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A simple block diagonal precoding for multiuser MIMO broadcast channels
EURASIP Journal on Wireless Communications and Networking volume 2014, Article number: 95 (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.
1 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].
Theorem 1 A 4 × 4 matrix $\mathbb{J}$ is a Jacket matrix similar to the diagonal matrix if and only if $\mathbb{J}$ has the following form:
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
We consider the downlink MIMO broadcast channel base station (BS) to K mobile users as shown in Figure 1. The MIMO channel of each user is assumed to be flat fading with distribution $\mathcal{CN}\left(0,\mathit{I}\right)$, where the BS has N_{T} transmitter antennas, and each user has N_{R} receiver antennas. In this linear precoding scheme, the precoded signal vector for the k th user can be written as
The received signal for the k th user can be represented as
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 $\sum _{\mathit{k}=1}^{\mathit{K}}{\mathit{N}}_{\mathit{k}}}\le {\mathit{N}}_{\mathrm{T}$.
Note that the precoding vectors are normalized to unity, i.e., ∥T_{ k }∥^{2} = 1 for k = 1,⋯, K. Furthermore, the power constraints are defined as tr(T_{ k }T_{ k }^{H}) ≤ P_{ k }, where P_{ k } is the total transmission power. The power constraint corresponding to the BS applies to the transmitters of k th BS. Therefore, a sum rate maximization problem with power constraints can be expressed as
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
The key idea of the BD algorithm is to employ the precoding matrix Τ to suppress the MUI completely. To eliminate all MUI, the following constraint is imposed.
${\tilde{\mathit{H}}}_{\mathit{k}}$ is defined as the channel matrix for all users other than the user k.
By applying the SVD, the following value for the channel is obtained
where Σ_{ k } is the diagonal matrix of which the diagonal elements are nonnegative singular values of ${\tilde{\mathit{H}}}_{\mathit{k}}$ and its dimension equals to the rank of ${\tilde{\mathit{H}}}_{\mathit{k}}$. V_{ k }^{(0)} contains vectors corresponding to the zero singular values, and V_{ k }^{(1)} consists of the singular vectors corresponding to nonzero singular values. Thus, V_{ k }^{(0)} is an orthogonal basis for the null space of ${\tilde{\mathit{H}}}_{\mathit{k}}$. In order to maximize the achievable sum rate of the BD, the water filling algorithm can be additionally incorporated. Define the SVD of ${\tilde{\mathit{H}}}_{\mathit{k}}{\tilde{\mathit{V}}}_{\mathit{k}}^{\left(0\right)}$ as
Thus, we define the total precoding matrix as
where Λ is a diagonal matrix of which the element λ_{ k } scales the power transmitted into each of columns of T^{BD}. To maximize the sum rate under a total power constraint at the BS, where the power allocation matrix is the solution to the following optimization, with T^{BD} chosen in Equation 9, the capacity of the BD [10, 15] is
where
The optimal powerloading coefficients of Λ are determined by using the water filling on the diagonal elements of Σ, assuming that P_{ k } is a total power constraint. A summary of the BD algorithm [10] in Algorithm 1.
3.2 Proposed QRbased BD method
In this subsection, we propose an alternative method to find vectors orthonormal to ${\tilde{\mathit{H}}}_{\mathit{k}}$ based on the QR decomposition. In order to compute the null space of ${\tilde{\mathit{H}}}_{\mathit{k}}$, we define a QR decomposition of ${\tilde{\mathit{H}}}_{\mathit{k}}$ as
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.
The pseudo inverse of the channel matrix H_{ k } = [H_{1}^{T}H_{2}^{T}⋯H_{ K }^{T}]^{T} is ${\overline{\mathit{H}}}_{\mathit{k}}={\mathit{H}}_{\mathit{k}}^{\mathit{H}}{\left({\mathit{H}}_{\mathit{k}}{\mathit{H}}_{\mathit{k}}^{\mathit{H}}\right)}^{1}=\left[{\overline{\mathit{H}}}_{1}\phantom{\rule{0.24em}{0ex}}{\overline{\mathit{H}}}_{2}\cdots {\overline{\mathit{H}}}_{\mathit{K}}\right]$. Then, we can show that
Clearly, ${\mathit{H}}_{\mathit{j}}{\overline{\mathit{H}}}_{\mathit{k}}=0$ when j ≠ k, which is called the zero interuser interference (IUI) constraint since it gets the IUI to be zero. By defining ${\tilde{\mathit{H}}}_{\mathit{j}}$ as ${\tilde{\mathit{H}}}_{\mathit{j}}={\left[{\mathit{H}}_{1}^{\mathit{T}}\cdots {\mathit{H}}_{\mathit{j}1}^{\mathit{T}}\phantom{\rule{0.12em}{0ex}}{\mathit{H}}_{\mathit{j}+1}^{\mathit{T}}\cdots {\mathit{H}}_{\mathit{K}}^{\mathit{T}}\phantom{\rule{0.12em}{0ex}}\right]}^{\mathit{T}}$, it is shown that the zero IUI constraint is satisfied such as ${\tilde{\mathit{H}}}_{\mathit{j}}{\overline{\mathit{H}}}_{\mathit{j}}=0$. The QR decomposition of ${\tilde{\mathit{H}}}_{\mathit{j}}$ is
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 ${\tilde{\mathit{H}}}_{\mathit{j}}{\mathit{Q}}_{\mathit{j}}=0$
Let G_{ k } = H_{ k }Q_{ k }^{1} and we apply the EVD of G_{ k } as
where ${\stackrel{\u2322}{\mathit{U}}}_{\mathit{k}}$ is a unitary matrix, and ${\stackrel{\u2322}{\mathit{\Sigma}}}_{\mathit{k}}$ is a diagonal matrix. Thus, we get the precoding matrix as
where Ψ is a diagonal matrix of which the elements scale the power transmitted into each of columns of T^{QR}. The capacity of the QREVD is
where
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).
