The document proposes a successive overrelaxation (SOR)-based linear precoding scheme to reduce the complexity of matrix inversion required for regularized zero-forcing (RZF) precoding in massive MIMO systems. The SOR-based precoding approximates the matrix inversion using an iterative SOR method, which can reduce complexity by about one order of magnitude compared to RZF precoding. It is also shown to converge within a few iterations and achieve performance close to RZF precoding. An empirical formula is provided to choose the optimal relaxation parameter for the SOR method in practical massive MIMO configurations.

1. A Low-Complexity Linear Precoding Scheme Based
on SOR Method for Massive MIMO Systems
Tian Xie, Qian Han, Huazhe Xu, Zihao Qi, and Wenqian Shen
Tsinghua National Laboratory for Information Science and Technology (TNList)
Department of Electronic Engineering, Tsinghua University, Beijing 100084, China
E-mail: razor9321@163.com
Abstract—Conventional linear precoding schemes in mas-
sive multiple-input-multiple-output (MIMO) systems, such as
regularized zero-forcing (RZF) precoding, have near-optimal
performance but suffer from high computational complexity
due to the required matrix inversion of large size. To solve
this problem, we propose a successive overrelaxation (SOR)-
based precoding scheme to approximate the matrix inversion
by exploiting the asymptotically orthogonal channel property in
massive MIMO systems. The proposed SOR-based precoding can
reduce the complexity by about one order of magnitude, and it
can also approach the classical RZF precoding with negligible
performance loss. We also prove that the proposed SOR-based
precoding enjoys a faster convergence rate than the recently
proposed Neumann-based precoding. In addition, to guarantee
the performance of SOR-based precoding, we propose a simple
way to choose the optimal relaxation parameter in practical
massive MIMO systems. Simulation results verify the advantages
of SOR-based precoding in convergence rate and computational
complexity in typical massive MIMO configurations.
I. INTRODUCTION
MIMO technology has played an essential role in cur-
rent wireless communications systems due to it can improve
the spectrum efficiency without additional requirements of
bandwidth or power [1]. However, the state-of-the-art MIMO
technology can not meet the exponentially increasing demand
of mobile traffic due to a very limited number of antennas is
usually used, e.g., only two transmit antennas are considered
for the next generation wireless broadcasting standard DVB-
T2 [2], and at most eight antennas can be adopted by LTE-A
celluar networks [3]. Recently, massive MIMO is proposed
to simultaneously improve the spectrum and energy efficiency
by several orders of magnitudes by using a large number of
antennas at the base station (BS) [4]. Thus, massive MIMO
is considered as a promising technology for 5G wireless
communications [5].
Realizing massive MIMO systems in practice has to deal
with several challenges, one of which is the low-complexity
and 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 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. Thanks
to the asymptotic orthogonality of massive MIMO channel
matrix, simple linear precoding (e.g., regularized zero-forcing
(RZF) precoding) can be used to achieve capacity-approaching
performance. Nevertheless, RZF precoding requires matrix
inversion of very large size, which exhibits prohibitively high
complexity. To reduce the complexity of matrix inversion of
large size, recently a Neumann-based precoding is proposed,
which can reduce the computational complexity in an iterative
method [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 inversion
for classical RZF precoding. This is motivated by the fact
that the matrix needs to be inversed in RZF precoding is a
Hermitian positive definite matrix and tends to be diagonal
dominant in massive MIMO systems [8], which provides the
potential to utilize SOR method. We also prove that SOR-
based precoding can enjoy a faster convergence rate than the
Neumann-based precoding. To guarantee the performance of
the proposed SOR-based precoding, we also provide a simple
method to determine the optimal relaxation parameter for
SOR method. Simulation results have shown that SOR-based
precoding can reduce the overall complexity by around one
order of magnitude compared with classical RZF precoding,
and it can approach the RZF precoding with only several
iterations (i.e., less than 4).
