A simple block diagonal precoding for multi-user MIMO broadcast channels

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1 Khan et al. EURASIP Journal on Wireless Communications and Networing 14, 14:95 RESEARC Open Access A simple bloc diagonal precoding for multi-user MIMO broadcast channels Md ashem Ali Khan, K M Cho, Moon o Lee * and Jin-Gyun Chung Abstract The bloc diagonalization (BD) is a linear precoding technique for multi-user multi-input multi-output (MIMO) broadcast channels, which is able to completely eliminate the multi-user interference (MUI), but it is not computationally efficient. In this paper, we propose the bloc diagonal Jacet matrix decomposition, which is able not only to extend the conventional bloc 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: Multi-user MIMO; Broadcast channel; Precoding; Bloc diagonalization; QR decomposition; Eigenvalue decomposition; Diagonal Jacet matrix 1 Introduction Recently, the research of the capacity region of the multi-user multi-input multi-output (MIMO) broadcast channels (BC) has been of concern. It is well nown 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 Jacet matrix decomposition where we believe that the required computational complexity in obtaining diagonal-similar matrices is smaller than that required in the conventional EVD. Definition 1 Let J N {a i,j }bean N matrix; then, it is n called a Jacet matrix when J 1 N ¼ o N 1 1 T,thatis, a i;j the inverse of the Jacet matrix can be determined by its element-wise inverse [-]. Definition Let A be an n n matrix. If there exists a Jacet matrix J such that A = J J 1, where Σ is a diagonal matrix, then we say that A is a Jacet matrix similar to the diagonal matrix. Moreover, we say that A is a Jacet diagonalizable [4]. * Correspondence: moonho@jbnu.ac.r Division of Electronics and Information Engineering, Chonbu National University, Jeonju , Republic of Korea Theorem 1 A4 4matrixJ is a Jacet matrix similar to the diagonal matrix if and only if J has the following form: J 4 ¼ ½AŠ ½BŠ ð1þ ½CŠ ½AŠ i.e., the entries of the main diagonal of a matrix are equal. Proof Refer to [4] for the proof. Multi-user diversity can significantly improve the performance of multiple antenna systems. The simplest ways to achieve the diversity gain in MIMO downlin communications are the zero forcing (ZF)-based linear precoding approaches. In [5,6], it was shown that the maximum sum rate in the multi-user MIMO broadcast channels can be achieved by dirty paper coding (DPC). owever, the high computational complexity of the DPC maes it difficult to implement in practical systems. A suboptimal strategy of the DPC [7], the Tomlinson-arashima precoding (TP) 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 bloc diagonalization (BD) method [1]. The ZF channel inversion scheme [8] can suppress co-channel interference (CCI) completely for the case where all users employ a single antenna. owever, its performance is degraded due to the effect of noise enhancement. Although the minimum mean-squared error (MMSE) channel inversion method [8] overcomes the 14 Khan et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly credited.

2 Khan et al. EURASIP Journal on Wireless Communications and Networing 14, 14:95 Page of 8 drawbac of the ZF, it is still confined to a single-receive antenna case. In the scenario where multiple antennas are located at both the mobile terminal and base station for each user, low-complexity BD methods have been proposed [8,11-1]. Moreover, the BD attempts to completely eliminate the multi-user interference (MUI) irrespective of the noise. The BD precoding has been proposed in [1] 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 low-complexity 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 reduction-aided RBD (LC-RBD-LR)- type precoding algorithm has been proposed in [11,1] based on the QR decomposition scheme. owever, 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 QR-based BD and Jacet matrix methods. We consider the channel matrix decomposition based on QR and Jacet matrices for the case where each user has multiple antennas. By using the QR decomposition to find the orthogonal complement, the complexity of the SVD-BD 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 bloc diagonal Jacet matrix decomposition because Jacet matrices are element-wise inverse matrices. Thus, we can calculate their complexity easily. The rest of this paper is organized as follows. In Section, we describe the system model. In Section, we discuss the BD method. In Section 4, we analyze the bloc diagonal Jacet decomposition of an equivalent channel matrix. In Section 5, we perform the complexity analysis. Finally, we draw meaningful conclusions in Section 6. System model We consider the downlin 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 CN ð; IÞ, 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 -th user can be written as x ¼ T s ðþ The received signal for the -th user can be represented as y ¼ X K j¼1 T j x j þ n ¼ T s þ XK þn ; ¼ 1; ; K; j¼1;j T j s j ðþ where and j are user indices, T C N TN is a precoding vector for the user, s represents the data symbol Figure 1 MIMO broadcasting system model.

