INTRODUCTION
The wireless communication system (Yang, 2005) coupled
with multiple transmit/receive antennas and Orthogonal FrequencyDivision Multiplexing
(OFDM), is regarded as a promising solution for enhancing the data rates of
nextgeneration wireless communication systems operating in frequencyselective
fading environments. Recent research result shows that MultipleInput and MultipleOutput
(MIMO) techniques (Nanda et al., 2005) could
be used to increase the capacity by a factor of the minimum number of transmit
and receive antennas compared with a SingleInput SingleOutput (SISO) system
with flat fading or narrowband channels, while OFDM (Van
Zelst and Schenk, 2004) can increase diversity gain and mitigate intersymbol
interference on a timevarying multipath fading channel.
Furthermore, when channel parameters are known at the transmitter, the capacity
of MIMO OFDM systems can be further increased by adaptively assigning transmitted
power to orthogonal eigenmodes according to the waterfilling rule (Telatar,
1999). At transmitters, the transmitted signals of different carriers are
usually eigenbeamformed independently to orthogonal modes of spatial channels
at every subchannel in MIMO OFDM systems (Willink, 2005b;
Stuber et al., 2004), which can be formed via
spatial filtering according to the Singular Value Decomposition (SVD) of channel
matrix at transmitters. However, these eigenmodes can not be used to steer the
data symbols encoded by spacetime codes, as one spacetime codeword is transferred
simultaneously by multiple carriers, while the eigenmodes are obtained at every
carriers. So, when coupled with adaptive power and bit allocation, these eigenmodes
have many disadvantages relative to their counterparts of a MIMO system in singlecarrier
transmission. Firstly, these eigenmodes ignore the effects of spacetime diversity
gains on the equivalent Signal Noise Ratios (SNR) of data symbols encoded by
a spacetime coder. Secondly, for a MIMO OFDM system configured with low rate
spacetime codes, it will be difficult to conduct adaptive power allocation
as a large number of eigenmodes exist, compared with data symbols carried by
one MIMO OFDM symbol. Thirdly, as one spacetime codeword is carried by many
eigenmodes at multiple carriers, it is also difficult to determine the modulation
order of data symbols encoded in a spacetime codeword. Therefore, these eigenmodes
can be viewed as the simple generalization of their counterparts of a MIMO system
in singlecarrier transmission for conveniently analyzing system capacity, but
not reflect the fact of one spacetime codeword being carried by multiple subchannels.
Here, we present a new approach of constructing orthogonal eigenmodes in MIMO
OFDM systems. For a MIMO OFDM system with leastsquared decoders, orthogonal
eigenmodes can be obtained by the SVD of equivalent channel matrix in system
models, where a general spacetime code is considered in a general view. The
result eigenmodes are correspondent to data symbols encoded in spacetime codewords
carried by one MIMO OFDM symbol. Thus, the novel eigenmodes can be used directly
to steer adaptive power allocation to data symbols and their bit allocations,
as usually do in a MIMO system with singlecarrier transmission.
Subsequently, based on novel eigenmodes, an improved waterfiling power allocation
scheme is also proposed to determine the power allocation and its correspondent
bit number for reason of complexity or poor performance of existing schemes.
The classical waterfiling power allocation scheme (Telatar,
1999) is only optimal to maximize system capacity but with a great deal
of residual power by reason of discrete modulation orders. Hence, many improved
waterfilling schemes (Ren et al., 2004; Van
den Bogaert et al., 2004; Liang et al.,
2003) or other power allocation schemes (Shi et al.,
2004; Codreanu et al., 2005; Quoting
et al., 2005; Ng et al., 2002) are
given in recent research literatures, such as iterative waterfiling algorithms
(Van den Bogaert et al., 2004), subchannel
group waterfilling scheme (Liang et al., 2003),
greedy algorithms (Codreanu et al., 2005), iterative
bits power allocation (Quoting et al., 2005),
and so on. Among these schemes, only greedy power allocation is optimal to maximize
the transported total bits but at a cost of calculation overhead. Moreover,
similar to greedy algorithm, other schemes are conducted in iterative wise,
and their convergences have significant effects on system performance. Whereas,
the improved waterfilling power allocation scheme can be done in two steps
and can also achieve the maximized transported bits. According to the scheme,
the classical waterfilling strategy is firstly adopted to determine the optimal
power allocation and corresponding bit number for every eignmode, followed by
a residual power reallocation to further determine additional bit number carried
by those eigenmodes.
