A Low-Complexity Linear Precoding Scheme Basedon SOR Method for Massive MIMO Systems

Tian Xie, Qian Han, Huazhe Xu, Zihao Qi, and Wenqian ShenTsinghua National Laboratory for Information Science and Technology (TNList)

Department of Electronic Engineering, Tsinghua University, Beijing 100084, ChinaE-mail: razor9321@163.com

Abstract—Conventional linear precoding schemes in mas-sive multiple-input-multiple-output (MIMO) systems, such asregularized zero-forcing (RZF) precoding, have near-optimalperformance but suffer from high computational complexitydue to the required matrix inversion of large size. To solvethis problem, we propose a successive overrelaxation (SOR)-based precoding scheme to approximate the matrix inversionby exploiting the asymptotically orthogonal channel property inmassive MIMO systems. The proposed SOR-based precoding canreduce the complexity by about one order of magnitude, and itcan also approach the classical RZF precoding with negligibleperformance loss. We also prove that the proposed SOR-basedprecoding enjoys a faster convergence rate than the recentlyproposed Neumann-based precoding. In addition, to guaranteethe performance of SOR-based precoding, we propose a simpleway to choose the optimal relaxation parameter in practicalmassive MIMO systems. Simulation results verify the advantagesof SOR-based precoding in convergence rate and computationalcomplexity in typical massive MIMO configurations.

I. INTRODUCTION

MIMO technology has played an essential role in cur-rent wireless communications systems due to it can improvethe spectrum efficiency without additional requirements ofbandwidth or power [1]. However, the state-of-the-art MIMOtechnology can not meet the exponentially increasing demandof mobile traffic due to a very limited number of antennas isusually used, e.g., only two transmit antennas are consideredfor the next generation wireless broadcasting standard DVB-T2 [2], and at most eight antennas can be adopted by LTE-Acelluar networks [3]. Recently, massive MIMO is proposedto simultaneously improve the spectrum and energy efficiencyby several orders of magnitudes by using a large number ofantennas at the base station (BS) [4]. Thus, massive MIMOis considered as a promising technology for 5G wirelesscommunications [5].

Realizing massive MIMO systems in practice has to dealwith several challenges, one of which is the low-complexityand near-optimal precoding scheme. Typical precoding meth-ods can be divided into nonlinear precoding and linear precod-ing. The optimal precoding is the nonlinear dirty paper precod-ing (DPC) [6], which can effectively eliminate the interferencebetween different users and achieve optimal performance.However, nonlinear precoding schemes usually suffer fromhigh complexity which makes them unpractical due to thehundreds of antennas in massive MIMO systems. Thanksto the asymptotic orthogonality of massive MIMO channel

matrix, simple linear precoding (e.g., regularized zero-forcing(RZF) precoding) can be used to achieve capacity-approachingperformance. Nevertheless, RZF precoding requires matrixinversion of very large size, which exhibits prohibitively highcomplexity. To reduce the complexity of matrix inversion oflarge size, recently a Neumann-based precoding is proposed,which can reduce the computational complexity in an iterativemethod [7]. But the required complexity is still unaffordable.

In this paper, we propose a successive overrelaxation (SOR)-based precoding to reduce the complexity of matrix inversionfor classical RZF precoding. This is motivated by the factthat the matrix needs to be inversed in RZF precoding is aHermitian positive definite matrix and tends to be diagonaldominant in massive MIMO systems [8], which provides thepotential to utilize SOR method. We also prove that SOR-based precoding can enjoy a faster convergence rate than theNeumann-based precoding. To guarantee the performance ofthe proposed SOR-based precoding, we also provide a simplemethod to determine the optimal relaxation parameter forSOR method. Simulation results have shown that SOR-basedprecoding can reduce the overall complexity by around oneorder of magnitude compared with classical RZF precoding,and it can approach the RZF precoding with only severaliterations (i.e., less than 4).

Notation: Lower-case and upper-case boldface letters denotevectors and matrices, respectively; (·)T , (·)H , (·)−1, tr(·), and|·| denote the transpose, conjugate transpose, matrix inversion,trace and absolute operators, respectively; ||·||F and ||·||2denote the Frobenius norm of a matrix and the 2-norm ofa vector, respectively; C denotes the set of complex numbers;Finally, IN is the N ×N identity matrix.

