A Bidirectional List-Sequential (BI-LISS) Equalizer for Turbo Schemes

Transcription

1 A Bidirectioal List-equetial BI-LI Equalizer for Turbo chemes Christia Kuh Lehrstuhl für achrichtetechik LT, Techische Uiversität Müche TUM 8090 Müche, Germay Phoe: , Fa: Abstract I the cotet of coded trasmissio over itersymbol iterferece chaels we ivestigate a bidirectioal list-sequetial equalizer BI-LI suitable for turbo schemes Especially for chaels with a umaageable high umber of states for a trellis based APP equalizatio this sequetial algorithm shows almost optimal performace with much smaller compleity Geerally, the list-sequetial LI equalizer is based o a decodig techique for covolutioal codes with high memory, amely sequetial decodig We itroduce a BI- LI equalizer which leads to a further reductio of the work amout ad provides the possibility of parallel processig For this purpose two idepedet LI equalizers are employed workig i the forward ad the backward directio respectively I ITRODUCTIO For coded trasmissio over chaels itroducig itersymbol iterferece II receivers based o the turbo priciple [, ] are kow to achieve a performace close to theoretical limits The equalizer ad decoder compoets have to be soft-i/soft-out algorithms calculatig a posteriori probabilities APP for the idividual bits I particular for chaels with a log delay spread a trellis based APP equalizer suffers from a high computatioal load ad has to be replaced with a algorithm of reduced compleity that approimates the a posteriori probabilities Followig the ideas of sequetial decodig for covolutioal codes with high memory [3] a tree based list-sequetial LI equalizer suitable for that task was itroduced i [4] Therefore, a modified Fao metric [5] is applied for eplorig a equalizer tree takig ito accout the a priori iformatio ad a appropriate legth bias for chaels with memory The tree-searchig is performed usig the stack algorithm [6] i order to fid a subset of the most probable sequeces trasmitted A soft-output is derived from those sequeces cotaied i the stack after carryig out a soft augmetatio This techique makes fully use of the a priori iformatio to improve the soft-output Basically, sequetial algorithms show a variable detectio effort Poor chael coditios reduce the effectivity of the stack algorithm ad it stops far away from the ed of the tree whe the stack is filled up Due to the soft augmetatio we are able to mitigate this problem for the uidirectioal LI equalizer but further improvemet ca be achieved usig a bidirectioal list-sequetial BI-LI equalizer I sequetial decodig origially all implemetatios aim at the hard decided maimum likelihood path to be foud after data data estimate FEC ecoder FEC decoder Fig Π Π m, a maimum umber of brach etetios A bidirectioal sequetial decodig algorithm for that purpose is suggested i [7] searchig the tree from the forward directio with a forward decoder ad idepedetly from the backward directio with a backward decoder Decodig stops wheever either the forward or the backward decoder reaches the ed of its tree Aother bidirectioal approach also workig with a forward ad a backward decoder is preseted i [8] This decodig algorithm is successful whe two appropriate partial paths of forward ad backward decoder, respectively, merge i a commo ecoder state Accordigly, the BI-LI equalizer performs a forward ad a backward LI equalizatio each with a soft augmetatio ad therefore the classical problem of ot fidig a mergig path after a fiite umber of brach etetios becomes obsolete i a turbo scheme The paper is orgaized as follows: I ectio II the system model ad basic defiitios are itroduced ectio III describes the BI-LI equalizer i detail I ectio IV simulatio results are preseted for the BI-LI equalizer ad the uidirectioal LI equalizer Fially, cocludig remarks are give i ectio V II YTEM MODEL We cosider the commuicatios system depicted i Fig Biary data is ecoded by a outer FEC ecoder typically usig a termiated or tailbited covolutioal code After imapper s II chael L e ˆ m, L ˆ m, Π BI LI y L m, Trasmissio system with iterative detectio terleavig subblocks =,,,, T etedig from time to with the vector elemets =,,, m,,, M, of legth M ad the biary digits m, = {+, } are formed Each bit sequece is mapped o a comple valued symbol s from the M -ary sigal costellatio, leadig to the symbol vector w

