IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 2, FEBRUARY

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1 IEEE JOURAL O SELECTED AREAS I COMMUICATIOS, VOL. 31, O. 2, FEBRUARY Spectrally Effcent Tme-Frequency Tranng OFDM for Moble Large-Scale MIMO Systems Lnglong Da, Zhaocheng Wang, and Zhxng Yang Abstract Large-scale orthogonal frequency dvson multplexng OFDM) multple-nput multple-output MIMO) s a promsng canddate to acheve the spectral effcency up to several tens of bps/hz for future wreless communcatons. One key challenge to realze practcal large-scale OFDM MIMO systems s hgh-dmensonal channel estmaton n moble multpath channels. In ths paper, we propose the tme-frequency tranng OFDM TFT-OFDM) transmsson scheme for largescale MIMO systems, where each TFT-OFDM symbol wthout cyclc prefx adopts the tme-doman tranng sequence TS) and the frequency-doman orthogonal grouped plots as the tmefrequency tranng nformaton. At the recever, the correspondng tme-frequency jont channel estmaton method s proposed to accurately track the channel varaton, whereby the receved tme-doman TS s used for path delays estmaton wthout nterference cancellaton, whle the path gans are acqured by the frequency-doman plots. The channel property that path delays vary much slower than path gans s further exploted to mprove the estmaton performance, and the sparse nature of wreless channel s utlzed to acqure the path gans by very few plots. We also derve the theoretcal Cramér-Rao lower bound CRLB) of the proposed channel estmator. Compared wth conventonal large-scale OFDM MIMO systems, the proposed TFT-OFDM MIMO scheme acheves hgher spectral effcency as well as the coded bt error rate performance close to the ergodc channel capacty n moble envronments. Index Terms large-scale MIMO, OFDM, spectral effcency, tme-frequency tranng TFT), tme-frequency jont channel estmaton. I. ITRODUCTIO ORTHOGOAL frequency dvson multplexng OFDM) and multple-nput multple-output MIMO) are wdely recognzed as two fundamental physcal layer technologes for future wreless communcatons due to the outstandng capablty to combat multpath fadng and hgh spectral effcency [1], [2]. Currently, most of the lterature as well as wreless standards e.g., IEEE n [3], IEEE m [4], 3GPP long term evoluton LTE) [5], etc.) manly address a small number of transmt antennas e.g., 2, or 4), whereby the spectral effcency of about 10 bps/hz or less can be acheved [1] [5]. However, large-scale MIMO Manuscrpt receved 3 February 2012; revsed 15 June Ths work was supported by atonal Key Basc Research Program of Chna Grant o. 2013CB329203), atonal atural Scence Foundaton of Chna Grant os , , ), Chna Postdoctoral Scence Specal Foundaton Grant o. 2012T50093), Scence and Technology Foundaton for Bejng Outstandng Doctoral Dssertaton Supervsor Grant o ), and Tsnghua Unversty-KU Leuven Blateral Scentfc Cooperaton Foundaton Grant o. BIL11/21T). The authors are wth Department of Electronc Engneerng as well as Tsnghua atonal Laboratory for Informaton Scence and Technology TLst), Tsnghua Unversty, Bejng , P. R. Chna e-mals: {dall, zcwang, yangzhx}@tsnghua.edu.cn). Dgtal Object Identfer /JSAC /13/$31.00 c 2013 IEEE systems wth tens of antennas could acheve the attractve spectral effcency up to several tens of bps/hz [6] [9]. For example, TT DoCoMo has demonstrated the feld experment of a large-scale MIMO system equpped wth antennas, whch approxmately acheves the spectral effcency of 50 bps/hz wth the transmsson rate of 4.92 Gbps over a 100 MHz channel [10]. The evoluton of WF standard called IEEE ac s now consderng the MIMO confguraton [11]. In addton, the measurement of MIMO channel has also been reported n [12]. The key challenges for realzng large-scale MIMO systems ncludes proper antenna placement to ensure ndependent channels, low-complexty sgnal detecton algorthms for practcal mplementaton, channel estmaton of the hghdmensonal MIMO channel matrx, etc. [6], [8]. Ths paper wll focus on channel estmaton for large-scale OFDM MIMO systems. Bascally, there are two categores of channel estmaton schemes for OFDM MIMO systems: frequency-doman estmaton and tme-doman estmaton. ormally, frequencydoman estmaton can be easly acheved by orthogonal plots, whch converts channel estmaton n MIMO systems to that n sngle-nput sngle-output SISO) systems [13]. However, the requred number of plots dramatcally ncreases when the number of transmt antennas becomes large. To mantan the plot overhead wthn a certan level n OFDM MIMO systems, t s common to reduce the plot densty to some extent, but poor channel estmaton performance would be caused, especally n large-scale MIMO systems. Alternatvely, tmedoman channel estmaton based on preamble could provde more relable estmaton over slow fadng channels, snce all subcarrers n the preamble can be used for channel estmaton. However, for large-scale OFDM MIMO systems, the overhead due to preamble s also hgh [10]. More mportantly, unless slow channel fadng s assumed, the preamble should be frequently nserted to track the channel varaton n moble envronments [14], [15], snce hgh-qualty servces are expected not only at home/offce, but also n moble cars, buses, etc. Furthermore, flat fadng channels are normally assumed for large-scale MIMO systems [8], [16], but actual wreless channels are usually characterzed by severe frequency-selectve fadng due to large delay spread, especally n urban/outdoor scenaros [17]. To solve the channelng problem of hgh-dmensonal channel estmaton for large-scale OFDM MIMO systems over frequency-selectve moble channels, we propose n ths paper the spectrally effcent tme-frequency tranng OFDM TFT- OFDM) transmsson scheme for large-scale MIMO systems.

