MMSE-Based Local ML Detection of Linearly Precoded OFDM Signals

Transcription

1 -Based Local Detecton of Lnearly recoded OFDM Sgnals L. Rugn,.Banell Dept. of Elect. and Inform. Eng. (D.I.E.I.) Unversty of eruga eruga, Italy {rugn, G. B. Gannaks Dept. of ECE Unversty of Mnnesota Mnneapols, MN Abstract Lnear precodng s a well known effectve technque to boost the performance of orthogonal frequency-dvson multplexng (OFDM) systems. A drawback of lnearly precoded OFDM (L-OFDM) systems s the hgh computatonal complexty requred by maxmum-lkelhood () detecton, whch s mandatory to capture all the channel dversty. Conversely, lowcomplexty technques, such as the lnear mnmum meansquared error () detecton, suffer from non-neglgble performance loss wth respect to the performance. In ths paper, we propose a detecton technque that performs a local () search n the neghborhood of the output provded by the detector. The trade-off between performance and complexty of the proposed - detector, whch fall between the ones of the and detectors, can be ncely adjusted by approprately settng the neghborhood sze. Smulaton results show that the - detector wth mnmum neghborhood sze outperforms a block decson-feedback equalzaton () approach, whle preservng a smlar complexty. eywords-ofdm; lnear precodng; ; local maxmumlkelhood I. INTRODUCTION Ths work was partally supported by the Italan Mnster of Unversty and Research under the project MC-CDMA: an ar nterface for the 4th generaton of wreless systems. Orthogonal frequency-dvson multplexng (OFDM) s a wdely employed technque for wreless communcatons over multpath Raylegh fadng channels [1]. OFDM systems generally adopt a cyclc prefx to transform the frequency-selectve fadng channel nto a set of parallel frequency-flat channels, whch facltates the decodng and the equalzaton steps. owever, the smplfed (scalar) equalzaton step s coupled wth a loss of multpath dversty, leadng to poor bt-error rate () performance. The multpath dversty s usually recovered by usng forward-error correcton (FEC) codng [1], at the expense of some data rate reducton. A dfferent approach, whch can also be combned wth standard FEC codng, conssts of explotng the multpath dversty by means of a lnear precoder. In a lnearly precoded OFDM (L-OFDM) system, multpath dversty s ntroduced by transmttng dfferent lnear combnatons of uncoded symbols over dfferent subcarrers []. Consequently, dfferently from an uncoded OFDM system, the presence of a deep fade n the frequency doman does not annhlate any transmtted symbol, but only affects the lnear combnaton transmtted n the faded subcarrer. Therefore, by explotng the fnte-alphabet property of the constellaton, the transmtted symbols can stll be recovered from the data receved on the other subcarrers, gvng rse to mproved performance wth respect to a non-precoded system. If a non-redundant precoder s chosen [3], the performance gan s obtaned wthout sacrfcng the data rate, as frstly proposed by [4] for sngle-carrer flat-fadng lnks. In order to recover the L-OFDM data, varous detecton technques can be appled. Each of these technques presents dfferent complexty versus tradeoffs. The maxmumlkelhood () detector s able to get both the dversty and codng gan furnshed by frequency-selectve Raylegh fadng channels, thus provdng good performance. owever, ts computatonal complexty s exponental n the precoder sze. On the contrary, lnear detectors and decson-drected schemes, such as the mnmum mean-squared error () detector and the decson-feedback equalzaton (), exhbt lower complexty, but suffer from performance loss wth respect to the detector. To reduce the performance gap between lnear and detectors, varous advanced technques could be employed, such as sphere decodng (SD) [5], semdefnte programmng (SD) [6], and probablstc data assocaton (DA) [7]. These technques approach the performance, wth a complexty that, although much smaller than for, s hgher than for lnear and decson-drected detectors. In ths paper, we ntroduce the local () detecton [8]-[10] for L-OFDM. The correspondng detector performs a complexty-constraned search n the neghborhood of an ntal estmate. We show that the output of the detector s a convenent choce for such an ntal estmate. By adjustng the neghborhood sze, the proposed - detector can ncely trade performance for complexty, fllng the gap between the and the detectors. Smulaton results n typcal wreless local area network (WLAN) channels show that the proposed - detectors outperform a approach, whle mantanng under control the ncrease of complexty. II. L-OFDM SYSTEM MODEL An OFDM system wth N subcarrers and a cyclc prefx of length L s consdered. Throughout the paper, we assume IEEE Communcatons Socety 370

