A Novel Reduced-Complexity Group Detection Structure in MIMO Frequency Selective Fading Channels

Transcription

1 A Novel Reduced-ompleity Group Detectio Structure i MIMO Freuecy Selective Fadig haels halid A. Qarae Departmet of Electrical Egieerig eas A&M Uiversity at Qatar Educatio ity, Doha, Qatar khalid.arae@atar.tamu.edu Narima Rahimia Departmet of Electrical ad omputer Egieerig eas A&M Uiversity ollege Statio, eas, USA arimarahimia@eo.tamu.edu Mohamed-Slim Alouii Electrical Egieerig Program AUS uwal, Saudi Arabia mohamed.alouii@kaust.edu.sa Abstract- I this paper a ovel reduced compleity detectio method amed modified symbol flippig method is itroduced ad its advatages o reducig the burde of the calculatios at the receiver compared to the optimum maimum likelihood detectio method o multiple iput- multiple output freuecy selective fadig chaels are eplaied. he iitial cocept of the symbol flippig method is derived from a prelimiary detectio scheme amed bit flippig which was itroduced i []. he detectio structure employed i this paper is ullig, detectio, ad cacellatio. O the detectio stage, the proposed method is employed ad the results are compared to the group maimum likelihood detectio scheme proposed i []. Simulatio results show that a 6 db performace gai ca be achieved at the epese of a slight icrease i compleity i compariso with the covetioal symbol flippig scheme. eywords-compleity, freuecy selective, group detectio, MIMO, symbol flippig I. INRODUION With the icreasig demad of high uality wireless multimedia services, the ew challege o research is to improve reliable high-speed wireless data lik while maitaiig good spectral efficiecy ad affordable detectio compleity. Foschii ad Gas [3] provided a iformatio theoretic aswer to this problem by usig multiple elemet atea arrays. As such, multiple iput- multiple output (MIMO) layered space-time architecture called Vertically- Layered Bell Laboratories Layered Space-ime (V-BLAS) was proposed by Foschii et al [4]-[6] to achieve the liearly icreasig capacity epected from the iformatio theory study. As most practical high data rate trasmissio systems are widebad, the arrowbad V-BLAS architecture must be reegieered for freuecy selective fadig chaels ad delay spread mitigatio techiues must be cosidered [7]. I [], a detectio method utilizig maimum likelihood seuece estimatio (MLSE) with successive cacellatio is proposed. First, the layered maimum likelihood detectio (L-MLD) scheme for V-BLAS i freuecy selective fadig chaels is proposed. It performs ullig, detectio ad cacellatio for a substream i a group of oe actual ad L virtual trasmit ateas, where L is the umber of chael taps. he the L-MLD scheme is eteded to a group detectio scheme that performs ML detectio withi the group, ad layered processig amog the groups. his detectio scheme is called group maimum likelihood detectio (G-MLD) scheme. he G-MLD scheme has much more compleity compared to the L-MLD scheme i two aspects. Firstly, sice the G-MLD scheme joitly detects the substreams withi the group, the detectio process is more complicated compared to the L-MLD scheme which detects oly oe substream i each layer. I additio, the detectio compleity grows epoetially with the umber of substreams per group. Secodly, to utilize the optimal group orderig i G-MLD, more compleity is epected compared to the use of optimal substream orderig i the L-MLD scheme.he total umber of groupig combiatios that should be take ito cosideratio for the G-MLD scheme is give i [8] as 3, where is the umber of trasmit ateas, is the umber of substreams per group ad! =. A reduced-compleity suboptimal orderig of ( )!! substreams which is applicable to both L-MLD ad G-MLD schemes is proposed i [8]. I this paper a ew detectio scheme is proposed i order to further reduce the compleity of G-MLD. his detectio scheme is amed modified symbol flippig (SF) detectio ad follows a simple idea i detectig the trasmitted sigals. II. PROBLEM SAEMEN he problem at had is a iteger least suares problem of the form: mi Y H Z m where m Y R, H R, ad Z deote the -dimesioal iteger lattice appear i may applicatios. I commuicatios, whe the chael is liear ad the oise is idepedet, idetically distributed (i.i.d.) Gaussia, maimum-likelihood (ML) decodig leads to a least-suares cost, i.e., if a lattice is used as a code for the Gaussia chael, maimum-likelihood decodig i the demodulator is a closest poit search. Whe the trasmitted symbols are from a fiite set, this ca be ofte cast as a iteger least-suares problem. I this case, the search space is a subset of m the ifiite lattice Λ Z ad we therefore have: mi Λ Z Y H Note that euatio () is the same as the cost fuctio of the ML detector for a MIMO system with trasmit ateas ad m receive ateas. () () //$5. row