MATERIALS AND METHODS
The MIMO OFDM system Model with universal spacetime codes: Firstly, a universal spacetime code can be defined as a rate T/K MxK design scheme over a complex subfield A of the complex field C, whose codeword matrix X is a MxK matrix with entries obtained from the Klinear combinations of T data symbols and their conjugates. If a codeword matrix X is represented as a column vector by stacking its columns, the column vector can be delineated as the linear transform of T data symbols and their conjugates, i.e:
where, vec(.) denotes the column vector by stacking the columns of a matrix
into one column vector, s is a column vector whose elements consist of T data
symbols and their conjugates, and the transform matrix Φ is denoted as
the generation matrix of the spacetime code design scheme.
Then, let us consider a MIMO OFDM system with M transmit and N receive antennas, and an OFDM modulation is conducted on K subcarriers, as illustrated in Fig. 1. A space time code is used to encode a data symbol vector s along spacetime directions with T data symbols and their conjugates, while a least squared spacetime decoder is used to restore the transmitted data symbols by decoding the received spacetime signals. In order to delineate the system model compactly, we omit the time indicator of MIMO OFDM symbols and neglect symbol timing errors and frequency offsets. Assume a MIMO OFDM only can carry one spacetime codeword, so the receive signals in a MIMO OFDM symbol period can be given as:
where, y (n,k) is the received data at the kth carrier of the nth receive antenna, H (n,m)(k) represents the fading coefficient at the kth carrier of the spatial channel between the nth receive antenna and the mth transmit antenna, x (m,k) denotes the element at mth row and kth column of a spacetime codeword matrix X, and w (n,k) is the channel noise at the kth carrier for the nth receive antenna.
Substituting the sum term in Eq. 2 by its matrix form, we rewrite Eq. 2 as:
where, x (:, k) is the kth column of spacetime codeword X, H[n,:] (k) is the nth column of the MIMO channel fading coefficient matrix H(k) at kth carriers, which can be delineated as:
Let y(n,:) denote the received signal vector at the nth receive antenna in a MIMO OFDM symbol period, we can rewrite (3) into matrix form as:

Fig. 1:  Discretetime
equivalent baseband model of a MIMO OFDM block transmission system. (a)
Transmitter and (b) Receiver 
where, w(n,:) is the channel white noise corresponding to y(n,:), and .
Thus, in a MIMO OFDM symbol period, assembling the received signals from all
the receive antennas into a matrix form, we can get:
where:
and
is given by:
According to Eq. 6, the least squared estimation of s can
be achieved as following:
where:
Then, since s consists of T data symbols and their conjugates, the transmitted
data can be derived from ,
completely. Moreover, for the scenario where a MIMO OFDM symbol can carry multiple
spacetime codewords, the similar results can also be derived in the same way.
A novel eignmode transmission coupled with space time codes: By singular
value decomposition, the eigenmodes concealed in Eq. 6, can
be disclosed in the same way as their counterparts in singlecarrier M IMO systems
(Ng et al., 2002). Let ,
it can be decomposed into orthogonal eigenmodes by singular value decomposition
as showed:
where, U and V denote the unitary matrices representing the left and right
eigenvectors of ,
respectively, and D is a diagonal matrix, whose elements are the ordered singular
values of ,
i.e., the corresponding fading coefficients of those orthogonal eigenmodes.
Then, according to Eq. 6 and 8, substituting
by its SVD, we can get:
Now, let , ,
,
and
Eq. 9 can be rewritten as:
Furthermore, it is also equivalent to:
where, r and
are the rank of
and its ith singular value, respectively.