3 Khan et al. EURASIP Journal on Wireless Communications and Networing 14, 14:95 vector, x ℂ N 1 is a transmit signal, ℂ N N T is a MIMO channel matrix, and n is a Gaussian noise with zero mean and variance σ. It is also assumed that all K X signals are detectable and N N T. ¼1 Note that the precoding vectors are normalized to unity, i.e., T = 1 for = 1,, K. Furthermore, the power constraints are defined as tr(tt ) P, where P is the total transmission power. The power constraint corresponding to the BS applies to the transmitters of th BS. Therefore, a sum rate maximization problem with power constraints can be expressed as X max logi þ T T ð4þ s:t: tr T T P ; ¼ 1; ; K ~ T ¼ ; ¼ 1; ; K 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 multi-user transmit optimization for broadcast channels. Bloc diagonalization method In this section, we represent a novel BD method for multi-user MIMO systems. The BD algorithm is an extension of the ZF method for multi-user 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]. The ey 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. ~ T ¼ ; ¼ 1; ; K ð5þ ~ is defined as the channel matrix for all users other than the user. ~ ¼ T T T T 1 1 þ1 K By applying the SVD, the following value for the channel is obtained h i ~ ¼ U Σ V ð1þ V ðþ ; T ð6þ ð7þ where Σ is the diagonal matrix of which the diagonal ele~ and its diments are non-negative singular values of ~. V() mension equals to the ran of contains vectors corresponding to the zero singular values, and V(1) consists of the singular vectors corresponding to nonzero singular values. Thus, V() is an orthogonal basis for the null ~. In order to maximize the achievable sum rate space of of the BD, the water filling algorithm can be additionally ~ V~ ðþ as incorporated. Define the SVD of h i ~ V ~ V ~ ðþ ¼ U ~ ð1þ V ~ ðþ : ~ Σ Thus, we define the total precoding matrix as h i ~ ðþ V ð1þ V~ ðkþ V ðk1þ Λ1= ; ~ ð1þ V ð11þ V T BD ¼ V ð8þ ð9þ where Λ is a diagonal matrix of which the element λ scales the power transmitted into each of columns of TBD. 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 TBD chosen in Equation 9, the capacity of the BD [1,15] is Σ Λ C BD ¼ max log I þ ; ð1þ Λ σ where.1 Bloc diagonalization Page of 8 Σ1 Σ¼4 5: ΣK ð11þ The optimal power-loading coefficients of Λ are determined by using the water filling on the diagonal elements of Σ, assuming that P is a total power constraint. A summary of the BD algorithm [1] in Algorithm 1.. Proposed QR-based BD method In this subsection, we propose an alternative method to ~ based on the QR defind vectors orthonormal to ~, composition. In order to compute the null space of ~ as we define a QR decomposition of R ~ ¼ ½Q Q ¼ Q R ; ð1þ where Q is an NT NT unitary matrix, so Q Q = I; R ℂ N T N R is an NT NR upper triangular matrix, and Q 1 1 is an NT (NR NT) matrix. Q ¼ Q Q ; where Q is an N column unitary matrix.

