for STBC-OFDM with outer channel coding is briefly introduced Using(2), the signal model in (1) can be expressed Section V contains the computer simulation results Xle While+ ailp IL. STBC-OFDM SYSTEM IN DISPERSIVE FADING CHANNELS with=diagwj We consider an STBC-OFDM system with K subcarriers, N trans mitter antennas and m receiver antennas, signaling through a (1),(kx-1),全hm)…,h(p frequency- and time-selective fading channel. As illustrated NLX In an STBC-OFDM system, the STBC proposed in [1, 13 lator; then the modulated MPSK symbols are encoded by an STBC is applied to data symbols transmitted at dfterent subcarriers in- coder. Each STBC code word consists of (PN)STBC symbols, which are transmitted from n transmitter antennas and across P tem involves the received signals over P consecutive OFDM slots consecutive OFDM slots at a particular OFDM subcarrier. The To simplify the problem, we assume that channel time respons STBC code words at different OFDM subcarriers are indeper hill, p P remain constant during one stbc code word (i.e, P consecutive OFDM slots). As will be seen, such an as- STBC code words [or(K PN) STBC code symbols] are transmit. sumption significantly simplifies the receiver design. And the sys tem model in(3)is further written as It is assumed that the fading process remains static during each OFDM word (one time slot) but it varies from one OFDM XWh;+三 (4) word to another, and the fading processes associated with differ ent transmitter-receiver antenna pairs are uncorrelated with y-y[1,.y:[PI 1 The signal model can be written as v=xH回+2 h;[1=h[ Moreover, by using the constant modulus property of the sym- PHIlp+aipl ols [a,bp, klli, p, k, and the orthogonality property ofSTBC codes ,p=1,,P (D [1], we get with XIpI=XI xi lpl (5)is the key equation in designing the low-complexity iterative IiLp, C H,会[m,q receivers for STBC-OFDM systems H,i[pl=Hi,iIp, 01, IIL. ML RECEIVER BASED ON THE EM ALGORITHM where Hi lal is the(NK)vector containing III-A. STBC Decoder based on the EM Algorithm the complex channel frequency responses between the i-th receiver Without channel state information(CSI), the maximum likelihood antenna and all N transmitter antennas at the p-th OFDM slot, (ML)detection problem is written as, x- arg maxx.