A simple block diagonal precoding for multiuser MIMO broadcast channels
 Md Hashem Ali Khan^{1},
 K M Cho^{1},
 Moon Ho Lee^{1}Email author and
 JinGyun Chung^{1}
https://doi.org/10.1186/16871499201495
© Khan et al.; licensee Springer. 2014
Received: 1 January 2014
Accepted: 24 May 2014
Published: 14 June 2014
Abstract
The block diagonalization (BD) is a linear precoding technique for multiuser multiinput multioutput (MIMO) broadcast channels, which is able to completely eliminate the multiuser interference (MUI), but it is not computationally efficient. In this paper, we propose the block diagonal Jacket matrix decomposition, which is able not only to extend the conventional block diagonal channel decomposition but also to achieve the MIMO broadcast channel capacity. We also prove that the QR algorithm achieves the same sum rate as that of the conventional BD scheme. The complexity analysis shows that our proposal is more efficient than the conventional BD method in terms of the number of the required computation.
Keywords
Multiuser MIMO Broadcast channel Precoding Block diagonalization QR decomposition Eigenvalue decomposition Diagonal Jacket matrix1 Introduction
Recently, the research of the capacity region of the multiuser multiinput multioutput (MIMO) broadcast channels (BC) has been of concern. It is well known that any algorithm requiring the eigenvalue decomposition (EVD) suffers from the high computational cost. In mobile wireless communication systems, in which MIMO technique is utilized, the channel characteristics may vary faster than the computation process of the precoding/decoding algorithm that is based on the EVD of the channel matrix that is changing instantaneously.
In [1], the authors proposed the MIMO channel precoding/decoding based on the Jacket matrix decomposition where we believe that the required computational complexity in obtaining diagonalsimilar matrices is smaller than that required in the conventional EVD.
Definition 1 Let J_{ N } ≜ {a_{i,j}} be a N × N matrix; then, it is called a Jacket matrix when ${\mathit{J}}_{\mathit{N}}^{1}=\frac{1}{\mathit{N}}{\left\{{\left({\mathit{a}}_{\mathit{i},\mathit{j}}\right)}^{1}\right\}}^{\mathit{T}}$, that is, the inverse of the Jacket matrix can be determined by its elementwise inverse [2, 3].
Definition 2 Let A be an n × n matrix. If there exists a Jacket matrix J such that A = J ∑ J^{−1}_{,} where Σ is a diagonal matrix, then we say that A is a Jacket matrix similar to the diagonal matrix ∑. Moreover, we say that A is a Jacket diagonalizable [4].
i.e., the entries of the main diagonal of a matrix are equal.
Proof Refer to [4] for the proof.
Multiuser diversity can significantly improve the performance of multiple antenna systems. The simplest ways to achieve the diversity gain in MIMO downlink communications are the zero forcing (ZF)based linear precoding approaches. In [5, 6], it was shown that the maximum sum rate in the multiuser MIMO broadcast channels can be achieved by dirty paper coding (DPC). However, the high computational complexity of the DPC makes it difficult to implement in practical systems. A suboptimal strategy of the DPC [7], the TomlinsonHarashima precoding (THP) algorithm which is based on nonlinear modulo operations, is still impractical due to its high complexity.