2 Each lattice poit i the rectagular -dimesioal lattice Z represets a cadidate of trasmittig matri ad is correspodigly mapped to a parallelepiped m -dimesioal m lattice. Note that the etries of H are o loger itegers. R Fig.. Geometrical iterpretatio of the iteger least-suares problem. A. ompleity of ML Detectio he total umber of cadidates tested i the cost fuctio give i euatio () i a thorough search is P, where P, is the trasmitted symbol costellatio size. Frame trasmissio is assumed for freuecy selective fadig chaels. he chael is assumed uasi-static with slow fadig, such that the chael gais will be static withi a frame but could vary from frame to frame. hese coefficiets are also assumed kow perfectly at the receiver, while i practice a chael estimator is reuired. If the chael is assumed to be freuecy selective with L chael taps ad each trasmit atea seds a frame with symbol, the there will be iter-symbol-iterferece (ISI) at the receiver. I this case the optimal detectio scheme is the joit Maimum Likelihood Seuece Estimatio (MLSE) ad the total umber of cadidate would be. O the other had, if the chael is memoryless ( L = ) ad each trasmit atea seds a frame with symbol, the there will be o ISI at the receiver. I this case there would be o eed for frame detectio ad symbol by symbol ML detector is optimal. I this case the total umber of cadidates reduces to P. B. Geometric Iterpretatio Whe performig joit MLSE at the receiver, the cost fuctio is: mi Λ Z Y H where is a matri that each of its etries ca be oe of the P possible symbols of the modulatio costellatio. herefore, there will be matrices to be tested i the cost fuctio. It should be oted that performig a thorough search amog all cadidates is a time cosumig process ad the compleity grows epoetially with the icrease i the umber of trasmit ateas ad frame legth. he iteger least-suares problem has a simple geometric iterpretatio. As the etries of ru over the itegers, spas the rectagular -dimesioal lattice Z, However, for ay give lattice-geeratig matri H, the m -dimesioal vector H spas a skewed lattice [9]. herefore, give the skewed lattice H ad give a vector Y R m, the iteger least-suares problem is to fid the closest lattice poit (i a Euclidea sese) to Y (Fig. ). (3) III. SYMBOL FLIPPING DEEION Each poit o the lattice ca be show with a vector of symbols. It should be oted that the dimesio of the modulatio costellatio is differet from the dimesio of the lattice i which the trasmittig cadidates lay. A. Geeralized Symbol Flippig Detectio Scheme I [] the cocept of symbol flippig detectio scheme is eplaied oly for the case of BPS modulatio. I this sectio the symbol flippig detectio is geeralized to all types of modulatio ad the modified symbol flippig detectio (MSF) is proposed i et sectio. I covetioal symbol flippig detectio scheme, the least suares solutio of the euatio Y = H is uatized ad is called, i.e., = Quat( H Y ) where Y is the received sigal. is cosidered as the iitial poit ad H is the Moore-Perose pseudo iverse of the chael matri H. he iitial poit is actually the Zero Forcig () solutio ad is oe of the cadidates i the rectagular - dimesioal lattice Z. Matri is a matri that each of its etries is oe of the P possible trasmittig symbols. I the covetioal SF, at the begiig, all the P possible symbols of the first etry of, i.e., are eamied ad put ito the cost, fuctio of euatio (), while all the other etries stay uchaged. is replaced by the symbol with the miimum, cost ad the ew matri is called. Net all the P possible symbols of the secod etry of, i.e., are eamied while, all the other etries stay uchaged. is the replaced by the, symbol with the miimum cost ad the resultig matri is called.his procedure repeats util the ( ) th etry of matri is also tested ad the fial matri ( ) is obtaied. is cosidered as the best cadidate or the trasmitted sigal. B. ompleity I SF detectio P possible cadidates are tested at the first etry of ad for each of the ( ) remaiig etries, P ew cadidates are tested as well. herefore the total umber of cadidates eamied for their cost i the cost fuctio of E. () is: P [( ) ] ( P ) = ( P ) (4)

