《OFDM技术开发系列》（英文版） space time code design in ofdm system

Space-Time Code Design in OFDM Systems Ben Lu and Xiaodong Wang Department of Electrical Engineering Texas A&M University, College Station, TX 7784 Er-mail:(benlu, wang]@ee. tamu.edu de ( STC)or- the pairwise error probability(Pep) of the considered thog n regegneycyedeiive adinltiphexing so M) asys- STC-OFDM system is analyzed. In Section II-C,the TC-OFDM systems can potentially provide a diversity II-D, following the proposed coding design principles, a nstructed for OFDM systems. selectivity order, and that the large effective length and In Section III, computer simulation results are present- n designing STC's for OFDM systems. Following the principles, we propose a new class of trellis-structured II. PERFORMANCE ANALYSIS s codes, our proposed STC's significa the A. System Mode rsity and the fr ive-fading diver I. INTRODUCTION Considerable amount of recent research has addressed Theodor the design and implementation of space-time coded (STC) systems for wireless fat-fading channels, e.g,Fig.. An STC-OFDM system withN=2 transmitter antennas 1],(2:[3]. The space-time coding methodologies in- and M= I receiver antenna tegrate the techniques of antenna array spatial diver- sity and channel coding, and can provide significan We consider an STC-OFDM system with K subcar t capacity gains in wireless channels. However, many riers, N transmitter antennas and M receiver antennas, which the STC design problem becomes a complicated Each STC code word consists of (NK)STC symbols issue. On the other hand, the orthogonal frequency and transmits simultaneously during one OFDM word lated flat-fading channels. Hence, in the presence of assumed that the fading process remains static during proposed in [4]. In this paper, by deriving the pairwise uncorrelated. As an example, an STC-OFDM I system error probability(PEP)of the OFDM system, we with N=2 transmitter antennas and M= 1 receiver number of transmitter antennas. m is the number of re- ceiver antennas. After matched filtering and symbo (i.e, the number of non-zero resolvable taps). Mean- applied to the received discrete-time signal to obtain while, we show that the large effective length [5] and the ideal interleaving are two most important design prin- yk]=Hk+z[k],k=1,…,K,(1) ciples in designing STC's for OFDM systems, Based on these two simple principles, a new class of bandwidth where H[k E CMXN is the matrix of complex chan The rest of this paper is organized as follows. In explained belbresponses at the k-th subcarrier, which is fficient STCs for OFDM systems is found Section II-A, an STC-OFDM system over frequency- tively the transmitted signal and the received signal at selective fading channels is described. In Section II-B. the k-th subcarrier; x[k]e cM is the ambient noise which is circularly symmetric complex Gaussian with Consider the channel response between the j-th transmitter antenna and the i-th receiver anten na. Fol- Texas Telecommunications Engineering Consortium(IxTEc). lowing [6], the time-domain channel impulse response 0-780364546/001000(C)2000

Space-Time Code Design in OFDM Systems Ben Lu and Xiaodong Wang Department of Electrical Engineering Texas A&M University, College Station, TX 77843 E-mail: {benlu, wangx}@ee.tamu.edu Abstract- We consider a space-time coded (STC) or￾thogonal frequency-division multiplexing (OFDM) sys￾tem in frequency-selective fading channels. By analyz￾ing the pairwise error probability (PEP), we show that STC-OFDM systems can potentially provide a diversity order as the product of the number of transmitter anten￾nas, the number of receiver antennas and the frequency selectivity order, and that the laTge effective length and the ideal inte~leawing are two most important principles in designing STC’S for OFDM systems. Following these principles, we propose a new class of trellis-structured STC’S. Compared with the conventional space-time trel￾lis codes, our proposed STC’S significantly improve the performance by efficiently exploiting both the spatial di￾versity and the frequency-selective-fading diversity. I. INTRODUCTION Considerable amount of recent research has addressed the design and implementation of space-time coded (STC) systems for wireless flat-fading channels, e.g., [1], [2], [3]. The space-time coding methodologies in￾tegrate the techniques of antenna array spatial diver￾sity and