Notation: Lower-case and upper-case boldface letters denote
vectors and matrices, respectively; (·)T
, (·)H
, (·)−1
, tr(·), and
|·| denote the transpose, conjugate transpose, matrix inversion,
trace and absolute operators, respectively; ||·||F and ||·||2
denote the Frobenius norm of a matrix and the 2-norm of
a 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 consisting
of L cells [9], [10]. In each cell, the BS equips M antennas to
simultaneously serve K single-antenna users. Here, we usually
have M K (e.g., M = 256 and K = 16).We assume that
time division duplex (TDD) protocols are used so that the
channel vectors are equal for both directions. The received
signal at mth user in jth cell is
yj,m =
L
l=1
hH
l,j,mxl + nj,m, (1)

2. where xl ∈ CM×1
is the transmit signal after precoding from
lth base station (BS) and hl,j,m ∈ CM×1
represents the ran-
dom channel vector from lth base to mth user in jth cell, while
nj,m represents the additive white Gaussian noise (AWGN)
following zero mean and unit-covariance complex Gaussian
random distribution. The channel vector form lth to mth user
in jth cell can be specified as :
hl,j,m = κl,j,mR
1/2
l,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 fading
vector zl,j,m follows circularly symmetric complex Gaussian
random distribution. Channel matrix in jth cell for all K users
is defined as
Hl,j = [hl,j,1hl,j,2...hl,j,k] ∈ CM×K
(3)
Assuming all BSs use linear precoding, the precoding vector
for mth user in the jth cell is gj,m ∈ CM×1
and data from
that 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 precoding
matrix in jth cell and sj = [sj,1sj,2...sj,K]T
∈ CK×1
is the
symbol vector in jth cell. We denote the total transmit power
constraints at BS in jth cell as
tr(GjGH
j ) = Pj, (5)
where Pj is the total transmit power in jth cell. The received
signal of mth user in jth cell can be expressed as
yj,m =
L
l=1
K
k=1
hH
l,j,mgl,ksl,k + nj,m. (6)
Based on the TDD protocol, uplink pilot transmissions are
utilized to acquire instantaneous channel state information
(CSI) in each BS [11] [12]. Due to limited coherence interval
of channel, the same set of orthogonal sequences is reused in
each cell. Thus the channel estimation of mth user might be
corrupted by pilot from neighboring cells, which is called pilot
contamination [4]. This procedure can be specified as follows:
ˆHi,i = Hi,i +
i=j
Hi,j +
1
√
ρtr
Np
i ΦH
, (7)
where ˆHi,i is the CSI acquired in BS in ith cell, Np
i denotes
the uplink channel AWGN matrix and Φ is the transmitted
pilot sequence matrix.
III. PROPOSED SOR-BASED PRECODING
In this section, we first give a brief review for the classical
RZF precoding. Then, the SOR-based precoding, together with
the discussion of the optimal relaxation parameter, is proposed.
After that, we prove that SOR-based precoding has a faster
convergence rate than Neumann-based precoding. Finally, we
provide the computational complexity analysis.
A. Classical RZF Precoding
The classical RZF precoding matrix in jth cell is [4]:
GRZF
j = βj
ˆHj,j( ˆHH
j,j
ˆHj,j + ϕjIK)−1
= βjWj, (8)
where βj is the power normalization factor which makes RZF
precoding satisfy the power constraints in (5), the regularized
parameter ϕj can be adaptively selected according to different
channel state information (CSI) [13], which can be achieved
by using the time-domain and/or frequency-domain training
pilot [14]–[19], and Wj = ˆHj,j( ˆHH
j,j
ˆHj,j + ϕjIK)−1
. We
can choose βj as [20]
βj =
K
tr(WjWj
H
)
. (9)
The SINR at the transmitter side can be computed as
γj,m =
(hH
j,j,mgj,m)
2
(
(l,k)=(j,m)
hH
l,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 size
K × K is required, where the complexity in cubic of the user
number K is high. However, the fact that massive MIMO
systems have the favorable asymptotically orthogonal channel
property inspires us to design a low-complexity precoding
scheme to solve this problem, which will be discussed in the
next section.
B. SOR-based Precoding
Denote Pj = ˆHH
j,j
ˆHj,j + ϕjIK, so we have Wj =
ˆHj,jPj
−1
. We can rewrite (4) in
xj = Gjsj = βj
ˆHj,jPj
−1
sj. (12)
The matrix inversion problem tj = Pj
−1
sj in (12) can be
described as
Pjtj = sj. (13)
Assuming an arbitrary nonzero vector q ∈ CM×1
, we have
qH
Pjq = qH ˆHH
j,j
ˆHj,jq + qH
ϕjIM q
= qH ˆHH
j,j(qH ˆHH
j,j)H
+ qH
ϕjIM q > 0,
(14)
and
Pj
H
= ( ˆHH
j,j
ˆHj,j)H
+ ϕjIH
K = 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 triangular
component, respectively.
