# Combined Transceiver Optimization for Uplink Multiuser MIMO with Limited CSI

Authors: Chia-Chang Hu; Chung-Lun Yang
Journal: ISRN Signal Processing
 (2011)

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## Abstract

Joint precoder and decoder optimization is considered for uplink multiuser multiple-input multiple-output (MU-MIMO) systems with limited channel state information (CSI) at both the transmitters and receivers. Instead of counting on complex iterative-based algorithms, an efficient and noniterative QR-based linear transceiver pair design is employed. In addition, an equal power distribution (EPD) scheme is applied to adjust transmit power allocation of each mobile station (MS) between its symbols under the total transmit-power constraint. Simulations are conducted to provide a comparative evaluation of the proposed QR-EPD algorithm with other transceiver designs on the sum mean-squared error (SMSE) and the averaged bit-error-rate (BER) performance.


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## Body

## 1. Introduction

Owing to the emerging demand on multimedia applications, next-generation wireless communication systems are expected to improve substantially both reliability and spectral efficiency. A key development in this regard is the use of multiple antenna arrays at both the base station (BS) and mobile station (MS) [1]. Due to its effectiveness in coping with multiple-access interference (MAI) and inter-stream interference (ISI), a linear precoder-decoder joint architecture is utilized based on the minimum sum mean-squared error (MSMSE) to improve multiuser multiple-input multiple-output (MU-MIMO) spatial multiplexing systems [2–4]. To further enhance system performance of MU-MIMO systems, one important strategy is to exploit feedback information at the transmitter. In particular, when channel state information (CSI) is fully known at the transmitter, spatial multiplexing is able to benefit significantly from the use of precoding at the transmitter [2]. In most previous research, the perfect CSI is available at both ends [5, 6]. However, the knowledge of CSI is in general incomplete in reality due to various kinds of channel imperfections, such as channel estimation error, feedback delay, and finite-rate channel equalization. In addition, the development of the transceiver depends primarily on the complex iterative-based algorithms, in which they either suffer from a slow convergence rate at high signal-to-noise ratio (SNR) or may even not converge to global optimum [7]. Therefore, instead of relying on complicated recursive-based algorithms, an efficient noniterative QR-based transceiver design is proposed for uplink MU-MIMO systems under limited CSI at both sides. Moreover, an equal power distribution (EPD) technique is applied to allocate uniformly the transmission power of each mobile user between its symbols under a fixed total transmit power. Thus, the superiority of the proposed QR-EPD scheme on the MSMSE and the averaged BER performance for uplink MU-MIMO systems can be achieved.The rest of the paper is organized as follows. Section2 describes the uplink multiuser MIMO system and channel model. In Section 3, the transceiver-pair design of the uplink multiuser MIMO systems is formulated as the constrained sum-MSE optimization problem. An efficient noniterative QR-based linear transceiver architecture is detailed for uplink multiuser MIMO systems with limited CSI in Section 4. A simple EPD scheme is applied to allocate uniformly the transmission power of each symbol of the mobile user under the sum-power constraint. Numerical results and conclusions are presented in Sections 5 and 6, respectively.Notation 1.
Symbols for matrices (vectors) are denoted by boldface upper (lower) case letters. The superscripts(·)⋆ and (·)H stand for complex conjugation and Hermitian transposition, respectively. E{·} denotes the expected-value operator. ∥·∥ indicates the matrix/vector Frobenius norm. IM is an M×M identity matrix. tr(·) and [·]i,j denote, respectively, the trace and the (i,j)th entry of a matrix. Finally, (x)+ stands for max(x,0).


## 2. Uplink MIMO System and Channel Model

Consider an uplink multiuser MIMO wireless system with anM-antenna BS and K mobile users each with N antennas, as shown in Figure 1. The uplink MIMO channel of the ith user is denoted by Hi∈CM×N, i=1,2,…,K. Two important constraints need to be satisfied in the scenario to guarantee resolvability. First of all, there must have at least as many receiver antennas at BS as the total number of transmit antennas of all mobile users, that is, KN≤M. As well, each user must have as many transmit antennas as the transmitted data streams. Suppose that user i has data streams, denoted by xi∈CN×1, i=1,2,…,K, which are assumed to be zero-mean white random vectors each with unit-energy E{xixiH}=IN, for all i and mutually statistically independent among users. A linear precoder is employed for each user, which is denoted by Fi∈CN×N, i=1,2,…,K. The received signal vector r∈CM×1 at the BS is given by  (1)r=∑i=1KHiFixi+n,
where the noise vector n is modeled as zero-mean white complex Gaussian variables, that is, CN(0,σn2IM). The data and the noise are assumed to be statistically independent. The received signal vector r is processed by all linear decoders Gj∈CN×M, for j=1,2,…,K. An estimate of the data vector for user j can be expressed as  (2)yj=Gjr=Gj(∑i=1KHiFixi+n).Figure 1
Block diagram of an uplink multiuser MIMO wireless system with anM-antenna BS and K mobile users each with N antennas.The MIMO channels among users are assumed to be independent and flat fading. The uplink CSI model at the BS can be expressed as(3)Hi=Ĥi+ΔHi,i=1,2,…,K,
where Ĥi is the estimated channel matrix and ΔHi is the CSI-error matrix. The entries of Hi, Ĥi, and ΔHi are identically independent distribution (i.i.d.) complex Gaussian variables each with CN(0,1), CN(0,1-σE2), and CN(0,σE2), respectively. Here, σE2 indicates the channel estimation-error variance for each mobile user. Furthermore, matrices Ĥi and ΔHi are assumed to be uncorrelated.