2 252 IEEE JOURAL O SELECTED AREAS I COMMUICATIOS, VOL. 31, O. 2, FEBRUARY 2013 Specfcally, the contrbutons of ths paper are summarzed as follows: 1) Wthout cyclc prefx CP) n standard OFDM whch s explctly referred as CP-OFDM n the sequel), we propose a novel OFDM transmsson scheme called TFT-OFDM, whch jontly adopts the one-sample shfted tme-doman tranng sequence TS) and the frequency-doman orthogonal grouped plots as the tmefrequency tranng nformaton for every OFDM data block. The correspondng tme-frequency jont channel estmaton scheme for TFT-OFDM MIMO s also proposed, whereby the receved tme-doman TS s drectly used for path delays estmaton wthout nterference cancellaton, whle the path gans are acqured by the frequency-doman grouped plots; 2) We further explot the channel property that path delays vary much slower than path gans to mprove the path delay estmaton accuracy by averagng dozens of receved TSs, and employ the sparse nature of wreless channels to acqure the path gans by only a very small amount of plots. Therefore, the proposed scheme acheves hgher spectral effcency than frequencydoman based channel estmatons where plenty of plots are requred to estmate both path delays and path gans; 3) We also propose the TFT-OFDM based transmsson frame composed of one preamble and the subsequent TFT-OFDM subframes, whereby the preamble s used to provde an ntal relable channel estmate. Unlke the preamble-based tme-doman channel estmaton schemes rely heavly on the preamble, channel trackng of the proposed TFT-OFDM MIMO scheme s acheved by tme-frequency jont channel estmaton realzed n every TFT-OFDM subframe. Thus, compared wth the tme-doman preamble based solutons, the preamble n the proposed scheme can be used much less frequently n moble envronments, leadng to the hgher spectral effcency of the TFT-OFDM MIMO scheme; 4) Moreover, the Cramér-Rao lower bound CRLB) of the proposed channel estmator s derved n ths paper, whch s approached by smulaton results. We also show that TFT-OFDM enjoys the near-capacty performance much better than conventonal schemes over doublyselectve channels as well as the robustness to moble channels. The rest of ths paper s organzed as follows. The system model of the proposed TFT-OFDM MIMO scheme s presented n Secton II. The preamble-based channel estmators n both the tme and frequency domans are dscussed n Secton III. Secton IV addresses the channel trackng and data detecton for TFT-OFDM MIMO systems. The performance analyss of the proposed scheme s provded n Secton V before smulaton results are presented n Secton VI. Fnally, conclusons are drawn n Secton VII. otaton: The boldface letters are used to denote matrces and vectors. The upper- and lower-case characters are used to represent quanttes n the frequency and tme domans, respectvely. F s the normalzed dscrete Fourer transform DFT) matrx wth the n +1,k +1)th entry CP Tme CP-OFDM Symbol Copy OFDM a) Cyclc ) Preamble OFDM TS OFDM TS Ext. M 1 Tme p x p) 1 CP The 1 st TFT-OFDM Symbol The Uth TFT-OFDM Symbol z p) 1 OFDM Data Plot Zero Grouped Plot Data Central Plot Zero M b) p) z U 1 Frequency p x U One-sample Shfted TS Frequency Tx p Tx p+1 Tx p Tx p+1 Fg. 1. Tme-frequency sgnal structure comparson: a) CP-OFDM MIMO wth frequency-doman tranng nformaton plots) only; b) The proposed TFT-OFDM MIMO wth both tme- and frequency-doman tranng nformaton for every OFDM data block. exp j2πnk/)/. I s the dentty matrx, and 0 M s the M zero matrx. means the crcular correlaton. ), ) T, ) H, ) 1, ) and denote the complex conjugate, transpose, conjugate transpose, matrx nverson, Moore-Penrose matrx nverson and absolute operatons, respectvely. Tr{ }, E{ },anddet{ } stand respectvely for trace, expectaton, and determnant operators. x means the estmate of x. Fnally, dag{u} s a dagonal matrx wth u at ts man dagonal. II. SYSTEM MODEL In ths secton, the tme-frequency sgnal structure of the proposed TFT-OFDM MIMO scheme s descrbed at frst, and then the system model s presented. A. Tme-Frequency Sgnal Structure of TFT-OFDM Fg. 1 compares the tme-frequency sgnal structure of CP- OFDM MIMO and the proposed TFT-OFDM MIMO. As shown n Fg. 1 b), the TFT-OFDM sgnals are transmtted frame by frame, whereby each frame s composed of one preamble wth ts cyclc extenson and the followng U TFT- OFDM symbols subframes). We assume t transmt antennas and r receve antennas n MIMO systems. In the tme doman, unlke CP-OFDM where the CP s utlzed as the guard nterval, the th TFT-OFDM symbol s p) 1 U) for the pth transmt antenna 1 p t ) s composed of the length- OFDM symbol x p) =[x p),0,xp),1,,xp), 1 ]T and the followed length-m z U p)

3 DAI et al.: SPECTRALLY EFFICIET TIME-FREQUECY TRAIIG OFDM FOR MOBILE LARGE-SCALE MIMO SYSTEMS 253 TS s p) = =[,0,zp),1 [ x p),,zp),m 1 ]T as below ] [ ] F = H Xp) +M) 1 +M) 1, 1) where X p) = F x p). Snce t has been proved n [18] that constant TS wthn the transmsson frame s not optmal for channel trackng, the TS of the th TFT-OFDM symbol wll be generated by cyclcally shftng the basc TS = [ 0,zp) 1,,zp) M 1 ]T by samples to the left accordng to [ ] 0M ) I = M z I 0 p), 2) M ) where for the pth transmt antenna s the Zadoff-Chu sequence also known as generalzed Chrp-lke GCL) sequence) [19] defned by =exp m j M 1 M πm2 r p ), 0 m M 1, 3) where r p s relatvely prme to M. The Zadoff-Chu sequences have deal autocorrelaton and the optmal crosscorrelaton equals to the theoretcal Sarwate bound [20]. In addton, the Zadoff-Chu sequences wth constant envelope both n the tme and frequency domans acheve the lowest peak-to-average power raton PAPR) of 0 db. Moreover, accordng to 2), we have,,zp),m 1,zp),0 ]T, 4) whch ndcates that the TSs and +1 for two adjacent TFT-OFDM symbols are cyclcally shfted to the left by one sample,.e., the frst M 1 samples of +1 s dentcal wth the last M 1 samples of. Thus, the one-sample shfted non-constant TSs { } U =1 n TFT-OFDM could also preserve the cyclc property of the transmtted data stream when constant TSs are used nstead. Ths property s useful for low-complexty equalzaton [21] as well as accurate tmng/frequency synchronzaton [22]. In the frequency doman, TFT-OFDM MIMO systems adopts G orthogonal plot groups randomly scattered wthn the sgnal bandwdth, where each plot group has only one non-zero central plot n the mddle surrounded by d zero plots on the left and rght sdes n Fg. 1 b), d = 1 s used as an example). Although frequency-doman plots are common n OFDM systems, the proposed plot pattern has the followng four dstnct features: 1) In most OFDM systems, the plots are regularly placed n the frequency doman, e.g., the equally-spaced comb-type plots are used to acheve the optmal channel estmaton performance [23], whle the proposed plots are randomly nserted nstead; 2) The 2d zero plots are used to allevate the potental nter-carrernterference ICI) mposed on the central plot over fast fadng channels. Snce ICI s domnantly caused by the neghborng subcarrers [24], d =1could be used for TFT-OFDM even the channel s varyng fast. ote that for large-scale MIMO systems typcally used n moble channels wth not very hgh speed, d =0can be adopted; 3) The subcarrer ndex set of the central plots can be denoted by +1 =[zp),1,zp),2 G p) =[g 0 + O p,g 1 + O p,,g G 1 + O p ] T, 5) where O p = d +1)p 1) s the subcarrer offset for the pth transmt antenna. To ensure the orthogonalty of the grouped plots assocated wth all t transmt antennas, t 1)2d +1) extra zero plots are usually requred for each grouped plot havng 2d +1 plots. However, due to the exstence of the 2d zero plots around the central plot, the grouped plots for adjacent two transmt antennas could overlap d zero plots, e.g., only t 1)d +1) nstead of t 1)2d +1)extra zero plots wll be requred; 4) The 2d zero plots naturally permt plot power boostng technque [1] because the power orgnally dedcated to them can be used by the central plot nstead,.e., the central plots could be X p),k = 2d +1 for k G p) f the average power of the data subcarrers s E{ X p),k 2 } =1for k / G p).however, wthout loss of generalty, the plot power s assumed to be the same as the useful data,.e., X p),k =1for k Gp) n ths paper. The length- p basc preamble c = F H p C s also a Zadoff- Chu sequence defned by 3), where p nstead of M) and r p =1are used. Based on basc preamble c, the preamble for the pth transmt antenna c p) =[c p) then generated by 0,cp) 1,,cp) p 1 ]T s c p) = F H p C p) =F H p dag {C} W p), 6) where W p) 2π j = [0,e p 1) 2π j t,,e p 1)p 1) t ] T, C p) = dag {C} W p) denotes the FT of c p), and p = t M s assumed. Fnally, the cyclc extenson c p) of length M 1 s generated by c p) = [ 0 M 1) 1 I M 1 ] c p). 7) One remark to make s that, although the preamble nserton s common n OFDM MIMO systems [1], [8], [14], [15] where the preamble-based channel estmaton wll be used to detect all the followng OFDM symbols over slowly tmevaryng channels 1, t wll be shown later n Secton V that the preamble n the proposed TFT-OFDM MIMO scheme s used only to acqure the ntal channel estmaton, whle the channel trackng for the subsequent TFT-OFDM symbols are realzed by explotng the tme-frequency tranng nformaton n every TFT-OFDM symbol. Therefore, the subframe number U n the transmsson frame could stll be large even when the channel s varyng fast. B. System Model of TFT-OFDM MIMO In MIMO systems, for a certan receve antenna 2, the channel mpulse response CIR) h p) assocated wth the pth transmt antenna durng the th TFT-OFDM symbol can be denoted by h p) =[h p),0,hp),1,,hp),l 1 ]T, 8) 1 Alternatvely, the tme-doman nterpolaton between separated preambles can be used to mprove the channel trackng capablty, but the preamble should be frequently nserted to ensure the relable performance over fast fadng channels. 2 As the tme-frequency tranng nformaton n every TFT-OFDM symbol wll be exploted by every receve antenna adoptng the dentcal processng, the receve antenna ndex wll be omtted from now on.

