Minimum BER Transmission for TDD-CDMA in frequency-selective channels

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1 Miimum BER Trasmissio for TDD-DMA i frequecy-selective chaels Ralf Irmer, Wolfgag Rave ad erhard Fettweis Dresde iversity of Techology, Mommsestrasse 13, 0106 Dresde, ermay {irmer,rave,fettweis}@if.et.tu-dresde.de Abstract A ovel approach to Multiuser Trasmissio (MT) for the DMA dowlik i frequecy-selective chaels is proposed. The key idea is to miimize the BER at the receivers directly by pre-distortio of the trasmitted symbols. The BER ca be predicted i the trasmitter by calculatig the sigal at the receiver (icludig iterferece) exactly ad treatig oly the additive oise at the receiver statistically. sig state-of-the-art oliear optimizatio methods like SQP (Sequetial Quadratic Programmig), the BER miimizig pre-distortio coefficiets ca be foud. Simulatios compare the proposed scheme with the RAE matched filter ad liear trasmitter preprocessig schemes, like Joit Trasmissio. The proposed MT scheme ca be applied for istace i the 3PP-TDD ad the TD-SDMA stadards, where the chael impulse respose is kow i the trasmitter due to chael reciprocity. I. INTRODTION Time Divisio Duplex (TDD)-DMA is a promisig cellular commuicatios cocept. For example, it is curretly used i the 3PP TDD ad the hiese TD-SDMA stadards [1]. New wireless multimedia applicatios are expected to have most traffic i the dowlik (D) directio. Also, mobile termials should be kept as simple as possible. Therefore, base-statio based sigal processig before trasmissio to reduce iterferece ad ehace the lik performace would be advatageous. The TDD scheme allows chael measuremets i the uplik () which ca be used for D preprocessig due to chael reciprocity. A crucial requiremet is that the chael remais costat betwee -IR measuremets ad D-trasmissio. This is fulfilled for slow or moderate speed. I this paper, perfect chael kowledge i the trasmitter ad receiver is assumed. Most of the proposed cocepts of multiuser trasmissio (MT) use a liear trasformatio, i.e. matrix multiplicatio or filterig i the trasmitter. For istace, the cocepts kow as Trasmitter Precodig [18], Joit Trasmissio [3], Joit Pre-Distortio [4] or Joit Sigal Precodig [5],[6] employ a Zero-Forcig optimizatio (TxZF). The chael coefficiets ad spreadig codes are used for the calculatio of the preprocessig coefficiets. The Trasmit Wieer filter [7] uses additioally the kowledge of the oise level at the receivers. The cocepts are compared i [8] ad []. ombied trasmitter ad receiver optimizatio for TDD-DMA is cosidered i [9], where the SNR at the detector is maximized. All above cocepts do ot exploit the kowledge of the actually trasmitted data symbols, which is however available i the trasmitter. This is a major differece betwee receiver processig, i.e. multiuser detectio (MD) ad D trasmitter processig (MT). I this paper, a cocept which icludes the kowledge about the trasmitted data sequece to calculate the preprocessor is proposed. The performace figure for the TDD-DMA D is the BER at the receivers for a limited total trasmit power. Therefore, istead of optimizig criteria like MMSE, we propose to miimize the BER directly. This fits actually also more to the detectio problem, which is solved here, ad ot a estimatio problem. sig the spreadig codes, the istataeous chael impulse respose, ad the receiver filter coefficiets (i.e. RAE), the symbol iteractio matrix ca be derived. sig the actually trasmitted symbols, which may be preprocessed by a symbol- ad user-specific