QR based MIMO Receivers: A Survey of Complexity ad Performace M.Karthikeya a ad D.Saraswady b, * a Podicherry Egieerig College, Puducherry. Abstract Multiple Iput Multiple Output (MIMO) becomes a key techology to achieve a higher data rates i wireless commuicatio. The theoretical limit of MIMO capacity ca be achieved by icreasig the umber of ateas at the trasmitter ad receiver. While icreasig the umber of ateas, receiver desig becomes very much challegig. Maximum Likelihood Detector (MLD) provides a optimal performace oly if the umber of ateas ad modulatio order is small, Otherwise it is impossible to implemet i real time. I this paper, we ivestigated maily QRM-MLD, LRAD ad SD based o Bit Error Rate (BER) ad complexity to compare with MLD. Our compariso will be helpful i uderstadig the complexity performace trade-off ivolved i various methods. Ad the diversity offered by various methods is discussed. This survey helps i choosig a appropriate method for the required chael coditios. Keywords: MIMO, MLD, SD, LRAD, QR, complexity, BER, SIC, ZF, MMSE. 1. Itroductio I wireless commuicatio, the demad of higher data rate with higher Quality of Service is growig day by day. Employig multiple ateas at both the trasmitter ad receiver is oe of the key techologies to achieve the higher data rate. The capacities of Multiple Iput Multiple Output (MIMO) systems grow liearly with the miimum of umber of trasmit ad receive ateas [1]. MIMO system ca achieve high spectral efficiecy ad higher data rate without the expese of badwidth []. owever the success of MIMO system maily depeds o the receiver desig. The receiver must be capable of idetifyig the trasmitted symbol accurately. I desigig the MIMO receiver, the trade-off betwee complexity ad performace is oe of the mai cosideratios. The geeral ad foremost problem to solve is elimiatio of Multi Stream Iterferece (MSI), where the multiple streams from the idividual trasmit atea iterfere with each other i the desig of receiver [3]. The Multi Stream sigal ca be detected i MIMO usig liear ad No-Liear processig of the received sigal. Liear equalizatio schemes such as Zero-Forcig (ZF) ad Miimum Mea Squared Error (MMSE) are preferred for their less complexity [4]. But these methods ted to suffer from poor Bit-Error Rate (BER) performace. Ad also these methods performace are very much iferior to Maximum Likelihood detector, which serves as a referece of optimum detectio. The No-Liear receiver amed Zero Forcig - Successive Iterferece Cacellatio (ZF-SIC) with optimal orderig was proposed i [5] superior to liear receivers. Later SIC method was exteded to MMSE receiver [6]. But SIC ted to suffer from error propagatio. The optimal receiver structure is the Maximum likelihood (ML) detector exhibits higher performace by computig the Euclidea distace betwee the received vector ad the etire possible trasmitted vectors ad choosig the oe with less distace. This exhaustive search complexity makes this detector impossible to be realized i practice for more tha two ateas at the trasmitter ad receiver. This complexity performace trade-off i receiver desig motivated the research towards sub-optimal receivers with reasoable complexity ad performace close to ML receivers. The three most popular methods with QR decompositio for achievig ear ML performace are cosidered i this paper for performace aalysis. First oe is sphere decodig (SD) [7] ad its variats such as LSD [8], FCSD[9] ad LISS[10] etc. Secod oe is QRM-MLD based o QR decompositio ad M parameter was proposed i [11]. With this method 1 Gb/s trasmissio speed was achieved i field trials [1]. * E-mail: karthikeyafu@gmail.com,dsaraswady@pec.edu 14
Third Lattice Reductio (LR) based detectio methods are proposed i [13],[14]. The LR orieted receivers ca achieve the full diversity. The rest of this paper is orgaized as follows. Sectio II describes the system model ad liear receivers. Sectio III describes about o-liear methods. Performace aalyses are carried out i sectio IV. Sectio V cocludes the paper.. Problem Defiitio Ad Overview Of Liear Receivers The received sigal of MIMO system with N T trasmit ad receive ateas is give by y x (1) Where is NT x N T chael matrix, x is NT x 1 trasmitted symbol vector, y is NR x 1 received symbol vector, is the complex oise vector whose compoets are idepedet ad idetically distributed (i.i.d.) Additive White Gaussia Noise(AWGN). This equatio ca be represeted i block as show i fig.1. The aim of ay receiver is solve (1) ad to fid a maximum likelihood of x. The simplest way of solvig the above defied problem is to make use of liear receivers such as Zero-Forcig (ZF) ad Miimum Mea Squared Receiver (MMSE). AWGN squared Euclidea distace betwee the estimate ˆx ad the origial received vector y. The estimate is obtaied usig 1 where W [ N ] MMSE xˆ W y (3) MMSE 0 Apart from the N0I term both the ZF ad MMSE are comparable. At high SNR MMSE equals ZF. 3. No Liear Receivers 3.1. Successive Iterferece Cacellatio Receiver This is the oe of the simple o liear method works by cacellig the effect of previously detected symbol i successive decodig. A improved versio of this was proposed i [16] with optimal orderig, where SIC is doe from strogest to weakest sigal thereby esurig less error propagatio. 