In this paper, we investigate achieving the full diversity order and power gains in case of using OSTBCs and quasiOSBCs in the x channel system with interference alignment with more than 2 antennas at each terminal. A slight degradation is remarked in the case of quasiOSTBCs. In terms of receiver structure, we show that due to the favorable structure of the channel matrices, the simple zeroforcing receiver achieves the full diversity order, while the interference cancellation receiver leads to degradations in performance. As compared to the conventional scheme, simulation results demonstrate that our proposed schemes achieve 14dB and 16.5dB of gain at a target bit error rate (BER) of 10
^{4}
in the case of OSTBCs with 3 and 4 antennas at each terminal, respectively, while achieving the same spectral efficiency. Also, a gain of 10dB is achieved at the same target BER in the case of quasiOSTBC with 4 antennas at each terminal.
Ⅰ. Introduction
In wireless communications systems, interference plays a major role in defining the achievable performance and capacity
[1
,
2]
. In conventional receivers, in multiuser scenarios, the interference is either ignored, hence considered as an additional noise, or jointly decoded via employing successive interference cancellation (SIC) detectors
[3

5]
. In both cases, the dimension of the interference remains the same, leading to degraded performance and diversity gain in the first case, while powerful algorithms should be employed in the case of SIC so as to avoid degradation in the performance.
Interference alignment is a transmission technique used to reduce the dimensions of the interference while maintaining the useful signals discernible at the intended receivers. This is achievable by precoding the transmit signals such that the interference is aligned at unintended receivers
[6]
. As such, interference is removed at the intended receivers using simple mathematical operations leading to an interferencefree system, where appropriate decoding algorithms can then be used to decode the useful signals. In
[6]
, Jafar and Shamai proposed a linear alignment algorithm for the twouser X channel, that achieves the maximum data rate of (4/3 ×
n_{T}
) symbols/channel use and a diversity gain of 1, with
n_{T}
denoting the number of antennas at each terminal.
In addition to the multiplexing gain, quantified by symbols/channel use, the diversity gain is an important measure of the system performance. When the channel is in deep fading, systems with unity diversity gain suffer from low signaltonoise ratio (SNR) at the receiver side, leading to degradation in the biterror rate (BER). Several diversity techniques have been proposed in the literature to explore further diversity gain
[7

9]
. In
[10]
, a technique that combines interference alignment in X channel and Alamouti diversity scheme with two transmit antennas has been proposed to achieve the maximum multiplexing gain of (
n_{T}
× 4/3 = 8/3) and the full diversity gain of 2, which is equal to the number of antennas at each of the four nodes. Furthermore, the proposed scheme inherits the spacetime orthogonality of the Alamouti algorithm, hence a simple linear receiver, that avoids computationally complex matrix inversion, is required to achieve the aforementioned gains.
In this paper, we propose several transmitter and receiver structures for the case of more than 2 transmit antennas aiming to increase the diversity and power gains. First, we examine the combination of orthogonal spacetime block codes (OSTBC) for
n_{T}
= 3 and 4 with code rate R
_{c}
= 1/2 built on the 2user X channel system with interference alignment. Due to the orthogonality of the codes, a modified interference cancellation receiver (ICR) is used to decouple the symbols transmitted from the two transmitters. Surprisingly enough, due to the special structure of the effective channel matrices, we show that the simple linear zeroforcing(ZF) receiver achieves the full diversity without requiring explicit matrix inversion. We extend this scenario to the quasiorthogonal STBCs (quasiOSTBCs) with
n_{T}
= 4, where in addition to proposing the interference alignment structure, we introduce a modified version of the ICR that takes into consideration the nonorthogonal nature of the codes. Note that the extension to the case of
n_{T}
= 2
^{(n+2)}
for any positive integer
n
is possible. Furthermore, we show once again that the simple ZF receiver is superior to the ICR in terms of power and diversity gains. Due to the nonorthogonality of the effective channel matrices, the achieved diversity in this later case slightly lacks the optimum value. Further research will be conducted to detail the reasons behind such a degradation and investigate methods to minimize it. It is fundamental to mention that in this paper we don't claim that our proposed techniques achieve the maximum multiplexing gain of (
n_{T}
× 4/3). However, at the same spectrum efficiency, quantified by bits/Hz/channel use, our proposed schemes achieve far better performance in terms of power and diversity gains.
