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3298 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 5, MAY 2017

Widely Linear Precoding for Large-Scale MIMO

with IQI: Algorithms and Performance Analysis

Wence Zhang, Rodrigo C. de Lamare, Senior Member, IEEE, Cunhua Pan,

Ming Chen, Jianxin Dai, Bingyang Wu, and Xu Bao

Abstract—In this paper, we study widely linear precoding

techniques to mitigate in-phase/quadrature-phase (IQ) imbal-

ance (IQI) in the downlink of large-scale multiple-input multiple-

output (MIMO) systems. We adopt a real-valued signal model,

which considers the IQI at the transmitter, and then develop

widely linear zero-forcing (WL-ZF), widely linear matched ﬁlter,

widely linear minimum mean-squared error, and widely linear

block-diagonalization (WL-BD) type precoding algorithms for

both single- and multiple-antenna users. We also present a

performance analysis of WL-ZF and WL-BD. It is proved that

without IQI, WL-ZF has exactly the same multiplexing gain and

power offset as ZF, while when IQI exists, WL-ZF achieves the

same multiplexing gain as ZF with ideal IQ branches, but with a

minor power loss, which is related to the system scale and the IQ

parameters. We also compare the performance of WL-BD with

BD. The analysis shows that with ideal IQ branches, WL-BD

has the same data rate as BD, while when IQI exists, WL-BD

achieves the same multiplexing gain as BD without IQ imbalance.

Numerical results verify the analysis and show that the proposed

widely linear type precoding methods signiﬁcantly outperform

their conventional counterparts with IQI and approach those

with ideal IQ branches.

Index Terms—IQ imbalance, large-scale MIMO, widely-linear

signal processing, downlink precoding.

Manuscript received August 29, 2016; revised December 29, 2016; accepted

February 26, 2017. Date of publication March 20, 2017; date of current

version May 8, 2017. This work was supported in part by the Research

Foundation for Advanced Talents of Jiangsu University under

Grant 16JDG065, in part by the China Scholarship Council, in part by

CNPq, in part by FAPERJ, in part by PUC-Rio, in part by the University

of York, in part by the National Nature Science Foundation of China

under Grant 61221002, Grant 61571125, Grant 61401399, Grant 61372106,

Grant 61502210, and Grant 61571211, in part by the National Science

and Technology Major Project under Grant 2016ZX03001016-003, in part

by the Postdoctoral Research Funding Plan in Jiangsu Province under

Grant 1501073B, in part by the Natural Science Foundation of Nanjing Uni-

versity of Posts and Telecommunications under Grant NY214108, and in part

by the China Postdoctoral Science Foundation under Grant 2015M570484.

Part of this paper was presented in Eusipco 2014 and ICC 2015. The associate

editor coordinating the review of this paper and approving it for publication

was R. Brown. (Corresponding author: Cunhua Pan.)

W. Zhang is with Jiangsu University, Zhenjiang 212013, China. He

was with CETUC, PUC-Rio, Rio de Janeiro 22451-900, Brazil. (e-mail:

wencezhang@ujs.edu.cn).

R. C. de Lamare is with the University of York, York YO10 5DD, U.K.,

and also with PUC-Rio, Rio de Janeiro, Brazil (e-mail: delamare@cetuc.

puc-rio.br).

C. Pan is with the School of Electronic Engineering and Computer

Science, Queen Mary University of London, London, U.K. (e-mail:

c.pan@qmul.ac.uk).

M. Chen and B. Wu are with the National Mobile Communications Research

Laboratory, Southeast University, Nanjing 210096, China (e-mail: chenming@

seu.edu.cn; wubingyang@seu.edu.cn).

J. Dai is with the School of Science, Nanjing University of Posts and

Telecommunications, Nanjing 210003, China (e-mail: daijx@njupt.edu.cn).

X. Bao is with Jiangsu University, Zhenjiang 212013, China (e-mail:

xbao@ujs.edu.cn).

Color versions of one or more of the ﬁgures in this paper are available

online at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/TWC.2017.2679706

I. INTRODUCTION

IRELESS communications systems have undergone

tremendous development during the past decades.

In order to meet the increasing demands for data services,

many new techniques have been proposed, among which

multiple-input multiple-output (MIMO) techniques play an

important role. In the 5th generation (5G) of mobile commu-

nication systems, one of the key techniques will be large-scale

MIMO, which employs a large number of antennas at the base

station (BS) with centralized or distributed antenna systems to

provide extremely high data rates with improved quality of

service (QoS) [1], [2].