4 Khan et al. EURASIP Journal on Wireless Communications and Networing 14, 14:95 The pseudo inverse of the channel matrix = ¼ 1 ¼ ½ 1 K. [T1 T KT]T is Then, we can show that 1 ¼4 5 1 K 1 1 ¼4 1 K K K I N R;1 1 5¼ 4 K K 5: I N R;K ð1þ ¼ when j, which is called the zero Clearly, j inter-user interference (IUI) constraint since it gets the h ~ j ¼ T T ~ j as IUI to be zero. By defining 1 j 1 Tjþ1 TK T, it is shown that the zero IUI constraint is j ¼. The QR decomposition of ~ j satisfied such as ~ j is ~ j ¼ Qj Rj for j ¼ 1; ; K : ð14þ ~ j Qj R j ¼. From the zero IUI constraint, we have ~ j Qj ¼ Since Rj is invertible, it is conjectured that Let G = Q1 and we apply the EVD of G as G ¼ U Σ U ; ð15þ where U is a unitary matrix, and Σ is a diagonal matrix. Thus, we get the precoding matrix as h i T QR ¼ Q11U 1 Q1U Q1KU K Ψ 1= ; ð16þ where Ψ is a diagonal matrix of which the elements scale the power transmitted into each of columns of TQR. The capacity of the QR-EVD is I þ Σ Ψ C QR EVD ¼ max log ð17þ ; σ Ψ where Σ1 Σ¼4 5: ΣK ð18þ The optimal power-loading coefficients of Ψ are determined by using the water filling on the diagonal ele ments of Σ, assuming that P is a total power constraint. Equation 1 and Equation 17 are the same as the chan- Page 4 of 8 nel capacity of the conventional BD and the QR-EVD decomposition (Algorithm ). Figure shows that the BD method has the same sum rate as the QR-EVD 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 Bloc diagonal Jacet decomposition of an equivalent channel matrix In this section, we introduce the bloc diagonal Jacet decomposition of an equivalent channel matrix. Assume that is an NR NT bloc diagonal matrix given by LΣL 1 ¼ 4 5; ð19þ LΣL 1 and its inverse is 1 LΣL ¼ 5: 1 LΣL 1 ðþ The channel matrix is decomposed into parallel singleinput single-output subchannels. A special Jacet matrix called a diagonal Jacet matrix can be defined as follows: J 1;1 ½J ¼ 4 5; and J ; ð1þ Its inverse matrix is ½J 1 ¼ 1=J 1;1 4 5: 1=J ; ðþ Obviously, the unitary matrices can be considered as the Jacet matrices. Let us denote B as a bloc matrix in the main diagonal of [1,17]. Then, Equation 19 can be written as ¼ I = B ; ðþ where B ¼ LΣL 1 ð4þ I/ is an identity matrix, and is the Kronecer product. It is worthwhile to note that each bloc in the diagonal of the matrix in Equation 19 is a matrix that satisfies the condition specified in Theorem 1, and

