IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 11, NOVEMBER 2015 4501 Widely Liner Estimtion for Spce-Time-Coded GFDM in Low-Ltency Applictions Mximilin Mtthé, Lucino Leonel Mendes, Nicol Michilow, Dn Zhng, nd Gerhrd Fettweis, Fellow, IEEE Abstrct This pper presents solution for chieving trnsmit diversity with generlized fruency division multiplexing (GFDM). Compred to previous works, the proposed solution significntly improves symbol error rte (SER) performnce nd ltency, where both spects re crucil for future 5G cellulr networks. It is shown tht widely liner estimtion t the receiver side cn jointly ulize nd demodulte the spce-time encoded GFDM signl. Moreover, mximum rtio combining cn further increse the SER performnce with multiple receive ntenns. SER performnce is evluted in Ryleigh fding multipth chnnels. Index Terms Estimtion, modultion, trnsmit diversity, GFDM, tctile internet. I. INTRODUCTION DURING the pst decdes, the evolution of mobile communiction networks hs incresed their cpcity in terms of number of users nd throughput. However, only higher throughput will not be enough to ddress future chllenges foreseen for the fifth genertion (5G) of mobile networks. Low ltency hs been pointed out s n importnt ruirement to trigger new services, such s the Tctile Internet 1], 2]. Besides low ltency, low out-of-bnd (OOB) emissions, frgmented spectrum lloction nd relxed ruirements for synchronicity will lso be importnt in 5G systems 3]. Orthogonl Fruency Division Multiplexing (OFDM) hs proven to be robust nd relible modultion scheme for high dt rte communiction systems. The circulrity introduced by cyclic prefix (CP), which needs to be longer thn the chnnel dely profile, llows for fruency domin uliztion with single tp per subcrrier nd increses the system performnce over fding multipth chnnels. Due to orthogonlity of the Mnuscript received April 13, 2015; revised July 7, 2015 nd August 7, 2015; ccepted August 7, 2015. Dte of publiction August 13, 2015; dte of current version November 13, 2015. This work ws supported in prt by CNPq-Brsil nd Finep/Funttel/Rdiocommuniction Reference Center under Grnt No. 01.14.0231.00 hosted by Intel nd ws performed in the frmework of the FP7 project ICT-619555 RESCUE, which is prtly funded by the Europen Union. The ssocite editor coordinting the review of this pper nd pproving it for publiction ws S. Gezici. M. Mtthé, N. Michilow, D. Zhng, nd G. Fettweis re with the Vodfone Chir Mobile Communiction Systems, Technische Universität, 01062 Dresden, Germny (e-mil: mximilin.mtthe@ifn.et.tu-dresden.de; nicol. michilow@ifn.et.tu-dresden.de; dn.zhng@ifn.et.tu-dresden.de; gerhrd. fettweis@ifn.et.tu-dresden.de). L. L. Mendes is with the Telecommuniction Engineering Deprtment, Intituto Ncionl de Telecomunicções, 37540-000 St. Rit do Spuci-MG, Brzil (e-mil: lucino@intel.br). Color versions of one or more of the figures in this pper re vilble online t http://ieeexplore.ieee.org. Digitl Object Identifier 10.1109/TCOMM.2015.2468228 subcrriers, OFDM cn be esily integrted with spce-time coding (STC) 4] to obtin trnsmit diversity over time-vrint fruency-selective chnnels. Nevertheless, OFDM hs some drwbcks tht might hmper its usge in the 5G physicl lyer (PHY). OFDM exhibits high OOB emission, restricted to 35 dbc, which demnds for chnnel filtering to meet the emission msks imposed by regultion gencies. This is significnt disdvntge in flexible or frgmented spectrum lloction scenrio 3]. Reducing the time durtion of the OFDM frme to the low-ltency ruirements of 5G lso poses chllenge. Due to the CP overhed, the size of the chnnel impulse response does not llow the use of short OFDM symbols efficiently. Moreover, short time domin OFDM symbols produce wide subcrriers in the fruency domin, which re subject to fruency selectivity per subcrrier. In such conditions, the ttrctive fruencydomin uliztion (FDE) cnnot be esily implemented since OFDM hs resolution of only one tp per subcrrier in the fruency-domin 5]. Severl new wveforms re being proposed to ddress the ruirements of 5G networks 3], where Generlized Fruency Division Multiplexing (GFDM) 6] is one wveform cndidte. In this technique, block of dt symbols is trnsmitted per subcrrier nd ech subcrrier is circulrly convolved with pulse shping filter. A CP cn be dded to the GFDM frme to provide low-complexity FDE. The self-interference throughout the dt symbols, cused by using non-orthogonl, wellloclized pulse shping filters, cn be mitigted by liner or itertive pproches t the receiver side. GFDM cn chieve low OOB emission 7] nd it cn reduce the ltency on the PHY 8]. Becuse ech GFDM subcrrier is represented by multiple fruency smples, FDE is possible even when the chnnel is fruency selective per subcrrier 6]. The shortening of GFDM subsymbols is not problemtic becuse only single CP is dded for the entire frme. These properties mke GFDM promising cndidte for the 5G PHY lyer. Trnsmit diversity is key feture for the next genertion of mobile communiction systems 9]. The STC proposed by Almouti is simple solution to chieve this property. In previous works 6], time-reversl spce-time coding (TR- STC) is shown s fesible solution for GFDM, since the block structure of the signl in time domin llows the encoding of the wveform smples insted of the dt symbols. This pproch chieves full diversity gin nd cn be used s fruency division multiple ccess scheme if single gurd subcrrier is used between multiple users 10]. However, TR-STC-GFDM ruires two GFDM frmes to build the 0090-6778 2015 IEEE. 