> ted from the j-th transmitter antenna at the k-th subcarrier and at hibitive complexity, we propose to use the expectation-maximization the p-th OFDM slot, yi Ip] is the K-vector of received signals from (EM) algorithm [8] to solve this problem the i-th receiver antenna and at the p-th time slot; ailp is the am- In the e-step of the EM algorithm, the expectation is taken bient noise, which is circularly symmetric complex Gaussian with with respect to the"hidden"channel response hi conditioned on covariance matrix o=I. In this paper, we restrict our attention to MPSK signal constellation,i.e,jnA∈全{e,cT and x( ) It is easily seen that, conditioned on y, andX( hi is complex Gaussian distributed [9, 14]. Using(4)and(5),its distribution is expressed as The channel frequency response between the j-th transmitter antenna and the i-th receiver ar the p-th time slot and at h:{,x)~Nh,∑h1),i=1,…,M,(6) the k-th subcarrier can be with h:=[(PK).+02Eh1-wx )y H,Lp,k)=>hi,s[4; ple-22xRl/k=w (k)hi, s(p),(2) ∑h4-(PK)(PK)·I+a2∑n]-∑h where Nc(p, 2) denotes the complex Gaussian distributed ran- i[;pl =ai,g(l;pr), T is the duration of one OFDM dom vector with mean A and variance 2, Eh, denotes the covari- slot, hi,i()=[aij(O; pT i,;(L-1: pr)I is the L- ance matrix of channel responses hi, and Eh, denotes the pseudo vector containing the time responses of all the taps and w(k)= inverse of Eh. According to the signal model in(2), Eh hih) ing dft coefficients where B, is the average power of the l-th tap associated withfor STBC-OFDM with outer channel coding is briefly introduced. Section V contains the computer simulation results. II. STBC-OFDM SYSTEM IN DISPERSIVE FADING CHANNELS We consider an STBC-OFDM system with✂ subcarriers, ✄ trans￾mitter antennas and ☎ receiver antennas, signaling through a frequency- and time-selective fading channel. As illustrated in Fig. 1, the information bits are first modulated by an MPSK modu￾lator; then the modulated MPSK symbols are encoded by an STBC encoder. Each STBC code word consists of ✆✞✝✟✄✡✠ STBC symbols, which are transmitted from ✄ transmitter antennas and across ✝ consecutive OFDM slots at a particular OFDM subcarrier. The STBC code words at different OFDM subcarriers are indepen￾dently encoded, therefore, during ✝ OFDM slots, altogether ✂ STBC code words [or ✆✞✂☛✝✟✄✡✠ STBC code symbols] are transmit￾ted. It is assumed that the fading