INTRODUCTION
High datarate wireless access is demanded by many applications. Traditionally,
more bandwidth is required for higher datarate transmission. However,
due to spectral limitations, it is often impractical or sometimes very
expensive to increase bandwidth. In this case, using multiple transmit
and receive antennas for spectrally efficient transmission is an alternative
solution. Multiple transmit antennas can be used either to obtain transmit
diversity, or to form MultipleInput MultipleOutput (MIMO) channels.
The combination of MIMO signal processing with Orthogonal FrequencyDivision
Multiplexing (OFDM) has gained considerable interest in recent years (Stuber
et al., 2004; Li et al., 2002). MIMO offers extraordinary
throughput without additional power consumption or bandwidth expansion
(Bolckei et al., 2002) and OFDM introduces overlapping but orthogonal
narrowband subchannels to convert a frequency selective fading channel
into a nonfrequency selective one. Moreover, OFDM avoids InterSymbol
Interference (ISI) by means of Cyclic Prefix (CP) (Engels, 2002). Hence,
in the presence of frequency selectivity, it is beneficial to consider
MIMO in the OFDM context (Lu et al., 2002).
Like single antenna OFDM, MIMOOFDM is very sensitive to frequency synchronization
and channel estimation errors (Ma et al., 2005). Carrier Frequency
Offset (CFO) induced by the mismatches of local oscillators in transmitter
and receiver causes InterCarrier Interferences (ICI), which may result
in significant performance degradation. Several carrier frequency synchronization
schemes for MIMOOFDM systems are reported in the literature (Schenk and
van Zelst, 2003; Priotti, 2004; Sun et al., 2005). Moreover, the
coherent detection of MIMOOFDM signals requires channel estimation to
mitigate amplitude and phase distortions in a fading channel. Various
channel estimation algorithms are also proposed for MIMOOFDM systems
(Ma et al., 2005; Wang et al., 2005; Minn and AlDhahir,
2006). In dealing with channel estimation, most investigators assume zero
frequency offset between the carrier and the local reference at the receiver.
In practice, this means that the offset is so small that the demodulated
signal incurs only negligible phase rotations. Using stable oscillators
is not a viable route to meet such conditions for; in general, the stability
requirements would be too stringent. Furthermore, even ideal oscillators
would be inadequate in a mobile communication environment experiencing
significant Doppler shifts. The only solution is to measure the CFO accurately
(Morelli and Mengali, 2000). The combination of CFO and channel estimation
leads to particularly complex problems in MIMOOFDM systems due to the
number of unknowns (Ma et al., 2005).
More recently, joint channel and frequency offset estimation issue have
received a lot of attentions (Ma et al., 2003; Cui and Tellambura,
2004) in OFDM context. The exact MaximumLikelihood (ML) solutions of
both frequency offset and Channel Impulse Response (CIR) is prohibitively
complex. Therefore, in (Ma et al., 2003), the ML estimate for only
frequency offset was obtained based on the Least Square (LS) CIR estimate.
In (Cui and Tellambura, 2004), an adaptive approach (i.e., steepest descent
algorithm) was employed to avoid the complexity of joint ML estimation,
where firstly the channel estimation is performed assuming that the frequency
offset is known and then the frequency offset is estimated assuming the
channel state is known. However, all of these estimators (Ma et al.,
2003; Cui and Tellambura, 2004) are designed for SISOOFDM systems rather
than MIMO systems.
In this study, we present a new reducedcomplexity scheme for ML estimate
of both CFO and CIR in multi antenna OFDM transmission, assuming that
a training sequence is available. As we shall see, the solution consists
of two separate steps: a CFO estimator and a channel estimator. It is
known that the ExpectationMaximization (EM) algorithm (McLachlan and
Krishnan, 2000; Moon, 1996) can provide the ML solutions in an iterative
manner for ML estimation problems (Georghiades and Han, 1997; Feder and
Weinstein, 1998). Therefore, to overcome the difficulty of ML estimation
of CFO, we resort to the EM algorithm and propose a novel EMbased CFO
estimator (first step). The CFO estimates are then exploited in the second
step to estimate the MIMO channel coefficients. Moreover, to benchmark
the performance of the proposed scheme, the CramerRao Bounds (CRBs) are
derived for both CFO and CIR estimators.
SYSTEM MODEL
We consider a MIMOcoded OFDM communication system with K subcarriers,
N transmit and M receive antennas, signaling through frequencyselective
fading channels in the presence of frequency offset. The system model
is showed in Fig. 1. It is assumed that the fading channel processes are
slowly time variant, such that the fading coefficients are assumed to
remain constant during each OFDM word (one time slot) but it varies from
one OFDM word to another and the fading processes associated with different
transmitreceive antenna pairs are uncorrelated. (However, in a typical
OFDM system, for a particular transmitterreceiver antenna pair, the fading
processes are correlated in both frequency and time).
At the receiver, the signals are received from M receive antennas. Accordingly,
the MIMO OFDM system model subject to CFO is given as (Sun et al.,
2005).
where, the normalized frequency offset ε is presented in the matrix
F(ε) given by:

