Precoded spatial multiplexing MIMO system with spatial component interleaver
 Xiang Gao^{1} and
 Zhanji Wu^{1}Email author
https://doi.org/10.1186/s1363801605588
© Gao and Wu. 2016
Received: 21 September 2015
Accepted: 9 February 2016
Published: 29 February 2016
Abstract
In this paper, the performance of precoded bitinterleaved coded modulation (BICM) spatial multiplexing multipleinput multipleoutput (MIMO) system with spatial component interleaver is investigated. For the ideal precoded spatial multiplexing MIMO system with spatial component interleaver based on singular value decomposition (SVD) of the MIMO channel, the average pairwise error probability (PEP) of coded bits is derived. Based on the PEP analysis, the optimum spatial Qcomponent interleaver design criterion is provided to achieve the minimum error probability. For the limited feedback precoded proposed scheme with linear zero forcing (ZF) receiver, in order to minimize a bound on the average probability of a symbol vector error, a novel effective signaltonoise ratio (SNR)based precoding matrix selection criterion and a simplified criterion are proposed. Based on the average mutual information (AMI)maximization criterion, the optimal constellation rotation angles are investigated. Simulation results indicate that the optimized spatial multiplexing MIMO system with spatial component interleaver can achieve significant performance advantages compared to the conventional spatial multiplexing MIMO system.
Keywords
1 Introduction
Coded modulation is one of the pivotal techniques of wireless communications. High spectral efficiency and link reliability are always the challenges and goals of wireless communication systems. Nowadays, bitinterleaved coded modulation (BICM) multipleinput multipleoutput (MIMO) technology has become one of the fundamental technologies and has been widely used in current wireless communication standards, such as IEEE 802.11ac, 3GPP LTE [1–5].
For the MIMO system, spatial multiplexing is an effective method to multiply channel capacity and spectral efficiency. A high data rate signal stream is mapped onto multiple layers and each lowrate substream is transmitted simultaneously [6–8]. Unfortunately, spatial multiplexing is sensitive to illconditioning of the channel matrix. Besides, MIMO can also be implemented to obtain the diversity gain. Precoding and spacetime coding (STC) are the most commonly used technologies. The basic idea of precoding is to use some form of channel state information (CSI) at the transmitter and receiver to customize the transmitted signal to the eigenstructure of the matrix channel [8–13]. The precoding based on singular value decomposition (SVD) with full CSI is known to achieve the MIMO channel capacity. After the SVD of channel matrix, the transmit precoding and receiver shaping transform the MIMO channel into M independent singleinput singleoutput (SISO) channels, where M is the number of the transmit layers [8]. The most serious drawback of SVD precoding is the requirement of complete CSI at both the transmitter and receiver. To reduce the data for the feedback channel, limited feedback (LF) precoding schemes [13–17] were proposed, where a finite set of predetermined unitary precoding matrices, referred to as the unitary codebook, are known to both the transmitter and the receiver. The receiver only needs to feedback the index of the precoding matrix as a function of the current CSI over a limited feedback channel. This practical approach significantly reduces the feedback overhead of MIMO systems.
However, a substantial tradeoff between spatial diversity and spatial multiplexing gains exists in a MIMO system [18–22]. How to get the maximum diversity gain and multiplexing gain at the same time is the main concern. Spacetime bitinterleaved coded modulation (STBICM) is a fullrate spacetime code to obtain high diversity and coding gain on MIMO channel [23, 24]. Through the serial concatenation of a channel encoder, an optimized bit interleaver, and a spacetime signal constellation mapper, the diversity order and the coding gain depend on the Hamming distance of certain coded bit subsequences. In order to exploit the maximum diversity gain, the joint design of the code, the bit interleaver, and the spacetime constellation mapper has great importance. A similar scheme named bitinterleaved coded multiple beamforming (BICMB) is proposed in [25]. It showed that for any convolutional code and any spatial demultiplexer, the maximum achievable diversity order is related with the product of the code rate and the number of streams. In fact, STBICM and BICMB can be viewed as the spacetime extension of the BICM concept. The optimal performance is based on the ideal interleaving condition or a optimized interleaver which enables a global optimization taking into account channel coding. Therefore, it put forward a very high request to the design of bit interleaver to achieve the ideal interleaving condition which is highly correlated with the channel coding and MIMO configuration.

Most of the proposed MIMO systems with SSD in existing literature are merely an extension of SISOSSD system, such as the proposed schemes in [36, 37]. It is simply a combination of spatial multiplexing MIMO and time domain component interleaving on each layer. In order to achieve the diversity gain, the variability of fading on time domain needs to be guaranteed. However, the spatial dimension provided by MIMO channel is unconsidered.

The optimal rotation angles in the current research mainly based on the optimization of uncoded system [34–36], such as the maximum product distance criterion introduced in [26], or based on the extensive computer simulation [38, 39]. As for the coded MIMO scheme, actual operating signaltonoise ratio (SNR) is quite low that it invalidates the results. Hence, the angle values that depend on the maximum product distance do not lead to the best error performance for the coded modulation MIMO schemes.

