Space Frequency Coded OFDM_Paulraj

8/12/2019 Space Frequency Coded OFDM_Paulraj

1/6

SPACE-FREQUENCY CODED BROADBAND OFDM SYSTEMS

Helmut Bolcskei a n d Arogyaswam i J. PaulrajInformation Systems Lab oratory, Stanf ord University

Packard 223, 350 Serra Mall, Stanford, CA 94305-9510Phone: (650)-724-3640, Fax: 650)-723-8473, email: bolcskei{paulraj}@rascals.stanford.edu

Abstract-Sp ace- time coding for fading channels is acommunication technique that realizes the diversity ben-efits of multiple tran smi t antennas. Previous work inthis area has focused on the narrowband flat fading casewhere spatial diversity only is available. In this paper,we investigate the use of space-time coding in OFDM-based broadband systems where both spatial and fre-quency diversity are available. We consider a strategywhich basically consists of coding across OFDM tonesand will therefore be called space-frequency coding. Fora spatial broadband channel model taking into accountphysical propagation parameters and antenna spacing, wederive th e design c ri teria fo r space-frequency codes andwe show that space-time codes designed to achieve fullspatial diversity in th e narro wband case will in general notachieve full space-frequency diversity. Specifically, we showthat the Alamouti scheme across tones fails to exploit fre-quency diversity. For a given set of propagation parame-ters and given antenna spacing, we e s ta b li s h th e m a x i m u machievable diversity order. Finally, we provide simulationresults studying the influence of delay sp read, propagationparameters, and antenna spacing on the performance ofspace-frequency codes.

1. I N T R O D U C T I O N A N D O U T L IN ETwo of the major impairments of wireless communi-

cations systems are fading caused by destructive additionof multipaths in the propagation medium and interferencefrom o ther users . Diversity provides the receiver with sev-eral (ideally independent) replica of the tr ansm itted sig-nal and is therefore a powerful means to combat fadingand interference. Common forms of diversity are time di-versity (due to Doppler spread) and frequency diversity(due to delay spread ). In recent years th e use of spa-tial or antenna) diversity has become increasingly pop-ular. Spatial diversity is particularly attr act ive since itcan be provided without loss in spectral efficiency. Re-ceive diversity, i.e., the use of multiple antennas on the

This work was suppor ted in part by FWF grants J1629-TECand J1868-TEC. H . Bolcskei is on leave from the Institut fiirNachrichtentechnik und Hochfrequenztechnik, Technische Uni-versitat Wien, Vienna, Austria. A . Paulraj is on part-time leaveat Gigabit Wireless Inc., 3099 N. First Street, San Jose, CA.

0-7803-6596-8/00/$10.00 000 IEEE

receive side is a well-studied subject [l]. Driven by mo-bile wireless applications, where it is difficult to deploymultiple antennas in the handset, t ransmi t d ivers i ty orequivalently the use of multiple antennas on the transmitside has become an active area of research [2]-[8]. Space-t im e coding evolved as one of the most promising transmitdiversity techniques [9]-[13].

Most of th e previous work on tr ansm it diversity hasbeen restricted to single-carrier systems operating over nar-rowband channels' where spatial diversity only is available.In thi s paper, we consider multi-antenna (multiple transmitand one or several receive antennas) broadband channels(i.e. channels with delay spread) where bot h spatial diver-sity (due t o multiple antenn as) and frequency diversity (dueto delay spread) are available. Orthogonal frequency divi-s ion mul t ip lex ing OFDM) [14]-[16]ignificantly reduces re-ceiver complexity in wireless broadband multi-antenna sys-tems [17, 181. We therefore study the use of space-timecoding in OFDM-based multi-antenna systems.

Con tri but ion s. We consider a strategy which basical-ly consists of employing a space-time code across OFDMtones and will therefore be called space-frequency coding.Our contributions are as follows.

For a spatial delay spread channel model taking intoaccount physical propagation parameters and anten-na spacing, we derive the design cri teria for space-frequency codes and we show that space-time codesdesigned to achieve full spatial diversity in the nar-rowband case will in general not yield full space-frequency diversity. The requirement of exploitingfrequency diversity as well imposes additional con-straints on the codes and makes the design consider-ably more complicated than in the narrowband case.

