Diversity Performance of Precoded OFDM with MMSE Equalization

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1 Uiversity of Wollogog Research Olie Faculty of Iforatics - Papers (Archive) Faculty of Egieerig ad Iforatio Scieces 7 Diversity Perforace of Precoded OFDM with MMSE Equalizatio Xiaojig Huag Uiversity of Wollogog, huag@uow.edu.au Publicatio Details This coferece paper was origially published as Huag, X, Diversity Perforace of Precoded OFDM with MMSE Equalizatio, 7th Iteratioal Syposiu o Couicatios ad Iforatio Techologies ISCIT 7, Sydey, 6-9 Oct, Research Olie is the ope access istitutioal repository for the Uiversity of Wollogog. For further iforatio cotact the UOW Library: research-pubs@uow.edu.au

2 Diversity Perforace of Precoded OFDM with MMSE Equalizatio Abstract Two sets of bit error rate (BER) lower bouds for precoded orthogoal frequecy divisio ultipleig (OFDM) systes usig iiu ea square error (MMSE) equalizatio over frequecy-selective ultipath fadig chaels are evaluated by Mote Carlo ethod i this paper. The first set represets the best perforace uder differet data group sizes used for precodig, whereas the secod set represets the best perforace uder differet chael ultipath diversity orders. These perforace bouds ca serve as the guidelies for syste desigers to decide proper data group sizes for precoded OFDM systes i order to achieve better trade-off betwee syste perforace ad copleity. Nuerical results also cofir the usefuless of these lower bouds for practical OFDM systes. Disciplies Physical Scieces ad Matheatics Publicatio Details This coferece paper was origially published as Huag, X, Diversity Perforace of Precoded OFDM with MMSE Equalizatio, 7th Iteratioal Syposiu o Couicatios ad Iforatio Techologies ISCIT 7, Sydey, 6-9 Oct, This coferece paper is available at Research Olie:

3 Diversity Perforace of Precoded OFDM with MMSE Equalizatio Xiaojig Huag School of Electrical, Coputer ad Telecouicatios Egieerig Uiversity of Wollogog, Northfields Aveue, Wollogog, NSW 5, Australia Tel: , Eail: Abstract Two sets of bit error rate (BER) lower bouds for precoded orthogoal frequecy divisio ultipleig (OFDM) systes usig iiu ea square error (MMSE) equalizatio over frequecy-selective ultipath fadig chaels are evaluated by Mote Carlo ethod i this paper. The first set represets the best perforace uder differet data group sizes used for precodig, whereas the secod set represets the best perforace uder differet chael ultipath diversity orders. These perforace bouds ca serve as the guidelies for syste desigers to decide proper data group sizes for precoded OFDM systes i order to achieve better trade-off betwee syste perforace ad copleity. Nuerical results also cofir the usefuless of these lower bouds for practical OFDM systes. I. INTRODUCTION Orthogoal frequecy divisio ultipleig (OFDM) is a data trasissio schee which odulates data sybols i parallel o orthogoal subcarriers [,]. With OFDM, the effect of itersybol iterferece (ISI) caused by chael tie spread ca be easily itigated. A OFDM trasitter ca be ipleeted by the iverse fast Fourier trasfor (IFFT) with cyclic prefi (CP) isertio or zero-padded suffi (ZP) appedig, ad siple frequecy doai chael equalizatio ca be applied at the receiver via fast Fourier trasfor (FFT). Due to these advatages, OFDM has bee widely used i today s digital couicatio systes such as wireless persoal/local/etropolita area etworks (WPANs/ WLANs/WMANs) ad digital audio/video broadcastig services (DAB/DVB) [3-6]. It is also a cadidate for future geeratio wireless obile couicatio systes [7]. However, the covetioal OFDM systes suffer fro soe ajor disadvatages ad cosiderable research has bee udertake to overcoe the over the past decades. First, the trasitted sigal wavefor has a large peak-to-average power ratio (PAPR), which reduces the power efficiecy of the OFDM systes [8-]. Secod, the receiver perforace is