The rest of the paper is as follows. In Section II we introduce the system model and review the LJJ scheme. In Sections III and IV, we introduce the OSTBCs and quasiOSTBCs built on the 2user X channel with interference alignment, respectively. In Section V we present the simulation results and in Section VI we draw the final conclusions.
Ⅱ. System model and related work
 2.1. System model
Consider a twouser X channel system as depicted in
Figure 1
, with each of the two transmitters equipped with
n_{R}
antennas and each of the receivers equipped with antennas, where in the deployed scenario
n_{T}
=
n_{R}
. Each transmitter has independent symbols intended for each of the receivers. These symbols are drawn independently from a finite modulation set Ω. Transmitter 1 has
intended for receiver 1 and receiver 2, respectively. In
, the superscript
k
denotes the index of the symbol, the first subscript denotes the index of the transmitter, and the second subscript denotes the index of the intended receiver. Likewise, transmitter 2 has
intended for receiver 1 and receiver 2, respectively. Vectors
s_{ij}
, for
i,j
= 1,2 are encoded by the STBC block to generate the
T
×
n_{T}
matrices
s_{ij}
, for
i,j
= 1,2, with
T
denoting the number of channel uses. Finally, encoded symbols are beamformed and linearly combined to generate the
T
×
n_{T}
block codes
X_{i}
, for
i
= 1,2. To denote the channels between the transmitters’ and the receivers’ antennas, we use H, G, A and B to denote the
n_{T}
×
n_{R}
matrices coupling transmitter 1 and receiver 1, transmitter 2 and receiver 1, transmitter 1 and receiver 2, transmitter 2 and receiver 2, respectively. Each entry in the channel matrices is i.i.d.CN(0,1), where CN(
μ
,
σ
^{2}
) denotes a complex Gaussian random variable with mean and variance of
μ
and
σ
^{2}
. The
T
×
n_{T}
signal matrices received at the antennas of receiver 1 and receiver 2, respectively, are given by:
채널 모델과 계통도 Fig. 1 Channel model and system diagram
The elements of the noise matrices
W
_{1}
and
W
_{2}
are i.i.d.
. The block codes are given as following:
The beamforming matrices
V_{ij}
for
i,j
= 1,2 are designed to align the interference at the unintended receivers. That is,
are aligned at receiver 2, whereas
are aligned at receiver 1. The zeroforcing precoding is a suitable choice to design the beamforming matrices
V_{ij}
for
i,j
= 1,2. As such, they are given by:
where the realvalued scalars
α_{A}
,
α_{H}
,
α_{B}
, and
α_{G}
satisfy the power constraint
, where tr(•) denotes the transpose operator. Hence,
for any matrix R.
 2.2. Review of the LJJ scheme
In the LJJ scheme
[10]
, each node is equipped with 2 antennas where each transmitter has
K
= 2 symbols intended for each of the two receivers. As such, at each channel use a total of 4 ×
K
= 8 symbols are transmitted. These symbols are encoded using Alamouti scheme as following:
for
i
= 1, 2, where i refers to the index of the transmitter. Note that the conjugated symbols are present in the same row of
S
_{i1}
and
S
_{i2}
. At the receivers, the corresponding rows in the receive block codes are conjugated so that the system can be rewritten as a function of conjugatefree symbols. The details of the decoding process which is partially similar to that depicted in
[10]
.
Ⅲ. Orthogonal STBC with Rc = 1/2 top of the xchannel With Interference Alignment
It is proven that Alamouti scheme is the only OSTBC with rate
R_{c}
= 1. For more than 2 transmit antennas, several orthogonal codes have been derived with lower rates (i.e.,
R_{c}
< 1). In the following section, we investigate the integration of OSTBC with
R_{c}
< 1 in the system depicted in
Figure 1
. Our goal is to increase the diversity gain
[12]
, while maintaining the same capacity, quantified by bits/channel use, as compared to the LJJ algorithm.
 3.1. Alignment design for orthogonal STBC withnT=nR= 3
For the case of
n_{T}
= 3, the matrices
S
_{i1}
and
S
_{i2}
for
i
= 1,2 are designed as follows.
and
with (•)
^{T}
denoting the transpose operator. The received signal matrices at receiver 1 and 2 are given in (1), where
X
_{1}
and
X
_{2}
are given in (2) with H, G, A, B and
V_{ij}
for
i,j
= 1,2 ∈
C
^{3×3}
, whereas
Y
_{1}
and
Y
_{2}
∈
C
^{12×3}
. Due to system symmetry, we focus in the sequel on the decoding at receiver 1. Let
y_{i}
, whose
jth
element denoted by
y_{i,j}
, be the
ith
column of
Y
_{1}
with conjugate operator applied to rows 9 to 12, then
where
H_{i}
and
G_{i}
have the same structure. Due to space limit, we give only
H_{i}
as follows:
The aligned interference coefficient matrices are given by:
where
, for
i
= 1,⋯,4.