0 For a given set of channel parameters and given an-tenn a spacing, we derive the ma xi mu m achievable d i -versi ty order and we discuss the impact of antennaspacing and propagation parameters on diversity.Using the design criteria established in thi s paper, weshow that the A l a m o u t i s c h e m e across tones fai ls t o

'Unless explicitly stated otherwise, throughout the paperwhen talking about the narrowband case we will actually meanthe frequency-flat slow fading case.
mailto:bolcskei%7Bpaulraj%[email protected]:bolcskei%7Bpaulraj%[email protected]


2/6

exploit the frequency-diversity available in the delayspread case.We provide simulation results demonstrating theperformance of known space-time codes employedas space-frequency codes under various propagationconditions.

Organizat ion of the paper. The rest of this paper isorganized as follows. In Section 2, we introduce th e channelmodel, we briefly describe broadb and OFDM-based multi-ante nna systems an d we discuss space-frequency coding. InSection 3, we derive the design criteria for space-frequencycodes and we discuss their relation to previously establisheddesign criteria for the narrowband (slow and fast fading)case. In Section 4, we provide some simulation results. Fi-nally, Section 5 contains our conclusions.

2. CHANNEL MODEL, OFDM, ANDSPACE-FREQUENCY CODINGWe shall first introduce the channel model, and then

briefly describe OFDM-based multi-antenna systems andspace-frequency coding.2.1. The Channel Model

In the following MT and M R denote the number oftransmit and receive antennas, respectively. We assumeth at the channel consists of L mat rix taps (each of size M Rby MT ith th e matrix-valued transfer function given by

I=OWe restrict ourselves to Rayleigh fading channels. Hencethe elements of th e Hi 1 = 0, 1, ..,L - 1) are (possibly cor-related) circularly symmetric zero mean complex gaussianrando m variables with variance 1. We furthermore assumetha t heavy scattering occurs around the transmitt er where-as the receiver is unobstructed. A schematic representationof the channel model is provided in Fig. 1. The 1-th tapin the channel impulse response is assumed to correspondto the I-th scatterer cluster with mean angle of arrival atthe receive array of 81 (derived from the position of thescatterer cluster), angular spread 61 (proportional to thescattering radius of the cluster), and a (complex) path gainp t Since heavy scattering occurs around the transmit ter,different transmit antennas will be uncorrelated which willbe taken into account by assuming that different columnsof Hl are uncorrelated. Due to the lack of scattering at th ereceiver different receive antennas will be correlated whichcorresponds to assuming th at different rows of HI are cor-related. We furthermore assume tha t the different scattererclusters are uncorrelated.We assume a uniform linear array at both the trans-mitter a nd the receiver. The relative ante nna spacing isdenoted as A = f, where d is the absolute antenna spac-ing and = c / fc is the wavelength of a narrowband signal

2 A circularly symmetric complex gaussian random variable isa random variable z = z+ jy CN 0,a 2 ) ,where x and y arei.i.d. N ( 0 , a 2 / 2 ) .

with center frequency fc. We furthermore assume that theangle of arrival for th e I-th pa th cluster a t th e receiver isgaussian distributed around the mean angle of-arriyal ,&e., he actual angle of arrival is given by3 BI = 81 +BI with0I - n/ O,ae,). The correlation matrix of the k-th columnof the zero mean matrix HI given by4 RI = &{hl,r;htk}with hl,r;= [h,(P,) it,) ... is independent of k, orequivalently the fading statistics are th e same for all trans-mit antennas. Defining p ~ ( s A , , e, ) = { h ~ r ~ h ~ r ~ ) * }for 1 = 0,1, ...,L - l , k = 0,1, ...,M T - 1 to be thefading correlation between two receive antenna elementsspaced S A wavelengths apart, the correlation matrixRI 1 = 0, 1, ...,L - 1) can be written as

[ ] m , n = IPIIPI ~m > A ,81OB,) , 2)where we have absorbed the power delay profile of the chan-nel into the correlation matrices. For small angular spreadthe correlation function in 2) is given by [19]

p l ( s ~ , I , e,) e - j 2 n 5 A c o s 8 r ) e - - 2 r r s A s i n 8 i ) o e l ) 2We note that although this approximation is accurate onlyfor small angular spread, it does provide the correct trendfor large angular spread, namely uncorrelated spatial fad-ing. It is interesting to observe tha t for oe = 0 the corre-lation matrix RI collapses to a rank-1 matrix and can bewritten as RI = I/3112a(&)aH(8i) with th e a rray responsevector of a uniform linear array given by

.(e) = [1 e j 2 ~ A c o s ( @ ej2r(MR-1)Acos(e) T... . 3)The rank of the correlation matrix RI ill therefore be driv-en by the antenna spacing at the receiver and the anglespread of the I-th scatterer cluster.