One of the main performance constraints of large-scale

MIMO systems comes from the impairments resulting from

hardware [3], [4]. Since large-scale MIMO systems employ a

large number of antennas, cheaper hardware is preferable in

order to reduce the cost, which may cause severe hardware

imperfection, e.g., the in-phase and quadrature-phase (IQ)

imbalance (IQI) [5]. Modern transceivers usually use the direct

conversion structure which contains two branches to process

the real and imaginary components of the baseband signals,

i.e., the in-phase (I) branch and the quadrature-phase (Q)

branch. The IQI exists when there is a gain difference between

the two branches and/or the phase difference is not exactly 90

◦

The IQI can be present at both the transmitter and the receiver,

according to many studies [6]–[8].

One way of handling IQI is to estimate the IQ parameters

and compensate for them (see [8]–[10] and references therein).

However, the IQ parameters are usually mingled with the

channel coefﬁcients, and thus are difﬁcult to obtain, especially

for large-scale MIMO systems, where the number of IQ

parameters is proportional to the system size and thus the esti-

mation and compensation for IQI can be very computationally

expensive.

Widely-linear approaches have long been used for

non-circular signal processing in MIMO systems [11]–[13]

and have been recently adopted to deal with IQI [5], [14]–[22].

In the uplink, the impact of IQI on the response pattern of large

antenna arrays is studied in [15]. The work in [16] describes an

equivalent interference model to study the impact on orthog-

onal frequency division multiple access (OFDMA) large-scale

MIMO systems and devised a receiver based on widely-linear

signal processing, which is extended in [22] to scenarios with

external interference. Kolomvakis et al. [20] investigated the

impact of IQI on the performance of uplink Massive MIMO

systems with maximum-ratio combining (MRC) receivers, and

showed that IQI can substantially degrade the performance

of MRC receivers. The study in [20] also proposed a low-

See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

ZHANG et al.: WIDELY LINEAR PRECODING FOR LARGE-SCALE MIMO WITH IQI 3299

complexity IQI compensation scheme. In order to suppress

the impact of IQI, a data-aided widely-linear minimum mean

square error (MMSE) receiver is proposed in [23], and an

IQI aware receiver was designed in [5] for the large-scale

MIMO uplink based on the minimum variance distortionless

response (MVDR) criterion. In [21], a widely-linear MMSE

receiver is proposed, the performance of which is shown to

be close to the linear MMSE receiver in an ideal system

without IQI.

To the best of the authors’ knowledge, there are only

a few related works in the downlink [14], [17]–[19]. The

study in [14] uses an augmented representation to maxi-

mize the power of the desired transmit signal when IQI is

present. Reduced-rank widely-linear precoders were devised

in [17] for single-antenna users to alleviate the impact

of IQI as well as to reduce the computational complex-

ity. A similar work was reported later in [18], which

studied the impact of IQI and proposed a widely-linear

regularized zero-forcing (RZF) precoding scheme. In [19],

we extended our previous work in [17] to scenarios with

multiple-antenna users in the large-scale MIMO downlink

and developed novel widely-linear block diagonal (BD) type

precoders.

In this paper, our previous work [17], [19] is extended

to give a comprehensive study on widely-linear precoding

algorithms for the large-scale MIMO downlink with IQI for

users with both single and multiple antennas. For large-

scale MIMO systems with single-antenna users, non-linear

precoding schemes, e.g., vector perturbation (VP) precoding

[24] and Tomlinson-Harashima precoding [25], [26], usually

have better performance than linear precoding schemes. How-

ever, linear precoding schemes, such as matched ﬁlter (MF)

(also referred to as maximum ratio transmission [27]), zero-

forcing (ZF) and MMSE [28], have much lower complexity

compared with the nonlinear precoding schemes and thus draw

great research interest [1], [2]. Generally, ZF and MMSE

perform better than MF, but with a comparatively higher

computational cost due to the matrix inversion involved.

Most current studies on large-scale MIMO downlink have

considered single-antenna users. However, it is well known

that with more receive antennas at the user equipment (UE),

the quality of service (QoS) of each user can be signiﬁ-

cantly improved. In fact, the long term evolution (LTE) and

LTE-Advanced (LTE-A) standards can support UEs with mul-

tiple antennas [29]. In terms of downlink precoding schemes,

block-diagonalization (BD) type precoding has been widely

considered for parallel transmission of multiple data streams

for users with multiple antennas [30]–[34]. In [32], the

regularized BD (RBD) precoding has been proposed which

outperforms conventional BD in [30] and [31], by taking into

account both the inter-user interference and noise. To reduce

the complexity, the work in [33] has devised the generalized

MMSE channel inversion (GMI) by replacing the singular

value decomposition (SVD) operation in BD and RBD with

a matrix inversion and QR decomposition. This scheme has

been further modiﬁed to obtain the simpliﬁed GMI (S-GMI)

technique in [34].