2) Compute the tj in such iterative steps:
t
(i+1)
j = (D+wL)
−1
(wsj+((1 − w)D−wLH
)tj
(i)
), (17)

3. 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 initial
solution vector t(0)
when 0 < w < 2 [21]. So without loss of
generality, we can set the initial solution t(0)
as a zero vector.
More specific discussion on w will be presented in the next
section.
C. Optimal Relaxation Parameter
The relaxation parameter affects the performance of SOR
method in a large extent. An optimal wopt for SOR method is
given by Young in [21, Theorem 7.2.3]:
wopt =
2
1 + 1 − ρ(BN )
2
, (18)
where ρ(BN ) = max
1≤i≤K
|λi(BN )| is the spectral radius of the
iteration matrix of Neumann method. This conclusion holds
for matrices with consistently ordered property [22, Definition
3.2.]. It is clear that diagonal matrices have the consistently
ordered property. When M goes to infinity and K keeps fixed
in massive MIMO systems, we have
lim
M→∞
( ˆHH ˆH + ϕIM) = lim
M→∞
ˆHH ˆH + lim
M→∞
ϕIM = MIM ,
(19)
which means that the diagonal elements of D tend to M and
elements of L converge to 0 [8]. So we can use (18) to get the
optimal w when the number of BS antennas goes to infinity.
0 5 10 15 20 25
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
M/K
SpectralRadiusOfIterationMatrix
Accurate optimal w
Optiaml w according to (18)
Optimal w according to (20)
Fig. 1. Comparison of spectral radius of the iterative matrix with accurate
optimal relaxation parameter and relaxation parameter from (18) and (20) in
different M/K.
However, in practical massive MIMO systems with finite M
and K, Pj does not have the consistently ordered property
because 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 obtained
through simulation and w acquired from (18) can not be
neglected. So we propose an empirical formula for the optimal
relaxation 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 comparison
among accurate optimal w and w acquired from (18) as well
as w acquired from (20) is shown in Fig. 1. , which shows that
our empirical formula achieves a close approximation to the
optimal relaxation parameter. In addition, when M/K goes
to infinity, the listed three ways to choose optimal w tend
to achieve the same spectral radius of iteration matrix, which
supports our discussion on the condition that the number of BS
antennas goes to infinity. An advantage of proposed empirical
formula is that spectral radius in (18) is dependent on CSI
and might be difficult to calculate in practical massive MIMO
systems, but (20) only depends on M/K and can be calculated
offline.
D. Convergence Rate
Since 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 iteration
matrix of SOR method. The approximation error can be
evaluated as
||t
(i+1)
j −tj||2 = ||B
i+1
SOR||F ||t
(0)
j −tj||2
≤ ||BSOR||
i+1
F ||t
(0)
j −tj||2.
(23)
We can observe from (23) that the final approximation error
is affected by the Frobenius norm of iteration matrix BSOR,
and a smaller ||BSOR||F means a faster convergence rate. The
following lemma 1 will verify that when the number of BS
antennas 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 infinity
in massive MIMO systems, the optimal relaxation parameter
w 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 M
and elements of L converge to 0 [8]. Thus lim
M→∞
D−1
L =
lim
M→∞
D−1
LH
= 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 infinity
in 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−1
L))
−1
((1 − w)D−wLH
)
= (I+wD−1
L)
−1
D−1
((1 − w)D−wLH
).
(25)

4. According to (19), we can use the polynomial expansion to
approximate (IK+D−1
L)−1
as [23, Theorem 2.2.3]
(IK + wD−1
L)−1
=
∞
k=0
(−1)
k
(wD−1
L)
k
= IK − wD−1
L +
∞
k=2
(−1)
k
(wD−1
L)
k
,
(26)
where we only keep the first two items, because
lim
M→∞
D−1
L = 0 . Based on (26), the iteration matrix
can be approached by
Bsor ≈ (IK − wD−1
L)D−1
((1 − w)D − wLH
)
= (IK − wD−1
L)((1 − w)IK − wD−1
LH
)
= (1 − w)IK + w2
D−1
LD−1
LH
− w(1 − w)D−1
L
− wD−1
LH
.
