This paper applies transmit antenna selection algorithms to spatialtemporal combiningbased spatial multiplexing (SM) ultrawideband (UWB) systems. The employed criterion is based on the largest minimum output signaltonoise ratio of the multiplexed streams. It is shown via simulations that the bit error rate (BER) performance of the SM UWB systems based on the twodimensional Rake receiver is significantly improved by antenna diversity through transmit antenna selection on a lognormal multipath fading channel. When the transmit antenna diversity through antenna selection is exploited in the SM UWB systems, the BER performance of the spatialtemporal combiningbased zeroforcing (ZF) receiver is also compared with that of the ZF detector followed by the Rake receiver.
I. INTRODUCTION
Ultrawideband (UWB) radio technology has attracted tremendous interest for applications requiring highdata rate communications over short ranges. UWB systems trying for higher data rates and higher quality of communications can employ multiple input multiple output (MIMO) techniques
[1]
. However, the implementation of MIMO suffers the main disadvantage of increasing hardware costs due to the radio frequency (RF) chains and analogtodigital converters needed. If we use an antenna selection technique, this handicap can be reduced
[2

5]
. In
[6]
, the transmit antenna selection problem in spatial multiplexing (SM) UWB MIMO systems with zeroforcing (ZF) detectors followed by Rake combiners (called a ZFRake), whose structure has been analytically examined in
[7]
, has been discussed. In
[8]
, the 2dimensional Rake architecture in the space and time domain followed by a ZF detector (called a 2RakeZF receiver) has been analytically examined to show that it can improve the error performance of the ZFRake receiver.
In this paper, a transmit antenna selection algorithm is applied to the spatialtemporal Rake combiningbased SM UWB systems proposed in
[8]
. The transmit antenna selection criterion employed for selecting a subset of the transmit antennas is based on the largest minimum postprocessing signaltonoise ratio (SNR) obtained on the basis of QR decomposition. Simulation results will show that the transmit antenna selection algorithm can significantly enhance the BER performance of SM UWB systems with the 2RakeZF receiver on a lognormal multipath fading channel. It will also be found that when the transmit antenna selection scheme is used in the SM UWB systems, the 2RakeZF receiver outperforms the ZFRake receiver and also achieves the diversity order of the full complexity antenna system.
II. SPATIALTEMPORAL COMBINING SYSTEMS USING TRANSMIT ANTENNA SELECTION
To examine the antenna diversity gain through transmit antenna selection, the spatialtemporal combiningbased SM UWB systems presented in
[8]
with
N_{T}
transmit antennas,
N_{R}
receive antennas, and a 1:
N_{S}
(
N_{T}
>
N_{S}
,
N_{R}
≥
N_{S}
) multiplexer are considered. The subset selection of
N_{S}
transmit antennas among the set of all possible
subsets of transmit antennas is carried out by a transmit antenna selection algorithm on the receiver. The receiver feeds back to the transmitter the determined subset. The input data to the multiplexer are serialtoparallel converted into
N_{S}
data streams and then sent to
N_{S}
transmit antennas for simultaneous transmission, which are modulated with the pulseamplitude of a UWB pulse with shortduration. The assumption that the pulse repetition interval is adequately larger than the channel delay spread can enable avoiding severe intersymbol interference. It is also assumed that the transmitted signals from the chosen transmit antennas pass through a multipath fading channel with lognormal distribution and the channel coefficients of the
L
resolvable paths are perfectly known to the receiver.
After the received signal at each receive antenna goes through a correlator, the resulting correlator outputs are spatially and temporally combined by the 2dimensional Rake receiver over each bit interval, as shown in
Fig. 1
. Then the discretetime spatial signal vector,
u
=
, is written as
[8]
,
where
E_{b}
is the average bit energy,
with the information bit from the nth data stream denoted by
b_{n}
∈ {+1, 1} ,
n
= 1,2,
^{…}
N_{S}
and
w
(
l
) = [
w
_{1}
(
l
)
w
_{2}
(
l
)
^{…}
w_{NR}
(
l
) ]
^{T}
with
w_{m}
(
l
) ,
m
= 1,2,
^{…}
N_{R}
, being the zeromean white Gaussian noise component of a variance of
N_{0}
/ 2 at the
m
th receive antenna. Here,
H
_{p}
is a channel matrix corresponding to a subset of best antennas selected from the larger set of available antennas and given by
Block diagram of 2RakeZF receiver with transmit antenna selection. ZF: zeroforcing, SNR: signaltonoise ratio.