process remains static during each OFDM word (one time slot) but it varies from one OFDM word to another; and the fading processes associated with differ￾ent transmitter-receiver antenna pairs are uncorrelated. The signal model can be written as ☞✍✌✏✎✑✓✒✕✔ ✖ ✗✙✘✛✚✢✜✗ ✎✑✓✒✤✣✌✦✥ ✗ ✎✑✓✒★✧✪✩ ✌ ✎✑✫✒ ✔ ✜ ✎✑✓✒✤✣✌ ✎✑✫✒✬✧✭✩ ✌ ✎✑✓✒✯✮ ✰ ✔✲✱✓✮✴✳✵✳✴✳✴✮ ☎✮☛✑✶✔✲✱✓✮✴✳✴✳✵✳✴✮ ✝ ✮ (1) with ✜ ✎✑✓✒✸✷✔ ✜ ✚ ✎✑✓✒✹✮✴✳✴✳✴✳✴✮ ✜ ✖ ✎✑✓✒ ✺✍✻✽✼✿✾✯✺✢❀❁✮ ✜ ✗ ✎✑✓✒✸✷✔ ❂✽❃✏❄❆❅ ❇ ✗ ✎✑★✮✦❈✓✒✹✮✵✳✴✳✴✳✵✮ ❇ ✗ ✎✑★✮ ✂❊❉ ✱✙✒ ✺✍✻❋✺●✮✏✣✌ ✎✑✓✒ ✷✔ ✣■❍✌❏✥ ✚ ✎✑★✮❑❈✫✒✹✮ ✳✴✳✴✳▲✮▼✣■❍✌❏✥ ✖ ✎✑★✮ ✂◆❉ ✱✙✒ ❍ ✼✿✾✯✺✢❀❆✻✽❖P✮✹✣✌✦✥ ✗ ✎✑✓✒ ✷✔ ◗✌✦✥ ✗ ✎✑★✮✏❈✓✒✹✮✴✳▲✳✵✳✴✮ ◗✌❏✥ ✗ ✎✑✬✮ ✂❘❉ ✱❆✒ ❙ ✺✍✻✽❖✍✮ where ✣✌ ✎✑✓✒ is the (✄❚✂)-vector containing the complex channel frequency responses between the ✰ -th receiver antenna and all ✄ transmitter antennas at the ✑ -th OFDM slot, which is explained below; ❇ ✗ ✎✑★✮▲❯✵✒ is the STBC symbol transmit￾ted from the ❱-th transmitter antenna at the ❯ -th subcarrier and at the ✑ -th OFDM slot; ☞✍✌❑✎✑✓✒ is the ✂-vector of received signals from the ✰ -th receiver antenna and at the ✑ -th time slot; ✩ ✌ ✎✑✓✒ is the am￾bient noise, which is circularly symmetric complex Gaussian with covariance matrix ❲❨❳❩❭❬. In this paper, we restrict our attention to MPSK signal constellation, i.e., ❇ ✗ ✎✑★✮▲❯✵✒✯❪❴❫ ✷✔✲❵✙❛❝❜✦❞❡✮❑❛❜❣❢❭❤ ✐ ❥❦✐ ✮✴✳✵✳▲✳✴✮ ❛❜❧❢✙❤✐ ❥❦✐✹♠♦♥ ♣q♥ r ✚✹s✏t ✳ The channel frequency response between the ❱-th transmitter antenna and the ✰ -th receiver antenna at the ✑ -th time slot and at the ❯ -th subcarrier can be expressed as ◗✌❏✥ ✗ ✎✑★✮▲❯✵✒✉✔ ✈ r ✚ ✇ ✘ ❞ ①✌❏✥ ✗ ✎②▼③▼✑✓✒④❛ r ❜ ❳✹⑤⑦⑥ ✇④⑧✙⑨ ✔❶⑩❍❷ ✆ ❯ ✠❏❸✌❏✥ ✗ ✆✑ ✠ ✮ (2) where ①✌❏✥ ✗ ✎②▼③▼✑✓✒ ✷✔❺❹✌❏✥ ✗ ✆ ②❻③✹✑❝❼✠ , ❼ is the duration of one OFDM slot; ❸ ✌❏✥ ✗ ✆✑ ✠ ✷✔❺✎❹✌✦✥ ✗ ✆❈✬③✹✑✓❼✠ ✮❶✳✴✳✴✳▲✮✪❹✌❏✥ ✗ ✆✞❽●❉ ✱✫③▼✑❝❼✠ ✒ ❙ is the ❽- vector containing the time responses of all the taps; and ⑩❷ ✆ ❯ ✠ ✷✔ ❛ r ❜✦❞✵✮✏❛ r ❜ ❳✹⑤❝⑥ ⑧✙⑨ ✮✴✳✵✳✴✳✴✮▼❛ r ❜ ❳✹⑤⑦⑥ ♠ ✈ r ✚✹s ⑧✙⑨ ❍ contains the correspond￾ing DFT coefficients. Using (2), the signal model in (1) can be expressed as ☞✍✌❑✎✑✓✒❾✔ ✜ ✎✑✓✒▼❿❸ ✌ ✎✑✓✒✬✧✭✩ ✌ ✎✑✫✒❧✮ ✰ ✔➀✱✓✮✵✳✴✳✴✳✴✮ ☎✮✪✑✶✔➀✱✓✮✵✳✴✳✴✳✴✮ ✝ ✮ (3) with ❿➁✷✔ ❂✽❃✏❄✙❅ ❿❷ ✮✴✳✴✳▲✳✴✮✵❿❷ ✼④✾✯✺✢❀❻✻✽✼④✾✯➂✴❀P✮✵❿❷ ✷✔ ⑩ ❷ ✆❈ ✠ ✮ ⑩❷ ✆ ✱ ✠ ✮✵✳✴✳✴✳▲✮✹⑩❷ ✆✞✂❶❉ ✱ ✠ ❍✺✍✻❋➂✽✮ ❸ ✌ ✎✑✓✒ ✷✔ ❸❍✌❏✥ ✚ ✆✑ ✠ ✮✴✳▲✳✴✳✴✮ ❸❍✌✦✥ ✖ ✆✑ ✠ ❍✼④✾✯➂✴❀❻✻✽❖ ✳ In an STBC-OFDM system, the STBC proposed in [1, 13] is applied to data