2 that the multpath channel s tme-nvarant, wth each path characterzed by a Raylegh statstc, and that the maxmum delay spread does not exceed the cyclc prefx duraton. We also assume tme and frequency synchronzaton at the recever end. As a consequence, the lth receved block, after the cyclc prefx removal and the fast Fourer transform (FFT), can be expressed as [11] y[] l = Dc[] l + w [] l, (1) where y [] l s a column vector of dmenson N, T D= N dag( F N[ h0, h1,..., hl 1,0,...,0] ) s the N N dagonal matrx that contans the channel coeffcents on the FFT grd, h s the th channel tap n the tme doman, F N s the untary N N FFT matrx, c [] l s the N 1 vector that represents the transmtted codeword, and w [] l stands for the addtve whte Gaussan nose (AWGN) n the frequency doman. In a L-OFDM system, the transmtted codeword c [] l s obtaned by lnear combnaton of dfferent uncoded symbols, as expressed by c [] l = Φ s [] l, where, focusng on non-redundant precodng, Φ s the N N precoder matrx, and s [] l s the N 1 vector that contans the data symbols (drawn from a generc constellaton of sze M ), whch are assumed to be ndependent and dentcally dstrbuted wth unt power. In order to reduce the system complexty, subcarrer groupng has been proposed n [3], where the subcarrers are grouped n subsets, and the precodng phlosophy s appled wthn each subcarrer subset. It turns out that the subcarrer groupng wth maxmally-separated subcarrers n each subset s the optmum one n order to preserve the maxmum dversty gan, whch s equal to the mnmum value between the precoder sze and the number of channel taps. By assumng N = B, where B s the number of subcarrer subsets, and s the precoder sze, by properly selectng the rows of the vector y [] l, expresson (1) can be splt n B equvalent equatons y [] l = D c [] l + w [] l, b= 1,..., B () b b b b l b where y [ ] s the 1 receved vector relatve to the bth subset, D b s the dagonal matrx contanng the subcarrer gans of the bth subset, and cb[] l = Θs b[] l represents the codeword transmtted on the bth subcarrer subset. The precoder Θ, whch s the same for all the subsets, can be ether untary or non-untary [3]. Snce the decodng technques consdered n ths paper do not depend on the subset ndex b and on the block ndex l, n order to smplfy the notaton, we rewrte () as y = Dc+ w = DΘs+ w = s+ w, (3) where = DΘ represents the aggregate effect of the channel and of the precoder on the uncoded symbol vector s. III. OVERVIEW OF DECODING TECNIQUES In order to explot all the performance advantages of lnear precodng (.e., collectng the dversty and codng gans offered by the channel), detecton should be performed at the recever sde. In ths case, due to the AWGN nature of w, the decson rule can be formulated as sˆ = arg max{ Λ( s )}, (4) s S where Λ () s = Re( s y) s s (5) s the log-lkelhood functon (LLF), and S s the set of all possble transmtted symbol vectors, wth cardnalty equal to M. owever, due to the hgh computatonal complexty nvolved n the evaluaton of M LLFs, some suboptmum de- tecton schemes have been suggested by the analogy between (3) and the nput-output relaton of equalzaton problems n sngle-carrer systems. Consequently, smpler lnear detecton technques can be appled at the recever to obtan a soft estmate of the transmtted symbol vector, as expressed by s = Gy. By employng the zero-forcng (ZF) or the crteron, the recever matrx G can be expressed by G =, and ZF 1 G = ( + σ w I ), (6) respectvely, where the superscrpt denotes Moore-enrose pseudonverse, and σ w s the varance of the elements of the AWGN vector w. The man drawback of lnear decodng technques s that they are unable to harness the maxmum possble dversty and codng gans, gvng rse to sgnfcant loss n performance wth