channel coding, and can provide significan￾t capacity gains in wireless channels. However, many wireless channels are frequency-selective in nature, for which the STC design problem becomes a complicated issue. On the other hand, the orthogonal frequency￾division multiplexing (OFDM) technique transforms a frequency-selective fading channel into parallel corre￾lated flat-fading channels. Hence, in the presence of frequency-selectivity, it is natural to consider STC in the (3FDM context. The first STC-OFDM system was proposed in [4]. In this paper, by deriving the pairwise error probability (PEP) of the STC-OFDM system! we show that the STC-OFDM system can potentially pro￾vide a diversity at the order of (NkfL), where N is the number of transmitter antennas, A4 is the number of re￾ceiver antennas and L is the frequency-selectivity order (i.e., the number of non-zero resolvable taps). Mean￾while, we show that the large effective length [5] and the ideal interleaving are two most important design prin￾ciples in designing STC’S for OFDM systems. Based on these two simple principles, a new class of bandwidth efficient STC’S for OFDM systems is found. The rest of this paper is organized as follows. In Section II-A, an STC-OFDM system over frequency￾selective fading channels is described. In Section II-B, This work was supported in part by the the U.S. National Sci￾ence Foundation under Grant CAREER CCR–9875314, and in part ‘by a .@ft .s=nt from the Motorola S1’s =d D Sp core l’=h￾nology Center. Ben Lu’s work was also supported in part by the Texas Telecommunications Engineering Consortim (TxTEC). the pairwise error probability (PEP) of the considered STC-OFDM system is analyzed. In Section II-C, the STC coding design principles are proposed. In Section II-D, following the proposed coding design principles, a new class of STC’S is constructed for OFDM systems. In Section III, computer simulation results are present￾ed. Section IV contains the conclusions. II. PERFORMANCE ANALYSIS A. System Model Fig, 1. An STC-OFDM system with N = 2 transmitter antennas and M = 1 receiver antenna. We consider an STC-OFDM system with K subcar￾riers, N transmitter antennas and M receiver antennas, signaling through a frequency-selective fading channel. Each STC code word consists of (NK) STC symbols and transmits simultaneously during one OFDM word. Each STC symbol is transmitted at a particular OFDM subcarrier and a particular transmitter antenna. It is assumed that the fading process remains static during each OFDM word; and the fading processes associat￾ed with different transmitter-receiver antenna pairs are uncorrelated. As an example, an STC-OFDM system with N = 2 transmitter antennas and Ill = 1 receiver antenna is illustrated in Figure 1, At the receiver, the signals are received from M re￾ceiver antennas. After matched filtering and symbol￾rate sampling, the discrete Fourier transform (DFT) is applied to the received discrete-time signal to obtain !l[k]= Iqk]x[k] + .%[k], k=l, . . ..K. (1) where .FI[k] c CM’~ is the matrix of complex chan￾nel frequency responses at the k-th subcarrierj which is explained below; x [k] c CN and y[k] s CM are respec￾tively the transmitted signal and the received signal at the k-th subcarrier; .z[k] c CM is the ambient noise, which is circularly symmetric complex Gaussian with unit variance. Consider the channel response between the j-th transmitter antenna and the i-th receiver antenna. Fol￾lowing [6], the time-domain channel impulse response 0-7803-6454-6/00/$10.00 (C) 2000

can be modeled as a tapped-delay line. With only the where y is the total signal power transmitted from all non-zero taps considered, it can be expressed as N transmitted antennas(Recall that the noise at, each receiver antenna is assumed to have unit variance); 7 全 hx(t)=∑a(1(7-),(2)h图,Ung(),(,) is given by where &()is the Kronecker delta function; L denotes d(1)=∑∑∑H13因 the number of non-zero taps; ai,j (4; t)is the complex =1k=0j=1 amplitude of the I-th non-zero tap, whose delay nI/A/, where nt is an integer and A, is the tone spac- ing of the OFDM system. In this paper, we restrict our hN1x2[W(=因 attention to the stc transmitted and received during one OFDM slot; for notational convenience, the time e"[k]W(k) ndex t is dropped henceforth NL)XI For OFDM systems with proper cyclic extension and sample timing, with tolerable leakage, the channel fre-=>h; Dhi