In linear processing systems, several practical precoding techniques have been proposed, typically as the channel inversion method [8, 9] and the block diagonalization (BD) method [10]. The ZF channel inversion scheme [8] can suppress cochannel interference (CCI) completely for the case where all users employ a single antenna. However, its performance is degraded due to the effect of noise enhancement. Although the minimum meansquared error (MMSE) channel inversion method [8] overcomes the drawback of the ZF, it is still confined to a singlereceive antenna case. In the scenario where multiple antennas are located at both the mobile terminal and base station for each user, lowcomplexity BD methods have been proposed [8, 11–13]. Moreover, the BD attempts to completely eliminate the multiuser interference (MUI) irrespective of the noise. The BD precoding has been proposed in [10] to improve the sum rate or reduce the transmitted power. A BD precoding algorithm has focused on how to implement the BD precoding algorithms with less computational complexity without the performance degradation. A lowcomplexity generalized ZF channel inversion (GZI) method has been proposed in [9] to equivalently implement the first singular value decomposition (SVD) operation of the original BD precoding, and a generalized MMSE channel inversion (GMI) method is also developed in [9] for the original regularized BD (RBD) precoding. Therefore, the performance of the BD scheme is poor at the low SNR regime, while preserving its good performance at high SNR. With the purpose of improving the performance of the BD, an RBD scheme [14] is proposed. The QR/SVD techniques require only low complexity to equivalently implement the BD precoding algorithms. As an improvement of the BD precoding algorithms, a lowcomplexity lattice reductionaided RBD (LCRBDLR)type precoding algorithm has been proposed in [11, 12] based on the QR decomposition scheme. However, the complexity of the RBD is too high, which is difficult to be implemented in practice. Owing to the SVD in the algorithm, the BD is not computationally efficient.
In this paper, we propose QRbased BD and Jacket matrix methods. We consider the channel matrix decomposition based on QR and Jacket matrices for the case where each user has multiple antennas. By using the QR decomposition to find the orthogonal complement, the complexity of the SVDBD can be reduced. As a new approach of the conventional BD scheme, the QR shows a significant improvement in computational complexity. In addition, we prove that the proposed QR algorithm has the same sum rate as the conventional BD scheme. We also discuss the block diagonal Jacket matrix decomposition because Jacket matrices are elementwise inverse matrices. Thus, we can calculate their complexity easily.
The rest of this paper is organized as follows. In Section 2, we describe the system model. In Section 3, we discuss the BD method. In Section 4, we analyze the block diagonal Jacket decomposition of an equivalent channel matrix. In Section 5, we perform the complexity analysis. Finally, we draw meaningful conclusions in Section 6.
2 System model
where k and j are user indices, ${\mathit{T}}_{\mathit{k}}\in {\u2102}^{{\mathit{N}}_{\mathrm{T}}\times {\mathit{N}}_{\mathit{k}}}$ is a precoding vector for the user k, s_{ k } represents the data symbol vector, ${\mathit{x}}_{\mathit{k}}\in {\u2102}^{{\mathit{N}}_{\mathit{k}}\times 1}$ is a transmit signal, ${\mathit{H}}_{\mathit{k}}\in {\u2102}^{{\mathit{N}}_{\mathit{k}}\times {\mathit{N}}_{\mathrm{T}}}$ is a MIMO channel matrix, and n_{ k } is a Gaussian noise with zero mean and variance σ^{2}. It is also assumed that all signals are detectable and $\sum _{\mathit{k}=1}^{\mathit{K}}{\mathit{N}}_{\mathit{k}}}\le {\mathit{N}}_{\mathrm{T}$.
The aforementioned problem is categorized as a convex optimization problem. Thus, it can be solved optimally and efficiently by using the water filling algorithm, which is proposed for the multiuser transmit optimization for broadcast channels.
3 Block diagonalization method
In this section, we represent a novel BD method for multiuser MIMO systems. The BD algorithm is an extension of the ZF method for multiuser MIMO systems where each user has multiple antennas. Each user's linear precoder and receiver filter can be obtained by twice SVD operations [15]–[16].
3.1 Block diagonalization
3.2 Proposed QRbased BD method
where Q_{ k } is an N_{T} × N_{T} unitary matrix, so Q_{ k }^{ H }Q_{ k } = I_{ k }; ${\mathit{R}}_{\mathit{k}}\in {\u2102}^{{\mathit{N}}_{\mathrm{T}}\times {\mathit{N}}_{\mathrm{R}}}$ is an N_{T} × N_{R} upper triangular matrix, and ${\overline{\mathit{Q}}}_{\mathit{k}}$ is an N_{T} × (N_{R} − N_{T}) matrix. ${\overline{\mathit{Q}}}_{\mathit{k}}^{\mathit{H}}=\left({\mathit{Q}}_{\mathit{k}}^{1}\phantom{\rule{0.24em}{0ex}}{\mathit{Q}}_{\mathit{k}}^{2}\right),$ where Q_{ k }^{1} is an N_{ k } column unitary matrix.