The aligned interference can be simply removed by constructing the set of vectors.
where the system becomes interferencefree and is represented by the following equation.
with the 8×4 matrices
being derived from
H_{i}
and
G_{i}
by removing their zero rows.
 3.2. Alignment design for orthogonal STBC withnT=nR= 4
For the case of
n_{T}
= 4, the matrices
S
_{i1}
and
S
_{i2}
for
i
= 1,2 are designed as follows.
and
The 12×4 received matrices
Y
_{1}
and
Y
_{2}
are given as in (1). Rows 9 to 12 are then conjugated and received matrices are remodeled as in (7) as follows.
where
H_{i}
and
G_{i}
have the same structure.
H_{i}
is given by:
In (16),
C
_{1}
to
C
_{3}
are given in (9) to (10), respectively, while
C
_{4}
is given by
Now, let the vectors
, for
i
= 1,⋯,4, be constructed as in (12), then the interferencefree system is given by
with the 8×4 matrices
being derived from
H_{i}
and
G_{i}
by removing their zero rows.
 3.3. Decoupling Symbols from Different Transmitters
In the following, we introduce two decoupling algorithms; the first is an interference cancellation receiver (ICR) based on the work given in
[10]
and
[11]
, whereas the second algorithm is the linear zeroforcing (ZF), where we shed light on the simplicity of the channel inversion due to the structure of the resulting interferencefree system.
 3.3.1. Interference cancellation receiver
Let the interferencefree system be represented by the following two equations.
where in the case of
n_{T}
= 3.
In the case of
n_{T}
= 4, the following holds.
To decouple the vector
s
_{11}
,
s
_{12}
is decoupled similarly, we construct the following two equations.
where ║
G
║
_{F}
is the Frobenius norm of
G
. Since the matrices in (20) are orthogonal, then subtracting (24) from (23) results in
Let
then
Due the operation completeness of the matrices in 20,
is also orthogonal. Then, the estimate of
s
_{11}
is given by
where
Q
(•) denoting the vector demodulation operator.
 3.3.2. Linear receiver
Let the system in (20) be reformulated as following.
The estimate of the symbols from the transmitters are then obtained using the linear ZF detector as follows.
with the estimate of si1 is given by
. Due to the structure of the matrices
, for
i
= 1,⋯,
n_{R}
, the following holds.
where the matrices C, D and E are realvalued with C and D being diagonal with diagonal elements equal to
, respectively.
Based on the block matrix inverse lemma,
Note that the matrix (
C

ED
^{1}
E^{H}
) is a scaled identity matrix, and hence calculating its inverse requires a single real division operation. The same holds for the inversion of the matrices C and D. As such, the complexity of the ZF detection reduces to simple real operations rather than fully computing the inverse of an 8×8 complexvalued matrix.
Ⅳ. Fullrate Quasiorthogonal STBC on top of the xchannel with interference alignment system
To achieve higher diversity gains employing a single receive antenna, the conventional Alamouti code
[7]
with two transmit antennas is extended to the 2
^{m}
transmit antennas case with
m
≥ 2
[13

15]
. In the following, we introduce alignment design and receiver structures for the case of the extended Alamouti with
n_{T}
=
n_{R}
= 4 built on the system depicted in
Figure 1
.
 4.1. Alignment design
In the current design, (2) and (3) still hold with the difference that the matrices
V_{ij}
for
i
= 1,2, A, B, G, and H are of size 4×4, and
S
_{i1}
and
S
_{i2}
, for
i
= 1,2, are respectively designed as
Based on (1), the 6×4 received signal matrices at receiver 1 and 2 are respectively written as
where
, for
i
= 1,⋯,4. Without loss of generality, we will focus on the decoding at receiver 1, where receiver 2 has the same decoding and performance due to system symmetry. The system in (33) is converted to the equivalent 24×1 vector
, where the elements of
are filled in rowwise. That is, the first 6 elements of
are the first row of Y1, elements 7 to 12 are the second row of Y1. The conjugate operator is applied to rows 3 and 4 before being inserted in
.
The aligned interference at receiver 1,
i.e.
