Blind equalization based on spatial and temporal diversity in block coded modulations

Blind Equaliza ion Based on Spa ial and Tempo al Di e si y in Block Coded Modula ions F ancesc Rey, Me i xell Lama ca, G ego i Vizquez Depa men o Signal Theo y and Communica ions, UPC Campus No d-Mbdul D5, c/ G an Capi i s/n, 08034 Ba celona (Spain) E-mail: { ey,xell,g ego i} @gps. sc.upc.es ABSTRACT Linea block codes can be applied in spa ial and/o empo al di e si y ecei e s in o de o de elop high pe o mance schemes o blind equaliza ion in mobile communica ions. The p oposed echnique uses he s uc u e o he encoded ansmi ed in o ma ion (wi h edundancy) o achie e equaliza ion schemes based on a de e minis ic c i e ion. Simula ions show ha he p oposed echnique is mo e e icien han o he schemes ha ollow simila equalize s uc u es. The esul is an algo i hm ha p o ides he design o blind channel equalize s in low EbNo scena ios. I. INTRODUCTION The p oposed app oach elies on he a ailabili y o spa ial and/o empo al di e si y in digi al ecei e s, which enables he de elopmen o blind channel equaliza ion echniques based on a de e minis ic design c i e ion. Blind equalize s can supp ess he IS1 and educe he impac o he inpu noise by he knowledge o he ansmi ed/ ecei ed da a s uc u e o i s p ope ies. In such sys ems no aining sequence is needed. Fo his eason he blind echniques a e o p ac ical in e es in mobile communica ions, whe e he channel is cons an ly changing and con inuous aining sequences can no be ansmi ed. I is known ha i he channel ou pu is o e sampled in he ime domain and/o in he spa ial domain (mul i-channel di e si y), channel compensa ion can be pe o med based on he ecei ed signal second-o de s a is ics. A ecen s udy [l] used he Bezou equa ion o in oduce a new blind equaliza ion c i e ion. Basically, i p oposed an algo i hm ha maximizes he signal- o-ISI-plus-noise a io (SINR) a he equalize ou pu . Un o una ely, his algo i hm o e ed high eliabili y only in mode a e o high SNR’s scena ios bu showed s abili y p oblems a lowe SNR’s. This pape enhances he esul s o he me hod p esen ed in [I], and enables he design o high-pe o mance blind channel equaliza ion schemes in mobile communica ion scena ios wi h low EbNo a es. The main idea is o combine he in o ma ion edundancy in oduced by a sys ema ic linea block coding echnique wi h he s uc u e inhe en o he ‘T ans o m Modula ion’ ([2]) ansmission echniques o imp o e he s a is ical s abili y o he equaliza ion echnique in p esence o noise. The esul is a mo e obus scheme ha can be applied o TDMA, DS-CDMA sys ems in equency selec i e mobile channels, and OFDM sys ems in equency la ading mobile channels ([3]). Mo eo e , he p oposed solu ion is a ailable o bo h spa ial and/o empo al di e si y on -ends, whe e i is possible o conside a single-inpu mul iple-ou pu o mula ion (SIMO) o he ansmission sys em. In some equaliza ion me hods ([4]) he ansmi e in oduces edundancy ha he ecei e uses o iden i y o equalize he channel. Al hough some addi ional edundancy is also added in ou me hod, he p oposed app oach can be conside ed a blind echnique because ha edundancy is no in oduced o equalize bu used o co ec de ec ion e o s. Thus he ecep ion is done in wo s eps. In he i s s ep he ecei e uses he s uc u e o he encoded ansmi