sesitive to carrier frequecy offset which causes itercarrier iterferece (ICI). Thus, coplicated frequecy sychroizatio is ecessary [,3]. Third, the ucoded OFDM syste oly achieves diversity order oe ad hece perfors poorly i frequecy-selective chaels. Chael codig has bee traditioally used to iprove the diversity across frequecy ad tie [4,5], ad recetly liear precodig ad block spreadig for OFDM systes are This research is supported by Australia Research Coucil Discovery Project DP itroduced to iprove the frequecy diversity perforace [6-]. Precoded OFDM divides a block of odulated data to be trasitted i a OFDM sybol ito groups ad applies a uitary atri to each data group to obtai differet liear cobiatios of the data sybols. After subcarrier appig, the data sybols are spread across the trasissio frequecy bad. Thus, if a subcarrier eperieces a deep fade after trasittig over a frequecy-selective ultipath chael, the data sybol ca be still recovered fro other received subcarriers so that the syste perforace is iproved due to the icreased diversity order. There are aily two factors which deterie the perforace of a precoded OFDM syste. Oe is the equalizatio/detectio ethod used at the receiver. The other is the precodig data group size. Regardig the equalizatio/detectio ethod, the aiu-likelihood (ML) detectio offers better perforace tha other liear equalizatio techiques such as zero-forcig (ZF) ad iiu ea square error (MMSE) equalizatios. However, the ML detectio requires higher coputatioal copleity especially whe the data group size is large. I practice, the liear equalizatio is preferable. Ituitively, the larger the data group size is, the better the syste perforace will be. However, larger data group size also iplies higher ipleetatio copleity. O the other had, if the data group size is too sall, the available diversity itroduced by the ultipath chael ca ot be fully eploited. Therefore, how to deterie a proper data group size accordig to the available ultipath diversity order is of sigificace to practical OFDM syste desig. I this paper, the bit error rate (BER) lower bouds of the MMSE equalizatio i precoded OFDM systes are evaluated for differet data group sizes uder the assuptio that the chael provides a full diversity, which idicate the best perforace the MMSE equalizatio could achieve for a give data group size. Further, assuig a sufficietly large data group size, the perforace lower bouds of the MMSE equalizatio uder differet ultipath diversity orders are also evaluated, which show the best perforace the MMSE equalizatio could achieve for a give ultipath diversity order. These perforace bouds ca serve as the guidelies for syste desigers to decide suitable group sizes for precoded OFDM systes i order to achieve the desired perforace with affordable copleity. The rest of the paper is orgaized as follows. I Sectio II, the precoded OFDM syste odels are preseted. I Sectio /7/$5. c 7 IEEE

4 III, the BER of the MMSE equalizatio is forulated as a fuctio of the data group size ad the ultipath diversity order. Sectio IV evaluates the two sets of BER lower bouds usig Mote Carlo siulatio ethod. Nuerical results of syste perforace for a precoded OFDM syste are also provided to cofir the evaluated bouds ad deostrate the usefuless of these bouds to syste desig. Fially, coclusios are draw i Sectio V. II. SYSTEM MODELS Referrig to the trasitter odel show i Fig. (a), let [] i, i,,, MN, deote MN data sybols ( M ad N are iteger powers of ), which are odulated fro the iforatio data bits after biary phase shift keyig (BPSK), quadrature phase shift keyig (QPSK) or ay other quadrature aplitude odulatio (QAM) costellatio appig. Before precodig, the MN data sybols are firstly divided ito N groups of size M with the th group deoted as a vector ( [ ] [ ] [ ]) T M, M,, M M,,,, N, () deotes atri traspositio, ad the epressed where () T as a vector (S/P). [] i CP Reoval