IN an effort to increase the achievable system capacity

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1 Miimum Bit Error Rate Mutiuser Detectio i Mutipe Atea Aided OFDM M. Y. Aias, A. K. Samiga, S. Che, L. Hazo Dept. of ECS., Uiv. of Southampto, SO7 BJ, UK. Te: , Fax: Emai: h@ecs.soto.ac.uk, Abstract I this cotributio, we propose a Miimum Bit Error Rate M mutiuser detector for Space Divisio Mutipe Access SDMA aided Orthogoa Frequecy Divisio Mutipexig OFDM systems. It is show that the M detector outperforms the Miimum Mea Squared Error MMSE detector, sice the M detector directy miimizes the, whie MMSE detector miimizes the mea-squared error MSE, which does ot guaratee achievig the miimum. Whe supportig two users, the proposed M scheme substatiay outperforms the cassic MMSE arragemet i the ivestigated propogatio sceario. I. INTRODUCTION IN a effort to icrease the achievabe system capacity of a OFDM system, atea arrays ca be empoyed for supportig mutipe users i a Space Divisio Mutipe Access SDMA commuicatios sceario [-4]. The beefit of this system is that i case of empoyig a sufficiety high umber of receiver ateas at the base statio, the degree of freedom provided by the umber of base statio receiver ateas ad L umber of trasmit ateas is higher tha ecessary for supportig L umber of simutaeous users. Hece, the remaiig degrees of freedom aow us to icrease the achievabe receiver diversity gai of the system ad therefore cotributes towards improvig the system s trasmissio itegrity. A variety of iear mutiuser detectors have bee proposed for performig the separatio of OFDM users based o their uique, user-specific, spatia sigature provided that their chae impuse respose was accuratey estimated [,4]. The most popuar desig strategy is costituted by the miimum mea-squared-error MMSE mutiuser detector MUD. However, as recogised i [-8], a better strategy is to choose the iear detector s coefficiets so as to directy miimize the error-probabiity or bit-error rate, rather tha the mea-squared error MSE. This is because miimizig the MSE does ot eccessariy guaratee that the of the system is aso miimized. The famiy of detectors that directy miimizes the is referred to as the miimum bit-error rate M detector [9,]. I this cotributio, we wi ivestigate the performace of the proposed M iear MUD i the cotext of a upik SDMA/OFDM system. II. SACE DIVISION MULTILE ACCESS SYSTEM MODEL The so-caed SDMA system is capabe of differetiatig L users trasmitted sigas at the base-statio BS ivokig their uique, user-specific spatia sigature created by the chae trasfer fuctios or chae impuse resposes CIR betwee the users sige trasmit atea ad the differet receiver ateas at the BS [], [4]. Figure portrays the atea array aided upik trasmissio sceario cosidered. I this figure, each of the L simutaeous users is equipped with a sige trasmissio atea, whie the receiver capitaizes o a - eemet atea frot-ed []. The set of compex sigas, x p [, k], p,..., received by the -eemet atea array i the k-th subcarrier of the -th OFDM symbo is costituted by the superpositio of the idepedety faded sigas associated with the L users sharigs the same space-frequecy resource [4]. The received siga was corrupted by the Gaussia oise at the array eemets. The idices [, k] have bee omitted for otatioa coveiece durig our forthcomig discourse, yiedig [4]: x = Hs + = x +, where the -dimesioa vector x of the received sigas, the vector of trasmitted sigas s ad the array oise vector, respectivey, are give by: x = x, x,..., x T, s = s, s,..., s L T, 3 =,,..., T. 4 Furthermore, x represets the oiseess compoet of x. The frequecy domai chae trasfer fuctio matrix H of dimesio L is costituted by the set of chae trasfer fuctio vectors of the L users: H = h, h,..., h L, each of which describes the frequecy domai chae trasfer fuctio betwee the sige trasmitter atea associated with a particuar user ad the receptio array eemets p,..., : h = h, h,..., h T. 6