Fig. 1. Space-time delay spread channel composed ofmultiple clustered paths. Each path cluster has a mean

angle of arrival 81 and an angle spread 61.Factoring the MR M R correlation matrix RI accord-ing to RI = R:2Ry2, where R: is of size M R x M R ,

the M R x MT matrices HI can be written asHI = R:2Hw,I, 1 = O , l , ...,L - 4)

3The variance ug, is proportional o the angular spread 61 and4The superscripts* and stand for transposition and con-hence the scattering radius of the l- th path cluster.jugate transposition, respectively.

2


3/6

/ \ith the Hw,ibeing uncorrelated MR x MT matrices withi.i.d. C N 0 , l ) entries.2.2. OFDM-based Multi-Antenna Systems

In an OFDM-based multi-antenna system the datastreams are OFDM-modulated before transmission. TheOFDM modulator applies an N-point IFFT to N con-secutive data symbols and then prepends the cyclic pre-fix (CP) (which is a copy of the last L samples of thesymbol) to the symbol, so that the overall OFDM sym-bol length is M = N + L . In the receiver, the individualsignals are passed through an OFDM demodulator whichfirst discards the CP and then applies an FFT. Organiz-ing the transmitted data symbols into frequency vectorsc k = [c f cr ... with c t ) denoting the datasymbol transmitte d from the i-th antenna on the k-th tone,the reconstructed d ata vector for the k-th tone is given by

k ]

r k = 6 H ( e j s k )k -I-nk, k = 0 , 1 , ...,N - , 5 )where nk is complex-valued additive white gaussian noisesatisfyingwith I M ~enoting the identity matrix of size MR. Thedata symbols c t are taken from a finite complex alphabetchosen such that the average energy of t he constellationelements is 1.2.3. Space-Fkequency Coding

E{nknp) = B ~ I M ~ B [ ~ 6)

The bit stream to be transmitted is encoded by thespace-frequency encoder into blocks of size MT x N. Onedata burst therefore consists of N vectors of size MT x 1or equivalently one spatial OFDM symbol. The channelis assumed to be constant over at least one OFDM sym-bol. Assuming perfect channel sta te information, the maxi-mum likelihood (ML) decoder computes t he vector sequence, k = 0 , 1 , ... N - 1 according to

N-1i% = argmin - i H ( e J k ) k1I 2 ,C k=O

where C = [CO1 ... cjv-11 and the minimization is overall possible codewords . We finally note that a n interest-ing experimental performance analysis of a space-frequencycoded OFDM system appears in [20].

3. DESIGN CRITERIAIn this section, we shall derive the design criteria for

space-frequency codes assuming tha t the receiver has per-fect channel state information.3.1. Pairwise Error Probability

Let c = [CO c ... CN-11 and E = [e- el ... enr 11 betwo different space-frequency codewords of size MT x Nand assume that C was transmitted. For a given channelrealization H(ej k),he probability that the receiver de-cides erroneously in favor of the signal E is given by [21]

P (C + EIH(ejPk))= Q ( / ~ d z ( C , E I H ( e j ~ k ) ) )where

N-1d2(C,EIH(ejgk))= IIH(e@)(Ck - k ) 1 l 2

denotes the squared Euclidean distance between the twocodewords C and E. Using the Chernoff bound Q(z) 5e-x2/2 we get

k=O

- dZ(C,EJH(e * ))C -+ EIH(ej k)) e 4 r n . 7)Next, we need to compute th e expected pairwise error prob-ability by averaging over all channel realizations taking intoaccount the channel model presented in Sec. 2.1 . For thiswe define Yk = H(egk)(Ck e h ) for k = 0 , 1 , ...,N-1 and

T Ty = [Y? Y T ... YN-11 .With th is notation we get d 2 (C , EIH(ejsk llY/I2andhence 7) can be rewritten asSince the Hi were assumed to be ii d . gaussian it followsfrom 1) hat the H( ej %k ) for k = 0 , 1 , ...,N - are gaus-sian aswell and hence the M R N x 1 vector Y s gaussian.The average over all channel realizations of the right-handside in (8) is fully characterized by the eigenvalues of thecovariance matrix of Y [22] defined as Cy = E { Y YH} n[23] it is shown that

