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. a0 benlu,wangx a1 @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 multiplex- ing (STBC-OFDM) systems in unknown wireless dispersive fad- ing 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 orthog- onality 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 esti- mate for both the pilot and the data transmission. Since the ac- tual 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 neg- ligible 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]. Mean- while, 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 prin- ciple [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 pro- posed 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 Foun- dation 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, in- cluding 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 re- cently, 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 du- ration of one STBC code word (or equivalently across several ad- jacent 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 inver- sion 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 aver- age 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 tem- poral filtering are adopted in the proposed EM-based ML receiver. Finally, for STBC-OFDM systems employing outer channel cod- ing, we propose a Turbo receiver, which iterates between the max- imum a posteriori (MAP)-EM STBC decoding algorithm and the MAP channel decoding algorithm to successively improve the re- ceiver 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 cod- ing is developed. In Section IV, an MAP-EM-based Turbo receiver 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 witha2 subcarriers,a3 trans- mitter antennas and a4 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 a5a7a6a8a3a10a9 STBC symbols, which are transmitted from a3 transmitter antennas and across a6 consecutive OFDM slots at a particular OFDM subcarrier. The STBC code words at different OFDM subcarriers are indepen- dently encoded, therefore, during a6 OFDM slots, altogether a2 STBC code words [or a5a7a2a11a6a8a3a10a9 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 a12a14a13a16a15a17a19a18a21a20 a22 a23a25a24a27a26a29a28 a23 a15a17a19a18a31a30 a13a33a32 a23 a15a17a19a18a35a34a37a36 a13 a15a17a38a18 a20 a28 a15a17a19a18a31a30 a13 a15a17a38a18a39a34a40a36 a13 a15a17a19a18a42a41 a43 a20a45a44a19a41a47a46a48a46a47a46a47a41 a4 a41a11a17a49a20a45a44a19a41a47a46a47a46a48a46a47a41 a6 a41 (1) with a28 a15a17a19a18a51a50a20 a28 a26 a15a17a19a18a52a41a47a46a47a46a47a46a47a41 a28 a22 a15a17a19a18 a53a14a54a56a55a58a57a42a53a29a59a60a41 a28 a23 a15a17a19a18a51a50a20 a61a56a62a16a63a65a64 a66 a23 a15a17a35a41a33a67a19a18a52a41a48a46a47a46a47a46a48a41 a66 a23 a15a17a35a41 a2a69a68 a44a25a18 a53a14a54a70a53a71a41a16a30 a13 a15a17a19a18 a50a20 a30a73a72a13a74a32 a26 a15a17a35a41a75a67a38a18a52a41 a46a47a46a47a46a76a41a77a30a73a72a13a74a32 a22 a15a17a35a41 a2a78a68 a44a25a18 a72 a55a58a57a42a53a29a59a65a54a56a79a80a41a52a30 a13a33a32 a23 a15a17a19a18 a50a20 a81 a13a33a32 a23 a15a17a35a41a16a67a19a18a52a41a47a46a76a46a48a46a47a41 a81 a13a74a32 a23 a15a17a39a41 a2a82a68 a44a65a18 a83a53a14a54a56a79a14a41 wherea30 a13 a15a17a19a18 is the ( a3a84a2 )-vector containing the complex channel frequency responses between thea43 -th receiver antenna and all a3 transmitter antennas at the a17 -th OFDM