As the equivalent channel matrix
includes the generation matrix of spacetime codes, the eigenmodes obtained
by Eq.10 can also reflect the corresponding spacetime diversity
gains of spacetime codes. Furthermore, these eigenmodes have unique corresponding
relations with the data symbols, transported by one MIMO OFDM symbol. Thus,
according to these eigenmodes and power allocation schemes, it’s very easy
to determine the modulation orders of these data symbols and their transmit
power. So, when compared with the classical eigenmodes for different carriers
as showed in reference (Nanda et al., 2005; Willink,
2005a; Stuber et al., 2004), adaptive spatial
processing could be performed conveniently with these novel eigenmodes.
Furthermore, as the relations between the equivalent channel matrix and the generation matrix of one spacetime code, system capacity is significantly effected on by the code rate of spacetime codes. For one spacetime code scheme with a unitary generation matrix, the spacetime diversity doesn’t change the corresponding system capacities without spacetime codes. That is, the number of data symbols in the novel eigenmodes should be identical to that in classical eigenmodes. So, there exists M=2*T/K for spacetime codes that could keep system capacity unchanged, and only Alamouti spacetime code exists for a transmitter with two antennas, when T/K is not more than one. For other spacetime codes, system capacity will increase in inverse proportion to spacetime code rates, i.e., the larger the transmission rate, the more the system capacity. In section V, this will be disclosed by numerical simulation results.
Generally speaking, the classical eigenmodes at different carriers for MIMO
OFDM systems can be viewed as simple extensions of the eigenmodes in MIMO systems
in singlecarrier transmission, which fit for the analysis of system capacity
other than link adaptation techniques. However, besides system capacity analysis,
the novel eignmode transmission can couple spacetime codes and link adaptation
techniques, perfectly. Furthermore, the number of novel eigenmodes is only limited
to the number of the transmitted data symbols carried by a MIMO OFDM symbol,
while M*M eigenmodes have to be disclosed to conduct power allocation to data
symbols transported in these eigenmodes.
Improved waterfilling power allocation algorithm: Based on adaptive modulation margin adaptive (MA) principals, link adaptation techniques can be implemented from two aspects, i.e., adaptive power allocation under total transmit power constraint for maximal transported bits, and adaptive bits allocation under total transmit bits for minimal transmit power, respectively. In this study, under the constraints of given total power and target Bit Error Ratio (BER), we only consider how to conduct power allocation to orthogonal eigenmodes in order to maximize transmit bits.
For given target BER P_{e}, the transmit power for an additive white
Gaussian noise (AWGN) channel to transmit c bits information with MQAM modulation,
is given by Codreanu et al. (2005):
where:
is denoted as complementary error function. Then, for given transmit power,
the number of bits transported by the AWGN channel, can be derived according
to Eq. 12, as showen in the following formula:
where, floor denotes the operator to round towards minus infinity.
In order to maximize the transported total bits, a improved waterfilling power
allocation scheme is given on the base of classical waterfilling schemes. According
to the scheme, the adaptive power and bit allocation are conducted in two steps.
Firstly, an initial power allocation is given by classical waterfilling scheme,
That is, the first step is executed to initially allocate the power for different
orthogonal eigenmodes according to the classical waterfilling scheme. Then,
after determining the transported bits at channel eigenmodes, the residual power
is reallocated among these eigenmodes to transport additional bits. Now, the
following, we present the details of the improved waterfilling scheme, which
is also illustrated its flow chart as showen in Fig. 2.

Fig. 2:  Flow
chart for the improved waterfilling power algorithm 
Initial power allocation based waterfilling scheme: For the eigenmodes
given by Eq. 11, the power allocation scheme can be described
as an optimal problem to maximize the system capacity under the constraint of
given total transmit power, i.e:
where, C denotes system capacity, while P is the given total transmit power. According to waterfilling power allocation algorithm, the optimal power allocation can be given by:
where:
and σ^{2} is noise variance of orthogonal eigenmodes, assumed
to have the same variance.
