Iterative Receivers for Space-time Block Coded OFDM Systems in Dispersive Fading channels Ben Lu, Xiaodong Wan Ye( Geoffrey)Li Department of Electrical Engineering School of Electrical and Computer Engineering Texas a&M Universit Georgia Institute of Technology College Station, TX 77843 Atlanta GA 30332 i benlu, wang @ee. tamu. edu liye(@ece gatech.edu Abstract- We consider the design of iterative receivers for drawback to any decision feedback system, when the decisions are space-time block coded orthogonal frequency-division multiplex ad. not accurate(e.g, in very fast fading channels ), the receivers in ing(STBC-OFDM) systems in unknown wireless dispersive fad- [5, 6] show error floors. In this paper, we approach the problem ing channels, with or without outer channel coding. First, we of receiver design without CSI by using iterative techniques, in- propose a maximum-likelihood(ML) receiver for STBC-OFDM cluding the expectation-maximization(E)algorithm [7 and the By assuming that the fading processes remain constant over the for sequence estimation in uncoded and coded systems. More re- duration of one STBC code word, and by exploiting the orthog- cently, in [9, 10], a receiver employing the Em algorithm, which onality property of the STbC as well as the OFDM modulation, exhibits a good performance but on the other hand a relatively high we show that the eM-based receiver has a very low computation complexity, is proposed for STC systems complexity, and that the initialization of the EM receiver is based In this paper, we focus on the design of iterative receivers on the linear minimum mean-square-error(MMSE) channel esti- for STBC-OFDM systems in unknown wireless dispersive fading mate for both the pilot and the data transmission. Since the ac- channels. We first derive the maximum likelihood(ML)receiver tual fading processes may vary within one STBC code word, we based on the EM algorithm for STBC-OFDM systems, under the also analyze the effect of a modelling mismatch on the receiver assumption that the fading processes remain constant over the du- performance, and show both analytically and through simulations ration of one STBC code word (or equivalently across several that the performance degradation due to such a mismatch is neg- jacent OFDM words contained in one STBC code word).Based ligible for practical Doppler frequencies We further propose a on such an assumption and the orthogonality property of the StB Turbo receiver based on the maximum a posteriori(MAP)EM as well as the OFDM modulation, we show that no matrix inver- algorithm for STBC-OFDM systems with outer channel coding. sion is needed in the EM algorithm. Therefore, the computational Compared with the previous non-iterative receiver employing a cost for implementing the EM-based ML receiver is low and the decision-directed linear channel estimator. the iterative receivers proposed here significantly improve the receiver performance and computation is numerically stable. Moreover, we show that the an approach the ML performance in typical wireless channels can achieve its minimum for both the pilot transmission mode and the data transmission mode(assuming correct decision feedback) well suited for real-time implementations Since the actual fading processes may vary over the duration of ne sTBC code word, we also analyze the effect of a modelling L INTRODUCTION imse as well as an upper bound on the instantaneous MSE Recently several studies addressing the design and applications of of the channel estimate in the initialization of the em algorithm ace-time coding(STC)have been conducted, e.g., [ 1, 2].Mean- ye show that the average mse due to a modelling mismatch is while, from the signal processing perspective, research on receiver ne negligible for practical Doppler frequencies. To further impre design for STC systems is also active. With ideal channel state the quality of the initialization step of the EM algorithm(andor to information(CSI), the iterative receivers based on the Turbo prin- reduce the computational complexity), following [5, 11, 12], the are al n concatenated STC systems [4]. When the CSI is not available poral filtering are adopted in the proposed EM-based mLreceiver [5, 6], the design of channel estimators and training sequences Finally, for STBC-OFDM systems employing outer channel cod vere studied and receiver structures based on the decision-directed g, we propose a Turbo receiver, which iterates between the max ast-square channel estimator or its simplified variant were pro- imum a posteriori(MAP)-EM STBC decoding algorithm and the sed for space-time trellis coded orthogonal frequency-division MAP channel decoding algorithm to successively improve the re- multiplexing (STTC-OFDM) systems. However, as a com ceiver performance in part by the U.s. National Science lation under grant CAREER CCR-98753 14 and grant CCR-9980599. The STBC-OFDM system is described. In Section Ill, an EM-based work of B. Lu was also supported in part by the Texas Telecommunications ML receiver for STBC-OFDM systems without outer channel cod Engineering Consortium(TxTEC) ing is developed In Section IV, an MAP-EM-based Turbo receiver