,
I_{i}
terms, is removed by performing simple addition and subtraction operations on the elements of
. To this end, let us define the following 4×1 vectors
where the vectors
for
i
= 1,⋯,4 are derived similarly from
Then,
with
Note that
, for
i
= 1,⋯,4, have the same structure of the extended Alamouti matrices, which implies that they have the same properties. After aligned interference cancellation, the system can be rewritten as
where the noise is still white with a covariance matrix
, where ⊗ denotes the Kronecker product and
I
_{4}
is the 4×4 identity matrix.
Before introducing the decoupling schemes, it is worthy to shed some light on the properties of the matrices given in (36), where these properties are essential in the design of the receiver. Let
F_{i}
for
i
= 1,⋯,
n
have the same structure of the matrices given in (36), then
1) The matrix
is realvalued, having the form
where
2) These matrices are complete in terms of matrix addition, matrix subtraction, and matrix multiplication. That is, the Gram matrix of
F_{i}F_{j}
, for
j ≠ i
, has the same structure explained in Property I. This can be simply proven using the block form of each matrix and perform each of the aforementioned operations.
3) The inverse of
given in Property I, has the following form
with
4) The sum of Gramians of the matrices defined in (36) is given by
with
Having that been said about the properties of the matrices given in (36), we introduce the properties of the Gramian matrix
which will be used in the decoupling algorithms to be introduced in the sequel. The (8×8) Gramian matrix is given by:
where
Note that the 4×4 matrices C and D have the structure explained in Property 4, while the 4×4 matrix E has the structure given in (36). As such, it comes with no surprise that the inverse of
H_{f}
has the same structure given (41), as will be explained later on.
 4.2. Decoupling symbols from different transmitters
In the literature, several detection techniques have been proposed
[10]
, which can be used to decouple the symbols from different receivers in (37). However, the effective channel matrix, or matrices, in (37) has a special structure that motivates the proposal of modified receivers that take into account the particular structure of these matrices. In the following, we introduce the details of two receiver structures.
 4.2.1. Interference cancellation receiver
The idea of the ICR is based on the work introduced in
[10]
and
[11]
, with the particularity that the system discussed here is not orthogonal. That is, the Gram matrix of
in (37) include both diagonal and skewdiagonal elements. As such, to decouple
s
_{11}
the terms which include
s
_{21}
should be canceled, where two IC stages are required rather than a single stage as in
[10]
.
A) Cancellation of the diagonal elements: Starting with (38), we construct the following four vectors.
Where based on property 1
where
diag
() is a skewdiagonal matrix. Subtracting the second equation, i.e.,
i
= 2 in (42), from the first one, and the fourth equation from the third one, cancels the diagonal elements of the channel matrix associated with
s
_{21}
, resulting in the following equations
where
B) Cancellation of the offdiagonal elements: Let (43) and (44) be divided by
θ
_{1}
and
θ
_{2}
, respectively, then subtract the second equation from the first one, results in
where
. It is worth mentioning that based on Property II, the structure of
is defined by Property I.
Finally, the ZF decoder can be used to recover the transmitted vector
s
_{11}
as following
where the estimate
. Note that
can be computed using Property III to reduce the computational complexity.
 4.2.2 Linear zeroforcing receiver
The system modeled in (37) is first filtered using
resulting in
where
, and the subscript f refers to filtered. As such, C and D have the structure given in Property 1, as well as satisfying Property 4, while E satisfies Property II. The transmitted vectors are recovered as follows
To avoid the inversion of the 8×8 matrix
H_{f}
, we introduce a costeffective method to compute the inverse without explicitly inverting
H_{f}
. Based on the block matrix inverse lemma, we make the following remarks:
1) C, D, (
C

ED
^{1}
E^{H}
), and (
D

E^{H}C
^{1}
E
) are realvalued and have the structure given in Property I, which implies that their inverses are simply computed using Property III. The inverse of each of these matrices requires only three operations  two real multiplication and one division operations.
2)
E^{H}C
^{1}
E
=
C
^{1}
E^{H}E
, where
E^{H}E
has the structure given in Property I. The same applies for
ED
^{1}
E^{H}
. Using Property II,
E^{H}E
can be obtained where 9 multiplication and 6 addition operations are required. Multiplying the result by
C
^{1}
requires 4 multiplication and 2 addition operations.
3)
H_{f}
is a Hermitian matrix, which implies that only the elements of an upper triangular matrix are computed.
Based on these remarks, it stems out that the computational complexity of the linear ZF detector is low due to the special structure of the effective channel matrices.