slot, which is explained below; a66 a23 a15a17a35a41a76a85a48a18 is the STBC symbol transmit- ted from the a86 -th transmitter antenna at the a85 -th subcarrier and at thea17 -th OFDM slot;a12a14a13a75a15a17a19a18 is thea2 -vector of received signals from the a43 -th receiver antenna and at the a17 -th time slot; a36 a13 a15a17a19a18 is the am- bient noise, which is circularly symmetric complex Gaussian with covariance matrix a87a89a88a90a92a91 . In this paper, we restrict our attention to MPSK signal constellation, i.e.,a66 a23 a15a17a35a41a76a85a48a18a42a93a95a94 a50a20a45a96a25a97a99a98a33a100a101a41a75a97 a98a103a102a92a104a105a106a107a105 a41a47a46a48a46a76a46a47a41 a97 a98a108a102a25a104a105a106a107a105a52a109a111a110a112a113a110a114 a26a52a115a16a116 a46 The channel frequency response between the a86 -th transmitter antenna and the a43 -th receiver antenna at the a17 -th time slot and at the a85 -th subcarrier can be expressed as a81 a13a74a32 a23 a15a17a35a41a76a85a48a18a117a20 a118 a114 a26 a119a24 a100 a120 a13a74a32 a23 a15a121a77a122a77a17a19a18a123a97 a114 a98 a88a52a124a126a125 a119a123a127a25a128 a20a130a129 a72a131 a5 a85 a9a74a132 a13a74a32 a23 a5 a17 a9 a41 (2) where a120 a13a74a32a23 a15a121a77a122a77a17a19a18 a50a20a134a133 a13a74a32a23 a5 a121a135a122a52a17a99a136 a9 , a136 is the duration of one OFDM slot; a132 a13a74a32 a23 a5 a17 a9 a50a20a134a15a133 a13a33a32 a23 a5 a67a39a122a52a17a19a136 a9 a41a130a46a47a46a47a46a76a41a37a133 a13a74a32 a23 a5a7a137a71a68 a44a38a122a77a17a99a136 a9 a18 a83 is the a137 - vector containing the time responses of all the taps; and a129 a131 a5 a85 a9 a50a20 a97 a114 a98a33a100a48a41a16a97 a114 a98 a88a52a124a99a125 a127a25a128 a41a47a46a48a46a47a46a47a41a77a97 a114 a98 a88a52a124a126a125 a109 a118 a114 a26a52a115 a127a25a128 a72 contains the correspond- ing DFT coefficients. Using (2), the signal model in (1) can be expressed as a12a14a13a75a15a17a19a18a138a20 a28 a15a17a19a18a77a139 a132 a13 a15a17a19a18a39a34a40a36 a13 a15a17a38a18a108a41 a43 a20a140a44a19a41a48a46a47a46a47a46a47a41 a4 a41a37a17a49a20a140a44a19a41a48a46a47a46a47a46a47a41 a6 a41 (3) with a139a141a50a20 a61a56a62a16a63a25a64 a139 a131 a41a47a46a47a46a76a46a47a41a48a139 a131 a55a123a57a42a53a29a59a135a54a56a55a123a57a42a142a47a59a80a41a48a139 a131 a50a20 a129 a131 a5a67 a9 a41 a129 a131 a5 a44 a9 a41a48a46a47a46a47a46a76a41a52a129 a131 a5a7a2a130a68 a44 a9 a72 a53a14a54a70a142a56a41 a132 a13 a15a17a19a18 a50a20 a132 a72a13a74a32 a26 a5 a17 a9 a41a47a46a76a46a47a46a47a41 a132 a72a13a33a32 a22 a5 a17 a9 a72 a55a123a57a42a142a47a59a135a54a56a79 a46 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 overa6 consecutive OFDM slots. To simplify the problem, we assume that channel time responses a132 a13 a15a17a19a18a52a41a16a17a49a20a140a44a99a41a48a46a76a46a47a46a47a41 a6 a41 remain constant during one STBC code word (i.e., a6 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 a12 a13 a20 a28 a139 a132 a13 a34a40a36 a13 a41 a43 a20a140a44a99a41a47a46a48a46a76a46a47a41 a4 a41 (4) with a12 a13 a20 a12 a72a13 a15a123a44a65a18a52a41a48a46a48a46a47a46a47a41a52a12 a72a13 a15a6 a18 a72 a55a58a143a29a53a29a59a135a54a56a79a14a41 a28 a50a20 a28 a72 a15a58a44a65a18a77a41a48a46a76a46a48a46a47a41 a28 a72a113a15 a6 a18 a72 a55a123a143a56a53a29a59a65a54a56a55a58a57a42a53a29a59a144a41a52a36 a13a145a50a20 a36a107a72a13 