L - 1Cy = [D(C- E)T(C-E)*DiH]C R I , 9)

I=Owhere D = diag{e-j*k}fz:, A B denotes the Kroneck-er product of the matrices A and B and the superscript *stan ds for elementwise conjugation. Denoting the nonzeroeigenvalues of CY as5 Xi(Cy) i = 0 , 1 , ... r(Cy)- 1) thefollowing result can be established [23]

where P(C -+ E) = EH {P (C EIH(eJsk))} s thepairwise error probability averaged over all channel real-izations. Next, using the following property of Kroneckerproducts

(A B)(F G) = (AF) (BG)and the factorizations Ri = R:/2Ry/2 1 = 0 , 1 , ... L - ) ,it can be shown [23] that

y = G(C,E) GH(C,E) 11)5r(A) denotes the rank of the matrix A.Here, E stands for the expectation operator.

3


4/6

with the N M R x M T M R L matrix = O l ...,L - . In this case we obtain from (9)G(C,E) = [(C - E)* @ R;I2 [D(C- E)*] @ R:I2 L-1CY= [D'(C- E)T(C-E)*DIH]@ I M ~l=O

\

... [DL-'(C- E)T]@ R2/_2,] 12) RYwhich implies that the rank of CY will be M R times theBased on lo ) , 11) and 12) we are now able to derive thedesign criteria for space-frequency codes.3.2. Design Criteria and Diversity Order

In the following, we assume t ha t N > MTL The de-sign criteria for space-frequency codes follow from 10) asthe well-known rank and determinant criteria derived in[ l l 91 for the single-carrier narrowband case with the ma-trix B(c,e)defined in Eq. 6 of [ l l] eplaced by GT(C,E).Note that for L = 1 and = I the matrix GT(C,E) e-duces to (C - E) @ 1 ~ ~ .ow, using the fact that everyeigenvalue of the MT x MT matrix (C E)(C E)* is aneigenvalue of the M T M R x M T M R matrix [(C- E)(C-E)*] @ 1h.1~ith multiplicity M R , it follows that the designcriteria in an OFDM system with no delay spread are equiv-alent to those in a single-carrier-based narrowband system.This is intuitively clear since for L = 1 there is no frequencydiversity

Let us next use 12) to establish some results on themaximum achievable diversity order in broadband space-frequency coded OFDM systems. Th e maximum rank ofthe N M R x M T M R L matrix G(C,E) s M T M R L . Sincer(R 12)= r(R) we can establish the following [23]:

Theorem 1 . The maximum achievable dzversaty order ina space-frequency coded broadband OFDM system is givenby

L-1dmaz = s C r ( R i ) T M R L ,

where s is the minimum rank of (C - E) over all pairs ofcodewords C and E.

From (12) it follows th at the I-th multipath can poten-tially add a diversity order of sr(R1). Since the rank of RIwill be governed by the angle spread of the I-th scatterercluster and the antenna spacing at the receiver, we con-clude that the achievable diversity order critically dependson the propagation environment and t he antenna spacing.In [23] it is shown that in order to achieve MTMRL-folddiversity it is necessary to have r(D'(C - E)*) = MTfor i = 0 , 1 , ..., L - 1 and for all pairs of codewords Cand E. Furthermore, it is required that ~ ( R I )M R for1 = 0, 1, ..,L - . These conditions guarantee that the in-dividual N M R x M T M R matrices [D'(C- E)*] @ R , i =0 , 1,...,L - ) have full rank . Finally, the space-frequencycode has to be designed such that the stacked matrixG(C,E) has full rank as well.

