Table of Contents
ISRN Signal Processing
Volume 2011, Article ID 735695, 6 pages
http://dx.doi.org/10.5402/2011/735695
Research Article

Combined Transceiver Optimization for Uplink Multiuser MIMO with Limited CSI

Department of Communications Engineering, National Chung Cheng University, 168 University Road, Min-Hsiung, Chiayi 621, Taiwan

Received 23 December 2010; Accepted 12 January 2011

Academic Editors: S. K. Bhatia and M. A. Nappi

Copyright © 2011 Chia-Chang Hu and Chung-Lun Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 [24]. 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. Section 2 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 an 𝑀-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 𝐇𝑖𝐶𝑀×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, 𝐾𝑁𝑀. As well, each user must have as many transmit antennas as the transmitted data streams. Suppose that user 𝑖 has data streams, denoted by 𝐱𝑖𝐶𝑁×1, 𝑖=1,2,,𝐾, which are assumed to be zero-mean white random vectors each with unit-energy 𝐸{𝐱𝑖𝐱H𝑖}=𝐈𝑁, for all 𝑖 and mutually statistically independent among users. A linear precoder is employed for each user, which is denoted by 𝐅𝑖𝐶𝑁×𝑁, 𝑖=1,2,,𝐾. The received signal vector 𝐫𝐶𝑀×1 at the BS is given by 𝐫=𝐾𝑖=1𝐇𝑖𝐅𝑖𝐱𝑖+𝐧,(1) where the noise vector 𝐧 is modeled as zero-mean white complex Gaussian variables, that is, 𝐶𝑁(0,𝜎2n𝐈𝑀). The data and the noise are assumed to be statistically independent. The received signal vector 𝐫 is processed by all linear decoders 𝐆𝑗𝐶𝑁×𝑀, for 𝑗=1,2,,𝐾. An estimate of the data vector for user 𝑗 can be expressed as 𝐲𝑗=𝐆𝑗𝐫=𝐆𝑗𝐾𝑖=1𝐇𝑖𝐅𝑖𝐱𝑖.+𝐧(2)