ed in o ma ion (wi h edundancy) o achie e he equaliza ion. In he second s ep he same edundancy is also used o co ec e o s. This pape s udies he i s pa o he whole ecei e , analyzing he pe o mance o he equalize using he edundancy symbols. I will be he aim o a la e s udy o choose linea codes de ined o e he complex ield ([5],[6]) ha , applying he s uc u e p esen ed in his pape , could be used o co ec decision e o s a he ou pu o he equalize . The nex sec ion illus a es he scheme and es ablishes he p oblem. Sec ion 3 desc ibes he linea block coding cha ac e is ics and shows how he s uc u e o he ansmi ed encoded da a can be used in he design o he equalize . Sec ion 4 applies hese esul s o imp o e he [l] cos unc ion. Finally sec ion 5 p esen s some simula ion esul s, whe e i is possible o see he imp o emen o he p oposed solu ion. * This wo k has been pa ially suppo ed by he Spanish Resea ch Council (TIC-95-1022-CO5-01) and by ACTS 11 TSUNAMI-I1 0-7803-4872-9/98/$10.00 0 1998 IEEE 1265 11. PROBLEM STATEMENT The p oposed app oach elies on he a ailabili y o spa ial and/o empo al di e si y in digi al ecei e s, which enables he use o single-inpu mul iple-ou pu o mula ion (SIMO) o he ansmission sys em. I consis s o an algo i hm o linea equaliza ion o he ecei ed da a ha is based on a de e minis ic design c i e ion. Figu e la shows a disc e e- ime model o a spa ial di e si y ecei e . The same in o ma ion signal T[k] is ansmi ed h ough B di e si y b anches. I is dis o ed by di e en channel esponses Ci[k/ and inally deg aded by AWGN e ms wl[k/. In a simila way, Figu elb shows he model o a empo al di e si y ecei e . The in o ma ion signal T[k] is ansmi ed h ough a channel, which dis o s he signal wi h a channel esponse C[k] and in oduces an AWGN, W[k]. The ecei ed signal is o e sampIed, a B sampIes pe symbol, and in oduced in B di e en b anches as in he p e ious s uc u e. 1 . / 47 CB kl : yBM,Fg Figu e la. Block diag am o he spa ial di e si y sys em. The equaliza ion p ocess o spa ial and empo al di e si y schemes is simila , and bo h can be combined in o he same ecei e sys em. Thei associa ed equa ions can be w i en in he z- ans o m domain as: (1) Simila equa ions can be de i ed in OFDM signals h ough equency- la ading channels in he ime domain (see [I], [7]). Y'(z) = T(z)C'(z)+ W'(z) i = 1, ..., B As shown in [l], he equaliza ion p ocess can be designed ollowing a blind c i e ion. The mul iple empo al o spa ial di e si y b anches a e combined by means o FIR il e s E'[k] o gene a e an ou pu R[k]: B R(z) = ZY'(z)E'(z) = i l B B = T(z)CC'(z)E'(z)+C~'(z)E'(z) ,=I ,=I The pe ec equaliza ion c i e ion equi es R(z) =T(z), and he e o e (neglec ing he noise e ec ): (3) The Bezou equa ion [8] gua an ees ha he p e ious equa ion has solu ion i and only i he B channel esponses ha e no common ze os, a esul well known in he li e a u e ([9]), and he e o e (igno ing he noise e m): T(z) = g.c.d.