EXIT CHATS FO TUBO ECEIVES IN MIMO SYSTEMS Césr Hermosill Universidd Técnic Federico Snt Mrí Vlpríso, Chile hermosil@inrs-telecomuquebecc Leszek Szczeciński Institut Ntionl de l echerche Scientifique-EMT Montrel, Cnd leszek@inrs-telecomuquebecc ABSTACT Itertive chnge of informtion between decoder nd front end receiver ws shown to improve the performnce when compring to non-itertive pproch This so-clled turbo processing my be described using Extrinsic Informtion Trnsfer (EXIT chrts which were lredy used in cse of turbo decoders nd turbo-equlizers This work tends the EXIT chrts technique to describe the behvior of MIMO nrrowbnd turbo receivers nd plins the min difficulties in the EXIT nlysis for yleigh fding chnnels INTOUCTION The growth of interest in the wireless ccess hve led reserchers to develop new trnsmission techniques tht cn ccommodte higher dt rtes in the limited bndwidth MIMO (Multiple Input, Multiple Output systems re ble to trnsmit multiple dt strems incresing significntly the spectrl efficiency, but the theoreticl cpcity of MIMO systems [] cn be pproched only if signl processing is introduced t the trnsmitter or the receiver In prticulr, so clled turbo processing hs been proposed for the MIMO receivers [2] A Turbo receiver is composed of front-end (FE receiver nd chnnel decoder which itertively chnge informtion Turbo receivers improve significntly the overll performnce mesured by Bit- or Block Error tes (BE, BLE when compring to non-itertive solutions Extensive simultions re commonly used for design nd for evlution of communiction systems but simple nlyticl or qusi-nlyticl methods re lwys useful to reduce the simultions time nd to yield optiml solutions In cse of turbo processing, Extrinsic Informtion Trnsfer (EXIT chrts hve been lredy pplied for evlution nd design of Turbo decoders [3] nd Turbo eceivers [4, 5] The ide behind the EXIT chrt is to reduce the description This work ws prtilly supported by the Chilen eserch Agency CONICYT, Project FONECYT #29 nd by NSEC, Cnd, projet #24974-2 of the whole itertive (turbo process to the chnge of one prmeter, clled mutul informtion (MI defined between bits nd their Logrithmic Likelihood tios (LL t the output nd the input of the SISO receiver/decoder [3] In this pper we describe how to use EXIT chrts for nlysis of turbo receivers in MIMO systems The min contributions when compring to previous work on this subject [3, 5] re: description of liner FE MIMO receivers using EXIT chrts nd definition of limittions of EXIT chrts for nlysis of fding chnnels 2 SYSTEM MOEL Consider multintenn trnsmitter, using Verticl Encoding (VE [6, 7], mening tht single dt strem is encoded nd demultipled to M trnsmitter ntenns (Fig while N ntenns re used for reception (cf Fig b A nrrow-bnd chnnel is considered which leds to the following trnsmission model [] r(n =Hs(n+η(n ( where r(n is the received vector t smple time n, H is N xm mtrix whose entries re unitry norm compl gussin, η(n is vector of compl normlly distributed noise with vrince ση 2 nd s(n =[s (n,s 2 (n,,s M (n] T is composed of sub-strem symbols which belong to 4QAM constelltion s k (n = 2 (c k, + jc k,, where c k, nd c k, re coded bits the resulting SN is then M/σ 2 η With nrrow-bnd ssumption, time ind n is dropped in the subsequent nlysis The simultions presented in this work re done for N=M=4 ntenns nd for /2 rte convolutionl encoder with octl genertor polynomil {5 7} 3 ECEIVE ALGOITHMS Turbo eceiver is composed of front-end (FE receiver nd decoder The FE receiver should produce soft outputs under the form of Logrithmic Likelihood tios (LLs nd the immedite cndidte is Mximum A Posteriori
t Strem encoder demux c s c c M s 2 s M T-MMSE The FE receiver minimizes MMSE between y k nd s k [2, 7, 8] Following the ides presented in [8], we implement its low-complity version A detiled description of the lgorithms my be found in [2, 7, 8] r r 2 r N Front end eceiver SISO eceiver M emux c M mux ( ( b (, Z SISO decoder (, b Figure MIMO trnsmitter with Verticl Encoding b Turbo receiver showing the chnge of mutul informtion between FE receiver nd decoder (MAP receiver (clled here T-MAP It produces the trinsic LLs of the j-th bit of the k-th substrem [4, 5]: Λ c k,j =ln P (c k,j = r P (c k,j = r Λ c k,j (2 using priori LL Λ c k,j obtined from the decoder in the previous itertion The complity of the lgorithm T-MAP grows ponentilly with the number of ntenns nd the modultion constelltion size, so less compl receivers using liner dt combining were proposed in the literture Such liner FE receivers consist of soft interference cnceller followed by liner combiner of resulting signl [2, 4, 7] y k = w H k (r b k b k = HE[s] h k E[s k ] (3 where h k