INTRODUCTION
The wireless communication system (Hongwei, 2005) coupled with multiple transmit/receive antennas (Nanda et al., 2005) and Orthogonal FrequencyDivision Multiplexing (OFDM) (Zelst and Schenk, 2004), is regarded as a promising solution for enhancing the data rates of nextgeneration wireless communication systems operating in frequencyselective fading environments. Channel parameters provide key information for the operation of wireless systems and need to be estimated accurately. So, many trainingbased MIMO OFDM channel estimation methods have been widely studied (Ogawa et al., 2004; Li and Wang, 2003; Stuber et al., 2004; Shenghao and Yuping, 2004), which could be put into two categories such as frequency domain (Stuber et al., 2004; Shenghao and Yuping, 2004) and time domain (Ogawa et al., 2004; Li and Wang, 2003) approaches. Nevertheless, for the scenarios with large numbers of users in cellular fast fading channels, there are two difficult problems (Barhumi et al., 2002; Jagannatham and Rao, 2006) that have to be figured out, i.e., the challenge to construct large numbers of orthogonal training sequences and the bandwidth overhead of channel estimation when the length of MIMO OFDM symbols is larger than that of wireless channel delay. So, it’s necessary to find a new time approach to overcome these drawbacks.
In fact, spatial channels for every receive antenna in MIMO links can be considered as MultipleInput SingleOut (MISO) channels, i.e., the equivalents of links in multiuser CDMA uplinks. So, we generalize the Steiner channel estimation in up CDMA wireless links (Steiner and Jung, 1994; Steiner and Baier, 1993) to estimate MIMO OFDM spatial channels. Training sequences of different transmit antennas, can be simply obtained by truncating the circular extension of one basic training sequence and the pilot matrix assembled by these training sequences is one circular matrix with good reversibility. Subsequently, high dimensional matrix inversion can be avoided via diagonalization of pilot matrix by unitary DFT matrices (Proakis, 1995). Furthermore, when the length of channel profiles is less than that of MIMO OFDM symbols, more bandwidth resources can be saved, as the training sequence only occupies a part of one MIMO OFDM symbol. Channel information is firstly estimated in time domain and then its frequency version at different subcarriers is obtained by its fourier transformation.
SYSTEM MODEL
For a MIMOOFDM system with M_{t} transmits and M_{r }receive
antennas, one OFDM modulator is employed on each transmit antenna. Firstly,
a data stream is divided into N parallel substreams, typical for any multicarrier
modulation scheme. Then, the kth substream of the nth symbol block transmitted
from the vth antenna, is denoted by .
An inverse DFT with N points is performed on each block and a guard interval
(GI) with N_{GI} samples is inserted in form of cyclic prefix. Subsequently,
these data are transmitted over spatial multipath fading channels. At the uth
receiver, the guard interval is removed from the received data symbols and then
a DFT is followed to obtain the estimation of transmitted data symbols. The
received data consists of superimposed data from M_{t} transmit antennas.
We assume carriers to be kept orthogonal via cyclic prefix and channels to be
constant over one OFDM symbol. Then, the received signal at the kth subchannel
of the vth receive antenna for the nth OFDM symbol is given by:
Where, ,
respectively denote the transmitted symbols at the kth carrier of the uth
transmit antenna for the nth OFDM symbol, the channel fading coefficient at
the kth carrier of the spatial channel between the vth receive antenna and
the uth transmit antenna and the Additive White Gaussian Noise (AWGN) with
zero mean and variance σ^{2}.
We consider a timevariant, frequency selective, Rayleigh fading channel, modeled
by a tapped delay line with L_{h} nonzero taps (Proakis, 1995). The
frequency fading coefficients for different carriers could be obtained via the
Fourier transform of Channel Impulse Response (CIR) for different transmit/receive
antenna pairs. So, the coefficient
is described by:
Where, 1/T is subcarrier spacing, channel gain
of the lth tap at time delay τ_{1}^{(v,u)} is a Wide Sense
Stationary (WSS), complex Gaussian random variable with zero mean. The channel
taps of the pairs between transmit and receive antennas, are assumed to be mutually
uncorrelated. Due to the motion of vehicles,
will be timevariant and bandlimited according to the maximum Doppler frequency
v_{max}.