3.. Sphere Decoder ML decoder works by searchig the etire vector costellatio for a vector havig small Euclidea distace with the received vector, but this process is achieved with NP-ard complexity. To reduce the complexity sphere decodig was proposed for MIMO ML receiver [17]. I SD searchig is limited to a subset of origial poits cotaied withi the sphere of radius z as show i fig.. x y z Fig. 1. MIMO system model.1. Zero Forcig Receiver The simplest possible MIMO detectio is to ivert the chael impulse respose. The ucostraied least square solutio ca be solved i ZF usig ˆx y () 1 where ( ) is the pseudo-iverse of. This method of detectig is kow as ZF equalizatio [15]... Miimum Mea Square Error Receiver The ZF ca be elimiatig the Iter Symbol Iterferece (ISI) ot the oise but the ultimate goal of ay receiver is to reduce the oise. The MMSE receiver is used to elimiate the oise ehacemet. It works by miimizig the average Fig.. Geometrical represetatio of sphere decodig. The two mai issues eed to be solved for implemetig SD are choosig a proper radius z ad idetifyig those poits lyig iside sphere. If we choose large z, the it will result i sphere with larger lattice poits ad results i high complexity. If we choose small z the o poits will lie iside the sphere. The above metioed problem ca be solved usig iterative process as follows. The SD problem ca formulated as 15
xˆ arg mi x z y x (3) where z is the subset of lattice poits. is the chose costellatio poits. Iitially pre-processig of chael matrix used to be doe usig QR decompositio for makig more orthogoal. The QR decompositio is expressed as R Q 0 Where R is NT x N T upper triagular matrix. 0 is ( NR NT ) x N T zero matrix ad Q is NR x NR orthogoal matrix. After applyig QR decompositio, (1) becomes Multiplyig by (4) y QRx (5) T Q yields The (3) ca be writte as (6) T T T y Q y Q QRx Q xˆ arg mi x z y x (7) z 1 z 1 xˆ ˆ 1 x x 1 x1 x r 1, 1 r 1, 1 (1) I the same way if we cotiued for ad so o to, we will obtai all the poits withi the sphere. 3.3. Lattice Reductio Aided Decoders (LRAD) The performaces of liear receivers suffer due to the orthogailty defect of the chael matrix. Liear receivers such as ZF ad MMSE will give good bit error rate performace if ad oly chael realizatio is orthogoal. Lattice reductio helps i achievig perfectly orthogoal, thereby we ca use the low complexity liear receivers to achieve high bit-error rate performace. Lattice reductio is used to achieve a more orthogoal i LR receivers. Various LR techiques ca be available ad a survey of this ca be foud i [18]. Amog these the most popular oe is LLL, which is havig good trade-off betwee results ad complexity. The system model of MIMO system with LR aided receivers ca be explaied with the block show i fig.3. AWGN The SD radius z ca be calculated as z y Rx (8) The Eq.(8) guaratees that the existece of at least a sigle lattice poit iside the sphere cetred at y. The solutio of (7) ca be obtaied usig tree search process usig iterative algorithm i SD. The radius ca be calculated usig ucostraied least square solutio proposed i [1] as follows. Where z R x x z z y sˆ ( ˆ ) (9).The eq. (9) ca be expaded as z r ( x x ) r ( x x r ( x x ))... 1,, ˆ 1, 1 1 ˆ 1 ˆ r 1, 1 ad the itervals of x ca be writte as (10) z z xˆ ˆ x x (11) r, r, Where. ad. represets the earest larger ad smaller elemets i the lattice. The for every x falls betwee the itervals i (11) fid x 1 by usig the iterval give below z Fig. 3. System model of LR Aided Decoder The the system model (1) ca be rewritte as y z (13) 1 where T, z T x ad T is a uimodular matrix. Both the x ad z are describig the same lattice poit but is more orthogoal whe compared to [19]. After this redefiitio of system model the zero forcig equalizatio will be applied to (13) as follows. ẑ y (14) From (14) the trasmitted sigal ca be detected as xˆ TzˆZF. The extesio of LR orieted detectio ca be exteded to MMSE by reducig the exteded chael matrix [ ] T wi ad the applyig () to the received vector [0]. Whe comparig to LR-ZF, LR y 16