4 254 IEEE JOURAL O SELECTED AREAS I COMMUICATIOS, VOL. 31, O. 2, FEBRUARY 2013 where h p),l s the path gan of the lth path wth the path delay τ p) l, L denotes the maxmum channel spread, and L = M s assumed to avod the nterference between two neghborng OFDM data blocks, so we have p = t M = t L.ote that although the delay spread L maybe large n frequencyselectve channels, the number of most sgnfcant taps or resolvable paths) Q s usually much smaller than the channel length L,.e., Q L, because of the sparse nature of wreless channels, especally for wdeband communcatons [25], [26]. For example, the ITU Vehcular B channel [27] wth the maxmum delay spread of 20 μs, whch s equvalent to L = 200 samples at the system samplng rate of 10 MHz, has only Q =6resolvable paths. Furthermore, t has been vary much slower than,l }L 1 l=0 ncludng phases and ampltudes) [28], [29], whch s caused by the fact that the duraton for the delay of a path to change by one tap s nversely proportonal to the sgnal bandwdth B, whle the coherence tme of the path gans s nversely proportonal to the carrer frequency f c.snceb f c for almost all of practcal wreless systems, path delays would change much slower than path gans 3. At the recever, the sgnals comng from dfferent transmt antennas wll mx together, and the receved OFDM data block y =[y,0,y,1,,y, 1 ] T after cyclcty reconstructon [21], [32] s proved that the path delays {τ p) l path gans {h p) t y = x p) } L 1 l=0 h p) + w 0, 9) where w = [w,0,w,1,,w, 1 ] T denotes the addtve whte Gaussan nose AWG) vector wth zero mean and the varance of σ 2 I. Applyng DFT to y above, the receved sgnal Y,k on the kth subcarrer could be presented by t Y,k = X p),k Hp),k + W,k, 0 k 1, 10) where H p) = [H p),0,hp),1,,hp), 1 ]T s the channel frequency response CFR) of h p), and we have H p) = F,L h p), 11) where F,L of sze L denotes the frst L columns of the DFT matrx F. In addton, we use H p) 0 and h p) 0 to denote the CFR and CIR durng the preamble, respectvely. III. PREAMBLE-BASED CHAEL ESTIMATIO Based on the preamble of the TFT-OFDM transmsson frame, the ntal channel estmaton can be acheved ether n the tme or frequency doman. Ther equvalence wll be also proved n ths secton. 3 Takng a typcal hgh-defnton televson HDTV) wreless system wth B =8MHz and f c = 770 MHz for example [30], the path delays reman constant over hundreds of OFDM symbols, although the path gans may vary after several OFDM symbols. Another example s that, all the channel models provdes by numerous communcatons standards employ the constant path delays [27], [31]. A. Tme-Doman Channel Estmaton The receved preamble d 0 = [d 0,0,d 0,1,,d 0,p 1] T n the tme doman at the receve antenna s mmune from the nter-block-nterference IBI) due to the protecton of the cyclc extenson, so d 0 can be expressed by t t d 0 = c p) h p) 0 + v 0 = c p) 0 hp) 0 + v 0 =c 0 h 0 + v 0, 12) where c p) 0 s the p L crculant [ matrx wth the frst ] column beng the preamble c p), c 0 = c 1) 0, c2) 0,, ct) 0 denotes the p t L tme-doman tranng matrx based [ ) T ) T ) ] T T on {c p) } t, h 0 = h 1) 0, h 2) 0,, h t) 0 presents the t L 1 equvalent total CIR for all t transmt antennas, and v 0 =[v 0,1,v 0,1,,v 0,p 1] T stands for the channel s AWG vector wth each element havng zero mean and the varance of σ 2. In 12), there are t L unknown parameters n h 0 and p observatons n d 0.If p t L, the tme-doman channel estmate ĥ0 can be obtaned by [33] ĥ 0 = c 0 d 0 = 1c c H 0 c H 0) 0 d 0. 13) } { c ) } H 1c H We have E {ĥ0 h 0 = E 0 c 0 0 v 0 = 0 tl 1 due to every element of v 0 has zero mean, so the mean square error MSE) of the unbased channel estmator 13) s { ) H ) MSE = E ĥ0 h 0 ĥ0 } h 0 { c ) H 1c H =tr 0 c 0 0 E { H v 0 v } ) 0 c 0 c H 1 } 0 c 0 14) { c = σ 2 ) H 1 } tr 0 c 0. Accordng to the proof n the Appendx, the mnmum MSE can be acheved by the followng optmal desgn crteron c H 0 c 0 = t LI tl. 15) The correspondng MSE n 14) s then derved as 4 { } 1 MSE mn = σ 2 tr t L I tl = σ 2. 16) It can be verfed that the proposed preamble 6) meets the optmal desgn crteron 15). Thus, the mnmum MSE 16) can be acheved, and the tme-doman channel estmator 13) s then smplfed by crcular correlaton [33] as ĥ 0 = 1 t L ch 0 d 0 = 1 t L c d 0. 17) B. Frequency-Doman Channel Estmaton The frequency-doman sgnal model 10) s also vald when p -pont DFT nstead of -pont DFT s used to produce the receved preamble D 0 =[D 0,0,D 0,1,,D 0,p 1] T n the frequency doman,.e., D 0 = C 0 H 0 + V 0, 18) 4 As wll be addressed n detal n the Appendx, the results here s consstent wth those n [13], but we use dstnct proof technque n ths paper.