The power allocated to orthogonal eigenmodes as showen in Eq. 15, is called waterfilling power to distinguish different power allocation results in the following. With the waterfilling power allocated the ith eignmode, its maximal bits carried can be given by Eq. 13, that is:
However, according to Eq. 12, the necessary power to transmit c_{i} bits is determined by:
which is called the expectation power to transmit c_{i} bits. Clearly, the waterfilling power for the ith eignmode is larger than its expectation power, and their difference is named as residual power, which will be further reallocated among these eigenmodes.
Reallocation of residual power: Subsequently, on the basis of the power allocation results in the first period, we calculate the additional power to transmit an additional bit at the ith eignmode. That is:
which is called additional power. Then, an accumulative sum sequence is obtained by a sorted version of additional power in ascending order at all the eigenmodes. The elements in the accumulative sum sequence, not more than the total residual power, can be found out, and the eigenmodes corresponding with these elements can transport an additional bit. So, the additional powers of these eigenmodes are allocated to these eigenmodes from the total residual power, whose residual power after reallocation, i.e., total overplus power, is averagely allocated to all the eigenmodes. At last, the bit number of these eigenmodes should be increased by one, respectively.
It’s very clear that the improved scheme as presented above can improve
the usage of power and transmit more bits than the classical waterfilling power
allocation scheme (Codreanu et al., 2005). Furthermore,
its simple implementation also has smaller delay than the greedy algorithm.
Hence, it is feasible for the MIMO OFDM systems with a larger number of carriers.
SIMULATION RESULTS
Here, we firstly consider a MIMO OFDM system with 2048 carriers at carrier
frequency of 4.5 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. Under spatially uncorrelated ITU vehicular A channels (Ng
et al., 2002) with Doppler frequencies of 200 Hz, we evaluate the
system capacities and throughputs with and without considering spacetime codes,
respectively. By the reason of terseness, the eigenmodes obtained for the two
scenarios are called spacetime eigenmodes and carrier eigenmodes, respectively.
At transmitter, the waterfilling power allocation algorithm (Telatar,
1999) is executed to adaptively adjust the transmit powers for all the eigenmodes
according to their fading coefficients.
Figure 3 shows the system capacities for different signal
noise ratios (SNR) when spacetime codes are Alamouti Code, SpaceTime Block
Code (STBC)
(Ng et al., 2002) and STBC
(Ng et al., 2002) with code rates of 1, 0.5 and
0.75, respectively. According to Fig. 3, the code rates of
space time codes have significant effects on the system capacity when the spacetime
eigenmodes are constructed, i.e., the smaller the code rates, the larger the
capacity difference between carrier eigenmodes and spacetime eigenmodes. As
the results pointed out in that, this phenomenon can owe to the increase of
spacetime diversity gains with the decrease of code rates, which lead to the
reduction of data symbol transmit rates. At the same time, the scales of system
capacities for the carrier eigenmodes and the spacetime eigenmodes are evaluated
by twenty time numerical simulation under the uncorrelated ITU indoor, pedestrian
and vehicular A channel scenarios, respectively. Then, these scale factors are
averaged out to 1.9526, 4.4807 and 3.5509, when Alamouti Code, SpaceTime Block
Code (STBC)
(Ng et al., 2002) and STBC
are considered, respectively. However, this doesn’t mean the throughputs
of carrier eigenmodes are larger than that of spacetime eigenmodes, as showed
in the following simulation results.
Furthermore, when spacetime codes are Alamouti Code, SpaceTime Block Code
(STBC)
and STBC
with code rates of 1, 0.5 and 0.75, respectively, the throughputs for different
signal noise ratios (SNR) are given in Fig. 4, where the modulation
order of data symbols in carrier eigenmodes is fixed unchanged in a spacetime
codeword. However, the adaptive modulation is only performed on the data symbols
of spacetime codes, for the case of spacetime eigenmodes other than carrier
eigenmodes, as the data symbols carried by spacetime codes in carrier eigenmodes
are transported by multiple carrier eigenmodes, simultaneously. Due to adaptive
modulation of data symbols for spacetime eigenmodes, the larger throughputs
are achieved than that of carrier eigenmodes, as showed in Fig.