For the coded MIMO system with SSD, the theoretical performance analysis is very difficult. Compared with SISO system, coded MIMO system with SSD lacks the theoretical analysis of the performance. The performance of proposed MIMOSSD systems in [34–39] are evaluated by simulations.

For the coded spatial multiplexing MIMO system, existing literatures mainly focus on the application and optimization of SSD with openloop MIMO or SVDprecoded closeloop MIMO. Few studies on the SSD technology concentrated on limited feedback precoding.

The optimum spatial Qcomponent interleaver design criterion for the proposed CISM scheme with SVD precoding is discussed.
An efficient spatial Qcomponent interleaver is proposed in [40] (Eq. (5)). It is one of the important foundation of SEP analysis and strongly associated with the conclusion in [40]. However, the proposed interleaver lacks theoretical basis. In this paper, we first investigate the performance of an ideal SVDprecoded BICM CISM scheme. Based on the analysis of average PEP of coded bits, the optimum spatial Qcomponent interleaver design criterion is provided to attain the minimum error probability. We proved that the optimum spatial Qcomponent interleaver is exactly the interleaver proposed in [40] and consummate the theoretical analysis of [40].

The optimal precoding matrix selection criterion that is suitable for the proposed MIMO scheme with LF precoding is discussed.
In [40], the CISM scheme with SVD precoding is studied. In this paper, the performance of the proposed coded CISM scheme with practical LF precoding is analyzed. Based on the linear zero forcing (ZF) receiver, in order to minimize a bound on the average probability of a symbol vector error, a novel effective SNRbased precoding matrix selection criterion and a simplified criterion that are suitable for the proposed MIMO scheme are proposed.

The optimal constellation rotation angle of the proposed coded MIMO scheme is investigated.
In [40], the optimal rotation angle obtained by SEP analysis is only applicable to the uncoded CISM scheme with SVD precoding. For the coded MIMO SSD system, the optimal angle depends on many factors, i.e., the number of antennas, the number of transmit layers, code rate, and modulation, which are usually ignored in present papers. Average mutual information (AMI) is an effective means to reflect the system performance and has been widely used in system optimization [32, 41–43]. A nonasymptotic spacetime block code (STBC) design criterion based on the bitwise AMI maximization at a specific target SNR is proposed in [43]. It establishes the relation curve between the optimal design parameter θ and SNR. Therefore, the operating SNR should be determined before selecting the optimal parameter. In this paper, the BICMAMI is used to search for the optimal constellation rotation angle. Based on the AMImaximization criterion, the relationship between the system achievable rate and the optimal angle is established. It provides a direct reference to choose the optimal angle without determination of SNR.
Simulation results verify the theoretical analysis and show that the optimized spatial multiplexing MIMO system with spatial component interleaver can achieve significant performance advantages.
Throughout this paper, we use bold letters to represent vectors or matrices. (·)^{ T } and (·)^{ H } represent transposition and conjugate transposition. tr(·) represents trace of a matrix. SNR = E _{r}/N _{0}, where E _{r} denotes the average symbol energy per receive antenna and N _{0}=2σ ^{2} denotes the variance of the complex Gaussian noise. Extensive literature has proven that Gray labeling is optimal for BICM system [44]. Therefore, Gray labeling is employed in this paper.
The paper is organized as follows. The spatial multiplexing MIMO scheme with spatial Qinterleaver is proposed in Section 2. Performance analysis about the proposed scheme with optimum SVD precoding is given in Section 3. The limited feedback unitary precoding for the proposed scheme is discussed in Section 4. Based on the AMI analysis, the optimal rotation angles for proposed systems are presented in Section 5. Simulation results are presented in Section 6 on Rayleigh fading channels. Concluding remarks are offered in Section 7.
2 System model
where \({s}_{k}^{n}\) denotes the kth symbol at the nth (n∈[ 1,M]) layer after the spatial Qcomponent interleaver. Obviously, Icomponents keep the same layerorder as before, and just Qcomponents change the layerorder.
where H is the N _{ R }×N _{ T } MIMO channel matrix. In this paper, the entries of H is supposed to be independent and identically distributed (i.i.d.) according to \({\mathcal {CN}}(0,1)\). F is the precoding matrix. \(\mathbf {n}=\left [n_{k}^{1},{n_{k}^{2}},\ldots,n_{k}^{N_{R}}\right ]^{T}\) denotes a column vector of N _{ R } complex Gaussian random variables with mean zero and variance \({\sigma ^{2}} = \frac {{{N_{0}}}}{2}\). Assuming the perfect CSI, after the MIMO detection and corresponding spatial Qcomponent deinterleaving, the received symbol on lth layer is reconstructed as \({y_{k}^{l}}={{y_{k}^{l}}(I)} + {\mathrm {j}} \cdot {{y_{k}^{l}}(Q)}\) that corresponds to \({x_{k}^{l}}\) in the transmitter. A serial concatenation of a softinsoftout rotated symbol demapper and a channel decoder is employed on each layer to approach the maximum likelihood (ML) receiver performance. For the lth layer, l∈[1,M], assume that the code sequence c _{ l } is transmitted and \(\hat {\mathbf {c}}_{l}\) is detected. The jth coded bits \({c_{l}^{j}}\) is mapped to the ith bit of kth symbol \({x_{k}^{l}}\). The soft demapper calculates the loglikelihood ratio (LLR) as follows.
where \({\chi }_{i}^{\alpha }\) denotes the subset of all signals x∈χ whose label has the value α∈{0,1} in position i. According to the LLRs, the information bits are decoded via channel decoder.