Figure 2 shows that the BD method has the same sum rate as the QREVD method and An's method [15] under condition that a MIMO broadcasting system consists of one base station and two users where the base station has four transmit antennas and each use has two receive antennas.
4 Block diagonal Jacket decomposition of an equivalent channel matrix
In this section, we introduce the block diagonal Jacket decomposition of an equivalent channel matrix. Assume that H_{ k } is an N_{R} × N_{T} block diagonal matrix given by
and its inverse is
The channel matrix is decomposed into parallel singleinput singleoutput subchannels. A special k × k Jacket matrix called a diagonal Jacket matrix can be defined as follows:
Its inverse matrix is
Obviously, the unitary matrices can be considered as the Jacket matrices.
Let us denote B_{2} as a 2 × 2 block matrix in the main diagonal of H_{ k }[1, 17]. Then, Equation 19 can be written as
where
I_{ k/2 } is an identity matrix, and ⊗ is the Kronecker product. It is worthwhile to note that each block in the diagonal of the matrix in Equation 19 is a 2 × 2 matrix that satisfies the condition specified in Theorem 1, and hence, we say that B_{2} can be decomposed by the EVD using Jacket matrices. In other words, B_{2} is able to be represented by
In addition, it is shown that H_{ k } is decomposed, which has the diagonal form as
Thus, we can write
where
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
In this subsection, we introduce a class of matrices of order 3 that can be factorized into EVD forms through Jacket matrices [1, 17]. A 3 × 3 matrix A is a Jacket matrix similar to a diagonal matrix Λ if and only if such a matrix can be factorized into the form of an EVD such as A = J Λ J^{−1}. Consider a special matrix, A, of which the elements in the first row are arbitrary, whereas the elements in the other rows are generated by cyclically shifting the previous row. One of its examples is given as follows.
The abovementioned matrix, A, can be decomposed as follows:
where ω = e^{−j 2π/n} (n is a matrix order). Note that ω^{3} = 1, and ω^{1} ≠ 1.
Consider a matrix A_{6} that is able to be decomposed via Jacket matrices as
where ⊗ is the Kronecker product. Then, the EVD of Equation 33 is given as
In general, a matrix of order n (n = 2^{k} × 3^{1}) can be decomposed via Jacket transform as follows:
The diagonal mobile communication channel matrix is given by Equation 23, where
A 4 × 4 block wise Jacket matrix is
Then, the capacity of a MIMO wireless communication system is given by
The channel matrix H_{ k } is also able to be decomposed by the EVD
Then, the EVD is obtained as
where QQ^{H} = Q^{H}Q = I_{N}, and Λ = dig(λ_{1}, λ_{2},⋯, λ_{ K }) with its diagonal elements given as
It is shown that the MIMO system capacity can be written as
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.
Based on the above analysis, two results of the SVDBD and the QR decomposition are shown in Figures 2 and 3, respectively. Figure 3 shows the required number of flops according to the number of transmit antennas, N_{T}, where n = 2 and k = 2. Figure 4 shows the required number of flops according to the number of users, K, where m = 24 and n = 2. From Figures 3 and 4, it is obvious that the QR decomposition can significantly reduce the number of flops compared with the BD algorithm. The larger values N_{T} and K have, the less number of flops the QR decomposition has. Figure 4 shows that the number of flops significantly decreases. In other words, the complexity highly declines.
The channel in Equation 27 can be decomposed by Jacket matrices, which has the diagonal form, where J is a unitary matrix. Therefore, Equations 8 and 15 are the same as Equation 27 because U and V are unitary matrices and a family of Jacket matrices, which are mathematically proved in the previous sections. Thus, the complexity analysis of Jacket matrices are the same as that of the QREVD decomposition as shown in Table 1. The complexity of the conventional EVD method and Jacketbased EVD method increases as the respective sizes of their matrices increase, as shown in Figure 5. In addition, we compare the performance of the conventionalbased EVD method and Jacketbased EVD method. Classes of these matrices, which are simply decomposed by the EVD based on Jacket transform, have been used to significantly reduce their computational complexity compared to the conventional EVD method.
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.
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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.
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Khan, M.H.A., Cho, K.M., Lee, M.H. et al. A simple block diagonal precoding for multiuser MIMO broadcast channels. J Wireless Com Network 2014, 95 (2014). https://doi.org/10.1186/16871499201495
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Keywords
 Multiuser MIMO
 Broadcast channel
 Precoding
 Block diagonalization
 QR decomposition
 Eigenvalue decomposition
 Diagonal Jacket matrix