5 DAI et al.: SPECTRALLY EFFICIET TIME-FREQUECY TRAIIG OFDM FOR MOBILE LARGE-SCALE MIMO SYSTEMS 255 where V 0 = F p v 0 denotes AWG, D 0 = F p d 0 presents the p -pont DFT of the tme-doman receved preamble d 0, C 0 = [ dag { C 1)}, dag { C 2)},, dag { C t)}] denotes the p t p frequency-doman tranng matrx based on {C p) } t,andthecfrh 0 durng the preamble can be related to the correspondng CIR h 0 by usng 11) as H 0 = F 0 h 0,where F p,l 0 p L F 0 = ) 0 p L F p,l t p tl Snce there are t p unknown parameters n H 0 and only p observatons n D 0, eq. 18) s an underdetermned problem wthout unque soluton. However, ths problem can be solved by usng the relatonshp between the CFR H 0 and the CIR h 0 as below D 0 = C 0 F 0 h 0 + V 0 = A 0 h 0 + V 0, 20) where A 0 = C 0 F 0, and the number of unknown parameters s reduced from t p n 18) to t L n 20). If p t L, the channel estmaton can be acheved by [13] ) 1C0 ĥ 0 = A 0 D 0 = C 0 F 0 ) H C 0 F 0 F 0 ) H D 0. 21) Then, the CFR can be obtaned by Ĥ0 = F 0 ĥ 0 = F 0 A 0 D 0. C. Unfcaton of The Tme- and Frequency-Doman Channel Estmators The tme-doman channel estmator 13) s based on tmedoman sgnals d 0 and c 0, whle the frequency-doman channel estmator 21) depends on the frequency-doman sgnals D 0 and C 0.SnceD 0 C 0 ) can be obtaned once d 0 c 0 ) s known, and vce versa, the tme- and frequency-doman channel estmators 13) and 21) can be drectly unfed by the extracted DFT matrx F 0 denoted by 19). Regardng to the optmal desgn crteron, we have n Secton III-A derved 15) for the tme-doman tranng matrx c 0, whle t has been proved n [13] that the optmal frequencydoman channel estmator 21) s subject to the followng optmal desgn crteron A H 0 A 0 = t LI tl. 22) Usng the well-known shft property of DFT, t can be derved that F p c 0 = C 0 F 0 = A 0. 23) Wth the help of 23), the unfcaton of the optmal desgn crtera 15) and 22) for the tme- and frequency-doman channel estmators, respectvely, can be revealed by c H 0 c 0 = F H p A 0 ) HF H p A 0 =A H 0 F p F H p ) A 0 =A H 0 A 0. 24) It reads clear from 24) that the dfferent desgn crtera 15) and 22) are essentally equvalent. Therefore, the tme- and frequency-doman channel estmators as well as ther correspondng optmal desgn crtera can be unfed under the same framework. IV. CHAEL TRACKIG AD DATA DETECTIO The preamble-based ntal channel estmaton becomes outdated to detect the subsequent TFT-OFDM symbols over fast tme-varyng channels. Ths secton addresses the cyclcty reconstructon and the channel trackng for every TFT-OFDM symbols wthn the transmsson frame to acheve relable data detecton. A. Cyclcty Reconstructon of the OFDM Symbol In CP-OFDM systems, the cyclcty property of the receved OFDM symbol s naturally restored due to the nserton of CP. In the proposed TFT-OFDM scheme, the CP s replaced by the known one-sample shfted TS to mprove the spectral effcency. However, the absence of CP would destroy the cyclcty property of the receved OFDM symbol over multpath fadng channels [21], [32], and cyclcty reconstructon s requred to acheve the low-complexty frequency doman equalzaton smlar to that wdely used n CP-OFDM. Due to the multpath propagaton, the receved OFDM symbol y of the th TFT-OFDM symbol can be expressed by t [ ] ) y = h p),isi x + h p) 0 M) 1 1,IBI + w, ) where h p),isi and h p),ibi present the Toepltz lower and upper trangular matrx wth the frst column [h p),0,hp),1,,hp),l 1, 0,, 0]T and the frst row [0,, 0,h p),l 1,hp),L 2,,hp),1 ]T, respectvely, and w = F H W denotes the AWG vector. Usng the overlap-add OLA) [34] algorthm by t tmes, the cyclc prefx reconstructon of y can be acheved by t [ ] ) 0 M) 1 ỹ = y t = h p) 1,IBI 1 [ ] d + t 0 M) 1 [ h p),cir xp) + w + h p),isi 1 [ ] v. 0 M) 1 0 M) 1 ] ) 26) where d s the tme-doman receved TS of the th TFT- OFDM symbol, h p),cir = hp),isi + hp),ibi means the crcular matrx wth the frst column [h p) 0,hp) 1,,hp) L 1, 0,, 0]T. In the frst equaton of 26), subtractng the second term on the rght means removng the IBI caused by the prevous TSs { 1 }t, and addng the thrd term ndcates restorng the tals bured n d )of{x p) } t over multpath channels. Snce any crcular matrx can be dagonalzed by the DFT matrx [35],.e., h p),cir = FH Hp) F, applyng DFT to ỹ would produce the sgnal Y n 10) as t Y = F ỹ = F F H Hp) F x p) + W + v t = H p) X p) + W + v = X H + W + v, 27)

6 256 IEEE JOURAL O SELECTED AREAS I COMMUICATIOS, VOL. 31, O. 2, FEBRUARY 2013 where [ { X = dag [ ) T H = H 1), X 1) H 2) } {, dag ) T,, X 2) } H t) { }],, dag X t), ) ] T T, W = F w, [ ] and v = F v T T 0 1 M) s the addtonal AWG ntroduced by the cyclcty reconstructon, whereby an extra nose term v bured n d has been nvolved. However, ths would lead to neglgble SR loss of 10log M+ ) 10 at the recever snce the TS length s usually much smaller than the OFDM data block length, e.g., the equvalent SR loss s 0.26 db when M = /16. Although smlar to the cyclcty reconstructon method for SISO tme-doman synchronous OFDM TDS-OFDM) systems n [32], [36], the proposed algorthm 26) dffers from those of [32], [36] n two aspects. Frstly, the conventonal method n SISO scenaros s extended to be used n MIMO scenaros. Secondly and more mportantly, the cyclcty reconstructon 26) s completed by only one step, whle [32], [36] requres teratve processng the teraton tmes s usually three or more). In practcal cyclcty reconstructon mplementaton, the actual CIR h p) 1 n 26) should be replaced by the estmate ĥp) 1, whch has been obtaned n the prevous 1)th TFT-OFDM symbol, and h p) n 26) can be ether smply approxmated by ĥp) 1 or predcted by the Kalman flter explotng the temporal correlaton nature of the channel as well as the prevous channel estmates {ĥp) u } 1 u=0 [37]. It s worth notng that the approxmaton of h p) for cyclcty reconstructon wound ndeed ncur some performance degradaton due to the mperfect matchng, but the performance loss s small due to the followng two reasons: 1) As wll be shown later, the tme-frequency jont channel trackng method could acheve the CIR estmate wth hgh accuracy n every TFT-OFDM symbol; 2) Snce large-scale MIMO technque s manly used n statc or slow tme-varyng channels [6], [8] [11], the CIR estmates n prevous TFT-OFDM symbols could be effcently exploted to produce a good approxmaton of the actual CIR n the current TFT-OFDM symbol. B. Tme-Frequency Jont Channel Estmaton In most OFDM MIMO systems, channel estmaton s acheved n the tme or frequency doman by usng the tmedoman preamble [8], [10] or the frequency-doman plots [2], [13], [23], respectvely. However, takng advantage of the tme-frequency tranng of TFT-OFDM, channel trackng for TFT-OFDM can be realzed by the followng tme-frequency jont channel estmaton composed of two sequental steps: the TS-based path delay estmaton and the plot-based path gan estmaton. The receved TS d =[d,0,d,1,,d,m 1 ] T of the th TFT-OFDM symbol s gven by t ) d = h p),isi zp) + h p),ibi xp), M: 1 + v, 28) where h p),isi and h p),ibi denote the M M Toepltz lower and upper trangular matrx wth the frst column,,hp),l 1, 0,, 0]T and the frst row [0,, 0,h p),l 1,hp),L 2,,hp),1 ]T, respectvely, and x p), M: 1 presents the last M elements of xp).theterm h p),ibi xp), M: 1 n 28) ndcate that the receved TS d s [h p),0,hp),1 contamnated by the IBI caused by the precedng OFDM data blocks. Beng dfferent from TDS-OFDM where teratve IBI cancellaton s exploted to acheve the tme-doman complete CIR estmaton based on the receved TS [36], by explotng the deal autocorrelaton property of the Zadoff-Chu sequences [19], we drectly correlate the local Zadoff-Chu sequence wth the contamnated TS d wthout nterference cancellaton to just estmate the path delays of the channel as below ĥ p) = 1 M zp) d = h p) +n p) +v p), 1 p t, 29) where v p) = 1 M zp) v, and n p) denotes the nterferences caused by the non-zero crosscorrelaton among dfferent Zadoff-Chu sequences as well as the IBIs from prevous OFDM symbols. As demonstrated later n Fg. 2, f only one receved TS s used, due to the absence of nterference cancellaton, the path delay nformaton hdden n ĥp) maybe not accurate when nterferences are severe, although the paths wth hgh gans can be dentfed well. Based on property of wreless channels that path delay vares much slower than path gans [28], [29], we further propose the averaged path delay estmaton to further mprove the accuracy, whereby the receved TSs wthn β adjacent TFT-OFDM symbols durng whch the path delays do not change obvously are averaged as below: h p) = 1 β u= β+1 ĥ p) u = 1 βm zp) u d u u= β+1. 