DESIGN TRAINING SEQUENCES
Denote L_{m} and W as the length of training sequence and spatial CIRs and L = W*M_{t} is the length of one basic sequence, which can be delineated as:
Its circular extension version is given by:
Where, the first L elements are consistent with corresponding elements of the basic sequence and other elements are determined by:
Then, training sequences for different transmit antennas are obtained by truncating the circular extended version as showed in Fig. 1. Furthermore, the pilot for the uth transmit antenna is presented as
Where:
If the design parameters showed above can be denoted as a quaternion (L_{m}, L, M_{t}, W), there exists the following relationship among these parameters, i.e.,
Where, Operator
denotes the largest integer not more than a given real number in the operator.

Fig. 1: 
Construction of training sequences from a circular extended
sequence, where the training sequence for every transmit antenna is obtained
via truncating a circular extended sequence according to its position in
the circular extended sequence 
However, the training sequences given by Eq. 6 are generally
described as binary sequences and should be converted into complex number. Firstly,
they are represented as bipolar sequences and then further converted into
complex numbers. Denote
as one bipolar sequence,
as its correspondent complex form, which can be determined by:
Where:
j 
= 
Unit of imaginary number. 
STEINER CHANNEL ESTIMATION
For the considering MIMO OFDM systems, one receive antenna must estimate all the spatial channels between the receive antenna and all the transmit antennas, simultaneously. Then, the channel information estimated by all receive antennas will be assembled together to coherently decode data symbols carried by one MIMO OFDM symbol. Therefore, the following algorithm will be described for one receive antenna, the same counterparts can be easily applied to all the receive antennas.
Provided that all spatial channels for MIMO radio links have the same delay profiles, the channel impulse response between the uth transmit antenna and the receive antenna, is described as:
Furthermore, training sequence transmitted by the uth transmit antenna is given by:
which is obtained by truncating the first L+W1 elements from designed training sequence as showed above. After spatial channel filtering, its received version is given by:
Where,
and
denote received training symbols and correspondent noise vector, respectively
and
is the pilot symbol matrix assembled by training sequence from the uth transmit
antenna, which can be delineated as:
According to linear convolution theory (Proakis, 1995), the received training symbols are the polluted version of training symbols transmitted by the uth transmit antenna. Hence, their first W1 elements and last W1 elements will be discarded away as they are interfered by the transmit signals before and after pilot sequence, respectively. So, the usable received data is given by:
At the same time, let ,
we could obtain following relationship based on Eq. 12
Where, G^{(u)} is one submatrix consisting of all the rows between
the Wth row and (W+L1)th row of pilot matrix ,
i.e.,
In fact, the received data at one receive antennas is the superimposed data from all transmit antennas, which can be described as:
which could be further expressed in matrix form according to Eq.
15, i.e.,
Simultaneously, other terms in Eq. 18 expression are given
by:
Assume the noise vector in Eq. 18 is a WideSense Stationary
(WSS) complex Gaussian vector with zero means and covariance matrix given by
R_{n} = σ^{2}I, where σ^{2} is noise variance
and I is norder unit matrix. Then, according to Eq. 18, the
spatial channels between all transmit antennas and the receive antenna, could
be estimated by:
If G is invertible, the above expression is further rewritten as:
Moreover, according to (16) and (19), G is actually a Lorder circular matrix,
which could be diagonalized by an unitary DFT matrix (Proakis, 1995), that is:
Where:
F 
= 
A Lorder DFT matrix, 
Δ 
= 
A Lorder diagonal matrix whose elements are given by the DFT of the basic
training sequence. 