respect to the approach [3]. Therefore, n order to mprove the performance of lnear detectors, whle mantanng the decodng complexty at a reasonable level, we can rely on decson-drected detectors. One possblty s to cancel the prevously-detected symbols by the block approach of [1][13], whch s desgned to mnmze the meansquared error (MSE) before the decson devce. A dfferent technque, alternatve to the seral cancellaton provded by the, s the parallel nterference cancellaton (IC) approach exploted n [14] for multuser detecton of drect-sequence code-dvson multple-access (DS-CDMA) sgnals. In ths case the other 1 symbols, estmated usng the detector, are consdered as nterference terms, and ther effect s subtracted n a parallel fashon. Although more powerful than lnear detectors, also decson-drected technques experence non-neglgble loss wth respect to the detector. As a consequence, advanced technques such as SD [5], SD [6], and DA [7], have been proposed n order to approach the performance wth a sgnfcantly reduced complexty. owever, such a complexty s stll hgh f compared wth lnear and decson-drected detectors. Another negatve aspect of certan technques, such as the SD, s that the worst-case complexty for a partcular receved block can be much hgher than the average complexty [15]. In order to overcome these drawbacks, we consder the technque, whch s characterzed by a fxed complexty, and enables performance-complexty tradeoffs between the and the detectors. IV. LOCAL DETECTION FOR L-OFDM SYSTEMS A. Bascs The key dea of detecton s that we can perform the search by explorng only a subset of S, thus reducng the computatonal complexty wth respect to the full-search detector. Indeed, f a frst estmate ŝ of the transmtted symbol vector s s avalable at the recever, and f such an estmate s IEEE Communcatons Socety 371

3 farly accurate, we have a hgh probablty to refne our estmate by restrctng the search only to those vectors that are close (n some sense) to ŝ. Such an dea s not new n telecommuncaton systems, and has already been exploted for multuser detecton n DS-CDMA (see, e.g., [8]-[10]). owever, as t wll be clarfed later, the consdered L-OFDM context leads to desgn choces that are dfferent from those of DS- CDMA systems. Gven a symbol vector ŝ and an nteger selected from {0,..., }, we defne the neghborhood of ŝ of sze as the set S () sˆ = { s S d (,) s s ˆ }, (7) where d (, ˆ ) ss denotes the ammng dstance between s and ŝ,.e., the number of entres of s that are dfferent from the entres of ŝ. We defne the detector of sze assocated wth ŝ as sˆ ( ) = argmax{ Λ( s )}, (8) s that s, the detector constraned to the restrcted set S () ˆ s. In other words, the detector evaluates all the LLFs assocated wth the vectors that dffer at most entres from the frst estmate ŝ, selectng the symbol vector ŝ that produces the hghest lkelhood among them. Although n (7) other dstance measures could be employed, e.g., the Eucldean dstance, the ammng dstance allows us to predct exactly the cardnalty of S () ˆ s, gven by C = ( M 1) = 0, (9) whch turns out to be ndependent of ŝ. Therefore, the number of LLFs to be evaluated n (8) can be easly controlled by a convenent choce of the neghborhood sze. Equaton (9) suggests that the computatonal complexty s polynomal n ( M 1), of order equal to the neghborhood sze, as t wll be detaled n Subsecton IV-C. One of the most nterestng propertes of the detectors s the followng. roperty 1: For any fxed ntal estmate ŝ, t holds true that r{ sˆ ( ) s} r{ sˆ ( ) s }, <, (10) S that s, the block-error probablty for the detector of sze s not hgher than the block-error probabltes of all the detectors of smaller sze. roof: Snce S () ˆ ˆ s S 1() s, t holds true that Λ( sˆ ( )) Λ( s ˆ ( 1)), and hence () sˆ r{ sˆ ( ) = s} r{ sˆ ( 1) = s }, (11) whch easly leads to (10). In partcular, for = 0, roperty 1 states that applyng an search to the output ŝ of any suboptmal detector does not produce a block-error probablty ncrease. Therefore, n most cases, also the wll be reduced, thus motvatng the approach. It should be ponted out that usng the output of an detector as the ntal estmate for another s qute common to many approaches for DS-CDMA [8]-[10]. Indeed, n DS-CDMA the number of