REVIEW Open Access

Linear precoding design for massive MIMObased on the minimum mean square erroralgorithmZhou Ge1* and Wu Haiyan2

Abstract

Compared with the traditional multiple-input multiple-output (MIMO) systems, the large number of the transmitantennas of massive MIMO makes it more dependent on the limited feedback in practical systems. In this paper, westudy the problem of precoding design for a massive MIMO system with limited feedback via minimizing meansquare error (MSE). The feedback from mobile users to the base station (BS) is firstly considered; the BS can obtainthe quantized information regarding the direction of the channels. Then, the precoding is designed by consideringthe effect of both noise term and quantization error under transmit power constraint. Simulation results show thatthe proposed scheme is robust to the channel uncertainties caused by quantization errors.

Keywords: Massive MIMO, Precoding, Minimum mean square error (MMSE), Limited feedback

1 IntroductionMultiple-input multiple-output (MIMO) techniqueshave gained considerable attention in modern wirelesscommunications since it can significantly improve thecapacity and reliability of wireless systems [1]. The es-sence of downlink multiuser MIMO is precoding, whichmeans that the antenna arrays are used to direct eachdata signal spatially towards its intended receiver. Unfor-tunately, the precoding design in multiuser MIMO re-quires very accurate instantaneous channel stateinformation (CSI) [2] which can be cumbersome toachieve in practice. To further achieve more dramaticgains as well as to simplify the required signal process-ing, massive MIMO techniques have been proposed in[3, 4] by installing a large number of antennas at basestations (BS), possibly in the order of tens or hundreds,which promises significant performance gains in termsof spectral efficiency and energy efficiency comparedwith conventional MIMO and is becoming a cornerstoneof 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 low-complexity and near-optimal precod-ing scheme [7, 8]. Generally, precoding approaches canbe classified into nonlinear precoding and linear precod-ing. The optimal precoding is the nonlinear dirty paperprecoding (DPC) [9], which can effectively eliminate theinterference between different users and achieve optimalperformance. However, nonlinear precoding schemesusually suffer from high complexity which makes themunpractical due to the hundreds of antennas in massiveMIMO systems. Since the asymptotic orthogonality ofmassive MIMO channel matrix, simple linear precoding(e.g., zero-forcing (ZF) precoding) can be used to achievecapacity-approaching performance. Nevertheless, ZFprecoding requires matrix inversion of very large size,which exhibits prohibitively high complexity. To reducethe complexity of matrix inversion of large size, aNeumann-based precoding is proposed in [10] to reducethe computational complexity in an iterative method,but the required complexity is still unaffordable. Re-cently, a low peak-to-average power ratio (PAPR) pre-coding based on the approximate message passing(AMP) algorithm [11] and a successive over-relaxation(SOR)-based precoding [12] are respectively proposed tominimize multiuser interference (MUI) in massive multi-user MIMO systems. The aforementioned works arebased on the assumption of a perfect CSI at the BS,

* Correspondence: zhougels@sina.com1Department of Information Engineering, Chongqing Youth Vocational andTechnical College, Chongqing, People’s Republic of ChinaFull list of author information is available at the end of the article

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Ge and Haiyan EURASIP Journal on Embedded Systems (2017) 2017:20 DOI 10.1186/s13639-016-0064-4

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which is somewhat too optimistic for practical applica-tions. As a result, it is essential to investigate the robustprecoding design in massive MIMO systems.Inspired by the abovementioned works, in this paper,

we study the precoding design for a single-cell downlinkmassive MIMO system with limited feedback, whileguaranteeing transmit power constraint. Each user ter-minal (UT) feeds back the quantized side information toBS to assist its transmission. We propose a linear preco-der design scheme via the minimum mean square error(MMSE) criteria with respect to the CSI imperfection.The proposed scheme is an improved approach, which isrobust to the channel uncertainties caused byquantization errors and the lack of channel quality infor-mation (CQI).The rest of this paper is organized as follows. In

Section 1, the system mode of massive MIMO is in-troduced and the problem is formulated. In Section 2,a linear precoder based on MMSE criteria is de-signed by considering the impact of the noise termand CSI quantization error. Numerical results arepresented in Section 3. Finally, concluding remarksare made in Section 4.Notations: Throughout this paper, boldface lowercase

and uppercase letters denote vectors and matrices, re-spectively. The transpose, conjugate transpose, trace,and Frobenius norm of a matrix A are denoted as AT,AH, tr(A), ‖A‖F, respectively. IM ×M denotes a M ×Midentity matrix. Ε[⋅] denotes the expectation operator.diag(⋅) stands for a diagonal matrix with the given ele-ments on the diagonal. Re(⋅) represents the real part ofthe input.