735695.fig.001
Figure 1: Block diagram of an uplink multiuser MIMO wireless system with an 𝑀-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 𝐇𝑖=𝐇𝑖+Δ𝐇𝑖,𝑖=1,2,,𝐾,(3) where 𝐇𝑖 is the estimated channel matrix and Δ𝐇𝑖 is the CSI-error matrix. The entries of 𝐇𝑖, 𝐇𝑖, and Δ𝐇𝑖 are identically independent distribution (i.i.d.) complex Gaussian variables each with 𝐶𝑁(0,1), 𝐶𝑁(0,1𝜎2E), and 𝐶𝑁(0,𝜎2E), respectively. Here, 𝜎2E indicates the channel estimation-error variance for each mobile user. Furthermore, matrices 𝐇𝑖 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 𝐫=𝐾𝑖=1𝐇𝑖+Δ𝐇𝑖𝐅𝑖𝐱𝑖+𝐧,(4) and the estimate of the data vector for user 𝑗 is expressed as 𝐲𝑗=𝐆𝑗𝐾𝑖=1𝐇𝑖+Δ𝐇𝑖𝐅𝑖𝐱𝑖.+𝐧(5) The MSE matrix for user 𝑗 is defined by MSE𝑗𝐲=𝐸𝑗𝐱𝑗𝐲𝑗𝐱𝑗H=𝐆𝑗𝐾𝑖=1𝐇𝑖𝐅𝑖𝐅H𝑖𝐇H𝑖+𝜎2n𝐈𝑀𝐆H𝑗+𝐆𝑗𝐾𝑖=1𝐸Δ𝐇𝑖𝐅𝑖𝐅H𝑖Δ𝐇H𝑖𝐆H𝑗𝐆𝑗𝐇𝑗𝐅𝑗𝐅H𝑗𝐇H𝑗𝐆H𝑗+𝐈𝑁.(6) Then it can be shown that MSE𝑗=𝐆𝑗𝐾𝑖=1𝐇𝑖𝐅𝑖𝐅H𝑖𝐇H𝑖+𝜎2n𝐈𝑀𝐆H𝑗+𝐆𝑗𝐾𝑖=1𝐅tr𝑖𝐅H𝑖𝜎2E𝐈𝑀𝐆H𝑗𝐆𝑗𝐇𝑗𝐅𝑗𝐅H𝑗𝐇H𝑗𝐆H𝑗+𝐈𝑁,(7) where the equality of 𝐸Δ𝐇𝑖𝐅𝑖𝐅H𝑖Δ𝐇H𝑖𝐅=tr𝑖𝐅H𝑖𝜎2E𝐈𝑀(8) is used in (7). The sum-MSE of all users is given by MSE=𝐾𝑗=1trMSE𝑗.(9) The uplink MU-MIMO problem is to minimize the sum-MSE subject to (s.t.) the sum-power constraint, which is given as follows: min(𝐅𝑗,𝐆𝑗)𝐾𝑗=1MSE=𝐾𝑗=1trMSE𝑗,(10)s.t.𝐾𝑗=1𝐅tr𝑗𝐅H𝑗=𝑃,(11) 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 [24] 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: min(𝐹𝑗,𝐺𝑗)𝐾𝑗=1,𝜇MSE+𝜇𝐾𝑗=1𝐅tr𝑗𝐅H𝑗.𝑃(12) 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 precoders 𝐅𝑖, 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] 𝐆𝑗=𝐅H𝑗𝐇H𝑗𝜌𝐈𝑀+𝐾𝑖=1𝐇𝑖𝐅𝑖𝐅H𝑖𝐇H𝑖1,(13) where 𝜎𝜌=2n+𝐾𝑖=1𝜎2E𝐅tr𝑖𝐅H𝑖(14) is used in (13). For convenience of notation, the matrix 𝐓 is defined as follows: 𝐓=𝜌𝐈𝑀+𝐾𝑖=1𝐇𝑖𝐅𝑖𝐅H𝑖𝐇H𝑖.(15) Then, substituting (13) into (9) derives a simplified SMSE expression given by 𝐓MSE=𝜌tr1+𝐾𝑗=1𝐈tr𝑁𝐈tr𝑀𝐈(16)=tr𝑀+1𝜌𝐾𝑖=1𝐇𝑖𝐅𝑖𝐅H𝑖𝐇H𝑖1+𝐾𝑁𝑀.(17) Therefore, the uplink MU-MIMO problem can be equivalently formulated as min𝐅𝑖𝐾𝑖=1𝐈MSE=tr𝑀+1𝜌𝐇𝐅𝐇H1𝐅+𝐾𝑁𝑀(18)s.t.tr=𝑃,(19) where the matrix 𝐇𝐅𝐇H in (18) is employed to represent the matrix 𝐾𝑖=1𝐇𝑖𝐅𝑖𝐅H𝑖𝐇H𝑖 in (17) and the relationships between both matrices are given by 𝐇𝐇=1𝐇2𝐇𝐾𝐅,(20)𝐅=1𝐅H1𝟎𝟎𝟎𝐅2𝐅H2𝟎𝟎𝟎𝐅𝐾𝐅H𝐾.(21) Note that no cooperation is assumed among mobile users in the MU-MIMO system. Therefore, the matrix 𝐅 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 𝐅, the block-diagonal constraint on 𝐅 needs to be relaxed for this moment. In what follows, the matrix of 𝐅opt is used to indicate the optimal solution of 𝐅. Instead of the singular value decomposition (SVD) used to decompose the matrix 𝐇 in [7], the computationally efficient QR factorization is employed to perform the decomposition of the matrix 𝐇H𝐇 as follows: 𝐐H𝐇𝐇H𝐇,𝐇=𝐑(22) where the 𝐾𝑁×𝐾𝑁 matrix 𝐐H𝐇 is a unitary matrix with 𝐐1𝐇=𝐐H𝐇 that takes the matrix 𝐇H𝐇 to an upper triangular form of 𝐑𝐇 with dimension 𝐾𝑁×𝐾𝑁. As a consequence, the QR-based scheme in [8, 9] can be utilized to find approximations of eigenvalues of 𝐇H𝐇, that is, 𝚲𝐇=𝐐H𝐇𝐇H𝐇.𝐇𝐐(23) Subsequently, the diagonal elements of Λ𝐇 is rearranged in a decreasing order. Finally, the matrix 𝐅opt can be attained by [5, 7, 10] 𝐅opt𝐇𝚺=𝐐𝑓𝐐H𝐇,(24) where the diagonal elements [𝚺𝑓]𝑖,𝑖, 𝑖=1,2,,𝐾𝑁, of the matrix 𝚺𝑓 are determined by [7] 𝚺𝑓𝑖,𝑖=𝜌𝜇𝚲𝐇𝑖,𝑖𝜌𝚲𝐇𝑖,𝑖+.(25) As the matrix 𝐅opt is derived, the 𝑖th block diagonal elements of 𝐅opt, for 𝑖=1,2,,𝐾, can be acquired by using 𝐅𝑖𝐅H𝑖=𝐅opt(𝑖1)𝑁+1𝑖𝑁,(𝑖1)𝑁+1𝑖𝑁,(26) 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 𝐅𝑖𝐅H𝑖, 𝑖=1,2,,𝐾. Here, a simple and novel scheme is proposed to perform factorization of 𝐅𝑖𝐅H𝑖 to obtain suboptimal precoders, denoted by 𝐅𝑖, 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, 𝐅tr(𝑖𝐅H𝑖)=tr(𝐅𝑖𝐅H𝑖), 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 𝐅𝑖𝐅H𝑖 in (26), there exists a transmitter matrix 𝐅𝑖̂𝐟=[1,̂𝐟2̂𝐟,,𝑁], such that the conditions of ̂𝐟𝑚2̂𝐟=𝑛2, for 𝑚,𝑛=1,2,,𝑁, can hold. In other words, each column vector ̂𝐟𝑚 of the matrix 𝐅𝑖 has equal power. Denote 𝛾 as the rank of the matrix 𝐅𝑖𝐅H𝑖. Here, 𝜆𝑘 and 𝐮𝑘 are employed to indicate the 𝑘th eigenvalue and the associated eigenvector of the matrix 𝐅𝑖𝐅H𝑖. Thus, the 𝑚th column vector of 𝐅𝑖 is obtained and given by ̂𝐟𝑚=𝛾𝑘=1𝜆𝑘𝑁𝑒𝑗(2𝜋𝑚𝑘/𝑁)𝐮𝑘.(27) 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.