In contrast to [5], [15], [16], and [20]–[23], in this work

we study the design and performance analysis for down-

link precoding in large-scale MIMO systems with transmitter

IQI, for both single- and multiple-antenna users. Although

duality exists in the uplink and downlink [35], these results

are different fort the case considered here in the down-

link. We ﬁrst adopt a useful mapping function reported

in [36] and [37], which transforms complex-valued vectors

and matrices into real-valued expressions and helps to set

up an equivalent real-valued signal model with consideration

of IQI. Based on this real-valued signal model, we develop

widely-linear ZF (WL-ZF) and widely-linear MF (WL-MF)

and widely-linear MMSE (WL-MMSE) precoders, which are

suited for single-antenna terminals. Unlike [14], [17], and [18]

where only single-antenna users are considered, we propose

widely-linear BD (WL-BD) type precoding algorithms, i.e.,

WL-BD, widely-linear RBD (WL-RBD) and widely-linear

S-GMI (WL-S-GMI), for users equipped with multiple anten-

nas. A performance analysis is carried out for the ZF and

BD versions of these proposed precoding schemes, which

captures the essential advantages of widely-linear precoding

approaches. By utilizing an afﬁne approximation of the sum

data rate developed in [38], the mathematical expression

for the sum data rate of WL-ZF is derived. Moreover, we

also compare ZF and WL-ZF in terms of both multiplex-

ing gain and power offset, where the IQ parameters are

treated as random variables in the analysis, which is different

from [18], [21], and [22], where the IQ parameters are

ﬁxed. For WL-BD, the sum data rate and multiplexing gain

are derived and compared with those of BD. We also give

simulation results to show the impact of IQI in large-scale

MIMO systems, as well as the performance of the proposed

widely-linear precoding schemes.

The main contributions of this paper are summarized as

follows:

• We extend previous work of WL-ZF, WL-MF and

WL-MMSE precoding for large-scale MIMO systems

with IQI and single-antenna users in [17] to cases with

multiple-antenna users and propose WL-BD, WL-RBD

and WL-S-GMI algorithms based on an equivalent

real-valued signal model.

• For WL-ZF, we show that it has the same multiplexing

gain as that of ZF with ideal IQ branches. The achieved

multiplexing gain equals the number of users. Compared

with ZF without IQI, WL-ZF has a power offset loss

around log

[1+4σ

β],whereβ is the ratio of the number

of users to that of the transmit antennas and σ

is the

variance of the gain difference between I and Q branches.

• For WL-BD, we prove that when there is no IQI,

it achieves the same sum data rate as BD, while the

WL-BD precoder in the presence of IQI has the same

multiplexing gain as that of BD with ideal IQ branches.

Moreover, compared with BD the increased complexity

of WL-BD is very small.

This paper is organized as follows. The system model is

described in Section II. In Section III, the proposed widely-

linear precoding algorithms are introduced. In Section IV, the

performance analysis of WL-ZF and WL-BD is carried out.

3300 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 5, MAY 2017

The numerical results are given in Section V and conclusions

are drawn in Section VI.

Notation: C

and C

N×M

denote the sets of N-dimensional

complex vectors and N × M complex matrices, respectively;

and R

N×M

denote the sets of N-dimensional real vectors

and N × M real matrices, respectively; (·)

∗

is the complex

conjugate; I

denotes an N × N identity matrix; CN (θ , )

denotes circularly symmetric complex Gaussian distribution

with mean θ and covariance ; U(a, b) denotes the uniform

distribution and N (a, b) denotes the Gaussian distribution,

where a and b are the mean and variance, respectively;

diag{a

, ···, a

} denotes a diagonal matrix with diagonal

entries given by a

, ···, a

; E{·} denotes the mathematical

expectation and Tr[·] denotes the trace of a matrix A.

