1 Signal Detection in MIMO Systems with Hardware Imperfections Message Passing on Neural

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Signal Detection in MIMO Systems with Hardware

Imperfections: Message Passing on Neural

Networks

Dawei Gao, Qinghua Guo, Guisheng Liao, Yonina C. Eldar, Fellow, IEEE, Yonghui Li, Fellow, IEEE, Yanguang

Yu, and Branka Vucetic, Fellow, IEEE

Abstract—In this paper, we investigate signal detection in

multiple-input-multiple-output (MIMO) communication systems

with hardware impairments, such as power ampliﬁer nonlinearity

and in-phase/quadrature imbalance. To deal with the complex

combined effects of hardware imperfections, neural network (NN)

techniques, in particular deep neural networks (DNNs), have

been studied to directly compensate for the impact of hardware

impairments. However, it is difﬁcult to train a DNN with limited

pilot signals, hindering its practical applications. In this work,

we investigate how to achieve efﬁcient Bayesian signal detection

in MIMO systems with hardware imperfections. Characterizing

combined hardware imperfections often leads to complicated

signal models, making Bayesian signal detection challenging. To

address this issue, we ﬁrst train an NN to ‘model’ the MIMO

system with hardware imperfections and then perform Bayesian

inference based on the trained NN. Modelling the MIMO system

with NN enables the design of NN architectures based on the

signal ﬂow of the MIMO system, minimizing the number of NN

layers and parameters, which is crucial to achieving efﬁcient

training with limited pilot signals. We then represent the trained

NN with a factor graph, and design an efﬁcient message passing

based Bayesian signal detector, leveraging the unitary approxi-

mate message passing (UAMP) algorithm. The implementation

of a turbo receiver with the proposed Bayesian detector is also

investigated. Extensive simulation results demonstrate that the

proposed technique delivers remarkably better performance than

state-of-the-art methods.

Index Terms—Hardware imperfections, I/Q Imbalance, power

ampliﬁer nonlinearity, multiple-input-multiple-output (MIMO),

neural networks (NNs), factor graphs, approximate message

passing (AMP), Bayesian inference.

I. INTRODUCTION

WE consider signal detection for multiple-input multi-

output (MIMO) communications in the presence of

hardware impairments, which arise, e.g., in millimeter wave

(mm-wave) communications, where mm-wave front ends suf-

fer from signiﬁcant hardware imperfections, compromising

Corresponding to Qinghua Guo (qguo@uow.edu.au).

Dawei Gao and Guisheng Liao are with the Hangzhou Institute of Technol-

ogy, Xidian University, Hangzhou 311200, China and also with the National

Laboratory of Radar Signal Processing, Xidian University, Xi’an 710071,

China (e-mail: gaodawei@xidian.edu.cn; liaogs@xidian.edu.cn).

Qinghua Guo and Yanguang Yu are with the School of Electrical, Computer

and Telecommunications Engineering, University of Wollongong, NSW 2522,

Australia (e-mail: qguo@uow.edu.au; yanguang@uow.edu.au).

Yonina C. Eldar is with the Faculty of Math and CS, Weizmann Institute

of Science, Rehovot, 7610001, Israel (email: yonina.eldar@weizmann.ac.il).

Yonghui Li and Brank Vucetic are with the School of Electrical and

Information Engineering, University of Sydney, Sydney, NSW 2006, Australia

(e-mail: yonghui.li@ sydney.edu.au; branka.vucetic@sydney.edu.au).

signal transmission quality and degrading system performance

[1]–[3]. A pronounced impairment is in-phase/quadrature (I/Q)

imbalance, i.e., the mismatch of amplitude, phase and fre-

quency response between the I and Q branches, which impairs

their orthogonality [4]. Power ampliﬁer (PA) nonlinearity leads

to nonlinear distortions to transmitted signals, which cannot be

overlooked, especially in mm-wave communications [2]. The

hardware imperfections need to be handled properly to avoid

inducing signiﬁcant system performance loss.