Fig. 1: 
MIMOOFDM system model 
and
Note that in Eq. 1 the additive white Gaussian noise
(AWGN) vector Z_{i}[p] contains independent zeromean complex
Gaussian random variables with power .
ML ESTIMATION OF CFO AND CIR
In practical OFDM applications, data transmission is organized in frames
and training blocks (carrying known symbols) are located at the beginning
of each frame. Our idea is to simultaneously make use of training symbols
for both CFO and CIR estimation. Therefore, in the sequel to this study,
we concentrate on a training block and omit the temporal index p for notational
simplicity.
The transmission model in (1) contains two unknown parameters: the CFO
ε and the channel parameters h_{i} (i = 1,..., M). The ML
estimates of ε and, h_{i} are given by minimizing the following
quadratic cost function:
Where:
Problem of Eq. 11 lies in the fact that we have to
estimate two parameters with only one cost function. We propose at first
to estimate the parameter ε. The ML estimation of ε leads to
the following mathematical development:
It is seen in Eq. 13 that the direct computation
of the optimal ML detection involves multipledimensional integral over
the unknown random vector h_{i} and hence, is of prohibitive complexity.
Instead of direct computation of Eq. 13, EMtype algorithms
provide an iterative and more easily implementable solution. The basic
idea of the EM algorithm is to solve problem Eq. 13
iteratively according to the following two steps
Step 1: Expectation (E)step: Compute
Step 2: Maximization (M)step: Solve
where, ε^{(k)} denotes the estimated CFO value at the kth
EM iteration. 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 (Poor, 1994).
In the Estep, the expectation is taken with respect to the hidden channel
response h_{i} conditioned on y_{i} and ε^{(k)}.
It is easily seen that, conditioned on y_{i} and ε^{(k)},
h_{i }is complex Gaussian distributed as (Stoica and Besson, 2003)
with
where, Σ_{z1} and Σ_{hi} denote respectively
the covariance matrix of the ambient white Gaussian noise z_{i}
and channel responses h_{i}. According to the above assumptions,
both of them are diagonal matrices as
where, is
the average power of the lth tap between the jth transmit and
ith receive antennas; if
the channel response at this tap is zero. Assuming Σ_{hi}
is known,
is defined as the pseudo inverse of Σ_{hi} as
As shown in Fig. 2, the diagonal elements of V^{H}Y
are equal to K/N. K/N is much larger than , which is inversely proportional
to the signaltonoise ratio of the fading channel. Therefore, we can
simplify Eq. 17 and 18 to
Hence, Eq. 14 reduces to
The second term is independent of ε and hence has no contribution
to the detector. Developing (24) and dropping terms irrelevant of ε,
we obtain:

Fig. 2: 
Evaluation of matrix products V^{H}V (K = 128,
N = 2) 
By denoting ri(n) = y*_{i}(n). M_{i}(n), the final expression
for (k+1)th estimate of ε is obtained as follows:
Where:
can be easily computed using the FFT algorithm. Better frequency resolution
can be achieved by zeropadding to effectively increase the length of
the symbol sequence.
When has
been obtained, it is straightforward, using (23), to estimate h_{i}.
The LS solution of (23) is also the ML channel estimate. The ML solution
can be determined by:
CRB ANALYSIS
To benchmark the performance of our estimators, we evaluate the CRBs
for the estimation of CFO and CIR. Remember that for each received antenna
we have: y_{i} = F(ε) V h_{i}+z_{i} (i =
1, ..., M). Stacking all the different vectors y_{i} in column
to form the vector y and doing the same thing with vectors h_{i}
to obtain h, we add together the contributions of all received antennas:
with
denoting the Kronecker product. Let η = [εh_{R} h_{l}]^{T}
denote the parameter vector of interest where h_{R} and h_{I}
stand for the real and imaginary parts of h, respectively. Under all the
made assumptions, the received signal y is a complexvalued circularly
symmetric Gaussian vector with mean μ = (1_{M}
F(ε)V) and covariance matrix For
this type of problem, the Fisher Information Matrix (FIM) for estimation
of is
block diagonal i.e., the estimation of is
decoupled from that of η. Therefore, we only consider the FIM for
η, which we denote by F. The latter is given by (Stoica and Besson,
2003):
The CRB is obtained as the inverse of the FIM F. Using the same principle
as those given in (Stoica and Besson, 2003), we obtain the final expression
for CRB as:
with the following definitions:
For the derivation of CRB for channel parameter estimation, we obtain
(Stoica and Besson, 2003):
Where:
RESULTS AND DISCUSSION The characteristics of the fading
channels, specifically; the system performance is simulated in Typical
Urban (TU) channel with six equalpower taps. In the following simulations,
the available bandwidth is 1 MHz and is divided into 128 subcarriers.
These correspond to a subcarrier symbol rate of 7.8 KHz and OFDM word
duration of 128 μsec . In each OFDM word, a cyclic prefix interval
of 32 μsec is added to combat the effect of ISI, hence, the duration
of one OFDM word 160 μsec . For all simulations, two transmitter
antennas and two receiver antennas are used. The information symbols are
drawn from a Quaternary PhaseShift Keying (QPSK) constellation. The simulated
system transmits data in a burst manner. Each data burst includes 10 OFDM
blocks. A preamble is applied at the beginning of each data burst for
synchronization purposes. MIMO channel estimates are also drawn from the
preamble. The performance of CFO estimator is evaluated with the meansquare
error (MSE) of the estimated frequency offset. In the case of channel
estimation parameters, is
plotted.
Example 1 (performance of CFO estimator): First, we test the effect
of the number of EM iterations on CFO estimation. The CFO is randomly
selected in the range [0.1, 0.1]. To start the iteration of proposed
algorithm, we set the initial estimate ε^{(0)} = 0. In Fig. 3, the MSE performance of the CFO estimator versus the number of EM iterations
is depicted. As it can be seen the MSE performance of the proposed estimator
converges after three iterations at the SNR of 5 dB, while it is improved
continually until the number of iterations increases to eleven for the
SNR values greater than 20 dB.
In Fig. 4, we depict the MSE performance versus SNR. The curves denoted
by EM Iter#4 and EM Iter#11 show the CFO estimator performance after the
fourth and eleventh EM iteration, respectively. The CRB derived in (30)
is also shown as a bench mark. As indicated in (30), CRB values are varying
according to the channel state (Morelli and Mengali, 2000) and for a given
SNR, the solid lines labeled Min/Max CRB indicate the minimum and maximum
CRB obtained in 10^{5} simulation runs. However, in high SNR above
20 dB, the proposed scheme needs eleventh iterations to maintain the improved
performance between CRBs in all ranges of SNR. From Fig. 4, it is also
deduced that proposed frequency estimator outperforms the one in (Sun
et al., 2005) in all ranges of SNR.
Example 2 (performance of channel estimator): In this example,
we test the performance of MIMO channel estimation with (N,M) = (2,2)
and CFO being also randomly selected in the range [0.1, 0.1].

Fig. 3: 
MSE performance of the CFO estimator versus number of EM iterations
for a 2x2 MIMOOFDM communication system 

Fig. 4: 
MSE of CFO estimation versus SNR for a 2x2 MIMOOFDM communication
system 
In Fig. 5, the simulated MSE performances of channel estimator in (27)
are presented and compared with the CRB in (33) and ideal case, where
the CFO is perfectly known. It is seen that the channel estimator of the
proposed algorithm shows almost ideal performance in all ranges of SNR.
To study the disturbance effects of CFO on channel estimation performance,
we also plot the MSE of channel estimator without estimating/compensating
CFO, in curve denoted by Without CFO E/C. It is not surprising to see
the significant performance loss due to CFO.

Fig. 5: 
Channel estimation performance of a 2x2 MIMOOFDM communication
system 
CONCLUSIONS
The problem of estimating the frequency offset and channel coefficients
in multi antenna OFDM transmission was investigated in this study. The
frequency estimates were obtained using an ML approach and the EM algorithm
was employed to reduce the computational complexity of ML solution. The
CFO estimates were then exploited to estimate the MIMO channel responses.
The performance of our estimators was benchmarked with CRBs and investigated
by computer simulation. Simulation results show that the proposed algorithm
achieves almost ideal performance compared with the CRBs for both channel
and frequency offset estimations.
The possible directions for the future research include verifying the
performance of our frequency offset and channel estimators for different
spacetime techniques such as the Bell labs layered spacetime (BLAST)
(Foschini, 1996), use of powerful coding schemes such as the low density
parity check (LDPC) and the turbo codes (Berrou and Glavieux, 1996) and
finally, extending the algorithm to the multiuser scenarios.