tab1
Table 1: The QR-EPD noniterative algorithm.

5. Numerical Results

In this section, computer simulations are conducted to demonstrate the performance of the proposed QR-EPD transceiver design as developed in Section 4. In the simulations, the notation of [𝐾,𝑀,𝑁,𝜎2E] 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, 𝜎2E 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 𝐇 into the matrix-product form as follows: 𝐇𝚺𝐇𝐕𝐇=𝐔H𝐇,(28) where both matrices 𝐔𝐇 and 𝐕𝐇 in (28) denote the 𝑀×𝑀 and 𝐾𝑁×𝐾𝑁 unitary matrices and 𝚺𝐇 is the diagonal matrix with singular values arranged in the decreasing order. The matrix 𝐅opt has been derived in [5, 7, 10], which is given by 𝐅opt𝐇𝚺=𝐕𝑓𝐕H𝐇,(29) where the diagonal elements [𝚺𝑓]𝑖,𝑖, 𝑖=1,2,,𝐾𝑁, of the matrix 𝚺𝑓 are determined by [7] 𝚺𝑓𝑖,𝑖=𝜌𝜇𝚺𝐇2𝑖,𝑖𝜌𝚺𝐇2𝑖,𝑖+.(30) 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 Figure 2, 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,𝜎2E=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.

fig2
Figure 2: (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 for 𝜎2E=0.

Results of Figures 3 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,𝜎2E] under the condition of limited CSI, that is, 𝜎2E{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 𝜎2E increases. Additionally, the performance deviation of these schemes between these two cases of 𝐾𝑁<𝑀 and 𝐾𝑁=𝑀 is increased as 𝜎2E 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.

fig3
Figure 3: (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 for 𝜎2E=0.01.
fig4
Figure 4: (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 for 𝜎2E=0.05.

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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