 Research
 Open Access
Lattice reduction aided with block diagonalization for multiuser MIMO systems
 Md Hashem Ali Khan^{1},
 JinGyun Chung^{1} and
 Moon Ho Lee^{1}Email author
https://doi.org/10.1186/s1363801504761
© Khan et al. 2015
 Received: 9 August 2015
 Accepted: 2 November 2015
 Published: 3 December 2015
Abstract
The block diagonalization (BD) precoding technique is a wellknown linear transmit strategy for multiuser multiinput multioutput (MUMIMO) systems. The MUMIMO broadcast channel is decomposed into multiple independent parallel single user MIMO (SUMIMO) channels and achieves the maximum diversity order at high data rates. The lattice reductionaided decoding (LRAD) features the reduced decoding complexity in MIMO communications. The LenstraLenstraLovasz (LLL) algorithm has been extensively used to obtain better bases of the channel matrix while the complex lattice reduction (CLR) is aimed at improving orthogonality of basis vectors and shortening them. The orthogonalization and size reduction work are left for the CLR algorithm so that a modification of the channel matrix is carried out, resulting in better precoding and detection performances. We also derive bounds for lattice decoding. Simulation results show that the bit error rate (BER) performance of our proposed algorithm is better than that of conventional ones and it reduces the complexity compared with the LLL algorithmbased schemes.
Keywords
 Complex lattice reduction
 Block diagonalization
 Multiuser MIMO
 Detection algorithms
 Proximity factors
 Low complexity
1 Introduction
Multipleinput multipleoutput (MIMO) systems have been proposed for the nextgeneration wireless communication systems to increase the transmission capacity, and therefore, a highperformance and lowcomplexity MIMO detector becomes an important issue. The maximum likelihood detector (MLD) is known to be an optimal detector; however, it is impractical for realization owing to its great computational complexity. Signal processing is performed on a percell basis in conventional wireless systems. The zeroforcing (ZF) and minimum meansquare error (MMSE) precoding are the wellknown linear precoding schemes. Although linear precoding techniques have considerably low computational complexity, they show relatively low performance due to the susceptible noise amplification, particularly when the channel matrix is illconditioned. The block diagonalization (BD) is one of the key processing techniques for multiuser MIMO (MUMIMO) systems. The MUMIMO downlink channel can be decomposed into multiple parallel single user MIMO (SUMIMO) channels with the use of BD which was first proposed in [1]. Because of no interference between the users after BD, the MUMIMO channel can be transformed into equivalent SUMIMO channels [2], and then the SUMIMO techniques can be applied. Two singular value decomposition (SVD) operations have to be implemented through the BD algorithm for the complete or full BD reported in [1, 3]. By using the first SVD, the multiuser interference (MUI) is forced to be zero and the second SVD is used to produce orthogonal parallel SUMIMO channels. By replacing the first SVD operation with a less complex solution to mitigate the MUI, a QR decompositionbased BD precoding scheme is presented in [4] for MUMIMO systems. QRBD utilizes a QR decomposition to the MUIMIMO channel to obtain the null space of MUI. Therefore, the complexity of SVD operation on BD precoding is reduced by QR operation in QRBD precoding. A generalized ZF channel inversion (GZI) precoding method is developed in [4], where the MUIMIMO channel is operated by pseudo inversion and QR decomposition to mitigate the MUI. Furthermore, the generalized MMSE channel inversion precoding scheme denoted as GMI is proposed in [4] to balance the MUI and the noise for each user effectively.
Lattice reduction (LR) is another preprocessing and detection technique that has recently attracted significant research efforts. Yao and Wornell used the LR algorithm in conjunction with MIMO detection techniques [5]. LR is a powerful concept for solving diverse problems involving point lattices. The LR has been successfully used in signal processing applications including global positioning system (GPS), frequency estimation, and particularly data detection and precoding in wireless communication systems. Besides linear detection schemes based on the ZF or the MMSE criterion, successive interference cancelation (SIC) is a popular way to detect the transmitted signals at the receiver side [6]. The LR has been proposed in order to transform the system model into an equivalent one with a betterconditioned channel matrix prior to lowcomplexity linear or SIC detection [6]. The symbol error rate (SER) curves can parallel those of the MLD algorithms by using LRaided detection schemes, which has devoted a great deal of interest to exploring the application of LR in MIMO systems. The LRaided detection schemes with respect to the MMSE criterion have been extended by Wuebben et al. [6]. In [7], both the LRaided SUMIMO detection and the LRaided SUMIMO precoding have been investigated. LRaided MIMO precoding for decentralized receivers was discussed in [8–12]. The aim of the complex LR (CLR) algorithm is to find a new basis which is shorter and nearly orthogonal as compared to the original matrix [12]. Therefore, if the second precoding filters for the equivalent SUMIMO channels after the first SVD were designed based on the latticereduced channel matrix, a better bit error rate (BER) performance can be achieved. Then, a CLRaided regularized BD (RBD) precoding algorithm is proposed, which not only has a lower complexity but also achieves a better BER performance than the RBD or QR/SVD RBD [12, 13].