2 User b Moduator s H H H Chae + x s ^ b ^ User User L b b L Moduator Moduator s L s H H H H L H L H L + + x x Mutiuser Detector ^ s ^L s b^ b^ L Fig.. Schematic of a atea array aided OFDM upik sceario, where each of the L users is equipped with a sige trasmit atea ad the BS s receiver is assisted by a -eemet atea frot-ed. The compex data siga, s, trasmitted by the -th user,,..., L ad the AWGN oise process, p, at ay atea array eemet p, p,..., are assumed to exhibit a zero mea ad a variace of σ ad σ for the data siga ad AWGN oise process, respectivey. The frequecy domai chae trasfer fuctios, H p of the differet array eemets p,..., for users,..., L are idepedet, statioary, ad compex Gaussia distributed processes with zero-mea ad uit variace. For iear mutiuser detectors, the estimate ŝ of the trasmitted siga vector s of the L simutaeous users is geerated by ieary combiig the sigas received by the differet atea eemets at the BS with the aid of the array weight matrix W, resutig i: ŝ = W H x. 7 By substitutig Equatio ito Equatio 7 ad cosiderig the -th user s associated vector compoet, we wi arrive at: ŝ = w H x, = w H Hs + w H = s + w H, L = w H H s + w H H i s i + w H, 8 i=,i where the weight vector w is the -th coum of the weight matrix W. The first term of Equatio 8 refers to the desired user s cotributio, whie the secod ad third term represet the iterferig users cotributios ad the Gaussia oise, respectivey. At the curret state-of-the-art, the most popuar MUD strategy is the MMSE desig, where w is chose as the uique vector miimizig the MSE expressed as MSE = E[ŝ s ], amey as [4]: w MMSE = HH H + σ I H, 9 where H is the -th coum of the system matrix H. III. ERROR ROBABILITY IN A BSK SYSTEM I this paper the term ad probabiity of error E are used iterchageaby. The ecoutered at the output of the MUD w of user may be expressed as []: w = r[sgb s w < ], = r[z < ], where z is the siged decisio variabe give by: z = sgb s w. The robabiity Desity Fuctio DF of the decisio variabe z is costituted by a mixture of the Gaussia distributio associated with each possibe combiatio of the trasmitted data symbos of a users. Uder the assumptio that a the oise-free siga states are equiprobabe, the DF of z is give by []: p z z = πσ w H w j= e z sgbj s j σ w Hw where is the umber of equiprobabe combiatios of the biary vectors of the L users, i.e. we have = L. Furthermore, s j, j,...,, deotes the oiseess siga at the output of the MUD reated to the -th user, whie b j, j,...,, is the trasmitted bit of user. The erroous decisio evets are associated with the area uder the DF curve i the iterva,, which is quatified as: E w = p z z ; w dz. 3 Upo usig the itegratio by substitutio techique ad itroducig the shorthad of y j = z sgb j s j σw H, 4 w the probabiity of error i Equatio 3 becomes: E w = π j= cjw exp y j dy j = Q[c j w ], j=,

3 Ampitude Symbo Idex Ampitude Symbo Idex where the step-size is represeted by µ, ad the update directio vector di at istace i is give by: di = w E [w i]. 9 I Equatio 9, w E [w i] is the gradiet of E [w i] with respect to w ad i idicates the iteratio idex. By expoitig the foowig idetity []: a CIR : user, atea b CIR : user, atea ct at fy dy = f[ct] ct f[at] at, Ampitude Symbo Idex c CIR 3: user, atea Ampitude Symbo Idex d CIR 4: user, atea Fig.. Four differet chae impuse resposes CIR recorded at the two receiver ateas for the two users supported. where c j w is give by: c j w = sgbj s j σ w H w = sgbj w H x j σ w H w,6 where x j, j,..., costitutes a possibe vaue of x defied i the cotext of Equatio. Note that the is ivariat to a positive scaig of the weight vector, i other words, the depeds oy o the vectoria directio of w, but ot o its magitude. IV. EXACT M MULTIUSER DETECTION The M soutio is defied as []: w M = arg mi w E w. 7 However, the compex, irreguar shape of the cost fuctio prevets us from derivig a cosed-form soutio for the M MUD weights. Therefore i practice a iterative strategy based o the steepest-descet gradiet method ca be used for fidig the M soutio []. Accordig to this method, the iear MUD s weight vector w is iterativey updated, commecig for exampe from the MMSE weights of Equatio 9, uti the weight vector that exhibits the owest is arrived at. I each