From the zero IUI constraint, we have ${\tilde{\mathit{H}}}_{\mathit{j}}{\mathit{Q}}_{\mathit{j}}{\mathit{R}}_{\mathit{j}}=0$. Since R_{ j } is invertible, it is conjectured that ${\tilde{\mathit{H}}}_{\mathit{j}}{\mathit{Q}}_{\mathit{j}}=0$
The optimal powerloading coefficients of Ψ are determined by using the water filling on the diagonal elements of $\stackrel{\u2322}{\mathit{\Sigma}}$, assuming that P_{ k } is a total power constraint. Equation 10 and Equation 17 are the same as the channel capacity of the conventional BD and the QREVD decomposition (Algorithm 2).
4 Block diagonal Jacket decomposition of an equivalent channel matrix
Obviously, the unitary matrices can be considered as the Jacket matrices.
Note that the size of each block element in the diagonal matrices (28), (29), and (30) is 2 × 2.
4.1 Eigenvalue decomposition of matrix of order 3
where ω = e^{−j 2π/n} (n is a matrix order). Note that ω^{3} = 1, and ω^{1} ≠ 1.
Therefore, the EVD can be also applied to block diagonal Jacket matrices.
5 Complexity analysis
In this section, we quantify the complexity of the QREVD decomposition algorithm and compare it with the conventional SVDBD schemes. The complexities of the alternative methods are usually compared by the number of floating point operations. A flop is defined as real floating operations, i.e., real additions, multiplications, divisions, and so on. One complex addition and multiplication elaborate two and six flops, respectively.
5.1 Complexity of matrix operations
For an m × n complexvalued matrix E ∈ ℂ^{m × n}, its multiplication with another n × p complexvalued matrix D ∈ ℂ^{n × p}, we use the total number of flops to measure the computational complexity of the existing algorithms [11, 13, 18, 19]. We summarize the total flops needed for the matrix operations as below:

Multiplication of m × n and n × p complex matrices is 8mnp flops.

When D = E^{∗}, the complexity is reduced to 4 nm (m + 1) flops, where D is a diagonal or block diagonal matrix.

The flop count for the SVD of realvalued m × n (m ≤ n) matrices is 4m^{2}n + 8mn^{2} + 9n^{3}. For complexvalued m × n (m ≤ n) matrices, we approximate the flop count as 24mn^{2} + 48m^{2}n + 54 m^{3} by treating every operation as the complex multiplication.

The QR decomposition on E using the GramSchmidt Orthogonalization (GSO) method takes 6 × 2m^{2}n flops.

The water filling operation is 2 m^{2} + 6 m flops for the water filling over m eigenvalues [18].
5.2 Complexity analysis for BD methods
For the conventional SVDBD method, obtaining the orthogonal complementary basis V_{ k }^{(0)} requires K times of SVD operations [19]. Hence, we consider GSO or QR decomposition methods. To calculate all, ${\tilde{\mathit{H}}}_{\mathit{k}}{\tilde{\mathit{V}}}_{\mathit{k}}^{\left(0\right)}$ requires K matrix multiplications while obtaining the singular vectors ${\tilde{\mathit{V}}}_{\mathit{k}}^{\left(1\right)}$ and the singular values λ_{ k } require another K SVD operations. The water filling is needed to find P_{ k }. The square root of the realvalued diagonal matrix P_{ k }^{1/2} needs to be calculated and multiplied by ${\tilde{\mathit{V}}}_{\mathit{k}}^{\left(0\right)}$ and ${\mathit{V}}_{\mathit{k}}^{\left(0\right)}$, respectively. Those operations repeat K times as well.