Iterative Receivers for Space-time Block Coded OFDM Systems in Dispersive Fading Channels Ben Lu, Xiaodong Wang Department of Electrical Engineering Texas A&M University College Station, TX 77843. benlu,wangx✁ @ee.tamu.edu Ye (Geoffrey) Li School of Electrical and Computer Engineering Georgia Institute of Technology Atlanta, GA 30332. liye@ece.gatech.edu Abstract – We consider the design of iterative receivers for space-time block coded orthogonal frequency-division multiplexing (STBC-OFDM) systems in unknown wireless dispersive fading channels, with or without outer channel coding. First, we propose a maximum-likelihood (ML) receiver for STBC-OFDM systems based on the expectation-maximization (EM) algorithm. By assuming that the fading processes remain constant over the duration of one STBC code word, and by exploiting the orthogonality property of the STBC as well as the OFDM modulation, we show that the EM-based receiver has a very low computational complexity, and that the initialization of the EM receiver is based on the linear minimum mean-square-error (MMSE) channel estimate for both the pilot and the data transmission. Since the actual fading processes may vary within one STBC code word, we also analyze the effect of a modelling mismatch on the receiver performance, and show both analytically and through simulations that the performance degradation due to such a mismatch is negligible for practical Doppler frequencies. We further propose a Turbo receiver based on the maximum a posteriori (MAP)-EM algorithm for STBC-OFDM systems with outer channel coding. Compared with the previous non-iterative receiver employing a decision-directed linear channel estimator, the iterative receivers proposed here significantly improve the receiver performance and can approach the ML performance in typical wireless channels with very fast fading, at a reasonable computational complexity well suited for real-time implementations. I. INTRODUCTION Recently several studies addressing the design and applications of space-time coding (STC) have been conducted, e.g., [1, 2]. Meanwhile, from the signal processing perspective, research on receiver design for STC systems is also active. With ideal channel state information (CSI), the iterative receivers based on the Turbo principle [3] are shown to be able to provide near-optimal performance in concatenated STC systems [4]. When the CSI is not available [5, 6], the design of channel estimators and training sequences were studied and receiver structures based on the decision-directed least-square channel estimator or its simplified variant were proposed for space-time trellis coded orthogonal frequency-division multiplexing (STTC-OFDM) systems. However, as a common This work was supported in part by the U.S. National Science Foundation under grant CAREER CCR-9875314 and grant CCR-9980599. The work of B. Lu was also supported in part by the Texas Telecommunications Engineering Consortium (TxTEC). drawback to any decision feedback system, when the decisions are not accurate (e.g., in very fast fading channels), the receivers in [5, 6] show error floors. In this paper, we approach the problem of receiver design without CSI by using iterative techniques, including the expectation-maximization (EM) algorithm [7] and the Turbo processing method [3]. In [8], EM receivers are studied for sequence estimation in uncoded and coded systems. More recently, in [9, 10], a receiver employing the EM algorithm, which exhibits a good performance but on the other hand a relatively high complexity, is proposed for STC systems. In this paper, we focus on the design of iterative receivers for STBC-OFDM systems in unknown wireless dispersive fading channels. We first derive the maximum likelihood (ML) receiver based on the EM algorithm for STBC-OFDM systems, under the assumption that the fading processes remain constant over the duration of one STBC code word (or equivalently across several adjacent OFDM words contained in one STBC code word). Based on such an assumption and the orthogonality property of the STBC as well as the OFDM modulation, we show that no matrix inversion is needed in the EM algorithm. Therefore, the computational cost for implementing the EM-based ML receiver is low and the computation is numerically stable. Moreover, we show that the mean-square-error (MSE) in the