step, the weight vector is updated accordig to a specific stepsize, µ, i the vectoria directio i which the cost fuctio decreases most rapidy, amey i the directio opposite to the gradiet of the cost fuctio give i Equatio. The steepest-descet gradiet agorithm that ca be used for fidig the M soutio is summarised as foows []: w i + = w i + µdi, 8 the gradiet of E w with respect to the MUD s weight vector w ca the be computed by: w E w = = = π j= e sgbj s j σ wh w sj c j w w σ e wh w sgb j πσ j= { } x j + w H w w H w x j w H w 3 w w H w H w I πσ w H w 3 e j= sj σ wh w sgb j x j. Observe i Equatio 6 that is idepedet of the magitude of the MUD s weight vector, ad the kowedge of the orietatio of the detector s weight vector is sufficiet for defiig the decisio boudary of the iear M detector. Therefore the M detector has a ifiite umber of soutios. It is desirabe i ay optimisatio probem to have a sige goba miimum. I the case of the proposed M, the MUD s goba miimum is foud by costraiig the detector s weight vector to have a uity magitude. This is achieved by itroducig the ormaisatio process i each iteratio accordig to: w = w w = w w H w. With the aid of this ormaisatio, the gradiet expressio of Equatio ca be simpified to []: w E w = πσ j= exp sj σ sgb j w s j x j, 3 where w is the MUD s ormaised weight vector evauated usig Equatio. Comparig the gradiet expressios of Equatio ad Equatio 3, we may cocude

4 a CTF : user, atea b CTF : user, atea Average MMSE Detector User M Detector User MMSE Detector User M Detector User Average SNR db Fig. 4. Average versus the average SNR expressed i db for the MMSE ad the M mutiuser detectors of user ad user supported by two receiver ateas usig 8 subcarrier OFDM commuicatig over the chae characterised by the CIRs ad CTFs show i Figure ad Figure 3, respectivey. c CTF 3: user, atea d CTF 4: user, atea Fig. 3. Chae trasfer fuctios CTF for the CIRs see i Figure a CTF, b CTF, c CTF 3, ad d CTF 4. that the costrait of Equatio imposed o the optimisatio probem of Equatio reduces the ifiite umber of M soutios to a sige soutio. I our previous discourse we assumed the expicit kowedge of the matrix H defied i Equatio. However, i practice H has to be determied o the basis of the chae impaired oisy vaue of x ad hece a umber of techiques have bee proposed i refereces [,8,9,] to this effect. V. RESULTS AND DISCUSSION I our quatitative ivestigatios we used the simpest possibe SDMA OFDM system supportig two users with the aid of two receiver ateas. As show i Figure, each user has a uique chae trasfer fuctio CTF with respect to each receiver atea. The four correspodig CIRs are show i Figure ad the resutat CTFs are depicted i Figure 3. The CIRs represet a threepath idoor type chae [3], where o fadig is experieced. Correspodigy, the time-ivariat CTF ad CTF are ecoutered by user at the first ad secod receiver atea, respectivey. Simiary, CTF 3 is ecoutered at the first receiver atea ad CTF 4 at the secod by user. The OFDM modem had 8 subcarriers. I our simuatios, we iitiaised the iterative M agorithm to the MMSE MUD weights give by Equatio 9. The resuts of our simuatios are show i Figures 4, ad 6. The average of user ad user recorded i the cotext of both the MMSE ad M detector is portrayed i Figure 4. We ca see from this figure that user has a better average i cojuctio with the MMSE detector compared to user for SNRs i excess of about db. By cotrast, the M detector of user outperforms that of user i terms of the average. We ca aso see that the M detectors of both users have a substatiay ower average compared to the MMSE detectors. Agai, as expected, this is because the MMSE is directy miimisig the MSE ad ot the. We may aso ote that the average differece betwee the MMSE ad M detectors is ot the same for both users. Specificay, the M MUD of user has a SNR advatage of amost db, whie that of user has about db SNR advatage. This is a cosequece of the uique combiatios of the chae trasfer fuctios of both users, sice it ca be see i Figure that the CIR of user exhibits a ower ratio betwee the mai ad the deayed CIR taps tha that