@' (z,} (4) A ma ix o mula ion o he me hod can be ound in [ 101 and is b ie ly summa ized he e in o de o app oach he p oblem. As shown in [lo], equa ion (2) can be w i en in ma ix no a ion as: - R=Z (5) whe e g is he equalize ou pu ec o , y is a gene alized Syl es e ma ix wi h he ecei ed da a and is he equalize weigh ec o . The pe ec equaliza ion case in (3) can be w i en as: - whe e he ecei ed da a ma ix y has been spli in wo pa s, 2: is he ansmi ed da a ec o and a is an a bi a y cons an . The op imum equalize coe icien s (i he noise e m is no conside ed) can be es ima ed wi h: and once he equalize has been es ima ed, he ecei ed da a can be il e ed o yield an es ima ion o he ansmi ed da a: An in e es ing algo i hm, o muIa ed in [ 11, ies o maximize he signal- o-ISI-plus-noise- a io (SINR) a he equalize ou pu , ha is: (9) and he equalize coe icien s a e designed o maximize he p e ious quo ien . In high EbNo scena ios he p e ious cos unc ion pe o ms co ec ly, bu in noisy en i onmen s many s abili y p oblems wi h he yo ma ix appea . Hence, he denomina o o (9) is c i ical and new design c i e ia shouid be ound. - - Figu e ib. Block diag am o he empo al di e si y sys em. id 1266 111. LINEAR BLOCK CODING STRUCTURES Equa ion (10) de ails he cons uc ion o linea block codes using ma ix no a ion. I desc ibes how o encode k symbol da a in o ma ion using he code gene a o ma ix G, and ob ain he n coded ou pu sequence: - - Conside ing a sys ema ic code, he encode ma ix becomes: 7 1 Lk 1 G, = ................ - iG =(n-!+k 1 and he code wo d is di ided in o wo pa s. The i s k-symbol pa i ion is always iden ical o he in o ma ion sequence o be ansmi ed, while in he second po ion each o he (n-k) symbols is a linea combina ion o he in o ma ion da a acco ding o G, : - - T i I The second s ep in he coding p ocess applies he pa i y check ma ix c, o e he mobile channel ou pu da a. This ma ix is de ined as: - - pa i y o edundancy symbols allow he ecei e o de ec he e ec s o he IS1 and noise, so new cons ain s can be in oduced o ind he equalize weigh ec o B. As in equa ion (8), i he coe icien s equalize co ec ly he ecei ed da a, he il e gi es a good es ima ion o he ansmi ed sequence: T'=Y,E (16) - - - hence, applying he check pa i y ma ix o a pe ec equaliza ion exp ession can be ound: The p e ious equa ion can also be seen as a :measu e o he quali y o he equalize . The close o ze o i is, he be e he equalize pe o ms, whe eas any non-null esidue in (17) will deno e a poo channel equaliza ion. IV. THE COST FUNCTION IMPROVEMENT As i was said be o e, equa ion (7) can exhibi poo pe o mance in e y low SNR scena ios. In his pape , a new me hod o design equaliza ion coe icien s and o o e come his p oblem, is p oposed. The cos unc ion equa ion ies o maximize he SINR a he ou pu , and we ha e p e iously discussed ha i is close o ze o i means ha he equalize is educing he IS1 e ec and il e ing he addi i e noise co ec ly. (13) 1 =("-k)d . =(n-k)x(n-k) L J and G, and c, ma ices a e de ined so ha : C, G, = 0 - Thus he pa i y check ma ix can de ec changes be ween he ansmi ed code and he ecei ed in o ma ion. I bo h sequences a e iden ical, he null ec o will be ob ained. On he con a y, he e ec o he noise and he IS1 channel in oduces di e ences in he ecei ed sequence, and he p oduc o ha sequence wi h he pa i y check ma ix di e s om he null ec o . Acco ding o his conside a ion he new cos unc ion is o mula ed as: Wi h his imp o emen he unc ion o be op imized is mo e obus in p esence o AWGN, i only needs he signal subma ix y, , a oiding he use o he esidual yo ma ix, which uses o be ill condi ioned. No ice ha he changes ha e been done in he denomina o , which was he c i ical pa o he quo ien : - - - - Equa ion (19) co esponds o a ypical Rayleigh quo ien o m (see [ll]), and he e o e he equalize (E ec o ) ha Because o he cons uc ion o G, and c, ma ices, any ull - - - - ank ma ix can be used as pa i y check symbol ma ix G, . Fu he mo e o ensu e ha he encoding p ocess main ains cons an he symbol ene gy o all he ansmi ed symbols we a e in e es ed on hose ans o m such ha : /&I2 =lg212 =.