3 b / b a..5.5 M P N L a / - d c O Fig. he rectagular -dimesioal lattice of cadidates. As it is observed, the SF detectio scheme reduces the compleity dramatically while for the joit MLSE detectio all the cadidates have to be tested. It is also oted the compleity of SF detectio icreases liearly with the icrease i the umber of trasmit ateas ad frame legth.. Performace Degradatio Despite havig much less compleity compared to the thorough search of ML detectio, SF detectio sometimes misses the ML solutio which leads to performace degradatio. I order to clarify this disadvatage, we ow provide a eample with geometric iterpretatio. Eample : For a system with = m =, P =, ad = there are P = 4 cadidates i total. he modulatio is assumed to be BPS. Hece the trasmit sigal matri has the dimesio of =. All the possible trasmittig matrices are: a = [,], b = [,], c = [, ], d = [, ] (5) where represets the traspose operatio. I this case, all the trasmittig cadidates lay o a rectagular - dimesioal discrete lattice Z. I Fig., this lattice is show. I real commuicatio applicatios with N trasmit ateas ad M receive ateas, where the chael matri coefficiets are assumed to be comple, the H ca be represeted as a ( m = M ) ( = N) real chael matri. For the case of simplicity ad represetig the skewed lattice i the -dimesioal space, without loss of geerality it is assumed that all the chael coefficiets are real. So the chael matri H is a m = real chael matri. If it is assumed that the coefficiets are comple the the skewed lattice would have the dimesio of m = M = 4 which is impossible to depict. Based o this assumptio, the skewed lattice has the dimesio of m = ad i fact is a parallelogram. I Fig.3 this parallelogram for a typical chael matri.3. H = is show. he poits a, b, c, d from the..5 rectagular lattice are mapped to a, b, c, d,respectively d / Fig.3 he skewed lattice of H for a real chael matri. I Fig.3, the dotted lies crossig the poits P, M, P, M are the perpedicular bisectors of the four sides. he dash-dotted lies P P, M M coect the mid-poits of the opposite sides ad the lie LL is the perpedicular bisector of segmet a d. here are areas that if the received sigal Y falls i, the the SF detectio may miss the ML solutio. I what follows, we preset those eamples. Eample - If the received sigal Y falls ito the triagle OL M, the Y is uatized to a which correspods with a i the rectagular -dimesioal lattice of cadidates. herefore a is cosidered as the uatized least suare solutio, a = = Quat( H Y ). I this case if the SF detectio is applied, oly ( P ) = 3 cadidates out of four cadidates would be eamied for the miimum cost. Symbol (bit) flippig at the first etry of a = [,] goes to cadidate b = [, ] which has higher cost compared to the cost of a ( b is further toy ) so = a. Symbol flippig at the secod etry of goes to cadidate c = [, ] which agai has higher cost compared to the cost of a ( c is further toy ) so = a ad is cosidered as the trasmitted cadidate. Sice the received sigal Y is at the left side of perpedicular bisector L L so d is the closest poit ad therefore d is the ML solutio. Hece, although there is a cadidate with the miimum cost amog the three cadidates ( a, b, c), but it is ot the ML solutio ad the closest poit is missed. Eample 3- If the received sigal Y falls ito the triagle OM N, the Y is uatized to b which correspods to b i the rectagular -dimesioal lattice of cadidates. herefore b is cosidered as the uatized least suare solutio, b = = Quat( H Y). Symbol flippig at the first etry of b = [,] goes to cadidate a = [,] that has lower cost ( a is closer to Y ) so = a. Symbol flippig at the secod etry of L M N P c /