quency response between the j-th transmitter antenna er, which is exactly the(;)+ th element of H(图]in(1),wth肉因-图 H1因会H13(k△)=∑he -J2TknI/K W(k)=diag{(k),…,(4)}(N)xN, D|∑w(wr where h;三a;(),h;至la;(1),…,a(L)is the L-sized vector containing the time responses of all he non-zero taps; and w,(k)=[e-32Tkni/k In(7),(elk]eLk)is a rank-one matrix, which equals to a zero matrix if the entries of codewords a and a e-22Tkni/k] contains the corresponding DFT coeffi- corresponding to the k-th subcarrier are same. Let D cients denote the number of instances when e[k]e[]+0, vk From( 3), it is seen that due to the close spacing of similarly as in [5], the minimum D over every two OFDM subcarriers, for a specific antenna pair(i,), the possible codeword pair is called the effective length of and hence they are correla t are DFT transformations the code. Denote r= rank(D), it is easily seen that hannel responses H specified by w/(k)] of the same random vector h, rS min(D, NL). Since w,(k)vary with different de lay profiles, the matrix D is also variant with different B. Pe channel environments. However, it is observed that D is a non-negative definite Hermitian matrix, by singular In this section, we analyze the pairwise error prob- value decomposition(SVD), it can be written as ability(PEP) of this system with coded modulation With perfect CSI at the receiver, the maximum likeli D=VAV, hood(ML decision rule of the signal model (1)is given where V is a unitary matrix and 4= diag( A1,..., Ar, al eigenv of D. Moreover, by assuming that all the(nml)el- =agm∑∑-∑13因 (4) ements of (hi, j Ji,j are i i d.(independent and identi I k=0 cally distributed) circularly symmetric complex Gaus- sian with zero-mean and equal-variance, (5) can be re- There the minimization is over all possible STC code- written as =ai[k], k. Assurning equal transmitted bound, the pep of transmitting a and deciding in favor P(a→)s(-8∑(),( of another codeword a at the decoder is upper bounded i=1 where ai()=(vhi), is the j-th element of VHh: P(x→clht)≤exp d'(2, a)),(5)Obviously, (@: ( ), are still.d. circularly symmetric 8N 0-7803-6454600/51000(C)2000

can be modeled as a tapped-delay line. With only the non-zero taps considered, it can be expressed as where 6 (.) is the Kronecker delta function; L denotes the number of non-zero taps; ai,j (/; t) is the complex amplitude of the l-th non-zero tap, whose delay is nl /A j, where nl is an integer and Af is the tone spac￾ing of the OFDM system. In this paper, we restrict our attention to the STC transmitted and received during one OFDM slot; for notational convenience, the time index t is dropped henceforth. For OFDM systems with proper cyclic extension and sample timing, with tolerable leakage, the channel fre￾quency response between the j-th transmitter antenna and the i-th receiver antenna and at the k-th subcarri￾er, which is exactly the (i, j)-th element of ll[k] in (1), can be expressed as 1=1 = h:jwj(k) , (3) where hi,j[l] s ai,j(l), h;,j S [a; ~(1), . . . . ai,j(L)]H is the L-sized vector contammg the \ime responses of all the non-zero taps; and Wf (k) 2 [e–~zm~n’l~, . . . , e-~2”’n’/K] T contains the corresponding DFT coeffi￾cients. From (3), it is seen that due to the close spacing of OFDM sub carriers, for a specific antenna pair (i, j), the channel responses {If;,j [k]}~ are DFT transformations [specified by Wf (k)] of the same random vector hi,j, and hence they are correlated in frequency. B. Performance Analysis In this section, we analyze the pairwise error prob￾ability (PEP) of this system with coded modulation. With perfect CSI at the receiver, the maximum likeli￾hood (ML) decision rule of the signal model (1) is given by M K–1 =arg*~~ Yi[~] - ~Hi,j[~]xj[~] 2, (4) j=l where the minimization is over all possible STC code￾word z = {~j [~]}j~k. Assuming equal transmitted power at all transmitter antennas, using the Chernoff bound, the PEP of transmitting x and deciding in favor of another codeword ii at the decoder is upper bounded by P(z -+ Z1’li) < exp (-d2(::)7) ~ ‘5) where y is the total signal power transmitted from all N transmitted antennas (Recall that the noise at each receiver antenna is assumed to have unit variance); H ~ {h;,j [k]}i,j,~. Using (3), dz(z, Z) is given by MK-l IN ,2 d2(x, ii)= ~ ~ ~ Hi,j [~l~j [1’cI i=l k=O j=l M K–1 =~~[h:, ... h:~] ,x(j..J,,) [Wf(fww ;=1 k=o =5h;Dhi , (6) i=l with ej [k] ~ Zj [k] – ~j [~] , e[k] ~ [e~[k], . . . . e~[k]]~x, , YVj(k) sdiag{wf(k),..., wf(k)}(~~),~ , [ K–1 D ~ ~ Wf(k)e[k]eH[k]’W~ (k) 1 (7) k=o (NL)x (NL) In (7), (e[k]eH [k]) is a rank-one matrix, which equals to a zero matrix if the entries of codewords x and x corresponding to the k-th subcarrier are same. Let D denote the number of instances when e[k]eH[k] # O, Vk; similarly as in [5], the minimum D over every two possible codeword pair is called the effective length of the code. Denote r ~ rank(D), it is easily seen that r < min(D, NL). Since w j (k) vary with different de￾lay profiles, the matrix D is also variant with different channel environments. However, it is observed that D is a non-negative definite Hermitian matrix, by singular value decomposition (SVD), it can be written as D = VAVH , (8) where V is a unitary matrix and A $ diag{~l, . . . . },, o , . ...0 }, where {Aj }~=1 are positive real eigenvalues of D. Moreover, by assuming that all the (NiML) el￾ements of {hi,j }i,j are i.i. d, (independent and identi￾cally distributed) circularly symmetric complex Gaus￾sian with zero-mean and equal-variance, (5) can be re￾written as where &i(j) ~ [VHki] j is the j-th element of VHhi. Obviously, {di(j) }~,j are still i.i.d. circularly symmetric 0-7803-6454-6/00/$10.00 (C) 2000

complex Gaussian with zero-mean and equal-variance, 3. The structure of D is partly decided by the chan and their magnitudes ai( nel delay profile [specified by (hi,j )i,j and w, (k)] tributed. By averaging the conditional PEP in(9 )over which causes that the eigenvalues are part- the Rayleigh pdf (probability density function), the ly decided by the channel delay profile; and the PEP of an STC-OFDM system over frequency-selective decoding performance may unfavorably vary with fading channels is finally written a can actually be alleviated by using an interleaver ble the stc symbols at the output of the src encoder, From the information theoreti P(x→x)≤ viewpoint, such an interleaver"whitens"the trans- ed STC symbols [ 7] ∏(1 In summary, in the system considered here, because of the diverse fading profiles of the wireless channels and the assumption that the Csi is (e.g, by ≤I() y-M.(10) search)is less helpful; instead, two general principles should be met in choosing STC codes in order to ro- bustly exploit the rich diversity resources in this system [ When deriving( 9), we assumed that the elements of namely, large efective length and ideal interleaving thijJi, have equal variances, or equivalently equal power. However, in practice, the multipath taps usu- D. Code Construction ally have different power, which causes the elements of In the previous section, based on the PEP analysis of lai ))i j are not mutually independent any more and an STC-OFDM system, we show that an STC should to some extent makes the upper bound in(10)opti- have the highest possible effective length and the ideal diversity order the STC-OFDM system considered here the spatial diversity and the frequency-selectivity di can provide is(NMD), i.e., the product of the number versity. In another aspect, the recent increasing inter- of transmitter antennas, the number of receiver anten- est in providing high data-rate services, such as video- nas and the number of selective-fading diversity order. conferencing, multimedia Internet access and wide area In other words, the attractiveness of the STC-OFDM network over wideband wireless channels, calls for the system lies in its ability to exploit all the available di- higher bandwidth efficiency. In the next, we propose a versity resources lass of bandwidth efficient STCs C. Designing Principles In addition to giving a very promising perspective of STC-OFDM systems,(10)also provides some implica- related to the structure of the code is r, the rank of 是名8世 the matrix D Rccall that r< min(D, NL), in or- 9 *q H+q iQ der to achieve the maximum diversity, clearly, the fective length of the code(which is the minimum Fig. 2. Encoder structure of the proposed STC D over every two possible codeword pair) must be larger than NL, the dimension of matrix D in(7). Before the invention of STC. there had been a lot Since L is associated with the channel characteris- of work done for analyzing and designing trellis-coded modulation(TCM) codes in fat-fading channels, such STC encoder)in advance. It is preferable to have as [5], [8]. For flat fading channels, the design criteria STC 