Ⅴ. Simulation Results and Discussion
In this section, we consider that each transmitter has full knowledge of
only
the channels coupling his transmit antennas with those of the two receivers. The elements in the channel matrices are independent and follow the circularsymmetric complex Gaussian distribution with mean and variance of zero and unity, respectively.
Figure 2
depicts the performance of OSTBC with code rate of 1/2 and
n_{T}
, referred to as
X
_{3}
, built on the system depicted in
Figure1
.
n_{T} = n_{R} = 3를 이용한 직교 STBC로 제안된 시스템의 BER 성능 Fig. 2 BER performance of the proposed system with n_{T} = n_{R} = 3 employing orthogonal STBC
To hold a fair comparison between the proposed scheme and the LJJ scheme, we fix the spectral efficiency to 8/3 bits/s/Hz, implying that the modulation schemes used for and LJJ schemes are QPSK and BPSK, respectively. Starting with the ICR, due the increased dimensionality of the interference, this receiver fails to achieve the full diversity gain of
n_{R}
=
n_{T}
= 3. Due to the special structure of the effective channel matrix, as explained in Section III, the simple linear ZF receiver achieves the full diversity order of
n_{T}
=
n_{R}
= 3 while attaining high gain in the bit error rate. At a target BER of 10
^{4}
, the proposed
X
_{3}
outperforms the conventional LJJ schemes by 14dB.
Figure 3
depicts the performance of OSTBC with code rate of 1/2 and
n_{T}
= 4, referred to as
X
_{4}
, built on the system depicted in
Fig. 1
. Again, for the same spectral efficiency of 8/3 bits/s/Hz, we employ QPSK and BPSK bittosymbol mapping for the proposed
X
_{4}
scheme and the conventional LJJ scheme, respectively. The ICR shows similar performance as in the case of the
X
_{3}
scheme. The simple linear ZF receiver achieves the full diversity order of
n_{T}
=
n_{R}
= 4 with a gain of about 16.5dB at a target BER of 10
^{4}
.
n_{T} = n_{R} = 4를 이용한 직교 STBC로 제안된 시스템의 BER 성능 Fig. 3 BER performance of the proposed system with n_{T} = n_{R} = 4 employing orthogonal STBC
Finally,
Figure 4
depicts the performance of the quasiOSTBC with
n_{T}
= 4, referred to as
T
_{4}
, built on the system depicted in
Figure 1
. Since the ICR employed a twostage SIC scheme to cancel the diagonal and skewdiagonal interference, the performance is therefore even worsen as compared to that in the cases of the
X
_{3}
and
X
_{4}
. Despite the quasiorthogonality of the effective channel matrices, the linear ZF receiver for the proposed
X
_{4}
scheme achieves a superior performance of about 10dB at a target BER of 10
^{4}
as compared to the LJJ algorithm both employing QPSK modulation, resulting in an equal spectral efficiency of 8/3 bits/s/Hz. Our proposed scheme performs close to the optimum diversity order of
n_{T}
=
n_{R}
= 4. The small degradation in the diversity order is due to the quasiorthogonality of the effective channel matrices. A further research will be conducted to optimize the code matrices so as to achieve the full diversity order, or at least to reduce the degradation.
n_{T} = n_{R} = 4를 이용한 준직교 STBC로 제안된 시스템의 BER 성능 Fig. 4 BER performance of the proposed system with n_{T} = n_{R} = 4 employing quasiorthogonal STBC
Ⅵ. Conclusion and Future work
In this paper, we introduced three schemes that achieve superior performances as compared to the conventional LJJ scheme in terms of BER and diversity order all at the same spectral efficiency. The first two proposed schemes, namely
X
_{3}
and
X
_{4}
, employ orthogonal STBCs to generate the code matrices, resulting in orthogonal effective matrices, hence achieving the optimum diversity order of 3 and 4, respectively. The third proposed scheme, referred to as
T
_{4}
, is superior to the conventional LJJ algorithm in terms of BER and diversity order, however, due to the quasiorthogonality of the effective channel matrices, the diversity order of this scheme slightly lacks the optimum one.
As a future work, we plan to investigate the implementation of several quasiOSTBCs with
n_{T}
> 4 built on the described system in this paper. Besides, a careful analysis of the diversity gain will be carried out and methods to achieve the optimum diversity order will be introduced.