4 goes to cadidate c [, ] to the cost of a ( c is further toy ) so = which has higher cost compared = a ad is cosidered as the trasmitted cadidate (the searchig path goes through the cadidates ( b, a, c) ). But if the symbol flippig starts from the secod etry of b ad eds to its first etry the the fial solutio would be differet. I this case, symbol flippig at the secod etry of b goes to cadidate d = [, ] that has lower cost, so = d ad symbol flippig at the first etry of goes to cadidate c = [, ] which has higher cost compared to the cost of d ( c is further toy ) so = d ad is cosidered as the trasmitted cadidate (the searchig path goes thorough the cadidates ( b, d, c) ). I aother epressio, startig the symbol flippig procedure from ay etry ofb, iitiates a ew searchig path thorough the lattice of cadidates. It should be oted that d is the closest poit toy. As such the SF detectio scheme may miss the closest poit if the symbol flippig procedure starts from the first etry of. herefore, the priority of startig the symbol flippig procedure from the etries of is essetial, i.e., startig from ay etry may lead to a differet searchig path. o summarize, the major disadvatages of the SF detectio scheme are: - his scheme is sesitive to the locatio of i the lattice of cadidates. If is too far from the ML solutio i the lattice, the the SF detectio is ot able to fid the ML solutio ad the closest poit is missed (eample ). - here are cases i which startig the symbol flippig procedure from ay etry of leads to a differet poit with miimum cost. We call this poit local miimum poit; however, the resultig poit may ot be the closest lattice poit (the ML solutio). We call the closest poit global miimum poit. I fact, global miimum poit has the miimum cost amog all the lattice poits (eample 3). IV. MODIFIED SYMBOL FLIPPING DEEION I this sectio, a modified SF detectio is proposed i order to rectify the metioed disadvatages i the previous sectio. he proposed SF detectio is the replaced with the MLSE used i the layered-group detectio scheme itroduced i [] for MIMO freuecy selective fadig chaels. As it is eplaied i [], if it is assumed that at each stage substreams are detected, the the euivalet system has ( L ) trasmit ateas where L is the umber of chael taps. Although the major disadvatage of usig the ullig, detectio, ad cacellatio structure is the eed for several receive ateas ( m ( )( L ) ) [], the umerous imagiary trasmit ateas create a substatial trasmit diversity which ca be eploited. I the other had, sice all the L trasmit ateas of a substream trasmit the delayed versios of that substream [], there is a correlatio amog all the L trasmit ateas. So the trasmittig matri for substream s i -th trasmitted by the group of L ateas is: si s i = < L > si < L> < L > which has the dimesio of ( L ) ( L) where s i is the trasmitted substream ad is the all zero row vector with a k legth k []. As it is eplaied i [], i order to avoid iter frame iterferece a guard iterval of L symbol period must be iserted (Zero Paddig).It is oticed that each of the rows of s i cotais the same origial symbol vector s i with a delay {,,, L}. Fially, the etire trasmitted symbol matri for substreams has the dimesio of ( L ) ( L) with the followig format []: g = s s s s si < L > s < L > where g is the first group of substreams to be detected ad cotais the s -th to s -th substreams, g = [ s, s,, s]. It is observed that both matrices of euatio (6), (7) have a kid of diagoal format. he modified SF detectio procedure takes the advatage of this diagoal format. I aother words, by eploitig the format of the trasmitted matri, the locatio of the iserted zeros i the rows of the trasmitted matri, ad the L degrees of trasmit diversity at the receiver, a attempt is made to make the iitial poit i the lattice of cadidates eactly determied. I this case, the matri Least = Quat( H Y ) has the dimesio of ( L ) ( L). By removig the etries of that correspod with the iserted Least zeros, the resultig matri, Z, is of dimesio of ( L ). Due to the uatizatio, the L rows of a substream i are Z ot the same ad may be differet. Sice each group of L trasmittig ateas related to a substream, trasmit the same substream but with differet delays, there are totally ( L ) poits that ca be take ito accout as the iitial poit. I aother epressio, for each substream there are L choices from the group of L trasmittig ateas, therefore i the matri Z s (6) (7)