II. SYSTEM MODEL

We consider a multi-cell massive MIMO system consistingof L cells [9], [10]. In each cell, the BS equips M antennas tosimultaneously serve K single-antenna users. Here, we usuallyhave M � K (e.g., M = 256 and K = 16).We assume thattime division duplex (TDD) protocols are used so that thechannel vectors are equal for both directions. The receivedsignal at mth user in jth cell is

yj,m =

L∑l=1

hHl,j,mxl + nj,m, (1)

where xl ∈ CM×1 is the transmit signal after precoding fromlth base station (BS) and hl,j,m ∈ CM×1 represents the ran-dom channel vector from lth base to mth user in jth cell, whilenj,m represents the additive white Gaussian noise (AWGN)following zero mean and unit-covariance complex Gaussianrandom distribution. The channel vector form lth to mth userin jth cell can be specified as :

hl,j,m = κl,j,mR1/2l,j,mzl,j,m, (2)

where κl,j,m denotes the large scale fading coefficient, Rl,j,m

is the channel covariance matrix, and the small scale fadingvector zl,j,m follows circularly symmetric complex Gaussianrandom distribution. Channel matrix in jth cell for all K usersis defined as

Hl,j = [hl,j,1hl,j,2...hl,j,k] ∈ CM×K (3)

Assuming all BSs use linear precoding, the precoding vectorfor mth user in the jth cell is gj,m ∈ CM×1 and data fromthat BS is sj,m ∈ C. So we have

xj =

K∑m=1

gj,msj,m = Gjsj , (4)

where Gj = [gj,1gj,2...gj,K ] ∈ CM×K is the precodingmatrix in jth cell and sj = [sj,1sj,2...sj,K ]T ∈ CK×1 is thesymbol vector in jth cell. We denote the total transmit powerconstraints at BS in jth cell as

tr(GjGHj ) = Pj , (5)

where Pj is the total transmit power in jth cell. The receivedsignal of mth user in jth cell can be expressed as

yj,m =

L∑l=1

K∑k=1

hHl,j,mgl,ksl,k + nj,m. (6)

Based on the TDD protocol, uplink pilot transmissions areutilized to acquire instantaneous channel state information(CSI) in each BS [11] [12]. Due to limited coherence intervalof channel, the same set of orthogonal sequences is reused ineach cell. Thus the channel estimation of mth user might becorrupted by pilot from neighboring cells, which is called pilotcontamination [4]. This procedure can be specified as follows:

Hi,i = Hi,i +∑i 6=j

Hi,j +1√ρtr

Npi Φ

H , (7)

where Hi,i is the CSI acquired in BS in ith cell, Npi denotes

the uplink channel AWGN matrix and Φ is the transmittedpilot sequence matrix.

III. PROPOSED SOR-BASED PRECODING

In this section, we first give a brief review for the classicalRZF precoding. Then, the SOR-based precoding, together withthe discussion of the optimal relaxation parameter, is proposed.After that, we prove that SOR-based precoding has a fasterconvergence rate than Neumann-based precoding. Finally, weprovide the computational complexity analysis.

A. Classical RZF Precoding

The classical RZF precoding matrix in jth cell is [4]:

GRZFj = βjHj,j(H

Hj,jHj,j + ϕjIK)−1 = βjWj , (8)

where βj is the power normalization factor which makes RZFprecoding satisfy the power constraints in (5), the regularizedparameter ϕj can be adaptively selected according to differentchannel state information (CSI) [13], which can be achievedby using the time-domain and/or frequency-domain trainingpilot [14]–[19], and Wj = Hj,j(H

Hj,jHj,j + ϕjIK)−1. We

can choose βj as [20]

βj =

√K

tr(WjWjH). (9)

The SINR at the transmitter side can be computed as

γj,m =(hH

j,j,mgj,m)2

(∑

(l,k)6=(j,m)

hHl,j,mgl,k)

2+ σ2

. (10)

Thus, the ergodic capacity of mth user in jth cell is

rj,m = log2 (1 + γj,m). (11)

We can observe from (8) that a matrix inversion of sizeK ×K is required, where the complexity in cubic of the usernumber K is high. However, the fact that massive MIMOsystems have the favorable asymptotically orthogonal channelproperty inspires us to design a low-complexity precodingscheme to solve this problem, which will be discussed in thenext section.