Subsequently, substitute Eq. 22 into 20,
then we will get:
Where, operator F(·) and F^{H}(·) could be explained
as performing DFT and Inverse Discrete Fourier Transformation (IDFT) to one
vector, respectively. So, we can rewritten Eq. 23 into an
alternative form as:
Where, operator dft(·) and idft(·) denote to perform DFT and
IDFT on a vector, respectively, while (·)./(·) denotes array right
division operator in elementwise. Note that
is the reverse version of basic sequence m, the result spatial channel are also
estimated in the reverse order.
BANDWIDTH OVERHEAD AND COMPLEXITY
Here, we use the time intervals when channel estimations are conducted as bandwidth overhead of channel estimation, which are quantitated into sample periods in corresponding MIMO OFDM system configurations and the number of multiplication as metric of complexity for channel estimation. The bandwidth overhead and complexity for different channel estimation approaches, are analyzed and compared with each other in the followings.
For the frequency approaches (Stuber et al., 2004; Shenghao and Yuping, 2004), pilot sequences transmitted by every transmit antenna are designed to be orthogonal to each other and carried by at least M_{t} OFDM symbols and the corresponding pilot patterns can be showed in Fig. 2. Following these schemes, M_{t} times NIFFT for OFDM modulation, M_{t} times NFFT for OFDM demodulation and M_{t}*N times division are needed to estimate all the spatial channels between all the transmit antennas and the receive antenna. However, M_{t}*L samples must be observed at one receive antenna to estimate the M_{t} spatial channel impulse responses in the time approaches (Ogawa et al., 2004; Li and Wang, 2003). The training sequences transmitted at different antennas only occupy M_{t}*L samples. Firstly, one M_{t} by L matrix inversion is involved to estimate all the CIRs of M_{t} spatial channels and then these spatial CIRs must be further converted into frequency domain via M_{t} times NFFT transforms to obtain corresponding fading coefficients at different carriers. As a result, M_{t} times NFFT and a M_{t}*L dimension matrix inversion calculation are used to obtain these MIMO OFDM channel coefficients. Without matrix inversion, it will take a M_{t}*L point IFFT, two M_{t}*L point FFT and M_{t} times NFFT for the Steiner scheme to obtain all MIMO OFDM subchannel coefficients with the same bandwidth overhead as the time approaches.
Their bandwidth overhead and complexity are delineated in Table
1, where the generalized Steiner approach is indicated to have smaller bandwidth
overhead and complexity when compared with frequency and time approaches, respectively.
Table 1: 
Bandwidth overhead and complexity for Steiner scheme, classical
frequency and time approaches, respectively 


Fig. 2: 
Two typical pilot patterns for channel estimation in MIMO
OFDM systems 
Thus, the scheme can make good tradeoff between classical frequency approaches
and classical time approaches with respect to complexity and bandwidth overhead.
NUMERICAL RESULTS
In order to evaluate the performance of the proposed Steiner channel estimation scheme, we consider a MIMO OFDM system with 2048 carriers at carrier frequency of 2.4 GHz, which has 20 MHZ bandwidth and a 1/4 OFDM symbols as guard intervals, which can eliminate intersymbol interference (ISI) caused by frequency selective channels. When the transceivers are equipped with 8x2 antennas as Base Stations (BS) and Mobile Stations (MS) respectively and system sample period is given as 48.828125ns, we figure out the corresponding bandwidth occupancy for the classical time and frequency channel estimation approaches under the case of the typical cellular wireless channel profiles with different maximal delay.
Following the design of TDSCDMA systems (Kan Zheng et al., 2005), the
interval of channel estimation is given as an fixed value of 675us. In case
of down links with 8 transmit antennas, the bandwidth occupancy of time approaches
is calculated to 0.3674, 0.4859 and 2.9748% for typical ITU indoor, pedestrian
and vehicular channel profiles with maximal delay of 310, 410 and 2510 ns, respectively,
while the result for frequency approaches would be 145.64%, i.e., the interval
is not sufficient to estimate channel information in down links. However, the
bandwidth occupancy for classical frequency approaches is 36.41% in up links
with 2 transmit antennas. The bandwidth occupancy of time domain approaches
in up links with 2 transmit antennas, is only one quarter of that in down links,
but the result for frequency methods is kept unchanged. According to the Table
1, the bandwidth occupancy in time domains is increased linearly with the
r.m.s of channel profiles and will not exceed its equivalent in frequency domains
if the length of channel profiles is still less than that of MIMO OFDM symbols.