coefficiet the symbol at the detector i the receiver ca be calculated. With the receiver oise variace, the BER is calculated. The BER is the miimized with respect to all preprocessig coefficiets, where a trasmit power costrait has to be fulfilled. fortuately, the preprocessor calculatio becomes oliear ad ca be solved oly umerically. State-of-the-art oliear optimizatio algorithms like Sequetial Quadratic Programmig (SQP) ca be used for this purpose. Furthermore, ot oly the optimizatio fuctio is available aalytically, but also its first (Jacobia) ad secod (essia) derivatives. Although a global optimum ca ot be guarateed, simulatios show sigificat performace improvemets usig this scheme compared to covetioal trasmissio without preprocessig ad also compared to liear preprocessig schemes. The liear MT schemes either employ oly a chip-matched filter i the receiver or RAE receivers. I this paper, RAE receivers are cosidered, sice mobiles are usually equipped with RAE receivers which are ecessary for cell-search, chael estimatio ad trackig, soft-hadover etc. The proposed scheme ca also easily be applied to chip-matched filter receivers, possibly i cojuctio with pre-rae trasmissio. The direct usage of BER as optimizatio criterio was proposed for the multiuser detectio problem i the receiver [10] [11] [1] [13] [14]. There, the perfect kowledge of the trasmitted symbols is usually ot available, whereas i multiuser trasmissio the data symbols are kow a priori. I [10], kow traiig symbols are used to adapt a multiuser detectio filter usig differet adaptatio algorithms. BER optimizatio i the trasmitter was recetly proposed i [15]. Aother MT oliear MT scheme is Tomliso-arashima Precodig (TP) [16], which is ot cosidered i this paper. Oe major differece betwee the metioed MD algorithms ad MT is, that i the former o trasmit power costrait has to be fulfilled. The paper is orgaized as follows. Sectio II describes the system model for a short code multiuser DMA dowlik trasmissio system icludig a frequecy-selective chael model. I sectio III, a expressio for the symbols at the decisio device icludig iterferece is derived for a RAE receiver. I sectio IV the BER i depedecy o the proposed preprocessig coefficiets is give, as well as its derivatives ad the trasmit power costrait. Oe method to optimize the BER umerically is explaied. Fially, i sectio V simulatio results for the proposed scheme are show, ad compared to a covetioal RAE ad liear preprocessig schemes.

2 II. SYSTEM AND ANNE MODE A DMA dowlik with active users is assumed, as show i fig. 1. QPS with ray labellig is used for modulatio. The block d v (k-1) d v (k) c v d 1 d α 1 s 1 s h 1 h η 1 r 1 h Rx,1 h Rx, 1 d 1 d ( 1 ) ( ) ϕ v u, d u (k) ( ) ( ) ϕ v u Fig.. Partial code cross-correlatio fuctios ϕ (1) v,u() ad ϕ () v,u(), * c u Fig. 1. d 3 Preprocessor 3 Spreader h 3 hael AWN h Rx,3 RAE 3 Despreader d 3 Detector System model for BER Processig of TDD-DMA dowlik sigals of the symbols d u(k) 1 {1j, 1j, 1 j, 1 j} of user u is orgaized i the vector d u of legth. The symbols of all users ca be stacked i d d T 1,.., d T T. The user specific short spreadig code c u [c u(1),.., c u()] T with legth (spreadig gai) is arraged i the ( ) spreadig matrix u blockdiag (c u,.., c u). The eergy of the spreadig code is ormalized, i.e. c u c u 1. The spread sequece for user u with legth () is s u ud u ad the trasmitted sigal of all users s su. The frequecy-selective dowlik chael for each user is modeled as chip-spaced tapped-delay lie with maximum legth. It is assumed to be costat durig oe block. The chael impulse respose (IR) of user u is h u [h u(1),.., h u()] T. The ( ) Toeplitz-structured chael matrix is h u(1) h u() h u(1)... 