## 3. The Uplink MIMO Problem Formulation

With the CSI model, the received signal vector at the BS is written as(4)r=∑i=1K(Ĥi+ΔHi)Fixi+n,
and the estimate of the data vector for user j is expressed as  (5)yj=Gj[∑i=1K(Ĥi+ΔHi)Fixi+n].
The MSE matrix for user j is defined by
(6)MSEj=E{(yj-xj)(yj-xj)H}=Gj(∑i=1KĤiFiFiHĤiH+σn2IM)GjH+Gj[∑i=1KE{ΔHiFiFiHΔHiH}]GjH-GjĤjFj-FjHĤjHGjH+IN.
Then it can be shown that  (7)MSEj=Gj(∑i=1KĤiFiFiHĤiH+σn2IM)GjH+Gj[∑i=1Ktr⁡(FiFiH)⋅σE2IM]GjH-GjĤjFj-FjHĤjHGjH+IN,
where the equality of  (8)E{ΔHiFiFiHΔHiH}=tr⁡(FiFiH)⋅σE2IM
is used in (7). The sum-MSE of all users is given by  (9)MSE=∑j=1Ktr⁡(MSEj).
The uplink MU-MIMO problem is to minimize the sum-MSE subject to (s.t.) the sum-power constraint, which is given as follows:
(10)min⁡{(Fj,Gj)}j=1KMSE=∑j=1Ktr⁡(MSEj),(11)s.t.∑j=1Ktr⁡(FjFjH)=P,
where P is the total transmit power constraint over all users, that is, the sum-power constraint. To find the joint optimum set of linear precoders and decoders, one can devise iterative-based algorithms that decrease monotonically the total MSE. Similar algorithms have been proposed in [2–4] with perfect CSI. In summary, such a set of transceivers are derived by using the Lagrange multipliers associated with the transmit power constraint in (11), which is given as follows:  (12)min⁡{(Fj,Gj)}j=1K,μMSE+μ[∑j=1Ktr⁡(FjFjH)-P].
The scalar μ in (12) is the Lagrangian multiplier. Similar to [7], a noniterative set of linear precoders and decoders are utilized by relaxing a nonconvex optimization problem in (10) into a simpler convex optimization problem. Thus, two traditional issues of the complexity cost and the convergence speed in most iterative-based algorithms can be avoided.


