Pilot-Symbol-Aided Turbo Receiver for Space-Time Block-Coded OFDM Systems
In Section 10.3 we have treated the problem of blind receiver design based on MCMC methods for OFDM systems. In this section we discuss the design of a pilot-symbol-aided receiver for OFDM communication systems operating over frequency-selective fading channels. Here we treat a general scenario where multiple transmit and receive antennas are employed. It is assumed that space-time block coding (STBC) (cf. Sections 5.5.2 and 6.7) is adopted at the transmitter end. The techniques in this section were developed in [292].
10.4.1 System Descriptions
We consider an STBC-OFDM system with Q subcarriers, N transmitter antennas, and M receiver antennas, signaling through frequency- and time-selective fading channels. As illustrated in Fig. 10.12, the information bits are first modulated by an MPSK modulator; then the modulated MPSK symbols are encoded by an STBC encoder. Each STBC code word consists of (PN) STBC symbols, which are transmitted from N transmitter antennas and across P consecutive OFDM slots at a particular OFDM subcarrier. The STBC code words at different OFDM subcarriers are independently encoded, therefore, during P OFDM slots, altogether Q STBC code words [or (QPN) STBC code symbols] are transmitted. It is assumed that the fading processes remain static during each OFDM word (one time slot), but it varies from one OFDM word to another, and that the fading processes associated with different transmitter–receiver antenna pairs are uncorrelated.
Figure 10.12. Transmitter and receiver structure for an STBC-OFDM system.
At the receiver, the signals are received from M receiver antennas. After matched filtering and symbol-rate sampling, the discrete Fourier transform (DFT) is then applied to the received discrete-time signals, to obtain
with
where Hi[p] is the NQ-vector containing the complex channel frequency
responses between the ith receiver antenna and
all N transmitter antennas at the pth OFDM slot, which is explained below; xj[p, k] is
the STBC symbol transmitted from the jth
transmitter antenna at the kth subcarrier and at
the pth OFDM slot; yi[p] is the Q-vector of received signals from the ith receiver antenna and at the pth time slot; zi[p] is the
ambient noise, which is circularly symmetric complex Gaussian with covariance
matrix 
. Here we restrict our attention to MPSK signal constellations
(i.e., 
).
Consider the channel response between the jth transmitter antenna and the ith receiver antenna. Following [396], the time-domain channel impulse response can be modeled as a tapped delay line, similar to (10.2), given by
where d(·) is the Kronecker delta
function; 
denotes the maximum number of resolvable taps, with tm being the
maximum multipath spread and Df being the tone spacing of the OFDM system;
ai,j(l; t) is the complex amplitude of the lth tap, whose delay is l/Df. For OFDM systems with proper cyclic extension
and sample timing, with tolerable leakage, the channel frequency response
between the jth transmitter antenna and the ith receiver antenna at the pth time slot and at the kth subcarrier can be expressed as [506]
where 
, T
is the duration of one OFDM slot; 
is the L-vector containing the time responses of all the taps;
and 
contains the corresponding DFT coefficients.
Using (10.36), the signal model in (10.34) can be further expressed as
with
As discussed in Section 6.7, an STBC is
defined by a P x N
code matrix 
, where N denotes the number of
transmitter antennas or the spatial transmitter diversity order, and P denotes the number of time slots for transmitting an
STBC code word, i.e., the temporal transmitter
diversity order. Each row of is a permuted and transformed (i.e.,
negated and/or conjugated) form of the N-dimensional vector of complex data symbols x. As a simple example, we consider a 2 x 2 STBC
(i.e., P = 2, N =
2). Its code matrix 
is defined by
The input to this STBC is the data vector x = [x1, x2]T. During the first time slot, the two symbols in
the first row [x1, x2] of are
transmitted simultaneously from the two transmitter antennas; during the second
time slot, the symbols in the second row 
of
are transmitted.
In an STBC-OFDM system, we apply the STBC encoder above to data
symbols transmitted at different subcarriers independently. For example, by
using the STBC defined by , at the kth subcarrier, during the first OFDM slot, two data
symbols [x1[1, k], x2[1,
k]] are transmitted simultaneously from two
transmitter antennas; during the next OFDM slot, symbols 
are
transmitted.
Simplified System Model
From the description above, it is seen that decoding in an STBC-OFDM system involves the received signals over P consecutive OFDM slots. To simplify the problem, we assume that the channel time responses hi[p], p = 1, ... , P, remain constant over the duration of one STBC code word (i.e., P consecutive OFDM slots). As will be seen, such an assumption simplifies the receiver design significantly. Using the channel model in (10.37) and considering the coding constraints of the STBC, the received signals over the duration of each STBC code word is obtained as
with
According to the definitions of W in (10.37) and X in (10.39), we have
where 
is an NQ x NQ matrix, which is
composed of N2 submatrices of
dimension Q x Q of
the form
where the last equality follows from the constant modulus property of the symbols {xj[p, k]}j, p, k, and the orthogonality property of the STBC [475] as well as that of the OFDM modulation. Hence, (10.40) reduces to
As will be seen in the following sections, (10.42) is the key equation in designing low-complexity iterative receivers for STBC-OFDM systems.