0 u... h u()... 0 h u() (1)... h u() h u(1) The legth () dowlik sigal at the receiver u is r u us u. () III. RAE REEIVER I DMA systems, the mobile statios are usually equipped with RAE receivers. Therefore, it is ecessary to ivestigate the decisio variable at the output of the RAE receiver, which icludes the udisturbed trasmitted symbol, the self iterferece (SI) ad the multiple access iterferece (MAI). The iterferece terms are ot treated i a statistical way, but specifically for each particular chael realizatio ad particular trasmit data symbols. First, the partial cross-correlatio fuctios of the spreadig codes are itroduced, 1 ϕ (1) v,u() c v( i)c u(i) (3) i0 1 ϕ () v,u() c v(i)c u(i ) (4) i0 where is the relative delay i chips betwee code sequeces of user v ad u. Eq. (3) ad (4) are visualized i Fig.. For the lag of chips betwee the users u ad v, ϕ (1) v,u() characterizes the impact of the data symbol d v(k 1) o symbol d u(k) ad ϕ () v,u() the impact of d v(k) o symbol d u(k). I the RAE receiver with maximum ratio combiig (MR), the receiver filter coefficiet h Rx,u is the time iverted cojugate complex of the chael coefficiet h u. Followig, it is assumed that the dowlik trasmissio is sychroous ad that the maximum delay is shorter tha the symbol duratio. This meas, that at the output of the RAE combier the symbol d u(k) is affected oly by the previous, curret ad ext symbol of all users v. The ifluece form the previous symbol of user v at the receiver for user u is γ a,v,u l1 ml1 h u(l)h v(m)ϕ (1) v,u(m l) (5) where l is the idex of the RAE figer of user u ad m is the chael coefficiet idex of user v. Similar to Eq. (5), the impacts of the curret ad ext symbols are γ b,v,u γ c,v,u h u(l)h v(m) l1 m1 ϕ () (m l) for m > l ϕ (1) ( 1 m l) l 1 l1 m1 for m l (6) h u(l)h v(m)ϕ () v,u( 1 m l) (7) The symbol iteractio coefficiets i Eq. (5)-(7) are data idepedet ad have to be calculated oly oce per block for short codes. For log codes, these coefficiets have to be recalculated for each symbol. The symbol k of user u at the decisio device is ow the sum of these symbol iteractio coefficiets weighted by the particular data symbols of all users, d u(k) v1 d v(k 1)γ a,v,u d v(k)γ b,v,u d v(k 1)γ c,v,u. (8) The oise η u(k) at the decisio device of user u has the variace σ u σ m1 h u(m)h u(m), (9) where the oise at each RAE figer is assumed to be idepedet ad σ N 0/. This leads to the decisio variable of symbol k of user u ˆd u(k) d u(k) η u(k). (10) IV. MINIMM BER TRANSMITTER The pricipal goal of trasmitter ad receiver basebad processig is to miimize the BER. sually, other criteria are used to achieve that, i.e. Miimum Mea Squared Error (MMSE). Such a criterio is more related to a estimatio tha a detectio problem. I the receiver, the trasmitted sigal is ot to be recostructed, but to be detected. owever, the expected BER ca be miimized directly. If the data symbols, the codes, the chael coefficiets ad the behavior of the receiver are kow i the trasmitter, oly the additive

