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Linear precoding design for massive MIMO based on the minimum mean square error algorithm
EURASIP Journal on Embedded Systems volume 2017, Article number: 20 (2017)
Abstract
Compared with the traditional multipleinput multipleoutput (MIMO) systems, the large number of the transmit antennas of massive MIMO makes it more dependent on the limited feedback in practical systems. In this paper, we study the problem of precoding design for a massive MIMO system with limited feedback via minimizing mean square error (MSE). The feedback from mobile users to the base station (BS) is firstly considered; the BS can obtain the quantized information regarding the direction of the channels. Then, the precoding is designed by considering the effect of both noise term and quantization error under transmit power constraint. Simulation results show that the proposed scheme is robust to the channel uncertainties caused by quantization errors.
Introduction
Multipleinput multipleoutput (MIMO) techniques have gained considerable attention in modern wireless communications since it can significantly improve the capacity and reliability of wireless systems [1]. The essence of downlink multiuser MIMO is precoding, which means that the antenna arrays are used to direct each data signal spatially towards its intended receiver. Unfortunately, the precoding design in multiuser MIMO requires very accurate instantaneous channel state information (CSI) [2] which can be cumbersome to achieve in practice. To further achieve more dramatic gains as well as to simplify the required signal processing, massive MIMO techniques have been proposed in [3, 4] by installing a large number of antennas at base stations (BS), possibly in the order of tens or hundreds, which promises significant performance gains in terms of spectral efficiency and energy efficiency compared with conventional MIMO and is becoming a cornerstone of future 5G systems [5, 6].
From a practical point of view, realizing massive MIMO systems has to deal with several challenges, one of which is the lowcomplexity and nearoptimal precoding scheme [7, 8]. Generally, precoding approaches can be classified into nonlinear precoding and linear precoding. The optimal precoding is the nonlinear dirty paper precoding (DPC) [9], which can effectively eliminate the interference between different users and achieve optimal performance. However, nonlinear precoding schemes usually suffer from high complexity which makes them unpractical due to the hundreds of antennas in massive MIMO systems. Since the asymptotic orthogonality of massive MIMO channel matrix, simple linear precoding (e.g., zeroforcing (ZF) precoding) can be used to achieve capacityapproaching performance. Nevertheless, ZF precoding requires matrix inversion of very large size, which exhibits prohibitively high complexity. To reduce the complexity of matrix inversion of large size, a Neumannbased precoding is proposed in [10] to reduce the computational complexity in an iterative method, but the required complexity is still unaffordable. Recently, a low peaktoaverage power ratio (PAPR) precoding based on the approximate message passing (AMP) algorithm [11] and a successive overrelaxation (SOR)based precoding [12] are respectively proposed to minimize multiuser interference (MUI) in massive multiuser MIMO systems. The aforementioned works are based on the assumption of a perfect CSI at the BS, which is somewhat too optimistic for practical applications. As a result, it is essential to investigate the robust precoding design in massive MIMO systems.
Inspired by the abovementioned works, in this paper, we study the precoding design for a singlecell downlink massive MIMO system with limited feedback, while guaranteeing transmit power constraint. Each user terminal (UT) feeds back the quantized side information to BS to assist its transmission. We propose a linear precoder design scheme via the minimum mean square error (MMSE) criteria with respect to the CSI imperfection. The proposed scheme is an improved approach, which is robust to the channel uncertainties caused by quantization errors and the lack of channel quality information (CQI).
The rest of this paper is organized as follows. In Section 1, the system mode of massive MIMO is introduced and the problem is formulated. In Section 2, a linear precoder based on MMSE criteria is designed by considering the impact of the noise term and CSI quantization error. Numerical results are presented in Section 3. Finally, concluding remarks are made in Section 4.
Notations: Throughout this paper, boldface lowercase and uppercase letters denote vectors and matrices, respectively. The transpose, conjugate transpose, trace, and Frobenius norm of a matrix A are denoted as A ^{T}, A ^{H}, tr(A), ‖A‖_{ F }, respectively. I _{ M × M } denotes a M × M identity matrix. Ε[⋅] denotes the expectation operator. diag(⋅) stands for a diagonal matrix with the given elements on the diagonal. Re(⋅) represents the real part of the input.