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4502 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 11, NOVEMBER 2015 codeword. Although this pproch presents good performnce in terms of symbol error rte (SER), it is clerly not optiml for low-ltency pplictions. To reduce ltency, STC cn be crried out on the dt symbols within single block. In previous works 11] liner receiver ws used to decode this spce-time code, leding to significnt SER performnce degrdtion due to remining chnnel intersymbol interference (ISI). The contribution of this pper is the ppliction of widely liner estimtion (WLE) 12] to minimize the effects of the ISI introduced by the multipth chnnel to the STC-GFDM SER performnce. This wy, both low ltency nd good SER performnce cn be chieved, however t the cost of incresed complexity t the receiver when compred to liner pproches 11]. Moreover, n pproch for using mximum rtio combining (MRC) with multiple receive ntenns is described, which further improves the SER performnce. Furthermore, the complexity of the proposed lgorithm is derived nd shown to be liner with the number of subcrries. The SER performnce of the proposed scheme is compred with STC-OFDM nd conventionl STC-GFDM over fding multipth chnnels bsed on the Extended Vehiculr A (EVA) 13] model. The WLE-STC-GFDM proposed in this pper is promising solution for pplictions where low ltency nd high robustness re needed nd computing power is vilble. One exmple scenrio is vehicle-to-vehicle communiction 14], where the terminls re not power-limited nd the short time window to estblish communiction between fst moving objects through double dispersive chnnel cn be chllenge. Note tht the present proposl is not limited to low-ltency pplictions. Insted, the ltency properties of the system depend on the frme structure of the GFDM signl. However, compred to the proposl in 10], only one GFDM block is necessry to chieve full trnsmit diversity, which is in prticulr suitble for low-ltency pplictions. The reminder of this pper is orgnized s follows: Section II presents the principles of GFDM. Section III shows how STC cn be pplied nd liner demodultor is shown, while Section IV derives the WLE for STC-GFDM nd nlyses its complexity. Section V provides the SER performnce nlysis over fruency-selective time-vrint chnnels nd, finlly, Section VI concludes this pper. II. GFDM BACKGROUND In GFDM block, N = MK dt symbols re trnsmitted on K subcrriers. Ech subcrrier is divided into M subsymbols. The complex-vlued dt symbols for one block re rrnged in mtrix D = (u 0 u 1... u M 1 ). (1) There, u m = (u 0,m u 1,m... u K 1,m ) T contins the K dt symbols tht re trnsmitted in the mth subsymbol. Ech symbol u k,m is trnsmitted on wveform g k,m n], which is derived from prototype pulse shping filter gn]. The prototype filter is circulrly shifted to the kth subcrrier nd mth subsymbol, such tht g k,m n] =g (n mk) mod N]exp (j2π kk ) n, (2) Fig. 1. Illustrtion of ISI nd ICI within GFDM block for K = 5, M = 5. Dt symbols re shown s overlpping dimonds, explining ISI nd ICI between trnsmitted dt. Note tht in ddition to the ISI nd ICI due to trnsmit filtering, multipth chnnel introduces more ISI, which cn be removed by fruency domin uliztion (FDE). For exmple, multipth chnnel would cuse d k,m 2 to interfere into the resource of d k,m. where n = 0, 1,...,N 1 is the time-index. Usully, gn] is rised cosine (RC) filter with rolloff α 6]. Accordingly, the trnsmit signl xn] is given by K 1 xn] = M 1 k=0 m=0 u k,m g k,m n]. (3) Eq. (3) cn be formulted to n uivlent mtrix expression x = Ad, (4) where the vector x = (xn]) T contins the smples of xn]. d = vec(d T ) holds the rows of D trnsposed nd stcked on top of ech other. A contins g k,m = (g k,m n]) T s its (km + m)th column. 1 The bove model cn be extended to the cse when only K on subcrriers re switched on with K on < K. In this cse, the columns corresponding to the switched off subcrriers re removed from A. Furthermore, then mtrix D is of dimension K on M. A CP with length N CP is dded to the trnsmitted signl to void interference between subsuent GFDM blocks. Since only one CP is needed for ll subsymbols, the CP overhed of GFDM is significntly reduced compred to OFDM 6]. On the receiver side, ssuming the chnnel impulse response shorter thn the CP, the received signl fter the CP removl is given by r = HAd + w, (5) where H is the circulnt chnnel mtrix nd w is the dditive white Gussin noise (AWGN) vector with vrince σw 2.The modultion process introduces intercrrier interference (ICI) nd ISI between the trnsmitted dt symbols, s shown in Fig. 1 nd the multipth chnnel further increses the ISI. The occuring interference needs to be combted t the receiver in order to provide cceptble SER performnce. For exmple, the trnsmitted dt symbols cn be estimted by liner opertion using zero-forcing receiver (ZFR) or minimum men squre error receiver (MMSER) 6]. Considering ZFR, the estimted dt symbols re given by ˆd = B ZF H 1 r, (6) 1 Throughout the pper we ssume zero-bsed indexing for mtrices.