or Overlap-Add N S/P r [] i S/P N Precodig after serial-to-parallel coversio U U U N r FFT Iterleavig R W MN Y (a) De-iterleavig IFFT y WMN Y Equalizatio P/S y [] i r R R R N d U CR (b) Fig.. Precoded OFDM syste odels: (a) trasitter ad (b) receiver. π π j ad j are the MN-poit MN WMN e MN MN WMN e MN MN MN MN iverse Fourier trasfor ad Fourier trasfor atrices respectively. CP or ZP Decisio ˆ P/S ˆ [] i The precodig process is to apply a M M uitary atri U, which satisfies the property U U U U I, where () deotes traspositio ad cople-cojugatio operatio ad I is the idetity atri of order M, to each vector to produce a precoded vector where each eleet is a liear cobiatio of the sybols i vector. To better eploit frequecy diversity, the precoded sybols are preferably apped oto subcarriers equally spaced across the trasitted badwidth. This is equivalet to a block iterleavig operatio aog N precoded vectors U,,,, N, ad the perforig IFFT of legth MN o the resultig precoded ad iterleaved vector Y. After IFFT ad parallel-to-serial coversio (P/S), a tie doai sequece y [] i, i,,, MN, is produced. To for a precoded OFDM sybol, either a CP or a ZP of sufficiet legth (loger tha the aiu chael ultipath delay i saples) are added to y [] i to avoid iterferece betwee adjacet precoded OFDM sybols ad tur the liear covolutio of the trasitted sigal with the chael ipulse respose ito a circular oe. The precoded OFDM sigal is the trasitted over a frequecy-selective ultipath fadig chael ad received at the receiver basebad. By reovig the CP or perforig a overlap-add operatio, MN -poit received precoded OFDM saples r [] i, i,,, MN, will be produced. After FFT ad de-iterleavig, the discrete-tie received sigal ca be epressed i the frequecy doai as R HU V,,,, N, () where R ( [] [ ] [( ) ]) T R, R N,..., R M N (3) is a vector of M eleets which are deciated fro R [] k, the MN -poit discrete Fourier trasfor (DFT) of r [] i, by a dow-saplig factor N, H diag( H[], H[ N ],..., H[ ( M ) N ] ) (4) is a M M diagoal atri with diagoal eleets deciated fro H [] k, the MN -poit DFT of the oralized discrete chael ipulse respose h [] i, ad V is a zero-ea Gaussia oise vector with covariace atri E{ V V } σ VI, where E {} deotes eseble average. To recover the trasitted data vector, equalizatio ad detectio ust be perfored o the received sigal R. Due to the copleity of the optiu ML detectio, oly the MMSE equalizatio is cosidered, sice it ca siply use a oe-tap equalizer for each subcarrier i the frequecy doai. The equalizatio ad detectio process ca be described as 7 Iteratioal Syposiu o Couicatios ad Iforatio Techologies (ISCIT 7) 83

5 follows. Let C [] k deote the oe-tap equalizer coefficiet to be applied to R [] k o the subcarrier k ad diag( C[], C[ N ],..., C[ ( M ) N ] ) C (5) deote a M M diagoal atri with diagoal eleets C [ ln ], l,,, M. First, applyig C to R produces the equalized precoded data vector C R. Secod, usig U to reove the precodig yields the decisio variable vector d U CR. Fially, a estiate of the trasitted data vector is obtaied after hard decisio. Repeatig the above process for,,, N, all the trasitted data sybols are retrieved. III. PERFORMANCE OF MMSE EQUALIZATION We first derive the post-equalizatio sigal-to-oise ratio (SNR) as a fuctio of the equalizer coefficiets C [ ln ] for the received sigal vector R. Accordig to the above described equalizatio process, the decisio variable vector ca be epressed as d U CR U CHU U CV. (6) Assue that the data sybols i are idepedet with average power σ so that E{ } σ I. The covariace atri of d ca be derived as U CH H C U σ V { d d } U C C U E σ. (7) Suppose that we wat to decide the th data sybol [ M ] i fro the th eleet i d. The useful sigal copoet ca be foud fro the first ter o M l the right-had-side of (6) as C[ ln ] H[ ln ] [ M ] u l,, where u l, is a eleet of U at the l th row ad the th colu, ad thus the useful sigal power