2 s =s,,s,,s T of legth which is appeded by L termiatig symbols ad trasmitted over the II chael The receiver observes the symbol vector y +L = y,,y,y +L T etedig from time to + L with elemets L y = h l s l + w l=0 where h = h 0,h,,h L T is the discrete time impulse respose of the chael with memory L ad uit eergy, L ie l=0 h l = The receivig process is corrupted by comple-valued additive white Gaussia oise AWG with iid oise samples w satisfyig the probability desity fuctio pdf pw = πσw e w σ w At the receiver we perform iterative equalizatio ad decodig Accordig to the turbo priciple reliability iformatio about the code bits is iterchaged betwee the soft-i/softout IO compoets The first oe is represeted by the equalizer which obtais a posteriori iformatio about the iterleaved code bits Lˆ m, usig the received symbols y +L together with the a priori iformatio L m, for all idividual bits Oly the etrisic iformatio L e ˆ m, is passed to the outer IO decoder after deiterleavig ormally, this decoder compoet applies the BCJR algorithm or oe of its approimatios [9] The etrisic part of the soft decoder output is fed back ad serves after iterleavig as a priori iput for the equalizer durig the et iteratio which is carried out usig the same set of received data Whe covergece is reached the iterative process stops ad the receiver delivers the data estimates III LIT-EQUETIAL TURBO EQUALIZATIO A APP soft-output Withi the cosidered turbo scheme the equalizer has to geerate the a posteriori log-likelihood ratio LLR Lˆ m, = l P m, =+ y +L P m, = y +L :m,=+ el P y+l = l :m,= el P y+l for each m, which ca be split up ito etrisic ad a priori parts Lˆ m, = l p y +L m, =+ p y +L m, = +lp m, =+ P m, = = L e ˆ m, +L m, 4 These LLRs ca efficietly be calculated operatig o a trellis usig the fiite state machie properties of the chael For chaels with high memory ad/or large symbol alphabets the umber of ML chael states causes a umaageable high computatioal compleity The basic idea is ow to approimate the a posteriori LLRs usig the observatio that the vast 3 majority of the cadidates cotributes a egligible amout to the total probability whe evaluatig the margializatio i 3 B APP path metric I order to fid a subset of the most probable cadidates we use a modified stack algorithm eplorig a equalizer tree, referred to as mai stack algorithm Evaluatig 3 oly for those requires their correspodig APP metric values i the log-domai which ca be stated as = lp y +L 5 = lp y +L +lp l p y +L ote that the chael part l p y +L as well as the a priori part l P of the metric ca be computed recursively i for a give sequece due to statistically idepedet oise samples w ad code bits m,, respectively Based o these properties the so-called legth bias part of the metric l p y +L ca also be determied i a sequetial maer Give the iitial ad fial chael state, the computatio of 5 ca be started either form the begiig of the block = +lp y L +lp l p y y 6 i the forward directio with = lp y L + l P l p y or from the ed of the block = + +lp y +L +L +lp l p y +L y +L +L+ 7 i the backward directio with =lp y+l +L + l P l p y +L These metric calculatios are carried out o a tree structure while performig brach etesios For a full legth path we calculate its APP metric 5 employig either the full legth forward or the full legth backward metric value +L = + =+ = [ l p y L l p y y ] = L [ + l p y L l p y y+ +L ] Durig the mai stack algorithm we perform metric computatios idepedetly i the forward ad backward directio for paths i the correspodig equalizer trees see Fig 3, solid paths I detail the chael part of the brach metric ca be epressed as l p y y L l=0 h ls l L = σw lπσw 8

3 ad for the a priori part of the brach metric we have [ M L m, l P = m, m= l e +Lm,/ + e Lm,/ ] 9 Fially, the bias parts of the forward ad backward brach metric ca be obtaied from l p y y =lp y l p y 0 ad l p y +L y +L +L+ =lp y +L +L l p y +L +L+ Although the bias term has o ifluece o the soft-output calculatio 3, it is absolutely ecessary to compare the metric values of the differet legth paths which occur while performig the mai stack algorithm Thus the legth bias is used to speed up the tree search Before presetig a procedure to determie the bias term we will give a detailed eplaatio of the mai stack algorithm C Mai stack algorithm with soft augmetatio Depedig o the equalizatio directio the two idepedet mai stack algorithms work o the mai stack or, respectively The TOP