Many techniques have been considered to mitigate the

impact of hardware imperfections. To handle PA nonlinearity,

Volterra series based techniques were proposed for nonlinear-

ity compensation at either transmitter or receiver [5], [6]. How-

ever, these techniques often need to determine a large number

of Volterra series coefﬁcients, which is a difﬁcult task. To

address this, some simpliﬁed methods such as those based on

memory polynomials [7], Hammerstein model [8] and Wiener

model [9] were proposed [7]. Addressing I/Q imbalance has

also attracted much attention [10]–[13]. In [12], a dual-

input nonlinear model based on a real-valued Volterra series

was proposed to model the I/Q imbalance, and its inverse

model was employed at the transmitter to pre-compensate

the I/Q imbalance. In [13], a single-user point-to-point mm-

wave hybrid beamforming system with I/Q imbalance at the

transmitter and its pre-compensation were considered. The

pre-compensation technique [13] assumes the availability of

instantaneous channel state information at the transmitter,

which can be difﬁcult to achieve in practical scenarios. With

higher orders, polynomial-based techniques have potential to

handle severer nonlinear distortions, which, however, are more

prone to numerical instability in determining their coefﬁcients

[14]–[16]. We also note that, most of the polynomial-based

algorithms in the literature deal with a single type of hardware

imperfections, i.e., either PA nonlinearity or I/Q imbalance.

However, hardware imperfections may occur at the same time,

leading to combined effects.

Neural networks (NNs) have recently emerged as a promis-

ing technique to deal with the nonlinear effects in communica-

tion systems [17]–[19]. In [20], a real-valued time-delay neural

network (RVTDNN) was proposed to model PA behaviors.

Various variants of RVTDNN were proposed [21], [22] to

address the combined effects of hardware impairments. In [21],

high-order signal components are applied to the RVTDNN to

pre-compensate both the PA nonlinearity and I/Q imbalance.

In [22], a deep NN (DNN) based technique was proposed to

arXiv:2210.03911v1 [eess.SP] 8 Oct 2022

mitigate combined PA nonlinearity and I/Q imbalance at the

transmitter of a MIMO system. In [23], a residual NN was

proposed for digital predistortion, where shortcut connections

are added between the input and output layer to improve the

performance of PA nonlinearity mitigation. These predistortion

based methods require feedback from the receiver, which can

be inconvenient or difﬁcult to implement, especially in the case

of time-variant environments. Post-compensation techniques

at the receiver have also been investigated [24], [25]. A

recurrent NN (RNN) was proposed in [24] to compensate

PA nonlinearity in a ﬁber-optic link. In [25], a deep-learning

(DL) framework that integrates feedforward NN (FNN) and

RNN was proposed to combat both the nonlinear distortion

and linear interference. However, these works do not consider

the impact of I/Q imbalance. Moreover, a signiﬁcant problem

with the DNN based techniques is that a large number of pilot

symbols are required to train the DNNs properly, leading to

unacceptable overhead and hindering their application espe-

cially in time-varying environments.

In this work, we investigate the issue of signal detection in

an uplink multi-user mm-wave MIMO system, where trans-

mitters (at users) suffer from combined distortions of PA

nonlinearity and I/Q imbalance due to the use of low-cost

mobile devices. To combat the combined effects of hardware

imperfections and multi-user interference, the conventional

approach is to design a DNN based detector with received

signal as input and predicated symbols as output (shown in Fig.

1), which we call direct detection. However, it is difﬁcult to

train the DNN with limited pilot symbols. Due to the superior

performance of Bayesian signal detection, in this work, we

investigate how to achieve efﬁcient Bayesian detection in

the presence of combined hardware imperfections. Bayesian

detection relies on a signal model. However, characterizing

combined hardware imperfections in a MIMO system leads

to a complicated signal model (which may also be subject to

modelling errors), making Bayesian signal detection challeng-

ing. We propose a new strategy, where we ﬁrst use an NN to

‘model’ the MIMO system (i.e., the NN serves as a substitute

for the signal model), which captures combined effects of

hardware imperfections and multi-user interference. Then we

perform Bayesian inference based on the trained NN. We call

this indirect detection. This strategy enables us to design the

NN architecture based on the signal ﬂow of the MIMO system

and minimize the number of layers and parameters of the NN,

making it possible to achieve efﬁcient training with limited

pilot symbols.

To perform Bayesian inference with the trained NN, we

represent it with a factor graph and develop message passing

based Bayesian signal detection. The presence of densely

connected factors due to the NN weight matrices makes the

Bayesian inference difﬁcult. The approximate message passing

(AMP) algorithm is promising in handling densely connected

factor graphs [26]. However, AMP works well for i.i.d (sub-)

Gaussian matrices, but suffers severe performance degradation

or easily diverges for a general matrix [26]. The work in [27]

shows that AMP can still work well in the case of a general

matrix when a unitary transform of the original model is used.

