EFFICIENT QUANTIZED CONSTANT ENVELOPE PRECODING FOR MULTIUSER DOWNLINK MASSIVE MIMO SYSTEMS Zheyu Wux Ya-Feng Liux Bo Jiangy and Yu-Hong Daix

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EFFICIENT QUANTIZED CONSTANT ENVELOPE PRECODING FOR MULTIUSER

DOWNLINK MASSIVE MIMO SYSTEMS

Zheyu Wu§,?, Ya-Feng Liu§, Bo Jiang†, and Yu-Hong Dai§

§LSEC, ICMSEC, AMSS, Chinese Academy of Sciences, Beijing, China

?School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China

†School of Mathematical Sciences, Nanjing Normal University, Nanjing, China

Email: {wuzy, yaﬂiu, dyh}@lsec.cc.ac.cn, jiangbo@njnu.edu.cn

ABSTRACT

Quantized constant envelope (QCE) precoding, a new transmission

scheme that only discrete QCE transmit signals are allowed at each

antenna, has gained growing research interests due to its ability of

reducing the hardware cost and the energy consumption of massive

multiple-input multiple-output (MIMO) systems. However, the dis-

crete nature of QCE transmit signals greatly complicates the precod-

ing design. In this paper, we consider the QCE precoding problem

for a massive MIMO system with phase shift keying (PSK) modu-

lation and develop an efﬁcient approach for solving the constructive

interference (CI) based problem formulation. Our approach is based

on a custom-designed (continuous) penalty model that is equivalent

to the original discrete problem. Speciﬁcally, the penalty model re-

laxes the discrete QCE constraint and penalizes it in the objective

with a negative `2-norm term, which leads to a non-smooth non-

convex optimization problem. To tackle it, we resort to our recently

proposed alternating optimization (AO) algorithm. We show that the

AO algorithm admits closed-form updates at each iteration when ap-

plied to our problem and thus can be efﬁciently implemented. Sim-

ulation results demonstrate the superiority of the proposed approach

over the existing algorithms.

Index Terms—Constructive interference, massive MIMO,

penalty model, QCE precoding.

1. INTRODUCTION

High hardware cost and power consumption have been widely rec-

ognized as the major issues for the practical deployment of massive

multiple-input multiple-output (MIMO) systems [1]. To deal with

this, various techniques have been proposed, including employing

low-resolution digital-to-analog converters (DACs) at the antenna ar-

rays [2] and reducing the peak-to-average power ratio (PAPR) of the

transmit signals (which enables the use of power-efﬁcient power am-

pliﬁers (PAs)) [3]. The combination of the above two techniques mo-

tivates a new transmission scheme in the framework of symbol-level

precoding [4, 5] called quantized constant envelope (QCE) trans-

mission, where each antenna is restricted to transmit coarsely QCE

signals.

The QCE precoding problem was ﬁrst formulated based on the

classical minimum mean square error (MMSE) criterion [6]–[9]. Re-

cently, it has been realized that incorporating the idea of constructive

interference (CI) into precoding design can greatly improve the bit

error rate (BER) performance of the system [5, 10]. Motivated by

this, the authors in [11] adopted the CI metric to formulate the QCE

precoding problem. In addition to the MMSE and CI metrics, some

works also considered the symbol error probability (SEP) criterion

directly (for quadrature amplitude modulation (QAM)) [12, 13]. In

this paper, we focus on the CI-based model due to its good BER per-

formance and its computational tractability. It is worthwhile men-

tioning a widely studied special case of QCE precoding, one-bit pre-

coding, where the ﬁnite phases allowed to be transmitted by each

antenna is {(2i−1)π

4, i = 1,2,3,4}. Although numerous algorithms

have been proposed for one-bit precoding [14]–[18], very few algo-

rithms have been designed for solving the general QCE precoding

problem. To the best of our knowledge, the only CI-based algorithm

designed in the QCE context is the maximum safety margin (MSM)

algorithm [11], which directly relaxes the discrete QCE constraint

and suffers from poor BER performance. We also remark here that

the direct extension of the existing algorithms for one-bit precoding

to solve the QCE precoding problem either is difﬁcult (due to the

more general and difﬁcult QCE constraint) or leads to a poor perfor-

mance. This remark will become clear in the algorithmic design and

simulation result sections.

In this paper, we consider the general QCE precoding design

problem for a downlink massive MIMO system with phase shift key-

ing (PSK) signaling and propose a new penalty approach to solve

the CI-based problem formulation. More speciﬁcally, we ﬁrst pro-

pose a penalty model exploiting the special structure of the discrete

QCE constraint and establish its global equivalence with the original

problem. Then, we apply the alternating optimization (AO) algo-

rithm proposed in our recent work [17] to solve the penalty model.

By carefully investigating the update rule, we ﬁnd that each iteration

of the AO algorithm admits closed-form solutions, making it suit-

able to solve large-scale problems arising from the massive MIMO

scenario. Simulation results show that, compared to the existing al-

gorithms, our proposed approach achieves better BER performance

with lower computational cost. In addition, the simulation also in-

dicates that the BER performance of the system can be signiﬁcantly

enhanced by slightly increasing the resolution of DACs from 1 bit to

2-4 bits.

2. PROBLEM FORMULATION

2.1. System Model

Consider a massive MIMO downlink transmission scenario, where

an N-antenna base station (BS) serves Ksingle-antenna users si-

multaneously. Let t∈CNdenote the transmitted signal vector from

the BS. Then the received signal at the users is given by

y=Ht +n,

where H= [h1,h2,...,hK]T∈CK×Nis the channel matrix be-

tween the BS and the users; n∼ CN(0, σ2I)models the additive

arXiv:2210.14534v3 [cs.IT] 20 Feb 2023

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EFFICIENTQUANTIZEDCONSTANTENVELOPEPRECODINGFORMULTIUSERDOWNLINKMASSIVEMIMOSYSTEMSZheyuWux;?,Ya-FengLiux,BoJiangy,andYu-HongDaixxLSEC,ICMSEC,AMSS,ChineseAcademyofSciences,Beijing,China?SchoolofMathematicalSciences,UniversityofChineseAcademyofSciences,Beijing,ChinaySchoolofMathematicalSciences,Nanjing...

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EFFICIENT QUANTIZED CONSTANT ENVELOPE PRECODING FOR MULTIUSER DOWNLINK MASSIVE MIMO SYSTEMS Zheyu Wux Ya-Feng Liux Bo Jiangy and Yu-Hong Daix

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