is the k-th column of mtrix H The symbol pecttions re computed from priori LLs delivered by the decoder [7, 8] The computtion of the vectors w k depends on the criterion used to design the FE receiver We considered the following receivers: T-MC The FE receiver is Mximum tio Combiner where w k = h k Errors introduced by imperfect interference cncelltion re ignored 4 EXIT CHATS The EXIT chrts technique plined in detils in [3, 4] llows us to nlyze the chnge of informtion between two devices (eg FE receiver nd decoder in the twoprmeters spce Mutul informtion I(Λ; x (MI between LLs nd the corresponding bits proved to be suitble for such description Ms obtined supposing tht the input nd output LLs of ech device re iid rndom vribles defined by their pdfs p Λ x (ξ x = b conditioned on the bits sent b=-, [3], I(Λ; x = 2 b=, p Λ (ξ x = b 2p Λ (ξ x = b log 2 dξ (4 p Λ (ξ x =+p Λ (ξ x = Since LL s pdf is not known it hs to be estimted from the dt Here we used simple method presented in [9] I(Λ; x N N n= log 2 ( + p( x n Λ n (5 where N is the number of bits trnsmitted in the simultion nd Λ n is the LL ssocited with the bit x n Ech device trnsforming LLs (ie the FE receiver or the decoder is chrcterized by the trnsfer function relting MI defined for the input nd the output Let Ii = I(Λ c ; c nd Io = I(Λ c ; c re input nd output Mf the receiver ssocited with respective LLs Then the receiver nd the decoder re described by their EXIT functions Io = fmi (I i nd = fmi ( where = Io nd Io = I(Λ, c ; c re the input nd output MI defined for the decoder s LLs The numericl procedure for obtining the EXIT trnsfer function cn be described s follows [4] ndomly generte input bits x n ϵ{, } nd nt LLs Λ n ccording to gussin distribution with vrince σi 2 nd men x n σi 2/2 Apply SISO device (decoder or FE receiver to the sequence Λ n From the obtined output LLs clculte MI using Eq(5 The input MI my be obtined vi numericl integrtion of Eq(4 (becuse pdf of LL is known to be gussin epeting the bove-defined procedure for different vlues of σi 2 results in pirs (I i,i o, (Ii, which re used s knots to interpolte functions fmi ( nd f MI ( Note
9 ecoder Chrcteristic MC Chrcteristic T MC Trjectory MMSE Chrcteristic T MMSE Trjectory MAP chrcteristic T MAP Trjectory 9 ecoder Chrcteristic MC Chrcteristic T MC Trjectory MMSE Chrcteristic T MMSE Trjectory MAP Chrcteristic T MAP Trjectory 8 8 =Ii 7 =Ii 7 6 6 5 5 2 3 4 5 6 7 8 9 =Io 4 2 3 4 5 6 7 8 9 =Io Figure 2 Exit chrts obtined for fixed mtrix H, SN=7[dB] Figure 3 Exit chrt for the yleigh fding chnnel H, SN=7[dB] tht for the FE receiver, received dt for given chnnel stte nd SN, hs to be simulted s well The output of one device is the input to the nother, so both trnsfer function re shown in the sme EXIT plne with xes (Ii = Io,Io = Ii The lines connecting MI points evluted during itertions re clled trjectory The gret dvntge of using EXIT chrts comes from the fct tht the men trjectory (verged over different reliztion of dt nd noise mesured for both devices ctully chnging informtion flls on their EXIT functions obtined independently [3, 4] ie I o,m = f MI(I o,m, I o,m = f MI(I o,m (6 where I o,m nd I o,m re output MIs produced in m-th itertion The reltion Eq(6 is usully well stisfied for prticulr chnnel reliztion s will be shown further on The convergence occurs t the crossing of EXIT functions of the decoder nd the FE receiver This grphicl representtion gives us the immedite insight into the number of itertions required to ttin the best performnce Also, becuse MI t the decoder s output my be mpped into the finl BE/BLE, the EXIT chrt llows us to compre turbo receivers in terms of their performnce [3, 4] (ie higher the finl Ms, lower the corresponding BE/BLE will be 4 EXIT for fding chnnels The EXIT chrts were pplied for fding chnnels in contt of turbo decoders [3] but we will show tht such ppliction requires prticulr ttention in cse of turbo-receivers To describe the process for fding chnnels let us nlyze the trnsition from the men MI t the FE receiver s output Ī o = E[Io ] to the men MI t the decoder s output Īo = E[Io ]=E[fMI (I o ]; in both equtions, pecttions re tken over different reliztion of the chnnel The men EXIT functions will define the trjectory if E[fMI (I o ] = fmi (E[I o ] which my be stisfied only when the function fmi ( is liner over the intervl of vritions of Io Therefore the EXIT nlysis for fding chnnels will be ccurte only if the distribution of Io is very nrrow function with men close to 5 where fmi ( is lmost liner with Ii When MI pproches, we should lwys pect the discrepncies becuse the function fmi ( become highly non-liner This nlysis is illustrted in the nt section 5 NUMEICAL ESULTS Figure 