a15a58a44a25a18a52a41a48a46a47a46a76a46a48a41a52a36a107a72a13 a15 a6 a18 a72 a55a58a143a29a53a29a59a135a54a56a79a95a41 a132 a13 a50a20 a132 a13 a15a58a44a25a18a42a20 a132 a13 a15a58a146a25a18a147a20a45a148a47a148a76a148a70a20 a132 a13 a15 a6 a18a42a46 Moreover, by using the constant modulus property of the sym- bols a96 a66 a23 a15a17a35a41a47a85a47a18 a116 a23 a32a149a101a32 a125 , and the orthogonality property of STBC codes [1], we get a139 a72 a28 a72 a28 a139 a20 a5a7a6a8a2a71a9 a148 a91 a46 (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, a150 a28 a20 a63a25a151a135a64a14a152a37a63a25a153a107a154 a155 a13 a24a113a26 a156a58a157a19a64 a17 a5 a12 a13a16a158 a28 a9 a46 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 a132 a13 conditioned on a12 a13 and a28 a109a123a159 a115 . It is easily seen that, conditioned on a12 a13 and a28 a109a123a159 a115 , a132 a13 is complex Gaussian distributed [9, 14]. Using (4) and (5), its distribution is expressed as a132 a13 a158 a5 a12 a13 a41 a28 a109a123a159 a115 a9a141a160 a161a163a162a65a5 a150a132 a13 a41 a150 a164a8a165a48a166 a9 a41 a43 a20a140a44a19a41a48a46a47a46a47a46a47a41 a4 a41 (6) with a150a132 a13 a20 a15a5a7a6a8a2a71a9 a148 a91 a34 a87 a88a90 a164a168a167a165a169a166a18 a114 a26 a139 a72 a28 a109a123a159 a115 a72 a12 a13 a41 a150 a164a8a165a48a166 a20 a164a8a165a48a166 a68a71a5a7a6a8a2a71a9 a15 a5a7a6a8a2a71a9 a148 a91 a34 a87 a88 a90 a164a168a167 a165a169a166a18 a114 a26 a164a8a165a169a166 a41 where a161 a162 a5a74a170 a41 a164 a9 denotes the complex Gaussian distributed ran- dom vector with mean a170 and variance a164 ; a164a8a165a48a166 denotes the covari- ance matrix of channel responses a132 a13 , and a164a171a167a165a48a166 denotes the pseudo inverse of a164a8a165a48a166 . According to the signal model in (2), a164a8a165a48a166 a50a20 a172 a5a74a132 a13 a132 a72a13 a9 a20 a61a56a62a16a63a65a64 a15a173 a88 a26 a32 a100 a41a47a46a48a46a47a46a76a41a16a173 a88 a26 a32 a118 a114 a26 a41a76a46a48a46a47a46a47a46a47a46a47a46a76a41a103a173 a88 a22 a32 a100 a41a47a46a47a46a47a46a76a41a52a173 a88 a22 a32 a118 a114 a26 a18 , where a173 a88a23 a32a119 is the average power of the a121 -th tap associated with the a86 -th transmitter antenna; a173 a88a23 a32a119 a20a82a67 if the channel response at this tap is zero. It is assumed that a164a8a165a48a166 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), a174a8a5 a28 a158 a28 a109a58a159 a115 a9 is computed as a174a175a5 a28 a158 a28 a109a123a159 a115 a9 a20 a176 a157a38a177a56a178a135a179 a46 a68 a44 a87 a88 a90 a155 a13 a24a113a26 a180 a149 a24a27a26 a128 a114 a26 a125 a24 a100 a181 a13 a15a17a35a41a47a85a47a18 a68a11a182 a72 a15a17a35a41a47a85a47a18a135a139a184a183a131 a5 a85 a9 a150a132 a13 a88 a34 a182 a72 a15a17a39a41a47a85a48a18 a150 a164a8a165a48a166 a5 a85 a9a74a182 a15a17a35a41a76a85a48a18 a185 a109a58a159 a115 a13 a5a74a182 a15a17a39a41a47a85a48a18 a9 a41 with a182 a15a17a35a41a47a85a47a18 a50a20 a15a66 a26 a15a17a35a41a47a85a47a18a52a41a48a46a47a46a47a46a47a41 a66 a22 a15a17a35a41a47a85a47a18a58a18a72 a57a14a54a56a79 a41 a139a134a183a131 a5 a85 a9 a50a20 a61a56a62a16a63a65a64 a129 a72a131 a5 a85 a9 a41a47a46a76a46a47a46a48a41a52a129 a72a131 a5 a85 a9 a57a14a54a56a55a123a57a42a142a47a59 a41 a150 a164a8a165a48a166 a5 a85 a9 a109 a13a58a186a16a32 a23 a186 a115 a50a20 a139 a150 a164a8a165a48a166 a139 a72 a109 a13a187a186 a114 a26a52a115 