Complexity comparison
Method  Computational complexity 

SVDBD  $6\mathit{K}\left(9{\left(\left(\mathit{K}1\right){\mathit{N}}_{\mathit{k}}\right)}^{3}+8{\left(\left(\mathit{K}1\right){\mathit{N}}_{\mathit{k}}\right)}^{2}{\mathit{N}}_{\mathrm{T}}+4\left(\mathit{K}1\right){\mathit{N}}_{\mathit{k}}{\mathit{N}}_{\mathrm{T}}^{2}\right)$ 
8KN_{ k }N_{ T }(N_{ T } − (K − 1)N_{ k })  
$6\mathit{K}\left(9{\mathit{N}}_{\mathit{k}}^{3}+8{\mathit{N}}_{\mathit{k}}^{2}\left({\mathit{N}}_{\mathrm{T}}\left(\mathit{K}1\right){\mathit{N}}_{\mathit{k}}\right)+4{\mathit{N}}_{\mathit{k}}{\left({\mathit{N}}_{\mathrm{T}}\left(\mathit{K}1\right){\mathit{N}}_{\mathit{K}}\right)}^{2}\right)$  
2K^{2}N_{ k }^{2} + 6KN_{ k }  
$\mathit{K}{\mathit{N}}_{\mathit{k}}+2\mathit{K}{\mathit{N}}_{\mathit{k}}^{2}+8\mathit{K}{\mathit{N}}_{\mathrm{T}}\left({\mathit{N}}_{\mathrm{T}}\left(\mathit{K}1\right){\mathit{N}}_{\mathit{k}}\right){\mathit{N}}_{\mathit{k}}$  
QRBD  $12\mathit{K}{\left(\mathit{K}1\right)}^{2}{\mathit{N}}_{\mathit{k}}^{2}\left({\mathit{N}}_{\mathrm{T}}\left(\mathit{K}1\right){\mathit{N}}_{\mathit{k}}/3\right)$ 
8KN_{ k }N_{ T }(N_{ T } − (K − 1)N_{ k })  
$6\mathit{K}\left(9{\mathit{N}}_{\mathit{k}}^{3}+8{\mathit{N}}_{\mathit{k}}^{2}\left({\mathit{N}}_{\mathrm{T}}\left(\mathit{K}1\right){\mathit{N}}_{\mathit{k}}\right)+4{\mathit{N}}_{\mathit{k}}{\left({\mathit{N}}_{\mathrm{T}}\left(\mathit{K}1\right){\mathit{N}}_{\mathit{k}}\right)}^{2}\right)$  
2K^{2}N_{ k }^{2} + 6KN_{ k }  
$\mathit{K}{\mathit{N}}_{\mathit{k}}+2\mathit{K}{\mathit{N}}_{\mathit{k}}^{2}+8\mathit{K}{\mathit{N}}_{\mathrm{T}}\left({\mathit{N}}_{\mathrm{T}}\left(\mathit{K}1\right){\mathit{N}}_{\mathit{k}}\right){\mathit{N}}_{\mathit{k}}$ 
6 Conclusion
In this paper, we propose the QR method to obtain the precoding matrix for MIMO broadcast downlink systems. In addition, the QR scheme that of achieves the same sum capacity as the SVDBD scheme. We show that the new method has the lower complexity than the conventional BD method through complexity analysis, and the efficiency improvement becomes significant when the base station or users have a large number of transmit antennas. These results also show that the QR decomposition algorithm requires much less complexity than the conventional BD method. Thus, the complexity analysis of Jacket matrices is the same as that of the QREVD decomposition. We believe that the amount of computation required to obtain diagonalsimilar matrices is much smaller than that of computation required in the conventional EVD. In addtion, by using the QR decomposition to find the orthogonal complement, it is shown that the complexity of the SVDBD can be significantly reduced. In addition, we show that EVD can be extended to the highorder matrices. These properties may be used for Jacket matrices to be applied to signal processing, coding theory, and orthogonal code design. The EVD can be used in the informationtheoretic analysis of MIMO channels.
Declarations
Acknowledgements
This work was supported by the MEST 2012–002521 and Brain Korea 21 (BK21) Plus Project in 2014, National Research Foundation (NRF), Republic of Korea.
Authors’ Affiliations
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