II. S

YSTEM MODEL

Consider the downlink of a large-scale MIMO system with

one BS and K users. The BS is equipped with N antennas, and

there are M

antennas at the k-th user. Deﬁne M =



k=1

as the total number of antennas at all the users. Consider

the IQI at all the N transmit antennas at BS,

and for the

n-th antenna, the transmit symbol x

is corrupted by IQI as

+ a

∗

,wherea

and a

are the IQ parameters of

the n-th antenna that are modeled as [6]:



1 + σ

[cos(θ

/2) + jg

sin(θ

/2)],



1 + σ

cos(θ

/2) − jsin(θ

/2)], (1)

where θ

∼ U(0,σ

) and g

∼ N (0,σ

) are the relative

phase and gain mismatches between the IQ branches of the

n-th transmit antenna, respectively. The IQ parameters are

normalized so that they do not change the average signal

power. A proof of the selection of the normalization factor

is given in Appendix A. In (1), θ

= 0

◦

and g

= 0represent

the ideal case with no IQI. Note that although the Gaussian

and uniform distributions are considered for modeling IQ

parameters, the proposed algorithms and the performance

analysis are valid for other distributions.

If we consider the IQI at BS, the received signal y

∈ C

at the k-th user is given by

= H

x + H

∗

+ n

, (2)

where H

∈ C

×N

is the downlink channel matrix of

the k-th user, the elements of which are independent and

identically distributed (i.i.d.) Gaussian random variables with

zero-mean and unit variance; A

= diag{a

, ···, a

} and

= diag{a

, ···, a

}; n

∼ CN (0,σ

) is the noise

vector at the receiver; x ∈ C

is the transmit signal vector

after precoding. Here we consider a narrow-band single-

carrier system for simplicity and the extension to multi-carrier

systems remains open for future work.

The IQI at the user’s receiver only degrades its own signal and can be

addressed individually by IQI compensation techniques [8]. In contrast, the

IQI at the BS affects all the users and is severe in large-scale MIMO systems

due to the potential use of cheap hardware for cost issues. Therefore, we only

consider the IQI at the BS in this paper.

Let L

be the number of data streams of user k and

∈ C

N×L

, s

∈ C

be the precoder and the transmit

signal vector for the k-th user, respectively. Denote P =

, ···, P

], s =[s

, ···, s

]

, and we have

x = Ps.

Note that in contrast to single-antenna users using ZF, MMSE

or MF, in the case of multiple-antenna users using BD type

precoders, a receive ﬁlter matrix is generally required to

decode the multiple streams which is designed together with

the precoder, as will be detailed later in Section III.

In this paper, we assume that the transmitter has perfect

channel state information and the transmit signals for different

users are i.i.d circularly symmetric Gaussian random variables

with zero-mean and unit variance, i.e., ∀k = j, E{s

}=0,

and E{s

}=I

. We also assume there is a transmit power

constraint, i.e.,

E{Ps

}=P

. (3)

It can be seen from (2) that the transmit signal vector is

corrupted by its complex conjugate. In the frequency domain,

a mirror frequency component is introduced due to the IQI.

One possible way of handling such IQI resorts to estimation

of the corresponding IQ parameters and pre-compensation

for the IQI [6], [7]. Since the signal model of (2) gives

rise to non-circular data which can be exploited by widely-

linear processing, IQI can also be tackled by widely-linear

approaches [5], [17].

In what follows, we will devise and carry out a performance

analysis of widely-linear precoding schemes, which are able

to mitigate the IQI without signiﬁcantly increasing the com-

putational complexity.

III. P

ROPOSED WIDELY-LINEAR

PRECODING ALGORITHMS

In this section, we employ a useful transformation,

i.e., the T -transform from [36] and [37], which represents

complex-valued matrices and vectors using their real-valued

equivalents. Then we employ the T -transform to develop an

equivalent real-valued signal model, which helps to design

widely-linear precoding schemes. Several widely-linear pre-

coding algorithms such as WL-ZF, WL-MMSE, WL-BD,

WL-RBD and WL-S-GMI are then developed.

A. Real-Valued Signal Model

A mapping function of C

→ R

and C

m×p

→ R

2m×2p

namely the T -transform, is deﬁned as:

T (x) =



Re(x)

Im(x)



, T (X) =



Re(X) −Im(X)

Im(X) Re(X)



, (4)

where Re(·) and Im(·) represent the real and imaginary parts

of a vector or a matrix, respectively. The T -transform sets

up a relationship between the complex-valued matrices and

their real-valued counterparts. It is very useful for design

and performance analysis of widely-linear precoders. Some

properties of the T -transform are summarized in Lemma 1,

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