30) ote that the number β could be very large e.g., β>10) due to the channel property mentoned above. Then, the path gans n h p) are drectly dscarded snce they maybe not accurate due to absence of nterference cancellaton, and only the path delays of the Q most sgnfcant taps of h p) are stored n the path delay set Γ p) = {τ p) l : h p),l 2 T th } L 1 l=0, 1 p t, 31) where T th s the power threshold could be determned accordng to [38], and Q s usually much smaller than the channel length L,.e., Q L [25], [26]. That s to say, the number of unknown parameters n the CIR h p) s substantally reduced from L to Q Q L) after the path delays Γ p) have been obtaned. otce that the path delay estmaton of the most sgnfcant taps could be very accurate snce β could be very large n 30) due to the fact that path delays vary much slower than path gans [28], and furthermore the IBIs caused by precedng OFDM data blocks may be averaged out due to the randomness nature of the data. After the path delays {Γ p) } t has been obtaned n 28), the number of unknown parameters of h p) s substantally

7 DAI et al.: SPECTRALLY EFFICIET TIME-FREQUECY TRAIIG OFDM FOR MOBILE LARGE-SCALE MIMO SYSTEMS 257 reduced from L to Q, thus only a small amount of frequencydoman grouped plots wll be suffcent to estmate the Q channel path gans. Due to the orthogonalty of the grouped plots among dfferent transmt antennas, the receved central plots Y p),k can be expressed by t Y,k = X p),k Hp),k + W,k = H p),k + W,k, k G p), 32) where the sgnal model 10), the central plots X p),k =1k G p) ), and X u),k =0u p, k Gp) ) have been utlzed. Eq. 32) can be rewrtten n a more compact matrx form as h p),γ Y p) Y p) = F p) hp),γ + Wp), 1 p t, 33) where = [Y,g0+O p,y,g1+o p,,y,gg 1+O p ] T G 1, h p),γ =[hp),τ 0,h p),τ 1,,h p),τ Q 1 ] T Q 1 denotes the selected CIR out of h p) accordng to the path delays Γ p) obtaned n 31), F p) presents the G Q extracted matrx generated by selectng the G p) rows and Γ p) columns out of the DFT matrx F, and W p) = [W p),g 0+O p,w p),g 1+O p,,w p),g G 1+O p ] T G 1 s the AWG vector. ote that the relatonshp 11) has been used to obtan 33). It s clear from 33) that only Q unknown path gans n have to be estmated by the G observatons n Yp).If G Q, theq not L, and usually Q L) path gans of h p) can be estmated by the receved central plots Y p) as ĥ p),γ = F p) ) Y p) = [ F p) ) HF p) ] 1 F p) ) HY p). 34) ote that the plot power boostng technque [1], whereby plots use hgher power than normal data, s more sutable for TFT-OFDM than CP-OFDM, snce no obvous equvalent sgnal-to-nose rato SR) degradaton wll be ntroduced n TFT-OFDM havng much fewer plots than CP-OFDM. Combnng the path delays {Γ p) } t obtaned n 31) based on the tme-doman receved TS d and the path gans {ĥp),γ }t acqured n 34) based on the frequencydoman receved plots {Y p) } t, the complete CIR estmates {ĥp) } t for all the t transmt antennas could be acheved by the proposed tme-frequency jont channel estmaton. C. Data Detecton for TFT-OFDM After cyclcty reconstructon and the tme-frequency jont channel estmaton have been completed, the obtaned OFDM symbol Y and the complete CIR estmates {ĥp) } t are fed nto the MIMO detector to recover the transmtted sgnals. In contrast to conventonal detecton algorthms for MIMO systems of small sze, e.g., maxmum-lkelhood ML) or maxmum a posteror MAP) detectors whose complexty ncreases exponentally wth the number of transmt antennas [8], we resort to the recently proposed low-complexty detecton algorthms specally desgned for large-scale MIMO systems [8], [16], [39] [42]. Among them, the frst category algorthms based on the local neghborhood search ncludes the lkelhood ascent search LAS) [8], [39] and reactve tabu search RTS) [16], whch are amed to acheve the near-ml performance. The second category algorthms approachng the near-map performance are based on message passng, e.g., the belef propagaton BP) algorthm [40], [41] and probablstc data assocaton PDA) scheme [8], [42]. They all enjoy the preferred property of large system behavor,.e., the detecton performance mproves when the number of transmt antennas t becomes large. Snce data detecton algorthm s not the focus of ths paper, we drectly adopt the near-capacty RTS scheme wth the low per-subcarrer complexty of O t r ). Please refer to [16] for more detals about the RTS algorthm. Fnally, the detected data are used by the channel decoder to ultmately recover the transmtted sgnal. V. PERFORMACE AALYSIS Ths secton addresses the performance analyss of the proposed scheme, ncludng the spectral effcency of proposed TFT-OFDM scheme, the CRLB as well as the computatonal complexty of the tme-frequency jont channel estmaton method. A. Spectral Effcency Due to the overhead caused by the tme-doman guard nterval ether CP n standard CP-OFDM or TS n the proposed TFT-OFDM) and the frequency-doman plots, the spectral effcency η 0 of the proposed TFT-OFDM MIMO scheme normalzed by the deal case wthout any overhead [21], [43] can be expressed n the percentage notaton as η 0 = U K) U + M)+ p + M 1, 35) where K = G 2d +1)+ t 1)d +1))s the number of used plots n each OFDM data block. For typcal wreless dgtal televson systems, large DFT sze, e.g., = 4096, s usually adopted [30]. Snce all channel models defned by ITU [27] and all channel models used for dgtal televson system evaluaton [31] have no more than sx resolvable paths, we could assume G = Q = 6 wthout loss of generalty. However, n practcal applcatons, the path number may be large,so we confgure G =10for system desgn wth some margn. As mentoned n Secton II-A, even we relax the quas-statc channel over the entre transmsson frame to only one subframe, d =0could be used for large-scale MIMO systems where the moble channels are not varyng very fast 5. Thus, for large-scale MIMO confguraton,.e., t = r =16, the number of used plots n TFT-OFDM s K = 160, whch s only 3.91% of the total subcarrer number = If the channel s varyng slowly, e.g., the grouped plots n TFT-OFDM transmsson frame can be nserted every three TFT-OFDM subframes the correspondng channel estmate update frequency s stll much hgher than the 5 ote that the proposed TFT-OFDM can be also used over fast varyng channels, whereby d =1could be confgured to allevate ICI caused by the rapd channel varaton [44].