where
h_{mn}
(
l
) is the channel fading coefficient of the
l
th resolvable path from the
n
th transmit antenna to the
m
th receive antenna and can be modeled as
h_{mn}
(
l
) =
ζ
_{mn}
(
l
)
g_{mn}
(
l
) where
g_{mn}
(
l
) is the lognormal fading magnitude and
ζ
_{mn}
(
l
) ∈{ +1,1} indicates the phase inversion with equal probability.
After the outputs of the spatialtemporal 2dimensional Rake combiners are performed by a ZF detector,
to spatially decorrelate
N_{S}
data streams, the decision variable is written as,
where (□)
^{†}
denotes the pseudoinverse and
Note that this work focuses on using ZF detection even if other SM MIMO detection techniques could be considered.
III. TRANSMIT ANTENNA SELECTION FOR SPATIALTEMPORAL COMBINING SYSTEMS
From expression (5), the SNR of the ZF detector output signal on the
k
th stream in the 2RakeZF receiver using the 2dimensional combining method in the space and time domain described in Section II can be described as,
In order to derive the postprocessing SNR expression in the 2RakeZF receiver, the channel matrix,
obtained after spatialtemporal combining is considered instead of using only
H
_{p}
. The
N_{S}
×
N_{S}
modified channel matrix
determined by the transmit antenna selection process can be factorized by QR decomposition
with unitary
N_{S}
×
N_{S}
matrix
and the upper triangular
N_{S}
×
N_{S}
matrix
. The left side of the
N_{S}
× 1 received signal vector
u
of the Eq. (1) is multiplied by the Hermitian matrix of
Then the received signal vector is modified to
The postdetection SNR of layer
k
after spatial and temporal combining can then be given by
where
is the
k
th diagonal element of the upper triangular matrix
. The elements of
can be calculated by the wellknown QR factorization algorithm based on a modified GramSchmidt method.
An antenna subset selection is determined by the maximum minimum postdetection SNR. It searches for a transmit antenna subset with the maximum value among
N_{search}
minimum postdetection SNRs computed for the set of all possible transmit antennas using the expressions (7) and (8). The selection criteria given by the expressions (7) and (8) are called a ZFbased selection criterion (ZFSC) and a QRSNRbased selection criterion (QRSNRSC), respectively. For every subset of transmit antennas, compute the corresponding
and
, respectively, which are defined as minimums of
, and
,
k
= 1,2, … ,
N_{S}
, using the SNR expressions (7) and (8). Then select the subsets with the maximum
and
values. Finally, the 2RakeZF receiver is operated for signal detection using the determined antenna subset. Thus the 2RakeZF receivers based on ZFSC and QRSNR SC are called ZF2RakeZF and QRSNR2RakeZF, respectively.
Even if the communication system with antenna selection considered in
[2]
and
[4]
is not a SM UWB one with a lognormal fading distribution, the antenna selection has been shown to maintain the same diversity advantage as the full complexity antenna system without antenna selection. To observe the antenna diversity gain of the ZF2RakeZF and QRSNR2RakeZF through antenna selection, this work considers the average BER performance of the 2RakeZF UWB system with no antenna selection in a lognormal fading channel examined in
[8]
, which is given by
where,
Here,
D_{2RakeZF}
(=
LN_{R}
–
N_{T}
+ 1) indicates the diversity order of the SM UWB system based on the 2RakeZF receiver and Γ(⋅) is the gamma function.