5 Khan et al. EURASIP Journal on Wireless Communications and Networing 14, 14:95 Page 5 of Eigenvalue decomposition of matrix of order In this subsection, we introduce a class of matrices of order that can be factorized into EVD forms through Jacet matrices [1,17]. A matrix A is a Jacet 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. a b c A ¼ 4 c a b5: ð1þ b c a Figure Comparison of the sum throughput of BD and QR. hence, we say that B can be decomposed by the EVD using Jacet matrices. In other words, B is able to be represented by B ¼ J Σ J 1 : ð5þ In addition, it is shown that is decomposed, which has the diagonal form as ¼ I = B ¼ I = J Σ J 1 ¼ I = J diag ð λ1 ; λ λ Þ I = J 1 ¼ JΣJ 1 : ð6þ Thus, we can write where ¼JΣJ 1 ; J J ¼ I = J ¼ 4 5 J Σ Σ ¼ I = Σ ¼ 4 5 Σ J 1 ¼ I = J 1 ¼ 4 J 1 J 1 ð7þ ; ð8þ 5 ; and ð9þ : ðþ Note that the size of each bloc element in the diagonal matrices (8), (9), and () is. The abovementioned matrix, A, can be decomposed as follows: a b c c a b5 ¼ 4 1 ω ω 5 b c a 1 ω ω a þ b þ c 4 a þ bω þ cω 5 a þ bω þ cω ω ω 5 ; 1 ω ω ðþ where ω = e jπ/n (n is a matrix order). Note that ω =1, and ω 1 1. Consider a matrix A 6 that is able to be decomposed via Jacet matrices as A 6 ¼ A A ¼ ðj J ÞðΛ Λ ÞðJ J Þ 1 ; ðþ where is the Kronecer product. Then, the EVD of Equation is given as A 6 ¼ a b a b c 4 c a b5 b a b c a a a ¼ ω ω a a 55 1 ω ω a þ b þ c 4 a þ b 4 a þ bω þ cω a þ b 55 a þ bω þ cω a a ω ω a a 55 : 1 ω ω ð4þ

6 Khan et al. EURASIP Journal on Wireless Communications and Networing 14, 14:95 Page 6 of 8 In general, a matrix of order n (n = 1 ) can be decomposed via Jacet transform as follows: A n ¼ A ¼ A l A l ¼ J Λ J 1 J lλ lj 1 l ¼ ðj J lþðλ Λ lþ J J l Þ 1 ð5þ The diagonal mobile communication channel matrix is given by Equation, where B ¼ cos45 i sin45 sin45 i cos45 p 1 i 1 i ¼ 1 ffiffiffi :8881 :51 þ :51i ¼ :51 þ :51i :8881 :9659 :588i :588 þ :9659i :8881 :51 :51i ¼ QΛQ :51 :51i :8881 : ð6þ A 4 4 bloc wise Jacet matrix is 1 4 ¼ ½BŠ 1 i ¼ 1 1 i pffiffi B C 1 i A 1 i ¼ 1 1 pffiffiffi 1 i ¼ ½Š I 1 1 i ½BŠ : ð7þ Then, the capacity of a MIMO wireless communication system is given by C ¼ log det I NR þ SNR N T bits=s=z ð8þ Figure The required flops versus the number of transmit antennas, N T. C ¼ XK ¼1 log 1 þ SNR λ bits=s=z: N T ð4þ Therefore, the EVD can be also applied to bloc diagonal Jacet matrices. 5 Complexity analysis In this section, we quantify the complexity of the QR- EVD decomposition algorithm and compare it with the conventional SVD-BD 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. The channel matrix is also able to be decomposed by the EVD ¼ QΛQ : ð9þ Then, the EVD is obtained as ¼ QΣΣ Q ¼ QΛQ ; ð4þ where QQ = Q Q = I N, and Λ = dig(λ 1, λ,, λ K ) with its diagonal elements given as λ ¼ σ ; if ¼ 1; ; ; K min ð41þ ; if ¼ K min þ 1; ; K: It is shown that the MIMO system capacity can be written as Figure 4 The required flops versus the number of users, K.