-=18n12 =1 (15) maximizes i co esponds o he gene alized eigen ec o associa ed wi h he maximum gene alized eigen alue: (20) - _. .I; .I;g = A. Y c, i c, ;E -- "2 __= - -- The equaliza ion pe o mance can be also imp o ed i a delay is allowed in R[k], and he bes equalize is selec ed as ha one whe e lgj I is he no m o he ow ec o s o G, . - - which yields he g ea es 2". Finally i is also wo h ema king ha he compu a ional load o he solu ion does no inc ease i adequa e ma ices a e selec ed o G, . In ou s udy Sub-Hadma d ma ices ha e been chosen. The in oduc ion o he edundancy P o ides a mo e ich s uc u e in he ansmi ed in o ma ion, which can be used in he design o he equalize . P e iously, we ha e seen how he = 1267 V. SIMULATIONS Figu es 2 o 6 compa e he pe o mance o he cos unc ion in [ 11 wi h he new cos unc ion p oposed in his pape . The simula ions display he pe cen age o eaIiza ions (IO00 we e a e aged) o which he equalize ou pu EbNo was highe han he alue indica ed in he x-axis. In all cases he ansmi ed in o ma ion consis ed o 118 QPSK da a symbols and 10- edundancy symbols gene a ed wi h a sub-Hadama d ma ix G, . Each b anch equalize had ou coe icien s and ou b anches (BA) we e simula ed. - - Figu es 2 and 3 show he pe o mance o he algo i hm using a TDMA bu s ansmission o e a equency selec i e channel wi h spa ial di e si y (Figu ela). The ou s a iona y channel esponses we e: c’(z) = (1+j)+(-O.1-0.2j)z-’+O.4~-~ +z-~ +0.5z4 C~(Z)=O.~+~~-~+~~Z-~+O.~Z-~+Z-~ C3(2) =0.1j+z-’-0.4jz-2+0.2z-3 -0.5~-~ C4(z) = (1+0.8j)-2jz-’ -0.4jz-* +O.2zS3 +(1-0.5j)~-~ No ice ha he ou channels ha e no common ze os. In o de o conside a di icul mobile scena io, he ou channels ha e been selec ed wi h high a enua ion in ce ain equencies (in special channels 1 and 3), wi h some close ze os and wi h a non-minimum phase beha io . linea block codes. In Figu e2 he channel EbNo is 12dB, while Figu e 3 illus a es he simula ion wi h EbNo=lSdB. In bo h cases he pe o mance o he imp o ed cos unc ion (I) is highe han he pe o mance o he algo i hm wi hou linea block codes (II). No ice ha he new me hod gua an ees ha wi h a EbNo highe han 12dB, mo e han he 95% o imes he ou pu EbNo is o e 7dB, which ep esen s a BER210”. 0 2 4 6 8 10 12 14 16 18 20 EbNo ldBl Figu e 3. Algo i hm pe ommce in spa id di e si y wi h EbNo=ISdB Compa ison o he algo i hm wi h ( ) and wi hou (II) he use o linea block codes. F equency espanse chnnnek .... .... .... .... .... . <. -*5 : ::; .. 