initialization of the EM algorithm can achieve its minimum for both the pilot transmission mode and the data transmission mode (assuming correct decision feedback). Since the actual fading processes may vary over the duration of one STBC code word, we also analyze the effect of a modelling mismatch on the receiver performance, by considering the average MSE as well as an upper bound on the instantaneous MSE of the channel estimate in the initialization of the EM algorithm. We show that the average MSE due to a modelling mismatch is negligible for practical Doppler frequencies. To further improve the quality of the initialization step of the EM algorithm (and/or to reduce the computational complexity), following [5, 11, 12], the techniques of significant-tap-catching linear estimation and temporal filtering are adopted in the proposed EM-based ML receiver. Finally, for STBC-OFDM systems employing outer channel coding, we propose a Turbo receiver, which iterates between the maximum a posteriori (MAP)-EM STBC decoding algorithm and the MAP channel decoding algorithm to successively improve the receiver performance. The rest of this paper is structured as follows. In Section II, the STBC-OFDM system is described. In Section III, an EM-based ML receiver for STBC-OFDM systems without outer channel coding is developed. In Section IV, an MAP-EM-based Turbo receiver
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 with
for 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, ✄ transmitter 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 modulator; 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 independently encoded, therefore, during ✝ OFDM slots, altogether ✂ STBC code words [or ✆✞✂☛✝✟✄✡✠ STBC code symbols] are transmitted. 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 different 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 transmitted 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 ambient 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 corresponding 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 independently. It is clear that decoding in an STBC-OFDM system 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 assumption significantly simplifies the receiver design. And the system 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 symbols ❵❇ ✗ ✎✑★✮✴❯✴✒ 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 prohibitive 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 random vector with mean ➟ and variance ↕ ; ↕✟➙✵➛ denotes the covariance 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
the j-th transmitter antenna; B ,=0 if the channel response at estimator also achieves the minimum mean-square error(MSE) this tap is zero. It is assumed that 2h is known(or measured MSE= ko2, (when assuming the data decision X is correct) with the aid of pilot symbols). It is seen that in the E-step, due to Note that in 6], a carefully designed optimal training sequence for the orthogonality property of the STBC(5), no matrix inversion is STC-OFDM systems can also achieve the minimum mse and does nvolved. Therefore, the computational complexity of the E-step not need matrix inversion. Recall that in the STBC-OFDM system significantly reduced and the computation is also numerically discussed above, to achieve the minimum mse and to avoid ma- more stable. Using(4),Q(XX )) is computed as trix inversion, P consecutive OFDM words (i.e, one STBC code word) need to be transmitted during the training stage [cf. Eq ( 5) Qxx)=cnst-n∑∑∑ In contrast, by employing the optimal training sequence as pre posed in 6], only one OFDM training word is needed. Therefore, 1p=1k=0 in order to improve the spectral efficiency, w rlvip, k]-aLp, kwy(k)h]+[az"p, kjEh,(k)ap, kl] training sequence in [ 6] and only transmit one(instead of P) pilot OFDM word at p= 0. In the above analysis, we assume that hile remain q(c{n,利]) for 1,..., P, whereas the actual channel may vary with a≌{x1b,周1…,mNx, these P OFDM slots For the(2 X 2)STBC[13, by assumi both the data X lp] and channe wr(k)= diag w (k),.,(k) quantities, the average MSE of the channel estimate is Nx(NL) ∑h2(k) △ ∑hW MSe- (far)2 (-1)K+k,(-1)K+k i=1,,N,j=1 where T is the duration of one OFDM slot; fa is the maximum Doppler frequency of the fading channel. The extra term(afaT) notes the(i',i')-th element of matrix A in (9)reflects the average mse due to the modeling mismatch For (7), the M-step proceeds as follow a practical normalized Doppler frequency(e.g, faT=0.01), the average mse due to the modeling mismatch is negligible. Further rmin,∑∑(pA)( 7) of the channel estimate is upper bounded It is seen from(7)that the M-step can be decoupled into K in- ent minimization problems, and the coding constraints of From the mse analysis in(9)and (10), in order to reduce [13] mator which only estimates the significant taps of the channel re- sponse, namely the significant-tap-catching least-square estimator III-B. Initialization of the eM Algorithm is adopted here As seen in (9)and(10), with the increase of maximum doppler ceiver)is closely related to the quality of the initial value of xo), a quency fa, the mismatch MSE of the channel estimate increases. ameliorate this problem, as indicated in [11, 12], a temporal fill i.e.. the initial value at the first EM iteration. The initial estimate ter is applied in addition to the least-square estimator to further of X is computed based on the method proposed in [11, 12] by the following steps. First, a linear estimator is used to estimate the exploit the time correlation of channel responses the em algorithm is l channel with the aid of the pilot symbols or the decision-feedback in Table 1 In Table 1, the ML detection in(*)takes into account of f the data symbols. Secondly, the resulting channel estimate is the STBC coding constraints of X F-filter denotes the significant- relation of the channel. Finally, based on the filtered channel estl- resents the pilot symbols and r )[mI, m=g where X[]rep- resents hard-decisions of the data symbols Xm which is pro- In(6), by assuming the perfect knowledge of thg hi is sim- vided by the EM algorithm after a total of I EM iterations. And response hi. When Eh, is not known to the receiver, a least-square T-filter denotes the tempo the time-domain correlation of the channel within one ofdm data estimator(LSE)can be applied to estimate the channel and to mea- burst (i. e, (Pa+ 1)OFDM slots], as in [11,12] The Lse hi is expressed IV. TURBO RECEIVER PK X ply lp In practice, in order to further exploit the frequency-selective fad- It is seen that in( 8), unlike a typical LSE, no matrix inversion is ing diversity embedded across all K OFDM subcarriers, it is com- involved here. Hence, its complexity is significantly reduced from mon to apply an outer channel code, (e. g, convolutional code O(NL )to only O(NL) and the computation is numerically Turbo code), in addition to the STBC. And we propose a Turbo ore stable, which is very attractive in systems using more trans- receiver employ ing the maximum a posteriori(MAP)-EM STB itter antennas (large N)and/or communicating in highly disper- decoding algorithm and the MaP outer-channel-code decoding al sive fading channels (large L). Moreover, following [11], such an gorithm for this concatenated STBC-OFDM system, as depicted
the ❱-th transmitter antenna; ➢ ❳ ✗ ✥ ✇ ✔❘❈ if the channel response at this tap is zero. It is assumed that ↕✟➙✵➛ is known (or measured with the aid of pilot symbols). It is seen that in the E-step, due to the orthogonality property of the STBC (5), no matrix inversion is involved. Therefore, the computational complexity of the E-step is significantly reduced and the computation is also numerically more stable. Using (4), ➤✟✆✜ ➒✜ ♠✿➓ s ✠ is computed as ➤➥✆✜ ➒✜ ♠④➓ s ✠ ✔ ➦➑✫➧✽➨❻➩ ✳ ❉ ✱❲❳❩ ➏ ✌ ✘q✚ ➫ ➉ ✘✛✚ ⑨ r ✚ ⑥ ✘ ❞ ➭✌ ✎✑★✮✴❯✴✒ ❉☛➯ ❍ ✎✑★✮✴❯✴✒❻❿➳➲❷ ✆ ❯ ✠❸➊ ✌ ❳ ✧ ➯ ❍ ✎✑✬✮✴❯✵✒ ➊↕✟➙✵➛ ✆ ❯ ✠❏➯ ✎✑★✮▲❯✵✒ ➵ ♠✿➓ s ✌ ✆❏➯ ✎✑✬✮✴❯✵✒ ✠ ✮ with ➯ ✎✑★✮✴❯✴✒ ✷✔ ✎❇ ✚ ✎✑★✮✴❯✴✒✹✮✵✳✴✳✴✳✴✮ ❇ ✖ ✎✑★✮✴❯✴✒✿✒ ❍✾✍✻✽❖ ✮ ❿❺➲❷ ✆ ❯ ✠ ✷✔ ❂✽❃✏❄❆❅ ⑩❍❷ ✆ ❯ ✠ ✮✴✳▲✳✴✳✵✮✹⑩❍❷ ✆ ❯ ✠ ✾✍✻✽✼④✾✯➂✴❀ ✮ ➊↕✟➙✵➛ ✆ ❯ ✠ ♠ ✌✿➸✏✥ ✗ ➸ s ✷✔ ❿ ➊↕✟➙✵➛ ❿❍ ♠ ✌➺➸ r ✚✹s ⑨❁➻ ⑥ ✥ ♠ ✗ ➸ r ✚✹s ⑨❁➻ ⑥ ✮ ✰ ➲ ✔➀✱❝✮✵✳▲✳✴✳✴✮ ✄ ✮ ❱ ➲ ✔➀✱❝✮▲✳✴✳✴✳✵✮ ✄ ✮ where ✎➼✶✒ ♠ ✌ ➸ ✥ ✗ ➸ s denotes the ✆✰ ➲ ✮ ❱ ➲ ✠ -th element of matrix ➼ . Next, based on (7), the M-step proceeds as follows ✜ ♠④➓➻ ✚✹s ✔ ⑨ r ✚ ⑥ ✘ ❞ ❄✙➋❻❅➽➌➾❃④➧ ➚➯➝➪➉❡✥ ⑥✙➶④➹✹➘ ➏ ✌ ✘✛✚ ➫ ➉ ✘✛✚ ➵ ♠④➓ s ✌ ✆❏➯ ✎✑★✮✴❯✴✒ ✠ ✳ (7) It is seen from (7) that the M-step can be decoupled into ✂ independent minimization problems, and the coding constraints of STBC are taken into account when solving the M-step, i.e., ➯ ✎✑★✮✴❯✴✒▼✮ ➴✑✬✮ are different permutations and/or transformations of ➯ ✎✿✱❝✮✵❯✴✒ [13]. III-B. Initialization of the EM Algorithm The performance of the EM algorithm (and hence the overall receiver) is closely related to the quality of the initial value of ✜ ♠ ❞ s , i.e., the initial value at the first EM iteration. The initial estimate of ✜ ♠ ❞ s is computed based on the method proposed in [11, 12] by the following steps. First, a linear estimator is used to estimate the channel with the aid of the pilot symbols or the decision-feedback of the data symbols. Secondly, the resulting channel estimate is refined by a temporal filter to further exploit the time-domain correlation of the channel. Finally, based on the filtered channel estimate, ✜ ♠ ❞ s is obtained through the ML detection. In (6), by assuming the perfect knowledge of ↕✟➙✵➛ , ❸➊ ✌ is simply the minimum mean-square estimate (MMSE) of the channel response ❸ ✌ . When ↕✟➙✵➛ is not known to the receiver, a least-square estimator (LSE) can be applied to estimate the channel and to measure ↕➙ ➛ . The LSE ❸➊ ✌ is expressed as, ❸➊ ✌ ✔ ✱ ✝✟✂ ❿❍ ➫ ➉ ✘✛✚❣✜❍ ✎✑✫✒♦☞P✌❑✎✑✫✒ ✳ (8) It is seen that in (8), unlike a typical LSE, no matrix inversion is involved here. Hence, its complexity is significantly reduced from ➷ ✆✞✄➄➬▼❽➠➬✵✠ to only ➷ ✆✞✄❚❽➮✠ and the computation is numerically more stable, which is very attractive in systems using more transmitter antennas (large ✄) and/or communicating in highly dispersive fading channels (large ❽). Moreover, following [11], such an estimator also achieves the minimum mean-square error (MSE), MSE✔ ✈⑨ ❲❩❳ , (when assuming the data decision ✜ is correct). Note that in [6], a carefully designed optimal training sequence for STC-OFDM systems can also achieve the minimum MSE and does not need matrix inversion. Recall that in the STBC-OFDM system discussed above, to achieve the minimum MSE and to avoid matrix inversion, ✝ consecutive OFDM words (i.e., one STBC code word) need to be transmitted during the training stage [cf. Eq.(5)]. In contrast, by employing the optimal training sequence as proposed in [6], only one OFDM training word is needed. Therefore, in order to improve the spectral efficiency, we adopt the optimal training sequence in [6] and only transmit one (instead of ✝) pilot OFDM word at ✑✶✔➄❈. In the above analysis, we assume that ❸ ✌ ✎✑✫✒ remain constant for ✑➱✔✃✱✓✮✴✳✴✳✵✳✴✮ ✝, whereas the actual channel may vary across these ✝ OFDM slots. For the ✆ ➆✍❐➝➆ ✠ STBC [13], by assuming that both the data ✜ ✎✑✓✒ and channel responses ❵❸ ✌ ✎✑✓✒ t ➉ are random quantities, the average MSE of the channel estimate is ❒☛❮➞➡ ✔ ✆✞❰❁Ï✫Ð❼ ✠ ❳ ➆ ✧ ❽ ✂❲❩❳ ✮ (9) where ❼ is the duration of one OFDM slot; Ï❡Ð is the maximum Doppler frequency of the fading channel. The extra term ✚ ❳ ✆✞❰❨Ï❡Ð❼ ✠ ❳ in (9) reflects the average MSE due to the modeling mismatch. For a practical normalized Doppler frequency (e.g., Ï❡Ð❼❘✔❶❈★✳❈✬✱ ), the average MSE due to the modeling mismatch is negligible. Furthermore, for a particular realization of ✜ ✎✑➲✒ , the instantaneous MSE of the channel estimate is upper bounded by ❒☛❮❡➡ Ñ ✆✞❰❨Ï❡Ð❼ ✠ ❳❽❳ ✧ ❽ ✂❲❩❳ ✳ (10) From the MSE analysis in (9) and (10), in order to reduce the computational complexity and to further improve the accuracy of the channel estimate, as indicated in [5], the least-square estimator which only estimates the significant taps of the channel response, namely the significant-tap-catching least-square estimator, is adopted here. Asseen in (9) and (10), with the increase of maximum Doppler frequency Ï❡Ð , the mismatch MSE of the channel estimate increases. To ameliorate this problem, as indicated in [11, 12], a temporal filter is applied in addition to the least-square estimator to further exploit the time correlation of channel responses. Finally, the procedure for initializing the EM algorithm islisted in Table 1. In Table 1, the ML detection in (Ò) takes into account of the STBC coding constraints of ✜ . F-filter denotes the significanttap-catching version of the least-square estimator, where ✜ ✎❈✫✒ represents the pilot symbols and ✜ ♠④Ó s ✎ÔÕ✒✹✮❣ÔÖ✔✲❈★✮✵✳✴✳✴✳▲✮ ➵ ❉ ✱P✮ represents hard-decisions of the data symbols ✜ ✎ÔÕ✒ which is provided by the EM algorithm after a total of × EM iterations. And T-filter denotes the temporal filter, which is used to further exploit the time-domain correlation of the channel within one OFDM data burst [i.e., (✝➵ ✧❶✱ ) OFDM slots], as in [11, 12]. IV. TURBO RECEIVER In practice, in order to further exploit the frequency-selective fading diversity embedded across all ✂ OFDM subcarriers, it is common to apply an outer channel code, (e.g., convolutional code or Turbo code), in addition to the STBC. And we propose a Turbo receiver employing the maximum a posteriori (MAP)-EM STBC decoding algorithm and the MAP outer-channel-code decoding algorithm for this concatenated STBC-OFDM system, as depicted
in Fig. 2. More specifically, the E-step of the MAP-EM algorithm Moreover, without CSI, after 4-5 Turbo iterations, the turbo re- is exactly the same as the E-step of the Em algorithm; but the ceiver performs close to the approximated ml lower bound in all M-step of the MAP-EM algorithm includes an extra term P(X), three types of channels with a Doppler frequency as high as 200Hz which represents the a priori probability of x that is fed back by As a final remark, the EM-based iterative receiver techniques the outer- channel-code decoder from the previous Turbo iteration proposed in this paper are also applicable to other space-time cod (For the details of the MAP-EM algorithm, see [7)) ing(STC)systems, such as the STTC-OFDM system [5], but at an increased receiver complexity