a128a60a188 a125 a32 a109 a23 a186 a114 a26a52a115a128a60a188 a125 a41 a43 a183 a20a140a44a99a41a48a46a76a46a47a46a47a41 a3 a41 a86 a183 a20a140a44a99a41a76a46a47a46a47a46a48a41 a3 a41 where a15a189a49a18 a109a13a186 a32a23 a186 a115 denotes the a5 a43 a183 a41 a86 a183 a9 -th element of matrix a189 . Next, based on (7), the M-step proceeds as follows a28 a109a123a159 a188 a26a52a115 a20 a128 a114 a26 a125 a24 a100 a63a25a151a135a64a190a152a191a62a123a177 a192 a182a168a193 a149a101a32 a125a25a194a123a195a52a196 a155 a13 a24a27a26 a180 a149 a24a27a26 a185 a109a123a159 a115 a13 a5a74a182 a15a17a35a41a47a85a47a18 a9 a46(7) It is seen from (7) that the M-step can be decoupled into a2 in- dependent minimization problems, and the coding constraints of STBC are taken into account when solving the M-step, i.e.,a182 a15a17a35a41a47a85a47a18a77a41 a197 a17a39a41 are different permutations and/or transformations of a182 a15a58a44a99a41a48a85a47a18 [13]. III-B. Initialization of the EM Algorithm The performance of the EM algorithm (and hence the overall re- ceiver) is closely related to the quality of the initial value of a28 a109a100 a115 , i.e., the initial value at the first EM iteration. The initial estimate of a28 a109a100 a115 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 cor- relation of the channel. Finally, based on the filtered channel esti- mate, a28 a109a100 a115 is obtained through the ML detection. In (6), by assuming the perfect knowledge of a164a8a165a48a166 , a150a132 a13 is sim- ply the minimum mean-square estimate (MMSE) of the channel responsea132 a13 . When a164a8a165a48a166 is not known to the receiver, a least-square estimator (LSE) can be applied to estimate the channel and to mea- sure a164 a165 a166 . The LSE a150a132 a13 is expressed as, a150a132 a13 a20 a44 a6a8a2 a139 a72 a180 a149 a24a27a26a103a28 a72 a15a17a38a18a111a12a80a13a75a15a17a38a18 a46 (8) It is seen that in (8), unlike a typical LSE, no matrix inversion is involved here. Hence, its complexity is significantly reduced from a198 a5a7a3a144a199a77a137a171a199a48a9 to only a198 a5a7a3a84a137a200a9 and the computation is numerically more stable, which is very attractive in systems using more trans- mitter antennas (largea3 ) and/or communicating in highly disper- sive fading channels (large a137 ). Moreover, following [11], such an estimator also achieves the minimum mean-square error (MSE), MSEa20 a118 a128 a87 a88 a90 , (when assuming the data decision a28 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 ma- trix inversion, a6 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 pro- posed 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 a6 ) pilot OFDM word at a17a49a20a144a67 . In the above analysis, we assume that a132 a13 a15a17a38a18 remain constant for a17a201a20a202a44a19a41a47a46a47a46a48a46a47a41 a6 , whereas the actual channel may vary across thesea6 OFDM slots. For the a5 a146a14a203a168a146 a9 STBC [13], by assuming that both the data a28 a15a17a19a18 and channel responses a96 a132 a13 a15a17a19a18 a116 a149 are random quantities, the average MSE of the channel estimate is a204a11a205a169a172 a20 a5a7a206a60a207a38a208 a136 a9 a88 a146 a34 a137 a2 a87 a88 a90 a41 (9) where a136 is the duration of one OFDM slot; a207a101a208 is the maximum Doppler frequency of the fading channel. The extra term a26 a88 a5a7a206a89a207a101a208 