3 Performance analysis of the proposed scheme with SVD precoding
where the N _{ R }×N _{ R } matrix U and the N _{ T }×N _{ T } matrix V are unitary matrices. Λ is a N _{ R }×N _{ T } diagonal matrix with singular values \({\lambda _{i}} \in {{\mathbb {R}}^ + }\) of H on the main diagonal in decreasing order. We denote \(\bar {\mathbf {U}}_{\left [M \right ]}\) and \(\bar {\mathbf {V}}_{\left [ M \right ]}\) as the first M column vectors of U and V, respectively. As introduced in [13, 40, 45], for the M layer spatial multiplexing MIMO system, the precoding and detection process can be expressed as linear transformations \({\textbf {z}}=\bar {\mathbf {U}}_{\left [ M \right ]}^{H} {\mathbf {H}} \bar {\mathbf {V}}_{\left [ M \right ]} \mathbf {s}+\bar {\mathbf {U}}_{\left [ M \right ]}^{H} {\mathbf {n}}\).
In this paper, convolutional code is employed as the channel code. The Hamming distance between c _{ l } and \(\hat {\mathbf {c}}_{l}\) is at least d _{free}. Without loss of generality, we assume \(d\left ({\textbf {c}_{l}},{\hat {\mathbf {c}}_{l}} \right) = {d_{{\text {free}}}}\). Thus, the elements in \(\chi _{i}^{{c_{l}^{\,j}}}\) and \(\chi _{i}^{\hat {c}_{l}^{\,j}}\) are equal for all j except for d _{free} distinct values of j.
Lemma 1.
For the SVDprecoded M layer spatial multiplexing MIMO system with spatial Qcomponent interleaver, the descendingorder eigenvalues of MIMO channel matrix are λ _{1}>λ _{2}>⋯>λ _{ M }. The optimum Qcomponent interleaving rule is f(l)=M−l+1, l=1,2,…,M, which brings \({\mathop {\min }\limits _{l \in [1,M]} \left ({{\lambda _{l}^{2}} + \lambda _{f(l)}^{2}} \right)}\) to its maximum value.
Proof.
See Appendix. □
From Lemma 1, we prove that the component interleaver used in [40] is optimal for the CISM scheme with SVD precoding.
4 Limited feedback precoding for the proposed scheme
The most serious drawback of SVD precoding is the requirement of complete CSI at both the transmitter and receiver. To reduce the data rate requirement for the feedback channel, LF precoding has drawn much attention and has been widely used in practical systems.
where H _{ p }=H F.
For the spatial multiplexing MIMO system without spatial Qcomponent interleaver, the optimal precoder selection criteria from a codebook were proposed in [13, 14]. It is shown that in order to minimize a bound on the average probability of a symbol vector error, the minimum substream SNR S N R _{min} must be maximized. A close approximation to maximizing the minimum SNR for ZF receiver is also provided under the assumption of M<N _{ T }. These two error ratebased percoding matrix selection criteria (SC) are summarized as follows.
where \({{SNR}_{l}^{{\textbf {F}_{k}}}}\) is the SNR for layer l corresponding to the precoding matrix F _{ k }.
where λ _{min}(H F _{ k }) is the minimum singular value of HF _{ k }.
where \(\overline {SNR}_{\min }^{{\text {eff}}} = \mathop {\min }\limits _{l \in [1,M]} \left ({{\overline {SNR}}_{l}^{{\text {eff}}}} \right)\). \({d_{\min }^{2}}\) is the squared minimum distance and N _{ e } is the average number of nearest neighbors of the perlayer constellation. Therefore, the performance of the proposed scheme is dominated by the minimum effective SNR of M layers. Thus, based on the optimal error performance, the optimal precoding matrix selection criterion that is applicable to the proposed scheme can be expressed as follows.
where \({\overline {SNR}_{l}^{{\text {eff}},{\textbf {F}_{k}}}}\) is the effective SNR for layer l corresponding to the precoding matrix F _{ k }. For each precoding matrix \(\textbf {F}_{k} \in {\mathcal {P}}\), the matrix with the largest \(\overline {SNR}_{\min }^{{\text {eff}}}\) is chosen.
SC 3 can provide the optimal error performance for the proposed scheme. However, to calculate \(\overline {SNR}_{\min }^{{\text {eff}}}\), it requires a search over all layers according to (14) based on the spatial Qcomponent interleaving rule for all alternative precoding matrix \({\textbf {F}_{k}} \in {\mathcal {P}}\). The computational complexity is much high, especially when M and the size of codebook are large. This motivates us to design a novel precoder selection criterion for the proposed scheme based on the error performance with computational reduction.