The variant of AMP is called unitary AMP (UAMP), which

was also known as UTAMP [27]–[29]. As NN weight matrices

are normally not i.i.d. (sub-) Gaussian, we adopt UAMP and

show that it plays a crucial role in achieving efﬁcient message

passing based Bayesian inference.

The contributions of this work are summarized as follows:

•A new strategy to achieve Bayesian signal detection for

a communication system with complicated input-output

relationship: We use an NN to model the behaviour

of the MIMO system, followed by Bayesian inference

based on the NN. This indirect detection strategy is

more efﬁcient than direct detection. Although this work

focuses on MIMO systems with I/Q imbalance and PA

nonlinearity, the developed method can be extended to

deal with a general system with complicated input-output

relationship.

•Signal-ﬂow-based NN architecture design: The architec-

ture of the NN is carefully designed based on the signal

ﬂow of the MIMO system, so that the number of layers

and parameters of the NN is minimized, which is crucial

to achieving efﬁcient training.

•Message passing based Bayesian inference on NNs: To

realize Bayesian signal detection based on an NN, we rep-

resent the NN as a factor graph and an efﬁcient UAMP-

based message passing inference algorithm (called MP-

NN) is developed.

•Iterative detection and decoding in coded systems: An-

other advantage of the new strategy is that the proposed

MP-NN Bayesian detector is able to work with a soft-in-

soft-out (SISO) decoder, leading to a much more powerful

turbo receiver. In contrast, it is unknown how to develop

a turbo receiver with existing DNN or polynomial based

direct detection techniques.

•Comparisons with existing techniques: We carry out

various comparisons with state-of-the-art methods and

demonstrate that the proposed approach delivers remark-

ably better performance.

The remainder of the paper is organized as follows. In

Section II, the signal model of MIMO communications with

combined hardware imperfections is given and existing tech-

niques are introduced. In Section III, with the new strategy,

we investigate the NN architecture design and training, and

develop a UAMP-based Bayesian detector by performing

message passing on the trained NN. The extension to turbo

receiver in a coded system is investigated in Section IV.

Simulation results are provided in Section V, followed by

conclusions in Section VI.

The notations used in this paper are as follows. Boldface

lower-case and upper-case letters denote vectors and matrices,

respectively. The superscript (·)∗represents the conjugate

operation. The notations (·)Tand (·)Hrepresent the transpose

and conjugate transpose operations, respectively. We use |x|

and ||x|| to denote the amplitude of xand the norm of x, and

use <{·} and ={·} to represent the real and imaginary parts

of a complex number, respectively. The notationhf(x)ip(x)

denotes the expectation of f(x)with respect to distribution

p(x).

II. SIGNAL MODEL AND EXISTING METHODS

A. Signal Model

We consider an uplink transmission of a multi-user mm-

wave MIMO system with Kusers. Considering the cost of

mobile devices, we assume that each user has a single antenna,

where low-cost modulators and PAs are used, resulting in I/Q

imbalance and PA nonlinear distortions during transmission

[30]. The base station (BS) is equipped with Nantennas.

The mth symbol of user kis denoted by xk(m)∈ A, where

Adenotes the symbol alphabet. The symbols of all users at

time instant mform a vector x(m). At the transmitter side, the

signal is up-converted to radio frequency through modulation,

and the mismatch between I and Q branches is characterized

as [22]

k(m) = ξkxk(m) + ζkx∗

k(m),(1)

where

ξk= cos(θk

2) + jλksin(θk

2),(2)

ζk=λkcos(θk

2) + jsin(θk

2)(3)

with real valued amplitude imbalance parameter λkand phase

imbalance parameter θk. The signal is then input to a PA.

The nonlinear distortion of PA can be characterized by the

amplitude to amplitude conversion A(|xa

k(m)|)and amplitude

to phase conversion φ(|xa

k(m)|)[31]:

A(|xa

k(m)|) = αa|xa

k(m)|

(1 + (αa|xa

k(m)|

xsat )2σa)1

2σa

,(4)

φ(|xa

k(m)|) = αφ|xa

k(m)|q1

1+(|xa

k(m)|

βφ)q2

,(5)

where αa,αφ,βφ,σa,xsat,q1and q2are model parameters.