2 System modelWe consider a single-cell downlink multiuser massiveMIMO system, as depicted in Fig. 1. For massive MIMOdownlink transmissions, a large number of antennas NTthat are equipped at the BS is serving K UTs with eachUT being equipped with nr antennas and the total

receive antennas of UTs is NR = Knr. Here, H∈CNT�NR

¼ H1;H2;…;HK½ � denotes the fast fading channel matrixfrom BS to the UTs, where each element is a zero meanunit variance independent and identical distributed (i.i.d)complex Gaussian. Let D ¼ HDμ1=2 , thus Dμ1=2 ¼ diagffiffiffiffiffi

μ1p

;ffiffiffiffiffiμ2

p;…;

ffiffiffiffiffiffiμK

p� �denotes slow fading diagonal

matrix.

Then, the total received signal y ¼ yT1 ; yT2 ;…; yTK� �T

atall UTs is given by

y ¼ DHWxþ n; ð1Þwhere x is the signals transmitted by the BS and n is anadditive white Gaussian noise with zero mean and vari-ance σ2. W = [W1,W2,⋯,WK] is the transmitting pre-coding matrix. Denoting PT as the power constraint atBS, the total transmit power at BS is limited bytr(WWH) ≤ PT.We assume that each UT can perfectly estimate the

downlink CSI and send it back to ST using localfeedback. All of the feedback channels are assumed tobe noiseless and delay free. To facilitate analysis, thechannel is decomposed into the channel direction in-formation (CDI) and CQI [11]. The kth UT estimatesthe CSI of channel Dk perfectly and quantizes theCDI ~Dk ¼ Dk= Dkk k to a unit norm vector D̂k :The acquisition of Dk at BS can be accomplished via

channel feedback. The quantized CDI D̂k is chosen froma predefined codebook C that consists of 2B codewordmatrices C1;C2;⋯;C2Bf g , where B is the bit size of thecodeword vector. Each UT quantizes its channel to thequantization vector that is closest to its channel vector,where closeness is measured in terms of the angle be-tween two vectors or, equivalently, the inner product[12, 13]. Thus, the quantization of kth UT is chosen ac-cording to the minimum angle criterion as

D̂k ¼ arg maxCi∈C ~DHk Ci

��� ���¼ arg min

Ci∈Csin2 ∠ ~Dk ;Ci

� � ; ð2Þ

and each UT feeds back the B bits codeword indices tothe BS. Due to simplicity and analytical tractability, weemploy random vector quantization (RVQ) for the code-book design where Cif g2

B

i¼1 are chosen independentlyand isotropically on the NT-dimensional unit sphere[14]. Throughout the paper, we assume that this CQI isknown perfectly to the BS, i.e., it is not quantized; thus,no CQI is fed back to the BS.

3 Linear precoder designIn this section, we introduce a linear precoder designscheme that considers the effect of both noise term andquantization error.

BS

UT UT

UT

UT

Fig. 1 System model

Ge and Haiyan EURASIP Journal on Embedded Systems (2017) 2017:20 Page 2 of 6

3.1 Channel modelIn the following, according to [15], the subspace of thetrue channel matrix can be decomposed as the weightedsum of the quantized channel and an independent andisotropic quantization error term.

~Dk ¼ D̂kXkYk þ SkZknr; ð3Þ

where Dk is an orthonormal basis for the subspacespanned by the columns of Dk. Xk∈Cnr�nr is a unitarymatrix; Yk∈Cnr�nr is upper triangular with positive diag-onal elements and satisfies YHk Yk ¼ Inr−ZHk Zk . Zk∈Cnr�nris upper triangular with positive diagonal elements andrepresents the quantization error, satisfying tr ZHk Zk

� ¼ sin2 ∠ ~Dk ; D̂k

� � . Sk is an orthonormal basis for an

isotropically distributed nr-dimensional plane in the leftnull space of D̂k .Therefore, the broadcast channel Dk can be decomposed

as

D ¼ D̂ INR−ZHZ� 1=2 þ SZh iR ¼ D̂Aþ SB; ð4Þ

where R ¼ diag ffiffiffiffiffiffiΛ1p ; ffiffiffiffiffiffiΛ2p ;…; ffiffiffiffiffiffiΛKp� and Z = diag(Z1,Z2,…, ZK) and Λk is a diagonal matrix that consists of nrnon-zero eigenvalues of DkDHk . A ¼ INR−ZHZ