S. No Receivers Table. 1- Compariso of MIMO Receivers Approximate BER values Low SNR (5 db) igh SNR (15 db) 1 ZF 0. 0.03 MMSE 0.08 0.00 Diversity order - N T +1 - N T +1 Very less Very less Complexity 3 ML 0.03 0.00001 NP-ard expoetial 4 SD 0.03 0.00004 Variable ad depeds o the radius 5 FCSD 0.03 0.00004 Fixed ad less tha SD 6 LRAD-ZF(LLL) 0. 0.003 7 LRAD-MMSE(LLL) 0.08 0.0007 8 LRAD-OSIC-ZF(LLL) 0. 0.000 9 LRAD-OSIC-MMSE(LLL) 0.08 0.00001 10 QRM-MLD(M=4) 0.3.003 11 QRM-MLD(M=16) 0.03 0.00001 Approaches ML sice M is high deotes oly at special case diversity order of NR is achieved. i.e NR NT 6. BER of 0. deotes bit error per 10 bits trasmitted ad all the approximate values are take by simulatig N N 4 with 4-QAM R T MMSE complexity is less, because the exteded chael matrix is more coditioed tha the ordiary chael matrix. 3.4. QRM-MLD This method is based o M-Algorithm predicts the trasmitted sigal by pre-processig the received sigal usig QR Decompositio (QRD). The method discussed i previous sectios also usig the same pre-processig. The differece is QRD was applied for real systems for those methods, but here it is applied for complex chael matrix as proposed i []. After QRD the received sigal is represeted as y x y QRx Q ( y QRx) y Rx (15) The above equatio ca be expaded to fid out the first layer metric values as follows NT NT, NT NT, i y r.ˆ x (16) where xˆ NT, i deotes all possible symbols i the chose costellatio with size (C). ece C times eq.(16) is evaluated ad values are calculated respectively. From that values M lowest metric values producig symbols are cosidered for the ext layer. Agai xˆ N T 1 ca be determied by the followig equatio y r.ˆ x xˆ (17) NT 1 NT 1, NT 1 NT 1, i NT, im where xˆ N 1, i represets all possible symbols i T costellatio. xˆ NT, i M represets all the M symbols. The M x C values are geerated ad agai M symbols correspodig to M lowest values are chose ad cosidered for the ext layer. This process is repeated Oe cadidate producig smallest metric value is chose ad i each layer from M symbol oe will be chose with lowest metric value. 4. Performace Compariso Of MIMO Receivers 4.1. BER ad Complexity trade-off The mai metric of ay receiver structure is Bit Error Rate (BER) performace. From the above discussios it is clear that No-Liear receivers exhibit higher bit error performace tha liear receivers because they suffers from less oise ehacemet due to miimum sigular value ( mi ). This effect for ZF ad MMSE receivers are give by [3] as 17
E E ZF (18) Z mi. (19) Z mi MMSE ( Z mi ) where Z is the statistical iformatio of oise. From (18) ad (19) it is very much clear that MMSE suffers from less oise ehacemet. O the other had whe cosiderig o liear receivers, ML receiver is the optimum receiver structure. It results i less BER ad serves as a referece to other sub optimal receivers. Due to its expoetial complexity, desig of receivers with applicable complexity is eeded. Three most successful methods i this directio are SD, LRAD ad QRM-MLD which we already discussed i the previous sectio. From the table it is almost clear that the sphere decodig almost approaches ML performace, but its complexity is variable ad at sometimes equal to ML. FCSD [9] was proposed to fix the complexity of sphere decoder. O the other had LRAD has provided less complexity sice after LR pre-processig liear detectio methods is employed. LRAD-OSIC- MMSE is the method of LRAD with successive iterferece cacellatio also achieves almost equal to the ML performace i the high SNR with less complexity. Whe cosiderig QRM-MLD, its performace ad complexity depeds o M. Whe M icreases, its performace also gets icreased at the expese of complexity. For example whe M=16 yields the same performace as like SD. But the advatage i choosig QRM-MLD is its fixed complexity whereas i SD, it depeds o SNR ad chael coditio umber. Due to this reaso QRM-MLD is simpler for real time implemetatio. 4.. Diversity order igher diversity ca be achieved by providig receiver with multiple idepedet copies of same sigal. Diversity order is a measures the umber of idepedet paths over which the data is received. The liear receivers always exhibit a poor diversity order of oe. But i joit spatial ecodig MIMO systems MMSE exhibits higher diversity order i low spectral efficiecies [5]. Whe comparig with the liear receiver, No liear receivers will provide higher diversity order. ML receiver provides a diversity order of N R, whereas suboptimum receivers achieves less tha that of ML with reasoable complexity. I some special cases these receivers achieves full diversity order. I [5] SD with fixed complexity achieves full diversity was proposed. With NR NT 6 LRAD provides full diversity order i all cases such as LRAD-ZF ad LRAD- MMSE [18]. 5. Coclusio ad Future work I this paper we preseted a comparative study of various MIMO detectio methods based o QR decomposed chael matrix. Table1. gives the differece i performace ad complexity of the most popular methods. Amog the methods discussed i this paper,lrad provides ear ML performace with acceptable complexity for real time implemetatio. Whe cosiderig LRAD, may ope problems are available for researchers such as ivestigatio of imperfect chael estimatio ad extesio of hard decisio to the soft output case. 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