MATTHÉ et l.: WIDELY LINEAR ESTIMATION FOR SPACE-TIME-CODED GFDM IN LOW-LATENCY APPLICATIONS 4503 Fig. 2. Block digrm of the proposed STC-WLE-trnsceiver. where B ZF = A + = (A H A) 1 A H (7) is the demodultion mtrix, contining the ZF filters γ k,m n]. Note tht H is digonlized by the Fourier trnsform nd hence FDE cn be efficiently performed. The MMSER jointly performs chnnel uliztion nd demodultion. Its demodultion mtrix is given by B MMSE = ( R w + A H H H HA ) 1 A H H H, (8) where ( ) H denotes conjugte trnspose nd R w is the noise covrince mtrix. The estimted symbols re given by ˆd = B MMSE r. (9) The MMSER reduces the noise enhncement for low signl to noise rtios (SNR) nd it cn mitigte the self-generted interference introduced in the modultion process when nonorthogonl wveforms re employed 15]. However, it is not unbised nd hence rescling opertion before constelltion detection is necessry. III. SPACE-TIME CODE FOR GFDM Trnsmit diversity is n importnt feture for mobile communiction systems to provide robustness ginst fruency nd time fding. Almouti hs proposed simple nd effective STC to obtin diversity in single crrier systems by using two trnsmit ntenns without reduction of the overll dt rte compred to single-ntenn trnsmission 4]. Almouti s STC scheme is esily integrted with orthogonl multicrrier systems 16], but it is chllenge to pply this technique to non-orthogonl systems 17]. In order to keep the overll ltency on the PHY low, no dditionl dely should be introduced by the STC. Hence, in the proposed solution, STC is crried out within single GFDM block by spce-time encoding the dt symbols trnsmitted by two subsuent subsymbols on ech subcrrier. This pproch ruires n even number M of subsymbols in ech GFDM block to be spce-time encoded. However, in 18] it ws shown tht GFDM ruires n odd number of subsymbols when common pulse shping filters re used. Therefore, the first subsymbol in the GFDM block is not encoded but left empty for pilots or synchroniztion signling. The corresponding loss in spectrl efficiency is compensted by the more efficient use of the CP in most useful pplictions. In fct, given number of subcrriers K, STC-GFDM with one empty subsymbol will be more fruency efficient thn STC-OFDM if M > K + 2N CP. (10) N CP Note tht pilots nd synchroniztion resources for OFDM re not considered here, which would further reduce its spectrl efficiency. The block digrm of the proposed STC trnsmitter is depicted in the left prt of Fig. 2. Let D s contin ll but the first column of D nd let D (1) s = ] u 1 u 2... u M 2 u M 1 = D s D (2) s = u 2 u 1... u M 1 u ] (11) M 2 = D s P s be the spce-time encoded dt to be trnsmitted by the two trnsmit ntenns. There, P s is given by ] 0 1 P s = I M 1, (12) 2 1 0 where I n is the n n identity mtrix nd denotes the mtrix Kronecker product. Let A s be shortened A, wherethek columns relting to the first subsymbol re discrded. Accordingly, the signl x (i) trnsmitted by the ith ntenn before CP ddition is given by ( x (i) = A s vec D (i) s T ). (13) Let F N nd FN H denote the N-point unitry discrete Fourier trnsform (DFT) nd its inverse opertion. The received signl t the jth receive ntenn fter removing the CP nd moving to fruency domin is given by r = Ĥ (1,j) ] Ĥ (2,j) d s P d s ] + w. (14) There, Ĥ (i,j) = F N H (i,j) A s, H (i,j) denotes the circulnt chnnel mtrix from the ith trnsmitting to the jth receiving ntenn, d s = vec(d T s ), P = I K P T s nd w is the fruency domin AWGN t the jth receiving ntenn. Liner spce-time decoding cn be used to recover the trnsmitted informtion, however it suffers from the ISI between GFDM subsymbols, introduced by the multipth chnnel, cf. Fig. 1, becuse no FDE cn be crried out prior to Almouti combining. Insted, the received time domin GFDM blocks re first demodulted with the ZFR yielding d = B ZF F H N r. (15)