2. Another factor in the PEP is Ilj=1 Ag, the prod- tance. Based on these criteria, in [51, a class of rate 2/3 Ict of eigenvalues of matrix D. Since D changes 8-PSK Ungerboeck codes was optimized for filat-fading IT=1 A; is not feasible. However conceptually close to our proposed design principles for in [1], the space-time trellis codes(STTC's)with STCs, we then extend the TCM codes designed there higher state numbers(and essentially larger effec- into the STC's (with two transmitter antennas)by split. tive length) have better performance, which sug- ting the original 8-PSK mapper into two QPsK map- gests that increasing the effective length of the STC pers, as depicted in Figure 2. Obviously the efective beyond the minimum requirerment(e. g, NL, in our length of the resulting STC code is the same as the o system)may help to improve the factor I;=1 A;. riginal TCM code, therefore a new class of STC's with 0-7803-6454600/10.00(C)2000

complex Gaussian with zero-mean and equal-variance, and their magnitudes { Ifii(j) I}Z,j are i.i. d. Rayleigh dis￾tributed. By averaging the conditional PEP in (9) over the Rayleigh pdf (probability density function), the PEP of an ST’c,-OFDM system over frequency-selective fading channels is finally-written as “ M /. , –M the recent increasing inter￾est in providing high data-rate services, such as video￾conferencing, multimedia Internet access and wide area network over wideband wireless channels, calls for the higher bandwidth efficiency. In the next, we propose a class of bandwidth efficient STC’S. D2 v~ $W; ‘: 4-?s. cl H: H :.* H:. H? H: Maw.. ?aI *, x; H ;.l H 4.2 Hi xi~ ~-p,x ., + + + + + ~o ~wp.= . . . ~: H ;-l ~ :_* ~~ Fig. 2, Encoder structure of the proposed STC. Before the invention of STC, there had been a lot of work done for analyzing and designing trellis-coded modulation (TCM) codes in flat-fading channels, such as [5], [8]. For flat fading channels, the design criteria of the TCM are large eflectiue length and product dis￾tance. Based on these criteria, in [5], a class of rate 2/3, 8-PSK Ungerboeck codes was optimized for flat-fading channels. Since the design criteria for TCM codes are conceptually close to our proposed design principles for STC’S, we then extend the TCM codes designed there into the STC’S (with two transmitter antennas) by split￾ting the original 8-PSK mapper into two QPSK map￾pers, as depicted in Figure 2. Obviously the eflecti~e length of the resulting STC code is the same as the o￾riginal TCM code, therefore a new class of STC’s with 0-7803-6454-6/00/$10.00 (C) 2000

effective length ranging from 2 to 6 [5] is immediately Performance in a single-tap fading channel constructed In Figure 3, we provide the performance of the Stc Following the ideal interleaving principle, two random for OFDM systems in a single-tap(or flat-)fading chan- interleavers are applied to the STC encoded symbols, as nel, which is conceptually equivalent to the qua the effective length of the STC, these two interleavers mum achievable diversity order is NMI66-sj-static illustrated in Figure 1. Note that, in order to preserve flat-fading channels in In th 2,(where are identical. Since OFDM signals are transmitted and N=2, M=1, L=1), which is exactly achieved by all three received in the block manner, therefore, using an in- STCs. Note that the interleaver, which operates in the terleaver does not introduce any additional processing frequency domain, has no impact in this particular flat- fading case. For clarity, here we omit the performance of three STC's without interleavers, which is the same III. SIMULATION RESULTS as what is shown in Figure 3 In this section, we provide computer simulation re- Performance in two-tap fading channel sults to illustrate the performance of our proposed new In Figure 4, we provide the performance of the STC class of STCs for multiple-antenna OFDM systems. in a two-tap equal-power fading channel, where the de The channel model is the same as described in Sec- lay spread between the two paths is 5us; while in Fig tion II-A.( Specifically, the fading processes comply ure 5, we show the performance in a two-path equal with the COST 207 model. In simulations the avail- power fading channel, where the delay spread between able bandwidth is 1 MHz and 256 sub-carrier tones are the two taps is 40us, From the figures, several conclu used for OFDM modulation. These correspond to a sions can be drawn. First, the use of the random in sub-channel separation of 3. 