4, while the throughputs also increase with the code rate of spacetime
codes, at the SNR above 15 dB. What’s more, the ratios between the numbers
of carrier eigenmodes and spacetime eigenmodes are 1, 3 and 2 for Alamouti
Code, SpaceTime Block Code (STBC)
and STBC
, respectively. Consequently, the link adaptation technique with a few eigenmodes
can be implemented effectively in spacetime eigenmodes.
Subsequently, the improved waterfilling power allocation scheme is tested
in the same system configuration scenarios as showed above, while other power
allocation algorithms such as classical waterfilling scheme (Telatar,
1999) and greedy algorithm (Codreanu et al.,
2005), are also evaluated as comparisons. Under given target BER 10^{3},
their throughput curves for the MIMO OFDM systems with two transmit antennas
and four receive antennas, are showed in Fig. 5, where the
Alamouti SpaceTime Code is used to encode the transported data symbols in spacetime
domain for the case of spacetime eigenmodes.

Fig. 3:  Capacity
curves of the MIMO OFDM systems when the carrier eigenmodes and spacetime
eigenmodes are conducted for the spacetime codes with different code
rates 

Fig. 4:  Throughput
curves of MIMO OFDM systems when the carrier eigenmodes and spacetime
eigenmodes are conducted for the spacetime codes with different code
rates, where different powers are allocated to different eigenmodes according
to waterfilling scheme with given target BER 106 
According to Fig. 5, the improved waterfilling power allocation
scheme can achieve good results than that the classical waterfilling scheme
achieves at SNR above 16 dB, as the power allocation strategy is introduced
to obtain additional transported bits at a cost of sensibility to noise. However,
the results in Fig. 5 also indicate the improved scheme is
actually a suboptimal scheme, whose performance is inferior to that of greedy
algorithm.

Fig. 5:  Throughputs
of the MIMO OFDM system with Alamouti SpaceTime Code when the spacetime
eigenmodes with waterfilling, equal power, greedy and improved waterfilling
power allocation scheme are conducted with given target BER 106 

Fig. 6:  Bit
error ratio (BER) of MIMO OFDM system with Alamouti SpaceTime Code when
spacetime eigenmodes with waterfilling, equal power, greedy and improved
waterfilling power allocation schemes are conducted with target BER 103 
At last, for given target BER 10^{3}, the system average BER curves
under different power allocation schemes, are showen in Fig. 6
for the MIMO OFDM systems with the same configuration as showed above, when
the Alamouti SpaceTime Code is adopted. As indicated in Fig.
6, the improved waterfilling scheme can obtain better system BER performance
than greedy algorithm and equal power schemes but inferior to classical waterfilling
scheme. Hence, the improved waterfilling scheme can work as an alternative
scheme of greedy scheme and waterfilling algorithm.
DISCUSSION
For a MIMO OFDM system configured with leastsquared spacetime decoders, the
proposed novel eignmode transmission with considering universal spacetime codes
can integrate generation matrix of spacetime codes with spatial MIMO channel
matrix. Compared with the classical eigenmodes at different carriers, these
novel eigenmodes can be used to conduct transmit power allocation for those
data symbols, as usually do in MIMO systems in singlecarrier transmission.
It would be more convenient to perform link adaptation technology with novel
eignmode transmission than that with classical eigenmodes. Furthermore, the
improved waterfilling power allocation scheme given in this study could obtain
comparable throughputs and system BER, when compared with greedy algorithm(Codreanu
et al., 2005), the classical waterfiling power allocation scheme
(Telatar, 1999), iterative waterfiling algorithms (Van
den Bogaert et al., 2004), subchannel group waterfilling scheme
(Liang et al., 2003), many improved waterfilling
schemes (Ren et al., 2004; Van
den Bogaert et al., 2004; Liang et al.,
2003), respectively.
CONCLUSION
In this study, an improved waterfilling scheme is proposed for determining the optimal transmit powers for orthogonal eigenmodes. Results indicate that the improved waterfilling scheme can obtain good tradeoff, with comparison to classical waterfilling schemes and greedy algorithms respectively. Compared with classical waterfilling scheme, it can also obtain larger throughputs via residual power allocation.