after equalizatio is M l C [ ln ] H[ ln ] σ q [, ] u l,. (8) The average power of the th eleet i d ca be also foud fro (7) as σ M C[ ln ] H[ ln ] ul, σv C[ ln ] l l [ ] M u l, q,. (9) Therefore, the output SNR after equalizatio ca be epressed as [, ] q[, ] [, ] q [, ]. () q Accordig to the MMSE criterio, C should be desiged so that E{ ( d )( d )} ( Ud U )( Ud U )} E{ ( CR U )( CR U )} () is iiized. Usig the orthogoality priciple, we have E {( C R U ) R } () ad cosequetly, C E U R E R R { }( { }) { } U H ( H UE{ } U H E{ V V }) U E H H H I (3) i σ where i is the iput SNR before equalizatio. σv Fro (3), the diagoal eleet is foud to be H [ ] [ ln ] C ln. (4) H[ ln ] Substitutig (4) ito (8) ad (9) ad usig (), the output SNR after MMSE equalizatio is fially epressed as [ ] M H ln ul, l H[ ln ] i [, ]. (5) M H[ ln ] ul, l H[ ln ] i If we oly cosider a class of uitary atrices satisfyig u l,, such as those adapted fro Fourier trasfor M atri, Hadaard atri ad rotated Hadaard atri [9], (5) ca be siplified as M H[ ln ] M l H[ ln ] i [] M M H[ ln ] M [ ] [ ] M l H ln l i H ln i (6) i 84 7 Iteratioal Syposiu o Couicatios ad Iforatio Techologies (ISCIT 7)

6 We see that the output SNR is deteried by the chael frequecy respose H [ k], or equivaletly, the chael ipulse respose h [] i. Assuig QPSK odulatio for data sybols ad akig a Gaussia distributio approiatio for ISI, the bit error probability of the equalizer for a realizatio of the chael ipulse respose ca be evaluated N as Q( [] ), where the Q-fuctio is defied as N t Q ( ) e dt. Also, assuig that the chael π ipulse respose has L idepedet paths, each of which is odelled as a idepedet cople Gaussia process, the average BER for such frequecy-selective fadig chael ca be evaluated as N ( ) P ( []) e M, L Eh Q (7) N where E h{} deotes the eseble averagig over all possible h [] i. We see that (7) is a fuctio of the data group size M ad the ultipath legth L. i.e., istead of evaluatig (8) usig the joit probability desity fuctio (pdf) of α, α,, α M, we geerate sufficiet realizatios of these idepedet cople Gaussia variables, evaluate the BER for each realizatio, ad the take a average. We see that the perforace is iproved as M icreases. Whe M, sice α l is chi-squaredistributed with two degrees of freedo ad pdf e, a ρ closed-for lower boud epressio ca be foud as Pe (, ) Q (9) ρ e dρ ρ i which is the best perforace the MMSE equalizatio could offer. IV. BER LOWER BOUNDS AND APPLICATION To show the relatioship betwee the syste perforace ad the group size as well as the relatioship betwee the syste perforace ad the chael diversity order, let s work out two sets of BER lower bouds usig the MMSE equalizatio. The first set represets the best possible perforace for a give block size M. We assue that the chael provides a full ultipath diversity, i.e., L >> M, so that H [ ln ] at differet l s becoe idepedet cople Gaussia variables with uit variace ad are alteratively deoted as α l for coveiece. The, the average BER ca be alteratively evaluated as Pe ( M, ) Eα Q (8) M M l i α l where E {} deotes the eseble average over α, α,, α α M. I (8), we ca also epress i as N E b for QPSK, where E b is the sigal eergy per bit ad N is the oise power spectral desity. Fig. shows the lower bouds of the MMSE equalizatio perforace uder differet block sizes M,, 4, 8, 6, 3, 64, 8, 56, 5, 4, ad by the Mote Carlo ethod, BER 3 4 M M E b /N (db) Fig.. Lower bouds of MMSE equalizatio perforace uder differet data group sizes. The curves fro right to left correspod to M,, 4, 8, 6, 3, 64, 8, 56, 5, 4, ad respectively. The secod set of lower bouds idicates the best perforace for a give uber of chael ultipath L (referred to as ultipath diversity order) with sufficietly large data block size M. Let M, the average BER ca be evaluated as ( ) P L E Q e h (), π dω π jω ( ) i H e jω where H ( e ) is the Fourier trasfor of [] i h. Fig. 3 shows this set of lower bouds for L,, 4, 8, 6, 3, 64, 8, 56, 5, 4, ad by the Mote Carlo ethod, i.e., we geerate sufficiet realizatios of the chael h [] i, 7 Iteratioal Syposiu o Couicatios ad Iforatio Techologies (ISCIT 7) 85