a136 a9 a88 in (9) reflects the average MSE due to the modeling mismatch. For a practical normalized Doppler frequency (e.g., a207a101a208 a136a82a20a130a67a35a46a67a39a44 ), the average MSE due to the modeling mismatch is negligible. Further- more, for a particular realization of a28 a15a17 a183a18 , the instantaneous MSE of the channel estimate is upper bounded by a204a11a205a101a172 a209 a5a7a206a89a207a101a208 a136 a9 a88 a137 a88 a34 a137 a2 a87 a88 a90 a46 (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 esti- mator which only estimates the significant taps of the channel re- sponse, namely the significant-tap-catching least-square estimator, is adopted here. As seen in (9) and (10), with the increase of maximum Doppler frequency a207a101a208 , the mismatch MSE of the channel estimate increases. To ameliorate this problem, as indicated in [11, 12], a temporal fil- ter 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 is listed in Table 1. In Table 1, the ML detection in (a210 ) takes into account of the STBC coding constraints of a28 . F-filter denotes the significant- tap-catching version of the least-square estimator, where a28 a15a67a38a18 rep- resents the pilot symbols and a28 a109a123a211 a115 a15a212a213a18a52a41a103a212a214a20a45a67a35a41a48a46a47a46a47a46a76a41 a185 a68 a44a80a41 rep- resents hard-decisions of the data symbols a28 a15a212a213a18 which is pro- vided by the EM algorithm after a total of a215 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., (a6 a185 a34a130a44 ) OFDM slots], as in [11, 12]. IV. TURBO RECEIVER In practice, in order to further exploit the frequency-selective fad- ing diversity embedded across alla2 OFDM subcarriers, it is com- mon 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 al- gorithm for this concatenated STBC-OFDM system, as depicted 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 a6a49a5 a28 a9 , which represents the a priori probability of a28 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 STBC- OFDM systems, with or without outer channel coding. The char- acteristics of the fading channels are described in Section II; specif- ically, the receiver performance is simulated in three typical chan- 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- width is 800 KHz and is divided into a44a19a146a38a216 subcarriers. These cor- respond to a subcarrier symbol rate of 5 KHz and OFDM word duration of a44a19a217a65a67a99a218 s. In each OFDM word, a guard interval ofa219 a67a19a218 s is inserted, hence the duration of one OFDM word a136a220a20a221a146a65a67a19a67a19a218 s. For all simulations, two transmitter antennas and two receiver an- tennas are used; and thea222 a26 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, a223 a44a19a146 infor- mation bits are transmitted from a44a19a146a19a216 subcarriers during two (a6 a20 a146 ) 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 EM- based ML receiver as derived in Section III is adopted; further- more, as in [12], the 7-tap significant-tap-catching scheme is ap- plied 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 im- proved through the EM iterations. Furthermore, although the re- ceiver 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 per- formance 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 iter- ations 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 simu- lations) between