� 1=2R and

B = ZR.The achievable rate of the kth UT can be expressed as

R ¼ Ε½ log2 INR þXKi¼1DHk WiWHi Dk����� �����INRþ

XKi¼1;i≠k

DHk WiWHi Dk

�����������

¼ Ε log2 INR þXKi¼1

DHk WiWHi Dk

����������

" #

− Ε log2 INR þXK

i¼1;i≠kDHk WiW

Hi Dk

����������

" #:

ð5Þ

3.2 Linear precoder designThe precoders can be designed to alleviate multiuserinterference, maximize the received desired signalpower, or minimize the MSE of the received signals.We in the following will take the MMSE precoder asan example. Then, MMSE cost function can be de-fined as [15]

minβ;W Ε x−β−1y

2F

� �s:t: Ε Wxk k2F

� ≤PT : ð6Þ

Substituting (4) into (6), the conditional expectationbecomes

E x−β−1y

2

F

� �¼ ΕA;B;x x−β−1AHD̂HWx−β−1BHSHWx−β−1n

2F

�

¼ β−2ΕA tr WHD̂AAHD̂HW� �h i

þβ−2ΕB;S tr WHSBBHSHW� � �

þ β−2 þ 1� INRk k2F −β−1ΕA tr WHD̂A� � �−β−1ΕA tr AHD̂

HW

� �h ið7Þ

The first term in (7) can be further calculated as

β−2 ΕA tr WHD̂AAHD̂HW

� �h i¼ β−2ΕR;Z tr R INR−ZZH

� WHD̂D̂

HW

� �h i¼ β−2 NT−NTΔNR

�tr WHD̂D̂

HW

� �; ð8Þ

where Ε RRH� � ¼ Ε diag Λ1;Λ2;…;ΛKf g½ � ¼ NT INR ; Ε

ZZH� � ¼ ΔNR INR : Here, Δ denotes the quantization errorthat can be approximated as [16]

Δ ¼ Γ1T

� T

Φ−1T2−

BT ; ð9Þ

where T = nr(NT − nr) and Φ ¼ 1T !Ynrm¼1

NT−mð Þ!nr−mð Þ! . Γ(⋅) de-notes gamma function.

The second item in (7) can also be calculated as

β−2 ΕB;S tr WHSBBHSHW� � �

¼ β−2PT NTΔNT−nr

�−β−2

NTnrΔNR NT−nrð Þ

�: ð10Þ

By substituting (8) and (10) into (7), the optimizationproblem can be rewritten by

Ε x−β−1y

2

F

� �¼ β−2 NT− N

2TΔ

NR NT−nrð Þ

�

tr WHD̂D̂HW

� �þ β

−2NTΔNT−nr

þ β−2 þ 1� NR−2αβ−1Re tr WHD̂� � �; F W; βð Þð11Þ

where αINR ¼ Ε A½ � ¼ Ε INR−ZHZ� 1=2

Rh i

:

Therefore, the optimization problem in (6) is equiva-lent to

minβ;W F W; βð Þ s:t: tr WHW�

≤PT : ð12Þ

Obviously, problem (12) is a convex optimizationproblem. Hence, problem (12) can be solved with aclosed-form solution by exploiting the Karush-Kuhn-Tucker (KKT) conditions [17]. Constructingthe Lagrangian function, we have

Ge and Haiyan EURASIP Journal on Embedded Systems (2017) 2017:20 Page 3 of 6

L W; β; λð Þ ¼ F W; βð Þ þ λ tr WHW� −PT� �: ð13ÞThe optimal solution to W can be calculated by taking

the first-order derivative of (13) with respect to W and

setting it to zero, i.e., ∂L W;β;λð Þ∂W ¼ 0: We can easily get

W ¼ ρ D̂D̂H þ ηINR� �−1

D̂; ð14Þ

where ρ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

PT

ρ D̂D̂HþηINR

� −1D̂

r(ρ is normalized power

factor) and η ¼ PTNTΔþNR NT−nrð ÞPT NT−nrð Þ−N2TΔ½ � :

Updating the Lagrange multiplier λ, we have

λ t þ 1ð Þ ¼ λ tð Þ þ a1 tr WHW�

−PT� � �þ

; ð15Þwhere [X]+ = max{X, 0}. a1 is the step size which is

positive and t is the step time.