In order to achieve the maximum \(\overline {SNR}^{{\text {eff}}}\), it has to make sure that (19) has the minimum value. Therefore, for the proposed scheme with M=2, the criterion that choosing the precoding matrix with the minimum value of \(\sum \limits _{k = 1}^{2} {\frac {1}{{{\lambda _{k}^{2}}\left ({{\textbf {H}_{p}}} \right)}}}\) leads to the optimal error performance.
Through the above analysis, we find a simple method to calculate the minimum effective SNR \(\overline {SNR}_{\min }^{{\text {eff}}}\) for SC 3. Instead of the calculation of effective SNR for each layer that is related to \({\left [ {\textbf {H}_{p}^{H}{\textbf {H}_{p}}} \right ]^{ 1}}\), the simplified method only need to compute the singular value of H _{ p }. We define that \(g\left ({{\textbf {F}_{k}}} \right) \buildrel \Delta \over = \sum \limits _{l = 1}^{M} {\frac {1}{{{\lambda _{l}^{2}}\left (\textbf {HF}_{k} \right)}}}\). A precoding matrix selection criterion with low complexity can be expressed as follows.
For each precoding matrix \(\textbf {F}_{k} \in {\mathcal {P}}\), the matrix with the minimum g(F _{ k }) is chosen.
It is worth noting that for the proposed scheme with M=2, SC 4 are equal to SC 3.
5 Optimal rotation angle based on AMI analysis
5.1 BICMAMI and motivation of optimization
In the practical communication system, the transmitted signal \({\mathbf {x}}={\left [{x_{k}^{1}},{x_{k}^{2}},\ldots,{x_{k}^{M}}\right ]^{T}}\) usually takes on a discrete finite alphabet (constellation signal set). Assuming equiprobable inputs, the AMI between the signal after rotated constellation mapping and the signal after the soft demapper is named BICMAMI [1].
where \({g_{I}}\left (x \right) = \frac {{{e^{x}}}}{{1 + {e^{x}}}}{\log _{2}}\frac {{{e^{x}}}}{{1 + {e^{x}}}} + \frac {1}{{1 + {e^{x}}}}{\log _{2}}\frac {1}{{1 + {e^{x}}}}\).
For the proposed scheme, the optimal constellation rotation angle is one of the most important problems for SSD. For the coded MIMO system, a number of factors (the number of antennas, precoding, modulation, code rate, and so on) will impact the optimal angle. As a result, the selection of the optimum angle becomes much more challenging. Maximizing BICMAMI is an effective way to optimize the system performance. In [43], the BICMAMI is used to design the optimal rotation angle of traceorthonormal STBC by establishing the relationship between the optimal angle and SNR. It inspires us to optimize the rotation angle of the proposed scheme in the same way. In this paper, BICMAMI is first used to evaluate the performance advantages of the optimal component interleaver and precoding matrix selection criterion proposed in Sections 3 and 4. Based on AMI maximization criterion, determination of the optimal rotation angle by establishing the relationship curve between optimal angle and corresponding AMI is investigated.
5.2 BICMAMI of the proposed schemes
The numeric results are shown in Fig. 2. The optimum rotation angle for SVDprecoded MIMO system with optimal Qcomponent interleaver is 29°, while it is 32° for the proposed scheme with cyclic Qcomponent interleaver. It is worth noting that the AMI of the proposed scheme with optimum Qcomponent interleaver is larger than that of the proposed scheme with cyclic Qcomponent interleaver. This observation means that the optimum Qcomponent interleaver proposed in Lemma I outperforms the commonly used cyclic Qcomponent interleaver.
For the LF precoded MIMO system, the AMIs of proposed systems with SC 1 and SC 4 are plotted in the same figure. As can be seen from the Fig. 3 a, for both percoding matrix selection criteria, the optimal rotation angles of the proposed MIMO system with N _{ R }=N _{ T }=4, M=2 are different at SNR = −2.6 dB, which are about 45° and 31° for SC 1 and SC 4, respectively. For N _{ R }=N _{ T }=8, M=4 MIMO system based on LF precoding with the cyclic Qcomponent interleaver at SNR = −3 dB, the optimal angles for SC 1 and SC 4 are both 0°. From Fig. 3, it can be observed that BICMAMI for SC 4 is always not less than that of SC 1. This observation indicates that the proposed MIMO scheme with SC 4 has a better performance than the proposed scheme with SC 1.