7 evaluate BER for each realizatio, ad the take a average. We see that as the chael diversity order icreases the perforace is also iproved. Whe L, the average BER approaches the sae best perforace epressed by the closed-for equatio (9). (i.e., N ), the perforace is already very close to this lower boud. BER 3 M N8 BER 3 L 4 (,3) M8 N 4 L E b /N (db) Fig. 3. Lower bouds of MMSE equalizatio perforace uder differet ultipath diversity orders. The curves fro right to left correspod to L,, 4, 8, 6, 3, 64, 8, 56, 5, 4, ad respectively. Coparig Fig. with Fig. 3, we otice that e (, ) (,) P P e, i.e., whe there is o preccodig ( M ), the best perforace a covetioal OFDM syste ca achieve is the sae as the perforace with diversity order oe ( L ). This is cosistet with what we have kow about the covetioal OFDM. Whe < M L <, we have ( M, ) > (, L). This eas that for a give data group size M the precoded OFDM syste ca ot achieve the perforace which the syste could potetially offer with diversity order L M. However, as M becoes larger, the perforace gradually approaches the best perforace that the syste provides for a give diversity order. To cofir the above evaluated BER lower bouds ad also deostrate their usefuless, the perforace for a precoded OFDM syste with MN 8 is tested by uerical siulatio usig a precodig atri adapted fro Fourier trasfor atri uder chael ultipath diversity order L 3. Fig. 4 shows the perforace usig MMSE equalizatio for differet group sizes M,, 4, 8, 6, 3, 64, ad 8 (cosequetly N 8, 64, 3, 6, 8, 4,, ad respectively). We see that whe M 3 (i.e., the data group size is less tha or equal to the ultipath diversity order), the perforace agrees with the lower boud ( M, ) for the give M. Whe choosig the aiu data group size M 8 (i.e., N ), the perforace approaches the lower, L for L 3. Also ote that whe M 64 boud ( ) E b /N (db) Fig. 4. Perforace of MMSE equalizatio for precoded OFDM syste with MN8 uder chael diversity order L3 ad differet group sizes M,, 4, 8, 6, 3, 64, ad 8 (cosequetly N8, 64, 3, 6, 8, 4,, ad respectively). Dashed lie is the lower boud for L3. The above eaple shows that we ca use the evaluated lower bouds to deterie the data group size for a proper, L we precoded OFDM syste desig. First, fro ( ) ca predict the best perforace the syste could potetially offer oce the ultipath diversity order is give. The, fro M, we ca copare syste perforace uder ( ) differet precodig sizes ad decide a suitable M subject to soe copleity costrait. We ca also estiate the perforace degradatio for a chose M. Applyig the above guidelies to a practical syste, the ultibad (MB) OFDM for ultra-widebad applicatios [4], it is of iterest to reveal that the syste desig sees iappropriate regardig the dual-carrier odulatio (DCM) for data rate over 3 Mbps. The MB-OFDM syste uses 8 subcarriers with a ZP of legth 3 which correspods to a potetial ultipath diversity of order 3. Sice the DCM is equivalet to a precodig with oly group size two, it is easily see that the ultipath diversity potetial is ot fully eploited. V. CONCLUSIONS We have show that the perforace of the precoded OFDM systes usig MMSE equalizatio is deteried by the precodig data group size ad the diversity order that the ultipath chael ca provide. Two sets of BER lower bouds are evaluated by the Mote Carlo ethod assuig full diversity order ad sufficietly large group size respectively. A closed-for lower boud epressio is also derived to defie the theoretical perforace liit for the MMSE equalizatio. These lower bouds ca serve as the guidelies for precoded OFDM syste desig Iteratioal Syposiu o Couicatios ad Iforatio Techologies (ISCIT 7)

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