2 shows the EXIT chrts obtined by mens of turbo receivers defined in Section 3 (T-MAP, T-MC, T-MMSE, for fixed chnnel mtrix H with entries: [926-87i -24-535i 35 +84i 483-852i; -432-235i 448 +22i -32-7i 57 +47i; -2-877i 649 +294i 36-29i -969-32i; -98-688i -54 +4i 44 +37i 948-34i] Solid lines represent the EXIT function obtined for ech device using the method presented in section 4 shed lines re trjectories whose vertices re computed when running ctul simultions of the Turbo eceiver We observe tht they fll ccurtely on the EXIT functions of the corresponding devices; this illustrte the usefulness of EXIT chrt nlysis The EXIT chrts obtined for studied turbo receivers in fding chnnels re shown in Fig3 In this cse the EXIT trjectory nd the FE receiver EXIT function re obtined through verging over different chnnel reliztions We my observe tht EXIT chrt looses its useful interprettion, especilly for T-MC, becuse some of the vertices in the trjectory do not coincide with the decoder s EXIT function To illustrte our plntion of this very fct (cf Section
Output Informtion from the decoder ( 9 8 7 6 5 4 3 2 Output Informtion Men Vlues ecoder Chrcteristic ecoder st itertion 2nd itertion Input Informtion Output Informtion from the decoder ( 9 8 7 6 5 4 3 2 Output Informtion ecoder st itertion 2nd itertion ecoder Chrcteristic Men Vlues Input Informtion 2 4 6 8 Input Informtion to the decoder (I i 2 4 6 8 Input Informtion to the decoder (I i Figure 4 eltionship between input nd output pdf of the decoder s Mbtined for the receiver T-MC in the first nd the second itertions Figure 5 eltionship between input nd output pdf of the decoder s Mbtined for the receiver T-MMSE in the first nd the second itertions 4 we show in Fig4 the pdf of Ii (obtined by mens of histogrms for the first nd the second itertion of the lgorithm T-MC; the decoder s EXIT function fmi ( is lso show the indicte non-linerity regions From the Fig4 we my see tht the pdf obtined in the first itertion flls in liner rnge of the decoder EXIT function (ie Ii ϵ(3, 8 Then, s plined in Section 4, fmi (E[I o ] E[fMI (I o ] In the second itertion however, the pdf is shifted to the right (s pected, since the tendency is to move to higher vlues of MI nd enters nonliner region of fmi ( ; then of course Ī o fmi (Ī i We observe in Fig3 tht the error between the trjectories nd the EXIT function is lower for receivers T-MMSE nd T-MAP, so in Fig5 we show pdfs obtined for T-MMSE (similr to those presented in Fig4 for T-MC It my be seen tht vrince of output Ms smller thn in cse of the receiver T-MC Then, the liner pproximtion of the function fmi ( holds resonbly well t lest for visul nlysis; the discrepncy will be noticeble only when zooming-in the it chrt close to Io (ie in highly nonliner region of fmi ( 6 CONCLUSIONS In this pper, EXIT chrts were pplied to nlyze Turbo eceivers in MIMO systems It hs been shown tht in sttic chnnels, the EXIT chrt is useful tool for prediction of turbo itertions in liner nd non-liner FE receivers It is lso plined wht re the min obstcles in pplying EXIT chrt in fding chnnels We conclude tht the EXIT nlysis is ccurte if the FE receiver produces mutul informtion reltively insensitive to the chnnel stte This ssumption is well stisfied by the T-MMSE nd T- MAP receivers, but not in the T-MC cse As promising reserch venue we indicte the ppliction of EXIT chrts for different types of receivers (eg V-BLAST [6] nd trnsmitters 7 EFEENCES [] G J Foschini nd M J Gnz, On the limits of wireless communictions in fding environment when using multiple ntenns, Wireless Personnl Communictions, vol 6, pp 3 335, Mr 998 [2] M Sellthuri nd S Hykin, Turbo-blst for wireless communictions: theory nd periments, IEEE Trns Signl Processing, vol 5, no, pp 2538 2546, Oct 22 [3] S ten Brink, Convergence behviour of itertively decoded prllel conctented codes, IEEE Trns Commun, vol 49, no, pp 727 737, Oct 2 [4] M Tüchler, Koetter, nd A C Singer, Turbo equliztion: principles nd new results, IEEE Trns Commun, vol 5, no 5, My 22 [5] A vn Zelst, vn Nee, nd G Atwter, Turbo BLAST nd its performnce, in IEEE Vehiculr Technology Conf, 2, 2, pp 283 286 [6] X Li, H Hung, G J Foschini, nd A Vlenzuel, Effects of itertive detection nd decoding on the performnce of BLAST, in IEEE Globl Telecommun Conf, no, Nov 2, pp 6 66 [7] C Hermosill nd L Szczecinski, Turbo receivers for nrrow-bnd mimo systems, in IEEE Interntionl Conf on Acoust, Speech, Signl Processing, Apr 23 [8] M Tüchler, Koetter, nd A C Singer, Minimum men squred error using priori informtion, IEEE Trns Signl Processing, vol 5, no 3, Mr 22
[9] M Tüchler nd J Hgenhuer, EXIT chrts of irregulr codes, in 22 Conf on Informtion Sciences nd Systems, Mr 22