of user. I Figure ad Figure 6, we ca see that the of the MMSE ad M MUD is differet for every OFDM subcarrier. This is because the particuar combiatio of the CTFs is uique for the differet OFDM subcarriers. These CTF differeces wi resut i a time-variat system matrix, H, for each OFDM subcarrier, thus imposig a direct ifuece o the cacuatio of the MUD s weight vaues, as suggested by Equatio 9 ad Equatio 7 for the MMSE ad M MUD, respectivey. By comparig the pots of Figure ad Figure 6 recorded for user ad user respectivey, we ca see that the peaks of the dramaticay atteuated subcarriers of Figure 3 are substatiay higher for the MMSE MUD. VI. CONCLUSION I this paper, we have preseted the ove cocept of M OFDM mutiuser detectio that directy miimises the i a SDMA OFDM system. We have show that the M detector outperforms the MMSE detector, because the MMSE detector miimises the MSE, which does ot aways guaratee attaiig the miimum. We have aso show that sice differet users of a SDMA

5 Average SNR db Average SNR db a MMSE b M Fig.. versus the average SNR for every OFDM subcarrier for the a MMSE, ad b M mutiuser detector of user whe supportig two users with the aid of two receiver ateas usig 8 subcarrier OFDM commuicatig over the chae characterised with the aid of the CIR ad CTF show i Figure ad Figure 3, respectivey Average SNR db Average SNR db a MMSE b M Fig. 6. versus the average SNR for every OFDM subcarrier for the a MMSE, ad b M mutiuser detector of user whe supportig two users with the aid of two receiver ateas usig 8 subcarrier OFDM commuicatig over the chae characterised with the aid of the CIR ad CTF show i Figure ad Figure 3, respectivey. OFDM system wi experiece differet uique combiatios of the chae trasfer fuctios i the cotext of the differet ateas, their performace aso varied. This is aso true i the cotext of the subcarriers s due to the frequecy seective ature of the mutipath chaes ecoutered. Our future research wi study the iteractio of M MUDs ad chae codig, whe commuicatig over time-variat fadig chaes. REFERENCES [] B. Suard, G. Xu, H. Liu, ad T. Kaiath, Upik Chae Capacity of Space-Divisio-Mutipe-Access Scheme, IEEE Trasactios o Iformatio Theory, vo. 44, o. 4, pp , Juy 998. []. Vadeameee, L. Va Der erre, M. G. E. Eges, B. Gyseickx, ad H. J. De Ma, A Combied OFDM/SDMA Approach, IEEE Jouras of Seected Areas i Commuicatios, vo. 8, o., pp. 3 3, November. [3] H. Böcskei, D. Gesbert, ad A. J. auraj, O The Capacity of OFDM-Based Spatia Mutipexig Systems, IEEE Trasactios o Commuicatios, vo., o., pp. 34, February. [4] L. Hazo, M. Müster, B. J. Choi ad T. Keer, OFDM ad MC- CDMA, Joh Wiey ad IEEE ress, West Sussex, Egad, 3. [] A. K. Samiga, S. Che, ad L. Hazo, Adaptive Miimum Symbo Error Rate CDMA Mutiuser Detectio for use Ampitude Moduatio, i roceedigs of IEEE VTC Sprig, Jeju, Korea, 3, pp [6] X. Wag, W.-S. Lu, ad A. Atoiou, Costraied Miimum- Mutiuser Detectio, IEEE Trasactios o Siga rocessig, vo. 48, o., pp , October. [7] R. C. de Lamare ad R. Sampaio-Neto, Adaptive M Decisio Feedback Mutiuser Receivers i Frequecy Seective Fadig Chaes, IEEE Commuicatios Letters, vo. 7, o., pp. 73 7, February 3. [8] S. Che, L. Hazo, ad N. N. Ahmad, Adaptive Miimum Bit Error Rate Beamformig Assisted Receiver for Wireess Commuicatios, i roceedigs of IEEE Iteratioa Coferece of Acoustics, Speech ad Siga rocessig ICASS, Hog Kog, Chia, 3, vo. IV, pp [9] B. Mugrew ad S. Che, Adaptive Miimum- Decisio Feedback Equaizers for Biary Sigaig, EURASI Siga rocessig Joura, vo. 8, o. 7, pp ,. [] C.-C. Yeh ad J. R. Barry, Adaptive Miimum Bit-Error Rate Equaizatio for Biary Sigaig, IEEE Trasactios o Commuicatios, vo. 48, o. 7, pp. 6 3, Juy. [] M. Müster ad L. Hazo, Co-chae Iterferece Caceatio Techiques for Atea Array Assisted Mutiuser OFDM Systems, i 3G-, Lodo, UK, March, pp [] S. Che, A. K. Samiga, B. Mugrew, ad L. Hazo, Adaptive Miimum- Liear Mutiuser Detectio for DS-CDMA Sigas i Mutipath Chaes, IEEE Trasactios o Siga rocessig, vo. 49, pp. 4 47, Jue. [3] L. Hazo, W. Webb, ad T. Keer, Sige ad Muti-carrier Quadrature Ampitude Moduatio, Joh Wiey, New York,.