3 u oise remais a radom variable. Figure 3 illustrates the proposed cocept. The QPS data symbol d(k) is moved by iterferece to positio d(k) at the decisio device. The squared distace to the decisio threshold I{d(k)} is decreased, whereas the squared distace R{d(k)} is icreased. The MMSE criterio would move the received symbol d(k) back to d(k), whereas the miimum BER criterio moves d(k) as far away as possible from both decisio thresholds, as log as the trasmit power costrait is obeyed. Subsequetly, ukow oise with kow pdf ad variace σu is added at d(k). I this paper, the preprocessig i the trasmitter is doe with Fig. 3. Q/Im du ( k ) ( k ) du Iterferece I/Re Received QPS symbol at decisio device the data symbols. Each symbol is multiplied with a preprocessig factor α u(k). These factors are combied i the preprocessig vectors [α u(1),.., α u()] T T 1,.., ] T. The oise-free symbol at the decisio device (8) becomes d u(k) v1 ad [ T α v(k 1)d v(k 1)γ a,v,u (11) α v(k)d v(k)γ b,v,u α v(k 1)d v(k 1)γ c,v,u. It would also be possible to modify the trasmitted chips s u, but this would make the BER optimizatio more difficult. For liear preprocessig schemes it was show i [6] that symbol preprocessig is equivalet to chip preprocessig. A. BER calculatio First, the bit error probability P e,u for the proposed preprocessig scheme i the trasmitter ad a RAE receiver of user u is derived, where SI ad MAI is kow istataeously ad oise η u(k) is modelled as additive aussia oise with variace σ u (9). The QPS modulatio with ray labellig ca be cosidered as two idepedet BPS data streams. For a hard decisio detector the average bit error probability of oe user is, P e,u 1 (1) k1 Pr sg R du(k) η u(k) sg (R (d u(k))) Pr sg I du(k) η u(k) sg (I (d u(k))) ad the average bit error probability of all users is P e 1 P e,u. (13) The distaces to the respective decisio thresholds are I,u(k) R du(k) sg (R (d u(k))) ad (14) Q,u(k) I du(k) sg (I (d u(k))). (15) With the aussia oise assumptio, (13) becomes ow P e 1 4 k1 erfc I,u(k) Q,u(k). (16) The BER i (16) icludes with (11), (14) ad (15) the ifluece of the data preprocessig vector. The depedecy of P e o oe particular data preprocessig factor α v(k) ca be calculated aalytically ow, as P e(α v(k)) cost. 1 4 erfc I,u(k 1) I,u(k) I,u(k 1) Q,u(k 1) Q,u(k 1) Q,u(k). (17) The partial derivative of the BER with respect to oe symbol preprocessig coefficiet is 1 σ u sg(r(d u(k 1)))γc,v,ue P e α d v(k) v(k) π I,u (k 1) sg(r(d u(k)))γb,v,ue I,u (k) σ u σ u sg(r(d u(k 1)))γa,v,ue I,u (k1) j sg(i(d u(k 1)))γc,v,ue Q,u (k 1) σ u j sg(i(d u(k)))γb,v,ue Q,u (k) j sg(i(d u(k 1)))γa,v,ue Q,u (k1) σ u (18) Now, the vector of all partial derivatives of P e (Jacobia) ca be provided, as well as the secod derivatives (essia) ca be derived (ot show here). This is very useful to optimize P e( ) umerically i a efficiet way. B. Trasmit Power ostrait The allowed trasmit power of oe data block is E Bl. The trasmit power costrait for the composed sigal of all users s su is hece g( ) s! s E Bl 0. (19) If o Pre-RAE filter i the trasmitter is used, the sigal eergies of the symbols k are idepedet, i.e. E Bl k1 E Bl,k. The

4 opt trasmit eergy of symbol k is E Bl,k s k x α v(k)d v(k)c v(x) v1 α u(k)d u(k)c u(k) α u(k)d u(k)c u(x) α v(k)d v(k)c v(k),v u c u(x) α u(k) d u(k) 1 1 α u(k)d u(k) α v(k)d v(k),v u 0 for orthogoal codes c u(x)c v(x) α u(k)α u(k) (0) With (0) ad orthogoal codes, the costrait (19) is ow g( ) E Bl, (1) with the partial derivatives g( ) αu(k). () α u(k). BER Optimizatio fuctio The costraied optimizatio task is arg mi P e( ) s.t. g( ) 0. (3) sig a agrage multiplier, the ucostraied optimizatio fuctio is ow F (, λ) P e( ) λg( ). (4) The miimum BER solutio is characterized by E Bl! 0 (5) F (, λ) λ F (, λ) Pe( )! λ 0. (6) D. Numerical BER Optimizatio sig oliear optimizatio methods, local miima for P e( ) ca be searched. ike for may oliear optimizatio problems, there