EEE TRANSACTIONS ON COMMUNICATIONS. VOL- 50. NO. 1 JANUARY 2002 XpFEX1P1. x2pllKx(2k XwHXOHXOWEH X, diag=[, 0], a Pp,]. p, K-1JKXK where e and E denote respectively the covariance ma. 1.2 trix of the ambient white Gaussian noise z and channel W三 diaglWf, wrI2x)×(x2) ns in Section Il both of them are diagonal matrices as Z=E(zz)=Iand W三四r(O),wf(1)…,f(K-1) E(M)=dig21,,on3x…, hp FMi(p), hi 2 (p)2a/>x (17,, is the average power of the lth tap related with the jth where yp] and alp] are K-sized vectors which contain respec- is zero. Assuming that Eh is known(or measured with the aid K subcarriers and at the pth time slot; the diagonal elements of defined as the pseudo inverse of >h as/'23 72,L」1s xlp] are the K STC symbols transmitted from the jth trans- 2t≠0 mitter antenna and at the pth time slot. 7,t Lf,j=1,2.(2) Without CSI, the maximum a posteriori(MAP) detection problem is written as Using(17)and(21),Q(XXO)is computed as shown in(23) XTp]=argmax logp(XpllyIp)), p=1, 2, .., Pq.(18)at the bottom of the next page, where [WEnw Iaij) denotes (Recall that X[o] contains pilot symbols. The optimal solution Next, based on(23), the M-step proceeds as follows to(18)is of prohibitive complexity. We next propose to use the expectation-maximization(EM)algorithm 31] to solve(18) ri+ y rargmax e (xIx)+log P(X) The basic idea of the MAP-EM algorithm is to solve(18) teratively according to the following two steps(for notational -argan x[) og P(EkD convenience, we temporarily drop the time index p, with the 人=0 understanding that the MAP-EM algorithm discussed below is applied to each OFDM word in the data burst) K-1 argmin dalk)-log P(akI 1. E-step: Compute Q(xIxo) k=0 =E[ogp(yX, m)lv,x(Oy (19 or 2(+ [v =arg zin a (e= k-J)-log P(=k 2. M-step: Solve X(i+-1 augmaxQ(xx)+logP(x)(20) where(24)follows from the assumption that X contains inde- pendent symbols. It is seen from(25)that the M-step can be where xo denotes hard decisions of the data symbols at the ith decoupled into K independent minimization problems, each of EM iteration and P(X represents the a priori probability of x which can be solved by enumeration over all possible T E Q2x Q which is fed back by the LDPC decoder from the previous turbo total complexity of the maximization step is O(KQ2).Note that, unlike in [19], here the maximization in the M-ster is nondecreasing and under regularity conditions the EM algo- ried out without taking the LDPC coding constraints into con- rithm converges to a local stationary point [321 iderations, i.e., the symbols in X are treated as uncoded sym In the E-step, the expectation is taken with respect to the bols. The LDPC coding structure is exploited by the turbo iter- hidden"channel response h conditioned on y and X It ation as well as the ldpc decoder is easily seen that, conditioned on y and x h is complex Within each turbo iteration, the above E-step and M-step are iterated I times. At the end of the Ith EM iteration, the ex h(y xo)Ne(h, En) trinsic a posteriori LLRs of the LDPC code bits are computed and then fed to the soft ldpc decoder. at each ofdm sub with h(wxoHz-xOw+Eh) carrier, two transmitter antennas transmit two STC symbols, which correspond to(2 log2 2)LDPC code bits Based on(25 ), after I EM iterations, the extrinsic a posteriori llr of the jth WHxOHxow+Eh)WHxaHy, d() is computed at the output of the map-em demodulator L,eEh-wHXOHx-Ixow+Er) as follows: WHx()z-X(wΣ =kg()=+-P((82 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 1, JANUARY 2002 with (17) where and are -sized vectors which contain respec￾tively the received signals and the ambient Gaussian noise at all subcarriers and at the th time slot; the diagonal elements of are the STC symbols transmitted from the th trans￾mitter antenna and at the th time slot. Without CSI, the maximum a posteriori (MAP) detection problem is written as 1 2 (18) (Recall that 0 contains pilot symbols.) The optimal solution to (18) is of prohibitive complexity. We next propose to use the expectation-maximization (EM) algorithm [31] to solve (18). The basic idea of the MAP-EM algorithm is to solve (18) iteratively according to the following two steps (for notational convenience, we temporarily drop the time index , with the understanding that the MAP-EM algorithm discussed below is applied to each OFDM word in the data burst): E-step: Compute (19) M-step: Solve (20) where denotes hard decisions of the data symbols at the th EM iteration and represents the a priori probability of , which is fed back by the LDPC decoder from the previous turbo iteration. It is known that the likelihood function is nondecreasing and under regularity conditions the EM algo￾rithm converges to a local stationary point [32]. In the E-step, the expectation is taken with respect to the “hidden” channel response conditioned on and . It is easily seen that, conditioned on and , is complex Gaussian distributed as with (21) where and denote respectively the covariance ma￾trix of the ambient white Gaussian noise and channel responses . According to the assumptions in Section II, both of them are diagonal matrices as and , where is the average power of the th tap related with the th transmitter antenna; 0 if the channel response at this tap is zero. Assuming that is known (or measured with the aid of pilot symbols), is defined as the pseudo inverse of as 1 0 0 0 1 1 2 (22) Using (17) and (21), is computed as shown in (23), at the bottom of the next page, where denotes the ( )th element of the matrix . Next, based on (23), the M-step proceeds as follows: (24) or (25) where (24) follows from the assumption that contains inde￾pendent symbols. It is seen from (25) that the M-step can be decoupled into independent minimization problems, each of which can be solved by enumeration over all possible (recall that denotes the set of all STC symbols). Hence, the total complexity of the maximization step is . Note that, unlike in [19], here the maximization in the M-step is car￾ried out without taking the LDPC coding constraints into con￾siderations, i.e., the symbols in are treated as uncoded sym￾bols. The LDPC coding structure is exploited by the turbo iter￾ation as well as the LDPC decoder. Within each turbo iteration, the above E-step and M-step are iterated times. At the end of the th EM iteration, the ex￾trinsic a posteriori LLRs of the LDPC code bits are computed and then fed to the soft LDPC decoder. At each OFDM sub￾carrier, two transmitter antennas transmit two STC symbols, which correspond to (2 ) LDPC code bits. Based on (25), after EM iterations, the extrinsic a posteriori LLR of the th ( 1 2 ) LDPC code bit at the th subcarrier is computed at the output of the MAP-EM demodulator as follows: