coe-ie opertion), the crrier synchronizer cnnot istinguish between ngles n + p, with = ±1, ±2,... As result, the crrier synchronizer hs infinitely ny stble opertion points, which re spce by p. The se goes for coe-ie opertion. Although the rottionl invrince is estroye by the coe structure, the usul p estition biguity ue to the syetry of the constelltion reins pprent uring coe-ie opertion s cn be foun in [2]. Most of the tie, the crrier phse estite exhibits sll rno fluctutions bout stble operting point. Occsionlly, noise or other isturbnces push the estite wy fro the current stble operting point, into the oin of ttrction of neighboring stble operting point. This phenoenon is clle cycle slip. After this, the estite reins for long tie in the close vicinity of the new operting point, until the next slip occurs. A cycle slips in the ist of coewor will of corse prevent error-free ecoing. The probbility of such n event increses exponentilly with the phse error vrince n linerly with the coewor length. Powerful error correcting coes usully wor t low SR n often hve lrge bloc lengths; therefore, cycle slipping is jor perfornce liiting fctor of our T-FB crrier synchroniztion syste. The occurrence n the irection of slip cn be etecte by onitoring nown synchroniztion wor (SW), which is, t regulr intervls, inserte into the sybol stre to be trnsitte. After crrier cycle slip, the nown sybols of the SW re foun to hve the wrong phse. The phse of ll sybols following the synchroniztion wor is correcte ccoringly, before coputing the SDs. Hence, the effect of the slip extens until the SW following the slip. III. UMERICAL RESULTS We consier rte 1/3 turbo coe consisting of the prllel conctention of two ienticl non-recursive systetic convolutionl coes with genertor polynoils (21) 8 n (37) 8 in octl nottion, seprte by pseuo-rno interlever of size 3333 bits. The encoer output is ppe onto sequence of =9999 BPSK sybols. Bursts re trnsitte with preble of 32, n postble of 16 pilot sybols. A SW of 16 pilot sybols ws inserte into the sybol stre once every 256 coe sybols. The totl nuber of pilot sybols ounts p =672. The loop preters fro (5) re selecte such tht the loop bnwith stisfies B L T=.75 n the ping fctor ζ equls.77, when the soft t ecisions woul equl the ctul t sybols 1. This choice of the loop bnwith is tre-off: sll enough to reuce the effect of AWG on the phse error, but lrge enough to trc phse noise n to liit the cquisition tie cuse by frequency error t the first itertion. In orer to evlute the synchronizer s perfornce, two criteri re use; bit error rte (BER), n en squre error (MSE) of the phse estite. Figures 2 n 3 show the MSE s obtine with the uniirectionl n the bi-irectionl T-FB synchronizer, E b E s 1 p + respectively. The SR equls = = 1B, r the frequency offset FT equls 1-3, n Θ is oele s tie-invrint uniforly istribute rno vrible in [-π,π[ (no phse noise). For resons of clrity we hve chosen not to epict the MSE fter itertion i=1 (perfore in the forwr irection). The qusi perioicl behvior of the MSE (with perio equl to ) is result of the slip etection/ correction process. Further, we observe tht The MSE ssues very lrge vlues uring the cquisition trnsient in ech itertion of the uni-irectionl lgorith (Figure 2). The MSE ssues very lrge vlues uring the cquisition trnsient in the initil forwr recursion of the bi-irectionl lgorith (Figure 3, curve itertion i= forwr ). Thns to the itionl bcwr initiliztion stge of the bi-irectionl T-FB synchronizer, this trnsient is not present in the finl () estites { ˆ }, which re use by the ecoer to copute the SDs { (1) } uring itertion i=1 (Figure 3, curve itertion i= bcwr ). The MSE of the bi-irectionl T-FB lgorith exhibits trnsient ner the strt of the t bloc in the forwr (o) itertions, n ner the en of the bloc in bcwr (even) itertions. This is result of (i) the reusing of the sples r, 1 The soft ecisions re close to the true t sybols when the synchronizer/ecoer itertions hve converge. During initil itertions the soft ecisions re less relible, which yiels sller loop bnwith n lower ping fctor.

4 .45.4 itertion i= (forwr) itertion i=2 (forwr) itertion i=3 (forwr) itertion i=4 (forwr).35 MSE Figure 2. MSE obtine with secon-orer uni-irectionl T-FB phse trcing loop s function of the tie inex.45.4 itertion i= forwr itertion i= bcwr itertion i=2 (bcwr) itertion i=3 (forwr) itertion i=4 (bcwr).35 MSE Figure 3. MSE obtine with secon-orer bi-irectionl T-FB phse trcing loop s function of the tie inex n (ii) the trnsition fro positive phse reference slope to negtive phse reference slope or vice vers. This effect cn be circuvente by ting s finl phse estites the estites t instnts =,..., (L/2)-1 fro the lst forwr itertion n those with inices = L/2,..., L-1 fro the lst bcwr itertion. The stey-stte MSE of the T-FB lgorith is sller thn tht of the non-coe-ie FB lgorith use for initiliztion, n ecreses s the nuber of itertions increses. This is becuse of the increse ccurcy of the SDs. After convergence of the itertions, the MSE is very close to the stey stte MSE for DA FB opertion, i.e. B L /E s =.19, which illustrtes the ner-optility of the propose synchronizer. In Figure 4, the perfornce of the bi-irectionl itertive T-FB synchronizer is ssesse in ters of BER versus Eb E s 1 =. 2 A totl of 1 synchronizer/ecoing itertions r re crrie out. The phse noise is oele s iscrete-tie Wiener process Θ = Θ 1 + ϑ, where Θ is uniforly istribute rno vrible in [-π,π[, n { ϑ } re zero-en 2 Here, we e bstrction of the pilot sybol insertion loss + of log( 1+ ) p 1.28 B. As consequence, the gp between the BER of the perfectly synchronize syste n the other BER curves is uniquely ue to phse estition/copenstion inccurcy.