System model
We consider a singlecell downlink multiuser massive MIMO system, as depicted in Fig. 1. For massive MIMO downlink transmissions, a large number of antennas N _{ T } that are equipped at the BS is serving K UTs with each UT being equipped with n _{ r } antennas and the total receive antennas of UTs is N _{ R } = Kn _{ r }. Here, \( \mathbf{H}\in {C}^{N_T\times {N}_R}=\left[{\mathbf{H}}_1,{\mathbf{H}}_2,\dots, {\mathbf{H}}_K\right] \) denotes the fast fading channel matrix from BS to the UTs, where each element is a zero mean unit variance independent and identical distributed (i.i.d) complex Gaussian. Let \( \mathbf{D}=\mathbf{H}{\mathbf{D}}_{\mu}^{1/2} \), thus \( {\mathbf{D}}_{\mu}^{1/2}=\mathrm{diag}\left\{\sqrt{\mu_1},\sqrt{\mu_2},\dots, \sqrt{\mu_K}\right\} \) denotes slow fading diagonal matrix.
Then, the total received signal \( \mathbf{y}={\left[{\mathbf{y}}_1^{\mathrm{T}},{\mathbf{y}}_2^{\mathrm{T}},\dots, {\mathbf{y}}_K^{\mathrm{T}}\right]}^{\mathrm{T}} \) at all UTs is given by
where x is the signals transmitted by the BS and n is an additive white Gaussian noise with zero mean and variance σ ^{2}. W = [W _{1}, W _{2}, ⋯, W _{ K }] is the transmitting precoding matrix. Denoting P _{ T } as the power constraint at BS, the total transmit power at BS is limited by tr(WW ^{H}) ≤ P _{ T }.
We assume that each UT can perfectly estimate the downlink CSI and send it back to ST using local feedback. All of the feedback channels are assumed to be noiseless and delay free. To facilitate analysis, the channel is decomposed into the channel direction information (CDI) and CQI [11]. The kth UT estimates the CSI of channel D _{ k } perfectly and quantizes the CDI \( {\tilde{\mathbf{D}}}_k={\mathbf{D}}_k/\left\Vert {\mathbf{D}}_k\right\Vert \) to a unit norm vector \( {\widehat{\mathbf{D}}}_k. \)
The acquisition of D _{ k } at BS can be accomplished via channel feedback. The quantized CDI \( {\widehat{\mathbf{D}}}_k \) is chosen from a predefined codebook C that consists of 2^{B} codeword matrices \( \left\{{C}_1,{C}_2,\cdots, {C}_{2^B}\right\} \), where B is the bit size of the codeword vector. Each UT quantizes its channel to the quantization vector that is closest to its channel vector, where closeness is measured in terms of the angle between two vectors or, equivalently, the inner product [12, 13]. Thus, the quantization of kth UT is chosen according to the minimum angle criterion as
and each UT feeds back the B bits codeword indices to the BS. Due to simplicity and analytical tractability, we employ random vector quantization (RVQ) for the codebook design where \( {\left\{{C}_i\right\}}_{i=1}^{2^B} \) are chosen independently and isotropically on the N _{ T }dimensional unit sphere [14]. Throughout the paper, we assume that this CQI is known perfectly to the BS, i.e., it is not quantized; thus, no CQI is fed back to the BS.
Linear precoder design
In this section, we introduce a linear precoder design scheme that considers the effect of both noise term and quantization error.
Channel model
In the following, according to [15], the subspace of the true channel matrix can be decomposed as the weighted sum of the quantized channel and an independent and isotropic quantization error term.
where D _{ k } is an orthonormal basis for the subspace spanned by the columns of D _{ k }. \( {\mathbf{X}}_k\in {C}^{n_r\times {n}_r} \) is a unitary matrix; \( {\mathbf{Y}}_k\in {C}^{n_r\times {n}_r} \) is upper triangular with positive diagonal elements and satisfies \( {\mathbf{Y}}_k^{\mathrm{H}}{\mathbf{Y}}_k={\mathbf{I}}_{n_r}{\mathbf{Z}}_k^{\mathrm{H}}{\mathbf{Z}}_k \). \( {\mathbf{Z}}_k\in {C}^{n_r\times {n}_r} \) is upper triangular with positive diagonal elements and represents the quantization error, satisfying \( \operatorname{tr}\left({\mathbf{Z}}_k^{\mathrm{H}}{\mathbf{Z}}_k\right)={ \sin}^2\left(\angle \left({\tilde{\mathbf{D}}}_k,{\widehat{\mathbf{D}}}_k\right)\right) \). S _{ k } is an orthonormal basis for an isotropically distributed n _{ r }dimensional plane in the left null space of \( {\widehat{\mathbf{D}}}_k \).