8 258 IEEE JOURAL O SELECTED AREAS I COMMUICATIOS, VOL. 31, O. 2, FEBRUARY 2013 TABLE I SPECTRAL EFFICIECY I MIMO SYSTEMS. Guard Interval Length Frequency-doman plots based MIMO [45] Tme-doman preamble based MIMO [8] Proposed TFT-OFDM MIMO M = /16 75% 80% 88.67% tme-doman preamble based scheme [8]), the plot occupaton rato wll be reduced to 1.30%. On the contrary, the Karhunen- Loeve theorem [24] requres that the number of frequencydoman plot should be not smaller than the channel length L for frequency-doman channel estmaton, and the plot number ncreases lnearly to be t L n CP-OFDM MIMO systems due to the orthogonalty requrement. Ths means that t L = 4096 plots occupyng 100% of the total subcarrers are requred n CP-OFDM MIMO systems when the typcal guard nterval length M = /16 s appled. o extra subcarrer s avalable f the number of transmt antennas t > 16. In practcal applcatons [3], [5], [45], fewer plots can be used due to nterpolaton can be used to estmate the CFR at data subcarrers at the cost of performance degradaton. However, the plot occupaton rato should be above a certan threshold to ensure relable performance for large-scale MIMO systems. One typcal threshold recommenced by LTE s that 25% subcarrers are used as plots when t > 4 [5]. Therefore, we can conclude that the proposed TFT-OFDM MIMO scheme has much hgher spectral effcency than standard CP-OFDM MIMO systems. Compared wth the tme-doman preamble based schemes [8], [10] where the preamble should be frequently nserted or equvalently, small subframe number wll be adopted) n moble envronments, the subframe number U n the TFT-OFDM MIMO transmsson frame can stll be large due to the good channel trackng capablty of the tme-frequency jont channel estmaton n every TFT-OFDM symbol. For example, 4 out of the total 29 OFDM symbols n a frame are dedcated for channel estmaton n the MIMO experment wth the moble speed of 10 km/h [10], whle the smulaton results n Secton VI demonstrate that large U = 50 could stll ensure near-capacty performance for TFT-OFDM MIMO scheme over 30 km/h tme-varyng channel, whereby U = 4 should be used for the tmedoman preamble based teratve channel estmaton/data detecton system [8]. Although the actual spectral effcency of practcal systems may vary wth dfferent confguratons, wth those data addressed above, Table I clearly ndcates that the proposed TFT-OFDM MIMO scheme outperforms ts conventonal counterparts n spectral effcency. Snce every receve antenna could use the tme-frequency jont channel estmaton method to dstngush the channels between dfferent transmt antennas and ths specfc receve antenna, the proposed TFT-OFDM transmsson scheme could be used n both scenaros havng small even a sngle) or large number of receve antennas, and hgher spectral effcency than conventonal solutons could be acheved n both cases. B. Cramér-Rao Lower Bound The CRLB bound s the theoretcal bound to evaluate the performance of practcal estmaton methods [33]. Snce the AWG vector W p) n 33) s subject to the dstrbuton of C ) 0,σ 2 I G, the condtonal probablty densty functon PDF) of Y p) wth the gven h p),γ s p Y p) h p),γ Y p) 1 = 2πσ 2 ) ) ; h p),γ G/2 exp { 1 Y p) 2σ 2 F p) hp),γ 2}. 36) The Fsher nformaton matrx [33] of 33) can then be derved as ) ) 2 ln p p) Δ Y Y p) h p) ; h p),γ,γ [J] m,n = E h p),γ,m hp),γ,n 37) = 1 [ ) ] HF σ 2 F p) p) where h p),γ,m h p),γ and hp),γ,n, m,n denotes the mth and nth entry of, respectvely. Fnally, accordng to the vector estmaton theory [33], the CRLB of the unbased estmator ĥp) { CRLB = E ĥ p) = σ 2 Tr,Γ hp),γ { F p) 2} Tr { J 1} ) ) } HF 1 p).,γ s 38) Let {λ } Q =0 beng the Q egenvalues of the matrx ) HF F p) p), then, we have the followng result accordng to the elementary lnear algebra [46] { ) ) } HF 1 Q Q ) Tr F p) p) = λ 1 = Q λ 1 /Q =1 =1 ) Q Q Q Q/ λ = { 2 ) HF }, =1 Tr F p) p) 39) where the equalty holds f and only f λ 1 = λ 2 = = λ Q, whch means that the matrx F p) extracted from the standard DFT matrx F should ) have orthogonal columns. Obvously, HF the Q Q matrx F p) p) has dentcal dagonals equal { ) } HF to G,.e., Tr F p) p) = GQ, so the CRLB of the proposed tme-frequency jont channel estmator becomes { CRLB = E ĥ p),γ hp),γ 2} = Qσ2 G. 40)

9 DAI et al.: SPECTRALLY EFFICIET TIME-FREQUECY TRAIIG OFDM FOR MOBILE LARGE-SCALE MIMO SYSTEMS 259 It s worth notng that f the matrx F p) does not have orthogonal columns, the MSE of the practcal channel estmator wll not acheve the CRLB 40). However, due to the random postons of the central plots n TFT-OFDM, the extracted matrx F p) n the proposed scheme has mperfect ) but approxmate orthogonal columns or equvalently, HF F p) p) GI Q), so the CRLB could be asymptotcally approached, whch wll be valdated by the smulaton results n Secton VI. Also note that 40) mples that the ncreased number of transmt antennas t has lttle mpact on the channel trackng performance, but the loss n spectral effcency has to be pad accordng to 35). C. Computatonal Complexty The computatonal complexty of the proposed tmefrequency jont channel trackng scheme can be evaluated n terms of how many multplcatons are requred. The crcular correlaton of the preamble-based ntal channel estmate ĥ0 n 17) can be effcently realzed by one p -pont DFT note that the DFT of the basc preamble c can be prestored at the recever) and one p -pont nverse DFT IDFT) plus p multplcatons. In the proposed scheme, p = t M. The cyclcty reconstructon 26) requres t tmes of M- pont lnear convoluton to compute the IBI between the TS and OFDM symbol to obtan ỹ, where each M-pont lnear convoluton can be mplemented by one 2M-pont DFT note that the DFT of the local TSs and 1 are also prestored at the recever) and one 2M-pont IDFT plus 2M multplcatons. Then, the -pont DFT s used to produce the frequency-doman OFDM symbol Y n 27). To acqure the TS-based path delay estmates {Γ p) } t n 31), {hp) } t n 30) requres t tmes of M-pont crcular correlaton, whch can be realzed by one M-pont DFT of d and t tmes of M-pont IDFT snce all the DFT of local TSs } t are known to the recever. The plot-based path { gan estmates {ĥp),γ }t for all t transmt antennas n 34) requre t GQ 2 +2Q 3 ) multplcatons to compute the Moore- Penrose nverse matrx of F p) and tqg multplcatons for matrx product. Fnally, based on the path delays and path gans for all the t transmt antennas, t tmes of -pont DFT s used to convert the CIRs {ĥp) } t to the correspondng CFRs {Ĥp) } t. Therefore, the overall complexty of proposed channel estmaton s O t log 2 )+Oα t ) where α = GQ 2 +2Q 3 + QG, whch s a lttle hgher than the complexty O 2 t r )+O 3 t ) that of the tme-doman based scheme [8] and the complexty O t Mlog 2 M)+ O t ) that of the frequency-doman soluton [23]. VI. SIMULATIO RESULTS Ths secton nvestgates the performance of the proposed TFT-OFDM scheme for large-scale MIMO systems n moble multpath channels. For performance comparson, the proposed TFT-OFDM scheme for large-scale MIMO systems s compared wth ts conventonal counterparts based on CP- OFDM wth frequency-doman comb-type plots [45] and tme-doman preamble based teratve channel estmaton/data detecton scheme n [8], respectvely. ote that the spectral effcency of those three dfferent schemes has been specfed n Table I. All those three systems have the same MIMO confguraton workng wth the sgnal bandwdth of 7.56 MHz at the central rado frequency of 770 MHz. The system parameters for TFT-OFDM MIMO scheme are consstent wth those specfed n Secton V-A, e.g., = 4096, M = 4096/16 = 256, U =10, G =10, d =0. For all schemes, we adopt the spectrally-effcent non-orthogonal space-tme block code STBC) [47], snce t could smultaneously acheve full transmt dversty as well as full rate. As mentoned before, the RTS algorthm [16] s employed for data detecton. Snce almost all practcal OFDM systems use channel codng for relable performance, we adopt the powerful low-densty party-check LDPC) code wth the block length of 64, 8000 bts and code rate of 2/3 as specfed by the standard [30]. The well-known teratve decodng algorthm called belef propagaton BP) [48] s used wth the maxmum teraton number of 50. The quadrature phase shft keyng QPSK) modulaton scheme s smulated. Two typcal 6-tap multpath channel models named Brazl D [31] and Vehcular B [27] are used. The frst channel has the maxmum delay spread of 20 μs, whle the later one havng the 0 db echo at the delay of 5.86 μs s deeply frequency-selectve characterzng the sngle frequency network SF) envronment. The channels assocated wth dfferent transmt-receve antenna pars are assumed ndependent. The moble veloctes of 5 km/h outdoor walkng speed) and 30 km/h vehcle speed n urban areas) are consdered 6. For the preamble-based transmsson scheme wth channel estmaton errors, the capacty lower bound C has been theoretcally proved as [49] [ )] γ 2 M C η 0 E log det I t + ĤĤH t 1+γ)+γM t σ 2, 41) Ĥ [ ] where σ 2 = 1/ Ĥ t r E trĥĥh ), and γ s the average receved SR. ote that 41) s dfferent from the deal capacty bound under perfect channel state nformaton CSI) at the recever [39], and the mnmum SR for a gven capacty C can be obtaned from 41). Frstly, we evaluate the averaged path delay estmaton over multpath channels. Fg. 2 and Fg. 3 show the smulaton results wth the SR of 5 db over the Brazl D [31] and Vehcular B channel [27], respectvely. The actual channel s also plotted for comparson. We can observe that although the rough channel estmates are not accurate due to the absence of nterference cancellaton, the path delay nformaton of the actual channel could be preserved well, especally when β becomes large as dscussed n Secton II-B, β could be very large, e.g., β =10sadopted n the smulaton). It should be ponted out that for channels whose actve taps are all of relatvely hgh gans e.g., the Brazl D channel n Fg. 2), the proposed scheme could accurately dentfy the path delays even when β =1, whle for channels havng very small actve taps e.g., the Vehcular B channel n Fg. 3, whereby the smallest path has 22 db lower power than the largest one), large β s requred for accurate path delay estmaton. ote 6 The moble speed of 10 km/h s evaluated n the large-scale MIMO experment n [10].