compared with that of the STBC V. SIMULATION RESULTS receivers developed here vide computer simulation results to illustrate REFERENCES the performance of our proposed iterative receivers for STBC OFDM systems, with or without outer channel coding. The char- [1 V. Tarokh, H Jafarkhani, and A R. Calderbank, "Space-time block codes from acteristics of the fading channels are described in Section Il; specif- rthogonal designs, " IEEE Trans. Inform. Theary, voL 45, Pp. 1456-1467, July ically, the receiver performance is simulated in three typical chan- [2] D. Agrawal, V Tarokh, A Naguib, and N Seshadri, "Space-time coded OFDM nel models with different delay profiles, namely the two ray and the typical urban(TU) model with 50Hz and 200Hz Doppler fre- quencies [12]. In the following simulations the available band- 3】]J The Turbo principle: Tutorial introduction and state of the art, width is 800 KHz and is divided into 128 subcarriers. These cor- rance, Sept 19g>al Symposium on Turbo Codes and Related Topics, Brest, respond to a subcarrier symbol rate of 5 KHz and OFDM word (41 [4] G Bauch,"Concatenation of space-time block codes and"Turbo-TCM, "in duration of 160us. In each OFDM word, a guard interval of 40p Proc. 1999 Internatiomal Conference on Communications. ICC99, Vancouver is inserted, hence the duration of one OFDM word T=200, For all simulations two transmitter antennas and two receiver an- [5]Y. Li, N. Seshadri, and S. Ariyavisitakul, "Channel estimation for OFDM sys- tennas are used; and the g1 StBC is adopted [13]. The modulator bile wireless channels, IEEEJ. sele Areas Commun, vol 17, pp 461-471, Mar. 1999 uses QPSK constellation V-A. Performance of eM-ML Receiver [71 G.J. McLachlan and T. Krishnan, The EM Algorithm and Extensions, John Wiley Sons, Inc, New York, NY, 1997 In an STBC-OFDM system without outer channel cod, 512 infor- [8] C N. Georghiades and J C Han, Sequence estimation in the presence of crs via the EM algorithm, IEEE Trans. CoNm mation bits are transmitted from 128 subcarriers during two(P 300-308,Mar.1997. 2)OFDM slots, therefore the information rate is 1.6 bits/sec/Hz. [9] C Cozzo and B. L. Hughes, "Joint detection and estimation in space-time cod- In Fig. 3-4, Mnen idea channel state information( CSI) is assumed ailable at the receiver side, the ml performance is shown in Computers, Sydney, Oct. 1999, Pp 613-617 Conference on Signals.Systems dashed lines, denoted by Ideal CSI. Without the CSl. the EM- [10]Y Li, C.N. Georghiades, and G. Huang,"EM-based sequence estimation for based ML receiver as derived in Section Ill is adopted; further- Theory, Sorrento, Italy, June 2000. psm::mPmt上:1m单hmam and the initialization of the EM algorithm [cf. Eq 8). From the figures, it is seen that the receiver performance is significantly [12] Y. Li and N.R. Sollenberger, Adaptive antenna arrays for OFDM systems ih cochannel interference, IEEE Trans. Commun., vol. 47, pp. 217- roved through the EM iterations. Furthermore, although the re ever is designed under the assumption that the channel remains 13]s.M. Alamouti“As static over one StBC code word ( whereas the actual channel varies cations,IEEE J. Select. Areas Commnz, vol. 16, pp. 1451-1458, O during one STBC code word), it can perform close to the ML per- [14] H V. Poor, An Introduction to Signal Detection and Estimation, formance with ideal CSi after two or three em iterations for al three types of channels with a Doppler frequency as high as 200Hz. V-B. Performance of MAP-EM-Turbo Receiver nE MPSK B⊥ tsModu1at ]回了 A 4-state, rate-1/2 convolutional code with generator(5,7)in octal notation is adopted as the outer channel code, as depicted in Fig. 2 The overall information rate for this system is 0. 8 bit/sec/Hz. Fig 5-6 show the performance of the Turbo receiver employing the EM STBCDecis⊥ons MAP-EM algorithm as derived in Section IV, for this concatenated Decc STBC-OFDM system. During each Turbo iteration, three EM iter- ations are carried out in the map-em stbc decoder. ldeal CSI denotes the approximated ML lower bound, which is obtained by performing the MAP STBC decoder with ideal CSI and iterating In⊥t⊥a1 sufficient number of Turbo iterations(six iterations in our simu- lations)between the MAP Stbc decoder and the map convolu- nal decoder. From the simulation results, it is seen that by em- Figure 1: Transmitter and receiver structure for an STBC-OFDM ploying an outer channel code, the receiver performance is signif- antly improved(at the expense of lowering spectral efficiency)