(27)
Then the Frobenius norm of BSOR satisfies
||BSOR||F ≤ ||(1 − w)IK − w(1 − w)D−1
L
− wD−1
LH
||F + ||w2
D−1
LD−1
LH
||F
≈ ||(1 − w)IK||F + |w(1 − w)| × ||D−1
L||F
+ |w| × ||D−1
LH
||F
= |1 − w|
√
K + |w(1 − w)| × ||D−1
L||F
+ |w| × ||D−1
LH
||F ,
(28)
where w2
D−1
LD−1
LH
can be neglected compared with
other items due to lim
M→∞
D−1
L = 0. Based on Lemma 1
and (28), we have ||BSOR||F ≤ ||D−1
LH
||F . Note that
||BN ||F = ||D−1
(L + LH
)||F = (
K
m=1
K
k=1,k=m
|
pkm
pmm
|2
)
1
2
= (2
K
m=1
K
k=1,k<m
|
pkm
pmm
|2
)
1
2
=
√
2||D−1
LH
||F ,
(29)
where pmk denotes the element in mth row and kth column
of Pj. Thus lemma 2 can be concluded.
E. Computational Complexity Analysis
We evaluate the computational complexity in terms of
required number of complex multiplications which is more
dominant than the other operations for the total computational
complexity. When the matrix inversion problem is solved, the
rest computation is the same for SOR-based, Neumann-based,
and RZF precoding. So we can just compare the complexity of
matrix inversion which originates from solving the following
linear 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 be
computed. 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 I
COMPUTATIONAL COMPLEXITY
Iterative
number
Neumann-based
precoding [7]
SOR-based
precoding
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-based
precoding is O(K3
) when i > 2, which means the reduction in
the 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 of
Neumann-based precoding reduces to O(K2
), but it is still
higher than that of SOR-based precoding. Thus, SOR-based
precoding has lower computational complexity than Neumann-
based precoding, especially when i is large.
IV. SIMULATION RESULTS
We provide simulation results of the achievable channel
capacity of the proposed SOR-based precoding in a 256 × 16
massive MIMO system as well as a 256 × 32 massive MIMO
system. The number of cells is set to 3. For convenience, we
assume 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 log
normal distribution, and rl,j,m is the distance between BS in
lth cell and mth user in jth cell, while R0 is reference distance
and ξ is the path loss exponent [24]. Users in the same cell are
uniformly distributed in an annulus which has an outer radius
of 450m and an inner radius of 100m in a 500m cell. The Path
loss exponent ξ is chosen as 3.3, reference distance is 100m,
and pilot transmit power in (7) is 0 dB. The RZF precoding
with exact matrix inversion is also included as benchmark. We
pick up one cell as the target cell and the signal from the other
cells are considered as interference.
Fig. 2 compares the capacity between Neumann-based pre-
coding, SOR-based precoding and RZF precoding in a 256×16
massive MIMO system. From Fig. 2, we can observe that in
practical massive MIMO systems, SOR-based precoding can
achieve 98% capacity of RZF precoding when i = 2, while
Neumann-based precoding only has 80% performance with
the same i. In fact, SOR-based precoding can achieve the
almost capacity of RZF precoding with only a small number
of iterations, which ensures a low computational complexity.
Fig. 3 compares the capacity between Neumann-based
precoding, SOR-based precoding, and RZF precoding in a
256 × 32 massive MIMO system. It provides a similar trend
comapred with Fig. 2, but shows a decrease in capacity as K
increases. For example, SOR-based precoding only achieves
90% capacity of exact matrix inversion RZF precoding when

5. 0 5 10 15 20 25 30
1.5
2
2.5
3
3.5
4
4.5
SNR (dB)
UserAverageCapacitybps/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 30
0.5
1
1.5
2
2.5
3
SNR (dB)
UserAverageCapacitybps/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-based
precoding only achieves 88% capacity of RZF precoding. This
is because due to the decrease in M/K, the spatial freedom
degree of massive MIMO systems has been reduced.
V. CONCLUSIONS
In this paper, we exploit the special channel property of
massive MIMO systems to propose the SOR-based precoding
scheme 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-based
precoding. Simulation results show that SOR-based precoding
achieves 90% of benchmark user average capacity when i = 2,
while Neumann-based precoding achieves 88% performance
when i = 4 for a 256 × 32 massive MIMO system. In
addition, we observe that the optimal relaxation parameter
decreases roughly exponentially to M/K, and then propose
a simple way to choose the relaxation parameter for practical
massive MIMO systems. Our next work will concentrate on
deriving a exact mathematical relationship between M/K and
the optimal relaxation parameter.
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