9 KHz and OFDM frame terleaver does bring obvious performance improvement duration of 256us. To each frame, a guard interval of moreover, it makes the performance robust(or consis- 40us is added to combat the effect of inter-symbol inter- tent) against different channel delay profiles. Secondly ference. N=2 transmitter antennas and M=1 receiver with the larger effective length, the 256-state STC-I antenna are used in the OFDM system viterbi decoding SNrs it can achieve the m灯四1) gorithm is used to decode the STC. For the purpose order NMl=4, (where N=2, M=1, L=2). Thirdl of comparison, we also include the performance of the the 16-state STC-I performs close to the 16-state STo STC-OFDM system proposed in [4, which is denoted which implies that the effective length can also be by STC-l; while the STC proposed in this paper is de- used to roughly evaluate the performance of the StC noted by STC-Il. It is clear that both two types of STC are based on the trellis-tree structure, where at the be- Performance in a six-tap fading channel to zero-state by properly padding the last several tailing power fading channel, where the six paths are equally coded by the STC encoder; and the two streams of 256 able diversity order increases to N ML= 12,(where STC coded QPSK symbols are interleaved and trans- N=2, M=1, L=6), it is seen that all three STC's im mitted from 2 transmitter antennas. Then the spectral efficiency of the STC-OFDM system considered here is prove their performance compared with that in two-tap w2X296x 508=1.72 bits/sec/Hz.(The approxima- the diversity resources in the system. It is also ob tion of the spectral efficiency is due to the fact that the served that due to the relatively small effective length number of tailing bits is related with the state number the performance improvement of two 16-state STC's is of the STC. less than the erformance improvement of the 256-state STC-l1. It is expected that the STC with larger effec- A. Performance of STC-OFDM in channels with dif- tive length can achieve even better performance in this ferent delay profiles system, although the increase of effective length gen erally leads to the corresponding increase of the STC Figures 3-6 show the performance of three STC's, complexity i. e, 16-state STC-l ( with effective length 3), 16-state STC-ll (with effective length 3) and 256-state STC-ll IV. CONCLUSIONS (with effective length 5), in channels with different de- In this paper, we have studied the stc design in lay profiles, where"w/o intl"denotes the performance OFDM systems. By analyzing the PEP, we have shown without interleaving and"w intl"denotes that with in- that in frequency-selective fading channels, the STC- terleaving. The performance is shown in terms of the OFDM systern can potentially provide a diversity order OFDM word error rate(WER)versus the signal-to- as the(NML), where N is the number of transmitter noise ratio(Snr)y antennas, M is the number of receiver antenn 0-7803-6454600/510.00(C)2000

effective length ranging from 2 to 6 [5] is immediately constructed. Fc]llowing the ideal interleaving principle, two random interleaves are applied to the STC encoded symbols, as illustrated in Figure 1. Note that, in order to preserve the effective length of the STC ~these two interleaves are identical. Since OFDM signals are transmitted and received in the block manner, therefore, using an in￾terleave does not introduce any additional processing delay. III. SIMULATION RESULTS In this section, we provide computer simulation re￾sults to illustrate the performance of our proposed new class of STC’S for multiple-antenna OFDM systems. The channel model is the same as described in Sec￾tion II-A. (Specifically, the fading processes comply with the COST 207 model. ) In simulations the avail￾able bandwidth is 1 MHz and 256 sub-carrier tones are used for OFDM modulation. These correspond to a sub-channel separation of 3.9 KHz and OFDM frame duration of 256ps, To each frame, a guard interval of 40ps is added to combat the effect of inter-symbol inter￾ference. N=2 transmitter antennas and Lf=l receiver antenna are used in the OFDM systems. With the per￾fect CSI at the receiver, the optimal Viterbi decoding algorithm is used to decode the STC. For the purpose of comparison, we also include the performance of the