B. SOR-based Precoding

Denote Pj = HHj,jHj,j + ϕjIK , so we have Wj =

Hj,jPj−1. We can rewrite (4) in

xj = Gjsj = βjHj,jPj−1sj . (12)

The matrix inversion problem tj = Pj−1sj in (12) can be

described asPjtj = sj . (13)

Assuming an arbitrary nonzero vector q ∈ CM×1, we have

qHPjq = qHHHj,jHj,jq + qHϕjIMq

= qHHHj,j(q

HHHj,j)

H + qHϕjIMq > 0,(14)

andPj

H = (HHj,jHj,j)

H + ϕjIHK = Pj , (15)

which means Pj is a positive definite Hermitian matrix.The SOR method to solve (13) can be described in 2 steps:1) Decomposing Pj as

Pj= D + L+LH , (16)

where D,L are the diagonal component and lower triangularcomponent, respectively.

2) Compute the tj in such iterative steps:

t(i+1)j = (D+wL)

−1(wsj+((1− w)D−wLH)tj

(i)), (17)

where the superscript i denotes the times of iteration.It has been proved that for a positive definite Hermitian

matrix Pj in (13), SOR method converges in any initialsolution vector t(0) when 0 < w < 2 [21]. So without loss ofgenerality, we can set the initial solution t(0) as a zero vector.More specific discussion on w will be presented in the nextsection.

C. Optimal Relaxation Parameter

The relaxation parameter affects the performance of SORmethod in a large extent. An optimal wopt for SOR method isgiven by Young in [21, Theorem 7.2.3]:

wopt =2

1 +

√1− ρ(BN )

2, (18)

where ρ(BN ) = max1≤i≤K

|λi(BN )| is the spectral radius of the

iteration matrix of Neumann method. This conclusion holdsfor matrices with consistently ordered property [22, Definition3.2.]. It is clear that diagonal matrices have the consistentlyordered property. When M goes to infinity and K keeps fixedin massive MIMO systems, we have

limM→∞

(HHH+ ϕIM) = limM→∞

HHH+ limM→∞

ϕIM =MIM ,

(19)which means that the diagonal elements of D tend to M andelements of L converge to 0 [8]. So we can use (18) to get theoptimal w when the number of BS antennas goes to infinity.

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

M/K

Sp

ectr

al R

adiu

s O

f It

erat

ion

Mat

rix

Accurate optimal w

Optiaml w according to (18)

Optimal w according to (20)

Fig. 1. Comparison of spectral radius of the iterative matrix with accurateoptimal relaxation parameter and relaxation parameter from (18) and (20) indifferent M/K.

However, in practical massive MIMO systems with finite Mand K, Pj does not have the consistently ordered propertybecause of the random interference in cells. It can be con-cluded from simulation results that when M/K is small (i.e.M/K < 10), the gap between accurate optimal w obtainedthrough simulation and w acquired from (18) can not beneglected. So we propose an empirical formula for the optimalrelaxation parameter with respect to M/K as follows

wopt = ae−b(M/K) + c, (20)

where a = 0.404, b = 0.323, and c = 1.035. A comparisonamong accurate optimal w and w acquired from (18) as wellas w acquired from (20) is shown in Fig. 1. , which shows thatour empirical formula achieves a close approximation to theoptimal relaxation parameter. In addition, when M/K goesto infinity, the listed three ways to choose optimal w tendto achieve the same spectral radius of iteration matrix, whichsupports our discussion on the condition that the number of BSantennas goes to infinity. An advantage of proposed empiricalformula is that spectral radius in (18) is dependent on CSIand might be difficult to calculate in practical massive MIMOsystems, but (20) only depends on M/K and can be calculatedoffline.

D. Convergence RateSince SOR method is convergent when 0 < w < 2 as

proved in [10], we have:

tj= (D+wL)−1

(wsj+((1− w)D−wLH)tj ), (21)

where xj denotes the accurate solution to (16). From (17) and(21), we can observe that :

t(i+1)j −tj = BSOR(t

(i)j −tj) = Bi+1

SOR(t(0)j −tj), (22)

where BSOR = (D + wL)−1

((1−w)D−wLH) is the iterationmatrix of SOR method. The approximation error can beevaluated as

||t(i+1)j −tj ||2 = ||Bi+1

SOR||F ||t(0)j −tj ||2

≤ ||BSOR||i+1F ||t(0)j −tj ||2.