4504 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 11, NOVEMBER 2015 Note tht FN H is only introduced for nottion purposes nd not necessry in rel implementtions, s the signl is vilble in time domin nywy. In order to chieve diversity gin, d is restructured into K (M 1) mtrix with columns ũ m which re combined ccording to 11] Jj=1 H (1,j) K ũ m û m = + H(2,j) K ũ m+1 Jj=1 H (1,j) 2 + H (2,j) 2 m odd, û m = K Jj=1 H (1,j) K Jj=1 H (1,j) K ũ m H(2,j) K ũ m 1 K 2 + H (2,j) K 2 m even, (16) where J is the number of receive ntenns, H (i,j) K is the K-point Fourier trnsform of the chnnel impulse response between the ith trnsmit nd jth receive ntenn nd opertions re to be understood element-wise. Note tht the chnnel-imposed ISI between the subsymbols trnsmitted on ech subcrrier is not removed by combining the STC fter the GFDM demodultion. Thus, the estimted subsymbols û m obtined by (16) hve residul ISI tht cn severely reduce the system performnce depending on the chnnel impulse response, leding to n error floor in the SER performnce curve 11]. IV. WIDELY LINEAR EQUALIZATION FOR STC-GFDM In order to improve on the STC-GFDM performnce, the multipth chnnel ISI needs to be ulized. This cn be ccomplished by jointly demodulting, ulizing nd combining the received signl with the help of widely liner estimtor, where the block digrm of the processing chin is given in Fig. 2. Widely liner estimtion outperforms conventionl liner estimtion when either the estimnd or the mesurement is n improper process 12]. The following section shows tht the received signl is improper nd the widely liner estimtor for STC-GFDM is derived. Subsuently, the structure of the resulting ution system is nlyzed nd low-complexity solution is explined. A. Derivtion of Widely Liner Estimtors Mpping n i.i.d. bit suence onto rottionlly invrint constelltion with unit verge symbol energy provides E d s d H ] s = IK(M 1) E d s d T ] s = 0K(M 1), (17) where 0 n is n n null mtrix. Further, ssuming the chnnel impulse responses between the ith trnsmit nd jth receive ntenns re invrint during the trnsmission of one GFDM block, which is resonble for short block durtions, the utocorreltion Ɣ of r is given by Ɣ = E r r H] (18) = Ĥ (1,j) ] ] Ĥ (2,j) Ĥ (1,j)H P P H Ĥ (2,j)H + σ 2 w I MK. (19) Similrly, the pseudoutocorreltion C is given by C = E r r T] (20) ] ] = Ĥ (1,j) Ĥ (2,j) P T Ĥ (2,j)T P. (21) Ĥ (1,j)T Note tht PP H = PP T = I. SinceC = 0, r is n improper (non-circulr) process nd WLE of d s cn improve the estimtion performnce. Compred to liner estimtor, widely liner estimtor jointly processes the received signl nd its conjugte to estimte the trnsmitted dt by ˆd s = ] U H ] r. (22) V r The filter coefficients U nd V re chosen to minimize the men squred error (MSE) between d s nd ˆd s nd re solutions to the liner system 12] Ɣ C ] ] ] U C Ɣ V =, (23) F where nd ] = E r d H s = Ĥ (1,j) (24) ] = E r d T s = Ĥ (2,j) P. (25) The solution of (23) is given by U = S 1 ( C Ɣ 1 ) V = S 1 ( C Ɣ 1 ), (26) where S = Ɣ C Ɣ 1 C is the Schur complement of Ɣ in F. Since Ĥ (i,j) is tll mtrix, Ɣ becomes singulr when σ w 0 nd hence zero forcing (ZF) estimtion cnnot be directly derived. Insted, the system model (14) for the widely liner estimtion problem is reformulted to liner estimtion problem of double size ccording to ] r r r = Ĥ ds d s ] d w + w w ], (27) where Ĥ = Ĥ(1,j) Ĥ (2,j) ] P Ĥ (2,j) P Ĥ (1,j). (28) The liner minimum men squre error (LMMSE) estimtor for d in (27) is given by ˆd (Ĥ,MMSE = ĤH ĤH ) 1 + σw 2 I B MMSE r. (29)
MATTHÉ et l.: WIDELY LINEAR ESTIMATION FOR SPACE-TIME-CODED GFDM IN LOW-LATENCY APPLICATIONS 4505 Direct clcultion shows tht Ĥ ĤH + σw 2 I = F from (23) nd hence (29) is uivlent to (22) nd (23). Writing the LMMSE estimtor in (29) to its lternte form 19] results in ˆd (ĤH,MMSE = Ĥ ) 1 + σ w 2 I ĤH r. (30) This reformultion llows the derivtion the ZF estimtor by ˆd (ĤH,ZF = Ĥ ) 1 ĤH r = B ZF r, (31) where B ZF = Ĥ+ Ĥ is the Moore-Penrose pseudo inverse of. From the Guss-Mrkov theorem 19] it is known tht (31) is the best liner unbised estimtor (BLUE) for d nd is hence the best widely liner unbised estimtor for d s. T Note tht ˆd,( ) = ˆd T H s,( ) ˆd s,( )] contins redundnt informtion nd hence only the upper hlf of B ( ), denoted by B s,( ), is ruired in ctul processing. Anlogously, widely liner MMSE nd ZF estimtors cn be derived when J receive ntenns re jointly combined. The system model chnges to r (1) Ĥ (1) r (2) ạ Ĥ (2) w (1) w (2) = d +.. (32). r (J) r Ĥ (J) Ĥ w (J) w nd widely liner MMSE nd ZF estimtors re given by ˆd,MMSE = Ĥ H (Ĥ Ĥ H + σ 2 w I ) 1 r ˆd,ZF = Ĥ + r. (33) This pproch considerbly increses the system size nd computtionl complexity. Alterntively, the ˆd cn be estimted seprtely