5 there are ( L ) ( L )...( L ) = ( L ) combiatios from times substreams that ca be cosidered as the iitial poit o the lattice of cadidates. Note that each of these combiatios form a matri with dimesio ( L ) that all the L correspodig rows of a substream are the same. Isertig zeros to their previous locatios leads to the trasmittig matri,. his matri is actually the Zero Forcig () solutio with dimesio ( L ) ( L) that is cosidered as oe possible iitial poit o the lattice of cadidates. By calculatig the cost ( Y H ) of all ( L ) possible solutios (iitial poits), the combiatio with the miimum cost is chose as the best iitial poit ad is called.,, best Oce, the iitial poit has bee chose, the SF detectio starts. As it was previously eplaied, startig the SF procedure from ay etry of the matri leads to a,, best local miimum which may ot be the ML solutio. herefore, the priority of startig from ay etry of is,, best importat. o mitigate the secod disadvatage through the modified SF detectio, the SF detectio is repeated times. It should be oted that although there are ( L ) ( L) etries i, at each row there are L zeroes ad all the L,, best rows of a substream have the same o-zero etries. herefore, there will be o-zero ad distict etries to be flipped. I this case, each time the SF procedure starts from oe of the distict o-zero etries of ad tests,, best all etries of. For istace, the first time it,, best starts from the first etry ad eds i the -th etry; et at the secod time it starts from the secod etry ad eds i the first etry of. his procedure repeats,, best times. Oce the procedure is fiished, i the worst case, if all the local miimum poits are differet, there will be local miimum poits. Net amog all the local miimum poits the poit with miimum cost is chose as the fial solutio. I aother epressio amog all the local miimum poits, the poit with the miimum cost is cosidered as the trasmitted sigal. Oce the detectio of substreams i the first group is doe, the cotributio of the detected substreams ad their delayed elemets are the subtracted out from the received sigal ad the substreams of the secod group are detected. V. ompleity of Modified Symbol Flippig Detectio I the modified SF detectio the procedure is performed for additioal ( ) times. I every additioal time the umber of ew cadidates tested is: ( P ) (8) herefore, the total umber of cadidates tested i the modified SF detectio has the followig upper boud: ( P ) [ ( P = ( ) ( P ) ) ] ( ) Sice some of cadidates are tested more tha oce, the umber of cadidates tested is fewer tha the metioed upper boud. Although this may lead i reducig the efficiecy of the scheme, the compleity is still of a polyomial of degree ad is much less tha the epoetial compleity imposed by the ML detectio. Hece, based o the iformatio of havig a diagoal trasmitted matri ad the kowledge of the locatios of the iserted zeros at the receiver ad also eploitig the trasmit diversity of degree L, the best iitial poit is chose (first,, best modificatio). he by employig the SF detectio startig from all the o-zero ad distict etries of, the,, best probability of missig the ML solutio is decreased (secod modificatio). Note that although for the first modificatio all the ( L ) combiatios of the iitial poit are tested, i real MIMO freuecy selective systems <<. his umber is several times smaller tha (9). As such, the total compleity of the modified SF detectio scheme is still much less tha that of the optimal ML detectio. VI. SIMULAION RESULS I this sectio, the performace of the proposed scheme i freuecy selective fadig chael is simulated usig Mote-arlo simulatios ad the pair wise error probability is evaluated. he bad pass modulatio scheme used is QPS. he chael has = 6 trasmit ateas, three symbol spaced paths ( L = ) ad their gais are assumed to be idepedetly Rayleigh faded ad uasi-static. All chael path gais have eual eergy, ad also eual variace per dimesio. he simulatios are accomplished for group detectio structure for =, 3. Accordig to [], the miimum umber of receive ateas is m ( ) ( L ). I order to compare the results i idetical situatio, the umber of receive ateas is fied for the case of = ; so m = ( 6 ) ( ) = 3. I Fig.4 the performace is show for a ( 6,3) system with group detectio =. Fig.5 shows the performace for the same MIMO system with group detectio = 3. he performace of the modified SF detectio is compared to the covetioal SF detectio, Zero Forcig () detectio ad ML detectio. I the simulatio results represets oe of the ( L ) solutios as,, with Fist Modificatio represets solutio, SF with First Modificatio is the same as,, best covetioal SF detectio after fidig the best iitial poit, ad fially SF with First & Secod Modificatio is our,, best proposed scheme as modified SF detectio.