Therefore, the broadcast channel D _{ k } can be decomposed as
where \( \mathbf{R}=\operatorname{diag}\left(\sqrt{\varLambda_1},\sqrt{\varLambda_2},\dots, \sqrt{\varLambda_K}\right) \) and Z = diag(Z _{1}, Z _{2}, …, Z _{ K }) and Λ _{ k } is a diagonal matrix that consists of n _{ r } nonzero eigenvalues of \( {\mathbf{D}}_k{\mathbf{D}}_k^{\mathrm{H}} \). \( \mathbf{A}={\left({\mathbf{I}}_{N_R}{\mathbf{Z}}^{\mathrm{H}}\mathbf{Z}\right)}^{1/2}\mathbf{R} \) and B = ZR.
The achievable rate of the kth UT can be expressed as
Linear precoder design
The precoders can be designed to alleviate multiuser interference, maximize the received desired signal power, or minimize the MSE of the received signals. We in the following will take the MMSE precoder as an example. Then, MMSE cost function can be defined as [15]
Substituting (4) into (6), the conditional expectation becomes
The first term in (7) can be further calculated as
where \( E\left[\mathbf{R}{\mathbf{R}}^{\mathrm{H}}\right]= E\left[\operatorname{diag}\left\{{\varLambda}_1,{\varLambda}_2,\dots, {\varLambda}_K\right\}\right]={N}_T{\mathbf{I}}_{N_R}, \) \( E\left[\mathbf{Z}{\mathbf{Z}}^{\mathrm{H}}\right]=\frac{\varDelta}{N_R}{\mathbf{I}}_{N_R}. \) Here, Δ denotes the quantization error that can be approximated as [16]
where T = n _{ r }(N _{ T } − n _{ r }) and \( \varPhi =\frac{1}{T!}{\displaystyle \prod_{m=1}^{n_r}\frac{\left({N}_T m\right)!}{\left({n}_r m\right)!}} \). Γ(⋅) denotes gamma function.
The second item in (7) can also be calculated as
By substituting (8) and (10) into (7), the optimization problem can be rewritten by
where \( \alpha {\mathbf{I}}_{N_R}= E\left[\mathbf{A}\right]= E\left[{\left({\mathbf{I}}_{N_R}{\mathbf{Z}}^{\mathrm{H}}\mathbf{Z}\right)}^{1/2}\mathbf{R}\right]. \)
Therefore, the optimization problem in (6) is equivalent to
Obviously, problem (12) is a convex optimization problem. Hence, problem (12) can be solved with a closedform solution by exploiting the KarushKuhnTucker (KKT) conditions [17]. Constructing the Lagrangian function, we have
The optimal solution to W can be calculated by taking the firstorder derivative of (13) with respect to W and setting it to zero, i.e., \( \frac{\partial L\left(\mathbf{W},\beta, \lambda \right)}{\partial \mathbf{W}}=0. \) We can easily get
where \( \rho =\sqrt{\frac{P_T}{\rho {\left(\widehat{\mathbf{D}}{\widehat{\mathbf{D}}}^{\mathrm{H}}+\eta {\mathbf{I}}_{N_R}\right)}^{1}\widehat{\mathbf{D}}}} \) (ρ is normalized power factor) and \( \eta =\frac{P_T{N}_T\varDelta +{N}_R\left({N}_T{n}_r\right)}{P_T\left[\left({N}_T{n}_r\right){N}_T^2\varDelta \right]}. \)
Updating the Lagrange multiplier λ, we have
where [X]^{+} = max{X, 0}. a _{1} is the step size which is positive and t is the step time.