## 4. MU-MIMO Transceiver Design

Given a set of precodersFi, i=1,2,…,K, the optimal linear decoder Gj can be obtained by taking derivative of (9) with respect to Gj⋆, j=1,2,…,K, and then equating it to zero. Therefore, the optimal linear decoder Gj is obtained by [3]  (13)Gj=FjHĤjH(ρIM+∑i=1KĤiFiFiHĤiH)-1,
where  (14)ρ=[σn2+∑i=1KσE2⋅tr⁡(FiFiH)]
is used in (13). For convenience of notation, the matrix T is defined as follows:  (15)T=ρIM+∑i=1KĤiFiFiHĤiH.
Then, substituting (13) into (9) derives a simplified SMSE expression given by
(16)MSE=ρ⋅tr⁡(T-1)+∑j=1Ktr⁡(IN)-tr⁡(IM)(17)=tr⁡{(IM+1ρ∑i=1KĤiFiFiHĤiH)-1}+KN-M.
Therefore, the uplink MU-MIMO problem can be equivalently formulated as
(18)min⁡{Fi}i=1KMSE=tr⁡{(IM+1ρH̃F̃H̃H)-1}+KN-M(19)s.t.tr⁡(F̃)=P,
where the matrix H̃F̃H̃H in (18) is employed to represent the matrix ∑i=1KĤiFiFiHĤiH in (17) and the relationships between both matrices are given by  (20)H̃=[Ĥ1Ĥ2⋯ĤK],(21)F̃=[F1F1H0⋯00F2F2H⋯0⋮⋮⋱⋮00⋯FKFKH].
Note that no cooperation is assumed among mobile users in the MU-MIMO system. Therefore, the matrix F̃ possesses a block diagonal structure as defined in (21). Unfortunately, no simple closed-form solution can be achieved for such a block diagonal constraint. In order to obtain the optimal solution of F̃, the block-diagonal constraint on F̃ needs to be relaxed for this moment. In what follows, the matrix of F̃opt is used to indicate the optimal solution of F̃. Instead of the singular value decomposition (SVD) used to decompose the matrix H̃ in [7], the computationally efficient QR factorization is employed to perform the decomposition of the matrix H̃HH̃ as follows:  (22)QH̃HH̃HH̃=RH̃,
where the KN×KN matrix QH̃H is a unitary matrix with QH̃-1=QH̃H that takes the matrix H̃HH̃ to an upper triangular form of RH̃ with dimension KN×KN. As a consequence, the QR-based scheme in [8, 9] can be utilized to find approximations of eigenvalues of H̃HH̃, that is,  (23)ΛH̃=QH̃HH̃HH̃QH̃.
Subsequently, the diagonal elements of ΛH̃ is rearranged in a decreasing order. Finally, the matrix F̃opt can be attained by [5, 7, 10]  (24)F̃opt=QH̃𝚺fQH̃H,
where the diagonal elements [Σf]i,i, i=1,2,…,KN, of the matrix Σf are determined by [7]  (25)[𝚺f]i,i=(ρμ[ΛH̃]i,i-ρ[ΛH̃]i,i)+.
As the matrix F̃opt is derived, the ith block diagonal elements of F̃opt, for i=1,2,…,K, can be acquired by using  (26)FiFiH=[F̃opt](i-1)N+1:iN,(i-1)N+1:iN,
while the condition of the total sum-power constraint has to be preserved simultaneously. In general, the precoders Fi, i=1,2,…,K, are obtained by performing Cholesky factorizations of the matrix FiFiH, i=1,2,…,K. Here, a simple and novel scheme is proposed to perform factorization of FiFiH to obtain suboptimal precoders, denoted by F̂i, i=1,2,…,K. Note that every multisymbol transmission of each user can be viewed as multiple single symbol transmissions of each user and each symbol has its corresponding column in the linear precoder matrix. Thus, our aim is now to directly impose power constraints on the columns of the precoder matrix. The question is how to obtain the precoder matrix that satisfies the predetermined total power constraint of each mobile user, that is, tr(F̂iF̂iH)=tr(FiFiH), for i=1,2,…,K. Fortunately, there exists a method that guarantees the existence of at least one precoder matrix satisfying the given power distribution constraint. This scheme is called the EPD, which is presented as follows: for a given matrix FiFiH in (26), there exists a transmitter matrix F̂i=[f̂1,f̂2,…,f̂N], such that the conditions of ∥f̂m∥2=∥f̂n∥2, for m,n=1,2,…,N, can hold. In other words, each column vector f̂m of the matrix F̂i has equal power. Denote γ as the rank of the matrix FiFiH. Here, λk and uk are employed to indicate the kth eigenvalue and the associated eigenvector of the matrix FiFiH. Thus, the mth column vector of F̂i is obtained and given by  (27)f̂m=∑k=1γλkN⋅ej(2πmk/N)⋅uk.
Thus, the transmit power of each user is distributed equally between its symbols. The procedures of the proposed QR-EPD algorithm are summarized in Table 1.Table 1
The QR-EPD noniterative algorithm.
(1) First, the QR factorization is employed to perform the decomposition of  the matrixH̃HH̃ as follows:  H̃HH̃=QH̃RH̃.(2) A simple QR-based scheme [8] is utilized to find approximations of eigenvalues of H̃HH̃, that is, ΛH̃=QH̃HH̃HH̃QH̃. The diagonal elements of ΛH̃ are rearranged in a decreasing order.(3) The matrixF̃opt is attained by F̃opt=QH̃ΣfQH̃H, where [Σf]i,i=(ρμ[ΛH̃]i,i-ρ[ΛH̃]i,i)+.(4) As the matrixF̃opt is derived, the ith block diagonal elements of F̃opt, for i=1,2,…,K, need to be acquired, that is, FiFiH=[F̃opt](i-1)N+1:iN,(i-1)N+1:iN.(5) Finally, the EPD scheme is utilized to perform factorization ofFiFiH to obtain suboptimal precoders F̂i, i=1,2,…,K, F̂i=[f̂1,f̂2,…,f̂N], where the column vector of F̂i is given by  f̂m=∑k=1γλkN·ej(2πmk/N)·uk.