5.3 Determination method of optimal rotation angle
The optimal rotation angles for the proposed schemes with Gray labeled QPSK
N _{ R }  N _{ T }  M  R  Modulation  Precoding  θ ^{opt} 

8  8  4  3/4  QPSK  SVD  45° 
8  8  6  3/4  QPSK  SVD  27° 
8  8  8  3/4  QPSK  SVD  25° 
4  4  2  1/2  QPSK  LF SC 4  0° 
4  4  2  3/4  QPSK  LF SC 4  45° 
8  8  2, 4, 6  1/2  QPSK  LF SC 4  0° 
6 Simulation result
In this section, simulation results are provided to illustrate the performance of the proposed spatial multiplexing MIMO system with spatial component interleaver. In the simulation, 1/2rate 64state BCC with the generator of (133,171)_{8} is used as the channel code. The high code rate R=3/4 is derived from it by employing “puncturing” as introduced in [49]. The coded bit length N=1200. The decoding of BCC is the standard BahlCockeJelinekRaviv (BCJR) algorithm. The i.i.d. Rayleigh MIMO fading channel is employed and the channel is changed independently from each block of N coded bits.
6.1 Results of SVDprecoded scheme
6.2 Results of LF precoded scheme
7 Conclusions
In this paper, precoded BICM spatial multiplexing MIMO system with spatial component interleaver is discussed. For the SVDprecoded spatial multiplexing MIMO system with spatial component interleaver, the PEP of coded bits is analyzed. Based on the bound of PEP, the optimum spatial Qcomponent interleaver design criterion and a optimum spatial Qcomponent interleaving rule are proposed. The performance of conventional spatial multiplexing MIMO system is dominated by the smallest singular value of the channel matrix (the weakest layer), while the proposed scheme can improve the performance of the weakest layer through the spatial component interleaving. In addition, the LF precoded BICM spatial multiplexing MIMO system with spatial component interleaver and ZF receiver is studied. Based on the minimum average probability of symbol vector error criteria, the optimal effective SNRbased precoding matrix selection criterion and a simplified criterion are proposed. The simplified criterion has the same performance for twolayer transmission and small performance loss for M>2. Moreover, the method of determining the optimal rotation angles for the proposed schemes via maximizing the BICM AMI is also presented. The AMI analysis also illustrates that the optimal rotation angle is more crucial for SVDprecoded MIMO scheme with spatial component interleaver. For the proposed scheme with LF precoding, performance gain brought by constellation rotation is very limited. The simulation results validate all the theoretical analyses. Through the above optimization, the performance of the proposed precoded BICM spatial multiplexing MIMO system with spatial component interleaver outperforms that of conventional precoded spatial multiplexing MIMO system.
8 Appendix
8.1 Proof of Lemma 1
Eigenvalue sequences for proposed optimum spatial Qcomponent interleaver in Lemma 1
ρ _{ l }  ρ _{1}  ρ _{2}  ⋯  ρ _{ i−1}  ρ _{ i }  ρ _{ i+1}  ⋯  ρ _{ M−1}  ρ _{ M } 

ρ _{ f(l)}  ρ _{ M }  ρ _{ M−1}  ⋯  ρ _{ M−i+2}  ρ _{ M−i+1}  ρ _{ M−i }  ⋯  ρ _{2}  ρ _{1} 
From Table 2, the sum of nth (\(n \in \left [ {1,\left \lfloor {\frac {M}{2}} \right \rfloor } \right ]\)) pair of eigenvalues (the nth column) is equal to that of (M−n+1)th pair of eigenvalues (the (M−n+1)th column). Without loss of generality, we assume that the sum of the ith (\(1 \le i \le \left \lfloor {\frac {M}{2}} \right \rfloor \)) pair of eigenvalues has the minimum value, that is η _{ f }=ρ _{ i }+ρ _{ M−i+1}.
Eigenvalue sequences for new spatial Qcomponent interleaver rule f ^{′}
ρ _{ l }  ⋯  ρ _{ i }  ⋯  ρ _{ M−k+1}  ⋯ 

\(\boldsymbol {\rho }_{f^{\prime }(l)}\phantom {\dot {i}\!}\)  ⋯  ρ _{ k }  ⋯  ρ _{ M−i+1}  ⋯ 
In the case of k>M−i+1, we can get \({\eta _{{f}^{\prime }}} \le {\rho _{i}} + {\rho _{k}} < {\rho _{i}} + {\rho _{M  i + 1}}\phantom {\dot {i}\!}\). Therefore, it always has \({\eta _{{f}^{\prime }}}<{\eta _{f}}\phantom {\dot {i}\!}\).
If k<M−i+1, M−k+1>i. For the new spatial Qcomponent interleaver rule f ^{′}, \({\rho _{i}} + {\rho _{k}} > {\rho _{M  k + 1}} + {\rho _{M  i + 1}} \ge {\eta _{{f}^{\prime }}}\phantom {\dot {i}\!}\). Because \({\rho _{i}} + {\rho _{M  i + 1}} > {\rho _{M  k + 1}} + {\rho _{M  i + 1}} \ge {\eta _{{f}^{\prime }}}\phantom {\dot {i}\!}\), it always has \({\eta _{{f}^{\prime }}}<{\eta _{f}}\phantom {\dot {i}\!}\).