This results indicate that the channel estimation schemes in time domain can
save a lot of bandwidth resources when compared with its equivalents in frequency
domains.
Furthermore, as showed in the analysis of computational complexity for the proposed steiner method and the typical time and frequency approaches, their computational complexity can be numerical as followings. For down links of current MIMO OFDM systems, 1.9126e+005, 2.0163e+005 and 1.7484e+006 multiplication operations are involved in classical time domain approaches for typical ITU indoor, pedestrian and vehicular channel profiles with maximal delay of 310, 410 and 2510 ns, respectively, while 4.5469e+004, 4.5857e+004 and 1.0374e+005 operations for up links. The results of the proposed steiner methods, are given as 1.8063e+005, 1.8084e+005 and 1.8724e+005 operations for down links and 4.5158e+004, 4.5211e+004, 4.6809e+004 operations for down links. However, the computational costs of classical frequency schemes are 3.93216e+005 and 9.8304e+004 multiplication operations for down and up links respectively.
Numerical results of bandwidth overhead and computational complexity are shown in Table 2 which indicate that the proposed steiner approach consume consume less channel resources with less complexity than the classical time and frequency methods, when typical cellular wireless channel scenarios are considered.
Subsequently, their channel estimation accuracys with the same system configuration
parameters, are further validated the typical ITU vehicle channel profiles with
maximal doppler frequency of 200Hz, where the 2norm of spatial channel matrix
is used as the accuracy metric. Here, pseudo noise (PN) sequences are used as
training sequences for different transmit antennas in classical time domain
schemes and orthogonal Hadamard sequences as frequency pilot symbols.
Table 2: 
Channel estimation cost of the proposed Steiner method, the
classical time and frequency approaches in the scenarios of MIMO OFDM systems
with 8x2 antennas configuration at BS and MS, where the typical ITU indoor,
pedestrian and vehicular channel profiles are used to numerically evaluate
channel estimation cost 


Fig. 3: 
Accuracy of the proposed steiner scheme, the classical time
and frequency domain approaches in down links of MIMO OFDM with 8 transmit
antennas for typical ITU vehicle profiles at 200 Hz Doppler frequency 

Fig. 4: 
Accuracy of the proposed steiner scheme, the classical time
and frequency domain approaches in up links of MIMO OFDM with 2 transmit
antennas for typical ITU vehicle profiles at 200 Hz Doppler frequency 
Figure 3 and 4 show the numerically simulated
results in down and up links respectively. Clearly, the proposed Steiner channel
estimation approach can obtain better channel estimation accuracy than that
of classical time approach, as the channel pilot matrix constructed by the proposed
Steiner method has better reversibility than that by PN sequences. But its different
performance from the classical frequency methods can be compensated by its simply
implementation with less bandwidth overhead.
CONCLUSION
In this study, aiming at two problems in MIMO OFDM systems, i.e., the bandwidth overhead of channel estimation and the challenge to construct large numbers of orthogonal training sequences, we adopt the Steiner channel estimation method for multiuser CDMA uplink radios for the scenarios of MIMO OFDM systems. The training sequences for different transmit antennas, are obtained by truncating the circular extension version of a basic sequences, which simplifies the construction of training sequences in time domain approaches. Furthermore, the pilot matrix assembled by these training sequences is a circular matrix and the high dimension matrix inversion operation can be avoid by FFT of the basic sequences. By reason of the good reversibility of the last pilot matrix, the proposed scheme has better channel estimation accuracy than the classical time approaches. Although its different performance from the classical frequency methods, this can be compensated by its simple implementation with less bandwidth overhead.