The distorted signal can then be expressed as

sk(m) = f(xa

k(m)) = A(|xa

k(m)|)ej(angle(xa

k(m))+φ(|xa

k(m)|)),

(6)

where angle(xa

k)denotes the phase of the complex signal xa

The received signal at time instant mis represented as

y(m) = Hs(m) + ω(m),(7)

where H∈CN×Kis the MIMO channel matrix, y(m) =

[y1(m), y2(m), . . . , yN(m)]T,s(m) = f(xa(m)) with

xa(m) = [xa

1(m), xa

2(m), . . . , xa

K(m)]Tbeing the length-K

vector, and ω(m)denotes a white Gaussian noise vector. Note

that the vectors and matrix in (7) are all complex-valued,

which can be rewritten as the following real model:

<{y(m)}

={y(m)}

| {z }

y0(m)

=<{H} −={H}

={H} <{H}

| {z }

<{s(m)}

={s(m)}

| {z }

s0(m)

+<{ω(m)}

={ω(m)}

| {z }

ω0(m)

(8)

Due to the combined effects of I/Q imbalance and PA non-

linearity, the input-output relationship of the MIMO system is

complex, and is denoted as

y0(m) = S(x(m)) + ω0(m),(9)

where S(·)is the system transfer function.

We assume that the channel matrix and the parameters of

I/Q imbalance and PA nonlinearity models are unknown. Each

user transmits a pilot signal followed by data. The aim of the

receiver at the BS is to detect the transmitted data symbols of

all users. To achieve this, there are two approaches.

•Direct detection: A symbol detector is trained directly

using pilot symbols, where the input is the received signal

and the output is the predicated symbols. As the system

transfer function S(·)is complicated, direct detection

seems sensible. To deal with the nonlinearity, polyno-

mial and DNN based techniques have been used in the

literature. However, low order polynomials have limited

capability to combat the nonlinearity. Although, high

order polynomials have better capability, it is difﬁcult to

determine the polynomial coefﬁcients due to numerical

instability. The DNN techniques are more effective to

deal with the nonlinearity, but it is difﬁcult to train a

DNN with a limited number of pilot symbols.

•Indirect detection: With the pilot symbols, the system

function S(·)is ﬁrst identiﬁed, then a symbol detector

is developed based on the system function. This strategy

allows the design of powerful Bayesian detectors, but

the implementation of indirect detection is challenging.

First, to identify S(·)with pilot symbols, we need to

estimate the parameters of the I/Q imbalance and PA

nonlinearity models and the MIMO channel at the same

time, which is a difﬁcult task due to the nonlinearity.

Second, even if we assume that S(·)is known, it is still

difﬁcult to develop a detector, especially a Bayesian one,

due to the nonlinearity of S(·). The aim of this work is to

develop a Bayesian detector by using NN and factor graph

techniques, which is more powerful than direct detection

proposed in the literature.

B. Existing Detection Methods

1) Polynomial Based Direct Detection: A real-valued mem-

ory polynomial (RMP) model was developed in [12], where

the I/Q branches after modulation are applied to the RMP

model in order to compensate the I/Q imbalance. The work

was extended to MIMO systems to address the joint effect of

I/Q imbalance and PA nonlinearity in [30].

RMP can be used to directly compensate the hardware im-

perfections and deal with multi-user interference. The detector

(for the kth user) can be expressed as

˜xk(m) = argminλa∈A|ˆxk(m)−λa|(10)

with

ˆxk(m) = ˆxQ

k(m) + jˆxI

k(m)(11)

ˆxQ

k(m)=

n=1

p=1

l=0

p,l,k<{yn(m−l)}p+bQ

p,l,k={yn(m−l)}p

(12)

ˆxI

k(m)=

n=1

p=1

l=0

p,l,k<{yn(m−l)}p+bI

p,l,k={yn(m−l)}p,

(13)

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1SignalDetectioninMIMOSystemswithHardwareImperfections:MessagePassingonNeuralNetworksDaweiGao,QinghuaGuo,GuishengLiao,YoninaC.Eldar,Fellow,IEEE,YonghuiLi,Fellow,IEEE,YanguangYu,andBrankaVucetic,Fellow,IEEEAbstractInthispaper,weinvestigatesignaldetectioninmultiple-input-multiple-output(MIMO)communic...

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