We propose complex lattice reduction aided with block diagonalization for MUMIMO systems.

A BDbased precoding algorithm is able to separate several SUMIMO channels from the MUMIMO downlink channel as well as achieve the maximum diversity order at high data rates and reduce the interference.

To reduce the complexity of precoding scheme, we employ the CLR to replace the SVD of conventional BDbased precoding algorithm by introducing a combined channel inversion to eliminate the MUI.

The LLL algorithm has been used to obtain better bases of the channel matrix, while the CLR is aimed at improving orthogonality of basis vectors and shortening them. We also derive the bounds for lattice decoding.

The simulation results show that the BER performance of our proposed algorithm is better than that of conventional algorithms and the complexity is reduced compared with the LLL algorithmbased schemes.
This paper is organized as follows. A system model is introduced in Section 2. In Section 3, we present precoding techniques in detail. In Section 4, we describe complex LRaided block diagonalization. In Section 5, MIMO detection algorithms are presented. In Section 6, we introduce performance bounds for lattice decoding, and complexity analysis is described in Section 7. Simulation results are presented in Section 8, and conclusions are drawn in Section 9.
2 System model
where \( \mathbf{D}\in {\mathbb{C}}^{r\times {N}_R} \) is the detection matrix, \( \mathbf{H}\in {\mathbb{C}}^{N_R\times {N}_T} \) is the complex Gaussian channel matrix with zero mean and unit variance, \( \mathbf{W}\in {\mathbb{C}}^{N_T\times r} \) is the precoding matrix, s ∈ ℂ ^{ r × 1} is the data vector, and n ∈ ℂ ^{ r } is the Gaussian noise with independent and identically distributed (i.i.d) entries of zero mean and variance N _{0}.
3 Precoding technique
In this section, we discuss conventional BD and CLR algorithms. This drawback would be more serious when the channel is highly correlated. One solution for this problem is known as BD which was first proposed in [3].
3.1 Block diagonalization
The product of \( {\mathbf{V}}_i^{(1)} \) and \( {\tilde{\mathbf{V}}}_i^{(0)} \) can yield an equivalent SUMIMO channel with orthogonal bases. Orthogonality can be measured by the coefficients \( {\mu}_{i,j}=\frac{\left\langle {h}_i,hj\right\rangle }{{\left\Vert {h}_j\right\Vert}^2} \), where h _{ i }, h _{ j } are the columns of the equivalent channel \( {\mathbf{H}}_i{\tilde{\mathbf{V}}}_i^{(0)}{\mathbf{V}}_i^{(1)} \).
3.2 CLLL reduction algorithm
where \( \mathcal{R}\left(\cdot \right),\mathcal{J}\left(\cdot \right) \) is the real and imaginary part, respectively.
The LR algorithm aims to find a new basis H _{ LR } for a given ℒ(H) which is shorter and nearly orthogonal compared with the original matrix H. Let the orthogonal factor be represented as \( {\mu}_{i,j}=\frac{\left\langle {h}_i,{h}_j^{*}\right\rangle }{{\left\Vert {h}_j^{*}\right\Vert}^2} \), where \( {h}_j^{*} \) represents the vectors generated by the GSO procedure.
where δ ∈ (1/2, 1) is a factor chosen to achieve a good performance with lower complexity. If only Eq. (9) is satisfied, this basis is the sizereduced basis as well. The parameter δ influences the quality of the reduced basis. Throughout this paper, δ = 3/4 as in [14].
4 Proposed complex LRaided BD
In this section, we combined the BD and CLR techniques. To cancel the MUI, we took the similar design concept from BD and thus the MUMIMO channel can be transformed into equivalent SUMIMO channels. We assume that the channel information is perfectly known both at the transmit side and the receiving side. We remark that a performance study of the proposed scheme with imperfect channel information and limited feedback can be considered.
where z = T ^{− 1} s and H _{ LR } possesses a better channel quality, and we can design the detector based on the better detector performance which can be achieved due to less noise enhancement increased by H _{ LR }. The basic idea behind approximate lattice decoding (LD) is to use LR in conjunction with traditional lowcomplexity decoders. With LR, the basis B is transformed into a new basis consisting of roughly orthogonal vectors. And the complexity is reduced also compared to the SVD technique.
5 MIMO detection algorithms
5.1 ZF and MMSE detection algorithms
5.2 Latticereductionaided linear detection
5.3 Latticereduction aided SIC
where the newly defined noise term η also incorporates residual interference. The detection procedure equals that of LRaided ZFSIC.
6 Performance bounds for lattice decoding
In this section, we shall introduce an analytic tool for approximate LD. However, such results do not directly translate into how close approximate LD is to LD in terms of the minimum distance, which is more useful in digital communications [18].
Consider a fixed but arbitrary nD complex lattice Λ. The decision regions of ZF and SIC have 2n faces. We only have to study n distances due to symmetry. The ith distance of ZF is d _{ i,ZF } = (1/2)‖h _{ i }‖ sin θ _{ i }, for i = 1, …, n, where θ _{ i } denotes the acute angle between and the linear space spanned by the other n − 1 basis vectors h _{1}, …, h _{ i − 1}, h _{ i + 1},...., h _{ n }. For the SIC detector, the ith distance is given by \( \left(1/2\right)\left\Vert {h}_i^{*}\right\Vert \).
The relations Eq. (29) and Eq. (30) hold irrespective of fading statistics, and similar relations exist for SIC. They reveal, in a quantitative manner, that approximate LD performs within a constant bound from LD. The mere effect on the error rate curve is a shift from that of LD, up to a multiplicative factor n, which obviously does not change the diversity order. In other words, the diversity order is the same as that of LD [18]. Therefore, existing results on the diversity order of LD can be extended to approximate LD. Moreover, since LD achieves full receive diversity in the uncoded VBLAST system [19], approximate LD also achieves full diversity. This provides an alternative way of showing the diversity order of LRaided decoding given in [19, 20].
7 Complexity analysis