In the following, in order to simplify the discussion werestrict our attention to the case of no spatial fading corre-lation and a uniform power delay profile, i.e., Ri = 1~~ for

1=0

rank of the N x N matrixRy = F(C,E) FH(C,E)

with the N x M T L matrixF(C,E)= [(C-E)T D(C-E)* ... DL-'(C-E)*]. 13)

The coding gain in this case can be obtained by makinguse of the fact that every eigenvalue of Ry is an eigen-value of CY with multiplicity MR The question of whatdiversity order a given space-frequency code achieves nowreduces to th e question of finding the minimum rank ofthe N x M T L matrix F(C,E) over the set of all possiblecodewords C and E. In order to achieve full (i.e. M T M R L -fold) diversity the matrix F(C,E)has to be full rank forevery pair of codewords C and E. This is the case when(C -E) has rank MT for all codewords C and E and eachof the blocks B; = Di(C- E ) T is linearly independent ofthe other Bi with 1 i for every pair C and E. Whilea space-time code designed to achieve full diversity in thenarrowband case will have full rank (C -E) and hence fullrank B, i = 0,1, ...,L - 1) for all codewords C and E, thelinear independence of the blocks B, will not be guaran-teed. Note tha t ensuring linear independence of the blocksB, amounts to ensuring t ha t the space-frequency code ex-ploits the available frequency diversity as well. We can ,however, make t he following statement:

Theorem 2 . A space-time code designed to provide adiversity order of s in a single-carrier-based narrowbandsystem provides at least th e same diversity order in a broad-band OFDM system.

The proof of Theorem 2 follows immediately by not-ing that if (C - E) has minimum rank s over the setof all possible codewords C and E, then the minimumrank of F(C,E) will be at least s. A result similar toTheorem 2 was reported in [24] for single-carrier systemsoperating in delay-spread environments. We emphasize,however, that a space-time code achieving full spatial di-versity in a narrowband environment will in general notachieve full space-frequency diversity in the OFDM broad-band case. Th e next subsection provides an example cor-roborating this statement. In [23] it is furthermore showntha t space-time codes designed for the narrowband fas t fad -ing case will in general not be guaranteed to achieve fullspace-frequency diversity in the OFDM broadband case. Infact they may not even achieve full spat ial diversity [23]. Wenote, however, that even though space-time codes designedfor the fa st fading case and s mart greedy space-time codes[ l l]will be sub optimu m in a space-frequency setting, theycan be expected to exploit at least some of the availablefrequency diversity. Summarizing, we conclude tha t usingexisting space-time codes in a space-frequency setting willin general be suboptimum. New designs are needed takinginto account the criteria derived in this paper. Some first

4


5/6

designs of space-frequency codes are provided in [23]. Thesystematic design of good space-frequency codes remains animportant open research problem.3.3. Space-Frequency Alamouti Scheme

In the previous subsection we found that space-timecodes designed to achieve full spatial diversity in thenarrowband case will in general not achieve full space-frequency diversity. We shall next provide an example il-lustrating this effect. Specifically, we consider the case oftwo transmit antennas M R is arbitrary) and employ theAlamouti scheme [lo] across OFDM tones, i.e., the matri-ces (C -E) have the following stru ctur e

with di = ci - i i = 0,1, ...,N - 1). FromN - 1. -

(C E)(C- E)H= ldiI2 1 2i = O

it follows tha t all possible code difference matrices C and Ewill be of full rank and hence th e code achieves full spatialdiversity in the narrowband case. This is a well-known factand has been proved in [lo]. Next, take the minimum dis-tance error event where only one out of the N di is nonzerosay d o . In that case the matrix (C-E)Twill have nonzeroentries only in the two top rows. Therefore, all the matri-ces Di(C-E)T i = 0,1, ...,L- 1) will have nonzero entr iesin the two top rows only and hence it follows from (13)that th e matrix F C,E) will have rank 2 irrespectively ofthe amou nt of delay spread (an d hence frequency diversi-ty) in the channel. We conclude tha t while the Alamoutischeme continues to achieve second-order diversity in thebroadband OFDM case, it fails to exploit the additionallyavailable frequency diversity. This once again shows thatthe broadband OFDM case calls for new code designs.

4. SIMULATION RESULTSIn this section, we provide simulation results demon-

stra ting the performance of known space-time codes em-ployed as space-frequency codes and studying t he influenceof physical propagation parameters on the performance ofspace-frequency codes. We simulated an OFDM systemwith MT = 2, N = 64 tones and C P of length 16 usingthe two transmit antenna 16-state 4-PSK code proposedin [ll]. The signal-to-noise-ratio (SNR) was defined asSNR = 10 log 2). ll results were obtained by averag-ing over 10,000 independent Monte Carlo tria ls, where eachrealization consisted of one burst (i.e. one OFDM symbol).

