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.
STBC-OFDM Receiver Based on the MAP-EM Algorithm
Without CSI, the MAP detection problem can be written as
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
dj(k) is computed
at the output of the MAP-EM STBC decoder as follows:
Equation 10.62
where 
is the set of 
for which
the jth outer-channel-code bit is "+1," and 
is similarly defined; 
satisfy the STBC coding constraints.
The extrinsic a priori LLRs 
are
provided by the MAP outer-channel-code decoder at the previous turbo iteration.
Finally, the extrinsic a posteriori LLRs 
are sent to the MAP outer-channel-code decoder, which in turn computes the
extrinsic LLRs 
and then feeds them back to the MAP-EM STBC decoder, and thus
completes one turbo iteration. At the end of the last turbo iteration, hard
decisions of the information bits are output by the MAP outer-channel-code
decoder.
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.
Simulation Examples
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 fd =
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.
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