is o guaratee that a local miimum is the global optimum, but ivestigatios have show satisfactory results. The algorithm used here is Sequetial Quadratic Programmig (SQP) which is described for istace i [17]. SQP is cosidered as state of the art i oliear optimizatio. The aalytical expressios for the fuctio ad the first derivatives ad a approximatio of the essia matrix are used to solve a Quadratic Programmig problem. The approximatio of the essia matrix is determied usig a Quasi-Newto method like BFS. The, a lie-search i the calculated directio is performed. This is repeated iteratively util a satisfactory solutio is foud. The algorithm requires a startig vector. ood results ca be achieved with arbitrary startig vectors (like all parameters set to oe), however a careful selectio ca miimize the ecessary umber of iteratios. I the eighborhood of the solutio, this algorithm has superliear covergece speed. The order of the algorithm complexity is cubic with the umber of users ad block legth. A closer examiatio of this optimizatio algorithm ad possible alteratives is beyod the scope of this paper ad remais a further field of studies. V. PERFORMANE EVAATION The performace of the miimum BER trasmitter is ivestigated umerically. As a referece, covetioal RAE receiver with o trasmitter preprocessig, a liear zero forcig (ZF) joit trasmitter [18] with RAE receiver ad a liear MMSE/Wieer filter [7] with RAE receiver are used. For the liear block trasmitter processig schemes o latecy time optimizatio was performed, which could further improve their performace. The frequecy-selective chael model is based o the 3PP multipath propagatio case 3. The chael taps are chip-spaced with ucorrelated fadig betwee the taps ad users. The power delay profile is [ ]dB. The chaels are costat for oe block. Short codes with spreadig factor 16 for 3PP-TDD dowlik are used, i.e. Walsh-adamard chaelisatio codes ad oe cell-specific scramblig code. The modulatio is QPS. Both, semi-aalytical predictio of the BER with (16) ad simulatio results are show. The good agreemet of both results shows that the approach of aalytical determiatio of the received sigal icludig iterferece followed by the predictio of the BER usig the error fuctio is feasible. Furthermore, eve low BER values for high SNRs ca be reliably predicted for all Multiuser Trasmissio schemes without the effort of simulatig a high umber of bits i a trasmissio system. raw BER Rx RAE 1 N10 /Q/1/1/4 ovetioal (Sim) 10-4 ovetioal (Aa) iear/zf (Sim) iear/zf (Aa) iear/mmse (Sim) 10-5 iear/mmse (Aa) Mi BER (Sim) Mi BER (Aa) E b,tx /N Fig. 4. Symbol based Multiuser Trasmissio with RAE receiver, 1 users, 4 taps Figure 4 shows the BER vs. E b /N 0 for differet Multiuser Trasmissio schemes with RAE receivers i the mobiles. It ca be see, that the BER ca be improved cosiderably usig the proposed Miimum BER Trasmissio scheme, especially for higher SNRs. The required SNR to achieve a BER of P e 10 3 vs the umber of active users is show i fig. 5. The covetioal trasmitter ca ot achieve this BER for > 1, whereas the liear MT ad the proposed Miimum BER ca achive this performace. I cotrast to liear TxZF ad TxWF, the proposed Miimum BER MT requires almost o higher trasmit power for a higher system load, eve for full load 16. The drop of required SNR of the proposed scheme ad the TxMF is due to the fact, that the total available trasmit power i the BS scales with the umber of users. This power ca be efficietly utilized by the TxWF ad the Miimum BER MT. This correspods with results from [6].

5 0 18 required E b /N 0 for BER1e-03 [db] Rx RAE 1 N10 /Q/1/1/4 ovetioal Rake iear/mmse Mi BER iear/zf umber of users Fig. 5. Required E b,t x /N 0 to achieve P e 10 3,with RAE receiver, hael 4 taps ad differet MT methods VI. ONSIONS A ovel approach to Multiuser Trasmissio (MT) for the DMA dowlik i frequecy-selective chaels was proposed. The key idea is to miimize the BER at the receivers directly. It ca be predicted by calculatig the sigal at the receiver (icludig iterferece) exactly ad treatig oly the additive oise at the receiver statistically. For this, the chael impulse resposes, the data sequece ad the oise variace have to be kow i the trasmitter. sig state-of-theart oliear optimizatio methods like SQP, the BER miimizig coefficiets for the symbols preprocessig ca be foud. Simulatios have show, that the proposed approach outperforms Matched filter, Zero Forcig ad MMSE approaches cosiderably. The extesio of the proposed scheme to multiple ateas is possible. A low complexity implemetatio of the oliear optimizatio algorithm remais a field for further studies. [9] R. Irmer ad. Fettweis, ombied trasmitter ad receiver optimizatio for multiple-atea frequecy-selective chaels, i Proc. IEEE Wireless Persoal Multimedia ommuicatios (WPM 0), oolulu, awaii, Oct. 00, pp [10] S. he, A. Samiga, B. Mulgrew, ad. azo, Adaptive miimum- BER liear multiuser detectio for DS-DMA sigals i multipath chaels, IEEE Tras. Sigal Processig, vol. 49, o. 6, pp , 001. [11] R. de amare ad R. Sampaio-Neto, Adaptive multiuser receivers for DS-DMA usig miimum BER gradiet-newto algorithms, i Proc. IEEE PIMR, 00, pp [1] X. Wag, W. u, ad A. Atoiou, ostrait miimum-ber multiuser detector, IEEE Tras. ommu., vol. 48, o. 10, pp , 000. [13] I. Psaromiligkos, S. Batalama, ad D. Pados, O adaptive probability of error liear filter receivers for DS-DMA chaels, IEEE Tras. ommu., vol. 47, o. 7, pp , [14] N. Madayam ad V. Aazhag, radiet estimatio for sesitivity aalysis ad adaptive multiuser iterferece rejectio i code-divisio multiple-access systems, IEEE Tras. ommu., vol. 45, o. 7, pp , [15] T. Weber, M. Meurer, ad A. Sklavos, Optimum oliear joit trasmissio, i Proc. OST 73 TD(03)008, Barceloa, 003. [16]. Widpassiger, R. Fischer, T. Vecel, ad J. uber, Precodig i multi-atea ad multi-user commuicatios, IEEE Tras. Wireless, 003, accepted. [17] J. Nocedal ad S. Wright, Numerical Optimizatio, Spriger, New York, [18] B. Vojcic ad W. Jag, Trasmitter Precodig i Sychroous Multiuser ommuicatios, IEEE Tras. ommu., vol. 46, o. 10, pp , Oct ANOWEDEMENTS This work was supported by the erma Sciece foudatio (DF), cotract Fe 43/4-. The authors thak Robert Jäschke ad Ree abedorf for makig some of the simulatios. REFERENES [1] R. Esmailzadeh ad M. Nakagawa, TDD-DMA for wireless commuicatios, Artech ouse, Bosto, 003. [] F. Watha, R. Irmer, ad. Fettweis, O trasmitter-based iterferece mitigatio i TDD-dowlik with frequecy-selective fadig chael eviromet, i Proc. Asia-Pacific oferece o ommuicatios (AP), Badug, Idoesia, Sept. 00, pp [3] P. W. Baier, M. Meurer, T. Weber, ad. Tröger, Joit Trasmissio (JT), a alterative ratioale for the dowlik of Time Divisio DMA usig multi-elemet trasmit ateas, Proc. IEEE ISSSTA, pp. 1 5, 000. [4] Bosch Proposal for 3PP, TS RAN Wg1 #6(99)918: Tx Diversity with Joit Predistortio, 1999, [5] A. N. Baretto ad. Fettweis, apacity icrease i the dowlik of spread spectrum systems through joit sigal precodig, Proc. IEEE I, vol. 4, pp , 001. [6] A. Noll Barreto ad. Fettweis, Joit sigal precodig i the dowlik of spread-spectrum systems, IEEE Tras. Wireless ommu., vol., o. 3, pp , May 003. [7] M. Joham,. usume, M. zara, W. tschick, ad J. Nossek, Trasmit Wieer filter for the dowlik of TDD DS-DMA systems, i Proc. IEEE ISSSTA, New Jersey, SA, Sept. 00, vol. 1, pp [8] S. eorgoulis ad D. ruickshak, Pre-Equalisatio, Trasmitter Precodig ad Joit Trasmissio Techiques for Time Divisio Duplex DMA, i Proc. 3 ommuicatio Techologies, IEE oferece Publicatio 477, 001, pp

A New Space-Repetition Code Based on One Bit Feedback Compared to Alamouti Space-Time Code

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