(1) If Δ = 0, that is perfect CSI. Here, η ¼ NRPT . Theprecoding matrix can be rewritten as

W ¼ ρ D̂D̂H þ ηINR� �−1

D̂ ð16Þ

(2) If Δ ≠ 0, that is imperfect CSI. Here, nr = 1 and NT=NR = K for simplicity, Δ ¼ NT−1NT 2

− BNT −1:. The precodingmatrix can be rewritten as

W ¼ ρ D̂D̂H þ PT2−

BNT −1 þ NT

PT 1−NT2− BNT −1

� �INR

24

35−1

D̂: ð17Þ

Here, we use the sub-gradient algorithm to solvethe problem. Using a constant step length t1 and t2,the sub-gradient algorithm can converge to the opti-mal point of convex problems within a small range.To summarize, the procedure of the sub-gradient algo-

rithm 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 monoton-ically reduces the MMSE in (12), which is lowerbounded by zero. Algorithm 1 can converge to theoptimal point of the problem (6) within a small

range. Although the precoder design depends on in-accurate CSI feedback, it may not always satisfy thetransmit power constraint. However, we can assume aprocedure that a feedforward link exists between UTsand BS. Each UT sends information from the preco-der W to the BS via the feedforward link, then BS es-timates the received power to satisfy the transmitpower constraint.

3.3 Analysis of computational complexityIn this subsection, we analyze the computationalcomplexity of the proposed precoder and compare itto the complexity of ZF precoder. We express thecomputational complexity in terms of the number offloating point operations (FLOPs). Following [18],the complexity of our proposed scheme can be cal-culated as

τ K þ 1ð Þ 2K−1ð ÞNT þ 2NT−1ð ÞK½ �; ð18Þ

where τ is the number of transmit symbols per user, K isthe number of UTs, and NT is the transmit antennas atBS. As the simplest precoding scheme, the computa-tional complexity of the ZF precoder can be calculatedas τNT(2K − 1) FLOPs.

4 Simulation resultsIn this section, the performance of the proposed schemeis evaluated by a computer simulation. In our simula-tions, the elements of all the signaling channel matricesare assumed to be i.i.d. complex Gaussian variables withzero means and unit variance. We assume that the num-ber of UTs is K = 30, the total transmit power is PT =20 dB, and the background noise is σ2 = 1.For simplicity, we assume that the received number of

antennas at each UT is nr = 1.

Table 1 The proposed sub-gradient algorithm

0 50 100 150 200 250 3000

1

2

3

4

5

6

7

8

9

Sum

rat

e (b

/s/H

z)

Number of antennas

Proposed scheme

Fig. 2 Sum rate versus the number of transmit antennas NT

Ge and Haiyan EURASIP Journal on Embedded Systems (2017) 2017:20 Page 4 of 6

Figure 2 depicts the effect of the number of anten-nas on the sum rate of the proposed algorithm withK = 6. As shown in Fig. 2, the sum rate increases withthe increasing of the number of transmit antennas. Itis observed that when the number of transmit anten-nas is more than 250, the sum rate gradually trendsto 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 increasesas the number of the number of UTs increases.Clearly, there is a narrow gap between the proposedlimited feedback scheme and perfect CSI.Figure 4 shows the sum rate performance of differ-

ent schemes under increasing number of feedbackbits B under transmit antennas NT = 160 and nr = 1.We take the traditional BD-ZF and conventionalMMSE (C-MMSE) into comparison. It is clearly ob-served that our proposed scheme achieves a highersum rate than other schemes, expect for perfecttransmit CSI. We also find that the proposed schemeovercomes the sum rate degradation problem athigh-SNR regions that the C-MMSE scheme hasencountered.

5 ConclusionsIn this paper, we investigated the problem of linearprecoding design for massive MIMO system in a sin-gle cell based on MMSE criteria under transmitpower constraint. The proposed scheme was robustto the uncertainties in the CSI as it taken into ac-count the effect of quantization errors and noiseterm. Simulation results show the superiority of ourproposed quantization scheme. In the future work,we plan to study the partial feedback of CSI formulticell massive MIMO systems.

0 10 20 30 40 50 60 70 80 90 100

100

101

K

Sum

rat

e (b

it/s/

Hz)

Perfect CSI

Limited feedback

Fig. 3 Sum rate versus the number of UTs K

Fig. 4 Performance comparison of the proposed scheme with different schemes under B = 6 bit and B = 16 bit

Ge and Haiyan EURASIP Journal on Embedded Systems (2017) 2017:20 Page 5 of 6

Competing interestsThe authors declare that they have no competing interests.

Author details1Department of Information Engineering, Chongqing Youth Vocational andTechnical College, Chongqing, People’s Republic of China. 2Department ofPolice Management Railway Police College, Zhengzhou, Henan 450003,People’s Republic of China.

Received: 10 March 2016 Accepted: 29 November 2016

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Ge and Haiyan EURASIP Journal on Embedded Systems (2017) 2017:20 Page 6 of 6

AbstractIntroductionSystem modelLinear precoder designChannel modelLinear precoder designAnalysis of computational complexity

Simulation resultsConclusionsCompeting interestsAuthor detailsReferences