(1) If Δ = 0, that is perfect CSI. Here, \( \eta =\frac{N_R}{P_T} \). The precoding matrix can be rewritten as

$$ \mathbf{W}=\rho {\left(\widehat{\mathbf{D}}{\widehat{\mathbf{D}}}^{\mathrm{H}}+\eta {\mathbf{I}}_{N_R}\right)}^{1}\widehat{\mathbf{D}} $$(16)

(2) If Δ ≠ 0, that is imperfect CSI. Here, n _{ r } = 1 and N _{ T } = N _{ R } = K for simplicity, \( \varDelta =\frac{N_T1}{N_T}{2}^{\frac{B}{N_T1}}. \). The precoding matrix can be rewritten as

$$ \mathbf{W}=\rho {\left[\widehat{\mathbf{D}}{\widehat{\mathbf{D}}}^{\mathrm{H}}+\frac{P_T{2}^{\frac{B}{N_T1}}+{N}_T}{P_T\left(1{N}_T{2}^{\frac{B}{N_T1}}\right)}{\mathbf{I}}_{N_R}\right]}^{1}\widehat{\mathbf{D}}. $$(17)
Here, we use the subgradient algorithm to solve the problem. Using a constant step length t _{1} and t _{2}, the subgradient algorithm can converge to the optimal point of convex problems within a small range.
To summarize, the procedure of the subgradient algorithm based on updating (15) is shown in Table 1.
Since MMSE function in (12) is convex on a single precoder W, updating W at each iteration monotonically reduces the MMSE in (12), which is lower bounded by zero. Algorithm 1 can converge to the optimal point of the problem (6) within a small range. Although the precoder design depends on inaccurate CSI feedback, it may not always satisfy the transmit power constraint. However, we can assume a procedure that a feedforward link exists between UTs and BS. Each UT sends information from the precoder W to the BS via the feedforward link, then BS estimates the received power to satisfy the transmit power constraint.
Analysis of computational complexity
In this subsection, we analyze the computational complexity of the proposed precoder and compare it to the complexity of ZF precoder. We express the computational complexity in terms of the number of floating point operations (FLOPs). Following [18], the complexity of our proposed scheme can be calculated as
where τ is the number of transmit symbols per user, K is the number of UTs, and N _{ T } is the transmit antennas at BS. As the simplest precoding scheme, the computational complexity of the ZF precoder can be calculated as τN _{ T }(2K − 1) FLOPs.
Simulation results
In this section, the performance of the proposed scheme is evaluated by a computer simulation. In our simulations, the elements of all the signaling channel matrices are assumed to be i.i.d. complex Gaussian variables with zero means and unit variance. We assume that the number of UTs is K = 30, the total transmit power is P _{ T } = 20 dB, and the background noise is σ ^{2} = 1.
For simplicity, we assume that the received number of antennas at each UT is n _{ r } = 1.
Figure 2 depicts the effect of the number of antennas on the sum rate of the proposed algorithm with K = 6. As shown in Fig. 2, the sum rate increases with the increasing of the number of transmit antennas. It is observed that when the number of transmit antennas is more than 250, the sum rate gradually trends to saturation.
Figure 3 shows the sum rate as a function of the number of UTs under perfect CSI and limited feedback. It is observed that the sum rate increases as the number of the number of UTs increases. Clearly, there is a narrow gap between the proposed limited feedback scheme and perfect CSI.
Figure 4 shows the sum rate performance of different schemes under increasing number of feedback bits B under transmit antennas N _{ T } = 160 and n _{ r } = 1. We take the traditional BDZF and conventional MMSE (CMMSE) into comparison. It is clearly observed that our proposed scheme achieves a higher sum rate than other schemes, expect for perfect transmit CSI. We also find that the proposed scheme overcomes the sum rate degradation problem at highSNR regions that the CMMSE scheme has encountered.
Conclusions
In this paper, we investigated the problem of linear precoding design for massive MIMO system in a single cell based on MMSE criteria under transmit power constraint. The proposed scheme was robust to the uncertainties in the CSI as it taken into account the effect of quantization errors and noise term. Simulation results show the superiority of our proposed quantization scheme. In the future work, we plan to study the partial feedback of CSI for multicell massive MIMO systems.
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Ge, Z., Haiyan, W. Linear precoding design for massive MIMO based on the minimum mean square error algorithm. J Embedded Systems 2017, 20 (2017). https://doi.org/10.1186/s1363901600644
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DOI: https://doi.org/10.1186/s1363901600644
Keywords
 Massive MIMO
 Precoding
 Minimum mean square error (MMSE)
 Limited feedback