ML Receiver Based on the EM Algorithm
We next consider ML receiver design for STBC-OFDM systems. With
ideal channel state information (CSI), the optimal decoder has been derived in
[476]. However, in
practice, CSI must be estimated by the receiver. We develop an EM-based ML
receiver for STBC-OFDM systems operating in unknown fast-fading channels. As in
a typical data communication scenario, communication is carried out in a bursty
manner. A data burst is illustrated in Fig.
10.13. It consists of (Pq + 1) OFDM words,
with the first OFDM word (p = 0) containing known
pilot symbols and the remaining (Pq) OFDM words
spanning the duration of q STBC code words.
EM-Based STBC-OFDM Receiver
Without CSI, the maximum likelihood detection problem is
written as
Equation 10.43
where the summation of log probabilities from all M receiver antennas follows from the assumption that
the ambient noise processes at different receiver antennas are independent. It
is seen in (10.43) that the direct
computation of optimal ML decisions involves multidimensional integration over
the unknown random vector hi, and hence is of prohibitive complexity. Next,
we turn to the EM algorithm to solve (10.43).
As discussed in Section 9.3.1, the basic
idea of the EM algorithm is to solve problem (10.43) iteratively according to the following two
steps:
Equation 10.44
Equation 10.45
where X(k) contains hard decisions on the data symbols
at the kth EM iteration
and X(k) satisfies the STBC coding constraints. It is
known that the likelihood function 
is nondecreasing as a function of
k, and under regularity
conditions the EM algorithm converges to a local stationary point [315].
In the E-step, the expectation is taken with respect to the
"hidden" channel response hi conditioned on yi and X(k). It is easily seen that conditioned on yi and X(k), hi has
a complex Gaussian distribution. Using (10.39) and (10.42),
this distribution can be expressed as
Equation 10.46
with
Equation 10.47
Equation 10.48
where Sz and 
denote, respectively, the covariance matrix of the ambient white Gaussian noise
zi and channel responses hi.
According to the assumptions made above, both of these are diagonal matrices,
given as
Equation 10.49
Equation 10.50
where 
is the average power of the lth tap associated with the jth transmitter antenna; = 0 if the
channel response at this tap is zero. Assuming that is known
(e.g., measured with the aid of pilot symbols), then
Equation 10.51
with
Equation 10.52
It is seen that in the E-step, due to the orthogonality
properties of the STBC and the OFDM modulation (10.42), no matrix inversion is involved. Therefore, the
computational complexity of the E-step is reduced from 
to 
and the computation is also numerically more stable. Using (10.39) and (10.46),
Q(X|X(k)) can be computed via
Equation 10.53
with
where tr(A) denotes the
trace of the matrix A, and [A](i',j') denotes the (i',j')th element of the matrix A.
Next, based on (10.53),
the M-step in (10.45) proceeds as
follows:
Equation 10.54
It is seen from (10.54)
that the M-step can be decoupled into Q
independent minimization problems, each of which can be solved by enumerating
over all possible x[p, k] 
WN, 
p; and the coding
constraints of STBC are taken into account when solving the M-step [i.e., x[p, k], p, are different
permutations and/or transformations of x[1,
k] as defined in (10.38)]. Hence the complexity of the M-step is 
;
and the total complexity of the EM algorithm is [
per EM
iteration.
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 X(0) [cf.
Eq. (10.44)]. The initial estimate of
X(0) is computed based on the
method proposed in [260, 263] by the following steps. First, a
linear estimator is used to estimate the channel with the aid of pilot symbols
or decision feedback of the data symbols. Second, the resulting channel estimate
is refined by a temporal filter to further exploit the time-domain correlation
of the channel. Finally, conditioned on the temporally filtered channel
estimate, X(0) is obtained through ML
detection. We next elaborate on the linear channel estimator as well as the
temporal filtering.
Least-Squares Channel
Estimator In (10.47), by assuming
perfect knowledge of , 
is simply
the minimum mean-square-error estimate of the channel response hi.
When is not known to the receiver, a least-squares estimator can be applied
to estimate the channel and to measure as well. We next derive the
least-squares channel estimator for STBC-OFDM systems. By treating hi as
an unknown vector without any prior information and using (10.39) and (10.42),
the least-squares estimate can be expressed as
Equation 10.55
with
It is seen that in (10.55), unlike a typical least-squares estimator, no
matrix inversion is involved here. Hence, its complexity is reduced from 
to only 
and the computation is numerically more stable, which is very
attractive in systems using many transmitter antennas (large N) and/or communicating in highly dispersive fading
channels (large L).
Finally, the procedure for initializing the EM algorithm is
listed in Table 10.1. Here, the ML
detection in (*) takes into account the STBC coding constraints of X.
Freq-filter denotes the least-squares estimator, where X[0] represents the pilot symbols and X(I)[m], m = 0, ..., q – 1,
represent hard decisions of the data symbols X[m] which
are provided by the EM algorithm after a total of I EM iterations. Temp-filter denotes the
temporal filter [260, 263], which is used to further
exploit the time-domain correlation of the channel within one OFDM data burst
[i.e., (Pq + 1) OFDM slots]:
Equation 10.56
where 
, is computed from (**); 
denotes the coefficients of an I-length (I 
Pq) temporal filter, which can be precomputed by
solving the Wiener equation or from robust design as in [260, 263]. Furthermore, as suggested in
[263], after
receiving all (Pq + 1) OFDM words in a burst, an
enhanced temporal filter can be applied as
Equation 10.57
Table 10.1. Procedure for Computing X(0) for the EM Algorthim
|
|
|
|
where Temp-filterp computes 
by
temporally filtering the "past" channel estimate 
, the
"current" channel estimate 
and the "future" channel estimate
. From the discussions above, it is seen that the computation involved
in initializing X(0) consists mainly of the ML
detection of X(0) in (*) and the estimation
of 
in (**), with a total complexity 
.
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