10 260 IEEE JOURAL O SELECTED AREAS I COMMUICATIOS, VOL. 31, O. 2, FEBRUARY 2013 Correlaton Results Drect Correlaton β=1) Actual Channel CP OFDM MIMO wth Comb Type Plots Iteratve Channel Estmaton and Data Detecton [8] Proposed Tme Frequency Jont Channel Esstmaton Theoretcal CRLB 40) Path Delay Samples) MSE 10 2 Correlaton Results Averaged Correlaton β=10) Actual Channel Path Delay Samples) SR db) Fg. 2. Averaged path delay estmaton over Brazl D channel. Fg. 4. MSE performance comparson between the proposed tme-frequency jont channel estmaton method for TFT-OFDM wth the conventonal schemes. Correlaton Results Correlaton Results Fg Path Delay Samples) Drect Correlaton β=1) Actual Channel Averaged Correlaton β=10) Ideal Channel Path Delay Samples) Averaged path delay estmaton over Vehcular B channel. that f the power of certan path s too small, t can be vewed as nose and wll be gnored. We then nvestgate n Fg. 4 the MSE performance of the proposed tme-frequency jont channel estmaton for TFT- OFDM n large-scale MIMO systems over the Brazl D channel wth the recever velocty of 5 km/h. For comparson, we also nclude the MSE performance of CP-OFDM wth comb-type plots and the tme-doman preamble based teratve channel estmaton/data detecton scheme for large-scale MIMO systems [8]. In addton, the theoretcal CRLB derved n 40) s also plotted as the benchmark for comparson. It s clear that TFT-OFDM outperforms CP-OFDM and TDS- OFDM by about 5 db when the channel estmaton MSE 10 2 s consdered, and performs 3 db better than [8]. Also we could observe that the proposed channel estmaton performs closely to the theoretcal CRLB wth a small SR gap, whch s caused by the fact that the extracted DFT matrx F p) has mperfect but approxmate orthogonal columns. Fg. 5 compares the LDPC coded bt error rate BER) performance of TFT-OFDM MIMO system wth ts counterparts over the Brazl D channel wth the recever velocty of 5 km/h. The BER wth perfect deal channel state nformaton CSI) s also ncluded as the performance lower bound of practcal systems. In addton, as the performance lmt, the mnmum SR of 7.1 db derved to attan the theoretcal ergodc capacty 41) for tranng-based transmsson [8] s also plotted as the benchmark for comparson. It s clear that the proposed TFT-OFDM scheme obvously outperforms the conventonal schemes dependent on only tme or frequency-doman tranng nformaton. For example, at the coded BER of 10 4, the proposed TFT-OFDM MIMO scheme outperforms CP-OFDM MIMO by the SR gan of 1.9 db, and performs 1.4 db better than the large-scale MIMO scheme wth tme-doman preamble [8]. We can also fnd that TFT-OFDM wth the proposed tme-frequency jont channel estmaton s about 1.2 db away from the deal CSI case. Moreover, TFT-OFDM performs only about 2.1 db away from the theoretcal bound, whch ndcates the near-capacty performance of the proposed scheme. Fg. 6 shows the coded BER performance over the Vehcular B channel wth the moble speed of 30 km/h, whch emulates the doubly-selectve fadng channel. Unlke the conventonal large-scale MIMO scheme [8] whose performance degrades about 1.0 db at the BER of 10 4 when moble speed s ncreased from 5 km/h to 30 km/h, only 0.1 db penalty wll be pad by the proposed scheme under the same condton. Ths s caused by that [8] requres the quas-statc channel over the entre transmsson frame of large duraton, whle our proposal could support the applcaton scenaros where the channel s varyng not only wthn the large-sze frame, but also wthn n the small-sze subframe. Therefore, we can conclude that the proposed TFT-OFDM scheme for large-scale

11 DAI et al.: SPECTRALLY EFFICIET TIME-FREQUECY TRAIIG OFDM FOR MOBILE LARGE-SCALE MIMO SYSTEMS CP OFDM MIMO wth Frequency Doman Plots Tme Doman Preamble Based MIMO [8] Proposed TFT OFDM MIMO wth Tme Frequency Tranng Ideal Chanel State Informaton Theoretcal Mnmum SR 41) CP OFDM MIMO wth Frequency Doman Plots Tme Doman Preamble Based MIMO [8] Proposed TFT OFDM MIMO wth Tme Frequency Tranng Ideal Chanel State Informaton Theoretcal Mnmum SR 41) BER BER SR db) SR db) Fg. 5. BER performance comparson between the proposed TFT-OFDM MIMO scheme and ts counterparts over the Brazl D channel wth the recever velocty of 5 km/h. Fg. 6. BER performance comparson between the proposed TFT-OFDM MIMO scheme and ts counterparts over the Vehcular B channel wth the recever velocty of 30 km/h. MIMO systems s robust to fast channel varaton, and more obvous performance gan could be expected when the channel s varyng faster. The proposed TFT-OFDM MIMO scheme has superor performance to CP-OFDM MIMO because the tme-frequency jont channel estmaton could acheve the complete CIR nformaton wth hgh accuracy, whle the plot-based CFR estmaton n CP-OFDM MIMO would ncur some errors due to nterpolaton, especally when the channel s deeply frequency-selectve and only a small amount of frequencydoman plots can be used n large-scale MIMO systems. The reason for the performance gan over the teratve channel estmaton/data detecton scheme n [8] because the latter one assumes quas-statc channel durng the entre transmsson frame, whch devates a lot from the actual moble channels. In contrast to the result that CP-OFDM wth comb-type plots performs worse than the tme-doman preamble based teratve channel estmaton/data detecton scheme when the moble speed s low e.g., 5 km/h n Fg. 5), CP-OFDM outperforms the preamble based scheme [8] because the frequency-doman plots wthn every OFDM symbol could update the CSI more frequently than the tme-doman preamble based scheme when the channel s varyng fast e.g., 30 km/h n Fg. 6). Snce the spectral effcency of TFT-OFDM MIMO s hgher than ts counterparts as addressed n Secton V-A, the proposed scheme acheves hgh spectral effcency as well as relable performance due to the tme-frequency jont processng. VII. COCLUSIOS In ths paper, we propose the spectrally effcent TFT- OFDM transmsson scheme for large-scale MIMO systems to solve the hgh-dmensonal channel estmaton ssue n moble envronments. Wthout cyclc prefx, TFT-OFDM has tranng nformaton n both