7 Khan et al. EURASIP Journal on Wireless Communications and Networing 14, 14:95 Page 7 of 8 Table 1 Complexity comparison Method Computational complexity SVD-BD 6K 9ððK 1ÞN Þ þ 8ððK 1ÞN Þ N T þ 4ðK 1ÞN N T 8KN N T (N T (K 1)N ) 6K 9N þ 8N ð N T ðk 1ÞN Þþ4N ðn T ðk 1ÞN K Þ K N +6KN QR-BD KN þ KN þ 8KN TðN T ðk 1ÞN ÞN 1KðK 1Þ N ð N T ðk 1ÞN =Þ 8KN N T (N T (K 1)N ) 6K 9N þ 8N ð N T ðk 1ÞN Þþ4N ðn T ðk 1ÞN Þ K N +6KN KN þ KN þ 8KN TðN T ðk 1ÞN ÞN 5.1 Complexity of matrix operations For an m n complex-valued matrix E C m n, its multiplication with another n p complex-valued matrix D C n p, we use the total number of flops to measure the computational complexity of the existing algorithms [11,1,18,19]. 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 bloc diagonal matrix. The flop count for the SVD of real-valued m n (m n) matrices is 4m n +8mn +9n.For complex-valued m n (m n) matrices, we approximate the flop count as 4mn +48m n +54m by treating every operation as the complex multiplication. The QR decomposition on E using the Gram-Schmidt Orthogonalization (GSO) method taes 6 m n flops. The water filling operation is m +6m flops for the water filling over m eigenvalues [18]. 5. Complexity analysis for BD methods For the conventional SVD-BD method, obtaining the orthogonal complementary basis V () requires K times of SVD operations [19]. ence, we consider GSO or QR decomposition methods. To calculate all, ~ ~V ðþ requires K matrix multiplications while obtaining the singular vectors ~V ð1þ and the singular values λ require another K SVD operations. The water filling is needed to find P. The square root of the real-valued diagonal matrix P 1/ needs to be calculated and multiplied by ~V ðþ and V ðþ, respectively. Those operations repeat K times as well. Based on the above analysis, two results of the SVD- BD and the QR decomposition are shown in Figures and, respectively. Figure shows the required number of flops according to the number of transmit antennas, N T, where n = and =. Figure 4 shows the required number of flops according to the number of users, K, where m = 4 and n =. From Figures 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 1 1 Conventional EVD Jacet EVD Capacity,C(bps/z) Figure 5 Capacity versus SNR at different sizes of matrix SNR[dB]

8 Khan et al. EURASIP Journal on Wireless Communications and Networing 14, 14:95 Page 8 of 8 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 7 can be decomposed by Jacet matrices, which has the diagonal form, where J is a unitary matrix. Therefore, Equations 8 and 15 are the same as Equation 7 because U and V are unitary matrices and a family of Jacet matrices, which are mathematically proved in the previous sections. Thus, the complexity analysis of Jacet matrices are the same as that of the QR- EVD decomposition as shown in Table 1. The complexity of the conventional EVD method and Jacet-based EVD method increases as the respective sizes of their matrices increase, as shown in Figure 5. In addition, we compare the performance of the conventional-based EVD method and Jacet-based EVD method. Classes of these matrices, which are simply decomposed by the EVD based on Jacet 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 downlin systems. In addition, the QR scheme that of achieves the same sum capacity as the SVD-BD 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 Jacet matrices is the same as that of the QR-EVD decomposition. We believe that the amount of computation required to obtain diagonal-similar 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 SVD-BD can be significantly reduced. In addition, we show that EVD can be extended to the high-order matrices. These properties may be used for Jacet matrices to be applied to signal processing, coding theory, and orthogonal code design. The EVD can be used in the information-theoretic analysis of MIMO channels. Received: 1 January 14 Accepted: 4 May 14 Published: 14 June 14 References 1. M Lee, MM Matalgah, W Song, Fast method for precoding and decoding of distributive multi-input multi-output channels in relay based decodeand-forward cooperative wireless networs. IET Commun 4(), (1). M Lee, A new reverse Jacet transform and its fast algorithm. IEEE Trans Circuits Syst II 47(1), 9 47 (). 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IEEE Commun Lett 15(6), (11) 17. M Lee, Jacet Matrices - Construction and Its Application for Fast Cooperative Wireless Signal Processing (LAP LAMBERT Academic Publishing, Germany, 1) 18. G Golub, CF Van Load, Matrix Computations, rd edn. (The John opisns University Press, Baltimore and London, 1996) 19. W Li, M Latva-aho, An efficient channel bloc diagonalization method for generalized zero forcing assisted MIMO broadcasting systems. IEEE Trans on Wireless Commun 1(), (11) doi:1.1186/ Cite this article as: Khan et al.: A simple bloc diagonal precoding for multi-user MIMO broadcast channels. EURASIP Journal on Wireless Communications and Networing 14 14:95. Competing interests The authors declare that they have no competing interests. Acnowledgements This wor was supported by the MEST 1 51 and Brain Korea 1 (BK1) Plus Project in 14, National Research Foundation (NRF), Republic of Korea.