In a word, we cannot find a new spatial Qcomponent interleaver rule f ^{′} which has \({\eta _{{f}^{\prime }}} > {\eta _{f}}\phantom {\dot {i}\!}\). f(l)=M−l+1 is the optimum Qcomponent interleaving rule to assure the maximum value of \({\mathop {\min }\limits _{l \in [1,M]} \left ({{\lambda _{l}^{2}} + \lambda _{f(l)}^{2}} \right)}\).
Declarations
Acknowledgements
This work is sponsored by the National Natural Science Fund (61171101), the National Great Science Specific Project (2009ZX0300301103) of People’s Republic of China, the Fundamental Research Funds for the Central Universities, and the 2014 BUPT Excellent Ph.D. Students Foundation (CX201426).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 G Caire, G Taricco, E Biglieri, Bitinterleaved coded modulation. IEEE Trans. Inf. Theory. 44(3), 927–946 (1998).View ArticleMathSciNetMATHGoogle Scholar
 A Martinez, A Guillen i Fabregas, G Caire, Error probability analysis of bitinterleaved coded modulation. IEEE Trans. Inf. Theory. 52(1), 262–271 (2006).View ArticleMathSciNetMATHGoogle Scholar
 B Vucetic, J Yuan, SpaceTime Coding (John Wiley & Sons, Inc., England, 2003).View ArticleGoogle Scholar
 IEEE Std. P802.11ac/D6.0: Part 11: Wireless LAN medium access control (MAC) and physical layer (PHY) specifications. Amend. 4: Enhancements for Very High Throughput for Operation sin Bands Below 6 GHz (2013).Google Scholar
 3rd Generation Partnership Project; Technical specification group radio access network; Evolved universal terrestrial radio access (EUTRA); Physical channels and modulation (release 12). 3GPP TS 36.211, version 12.2.0 (2014). http://www.3gpp.org/ftp/Specs/archive/36_series/36.211/.
 E Telatar, Capacity of multiantenna gaussian channels. Eur. Trans. Telecommun.10:, 585–596 (1999).View ArticleGoogle Scholar
 GJ Foschini, Layered spacetime architecture for wireless communication in fading environment when using multiple antennas. Bell Labs. Techn. J.6(2), 41–59 (1996).Google Scholar
 A Goldsmith, Wireless Communications (Cambridge University Press, England, 2004).Google Scholar
 D Love, R Heath, T Strohmer, Grassmannian beamforming for multipleinput multipleoutput wireless systems. IEEE Trans. Inf. Theory. 49(10), 2735–2747 (2003).View ArticleMATHGoogle Scholar
 E Akay, E Sengul, E Ayanoglu, Bit interleaved coded multiple beamforming. IEEE Trans. Commun.55(9), 1802–1811 (2007).View ArticleGoogle Scholar
 E Sengul, J Hong, E Ayanoglu, Bitinterleaved coded multiple beamforming with imperfect CSIT. IEEE Trans. Commun.57(5), 1505–1513 (2009).View ArticleGoogle Scholar
 L Boyu, E Ayanoglu, Multiple beamforming with perfect coding. IEEE Trans. Commun.60(6), 1575–1586 (2012).View ArticleGoogle Scholar
 D Love, R Heath, V Lau, D Gesbert, B Rao, M Andrews, An overview of limited feedback in wireless communication systems. IEEE J. Selected Areas Commun.26(8), 1341–1365 (2008).View ArticleGoogle Scholar
 D Love, R Heath, Limited feedback unitary precoding for spatial multiplexing systems. IEEE Trans. Inf. Theory. 51(8), 2967–2976 (2005).View ArticleMathSciNetMATHGoogle Scholar
 H Wang, L Li, Y Zhang, M Juntti, in Proceedings of the 2012 IEEE Consumer Communications and Networking Conference (CCNC): 1417 Jan. 2012. Unitary precoder design for mimo spatial multiplexing systems with limited feedback (IEEELas Vegas, NV, 2012), pp. 838–842. http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6181175%26newsearch=true%26queryText=Unitary\%20precoder\%20design\%20for\%20mimo\%20spatial\%20multiplexing\%20systems\%20with\%20limited\%20feedback.View ArticleGoogle Scholar
 L Ding, R Liu, B Jiang, X Gao, in Proceedings of 2010 International Conference on Wireless Communications and Signal Processing (WCSP): 2123 Oct. 2010. Limited feedback unitary precoding using improved euclidean distance metrics for spatial multiplexing systems (IEEESuzhou, 2010), pp. 1–6. http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=5633557&newsearch=true&queryText=Limited\%20feedback\%20unitary\%20precoding\%20using\%20improved\%20euclidean\%20distance\%20metrics\%20for\%20spatial\%20multiplexing\%20systems.Google Scholar
 P Cheng, Z Chen, Y Rui, Y Guo, Limited feedback unitary precoding for mimo full stream transmission. IEEE Trans. Veh. Technol.63(8), 4092–4096 (2014).View ArticleGoogle Scholar
 L Zheng, DN Tse, Diversity and multiplexing: a fundamental tradeoff in multiple antenna channels. IEEE Trans. Inf. Theory. 49(5), 1073–1096 (2003).View ArticleMATHGoogle Scholar
 H Gamal, G Caire, M Damon, Lattice coding and decoding achieve the optimal diversitymultiplexing tradeoff of mimo channels. IEEE Trans. Inf. Theory. 50(6), 968–985 (2004).View ArticleMATHGoogle Scholar
 S Sfar, L Dai, KB Letaief, Optimal diversitymultiplexing tradeoff with group detection for mimo systems. IEEE Trans. Commun.53(7), 1178–1190 (2005).View ArticleGoogle Scholar
 E Sengul, E Akay, E Ayanoglu, in Proceedings of IEEE 61st Vehicular Technology Conference (VTC): 30 May1 June 2005. Diversity analysis of single and multiple beamforming (IEEE, 2005), pp. 1293–1296. http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1543517&newsearch=true&queryText=Diversity\%20analysis\%20of\%20single\%20and\%20multiple\%20beamforming.