in Fig. 2. More specifically, the E-step of the MAP-EM algorithm is exactly the same as the E-step of the EM algorithm; but the M-step of the MAP-EM algorithm includes an extra term ✝✶✆✜ ✠ , which represents the a priori probability of ✜ that is fed back by the outer-channel-code decoder from the previous Turbo iteration. (For the details of the MAP-EM algorithm, see [7].) V. SIMULATION RESULTS In this section, we provide computer simulation results to illustrate the performance of our proposed iterative receivers for STBCOFDM systems, with or without outer channel coding. The characteristics of the fading channels are described in Section II; specifically, the receiver performance is simulated in three typical channel models with different delay profiles, namely the two-ray and the typical urban (TU) model with 50Hz and 200Hz Doppler frequencies [12]. In the following simulations the available bandwidth is 800 KHz and is divided into ✱✓➆✫Ø subcarriers. These correspond to a subcarrier symbol rate of 5 KHz and OFDM word duration of ✱✓Ù❆❈❝Ús. In each OFDM word, a guard interval of Û ❈✓Ús is inserted, hence the duration of one OFDM word ❼Ü✔Ý➆❆❈✓❈✓Ús. For all simulations, two transmitter antennas and two receiver antennas are used; and the Þ✚ STBC is adopted [13]. The modulator uses QPSK constellation. V-A. Performance of EM-ML Receiver In an STBC-OFDM system without outer channel cod, ß ✱✓➆ information bits are transmitted from ✱✓➆✓Ø subcarriers during two (✝ ✔ ➆ ) OFDM slots, therefore the information rate is 1.6 bits/sec/Hz. In Fig. 3–4, when ideal channel state information (CSI) is assumed available at the receiver side, the ML performance is shown in dashed lines, denoted by Ideal CSI. Without the CSI, the EMbased ML receiver as derived in Section III is adopted; furthermore, as in [12], the 7-tap significant-tap-catching scheme is applied to simplify the implementation of the E-step [cf. Eq.(6)] and the initialization of the EM algorithm [cf. Eq.(8)]. From the figures, it is seen that the receiver performance is significantly improved through the EM iterations. Furthermore, although the receiver is designed under the assumption that the channel remains static over one STBC code word (whereas the actual channel varies during one STBC code word), it can perform close to the ML performance with ideal CSI after two or three EM iterations for all three types of channels with a Doppler frequency as high as 200Hz. V-B. Performance of MAP-EM-Turbo Receiver A 4-state, rate-1/2 convolutional code with generator (5,7) in octal notation is adopted as the outer channel code, as depicted in Fig. 2. The overall information rate for this system is 0.8 bit/sec/Hz. Fig. 5–6 show the performance of the Turbo receiver employing the MAP-EM algorithm as derived in Section IV, for this concatenated STBC-OFDM system. During each Turbo iteration, three EM iterations are carried out in the MAP-EM STBC decoder. Ideal CSI denotes the approximated ML lower bound, which is obtained by performing the MAP STBC decoder with ideal CSI and iterating sufficient number of Turbo iterations (six iterations in our simulations) between the MAP STBC decoder and the MAP convolutional decoder. From the simulation results, it is seen that by employing an outer channel code, the receiver performance is significantly improved (at the expense of lowering spectral efficiency). Moreover, without CSI, after 4-5 Turbo iterations, the Turbo receiver performs close to the approximated ML lower bound in all three types of channels with a Doppler frequency as high as 200Hz. As a final remark, the EM-based iterative receiver techniques proposed in this paper are also applicable to other space-time coding (STC) systems, such as the STTC-OFDM system [5], but at an increased receiver complexity compared with that of the STBC receivers developed here. REFERENCES [1] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inform. Theory, vol. 45, pp. 1456–1467, July 1999. [2] D. Agrawal, V. Tarokh, A. Naguib, and N. Seshadri, “Space-time coded OFDM for high data-rate wireless communication over wideband channels,” in IEEE Vehicular Technology Conference, 1998. VTC’98., May 1998. [3] J. Hagenauer, “The Turbo principle: Tutorial introduction and state of the art,” in Proc. International Symposium on Turbo Codes and Related Topics, Brest, France, Sept. 1997. [4] G. Bauch, “Concatenation of space-time block codes and ‘Turbo’-TCM,” in Proc. 1999 International Conference on Communications. ICC’99, Vancouver, June 1999. [5] Y. Li, N. Seshadri, and S. Ariyavisitakul, “Channel estimation for OFDM systems with transmitter diversity in mobile wireless channels,” IEEE J. Select. Areas Commun., vol. 17, pp. 461–471, Mar. 1999. [6] Y. Li, “Simplified channel estimation for OFDM systems with multiple transmit antennas,” submitted to IEEE J. Select. Areas Commun., Nov. 1999. [7] G. J. McLachlan and T. Krishnan, The EM Algorithm and Extensions, John Wiley & Sons, Inc, New York, NY, 1997. [8] C. N. Georghiades and