users, whch plays the role of the precoder sze n L-OFDM, can be very hgh, and therefore the neghborhood sze s forced to a value = 1 to lmt the complexty. As a consequence, nstead of ncreasng, detectors for DS-CDMA try to mprove the performance by teratng the detecton wth = 1. On the contrary, n OFDM for WLAN applcatons, the precoder sze may be very small, because the maxmum dversty gan can be acheved wth a precoder of sze = L ( L = 16 n WLAN scenaros [16]). Therefore, f the constellaton sze s not very hgh, detectors wth = are not very complex. B. - Detector In the maxmzatons of non-convex functons, the ntalzaton pont s often a crucal step, because an unlucky startng pont can lead to a local maxmum that s located very far from the global one. As a consequence, for an technque of sze, we would lke an ntal estmate wth at most errors. owever, such a condton s qute hard to guarantee even when usng the detector. Therefore, as an alternatve crteron, we could ask for a detector whose soft output vector contans at least entres that are close to the transmtted ones. Ths way we can force the detector to confne ts search to those vectors that dffer from the estmated one only on the remanng entres. If we select the MSE as the measure of closeness, and we restrct the choce among the lnear detectors for complexty reasons, the detector we are lookng for s, of course, the decoder. Indeed, the detector mnmzes the MSE of each symbol n the data vector s, ndependently of the others. Instead of the detector, [8] adopts the ZF detector as a frst stage. owever, n L-OFDM systems, ths detector performs poorly n the presence of deep fades. Alternatvely, [10] suggests to use a approach. Although the for s typcally smaller than for detecton, the suffers from error propagaton, a phenomenon that tends to concentrate the errors n few blocks. For ths reason, the technques seem to be less effectve when decson-drected approaches are used as startng schemes. Ths fact wll be confrmed by smulatons. C. Computatonal Complexty Reducton For a constellaton sze M >, the number C of LLFs to be evaluated may be too hgh even when moderate values of the neghborhood sze and of the precoder sze are employed. A possblty s to exclude from the search those vectors whose entres are not adjacent to those of ŝ. We defne the constellaton neghborhood of ŝ of sze and dstance D as S ˆ ˆ ˆ D, ( s) = { s S( s ) de( s, s) D = 1,..., }, (1) where d (, ˆ E s s ) represents the Eucldean dstance between the th entry s of s and the th entry s ˆ of ŝ. We can then defne the reduced constellaton (RC) detector of sze and dstance D assocated wth ŝ as sˆ ( ) = arg max{ Λ( s )}. (13) RC- s S D, As an example, for quaternary phase-shft keyng (QS), the choce D = excludes from the constellaton neghborhood all the vectors wth s = sˆ, thus reducng the effectve sze of () sˆ IEEE Communcatons Socety 37

4 the constellaton from M = 4 to m = 3. As far as performance s concerned, assumng the as frst stage, the reduced constellaton approach excludes those vectors wth components characterzed by the hghest MSEs. Such symbols should be less lkely to be correct, and therefore the RC-- detector should present a small performance loss wth respect to the - detector. A second possblty to reduce the complexty s to explot the partcular structure of some precoders desgned for L- OFDM. In ths case, rather than reducng the number of vectors n the neghborhood set, we smplfy the computaton of the LLF. For nstance, we may assume that the precoder matrx Θ s chosen to be untary, and that all ts entres [ Θ ], j have modulus equal to 1/. Ths class of precoders ncludes those desgned for lnear detecton [17], and those desgned for detecton [3][4], whch are expressed by 1 Θ = F dag(1,,..., ), (14) α α where s a power of two and α satsfes the equaton α = 1. In ths case, the computatonal complexty of the detector wth = 1 can be sgnfcantly reduced. Ths fact s explaned n the followng for BS. Lettng sˆ + e denote the vector obtaned by flppng the th entry of ŝ, where e = [0,...,0, sˆ,0,...,0] T, t holds true that T T T Λ ( sˆ+ e) = Λ ( sˆ) + Re( e y e sˆ) e e. (15) Snce e s non-zero only n the th poston, e T e turns out to be equal to 4[ ],. owever, snce = DΘ and [ Θ ], j = 1/, [ ], does not depends on, and T e e = 4tr( D D )/. ence, n order to fnd the most lkely among the vectors { sˆ + e, = 1,..., }, t s suffcent to T look for the maxmum value assumed by Re( e ( ˆ y s )), for = 1,...,. By defnng the dagonal matrx E= dag( sˆ ) = [ e1, e,..., e ] T, the detector has only to fnd the maxmum value of the vector v = ERe( ( y s ˆ)), (16) and successvely, f max( v) > 4 tr( D D ) /, t has to flp the symbol of ŝ correspondng to the poston of max( v ). As a consequence of (16), the complexty of such an detector s comparable wth that of decson-drected detectors. For other constellatons, smlar consderatons hold true wth mnor modfcatons. Before elaboratng further on the computatonal complexty of the - detector, we frst hghlght that a untary precoder also smplfes the detector computaton, because the matrx to be nverted n (6) s dagonal. Moreover, we also pont out that the detector can equvalently maxmze the relatve LLF Λ ( sˆ+ e) Λ( s ˆ), whch s easer to be evaluated than Λ ( sˆ + e ), where, for a generc constellaton, e represents an error vector contanng at most non-zero values. Therefore, by pluggng e n (15), and explotng both = DΘ and (14), t can be shown that the number of complex multplcatons requred for each receved block y can be smartly reduced to Nmult = 4log ( + )( m 1) = 0, (17) whch s equal to Nmult = 3 + 4log + 7 when = and BS ( m = ) s used. Moreover, log addtonal complex multplcatons are requred at the begnnng to compute, whch has to be updated only when the channel changes. From (17), by focusng on the BS case and assumng /, the computatonal complexty ncreases as O ( ). Thus, for =, the complexty of the - detector s below the average complexty of SD, whch s 3 roughly O ( ) [15], and below the complextes of DA and SD, whch are O ( ) [7], and O ( ) [6], respectvely. Ths fact motvates the usefulness of the proposed algorthm wth for L-OFDM systems. When >, the complexty of the - detector becomes comparable to those of near- algorthms, and hence a performance comparson wth such technques would be requred (due to the lack of space, ths comparson s not pursued here). D. Effect of a lot-aded Channel Estmaton Technque So far, we have mplctly assumed that the dagonal channel matrx D, contaned nto, s known to the recever. In practce, only an estmated verson of D s avalable, and therefore the detector should be obtaned by replacng the exact wth ts estmate ˆ = DΘ ˆ. In ths subsecton, our am s to modfy the detector of the frst stage n order to take nto account the channel estmaton errors, and thereby mprove the overall performance of the - detector. We assume that N pl plot subcarrers, equally spaced [18], are transmtted n the frst OFDM block. Snce we have assumed that the channel can have at most L taps, we choose Npl = L. At the recever, we assume channel estmaton (E), whch acheves the Cramér-Rao lower bound (CRLB) [19]. In ths case, the covarance matrx of the frequencydoman channel estmaton error ε = dag( Dˆ D ) can be expressed by E{ εε } = σ 1 wnnplff, where F s the N Npl matrx obtaned by selectng the frst N pl columns of F N. It n turns out that f { Npl /, n, n< log Npl}, the covarance matrx of the channel estmaton error n each precoded block s E{ εε } = σ w I. Ths mples that the channel estmaton error can be nterpreted as an addtve whte Gaussan error wth the same power of the thermal AWGN represented by w. In such a stuaton, we can defne the modfed (M) detector as n (6), where and σ w are smply replaced by Ĥ and σ w, respectvely. By smlar consderatons, we can also defne the modfed (M), and the modfed IC (MIC), to be used when the channel s estmated by the ad of plot tones. V. SIMULATION RESULTS In ths secton, we present smulaton results n order to assess the performance of the - detectors. We consder an OFDM system wth cyclc prefx of length L = 16, and N = 64 subcarrers, spaced wth one another by 31.5 kz. The chosen precoder s the one n (14). We use the channel model C of the IEEE 80.11a standard [16]. In ths model, each tap suffers ndependent Raylegh fadng, wth an exponentally decayng power delay profle and an rms delay spread equal to 150 ns. Fgs. 1- show the performance of varous detectors versus Eb / N 0, averaged over 400 channel realzatons, when IEEE Communcatons Socety 373