 S Sfar, L Dai, KB Letaief, Diversity analysis of single and multiple beamforming. IEEE Trans. Commun.54(6), 990–993 (2006).View ArticleGoogle Scholar
 AM Tonello, in Proceedings of IEEE 52nd Vehicular Technology Conference (VTCFall): 2428 Sep. 2000. Spacetime bitinterleaved coded modulation with an iterative decoding strategy (IEEEBoston, MA, 2000), pp. 473–478. http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=886696&newsearch=true&queryText=Spacetime\%20bitinterleaved\%20coded\%20modulation\%20with\%20an\%20iterative\%20decoding\%20strategy.Google Scholar
 N Gresset, L Brunel, JJ Boutros, Spacetime coding techniques with bitinterleaved coded modulations for MIMO blockfading channels. IEEE Trans. Inf. Theory. 54(5), 2156–2178 (2008).View ArticleMathSciNetMATHGoogle Scholar
 Hong Ju Park, E Ayanoglu, Diversity analysis of bitinterleaved coded multiple beamforming. IEEE Trans. Commun.58(8), 2457–2463 (2010).View ArticleGoogle Scholar
 J Boutros, E Viterbo, Signal space diversity: a power and bandwidth efficient diversity technique for the rayleigh fadin channel. IEEE Trans. Inf. Theory. 44(4), 1453–1467 (1998).View ArticleMathSciNetMATHGoogle Scholar
 NF Kiyani, UH Rizvi, JH Weber, GJM Janssen, in Proceedings of IEEE Wireless Communications and Networking Conference: 1115 March 2007. Optimized rotations for ldpccoded mpsk constellations with signal space diversity (IEEEKowloon, 2007), pp. 677–681. http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4224375&newsearch=true&queryText=Optimized\%20rotations\%20for\%20ldpccoded\%20mpsk\%20constellations\%20with\%20signal\%20space\%20diversity.View ArticleGoogle Scholar
 NH Tran, HH Nguyen, T LeNgoc, Performance of BICMID with signal space diversity. IEEE Trans. Wireless Commun.6(5), 1732–1742 (2007).View ArticleGoogle Scholar
 NF Kiyani, JH Weber, in Proceedings of IEEE Symposium on Communications and Vehicular Technology in the Benelux: 1515 Nov. 2007. OFDM with BICMID and rotated MPSK constellations and signal space diversity (IEEEDelft, 2007), pp. 1–4. http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4436247&newsearch=true&queryText=OFDM\%20with\%20BICMID\%20and\%20rotated\%20MPSK\%20constellations\%20and\%20signal\%20space\%20diversity.View ArticleGoogle Scholar
 NF Kiyani, JH Weber, EXIT chart analysis of iterative demodulation and decoding of mpsk constellations with signal space diversity. J. Commun.3(3), 43–50 (2008).View ArticleGoogle Scholar
 M Zhenzhou, S Zhiping, Z Chong, Z Zhongpei, in Proceedings of International Conference on Communications, Circuits and Systems: 2527 May 2008. Design of signal space diversity based on nonbinary ldpc code (IEEEFujian, 2008), pp. 31–34. http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4657720&newsearch=true&queryText=Design\%20of\%20signal\%20space\%20diversity\%20based\%20on\%20840\%20nonbinary\%20ldpc\%20code.View ArticleGoogle Scholar
 Q Xie, J Song, K Peng, F Yang, Z Wang, Coded modulation with signal space diversity. IEEE Trans. Wireless Commun.10(2), 660–668 (2011).View ArticleGoogle Scholar
 Z Wu, M Peng, W Wang, Improved codingrotatedmodulation orthogonal frequency division multiplexing system. IET Commun.6(3), 272–280 (2012).View ArticleMathSciNetGoogle Scholar
 L Yueqian, M Salehi, in Proceedings of 46th Annual Conference on Information Sciences and Systems (CISS): 2123 March 2012. Coded MIMO systems with modulation diversity for blockfading channels (IEEEPrinceton, NJ, 2012), pp. 1–5. http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6310745&newsearch=true&queryText=Coded\%20MIMO\%20systems\%20with\%20modulation\%20diversity\%20for\%20blockfading\%20849\%20channels.View ArticleGoogle Scholar
 L Heunchul, A Paulraj, MIMO systems based on modulation diversity. IEEE Trans. Wireless Commun.58(12), 3045–3049 (2010).Google Scholar
 SU Hwang, J Choi, S Jeon, HJ Ryu, in Proceedings of IEEE International Symposium on Broadband Multimedia Systems and Broadcasting: 1315 May 2009. Performance evaluation of MIMOOFDM with signal space diversity over frequency selective channels (IEEEBilbao, 2009), pp. 1–5. http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=5133821&newsearch=true&queryText=Performance\%20evaluation\%20of\%20MIMOOFDM\%20with\%20signal\%20space\%20diversity\%20over\%20frequency\%20selective\%20channels.View ArticleGoogle Scholar
 S Hong, J Choi, SU Hwang, S Jeon, JS Seo, in Proceedings of IEEE 69th Vehicular Technology Conference (VTC): 2629 April 2009, ed. by IEEE. Interleaved spatial diversity transmission with coordinate interleaver for MIMOOFDM systems (IEEE, Barcelona, 2009), pp. 1–4. Weblink: http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=5073614&newsearch=true&queryText=Interleaved\%20spatial\%20diversity\%20transmission\%20with\%20coordinate\%20interleaver\%20for\%20MIMOOFDM\%20systems.