6 - covetioal SF detectio for more tha 6 db for = 3 at the 5 frame error rate of. It is oticeable that because of the idea of imagiary ateas, o additioal cost was imposed at the trasmitter side i order to create the trasmit diversity. P e Frame - -3 SF with First Modificatio -4 SF with First Modificatio ML SF with First & Secod Modificatio SNR Fig. 4. Performace compariso for group detectio (=) for a (6, 3) system ad 3-tap Rayleigh fadig chael P e Frame with First Modificatio SF SF with First Modificatio SF with First & Secod Modificatio ML SNR Fig. 5. Performace compariso for group detectio (=3) for a (6, 3) system ad 3-tap Rayleigh fadig chael It ca be see that a icrease i, leads to a icrease i the performace differece betwee the SF detectio scheme ad the ML detectio scheme. However the SF scheme has much less compleity compared to the ML detectio whe icreases. It ca also be see that a icrease i leads to a better performace for all types of detectio. It is also observed that the first modificatio o the covetioal SF detectio is more effective o improvig the performace compared to the secod oe. his cofirms the fact that the SF detectio scheme is very sesitive to the locatio of the iitial poit o the lattice of cadidates. he sesitivity of the performace to the locatio of the iitial poit is also obvious by comparig the performace curve of with First Modificatio with the curve. he icrease i the performace after applyig the first modificatio is resulted from the trasmit diversity of oe real ad L imagiary trasmit ateas for each substream. I the simulatio results the performace improvemet of the SF detectio after applyig both modificatios ca be easily see so that the modified SF detectio outperforms the VII. ONLUSIONS he advatages ad disadvatages of the SF detectio were eplaied ad the the modified SF detectio was proposed i order to rectify the covetioal SF detectio defects. I the proposed detectio scheme two modificatios were applied to the covetioal SF detectio. Although the modified SF detectio scheme has a higher compleity (of order ) tha that of covetioal SF detectio (liear), it still has a relatively low compleity i compariso with the optimal ML scheme ad outperforms covetioal SF by 6 db. Ackowledgemet his work was supported i part by Qatar Natioal Research Fud (QNRF), Qatar, ad i part by Qatar elecom (Qtel), Qatar REFERENES [] Y. Wu ad S. Y. ug, Sigal detectio for MIMO-ISI chaels: A iterative greedy improvemet approach IEEE ras. Sigal Processig, vol 5, o 3, pp. 73-7, March 4. [] D... So ad R. S. heg, Layered maimum likelihood detectio for MIMO systems i freuecy selective fadig chaels IEEE ras.wireless omm., vol. 5, o 4, pp , Aug. 6. [3] G. J. Foschii ad M. J. Gas, O the limits of wireless commuicatios i a fadig eviromet whe usig multiple ateas, WirelessPersoal ommuicatios, vol., o. 3, pp , Mar [4] G. J. Foschii, Layered space-time architecture for wireless commuicatio i a fadig eviromet whe usig multiple ateas, Bell Laboratories echical Joural, vol., o., pp. 4-59, Autum 996. [5] P. W. Woliasky, G. J. Foschii, G. D. Golde, ad R. A. Valezuela, V-BLAS: A architecture for realizig very high data rates over rich scatterig wireless chael, i Proc. URSI Iteratioal Symposium o Sigals, Systems, ad Electroics, Sep. 998, pp [6] G. J. Foschii, G. D. Golde, R. A. Valezuela, ad P. W. Woliasky, Simplified processig for high spectral efficiecy wireless commuicatio employig multi-elemet arrays, IEEE J. Select. Areas ommu., vol. 7, pp , Nov [7] N. Boubaker,. B. Letaief, ad R. D. Murch, Performace of BLAS over freuecy selective wireless commuicatio chaels, IEEE ras.ommu., vol. 5, pp , Feb.. [8] N. Rahimia, ad A. Iravai, Reduced compleity ovel orderig for group maimum likelihood detectio i MIMO freuecy selective fadig chaels i Proc. eth IEEE Iteratioal Symposium o Spread Spectrum echiues ad Applicatios (ISSSA 8) August 8,,pp [9] B. Hassibi ad H. Vikalo, O the sphere decodig algorithm part I: Epected compleity, IEEE ras. Sigal Processig, vol 53, o 8, pp.86-88, August 5.