STC-OFDM system proposed in [4], which is denoted by STC-I; while the STC proposed in this paper is de￾noted by ST C- II. It is clear that both two types of STC are based on the trellis-tree structure, where at the be￾ginning and the end of encoding, the encoders are forced to zero-state by properly padding the last several tailing bits. During each OFDM slot, 512 binary bits are en￾coded by the STC encoder; and the two streams of 256 STC coded QPSK symbols are interleaved and trans￾mitted from 2 transmitter antennas. Then the spectral efficiency of the STC-OFDM system considered here is x 2 x ~ x ~ = 1.72 bits/see/Hz. (The approxima- ~ 512 tion oft e spectral efficiency is due to the fact that the number of tailing bits is related with the state number of the STC. ) A. Performance of STC- OFDM an channels with dif￾ferent delay projiles Figures 3–6 show the performance of three STC’S, i.e., 16-state STC-I (with effective length 3), 16-state STC-1 I (with effective length 3) and 256-state STC-I I (with effective length 5), in channels with different de￾lay profiles, where “w/o intlv” denotes the performance without interleaving and “w intlv” denotes that with in￾terleaving. The performance is shown in terms of the OFDM word error rate (WER) versus the signal-to￾noise ratio (SNR) y. Performance in a single-tap fading channel In Figure 3, we provide the performance of the STC for OFDM systems in a single-tap (or flat-) fading chan￾nel! which is conceptually equivalent to the quasi-static flat-fading channels in [1]. In this case, the maxi￾mum achievable diversity order is iVikIL = 2, (where N=2,Jf=l ,L=l), which is exactly achieved by all three STC’S. Note that the interleaver, which operates in the frequency domain, has no impact in this particular flat￾fading case. For clarity, here we omit the performance of three STC’S without interleavers, which is the same as what is shown in Figure 3. Performance in two-tap fading channels In Figure 4, we provide the performance of the STC in a two-tap equal-power fading channel, where the de￾lay spread between the two paths is 5ps; while in Fig￾ure 5, we show the performance in a two-path equal￾power fading channel, where the delay spread between the two taps is 40ps, From the figures, several conclu￾sions can be drawn. First, the use of the random in￾terleave does bring obvious performance improvement; moreover, it makes the performance robust (or consis￾tent) against different channel delay profiles. Secondly, with the larger effective length, the 256-state STC- II performs the best out of all three STC’S, and at high SNR’S it can achieve the maximum available diversity order IVJ4L = 4, (where N=2,iM=l, L=2). Thirdly, the 16-state STC- I performs close to the 16-state STC- 1I, which implies that the effective length can also be used to roughly evaluate the performance of the STC. Performance in a six-tap fading channel Figure 6 shows the performance in a six-path equal￾power fading channel, where the six paths are equally spread at the dist ante of 6 .5~s, As the total avail￾able diversity order increases to IVLL5 = 12, (where N=2,A4=1 ,L=6), it is seen that all three STC’S im￾prove their performance compared with that in two-tap fading channels [cf. Fig. 4–5] by efficiently exploiting the diversity resources in the system. It is also ob￾served that due to the relatively small effective length, the performance improvement of two 16-state STC’S is less than the performance improvement of the 256-state STC-11. It is expected that the STC with larger effec￾tive length can achieve even better performance in this system, although the increase of effective length gen￾erally leads to the corresponding increase of the STC comple.xit y, IV. CONCLUSIONS In this paper, we have studied the STC design in OFDM systems. By analyzing the PEP, we have shown that in frequency-selective fading channels, the STC￾OFDM system can potentially provide a diversity order as the (N LfL), where N is the number of transmitter antennasj A4 is the number of receiver antennas and 0-7803-6454-6/00/$10.00 (C) 2000