the tme and the frequency domans for every OFDM symbol, and the frequency-doman grouped plots occupy much fewer subcarrers than that n common CP-OFDM MIMO systems. Ths s acheved by the tmefrequency jont channel estmaton method, whereby the path delays are frstly acqured by the tme-doman receved TSs wthout nterference cancellaton, then there remans much fewer channel parameters to be estmated by fewer frequencydoman plots. The transmsson frame structure composed of one preamble and the subsequent TFT-OFDM symbols could provde effcent channel trackng and data detecton n MIMO systems. Ths paper proves the unfcaton of the tme- and frequency-doman channel estmaton based on the preamble, and derves CRLB of proposed tme-frequency jont channel estmaton. Smulaton results ndcate that the proposed scheme enjoys the BER performance close to the theoretcal ergodc capacty. The proposed TFT-OFDM MIMO scheme can be also drectly appled n multple access systems n both the uplnk and downlnk, and the prncple of jont tme-frequency processng behnd TFT-OFDM can be adapted for other OFDM MIMO systems ncludng large- and smallscale systems) to acheve hgher spectral effcency as well as more relable performance over severe fadng channels. ACKOWLEDGMETS The frst author would lke to thank Prof. Fefe Gao for hs valuable support for ths paper. The authors would lke to thank the Guest Edtor and the anonymous revewers for ther helpful comments and suggestons to mprove the qualty of ths manuscrpt. APPEDIX PROOF OF 15) AD 16) We rewrte 14) as below: { c0 MSE = σ 2 tr H ) 1 } c 0. 42)

12 262 IEEE JOURAL O SELECTED AREAS I COMMUICATIOS, VOL. 31, O. 2, FEBRUARY 2013 Let A = ) [ ] 1, c H 0 c 0 sncec0 = c 1) 0, c2) 0,, ct) 0,the p p matrx A could be denoted as 7 ) Hc ) c 1) 1) Hc ) 0 0 c 1) 2) Hc 0 0 c 1) 1 t) 0 0 ) Hc ) c 2) 1) Hc ) 0 0 c 2) 2) Hc 0 0 c 2) t) 0 0 A= ) Hc ) c t) 1) Hc ) 0 0 c t) 2) Hc 0 0 c t) t) 0 0 p p By denotng the nth dagonal entry of A as An), 42) can be further expressed as p MSE = σ 2 An) σ 2 p p p An), 43) n=1 n=1 where the equalty holds f and only f all the dagonal entres of A are dentcal,.e., A1) = A2) = = A p ). 44) Accordng to the Hadamard nequalty [50], we have p n=1 An) det{a}, 45) where the equalty holds f and only f A s a dagonal matrx, whch ndcates that A 1 s also a dagonal matrx,.e., the off-dagonal elements of A should be zero: ) Hc c ) j) 0 0 = 0 M, j. 46) Due to the perfect autocorrelaton property of the Zadoff- Chu sequence [19], we have ) Hc c ) ) 0 0 = p I L, 1 t. 47) Combnng 46) and 47), we have c 0 H c 0 = A 1 = p I p = t LI tl. 48) So the optmal desgn crteron 15) s proved. Agan, by usng the Hadamard nequalty, we have Thus, det{a 1 } =det{a}) 1 p p. 49) ) p 1 det{a}, 50) where the equalty holds f and only f A s a dagonal matrx. Substtutng 45) and 50) nto 43), we fnally obtan the mnmum MSE of the channel estmator as below { } 1 MSE mn = σ 2 tr t L I tl = σ 2. 51) So 16) s proved. One remark to make s that, the optmal preamble desgn crteron 15) and the correspondng mnmum MSE 16) are consstent wth the results n [13], but the proof technque n ths paper dffers from that n [13]. Here we acqure the optmal desgn crteron based on mnmzng the MSE, and p 7 ote that p = tm = tl as descrbed n Secton II. the Hadamard nequalty s used to avod the nverson of the matrx wth large sze, whle [13] obtaned the optmal desgn crteron based on maxmzng the mutual nformaton, and the arthmetc-harmonc means nequalty are exploted for the proof. REFERECES [1] G. Stuber, J. Barry, S. Mclaughln, Y. L, M. Ingram, and T. Pratt, Broadband MIMO-OFDM wreless communcatons, Proc. 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Theory, vol. 49, no. 4, pp , Apr [50] C. L, Y. Ln, S. Tsa, and P. Vadyanathan, Optmzaton of transcevers wth bt allocaton to maxmze bt rate for MIMO transmsson, IEEE Trans. Commun., vol. 57, no. 12, pp , Dec Lnglong Da M 11) receved hs B.S. degree from Zhejang Unversty n 2003, M.S. degree wth the hghest honor) from Chna Academy of Telecommuncatons Technology CATT) n 2006, and Ph.D. degree wth the hghest honor) from Tsnghua Unversty n 2011, respectvely. He s now a Post Doctoral Fellow wth the Department of Electronc Engneerng, Tsnghua Unversty, Bejng, Chna. Hs research focuses on wreless and optcal communcatons. He has publshed over 20 journal and conference papers. He was awarded the 2011 Tsnghua Excellent Doctor of Electronc Engneerng, the 2011 Tsnghua Academc Star, and the 2012 Bejng Excellent Doctoral Dssertatons. Zhaocheng Wang M 06-SM 11) receved hs B.S., M.S. and Ph.D. degrees from Tsnghua Unversty n 1991, 1993 and 1996, respectvely. From 1996 to 1997, he was wth anyang Technologcal Unversty TU) n Sngapore as a Post Doctoral Fellow. From 1997 to 1999, he was wth OKI Techno Centre Sngapore) Pte. Ltd., frstly as a research engneer and then as a senor engneer. From 1999 to 2009, he worked at SOY Deutschland GmbH, frstly as a senor engneer and then as a prncpal engneer. He s currently a Professor at the Department of Electronc Engneerng, Tsnghua Unversty. Hs research areas nclude wreless communcatons, dgtal broadcastng and mllmeter wave communcatons. He holds 25 granted US/EU patents and has publshed over 70 techncal papers. He has served as techncal program commttee co-char/member of many nternatonal conferences. He s a Senor Member of IEEE and a Fellow of IET. Zhxng Yang M 00-SM 11) receved hs B.S. degree from the Department of Electronc Engneerng, Tsnghua Unversty, Bejng, Chna, n He s now a full professor at the Department of Electroncs Engneerng of Tsnghua Unversty, Bejng, Chna. He s the executve drector of the State Key Laboratory on Mcrowave and Dgtal Communcatons, Chna, and the executve drector of the development group of the dgtal televson terrestral broadcastng state standard for Chna. He receved several natonal awards and held dozens of patents. Hs research nterests are n hgh-speed data transmsson over broadband dgtal televson terrestral broadcastng, wreless lnks, wreless communcaton theory and communcaton systems desgn. He s a Senor Member of IEEE.