for every receiving ntenn nd then combined, weighted by the qulity of the chnnels. In the following the ZF MRC receiver is derived, the computtion of the MMSE MRC is strightforwrd but ruires hevier nottion. For the ZF receiver we find tht ˆd s,zf = d s + B s,zf w (34) nd thus the MSE of the estimted dt uls ( ( )( ) ]) e = dig E d s,zf d s d H s,zf d s (35) ( ) = σw 2 dig B s,zf BH s,zf. (36) The opertor dig( ) returns the digonl of mtrix rgument nd returns digonl mtrix for vector rgument. The Fig. 3. Smple uivlent chnnel H for M = 9, K = 8 using RC filter with α = 1. estimted dt symbols from the J receiving ntenns re now linerly combined, weighted by their inverse MSE s = dig (e )] 1 (37) ccording to 1 J ˆd s = s j=1 Note tht ˆd s = ˆd (1) s for J = 1. J s ˆd. (38) B. Complexity Anlysis According to (29), the widely liner MMSE nd ZF estimtors for d s ruire to solve liner ution system of size 2N. The ppliction of generl-purpose solvers for such systems ruires computtionl complexity of cubic order in both the number of subcrriers nd subsymbols which is prohibitively complex for low-ltency pplictions. In wht follows we nlyze the structure of the ution systems nd show tht, due to the sprsity of the trnsmit filter in the fruency domin, solution cn be found with liner complexity in the number of subcrriers, where the number of complex opertions is considered s figure of merit. Consider the uivlent chnnel mtrix Ĥ in (28). When using bnd-limited filter tht only hs B non-zero coefficients in the fruency domin, F N A s is sprse mtrix with B entries per column. For exmple, when using RC filter with rolloff α, (F N g k,m ) n N = 0when n km >(1 + α) M 2 nd B = (1 + α)m. Accordingly, j=1 F N H (i,j) A s = F N H (i,j) F H N F NA s (39) obeys the sme sprsity since the chnnel mtrix is digonlized by the DFT. Note tht due to periodicity properties of the DFT F N g 0,m hs contributions round n = 0ndn = N 1. Since P only opertes within isolted subcrriers, the blocks of Ĥ re ully sprse. An illustrtion is given in Fig. 3. s
4506 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 11, NOVEMBER 2015 Let T 2U be 2U 2U permuttion mtrix given by TABLE I CP EFFICIENCY AND NEF FOR OFDM AND GFDM T 2U =e 0 e U e 1 e U+1... e U 1 e 2U 1 ] T (40) where e i is zero column vector of length 2U with 1 t its ith position. With N on = K on (M 1) permuttion is pplied to (27) by T 2N r r = T 2N Ĥ TT 2N on } {{ } Q T 2Non d d +w (41) where Q is sprse mtrix of size 2N 2N on where ech column only hs 2B non-zero elements. Eq. (41) is now solved for d by ( ( ˆd ) ) = Q H 1 ( ) Q + σn 2 I Q H r (42) nd subsuently the inverse permuttion is pplied to cquire the originl d. However, s the present permuttions only describe the order of vribles in the system, they re not ccounted for complexity. On the other hnd, the clcultion of (Q ) H r ruires only 2BN complex multiplictions. (Q ) H Q is positive definite hermitin bnd digonl mtrix with periodic boundry conditions with S = 4(M 1) 1 superdigonls since only the subsymbols of djcent subcrriers overlp in the fruency domin. The periodic boundry condition exists due to the edge subcrriers, tht wrp round in the discrete fruency domin. Accordingly, such subcrriers simultneously hve fruency components t the lowest nd highest fruencies, mking them unsuitble for spectrl locliztion. Furthermore, similr to OFDM, empty edge crriers cn be used to reduce emission into djcent systems. But, due to good fruency-locliztion of GFDM, much fewer empty crriers would be needed for this purpose. Furthermore, without edge crriers, i.e. in cse of K on < K, the periodic boundries vnish since the lst subcrrier does not overlp with the first one. Then, the bnd digonl structure of (42) cn be solved with N on ((S 2 + 1) + 4S) complex multiplictions 20, LAPACK zpbsv]. If K = K on, the periodic boundry conditions cn be overcome with the ppliction of the Woodbury formul 21] nd the solution is ccomplished with n extr effort in the order of O(K M 3 ) complex multiplictions which is still liner with the number of subcrriers. Hence, by exploiting the structure of the ution system, the computtionl effort cn be significntly reduced to O(K M 3 ) compred to O(K 3 M 3 ) when generl-purpose solvers re employed. In the beneficil cse of empty edge crriers, even