5 sttisticlly inepenent Gussin increents with stnr evition σ (typiclly bout few egrees). For the se of coprison we lso show in Figure 4 the BER for perfectly synchronize syste, n the BER s obtine with the coeie FF joint phse n frequency estitor fro [1] operting on the se signl n with ienticl initiliztion. In Figure 4, rs stns for σ (in egrees). We e the following observtions: When the crrier phse is constnt ((F, σ ) = (, )), the receiver with FF synchroniztion yiels essentilly the se BER s the perfectly synchronize receiver, n outperfors the receiver with FB synchroniztion by bout.2 B in the wterfll region of the turbo coe. The BER egrtion of the ltter receiver cn be e vnishingly sll by reucing the loop bnwith, becuse the crrier phse is tie-invrint. The BER perfornce of the receiver with FF synchroniztion is consierbly egre in the presence of oertely tie-vrying phse (such s (FT, σ ) = (5.1-5, ) or (,.36 o )). As the phse estite in the first itertion is constnt over the fre, the tie-vritions of the ctul phse give rise to unrelible SDs tht cnnot be iprove in subsequent itertions. The receiver with FB synchroniztion cn hnle phse vritions (such s (FT, σ ) = (1-3, ) or (, 1.8 )) tht re too lrge for proper opertion of the receiver with FF synchroniztion. As (FT, σ ) = (1-3, ) n (FT, σ ) = (, ) yiel the se BER perfornce, it follows tht the BER egrtion s copre to perfect synchroniztion is cuse by the itive-noise coponent of the phse error (frequency offset cuses no stey-stte phse error, becuse the PLL is type-ii). A slightly lrger egrtion occurs for (FT, σ ) = (, 1.8 ), becuse phse-noise coponent is e to the phse error. IV. COCLUSIOS This contribution consiers the estition of tie-vrying crrier phse in coe systes. We propose new ML-bse coe-ie itertive FB type-ii phse-trcer which optilly exploits the coe structure without requiring oifiction of the stnr ecoer. When pplie to receiver with itertive MAP etection/ecoing, the propose phse estition/copenstion schee yiels very low itionl coplexity. As its erivtion oes not rely on the nowlege of the specific crrier phse sttistics, the propose lgorith exhibits high robustness in the presence of tie-vrying crrier phse. Siultion results show tht our coe-ie FB synchronizer cn hnle phse noise n frequency offsets with which its FF counterprt fro [1,2] cnnot cope. BER 1.E+ 1.E-1 1.E-2 1.E-3 1.E-4 1.E-5 FB, FT= or FT=1E-3, rs= FB, FT=, rs=1.8 perfectly synchronize FF, FT=, rs= FF, FT=5E-5, rs= FF, FT=, rs=.36 1.E Eb/ (B) Figure 4. BER perfornce of turbo receiver with T-FF or T-FB crrier synchroniztion REFERECES [1] ele oels, Heii Steen n Mrc Moenecley, On the Crer- Ro lower Boun n the perfornce of synchronizers for (turbo) encoe systes, in Proc. IEEE Worshop on Signl Processing Avnces in Wireless counictions, Lisbon, Portugl, July 11-14, 24 [2] ele oels, Vincenzo Lottici, Antoine Dejonghe, Heii Steen, Mrc Moenecley, Mrco Luise n Luc Vnenorpe, "A Theoreticl Frewor for Soft Infortion Bse Synchroniztion in Itertive (Turbo) Receivers," ccpete for publiction in EURASIP Journl on Wireless Counictions n etworing, specil issue on Avnce Signl Processing Algoriths for Wireless Counictions [3] L. Lu n S. W. Wilson, Synchroniztion of turbo-coe systes t low SR, in Proc. IEEE Int. Conf. Coun., Atlnt, GA, 1998, pp [4] W. Oh n K. Cheun, Joint ecoing n crrier recovery lgoriths for turbo coes, IEEE Coun. Lett., vol. 6, Sept. 21, pp [5] C. Lnglis n M. Helr, Phse crrier recovery for turbo coes over sttelite lin with the help of tenttive eisions, in Proc. Intern. Syp. on Turbo Coes & Relt. Topics, Brest, Frnce, Sept. 2, pp [6] A. Anstsopoulos n K.M.Chugg, Aptive itertive etection for phse trcing in turbo-coe systes, IEEE Trns. Coun., vol. 49, pp , Dec. 21 [7] G. Colvolpe, A. Brbieri n G. Cire, Itertive ecoing in the presence of strong phse noise, subitte to IEEE J. on Sel. Ares in Coun., vilble t www. eureco.fr/cire [8] L.R. Bhl, J. Coce, F. Jeline n J. Rviv, Optil ecoing of liner coes for iniizing sybol error rte, IEEE Trns. Infor. Theory, vol IT-2, Mrch 1974, pp [9] H. Meyr n G. Ashei, Synchroniztion In Digitl Counictions, ew Yor, Wiley, 199