## 5. Numerical Results

In this section, computer simulations are conducted to demonstrate the performance of the proposed QR-EPD transceiver design as developed in Section4. In the simulations, the notation of [K,M,N,σE2] is employed to denote a K-user uplink MU-MIMO system equipped with M antennas at the BS and N antennas at the MS. Moreover, σE2 is used to stand for channel estimation-error variance for each user. In addition, the 4-QAM modulation method is adopted in each user's data streams. The total transmit power equal to the number of mobile users, that is, P=K, is assumed. To emphasize the importance of the use of the EPD scheme, the SMSE and averaged BER performance of the QR-EPD and SVD-EPD schemes are evaluated and compared with the transceiver design proposed in [7], denoted as SVD non-EPD. In both the SVD-EPD and SVD non-EPD schemes, the SVD is applied to decompose the matrix H̃ into the matrix-product form as follows:  (28)H̃=UH̃𝚺H̃VH̃H,
where both matrices UH̃ and VH̃ in (28) denote the M×M and KN×KN unitary matrices and ΣH̃ is the diagonal matrix with singular values arranged in the decreasing order. The matrix F̃opt has been derived in [5, 7, 10], which is given by  (29)F̃opt=VH̃𝚺fVH̃H,
where the diagonal elements [Σf]i,i, i=1,2,…,KN, of the matrix Σf are determined by [7]  (30)[𝚺f]i,i=(ρμ[𝚺H̃]i,i2-ρ[𝚺H̃]i,i2)+.
Also, to check out our derivations of the closed-form solution, the QR-EPD scheme is compared with an iterative transceiver structure based on the Karush-Kuhn-Tucker (KKT) optimality conditions [11].In Figure2, the SMSE and the averaged BER comparisons between the QR-EPD, the SVD-EPD, the SVD non-EPD, and the KKT schemes are evaluated in terms of SNR(dB) with parameters [K,M,N,σE2=0], that is, the perfectly-known CSI is available. In both figures, it is observed the fact that the SMSE and the average BER performance of the proposed QR-EPD are close to the SVD-EPD and better than those of the SVD non-EPD and the KKT. Another important fact to be emphasized is that when the total number of transmit antennas of all mobile users is less than the number of receiver antennas at BS, that is, KN<M, the SMSE and the average BER performance of these four algorithms are much better than those of the same schemes with KN=M. This is reasonable because the utilization of KN<M is able to offer a larger spatial diversity gain.(a) The SMSE and (b) the averaged BER comparisons between the proposed QR-EPD, the SVD-EPD, the SVD non-EPD, and the KKT algorithms forσE2=0.
(a)(b)Results of Figures3 and 4 show that the SMSE and the averaged BER performance of the proposed QR-EPD, the SVD-EPD, the SVD non-EPD, and the KKT schemes for the MIMO communication systems with the use of parameters [K,M,N,σE2] under the condition of limited CSI, that is, σE2∈{0.01,0.05}. It can be observed from figures that the SMSE and the averaged BER performance of all these schemes degrade dramatically as σE2 increases. Additionally, the performance deviation of these schemes between these two cases of KN<M and KN=M is increased as σE2 increases. However, it can be seen from Figures 3 and 4 that the SMSE and the average BER performance of the proposed QR-EPD scheme are still close to the SVD-EPD and superior than those of the SVD non-EPD and the KKT techniques when the imperfect CSI is assumed.(a) The SMSE and (b) the averaged BER comparisons between the proposed QR-EPD, the SVD-EPD, the SVD non-EPD, and the KKT algorithms forσE2=0.01.
(a)(b)(a) The SMSE and (b) the averaged BER comparisons between the proposed QR-EPD, the SVD-EPD, the SVD non-EPD, and the KKT algorithms forσE2=0.05.
(a)(b)


## 6. Conclusions

In this paper, a joint linear precoder-decoder pair is considered for uplink multiuser MIMO wireless systems with limited CSI at both the transmitters and the receivers. Instead of counting on complex iterative-based algorithms, the computationally efficient noniterative QR-based linear transceiver pair is proposed. In addition, an EPD scheme is applied to adjust transmit power allocation of each mobile station between its symbols under the sum-power constraint. Simulation results, compared with those of the existing iterative-based transceiver designs, have shown that the proposed QR-EPD scheme is capable of achieving a superior SMSE and enhance the average BER performance simultaneously without compromising the computational complexity.



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*Source: 735695-2011-02-10.xml*