5 BS and a precoder sze = 8 are used. Fg. 1 shows that at = 10 3, the performance gan of the - detector wth = 1 wth respect to and detector s roughly 1.4 db and.8 db, respectvely, whle the loss wth respect to the detector s approxmately 1. db. Fg. 1 also ndcates that 1 teraton of the search gves small performance mprovement (0.3 db when = 10 3 ) wth respect to the non-teratve - detector, at the expense of doublng the complexty. More teratons are not effectve. Fg. shows that the performance obtaned by the - detector wth > 1 s very close to those of detecton. Although Fg. 1 suggests that the decson-drected detectors outperform the detector, Fg. ndcates that ther counterparts behave dfferently. Indeed, when = 1, the - detector outperforms both the detector wth ntalzaton (-) and wth IC ntalzaton (-IC), wth a gan of roughly 1 db when = Ths fact s clearly explaned by Fg. 3, whch plots versus k 6 the number of detected blocks wth k errors, when blocks are transmtted at Eb / N 0 = 10 db. Due to error propagaton, the produces a sgnfcant number of blocks wth more than one symbol error, whch are not recovered by a subsequent approach. On the contrary, most of the erroneous blocks of the detector contans only one error, and therefore n ths case the approach wth = 1 s qute effectve. Fg. 4 shows the performance, when QS wth Gray codng and = 16 are used. In ths case, the performance gans of the - detectors are smaller than n the prevous case. Ths s due to the fact that for QS the cardnalty of S s hgher than for BS. Ths means that n the QS case the search s performed n a smaller subset of the total set S, thus neglectng more possble solutons than for BS. owever, also n ths case the - detector wth = 1 outperforms the detector. Moreover, the RC- - detectors wth D = acheve the same performance of ther - counterparts, despte the smaller complexty (roughly half when = ). Fg. 5 compares the performance of,, and IC, wth the ones of ther modfed counterparts, n the presence of channel estmaton errors. BS, N pl = 16, and = 8 are used. When = 10 4, the M provdes 1. db gan wth respect to the. Fg. 6 depcts the of the - detector n the same smulaton scenaro of Fg. 5. It can be observed that the - approach seems to be effectve also n the presence of channel estmaton errors. VI. CONCLUSIONS In ths paper, we have consdered the detecton approach n L-OFDM systems. We have shown that the output of the detector s a convenent ntalzaton for the detector. The performance and complexty of the proposed - detector, whch fall between the ones of the and detectors, can be ncely adjusted by controllng the neghborhood sze. Smulaton results have demonstrated that the - detector wth mnmum (nontrval) neghborhood sze outperforms the approach, whle exhbtng comparable complexty. erformance mprovements obtaned by explotng the soft nformaton of the detector, as well as the thorough comparson wth near- technques, wll be the subject of future nvestgaton. REFERENCES [1] R. van Nee and R. rasad, OFDM for Wreless Multmeda Communcatons, Artech ouse, 000. [] Z. Wang and G. B. Gannaks, Complex-feld codng for OFDM over fadng wreless channels, IEEE Trans. Inform. Theory, vol. 49, pp , Mar [3] Z. Lu, Y. Xn, and G. B. Gannaks, Lnear constellaton precodng for OFDM wth maxmum multpath dversty and codng gans, IEEE Trans. Commun., vol. 51, pp , Mar [4] J. Boutros and E. Vterbo, Sgnal space dversty: a power- and bandwdth-effcent dversty technque for the Raylegh fadng channel, IEEE Trans. Inform. Theory, vol. 44, pp , July [5] E. Vterbo and J. Boutros, A unversal lattce code decoder for fadng channels, IEEE Trans. Inform. 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6 ZF IC - ( = 1) - ( = 1), 1 ter. - ( = 1), 7 ter Fgure 1. performance for BS ( = 1) -IC ( = 1) - ( = 1) - ( = ) - ( = 3) ZF IC - ( = 1) RC-- ( = 1) - ( = ) RC-- ( = ) Fgure 4. performance for QS. IC M M MIC Fgure. performance for BS Fgure 5. performance wth channel estmaton errors. Number of blocks wth k symbol errors IC M M MIC -M ( = 1) -M ( = ) k Fgure 3. Dstrbuton of the number of errors wthn the same data block Fgure 6. performance wth channel estmaton errors. IEEE Communcatons Socety 375