2w 400 600 800 1000 1200 -30 k Figu e 4. Channel 1 and 3 equency esponse. Figu es 5 and 6 show he pe o mance o he algo i hm in a TDMA bu s ansmission wi h empo al-di e si y (Figu e 1 b). Two di e en non-minimum phase channels we e simula ed: C’[z]; C3[z]. Thei equency esponses a e illus a ed in Figu e. 4. Figu e 5 illus a es he simula ion wi h EbNo=12dB, and Figu e 6 wi h EbNo=15dB. The dashed line cu es a e ela ed o channel 3, and he solid cu es o channel 1. As in he p e ious case I ep esen s he esul s using linea block codes and I1 he esul s wi hou hem. No ice ha he pe o mance wi h spa ial di e si y is be e han wi h empo al di e si y. Two easons can jus i y ha . Fi s , he spa ial di e si y uses B an ennas, and so he maximum ou pu EbNo could be: mm{(EbNo),,,,}= (ELINO),, lOlog(B) meanwhile in empo al di e si y: “DbNO).,,, I= (EbNO),” Second, when he Nyquis pulse is o e sampled (in empo al di e si y), addi ional IS1 is in oduced in o he ecei e , causing he pe o mance o he sys em o be educed. 1268 VI. CONCLUSIONS In his pape linea block codes in he complex ield ha e been in oduced o blind equaliza ion o mobile channels in spa ial and empo al di e si y ecei e s. The p oposed c i e ion, combining a blind equaliza ion echnique ([ 11) wi h he edundancy in oduced by a sys ema ic linea code, can be applied o e TDMA s uc u es wi h equency selec i e mobile channels, DS-CDMA sys ems and OFDM modula ion wi h equency la ading channels. Spa ial and empo al di e si y ecei e s o e TDMA s uc u es, in equency selec i e mobile channels, ha e been conside ed. The esul s show he pe o mance imp o emen o his new equaliza ion me hod o e he p e ious scheme p esen ed in [ 11. Be e esul s ha hose p esen ed in he cu en pape can be achie ed in oducing codes de ined o e he complex ields ([5],[6]) ha , ollowing he s uc u e p esen ed in sec ion 3, could be used o co ec mos o he e o s a he ou pu o he equalize . REFERENCES M. Lama ca, G. V izquez, ‘Di e si y echniques o blind channel equaliza ion in mobile communica ions’, P oceedings. o PIMRC’97, pag. 1079-1083, Helsinki (Finland), Sep . 1997. M. Lama ca, G. Vhzquez, ‘T ans o m modula ions in mobile communica ions’, P oc. o PIMRC’97, pag 462- 466, Helsinki (Finland), Sep . 1997. R S eele, Mobile Radio Communica ions, Pen ech P ess. A. Scaglione, G.B. Giannakis, S. Ba ba ossa ‘Sel - eco e ing mul i a e equalize s using edundan il e bank p ecode s”, P oceedings o ICASSP ’98 Sea le May 1998. T.G. Ma shall, ‘Coding o eal-numbe sequences o e o co ec ion: a digi al signal p ocessing p oblem’ IEEE Jou nal on Selec ed A eas in Communica ions, V01.2, Ma ch 1984. R.E. Blahu , ‘Algeb aic ields, signal p ocessing, and e o con ol’, P oceedings o he IEEE, Vo1.73, no 5, May 85. M. Lama ca, G. Vhquez, ‘Channel es ima ion o ans o m modula ions in mobile communica ions’, P oceedings o EUSIPCO’96, T ies e (I aly), Sep . 1996. Figu e 5. Algo i hm pe o mance in empo al di e si y wi h EbNo=lZdB, h ough di e en channels. Compa ison o he algo i hm wi h (I) and wi hou (U) he use o linea block codes. .--, Figu e 6. Algo i hm pe o mance in emponl di e si y wi h EbNo=lSdB, h ough di e en channels. Compa ison o he algo i hm wi h (I) and wi hou (II) he use o linea block codes. [8] T. Kaila h, Linea Sys ems, P en ice-Hall, 1980. [9] Y.Ii1, Z.Ding, ‘Blind channel iden i ica ion based on second o de cyclos aciona y s a is ics’, P oceedings. o ICASSP ’93 Minneapolis (USA) [lo] M. Lama ca, G. Vgzquez, ‘Mul ichannel ecei e s o OFDM and TDM in mobile communica ions’, P oc. o ICASSP ’97, Munich (Ge many), Ap il. 1997. [ 111 G.H. Golub, C.F. Van Loan, Ma ix Compu a ions, Johns Hopkins Uni . P ess, 1989. 1269