stack etry of each mai stack correspods to the path with highest metric which has ot yet reached full legth A flow chart is depicted i Fig ad i the followig we will oly poit out the differeces to the stack algorithm Origially, the stack algorithm works oly i Iitialize mai stack / with root ode metric has reached full legth I cotrast to that the mai stack algorithm stops whe the correspodig stack is filled up, ie the umber of paths i the stack reached a maimum value of ma/ ma, to use as may full legth sequeces as possible for the soft-output processig i 3 uder a give memory costrait Whe the stack is filled up with differet legth paths, a additioal improvemet of the soft-output is achieved by carryig out a full legth augmetatio without icreasig the stack size More precisely, we use the a priori iformatio L m, by meas of soft bits m, =E{ m, } =tahl m, / as well as soft symbols s =E{s } = b,,b M [ M sb,,b M m= +b m m, ], 3 b m {±}, for the augmetatio ad the required metric calculatios I Fig 3 a eample for the forward ad backward tree is preseted olid lies belog to hard decisios ad dashed lies represet the soft augmetatio These augmeted backward =+03 =+03 =+03 = 05 =+ = =+ + =+ + = = + = 05 =+ = + =+08 + =+ + = + =+08 forward Fig Eted TOP stack etry ad calculate metrics of successors Remove TOP etry from stack Isert the M ew etries i the stack which is ordered accordig to descedig metric values o tack filled up Eted all shorter paths to full legth ad update path metric usig soft values tore augmeted stack / for soft-output calculatio Flow chart for the stack algorithm with soft augmetatio Fig 3 Equalizer subtrees for BI-LI applied to BPK modulatio paths geerally cotai soft values m, ad hard decided values m, as show i Tab I k ma ma Lˆ TABLE I EXAMPLE FOR THE AUGMETED TACK OF A BI-LI EQUALIZER FOR BPK TRAMIIO WITH TACK IZE ma AD ma the forward directio ad stops whe the path with the highest

4 D Auiliary stack algorithm for bias estimatio The legth bias parts of the brach metrics 0 ad are essetial for the mai stack algorithm to be successful Evaluatig l p y =l e l py +l P 4 yields the eact value of the forward partial bias term ad l p y +L +L =l l py+l e +L +l P 5 the eact value of the backward partial bias term Ufortuately, the calculatio of 4 ad 5 is far too comple, but the correspodig sums are domiated oly by a small subset of sequeces of a particular legth These sequeces maimizig the forward partial auiliary metrics =lpy +lp or the backward partial auiliary metrics = lp y +L +L +lp, respectively, ca also be foud by eplorig the equalizer trees For this we use a slightly modified M-algorithm, referred to as auiliary stack algorithm I the forward directio this algorithm works o a auiliary stack with stack size ma = M ad recursively computes partial auiliary metric values = +lp y L +lp 6 usig the chael part 8 as well as the a priori part 9 of the brach metric, startig from =lp y L + l P Cosequetly, the forward partial bias term ca be estimated by l p y l β =l e 7 A flow chart for the auiliary stack algorithm is depicted i Fig 4 imilarly, we have for the backward directio a auiliary stack with size ma = M ad the partial auiliary metric = + +l p y+l +L +l P 8 startig from =lp y+l +L +lp A estimatio of the backward partial bias is obtaied by l p y +L +L l β =l e 9 Cotrary to the mai stacks, the auiliary stacks cotai oly sequeces of a particular legth Moreover, to reach the ed of the trees ad to guaratee limited stack sizes we have to discard paths while performig the auiliary stack algorithms for the tree depth > i the forward directio ad for i the backward directio Fig 4 Lˆ m, l Iitialize auiliary stack / with root ode Eted all stack etries ad calculate metrics of successors Remove all previous etries ad isert ew etries i the auiliary stack Compute bias estimate β / β o Ed of tree o tack filled up Order stack accordig to descedig metric values Remove all etries below positio M / M Bias complete Flow chart for the auiliary stack algorithm E oft-output processig We ow have available a set of augmeted paths i the forward mai stack ad i the backward mai stack Due to the soft augmeted part we have to modify 3 to cotrol the metric ifluece o umerator ad deomiator Therfore, we itroduce the weightig factors ± m, / which leads to + m, e + + m, e m, e + m, e 0 For the hard decided part of the tree we have almost the origial formula, but for the soft decided part we weight the metric values with the correspodig bit probabilities P m, = ± = ± m, A importat differece accordig to 3 is that we oly have the estimated bias parts l β +L ad l β L, which are i geeral differet from each other, istead of the eact value l p y +L which is valid for both directios I order to prevet BI-LI to privilege paths of the augmeted stack with the smaller estimated full legth bias, we have to use the metrics without the estimated bias terms = +l β +L, = +l β L for the evaluatio of 0 For that reaso it is oly ecessary to estimate the partial bias up to l β i the forward directio