IV. SIMULATION RESULTS
The average power of the path with index
l
=0 is assumed to be unity. The lognormal fading amplitude
g_{mn}
(
l
) can be represented by
g_{mn}
(
l
) =
e
^{θmn(l) }
, where
θ
_{mn}
(
l
) is a Gaussian random variable with mean
μ _{θmn(l)}
and variance
(independent of
l , m
, and
n
). It is assumed that the standard deviation of 20log
_{10}
g
_{mn}
(
l
) =
θ
_{mn}(l)
(20log
_{10}
e
) is 5 dB. In order to satisfy
E
[
g_{mn}(l)^{2}
]=
e^{ρl}
, it is required to be
where
σ_{θ}
is obtained as
σ _{θ }
= 5/(20log
_{10}
e
). Here the power decay factor of
ρ
= 0 is used. In the plots, (
N_{T}
,
N_{S}
,
N_{R}
,
L
) implies that
N_{T}
transmit antennas,
N_{S}
selected transmit antennas,
N_{R}
receive antennas, and
L
resolvable paths are used as antenna diversity system parameters. On the other hand, (
N_{T}
,
N_{R}
,
L
) represents no antenna selection (called No AS) systems.
Fig. 2
shows the BER performance as a function of
E_{b}
/
N_{0}
in decibels for the SM UWB (3,2,2,2) antenna selection diversity systems with the 2RakeZF receiver. The antenna diversity performances of the ZFRake receiver with ZFSNR and QRSNRbased antenna selection criterion (called ZFZFRake and QRSNRZFRake, respectively) are included for comparison. The transmit antenna selection scheme with one more transmit antenna is shown to significantly improve the performance of the
Bit error rate (BER) versus E_{b} /N_{0} in (3,2,2,2) 2RakeZFsystems. ZF: zeroforcing, No AS: no antenna selection, SNR: signaltonoiseratio.
(2,2,2) 2RakeZF receiver with no antenna selection. It is also found that the 2RakeZF receiver with antenna selection has much better performance than the ZFRake receiver. It is observed that the (3,2,2,2) 2RakeZF antenna diversity system outperforms the (4,2,2,2) ZFRake antenna diversity system. It is seen that the QRSNRSC provides much better performance than the ZFSC. The BER performance of the (3,2,2,4) 2RakeZF system with
L
= 4 resolvable multipath components versus
E_{b}
/
N_{0}
in decibels is evaluated in
Fig. 3
. It is shown that the antenna selection scheme enhances the performance of the 2Rake ZF receiver even with more resolvable paths, but the degree of performance improvement, which is the performance gap between the antenna selection diversity system and no antenna diversity system, is reduced.
Using expression (9), theoretical BER results of the 2RakeZF systems are also plotted in
Figs. 2
and
3
. It is shown that the (2,3,2) and (2,3,4) 2RakeZF MIMO systems can be regarded as full complexity antenna systems without antenna selection of the (3,2,2,2) and (3,2,2,4) 2RakeZF antenna selection systems, respectively. The ZF 2RakeZF has almost the same asymptotic slope as the full system, which means that it can provide the same diversity gain as the full system. Meanwhile, the QRSNR2RakeZF has a slightly larger slope than the full antenna system, which implies that it offers an additional SNR gain. Thus it is confirmed that the antenna selection achieves the diversity order of the full system.
Bit error rate (BER) versus E_{b}/N_{0} in (3,2,2,4) 2RakeZF systems. ZF: zeroforcing, No AS: no antenna selection.
V. CONCLUSIONS
The BER performance of a SM UWB system with transmit antenna selection has been evaluated over indoor lognormal fading channels. The employed receiver consists of maximal ratio combiners and multipath combiners that capture
LN_{R}
resolvable paths in the space and time domain and then a ZF detector to spatially process the
N_{S}
parallel transmitted data streams. It has been shown that the transmit antenna selection algorithms based on the largest minimum postdetection SNR significantly boost the BER performance of the 2RakeZF receiver when the number of multipath combined in time is relatively small. It has been observed that the 2RakeZF scheme outperforms the ZFRake even with the transmit antenna selection process. It has also been observed that the QRbased antenna selection criterion offers much better performance than the ZFbased selection one. Furthermore, it has been confirmed that the diversity advantage gained by antenna selection is similar to that of the full complexity antenna system.
Acknowledgements
This study was supported by research funds from DongA University.
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