J. C. Han, “Sequence estimation in the presence of random parameters via the EM algorithm,” IEEE Trans. Commun., vol. 45, pp. 300–308, Mar. 1997. [9] C. Cozzo and B. L. Hughes, “Joint detection and estimation in space-time cod- à ing and modulation,” in Thirty-Third Asilomar Conference on Signals, Systems Computers, Sydney, Oct. 1999, pp. 613–617. [10] Y. Li, C. N. Georghiades, and G. Huang, “EM-based sequence estimation for space-time coded systems,” in IEEE International Symposium on Information Theory, Sorrento, Italy, June 2000. [11] Y. Li, L. J. Cimini, and N. R. Sollenberger, “Robust channel estimation for OFDM systems with rapid dispersive fading channels,” IEEE Trans. Commun., vol. 46, pp. 902–915, July 1998. [12] Y. Li and N. R. Sollenberger, “Adaptive antenna arrays for OFDM systems with cochannel interference,” IEEE Trans. Commun., vol. 47, pp. 217–229, Feb. 1999. [13] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Select. Areas Commun., vol. 16, pp. 1451–1458, Oct. 1998. [14] H. V. Poor, An Introduction to Signal Detection and Estimation, SpringerVerlag, 2nd edition, 1994. IFFT . IFFT . . . . . . . . . . . Modulator MPSK STBC Bits Encoder Info. FFT FFT Decoder EM STBC EM Alg. Initial. X (0) Decisions Pilot (p=0) (p=0) o o Figure 1: Transmitter and receiver structure for an STBC-OFDM system
h间= F-filtery回,x间 h:[mP+n]=T-fiterhi[mP+n-1], hi[mP+n-2 hmP+n-d r#2,Fd=50 wimP+n'lX, hi[mP +n] f EM Algorithm eac3、Fd20t hi[mP+n]=F.. [m,x(mI end Figure 4: Typical urban(TU) fading channels with Doppler fre- quencies fa= 50Hz and fa=200Hz. Table 1: Procedure for computing X)for the EM algorithm Encode Modulator 二m了 入2 MAP Chan Turbo Iteral, Fd=200Hz 一 Turbo itert5,Fd200Hz In⊥tia1 Sianahtc-Noe。RaB Figure 2: Transm receiver structure for an STBC-OFDM Figure 5: STBC-OFDM systems employing outer convolutional system with outer code. Ii denotes the interleaver and code. Two-ray fading channels with Doppler frequencies fa II- denotes the nding deinterleaver 50Hz and a= 200Hz urbo iter5. Fd=200Hz 一 EM Iter.Fd=200Hz IdeAl CsI Figure 6: STBC-OFDM systems employing outer convolutional Figure 3: Two-ray fading channels with Doppler frequencies fa code. Typical urban(TU) fading channels with Doppler frequen 50Hz and fa= 200Hz cies fa= 50Hz and fa= 200Hz
âá ✌✹ãä✵å = F-filter æ ✌ ãä✵å✏ç✦è❶ãä✵å ç é✯ê➄ë➞ç✙ì❆ì✙ì✙ç✞íîç for ï êðä❋ç❻ë➞ç✙ì✙ì✙ì❆ç✞ñóò☛ë for ô ê◆ë➞ç♦õ❋ç❭ì❆ì✙ì❆ç✦ö â÷ ✌✏ãïö●ø ô å = T-filter âá ✌✹ãïö●ø ô ò✪ë✹å✏ç âá ✌✹ãïö●ø ô òùõ✵å✏ç✙ì❆ì✙ì❆ç âá ãïö●ø ô òùú✞å çûéqê➄ë➞ç❭ì❆ì✙ì❆ç❏íîç end è♠ ❞ s ãïå = arg max➎ ✌➏ ✘q✚ ➫ ü ➸ ✘✛✚ ý➺þ✵ÿ✁ æ ✌ ãïö●ø ô➲ å èç â÷ ✌✏ãïö❶ø ô➲ å ç ✂☎✄✝✆ è♠④Ó s ãïå = EM æ✌ ãïå ✌ ç❑è♠ ❞ s ãïå ç [cf. EM Algorithm] for ô ê◆ë➞ç♦õ❋ç❭ì❆ì✙ì❆ç✦ö âá ✌✏ãïö●ø ô å = F-filter æ✌ ãïå✏ç✦è♠④Ó s ãïå ç❏éqê➄ë➞ç❭ì❆ì❆ì✙ç❏íîç✞✂☎✄✞✄✝✆ end end Table 1: Procedure for computing ✜ ♠ ❞ s for the EM algorithm. . . . . . . . . . . . . IFFT IFFT Encoder Π Modulator MPSK STBC Encoder FFT FFT Decoder (0) X MAP Channel o (p=0) EM Alg. Initial. Pilot (p=0) o Decoder Channel λ MAP-EM STBC λ1 e e 2 Π Π -1 Figure 2: Transmitter and receiver structure for an STBC-OFDM system with outer channel code. ✟ denotes the interleaver and ✟r ✚ denotes the corresponding deinterleaver. 0 2 4 6 8 10 12 14 16 10−3 10−2 10−1 100 STBC−OFDM in two−path Fading Channels, without CSI OFDM Word Error Rate, WER Signal−to−Noise Ratio (dB) EM Iter#1, Fd= 50Hz EM Iter#2, Fd= 50Hz EM Iter#3, Fd= 50Hz EM Iter#1, Fd=200Hz EM Iter#2, Fd=200Hz EM Iter#3, Fd=200Hz Ideal CSI Figure 3: Two-ray fading channels with Doppler frequencies Ï❡Ð ✔ ß❈ Hz and Ï❡Ð ✔➀➆✙❈✫❈Hz. 0 2 4 6 8 10 12 14 16 10−3 10−2 10−1 100 STBC−OFDM in TU Fading Channels, without CSI OFDM Word Error Rate, WER Signal−to−Noise Ratio (dB) EM Iter#1, Fd= 50Hz EM Iter#2, Fd= 50Hz EM Iter#3, Fd= 50Hz EM Iter#1, Fd=200Hz EM Iter#2, Fd=200Hz EM Iter#3, Fd=200Hz Ideal CSI Figure 4: Typical urban (TU) fading channels with Doppler frequencies Ï✫Ð ✔ ß❈ Hz and Ï✫Ð ✔➀➆✙❈✓❈Hz. 0 1 2 3 4 5 6 10−3 10−2 10−1 100 STBC−OFDM in two−path Fading Channels, without CSI OFDM Word Error Rate, WER Signal−to−Noise Ratio (dB) Turbo Iter#1, Fd= 50Hz Turbo Iter#3, Fd= 50Hz Turbo Iter#5, Fd= 50Hz Turbo Iter#1, Fd=200Hz Turbo Iter#3, Fd=200Hz Turbo Iter#5, Fd=200Hz Ideal CSI Figure 5: STBC-OFDM systems employing outer convolutional code. Two-ray fading channels with Doppler frequencies Ï✫Ð ✔ ß❈ Hz and Ï✫Ð ✔➀➆✙❈✫❈Hz. 0 1 2 3 4 5 6 10−3 10−2 10−1 100 STBC−OFDM in TU Fading Channels, without CSI OFDM Word Error Rate, WER Signal−to−Noise Ratio (dB) Turbo Iter#1, Fd= 50Hz Turbo Iter#3, Fd= 50Hz Turbo Iter#5, Fd= 50Hz Turbo Iter#1, Fd=200Hz Turbo Iter#3, Fd=200Hz Turbo Iter#5, Fd=200Hz Ideal CSI Figure 6: STBC-OFDM systems employing outer convolutional code. Typical urban (TU) fading channels with Doppler frequencies ÏÐ ✔ ß❈ Hz and ÏÐ ✔➀➆✙❈✓❈Hz