Fig. 3. WER in a single-tap(or flat.)fading channel in a two-tap equal-power fading channel, whe the delay spread between two paths is 5, L is the frequency-selectivity order (or the number of non-zero resolvable taps). We have also proposed that the large effective length and the ideal built-in inter- leaver are two most important coding design principles for the STC in OFDM systems. By following these two principles, a new class of trellis-structured STC's is de- signed. Computer simulations have demonstrated the significant performance improvement of our proposed STCs over the conventional space-time trellis codes Further research on the Stc design for OFDM system s in frequency-selective and time-selective fading chan nels, the STC design based on the bit-interleaved coded modulation is now pursued o. 16-state STC- wo inMy ACKNOWLEDGMENT The authors would like to thank K.r. narayanan tate STC-l w ina his helpful comment REFERENCES g. 5. WER in a two-tap equal-power fading channel, where [1] V. Tarokh, N. Seshadri, and A. R. Calderbank, "Space-time the delay spread between two paths is 40us Ⅰ EEE Tran 2V. Tarokh, H. Jafarkhan A. R. Calderbank, "Spac IEEE J. Select. Areas Commun., vol 17, pp451 [3]V. Tarokh, A Naguib, N. Seshadri, and A. R. Calderb 4 ol.45,pp.1121-1128,May al. v, Tarokh,A N time coded OFDM for in ieee vehicular tech otello, Bandwidth cfficient coding ieeE J. Select. are [6]John G. Proakis, Digital Communications, McGraw-Hill channels: set partitioning for optimum Fig. 6. WER in a six-tap equal-power fading channel code design, IEEE Trans. Co77u7, yol. 36, p paths are equally spread at the distance of 6.5us Sept. 1988 0-780364546/00/$10.00(C)2000

L is the frequency-selectivit y order (or the number of non-zero resolvable taps). We have also proposed that the large ejfective length and the ideal built-in inter￾leave are two most important coding design principles for the STC in OFDM systems. By following these two principles, a new class of trellis-structured STC’S is de￾signed. Computer simulations have demonstrated the significant performance improvement of our proposed STC’s over the conventional space-time trellis codes. Further research on the STC design for OFDM system￾s in frequency-selective and time-selective fading chan￾nels, the STC design based on the bit-interleaved coded modulation is now pursued. ACKNOWLEDGMENT The authors would like to thank K. R. Narayanan for his helpful comments. [1] [2] [3] [4] [5] [6] [7] [8] REFERENCES V. Tarokhl N. Seshadri, and A. R. Calderbank, “Space-time codes for hmh data rate wireless communication: Performance criterion and code construction,>! IEEE Trans. Infomn. The￾ory, vol. 44, pp. 744–765, Mar. 1998. V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space￾time block coding for wireless communications: performance results,” IEEE J. Select. A7eas Commun., vol. 17, pp. 451– 4150, Mar. 1999. V. Tarokh, A. Naguib, N. Seshadri, and A. R. Calderbank, “Combined array processing and space-time coding,” IEEE Trans. Injorm. Theo~y, vol. 45, pp. 1121–1128, May 1999. D. Amawal. V. Tarokh. A. Namib. and N. Seshadri. l’S~ace￾time ‘coded’ OFDM for high ~ata~rate wireless cornm&ica￾tion over wideband channels ,“ in IEEE VehiculaT Techn ology Con.fe~ence, 1998. VTC’98., May 1998. C. Schlegel and D. J. Costello, “Bandwidth efficient coding for fading channels: Code construction and performance anal- ~is,” IEEE J. Select. Areas Com?nun., vol. 7, pp. 1356–1368, Dec. 1989. John G. Proakis, Digital Communications, McGraw-Hill, Inc, 3rd edition, 1995. E. Biglieri, J. Proakk, and S. Shamaij “Fading channels: Information-theoretic and communication aspects,” IEEE TTans. InfoTm. Theo~y, vol. 44, pp. 2619–2692, Oct. 1998. EI. Divsalar and M. K. Simon, ‘(The design of trellis cod￾ed MPSK for fading channels: set partitioning for optimm code design,” IEEE Trans. Gommun., vol. 36, pp. 1013–1021, Sept. 1988. 5!!.s ,ePe’ated tw-,q 5TC-OFDM t... . .:. !y><%$;:. ,,,.,,:,..,...,, .. ....{ I I 10~5 \ Y 10 20 S(Q”.1-tO-N& Raflo (@B) 25 Fig. 4. WER in a two-tap equal-power fading channel, where the delay spread between two paths is 51M. 4WS mmr.ted IWO-W STC-OFDM ,- C-- 256-state STC-11w/o intlv ~~+ 25S-state STC-11 w intlv i,-45 TO 20 SIOnal-tO-N& Rallo (dB] 25 Fig. 5. WER in a two-tap equal-power fading channel, where the delay spread between two paths is 40ps. IL lo~.,“-+”” .— .- u-.. ::+ .-o-- “ -4- ,,.4 G 16-date STC–I VA indv i 6–state STC–I w id. i 6-state STC-11wlo intlv 16-state STC-11 w inflv 256-state STC-11WIOintlv 256-state STC-11 w mtlv . slgnel-lo-N& mu. (dB) ‘“ . . Fig. 6. WER in a six-tap equal-power fading channel, where paths are equally spread at the distance of 6.5LLs. six 0-7803-6454-6/00/$10.00 (C) 2000