5 ad up to l β i the backward directio I coclusio the structure of BI-LI is preseted i Fig 5 y L m, M algorithm forward auiliary stack M algorithm backward auiliary stack l β l β stack algorithm forward mai stack stack algorithm backward mai stack m, m, soft output processig Lˆ m, BER o II BI-LI, it ma = ma = 4096 LI, it ma = 89 BI-LI, it ma = ma = 048 BI-Gai BI LI equalizer 0 4 Fig 5 tructure of the BI-LI equalizer IV IMULATIO REULT The ioosphere is a very challegig physical chael which suffers from a large delay spread ad is typically modeled by two propagatio paths with equal power Therefore, we cosider the 6-tap chael h =/, 0, 0,0, 0, / T for biary trasmissio [0] Blocks of = 8 symbols are arraged i a iterleaver frame of size 89 bits The outer rate / covolutioal code is termiated ad defied by the geerator g = [ + D + D 3 + D 4, +D 3 + D 4 ] I Fig 6 the BER after decodig for a turbo receiver employig BI-LI with ma= ma = 4096 ad ma= ma= 8 is depicted ote that the full tree has 8 paths The system performace is compared to coded trasmissio over a AWG chael with APP decodig We reach this performace boud i iteratio at a BER E / i db b 0 Fig 7 BER performace after iteratios equalizatio ad decodig for = 8, various ier LI based equalizers with ma + ma= 56 ad a outer BCJR decoder for the rate /, memory 4 cov code receiver employig BI-LI with equal computatioal compleity, ie ma= ma= 4096, reaches the performace boud at a 5dB lower E b / 0 Moreover, we reduced the mai stack sizes by factors of two for the BI-LI equalizer Eve i that case, BI-LI shows a better performace V COCLUIO We have itroduced a bidirectioal list-sequetial BI-LI equalizer which overcomes the problems of former bidirectioal sequetial decodig algorithms Furthermore, the idepedet compoets of BI-LI provide the possibility of parallel processig Especially i the low E b / 0 -rage BI-LI shows a better performace tha its uidirectioal complemet LI BER it 0 it it 4 it 6 it 8 it 0 it o II E / i db b 0 Fig 6 BER performace after the decoder for = 8, a ier BI-LI equalizer with ma = ma =4096, ma= ma =8 ad a outer BCJR decoder for the rate /, memory 4 covolutioal code of 0 3 I Fig 7 a compariso of differet LI based equalizers after iteratios equalizatio ad decodig is preseted For each equalizer we have fairly small auiliary stack sizes of ma+ ma = 56 Compared to the uidirectioal LI equalizer with ma = 89, a turbo REFERECE [] C Berrou, A Glavieu, ad P Thitimajshima, ear shao limit errorcorrectig codig ad decodig: Turbo codes, i Proc of the IEEE It Cof o Commuicatios ICC, vol, pp , May 993 [] J Hageauer, The turbo priciple: Tutorial itroductio ad state of the art, i Proc of the It ymposium o Turbo Codes & Related Topics, pp, ept 997 [3] J M Wozecraft ad B Reiffe, equetial Decodig Cambridge, Mass: MIT Press, 96 [4] J Hageauer ad C Kuh, Turbo equalizatio for chaels with high memory usig a list-sequetial LI equalizer, i Proc of the It ymposium o Turbo Codes & Related Topics, pp 9 3, ept 003 [5] R Fao, A heuristic discussio of probabilistic decodig, IEEE Trasactios o Iformatio Theory, vol 9, pp 64 73, Apr 963 [6] Li ad D J Costello, Error Cotrol Codig: Fudametals ad Applicatios Pretice-Hall, Eglewood Cliffs, d ed, 004 [7] G D Forey, Fial report o a codig system desig for advaced solar missios, i Rep A-3637, Cotract AA CR7367, AA Ames Res Ctr, 967 [8] Kallel ad K Li, Bidirectioal sequetial decodig, IEEE Trasactios o Iformatio Theory, vol 43, pp 39 36, July 997 [9] J Hageauer, E Offer, ad L Papke, Iterative decodig of biary block ad covolutioal codes, IEEE Trasactios o Iformatio Theory, vol 4, pp , Mar 996 [0] R Otes ad M Tüchler, Block IO liear equalizers for turbo equalizatio i serial-toe HF modems, i Proceedigs of the orwegia igal Processig ymposium ORIG, pp 93 98, Oct 00