BIO
이슬람 모하이센(Islam Mohaisen)
B.Eng., Islamic University of Gaza, Palestine
※관심분야 : MIMO detection, Multiuser precoding
이샛별(SaetByeol Lee)
한국기술교육대학교 전자공학과 학사
※관심분야 : MIMO detection, Multiuser precoding, social network analysis
마나르 모하이센(Manar Mohaisen)
Korea Tech, Korea, assistant professor
Ph.D., Inha University, Korea
M.Sc., University of Nice SophiaAntipolis, France
B.Eng., Islamic University of Gaza, Palestine
※관심분야 : MIMO detection, Multiuser precoding, social network analysis, game theory
하템 엘아이디(Hatem Elaydi)
Islamic University of Gaza, Palestine, associate professor
Ph.D., New Mexico State University, USA
M.Sc., New Mexico State University, USA
B.S., Colorado Technical University, USA
※관심분야 : MIMO systems, control systems, digital signal processing, quality assurance in higher education
Iyer A.
,
Rosenberg C.
,
Karnik A.
2009
“What is the right model for wireless channel interference,”
IEEE Trans. on Wireless Communications
8
(5)
2662 
2671
DOI : 10.1109/TWC.2009.080720
Chafekar D.
,
Kumar V.S.
,
Marathe V.
,
Parthasarathy S.
,
Srinivasan A.
2011
“Capacity of wireless networks under SINR interference constraints,”
Wireless Networks
17
(7)
1605 
1624
DOI : 10.1007/s1127601103672
Lee JH.
,
Toumpakaris D.
,
Yu W.
2011
“Interference mitigation via joint detection,”
IEEE J. on Selected Areas on Communications
29
(6)
1172 
1184
DOI : 10.1109/JSAC.2011.110606
Chen W.
,
Lataief KB.
,
Cao Z.
2009
“Network interference cancellation,”
IEEE Trans. on Wireless Communications
8
(12)
5982 
5999
DOI : 10.1109/TWC.2009.12.081576
Mohaisen M.
,
Chang KH.
“Maximumlikelihood cochannel interference cancellation with power control for cellular OFDM networks,”
NSW
in Communications and Information Technologies, 2007. ISCIT '07. International Symposium on
Sydney
2007
198 
202
Jafar S.
,
Shamai S.
2008
“Degrees of freedom region for the MIMO x channel,”
IEEE Trans. on Information Theory
54
(1)
151 
170
DOI : 10.1109/TIT.2007.911262
Alamouti S.
1998
“A simple transmitter diversity technique for wireless communications,”
IEEE Journal on Selected Areas in Communications
16
(8)
1451 
1458
DOI : 10.1109/49.730453
Tarokh V.
,
Jafarkhani H.
,
Calderbank A.
1999
“Spacetime block codes from orthogonal designs,”
IEEE Trans. on Information Theory
45
(5)
1456 
1467
DOI : 10.1109/18.771146
El Gamal H.
,
Hammons R.
2003
“On the design of the algebraic spacetime codes for MIMO blockfading channels,”
Information Theory, IEEE Transactions on
49
(1)
151 
163
DOI : 10.1109/TIT.2002.806116
Li L.
,
Jafarkhani H.
,
Jafar S.
“When Alamouti codes meet interference alignment: transmission schemes for twouser X channel,”
in Proceedings of Information Theory Proceedings (ISIT)
St. Petersburg
2011
2717 
2721
Naguib A.
,
Seshadri N.
,
Calderbank A.
“Applications of spacetime block codes and interference suppression for high capacity and high data rate wireless systems,”
in Proceedings of Asilomar Conference
Pacific Grove: CA
1998
1803 
1810
Narasimhan R.
,
Ekbal A.
,
Cioffi J.M.
“FiniteSNR diversitymultiplexing tradeoff of spacetime codes,”
in Proceedings of IEEE ICC
2005
458 
462
Rupp M.
,
Mecklenbrauker C.
“On extended Alamouti schemes for spacetime coding,”
in Proceedings of WPMC
2002
115 
119
Mecklenbrauker C.
,
Rupp M.
2004
“Generalized Alamouti codes for trading quality of service against data rate in MIMO UMTS,”
EURASIP Journal on Applied SignalProcessing
2004
662 
675
DOI : 10.1155/S1110865704310061
Badic B.
,
Rupp M.
,
Weinrichter H.
2004
“Adaptive channelmatched Alamouti spacetime code exploiting partial feedback,”
ETRI Journal
26
(5)
443 
450
DOI : 10.4218/etrij.04.0703.0006