the MAP STBC decoder and the MAP convolu- tional decoder. From the simulation results, it is seen that by em- ploying an outer channel code, the receiver performance is signif- icantly improved (at the expense of lowering spectral efficiency). Moreover, without CSI, after 4-5 Turbo iterations, the Turbo re- ceiver 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 cod- ing (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 sys- tems 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- ing and modulation,” in Thirty-Third Asilomar Conference on Signals, Systems a224 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 communi- cations,” IEEE J. Select. Areas Commun., vol. 16, pp. 1451–1458, Oct. 1998. [14] H. V. Poor, An Introduction to Signal Detection and Estimation, Springer- Verlag, 2nd edition, 1994. IFFT IFFT.. . .. . .. . .. . Modulator MPSK STBC EncoderBits 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. a225 a226 a13a52a227a228a48a229 = F-filter a230 a13 a227a228a48a229a16a231a33a232a130a227a228a48a229 a231 a233a42a234a144a235a169a231a25a236a65a236a25a236a25a231a7a237a238a231 for a239 a234a240a228a70a231a135a235a169a231a25a236a25a236a25a236a65a231a7a241a243a242a11a235 for a244 a234a78a235a169a231a111a245a70a231a92a236a65a236a25a236a65a231a33a246 a247 a226 a13a16a227 a239 a246a71a248 a244 a229 = T-filter a225 a226 a13a52a227 a239 a246a71a248 a244 a242a37a235a52a229a16a231 a225 a226 a13a52a227 a239 a246a71a248 a244 a242a249a245a48a229a16a231a25a236a65a236a25a236a65a231 a225 a226 a227 a239 a246a71a248 a244 a242a249a250a7a229 a231a251a233a113a234a144a235a169a231a92a236a65a236a25a236a65a231a74a237a238a231 end a232 a109a100 a115 a227 a239 a229 = arg maxa154 a155 a13 a24a113a26 a180 a252 a186 a24a27a26 a253a187a254a48a255a1a0 a230 a13 a227 a239 a246a71a248 a244 a183 a229 a232 a231 a247 a226 a13a16a227 a239 a246a130a248 a244 a183 a229 a231 a2a4a3a6a5 a232 a109a123a211 a115 a227 a239 a229 = EM a230 a13 a227 a239 a229 a13 a231a75a232 a109a100 a115 a227 a239 a229 a231 [cf. EM Algorithm] for a244 a234a78a235a169a231a111a245a70a231a92a236a65a236a25a236a65a231a33a246 a225 a226 a13a16a227 a239 a246a71a248 a244 a229 = F-filter a230 a13 a227 a239 a229a16a231a33a232 a109a123a211 a115 a227 a239 a229 a231a74a233a113a234a144a235a169a231a92a236a65a236a65a236a25a231a74a237a238a231a7a2a4a3a7a3a6a5 end end Table 1: Procedure for computing a28 a109a100 a115 for the EM algorithm. .. . .. . .. . .. . IFFT IFFT Encoder Π Modulator MPSK STBC Encoder FFT FFT Decoder (0)X MAP Channel (p=0)oEM Alg. Initial. Pilot(p=0)o Decoder Channel λ λMAP-EM STBC 1 e e 2 Π Π -1 Figure 2: Transmitter and receiver structure for an STBC-OFDM system with outer channel code. a8 denotes the interleaver and a8 a114 a26 denotes the corresponding deinterleaver. 0 2 4 6 8 10 12 14 1610 ?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 a207a101a208 a20 a223 a67 Hz and a207a101a208 a20a140a146a25a67a38a67 Hz. 0 2 4 6 8 10 12 14 1610 ?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 fre- quencies a207a38a208 a20 a223 a67 Hz and a207a38a208 a20a140a146a25a67a19a67 Hz. 0 1 2 3 4 5 610 ?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 a207a38a208 a20 a223 a67 Hz and a207a38a208 a20a140a146a25a67a38a67 Hz. 0 1 2 3 4 5 610 ?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 frequen- cies a207 a208 a20 a223 a67 Hz and a207 a208 a20a140a146a25a67a19a67 Hz.