Pilot-Symbol-Aided Turbo Receiver

In practice, in order to impose the coding constraints across the various OFDM subcarriers and further improve receiver performance, an outer channel code (e.g., a convolutional code or turbo code) is usually applied in addition to the STBC. As illustrated in Fig. 10.14, the information bits are encoded by an outer-channel-code encoder and then interleaved. The interleaved code bits are modulated by an MPSK modulator. Finally, the modulated MPSK symbols are encoded by an STBC encoder and transmitted from N transmitter antennas across the P consecutive OFDM slots at a particular OFDM subcarrier. During P OFDM slots, altogether Q STBC code words [or (QPN) STBC symbols] are transmitted.

In what follows we discuss a turbo receiver employing the maximum a posteriori probability (MAP)-EM STBC decoding algorithm and the MAP outer-channel-code decoding algorithm for this concatenated STBC-OFDM system, as depicted in Fig. 10.14. It consists of a soft MAP-EM STBC decoder and a soft MAP outer-channel-code decoder. The MAP-EM STBC decoder takes as input the fast Fourier transform (FFT) of the received signals from M receiver antennas, and the interleaved extrinsic log-likelihood ratios of the outer-channel-code bits [cf. Eq. (10.62)] (which is fed back by the outer-channel-code decoder). It computes as output the extrinsic a posteriori LLRs of the outer-channel-code bits [cf. Eq. (10.62)]. The MAP outer-channel-code decoder takes as input the deinter-leaved LLRs of the outer-channel-code bits from the MAP-EM STBC decoder and computes as output the extrinsic LLRs of the outer-channel-code bits as well as the hard decisions of the information bits at the last turbo iteration.

Equation 10.58

The MAP-EM algorithm solves (10.58) iteratively according to the following two steps:

Equation 10.59

Equation 10.60

Comparing the MAP-EM algorithm in (10.59)–(10.60) with the maximum likelihood EM algorithm in (10.44)–(10.45), we see that the E-step is exactly the same, but the M-step of the MAP-EM algorithm includes an extra term P(X), which represents the a priori probability of X, which is fed back by the outer-channel-code decoder from the previous turbo iteration.

Similar to (10.54), the M-step for the MAP-EM can be written as

Equation 10.61

where the second equality in (10.61) holds by assuming that the outer-channel-code bits are ideally interleaved and hence x[p, k], at different OFDM subcarriers are independent. The last equality in (10.61) follows from the fact that x[p, k], p = 2, ... P, are uniquely determined by x[1, k] according to the STBC coding constraints. Note that when computing the M-step in (10.61), we consider only the coding constraints of the STBC; the coding constraints induced by the outer channel code are exploited by the MAP outer-channel-code decoder and the turbo processing.

Within each turbo iteration, the E-step and M-step above are iterated I times. At the end of the Ith EM iteration, the extrinsic a posteriori LLRs of the outer-channel-code bits are computed, and then fed to the MAP outer-channel-code decoder. Recall that only the STBC symbols in the first OFDM slot are obtained from the MPSK modulation of outer-channel-code bits; the STBC symbols transmitted during the remaining (P - 1) OFDM slots are simply permutations and/or transformations of the STBC symbols in the first OFDM slot, as defined in (10.38). At each OFDM subcarrier, N transmitter antennas transmit N STC symbols, which correspond to (N log₂|W|) outer-channel-code bits. Based on (10.61), after I EM iterations, the extrinsic a posteriori LLR of the jth (j = 1, ... , N log₂|W|) outer-channel-code bit at the kth subcarrier d^j(k) is computed at the output of the MAP-EM STBC decoder as follows:

Equation 10.62

The MAP-EM algorithm needs to be initialized at each turbo iteration. Except for the first turbo iteration, is simply taken as given by (10.61) from the previous turbo iteration. The procedure for computing at the first turbo iteration is similar to that described in Table 10.1.

In this section we provide computer simulation results to illustrate the performance of the proposed iterative receivers for STBC-OFDM systems, with or without outer channel coding. The receiver performance is simulated in three typical channel models with different delay profiles: the two-ray, typical urban (TU), and hilly terrain (HT) model with 50-Hz and 200-Hz Doppler frequencies [263]. In the following simulations the available bandwidth is 800 kHz and is divided into 128 subcarriers. These correspond to a subcarrier symbol rate of 5 kHz and OFDM word duration of 160 ms. In each OFDM word, a cyclic prefix interval of 40 ms is added to combat the effect of intersymbol interference, hence the duration of one OFDM word T = 200 ms. For all simulations, two transmitter antennas and two receiver antennas are used; and the STBC is adopted [see (10.38)]. The modulator uses a QPSK constellation. The OFDM system transmits in a burst manner as illustrated in Fig. 10.13. Each data burst includes 11 OFDM words (q = 5, P = 2), the first OFDM word contains the pilot symbols and the remaining 10 OFDM words span the duration of five STBC code words. Simulation results are shown in terms of the OFDM word-error rate versus the SNR.

Performance of the EM-Based ML Receiver In an STBC-OFDM system without outer channel coding, 512 information bits are transmitted from 128 subcarriers during two (P = 2) OFDM slots; therefore, the information rate is 2x160/200 = 1.6 bits/s per hertz, with 160/200 being the factor induced by the cyclic prefix interval. In Figs. 10.15–10.17, when ideal CSI is assumed to be available at the receiver side, the ML performance is shown in dashed lines, denoted by Ideal CSI. (Note that the ML performance difference between the 50-Hz and the 200-Hz Doppler fading channels is unnoticeable; hence, we only present the ML performance when f_d = 50 Hz.) Without the CSI, the EM-based ML receiver is employed. The performance after each EM iteration is demonstrated in curves denoted by EM Iter#1, EM Iter#2, and EM Iter#3. From the figures it is seen that the receiver performance is significantly improved through EM iteration. Furthermore, although the receiver is designed under the assumption that the fading channels remain static over one STBC code word (whereas the actual channels vary within 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 200 Hz.

In Fig. 10.18, the performance of an EM-based ML receiver employing the causal temporal filtering scheme (denoted by C-T) is compared with that employing the noncausal temporal filtering scheme (denoted by N-T) in two-ray fading channels. It is seen that applying a second-round noncausal temporal filtering in addition to the first-round causal temporal filtering [263] does not bring much performance improvement to the EM-based ML receiver considered here, which is also true for the TU and HT fading channels. Because in the proposed EM-based ML receiver the performance improvement is achieved mainly by the EM iterations, we conclude that only causal temporal filtering is needed for initializing the EM algorithm.

Performance of the MAP-EM-Based Turbo Receiver A four-state rate-½ convolutional code with generator (5,7) in octal notation is adopted as the outer channel code, as depicted in Fig. 10.14. The overall information rate for this system is 0.8 bits/s per hertz. Figures 10.19–10.21 show the performance of the turbo receiver employing the MAP-EM algorithm for this concatenated STBC-OFDM system. The performance of the turbo receiver after the first, third, and fifth turbo iteration is demonstrated, respectively, in curves denoted by Turbo lter#1, Turbo lter#3, and Turbo lter#5. 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 a sufficient number of turbo iterations (three to four iterations are shown to be enough for the systems simulated here) between the MAP STBC decoder and the MAP convolutional decoder. From the simulation results, it is seen that by employing outer channel coding, the receiver performance is significantly improved (at the expense of lowering spectral efficiency). Moreover, without CSI, after three to five 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 200 Hz.