the ppliction of the Woodbury formul is not necessry t ll nd complexity reduces to O(K M 2 ) V. SER PERFORMANCE ANALYSIS An pproximtion for the SER performnce over fding multipth chnnels for STC-OFDM 22] nd STC-GFDM with QAM modultion nd ZFR 6] is given by ( μ ) (1 ) 2 2 1 ϕ 2J p e = 4 2 2 μ 2 2J 1 u=0 ( )( ) 2J 1 + u 1 + ϕ u, (43) u 2 where μ is the number of bits per symbol of the QAM constelltion nd 3R ϕ = CP σr 2 E s 2(2 μ 1)ξN 0 + 3R CP σr 2 (44) E s is the uivlent verge signl to noise rtio. E s nd N 0 re the verge energy of the QAM constelltion nd the noise spectrl density, respectively. R CP is the rte reduction due to CP with N CP smples or vcnt subsymbols nd ξ is the noise enhncement fctor (NEF). These prmeters tke different vlues for STC-OFDM nd STC-GFDM, s presented in Tble I. The uivlent prmeter for the multipth Ryleigh chnnel is given by I 1 σr 2 = σr 2 h 2 i, (45) where h is the chnnel impulse response with length I nd σ 2 r the prmeter of the Ryleigh distributed tps. Fig. 4() compres the performnce of uncoded STC-OFDM, STC-GFDM nd WLE-STC-GFDM with the configurtion presented in Tble II where (43) is used s reference for the simultions results. Tble III presents the non zero tps of the chnnel dely profile bsed on the EVA model 13], ssuming the smpling fruency presented in Tble II. Fig. 4() shows tht the theoreticl performnce of STC-OFDM nd STC-GFDM re similr. STC-GFDM mkes better use of the CP, but the vcnt first subsymbol reduces the overll efficiency to the sme level chieved by STC-OFDM. STC-GFDM hs poor performnce when conventionl combining is used fter demodultion due to ISI mong the subsymbols 11], which leds to cler error floor. The WLE cn remove the ISI during the demodultion nd combining process, leding to significntly improved performnce close to the theoreticl curve, but still slight degrdtion is present. Fig. 5 shows the rtio between the MSE per subcrrier of WLE-STC-GFDM nd STC-OFDM for rndom chnnel reliztions. Apprently, the rtio is not constnt nd there re i=0
MATTHÉ et l.: WIDELY LINEAR ESTIMATION FOR SPACE-TIME-CODED GFDM IN LOW-LATENCY APPLICATIONS 4507 Fig. 4. Simulted system performnce of STC-OFDM nd STC-GFDM over mobile chnnel. () Uncoded SER. (b) RS-encoded BER. TABLE II GFDM CONFIGURATION TABLE IV PARAMETERS OF THE REED SOLOMON CODE TABLE III CHANNEL DELAY PROFILE USED FOR THE PERFORMANCE ANALYSIS Fig. 5. Rtio of MSE per subcrrier between GFDM nd OFDM. The rtio ws evluted for 1000 chnnel reliztions, ech one shown in () s one line. The histogrm in (b) shows the number of occurrences of ech rtio, ignoring the outliers. chnnel reliztions where WLE-STC-GFDM hs lrger MSE thn STC-OFDM which is the reson, why WLE-STC-GFDM devites from the theoreticl curve. However, this devition only ppers for high SNR where the SER is below 10 3 nd cn be esily combted with forwrd error correction. Exemplry, we choose simple Reed Solomon (RS) code 23], whose prmeters from Tble IV hve been used to fit one GFDM block, such tht ech GFDM block cn be decoded independently from other GFDM blocks. As cn be seen from Fig. 4(b), its limited error correction cpbility is lredy sufficient to combt the performnce loss observed in the uncoded cse. This implies the usbility of more powerful chnnel codes such s turbo or LDPC codes with short nd vrible code-block length 24], 25] tht cn lso stisfy the low-ltency constrints. Given their wide ppliction in tody s communiction systems, they cn be more suitble thn RS codes due to higher efficiency nd simple configurtion for other pplictions. The simultion results presented in Fig. 4(b) show tht the chnnel coding is very effective to ddress the remining performnce loss of the WLE-STC-GFDM scheme nd the 0.3 db gp between STC-GFDM nd STC-OFDM due to the more efficient use of the CP (considering the empty subsymbol) is clerly present for high SNR. Fig. 4() lso shows the WLE-STC-GFDM performnce when two ntenns re used in the receiver side, where the signl on ech receiving ntenn hs been processed seprtely nd is then combined by (38). As shown in Fig. 4(), 2 2 STC-GFDM does not hve the diversity loss observed in the 2 1 cse. A reson is given by (38), where subcrriers with high MSE re outweighed by subcrriers with smller MSE, nd hence the influence of bd subcrriers is reduced.
4508 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 11, NOVEMBER 2015 VI. CONCLUSION This pper hs presented n dvnced pproch on demodulting spce-time encoded GFDM signl. Spce-time encoding ws crried out within GFDM block in order to keep the overll system ltency. This wy trnsmit diversity cn be chieved without incresing the PHY ltency compred to single ntenn trnsmission. At the receiver, the GFDM block is decoded with the help of widely liner estimtor which provides significnt gins compred to previous works 11]. The proposed scheme reches nerly-optiml performnce compred to OFDM, with slight degrdtions in high SNR regions (SER < 10 3 ). These devitions cn be combted with forwrd error coding. The scheme cn be combined with MRC pproch t the receiver where the symbols from the ntenns re linerly combined fter WLE hs been crried out per ntenn. This wy, the mtrix inversion complexity for the WLE is kept constnt regrdless of the number of receive ntenns. The complexity nlysis revels tht processing in the fruency domin llows the system to be solved with liner complexity in the number of subcrriers nd is s such suitble for low-ltency implementtions. ACKNOWLEDGMENT The computtions were performed on computing cluster t the Center for Informtion Services nd High Performnce Computing (ZIH) t TU Dresden. REFERENCES 1] G. P. Fettweis, The tctile internet: Applictions nd chllenges, IEEE Veh. Technol. Mg., vol. 9, no. 1, pp. 64 70, Mr. 2014. 2] G. Fettweis nd S. Almouti, 5G: Personl mobile internet beyond wht cellulr did to telephony, IEEE Commun. Mg., vol. 52, no. 2, pp. 140 145, Feb. 2014. 3] G. Wunder et l., 5GNOW: Non-orthogonl, synchronous wveforms for future mobile pplictions, IEEE Commun. Mg., vol. 52, no. 2, pp. 97 105, Feb. 2014. 4] S. Almouti, A simple trnsmit diversity technique for wireless communictions, IEEE J. Sel. Ares Commun., vol. 16, no. 8, pp. 1451 1458, Oct. 1998. 5] M. Qin nd R. Blum, Properties of spce-time codes for fruencyselective chnnels, IEEE Trns. Signl Process., vol. 52, no. 3, pp. 694 702, Mr. 2004. 6] N. Michilow et l., Generlized fruency division multiplexing for 5th genertion cellulr networks, IEEE Trns. Commun., vol. 62, no. 9, pp. 3045 3061, Sep. 2014. 7] M. Mtthé, N. Michilow, I. Gspr, nd G. Fettweis, Influence of pulse shping on bit error rte performnce nd out of bnd rdition of generlized fruency division multiplexing, in Proc. IEEE ICC Workshop, Sydney, NSW, Austrli, 2014, pp. 43 48. 8] I. Gspr et l., LTE-comptible 5G PHY bsed on generlized fruency division multiplexing, in Proc. ISWCS, Brcelon, Spin, 2014, pp. 209 213. 9] J. Östmn, W. Yng, G. Durisi, nd T. Koch, Diversity versus multiplexing t finite blocklength, in Proc. IEEE ISWCS, Aug. 2014, pp. 702 706. 10] M. Mtthe et l., Multi-user time-reversl STC-GFDMA for future wireless networks, EURASIP J. Wireless Commun. Netw., vol. 2015, no. 1, pp. 1 8, My 2015. 11] M. Mtthe, L. L. Mendes, nd G. Fettweis, Spce-time coding for generlized fruency division multiplexing, in Proc. Eur. Wireless 20th Eur. Wireless Conf., My 2014, pp. 1 5. 12] B. Picinbono nd P. Chevlier, Widely liner estimtion with complex dt, IEEE Trns. Signl Process., vol. 43, no. 8, pp. 2030 2033, Aug. 1995. 13] F. Rezei, M. Hempel, nd H. Shrif, LTE PHY performnce nlysis under 3GPP stndrds prmeters, in Proc. IEEE 16th Int. Workshop CAMAD, Kyoto, Jpn, Jun. 2011, pp. 102 106. 14] Y. Jeong, J. W. Chong, H. Shin, nd M. Win, Intervehicle communiction: Cox-fox modeling, IEEE J. Sel. Ares Commun., vol. 31, no. 9, pp. 418 433, Sep. 2013. 15] N. Michilow, S. Krone, M. Lentmier, nd G. Fettweis, Bit error rte performnce of generlized fruency division multiplexing, in Proc. 76th IEEE VTC Fll, Quebec City, QC, Cnd, Sep. 2012, pp. 1 5. 16] W. Xudong, W. Nn, nd G. Chunli, Generlised closed-form symbol error rte nlysis for orthogonl spce-time block coded ofdm system, Commun., Chin., vol. 10, no. 9, pp. 155 164, Sep. 2013. 17] C. Lélé, P. Siohn, nd R. Legouble, The Almouti scheme with CDMA-OFDM/OQAM, EURASIP J. Adv. Signl Process., vol. 2010, pp. 1 14, 2010. 18] M. Mtthe, L. Mendes, nd G. Fettweis, Generlized fruency division multiplexing in gbor trnsform setting, IEEE Commun. Lett., vol. 18, no. 8, pp. 1379 1382, Aug. 2014. 19] S. M. Ky, Fundmentls of Sttisticl Signl Processing: Estimtion Theory, ser. Prentice Hll Signl Processing Series. Englewood Cliffs, NJ, USA: Prentice Hll, 1993, no. 1. Online]. Avilble: http://books.google.de/books?id=fwesqaacaaj 20] The Numericl Algorithms Group, (NAG), Oxford, U.K., The NAG Librry. Online]. Avilble: www.ng.com 21] W. H. Press, S. A. Teukolsky, W. T. Vetterling, nd B. P. Flnnery, Numericl Recipes 3rd Edition: The Art of Scientific Computing, 3rded. New York, NY, USA: Cmbridge Univ. Press, 2007. 