Blind MCMC Receiver for Coded OFDM with Frequency-Selective Fading and Frequency
Offset
In practical OFDM systems, the existence of frequency offset,
which is caused by the mismatch between the oscillator in the transmitter and
that in the receiver, destroys the orthogonality among OFDM subcarriers and
leads to a performance degradation [374]. Several schemes of frequency
offset estimation in OFDM systems have been investigated in [84, 89, 243, 295, 341, 440, 500, 507]. For OFDM applications over
additive Gaussian white noise (AWGN) channels, the maximum-likelihood (ML)
frequency offset estimates are derived in [89, 243, 295, 507]. Given that wireless channels
typically exhibit frequency-selective fading, these methods designed for AWGN
channels are not applicable in wireless OFDM systems. On the other hand,
frequency offset estimators in frequency-selective fading channels are developed
in [84, 341, 440], which require some particular
form of data redundancy (e.g., data repetition [341] or pilot insertion [84, 440]). In [500], a blind subspace method for
frequency offset estimation is proposed.
In wireless OFDM systems, in addition to the frequency offset,
the frequency-selective fading channel states are also unknown to the receiver.
The problem of channel estimation in OFDM systems has been studied in many
previous works. The methods proposed in [260, 506] estimate the fading channel
based on the pilot symbols, while blind estimation schemes based on second- or
high-order statistics are proposed in [109, 345, 614]. Moreover, in [205, 367], subcarrier phase estimators are
proposed by employing the expectation-maximization (EM) algorithm.
As an important remark, we note that the ultimate objective of
the receiver is to recover the information-bearing data symbols from the
received signals. Although the prevailing receiver-design paradigm is to
estimate the unknown parameters first and then to use these estimated parameters
in the detector, such an "estimate-then-plug-in" approach is ad hoc and bears no
theoretical optimality. In this section we treat the problem of blind receiver
design for coded OFDM systems in the presence of unknown frequency offset and
frequency-selective fading, under the Markov chain Monte Carlo (MCMC) framework
for Bayesian computation (cf. Chapter 8) and the principle of turbo
processing (cf. Chapter
6). The techniques in this section were developed in [290].
10.3.1 System Description
Channel Model with Frequency Offset
When there is a carrier frequency offset in the OFDM channel,
the received time-domain signal in (10.4) becomes [341]
Equation 10.11
where 
is the relative
frequency offset of the channel (the ratio of the actual frequency offset to the
intercarrier spacing). Note that for practical purposes, we assume that the
absolute value of the frequency offset is no larger than half of the OFDM
subcarrier spacing (i.e., || < 0.5). That
is, any large frequency offset has already been compensated (e.g., by an
automatic frequency control circuit [120]) and what remains is the
residual frequency offset. We next write the signal model (10.11) in matrix form. Denote
Note that W is the DFT
matrix and (1/Q)WH is
the inverse DFT matrix [i.e., WWH/Q=WHW/Q=IQ].
Hence H[i]=Wh[i] and N[i]=Wn[i].
Then upon applying a DFT to {ym[i]}m in (10.4), we obtain
the signal model
Equation 10.12
For a better understanding of the effect of the frequency
offset, we now take a closer look at the matrix 
in (10.12). Since || < 0.5, after some simple algebra, the (i,j)th element of the matrix 
can be
expressed as
with
|y(i,
j)| 
1, 
i, j and |y(i, j)| 
|y(i', j')| if |i - j| |i' - j'|.
Hence 
I when
0; and the spillover to off-diagonal
elements of , which corresponds to the ICI [341], increases as increases.
Sections
