UNCORRECTED PROOF
Song et al. EURASIP Journal on Advances in Signal Processing
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However, this algorithm brings large complexity when the57
constellation size is large.58
Recently, an efficient scheme called lattice reduction59
[14–18] has been shown great potential in MIMO detec-60
tions. The lattice reduction (LR) algorithm attempts to61
change the orthogonality of the channel matrix, as the62
orthogonality of the channel matrix largely affects the per-63
formance of the MIMO system [15]. As the well-known64
LR algorithm, Lenstra, Lenstra, and Lovasz (LLL) algo-65
rithm has firstly been considered for LR-aided detection66
[16]. It allows suboptimum detectors, such as the LDs and67
the VL BAST detectors, to exploit all the available diversity68
[16], while it has poly nomial complexity. Many LR algo-69
rithms have been developed to i mprove the performance70
of the L LL algorithm. The LR algorithm improving the71
minimum Euclidean distance of the LR-aided LDs, which72
can improve the BER performance of the L R-aided LDs,73
is proposed in [17]. Element-based lattice reduction (ELR)74
algorithm [15], a low complexity LR algorithm, was pro-75
posed to reduce the diagonal elements of the noise covari-76
ance matrix. The ELR-aided detectors show better BER77
performance than other LR-aided de tectors, especially78
for large MIMO systems [15]. An improved ELR algo-79
rithm [18] is proposed to enhance the BER performance80
of the ELR algorithm; however, it brings large complexity81
increase. As it needs to solve a clos est vec tor problem by82
the sphere decoding method [4, 5], which requires expo-83
nential complexity, it will be complexity expensive and84
impractical when the number of the transmitted signal85
be comes moderate and large. Moreover, [19, 20] proposed86
some efficient ways to re duce the complexity of the LR87
algorithm.88
As shown above, the pre vious LR algorithms aim to89
use unimodular transformation to change the orthogo-90
nality of the channel matrix, such that the gap between91
the suboptimal detectors and the MLD is reduce d . How-92
ever, these LR algorithms do not take the received signal93
into consideration. In fact, the received signal is impor-94
tant information in the MIMO detection. It can partially95
refle ct the received noise and is also usef ul in enhanc-96
ing the BER performance of the LR-aided detectors. In97
this paper, we make a readjustment of the received sig-98
nal in the LR domain and propos e a scheme to improve99
the LR-aided detectors. The propo sed scheme is based100
on a new criterion called the log-likelihood-ratio (LLR)101
criterion, which utilizes both the received signal and the102
channel state information (CSI). Then, by the LLR crite-103
rion, we propose our LLR-based transformation algorithm104
(TA) which target s to use the unimodular transforma-105
tion to minimize the pairwise error probabilities (PEPs) of106
the symbols , while these PEPs are deduced exactly in this107
paper by the information of the CSI and the received sig-108
nal. We show that the PEPs affect the error propagation of109
the VBLAST detector, and decreasing the PEPs can reduce110
the error propagation and enhance the BER performance 111
of the VBLAST detector. In our proposed algorithm, a 112
standard LR algorithm such as LLL or E LR algorithm is 113
performed as the initial stage, where the LLL and the ELR 114
algorithm is the classical LR algorithms as shown above, 115
then an algorithm decre asing the PEPs of the symbols is 116
shown. The simulation results validate that our LLR-based 117
TA-aided VBLAST detectors can prov ide subst antial BER 118
performance gain over the p ervious LR-aided VBLAST 119
detectors, while only moderate computational complexity 120
is increa s ed. 121
Notation: (·)
T
is the transpose of (·),and(·)
†
is the 122
pseudo-inverse of (·).WewriteA
i,j
for the entry in the ith 123
row and j th column of the matrix A, a
i
for the ith entry in 124
a. a
i
and a
i
denote for the ith row and the ith column of 125
the matrix A,respectively. 126
2 Preliminary 127
2.1 System model 128
Consider a MIMO system with N
t
transmitted antennas
Q4
129
and N
r
received antennas as 130
r
c
= H
c
s
c
+ w
c
,(1)
where r
c
∈ C
N
r
is the received signal, H
c
∈ C
N
r
×N
t
is the 131
channel matrix. The entries of H
c
are represented as inde- 132
pendent and identically distributed variables drawn from 133
CN
0,
1
N
t
. s
c
∈ C
N
t
is the transmitted signal indepen- 134
dent and identically drawn from the MQAM constella- 135
tion, where M is the constellation size, and the covariance 136
matrix of s
c
is σ
2
s
c
I.w
c
∈C
N
r
is a zero-mean white Gaussian 137
random vector with covariance matrix σ
2
I.Moreover,the 138
number of the transmitted signal is assumed to be less 139
than or equal to the number of the received signal. 140
Note that (1) is equivalent to the real input-output 141
model [14] 142
r = Hs+ w,(2)
where
143
r =
R(r
c
)
T
, I(r
c
)
T
T
,
144
s =
R(s
c
)
T
, I(s
c
)
T
T
,
145
w =
R(w
c
)
T
, I(w
c
)
T
T
,
and
146
H =
R(H
c
) −I(H
c
)
I(H
c
) R(H
c
)
.
We know r ∈ R
K
, w ∈ R
K
, s ∈ R
N
,andH ∈ R
K×N
, 147
where K = 2N
r
and N = 2N
t
. 148
2.2 The LR-aided detectors 149
Note the orthogonality of the channel matrix largely 150
affects the performance of the MIMO detection and the 151