(23)

We can observe from (23) that the final approximation erroris affected by the Frobenius norm of iteration matrix BSOR,and a smaller ||BSOR||F means a faster convergence rate. Thefollowing lemma 1 will verify that when the number of BSantennas goes to infinity in massive MIMO systems, SOR-based precoding has a faster convergence rate than Neumann-based precoding.

Lemma 1. When the number of BS antennas goes to infinityin massive MIMO systems, the optimal relaxation parameterw in (18) converges to 1.

Proof: The iteration matrix of Neumann method is

BN = D−1(L + LH). (24)

According to (19), the diagonal elements of D tend to Mand elements of L converge to 0 [8]. Thus lim

M→∞D−1L =

limM→∞

D−1LH = 0. So ρ(BN ) tends to 0 as M goes infinity,

which means that wopt =2

1+√1−02 = 1.

Lemma 2. When the number of BS antennas goes to infinityin massive MIMO systems, we have ||BSOR||F ≤

1√2||BN ||F .

Proof: Note that BSOR can be rewritten as

BSOR = (D+wL)−1

((1−w)D−wLH)

= (D(I+wD−1L))−1

((1− w)D−wLH)

= (I+wD−1L)−1

D−1((1− w)D−wLH).

(25)

According to (19), we can use the polynomial expansion toapproximate (IK+D−1L)−1 as [23, Theorem 2.2.3]

(IK + wD−1L)−1 =

∞∑k=0

(−1)k(wD−1L)k

= IK − wD−1L +

∞∑k=2

(−1)k(wD−1L)k,

(26)where we only keep the first two items, becauselim

M→∞D−1L = 0 . Based on (26), the iteration matrix

can be approached by

Bsor ≈ (IK − wD−1L)D−1((1− w)D− wLH)

= (IK − wD−1L)((1− w)IK − wD−1LH)

= (1− w)IK + w2D−1LD−1LH − w(1− w)D−1L

− wD−1LH .(27)

Then the Frobenius norm of BSOR satisfies

||BSOR||F ≤ ||(1− w)IK − w(1− w)D−1L− wD−1LH ||F + ||w2D−1LD−1LH ||F

≈ ||(1− w)IK ||F + |w(1− w)| × ||D−1L||F+ |w| × ||D−1LH ||F

= |1− w|√K + |w(1− w)| × ||D−1L||F

+ |w| × ||D−1LH ||F ,(28)

where w2D−1LD−1LH can be neglected compared withother items due to lim

M→∞D−1L = 0. Based on Lemma 1

and (28), we have ||BSOR||F ≤ ||D−1LH ||F . Note that

||BN ||F = ||D−1(L + LH)||F = (

K∑m=1

K∑k=1,k 6=m

| pkmpmm

|2) 12

= (2

K∑m=1

K∑k=1,k<m

| pkmpmm

|2) 12

=√2||D−1LH ||F ,

(29)where pmk denotes the element in mth row and kth columnof Pj . Thus lemma 2 can be concluded.

E. Computational Complexity Analysis

We evaluate the computational complexity in terms ofrequired number of complex multiplications which is moredominant than the other operations for the total computationalcomplexity. When the matrix inversion problem is solved, therest computation is the same for SOR-based, Neumann-based,and RZF precoding. So we can just compare the complexity ofmatrix inversion which originates from solving the followinglinear equation

t(k+1)i,j = t

(k)i,j +

w

aii(si−

i−1∑l=1

aijtl,j(k+1)−

K∑l=i

aijtl,j(k)), (30)

where t(k+1)i,j denotes the ith element in tj . Solving (31) needs

K times of multiplication, and there are K elements need to becomputed. It is clear that the overall complexity is i×K2. Ta-ble I shows the computational complexity between Neumann-based precoding and proposed SOR-based precoding.

TABLE ICOMPUTATIONAL COMPLEXITY

Iterativenumber

Neumann-basedprecoding [7]

SOR-basedprecoding

i = 2 3K2 −K 2K2

i = 3 K3 +K2 3K2

i = 4 2K3 4K2

i = 5 3K3 −K2 5K2

According to Table I, the complexity of Neumann-basedprecoding is O(K3) when i > 2, which means the reduction inthe complexity of RZF precoding is not obvious. By contrast,the complexity of proposed SOR-based precoding is O(K2)for any iterative number i. When i = 2, complexity ofNeumann-based precoding reduces to O(K2), but it is stillhigher than that of SOR-based precoding. Thus, SOR-basedprecoding has lower computational complexity than Neumann-based precoding, especially when i is large.