Multiplication of m × n and n × p complex matrices: 8mnp

QR decomposition of an m × n(m ≤ n) complex matrix: 16(n ^{2} m − nm ^{2} + 1/3m ^{3})

SVD of an m × n(m ≤ n) complex matrix where only Σ and V are obtained: 32(nm ^{2} + 2m ^{2})

SVD of an m × n(m ≤ n) complex matrix where U, Σ, and V are obtained: 8(4n ^{2} m + 8nm ^{2} + 9m ^{3})

Inversion of an m × m real matrix: 2m ^{3} − 2m ^{2} + m
Complexity of LR algorithm
Steps  Operations  Flops  Case (2, 2, 2) × 6 

1  QR  \( 16K\left({N}_T^2{N}_i+{N}_T{N}_1^2+1/3{N}_1^3\right) \)  12544 
2  HW  \( 8{N}_R{N}_T^2 \)  1728 
3_{ ZF }  \( CLR{\left({H}_{LR}^T\right)}^T \)  \( 25.6K\left({N}_T^2{N}_i{N}_T{N}_i^2+1/3{N}_i^3\right) \)  3891 
4_{ ZF }  \( {H}_{LR}^T{\left({H}_{LR}{H}_{LR}^T\right)}^{1} \)  \( K\left(2{N}_1^32{N}_i^2+{N}_i+16{N}_T{N}_i^2\right) \)  1192 
4_{ MMSE }  \( {H}_{LR}^T{\left({H}_{LR}{H}_{LR}^T\right)}^{1} \)  \( K\left(18{N}_1^32{N}_i^2+{N}_i+16{N}_T{N}_i^2\right) \)  1566 
Computational complexity of QR/SVDBD [22]
Steps  Operations  Flops  Case 

1  H = QR  \( 16K\left({N}_T^2{N}_i+{N}_T{N}_i^2+1/3{N}_i^3\right) \)  12544 
2  H _{ eff } = HW  \( 8{N}_R{N}_T^2 \)  1728 
3  \( {\mathbf{H}}_{i, eff}={\mathbf{U}}_i{\boldsymbol{\Lambda}}_i{\mathbf{V}}_i^H \)  \( 64\left(9/8{N}_i^3+{N}_T{N}_i^2+1/2{N}_i^2{N}_i\right) \)  13248 
We focus on the computational complexity reduction of the alternative BD methods. The complexities of the alternative methods are usually compared by the number of floating point operations (flops). A flop is defined as a real floating operation, e.g., a real addition, multiplication, division, and so on. Based on the analysis, we summarize the computational costs of the alternative BD methods, where QRBD denotes the BD method similar as SVDBD but replacing the SVD operation with the fast Givens QR operation.
8 Simulations results
The analysis of probability of error is compared to the BER results of simulations.
9 Conclusions
In this paper, several detection schemes for multiple antenna systems are investigated, which make use of the LR algorithm proposed by [14]. It is shown that the performance of our proposed algorithm is better than that of conventional methods and the complexity is reduced compared with the LLLbased schemes. It is clear that the performance of BD precoding with LR is as almost similar to the LRMMSE detection. Aside from the improved performance, it is suggested that the MMSEbased LR has a significantly smaller complexity than the ZFbased LR. Simulation results evidence that our proposed algorithms have substantial performance gains compared to the existing MUMIMO linear precoding and BD detection.
Notes
Declarations
Acknowledgements
This work was supported by the MEST 2015R1A2A1A 05000977, NRF, Korea.
Open AccessThis 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
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