symbols transmitted at different subcarriers in￾dependently. It is clear that decoding in an STBC-OFDM sys￾tem involves the received signals over ✝ consecutive OFDM slots. To simplify the problem, we assume that channel time responses ❸ ✌ ✎✑✓✒✹✮✏✑✶✔➀✱❝✮✵✳▲✳✴✳✴✮ ✝✮ remain constant during one STBC code word (i.e., ✝ consecutive OFDM slots). As will be seen, such an as￾sumption significantly simplifies the receiver design. And the sys￾tem model in (3) is further written as ☞✌ ✔ ✜ ❿❸ ✌ ✧✭✩ ✌ ✮ ✰ ✔➀✱❝✮✴✳✵✳▲✳✴✮ ☎✮ (4) with ☞✌ ✔ ☞❍✌ ✎④✱❆✒✹✮✵✳✵✳✴✳✴✮✹☞❍✌ ✎✝✒ ❍ ✼✿➃✢✺✢❀❻✻✽❖✍✮ ✜ ✷✔ ✜ ❍ ✎✿✱❆✒▼✮✵✳▲✳✵✳✴✮ ✜ ❍q✎✝✒ ❍ ✼④➃✽✺✢❀❆✻✽✼✿✾✯✺✢❀➄✮✹✩ ✌➅✷✔ ✩❦❍✌ ✎✿✱✙✒✹✮✵✳✴✳▲✳✵✮✹✩❦❍✌ ✎✝✒ ❍ ✼✿➃✢✺✢❀❻✻✽❖❴✮ ❸ ✌ ✷✔ ❸ ✌ ✎✿✱✙✒✯✔ ❸ ✌ ✎✿➆✙✒➇✔✲➈✴➈▲➈❋✔ ❸ ✌ ✎✝✒✯✳ Moreover, by using the constant modulus property of the sym￾bols ❵❇ ✗ ✎✑★✮✴❯✴✒ t ✗ ✥ ➉❡✥ ⑥ , and the orthogonality property of STBC codes [1], we get ❿❍✜ ❍✜ ❿ ✔ ✆✞✝✟✂●✠ ➈ ❬ ✳ (5) (5) is the key equation in designing the low-complexity iterative receivers for STBC-OFDM systems. III. ML RECEIVER BASED ON THE EM ALGORITHM III-A. STBC Decoder based on the EM Algorithm Without channel state information (CSI), the maximum likelihood (ML) detection problem is written as, ➊✜ ✔ ❄✙➋❻❅✍➌✪❄✙➍❦➎ ✌➏ ✘q✚ ➐✿➑✓❅ ✑ ✆☞✌✏➒✜ ✠ ✳ Since the optimal solution of this problem is of pro￾hibitive complexity, we propose to use the expectation-maximization (EM) algorithm [8] to solve this problem. In the E-step of the EM algorithm, the expectation is taken with respect to the “hidden” channel response ❸ ✌ conditioned on ☞ ✌ and ✜ ♠④➓ s . It is easily seen that, conditioned on ☞✌ and ✜ ♠④➓ s , ❸ ✌ is complex Gaussian distributed [9, 14]. Using (4) and (5), its distribution is expressed as ❸ ✌ ➒ ✆☞ ✌ ✮ ✜ ♠④➓ s ✠➁➔ →↔➣❆✆❸➊ ✌ ✮ ➊↕✟➙✵➛ ✠ ✮ ✰ ✔➀✱✓✮✵✳✴✳✴✳✴✮ ☎✮ (6) with ❸➊ ✌ ✔ ✎ ✆✞✝✟✂●✠ ➈ ❬ ✧ ❲❩❳ ↕➝➜➙➞➛ ✒ r ✚ ❿❍✜ ♠④➓ s❍☞ ✌ ✮ ➊↕✟➙✵➛ ✔ ↕✟➙✵➛ ❉●✆✞✝✟✂●✠ ✎ ✆✞✝✟✂●✠ ➈ ❬ ✧ ❲❩❳ ↕➝➜➙➞➛ ✒ r ✚ ↕✟➙➞➛ ✮ where →➣ ✆❏➟✮ ↕ ✠ denotes the complex Gaussian distributed ran￾dom vector with mean ➟ and variance ↕ ; ↕✟➙✵➛ denotes the covari￾ance matrix of channel responses ❸ ✌ , and ↕➠➜➙✵➛ denotes the pseudo inverse of ↕✟➙✵➛ . According to the signal model in (2), ↕✟➙✵➛ ✷✔ ➡ ✆❏❸✌❸❍✌ ✠ ✔ ❂✽❃✏❄❆❅✎➢ ❳ ✚ ✥ ❞ ✮✴✳✵✳✴✳▲✮✏➢❳ ✚ ✥ ✈ r ✚ ✮▲✳✵✳✴✳✴✳✴✳✴✳▲✮❣➢❳✖ ✥ ❞ ✮✴✳✴✳✴✳▲✮✹➢❳✖ ✥ ✈ r ✚ ✒ , where ➢ ❳ ✗ ✥ ✇ is the average power of the ② -th tap associated with