IV. SIMULATION RESULTS

We provide simulation results of the achievable channelcapacity of the proposed SOR-based precoding in a 256× 16massive MIMO system as well as a 256× 32 massive MIMOsystem. The number of cells is set to 3. For convenience, weassume Rl,j,m in (2) to be IM , and κl,j,m =

zl,j,m(rl,j,m/R0)

ξ ,where zl,j,m is the shadow fading factor which abides a lognormal distribution, and rl,j,m is the distance between BS inlth cell and mth user in jth cell, while R0 is reference distanceand ξ is the path loss exponent [24]. Users in the same cell areuniformly distributed in an annulus which has an outer radiusof 450m and an inner radius of 100m in a 500m cell. The Pathloss exponent ξ is chosen as 3.3, reference distance is 100m,and pilot transmit power in (7) is 0 dB. The RZF precodingwith exact matrix inversion is also included as benchmark. Wepick up one cell as the target cell and the signal from the othercells are considered as interference.

Fig. 2 compares the capacity between Neumann-based pre-coding, SOR-based precoding and RZF precoding in a 256×16massive MIMO system. From Fig. 2, we can observe that inpractical massive MIMO systems, SOR-based precoding canachieve 98% capacity of RZF precoding when i = 2, whileNeumann-based precoding only has 80% performance withthe same i. In fact, SOR-based precoding can achieve thealmost capacity of RZF precoding with only a small numberof iterations, which ensures a low computational complexity.

Fig. 3 compares the capacity between Neumann-basedprecoding, SOR-based precoding, and RZF precoding in a256 × 32 massive MIMO system. It provides a similar trendcomapred with Fig. 2, but shows a decrease in capacity as Kincreases. For example, SOR-based precoding only achieves90% capacity of exact matrix inversion RZF precoding when

0 5 10 15 20 25 301.5

2

2.5

3

3.5

4

4.5

SNR (dB)

Use

r A

ver

age

Cap

acit

y b

ps/

Hz

RZF precoding

SOR−based precoding, i = 1

SOR−based precoding, i = 2

SOR−based precoding, i = 3

Nuemann−based precoding, i = 1

Neumann−based precoding, i = 2

Neumann−based precoding, i = 3

Neumann−based precoding, i = 4

Fig. 2. Capacity comparison for the 256× 16 massive MIMO system.

0 5 10 15 20 25 300.5

1

1.5

2

2.5

3

SNR (dB)

Use

r A

ver

age

Cap

acit

y b

ps/

Hz

RZF precoding

SOR−based precoding, i = 1

SOR−based precoding, i = 2

SOR−based precoding, i = 3

Nuemann−based precoding, i = 1

Neumann−based precoding, i = 2

Neumann−based precoding, i = 3

Neumann−based precoding, i = 4

Fig. 3. Capacity comparison for the 256× 32 massive MIMO system.

i = 2, and Neumann-based precoding achieves 70% perfor-mance of RZF precoding. Even when i = 4, Neumann-basedprecoding only achieves 88% capacity of RZF precoding. Thisis because due to the decrease in M/K, the spatial freedomdegree of massive MIMO systems has been reduced.

V. CONCLUSIONS

In this paper, we exploit the special channel property ofmassive MIMO systems to propose the SOR-based precodingscheme to reduce the computational complexity from O(K3)to O(K2). When M tends to infinity and K keeps fixed, SOR-based precoding is proven to converge to GS-based precoding,and enjoys a convergence rate faster than Neumann-basedprecoding. Simulation results show that SOR-based precodingachieves 90% of benchmark user average capacity when i = 2,while Neumann-based precoding achieves 88% performancewhen i = 4 for a 256 × 32 massive MIMO system. Inaddition, we observe that the optimal relaxation parameterdecreases roughly exponentially to M/K, and then proposea simple way to choose the relaxation parameter for practicalmassive MIMO systems. Our next work will concentrate onderiving a exact mathematical relationship between M/K andthe optimal relaxation parameter.

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