Simulation Example 1. In t he first simulation exam-ple we stu dy the impact of delay spread on the performanceof space-frequency codes. For M R = 2 and uncorrelatedspatial fading Fig. 2 shows the symbol error ra te as a func-tion of SNR for the low delay spread case (i.e. one path with/pol2= 1) and for the high delay spread case (i.e. six path swith IP;l2 = l), respectively. We can see that the presenceof delay spread drastica lly improves the performance of the

space-frequency code. From the slopes of the two curveswe can conclude th at th e space-frequency code does indeedseem to be able to exploit a t least some of the availablefrequency diversity, but certainly not all of it. Recall thatsince MT = M R = 2 and six paths with path gain 1 arepresent th e channel provides 24-th ord er diversity.

Simulation Example 2. In Sec. 3.2 we have demon-strated that the rank of the correlation matrices RI hascritical impact on the available diversity. Furthermore, weshowed in Sec. 2.1 that the rank of the correlation matri-ces is driven by the angle spread of the individual scattererclusters and by th e ante nna spacing at the receiver. In thissimulation example, we st udy t he influence of spatia l fadingcorrelation on the performance of space-frequency codes.We assumed the following power delay profile [l 0.77 0.561.Fig. 3 shows the symbol error rat e for M R = 2 and M R = 3and low and high spatial fading correlation, respectively.The solid curves correspond to M R = 2 whereas the dashed-dott ed curves correspond to M R = 3. T he upper two curvesshow the symbol error r ate in the case of high spatial fadingcorrelation which was achieved by making the cluster anglespreads small and choosing the relative antenna spacing tobe A = 1/4. Low spatial fading correlation was achieved bysetting A = 1 / 2 and choosing large cluster angle spreads.We can clearly see that in the case of low spatial fading cor-relation th e performance of th e code is significantly bettertha n in the case of high spatial fading correlation. This isconsistent with our findings (and intuition) that high spa-tial fading correlation reduces the diversity order and henceyields degraded performance in terms of symbol error rate.We can furthermore see that adding a receive antenna hasmuch less impact in the case of high spatial fading correla-tion. This is so, since due to th e small cluster angle spreadsand hence large spatial fading correlation only a small in-crease in spatial diversity can be expected from additionalreceive antennas.

5. CONCLUSIONWe studied space-frequency coded broadband OFDM

systems where both spatial and frequency diversity areavailable. Considering a strategy which consists of cod-ing across OFDM tones and employing a spatial broadbandchannel model taking into account physical propagation pa-rameters and antenna spacing, we derived the design cri te-r ia for space-frequency codes. We furthermore showed thatspace-time codes designed to achieve full spatial diversi-ty in the narrowband case will in general not achieve fullspace-frequency diversity in the broadband case. Space-time codes designed for the fast fading case and smartgreedy space-time codes [ll] an be expected to be able toexploit a t least some of th e available frequency diversity.Nevertheless, these codes will generally be suboptimumwhen employed as space-frequency codes. Hence, new de-signs taking into account the design criteria derived in thispaper are needed. We furthermore established the maxi-mum achievable diversi ty order in space-frequency codedOFDM systems and we studied the impact of spatial fadingcorrela tion on the performance of space-frequency codes.

5


6/6

Our simulation results showed that space-time Trellis codesdesigned for the narrowband case do seem to ex-ploit at least some of the available frequency diversity(but certainly not all of it) when employed as space-frequency codes. Acknowledgment

Th e auth ors would like to thank R. W. Heath Jr. forproviding part of the MATLAB code used to obtain thesimulation results in Sec. 4.

............................... .I o w a whbhdelm

. . . . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

f 10-

10- 2 3 4 5 7 8SNWdB

Fag.2. Impact of delay spread o n the performance ofspace-frequency codes.

n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,.....................10

1 2 3 4 5 8 7 8SNWdB

Fig.3. Impact of spatial fading correlation and M R on theperformance of space-frequency codes.

6. REFERENCESW. C. Jakes, Microwave mobile communications. New York:Wiley, 1974.A. Wittneben Base station modulation diversity for digi-tal SIMULCAhT, in Proc. IEEE VTC, pp. 505-511, May1993.V. Weerackody, Diversity for direct-sequence spread spec-trum system using multiple transmit antennas, in Proc.IEEE ICC, pp. 1775-1779, May 1993.