22] M. K. Simon, Digitl Communiction over Fding Chnnels, 2nd ed., ser. Wiley Series in Telecommunictions nd Signl Processing. Hoboken, NJ, USA: Wiley-Interscience, 2005. 23] W. E. Ryn nd S. Lin, Chnnel codes: Clssicl nd modern. Cmbridge, U.K.: Cmbridge Univ. Press, 2009. 24] M. Lndolsi, A comprtive performnce nd complexity study of shortlength LDPC nd turbo product codes, in Proc. 2nd Int. Conf. Inf. Commun. Technol., 2006, vol. 2, pp. 2359 2364. 25] IEEE 802.16e LDPC Encoder/Decoder Core, TurboBest, Kdim, Isrel, 2006. Online]. Avilble: http://www.turbobest.com/ WhitePper80216eLDPC.pdf Mximilin Mtthé received the Dipl.-Ing degree in electricl engineering from Technicl University Dresden (TU Dresden), Dresden, Germny, in 2013. He is currently pursuing the Ph.D. in the Vodfone Chir Mobile Communiction Systems t TU Dresden. During his studies, he focused on mobile communiction systems nd communiction theory. He performed his internship t Ntionl Instruments Dresden nd worked on the design nd implementtion of mesurement site for LTE test UEs. In his Diplom Thesis he concentrted on wveform design for flexible multicrrier trnsmission systems. His reserch focuses on the design nd evlution of MIMO rchitectures for future cellulr networks. Lucino Leonel Mendes received the B.Sc. nd M.Sc. degrees in electricl engineering from Intel, Brzil, in 2001 nd 2003, respectively. In 2007, he received the doctor s degree in electricl engineering from Unicmp, Brzil. Since 2001, he hs been Professor t Intel, Brzil, where he hs cted s Technicl Mnger of the hrdwre development lbortory from 2006 to 2012. He hs coordinted the Mster Progrm t Intel nd severl reserch projects funded by FAPEMIG, FINEP, nd BNDES. He is post-doc visiting resercher, sponsored by CNPq-Brsil, t Vodfone Chir Mobile Communictions Systems t the Technicl University Dresden since 2013. His min re of reserch is wireless communiction nd currently he is working on multicrrier modultions for 5G networks nd future mobile communiction systems.
MATTHÉ et l.: WIDELY LINEAR ESTIMATION FOR SPACE-TIME-CODED GFDM IN LOW-LATENCY APPLICATIONS 4509 Nicol Michilow received the Dipl.-Ing. degree in electricl engineering with focus on wireless communictions nd informtion theory from Technicl University Dresden, Dresden, Germny, in 2010. From 2008 to 2009, he worked in the R&D Deprtment of Ashi Ksei Corportion, Jpn, developing signl processing lgorithms for sensor dt nlysis. Since 2010, he hs been Reserch Associte t the Vodfone Chir t TU Dresden, pursuing Dr.-Ing. degree. His scientific interests re focused on flexible nd non-orthogonl multi-crrier wveforms for next genertion cellulr systems. During his time t the Vodfone Chir, he contributed to the FP7 projects 5GNOW, CREW, QOSMOS, EXALTED, nd ws prt of the RF Led User Progrm with Ntionl Instruments. Gerhrd Fettweis (F 09) received the Ph.D. degree from RWTH Achen, Germny, in 1990. Therefter he ws t IBM Reserch nd then t TCSI Inc., USA. Since 1994, he hs been Vodfone Chir Professor t Technicl University Dresden, Dresden, Germny, with his min reserch interest on wireless trnsmission nd chip design. He is n Honorry Doctorte t Technicl University Tmpere. As repet entrepreneur he hs co-founded 11 strtups so fr. He hs setup funded projects in size of close to EUR 1/2 billion, notbly he runs the Germn Science Foundtion s CRC HAEC nd COE cfaed. He is ctively involved in orgnizing IEEE conferences, nd hs been TPC Chir of ICC 2009 nd TTM 2012, Generl Chir of VTC Spring 2013, nd DATE 2014. Dn Zhng received the B.Sc. degree in electricl engineering from Zhejing University, Chin, in 2004, nd the M.Sc. nd the Dr.-Ing. degree in electricl engineering nd informtion technology from RWTH Achen University, Achen, Germny, in 2007 nd 2013, respectively. Her reserch interests re in the re of sttisticl signl processing, communiction theory, nd optimiztion theory. During her Ph.D. studies, she focused on itertive receiver designs. Since September 2014, she hs worked s Postdoctorl Resercher t the Vodfone Chir Mobile Communictions Systems t Technicl University Dresden, Dresden, Germny. She hs worked on vrious ntionl nd interntionl reserch projects with publiction in journls, conference proceedings, nd workshops. Her current reserch focus is on the trnsceiver design of the wveform GFDM (GFDM) with the objective of providing new ir interfce for 5G networks.