 X Gao, Z Wu, in Proceedings of IEEE 14th International Conference on Communication Technology (ICCT): 911 Nov. 2012. Joint coding and modulation diversity mimo DFTSOFDM scheme (IEEEChengdu, 2012), pp. 1103–1107. http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6511361&newsearch=true&queryText=Joint\%20861\%20coding\%20and\%20modulation\%20diversity\%20mimo\%20DFTSOFDM\%20scheme.Google Scholar
 Z Wu, X Gao, An improved MIMOOFDM scheme for the next generation WLAN. J. Syst. Eng. Electron.24(1), 52–59 (2013).View ArticleGoogle Scholar
 KV Srinivas, RD Koilpillai, S Bhashyam, K Giridhar, Coordinate interleaved spatial multiplexing with channel state information. IEEE Trans. Wireless Commun.8(6), 2755–2762 (2009).View ArticleGoogle Scholar
 Sungnam Hong, Min Sagong, Chiwoo Lim, Kyungwhoon Cheun, in Proceedings of IEEE Globecom Workshops: 913 Dec. 2013. FQAM: a modulation scheme for beyond 4G cellular wireless communication systems (IEEEAtlanta, GA, 2013), pp. 25–30. http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6824956&newsearch=true&queryText=FQAM:\%20a\%20modulation\%20scheme\%20for\%20beyond\%204G\%20cellular\%20wireless\%20871\%20communication\%20systems.Google Scholar
 A Barbieri, D Fertonani, G Colavolpe, Timefrequency packing for linear modulations: spectral efficiency and practical detection schemes. IEEE Trans. Commun.57(10), 2951–2959 (2009).View ArticleGoogle Scholar
 A El Falou, C Langlais, CA Nour, C Douillard, in Proceedings of IEEE Vehicular Technology Conference (VTC Fall): 36 Sept. 2012. Adaptive traceorthonormal STBC for MIMO system with capacity approaching FEC codes (IEEEQuebec City, QC, 2012), pp. 1–5. http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6399204&newsearch=true&queryText=Adaptive\%20traceorthonormal\%20STBC\%20for\%20MIMO\%20system\%20with\%20capacity\%20approaching\%20FEC\%20codes.View ArticleGoogle Scholar
 SY Goff, Signal constellation for bitinterleaved coded modulation. IEEE Trans. Inf. Theory. 49(1), 307–313 (2003).View ArticleMathSciNetMATHGoogle Scholar
 JG Proakis, Digital Communications, 5th edn. (McGraw Hill, New York, 2008).Google Scholar
 HJ Park, E Ayanoglu, in Proceedings of IEEE International Conference on Communications (ICC): 1418 June 2009. Diversity analysis of bitinterleaved coded multiple beamforming (IEEEDresden, 2009), pp. 1–9. http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=5198942&newsearch=true&queryText=Diversity\%20analysis\%20of\%20bitinterleaved\%20coded\%20multiple\%20beamforming.Google Scholar
 RW Heath, S Sandhu, A Paulraj, Antenna selection for spatial multiplexing systems with linear receivers. IEEE Commun. Lett.5(4), 142–144 (2001).View ArticleGoogle Scholar
 R, 1091246, codebook design for 8 tx transmission in ltea3GPP TSG RAN WG1 Meeting #56bis. Seoul, Korea (2009). http://www.3gpp.org/ftp/tsg_ran/WG1_RL1/TSGR1_56b/Docs/.
 IEEE 802.11n 2009. Part 11: Wireless LAN medium access control (MAC) and physical layer (PHY) specifications, Amendment 5: Enhancements for higher throughput (2009).Google Scholar