N. Seshadri and J. Winters, Two signaling schemes for im-roving the error performance of frequency-division-duplexFDD) transmission systems using transmitter antenna di-versity, Int. J . Wireless Information N etwor ks , vol. 1,no. 1, pp. 49-60, 1994.J. H. Winters, J. Salz, and R. D. Gitlin, The impact ofantenna diversity on the capacity of wireless communicationsystems, IEEE Dans. Comm., vol. 42, pp. 1740-1751, Feb.1994.G. W. Wornell and M. D. Trott, Efficient signal process-in techniques for exploiting transmit antenna diversity onf s i n g channels, IEEE hns. Signal Processing, vol. 45,no. 1,pp. 191-205, 1997.A. Hiroike, F. Adachi, and N. Nakaima, Combined ef-fects of phase sweepin transmitter diversity and channelcoding, IEEE h n s . eh. Technol., vol. 41, pp. 170-176,May 1992.A. Narula, M. D. Trott, and G. W. Wornell, Performancelimits zf coded diversity methods for transmitter antennaarrays, IEEE Dans. Inf. Theory, vol. 45, no. 7, pp. 2418-2433, 1999.J . Guey, M. Fitz, M. Bell, and W. Kuo, Signal design fortransmitter diversity wireless communication systems overRayleigh fading channels, in Proc. IEEE VTC, pp. 136-140, 1996.S. M. Alamouti, A simple transmit diversity techniquefor wireless communications, IEEE J . Sel. Areas Comm.,V. Tarokh, N. Seshadri, and A. R. Calderbank, Space-timecodes for high data rate wireless communication: Perfor-mance criterion and code construction, IEEE Trans. In .Theory, vol. 44, pp. 744-765, March 1998.V. Tarokh, H. Jafarkhani, and A. R. Calderbank, Space-time block codes from orthogonal designs, IEEE Trans.I n . Theory, vol. 45, pp. 1456-1467, July 1999.A. R. Hammons Jr. and H . El Gamal, On the theory ofspace-time codes for PSK modulation, IEEE Zkuns. Inf.Theory, vol. 46, pp. 524-542, March 2000.A . Peled and A. Ruiz, frequency domain data transmis-sion using reduced computational complexity algorithms,in Proc. IEEE ICASSP-80, (Denver, CO), pp. 964-967,1980.L. J. Cimini Analysis and simulation of a digital mobilechannel usink orthogonal frequency division multiplexing,IEEE %ns. Comm., vol. 33, pp. 665-675, July 1985.B. LeFloch, M. Alard, and C. Berrou, Coded orthogonalfrequency division multiplex, Proc. of IEEE, vol. 83, p-p. 982-996, June 1995.G. G. Raleigh and J . M. Cioffi, Spatio-temporal codingfor wireless communication, IEEE Tkans. Comm., vol. 46,no. 3,pp. 357-366, 1998.H. Bolcskei, D. Gesbert, and A. J. Paulraj, On the ca-pacity of wireless systems employing OFDM-based spatialmultiplexing, IEEE Dans. Comm., 1999. submitted.D. AsztBly, On antenna arrays in mobile communicationsystems: Fast fadin and GSM base station receiver algo-rithms Tech. Rep. fRR-S3-SB-9611,Royal Institute of Tech-nology, Stockholm, Sweden, March 1996.D. Agarwal, V. Tarokh, A. Naguib, and N . Seshadri, Space-time coded OFDM for high data rate wireless communi-cation over wideband channels, in Proc. IEEE VTC 98,pp. 2232-2236, May 1998.J. G. Proakis, Digital Communications. New York:McGraw-Hill, 3rd ed., 1995.G. L. Turin, The characteristic function of hermitianquadratic forms in complex normal variables, Biometrika,H . Bolcskei and A. J. Paulraj, Space-frequenc codedbroadband OFDM systems: Design criteria, pedrm an ceanalysis, and code design, IEEE 2s . I n . Theory. inpreparation.V. Tarokh, A. Naguib, N. Seshadri, and A. R. Calderbank,Space-time codes for high data rate wireless communica-tion: Performance criteria in the presence of channel esti-mation errors, mobility, and multiple paths, IEEE h n s .Comm., vol. 47, pp. 199-207, Feb. 1999.

vol. 16, pp. 1451-1458, Oct. 1998.

vol. 47, pp. 199-201, 1960.

6

Documents

Space Frequency Coded OFDM_Paulraj