1 Message Passing-Based Joint User Activity Detection and Channel Estimation for

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Message Passing-Based Joint User Activity

Detection and Channel Estimation for

Temporally-Correlated Massive Access

Weifeng Zhu, Member, IEEE, Meixia Tao, Fellow, IEEE, Xiaojun Yuan, Senior

Member, IEEE, and Yunfeng Guan

Abstract

This paper studies the user activity detection and channel estimation problem in a temporally-

correlated massive access system where a very large number of users communicate with a base station

sporadically and each user once activated can transmit with a large probability over multiple consec-

utive frames. We formulate the problem as a dynamic compressed sensing (DCS) problem to exploit

both the sparsity and the temporal correlation of user activity. By leveraging the hybrid generalized

approximate message passing (HyGAMP) framework, we design a computationally efﬁcient algorithm,

HyGAMP-DCS, to solve this problem. In contrast to only exploit the historical estimations, the proposed

algorithm performs bidirectional message passing between the neighboring frames for activity likelihood

update to fully exploit the temporally-correlated user activities. Furthermore, we develop an expectation

maximization HyGAMP-DCS (EM-HyGAMP-DCS) algorithm to adaptively learn the hyperparameters

during the estimation procedure when the system statistics are unknown. In particular, we propose to

utilize the analysis tool of state evolution to ﬁnd the appropriate hyperparameter initialization of EM-

HyGAMP-DCS. Simulation results demonstrate that our proposed algorithms can signiﬁcantly improve

the user activity detection accuracy and reduce the channel estimation error.

Index Terms

Temporally-correlated massive access, user activity detection, channel estimation, hybrid generalized

approximate message passing (HyGAMP), expectation maximization (EM).

This paper was presented in part at the IEEE International Conference of Communications (ICC) 2021 [1].

W. Zhu, M. Tao, and Y. Guan are with the Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai

200240, China (e-mail: {wf.zhu, mxtao, yfguan69}@sjtu.edu.cn).

X. Yuan is with the Center for Intelligent Networking and Communication (CINC), University of Electronic Science and

Technology of China, Chengdu 610000, China (e-mail: xjyuan@uestc.edu.cn).

arXiv:2210.12954v2 [cs.IT] 26 Jan 2023

I. INTRODUCTION

Massive machine-type communications (mMTC) is one of the main use cases of the ﬁfth-

generation (5G) cellular networks, for Internet of Things (IoT) applications [2]. It aims to provide

wireless connectivity to a massive number of IoT devices, whose trafﬁc is typically sporadic

[3], [4]. The main technical challenge of mMTC is to design scalable, efﬁcient, and low-latency

multiple-access schemes. Due to the massive number of device, the conventional grant-based

random access schemes suffer from excessive delay and signaling overhead, then the grant-free

random access scheme is considered as the promising solution to decrease the access delay and

reduce the control overhead for coordination [4].

In the grant-free random access protocol, a unique pilot sequence for identiﬁcation and channel

estimation is pre-allocated to each device. Note that, due to the massive number of the IoT devices

but the limited coherence time interval, the pilot sequences are usually non-orthogonal. When one

device is activated, it directly transmits its dedicated pilot followed by data without waiting for

the grant from the base station (BS). Meanwhile, the BS can perform joint user activity detection

and channel estimation based on the received signals. Given the sporadic trafﬁc generated from

IoT devices, the joint user activity detection and channel estimation problem by nature can

be cast into a large-scale sparse signal recovery problem, which can be reliably solved by the

compressed sensing (CS) techniques [4].

In many practical IoT systems, once a device is activated, it often transmits continuously

over multiple frames. This suggests that the device activities are often temporally-correlated. To

exploit the temporally-correlated user activity, we are motivated to detect the active users and

estimate their channels in multiple consecutive frames jointly. The joint user activity detection

and channel estimation problem can be formulated from the dynamic compressed sensing (DCS)

perspective [5], [6], which takes both the sporadic user activity and the temporal correlation of

user activities into account. Based on the probabilistic and graphical model of the signals, the joint

user activity detection and channel estimation can be realized by the standard message passing

(MP) algorithm [7]. To simplify the implementation of MP in large-scale systems, we propose to

leverage the hybrid generalized approximate message passing (HyGAMP) [8] framework which

employs a hybrid of AMP and MP for the graphical model. Considering that the perfect system

statistics are usually unknown, the expectation maximization (EM) technique is incorporated with

the algorithm to adaptively learn the hyperparameters in the estimation procedure. Compared

with the existing DCS-based algorithms, we show that the proposed algorithms can signiﬁcantly

improve the user activity detection accuracy and reduce the channel estimation error.

A. Related Works

To accomplish joint user activity detection and channel estimation for massive access, a

variety of advanced sparse signal recovery algorithms have been proposed for different wireless

communication systems. For the single-cell scenario, the works [9], [10] utilize the standard

CS algorithms of orthogonal matching pursuit (OMP) and basis pursuit denoising (BPDN).

However, these two kinds of algorithms usually suffer high computational complexity due to

the extremely large amount of devices. Then the works [11], [12] propose the computationally

efﬁcient AMP algorithms with Bayesian denoiser for user activity detection. In the work [13],

the message-scheduling generalized AMP algorithm is developed to further reduce the computa-

tional complexity without degrading the detection and estimation performance. The AMP-based

algorithms are also extended to perform data detection along with the activity detection in [14],

[15]. Besides the AMP-based algorithms, covariance matching pursuit [16], [17], dimension

reduction-based optimization [18] and deep learning methods [19]–[21] are also investigated

for performance enhancement. In particular, the covariance matching pursuit algorithm [16],

[17] and dimension reduced-based optimization algorithm [18] are especially designed for the

massive multiple-input multiple-out (MIMO) systems, which can signiﬁcantly outperform the

conventional CS-based methods. The deep learning methods [19]–[21] employ the artiﬁcial

neural networks and learn the network parameters based on the pre-collected data, which are

able to outperform the traditional hand-designed algorithms and to enjoy low computational

complexity. Moreover, a sparsity-constrained method is proposed for the non-ideal scenario

where different users have unknown frequency offsets in [22]. In the multi-cell systems, the

cooperative user activity detection and channel estimation is studied in [23], [24] based on the

AMP-based algorithms. Recently, unsourced random access as a new random access protocol is

also studied in [16], [25], where all users share a common codebook and the BS only needs to

detect the transmit codewords.

The aforementioned works all detect the user activity in each frame individually by assuming

the user activity is independent for each frame. Due to the low-rate transmission, the informa-

tion transmission of one active user may occupy multiple consecutive frames. Such temporal

correlation of user activity can be exploited for performance enhancement. Assuming the user

sparsity level and the channel state information (CSI) are available, the work [26] proposes a

DCS-based algorithm where the set of the detected active user in the last slot is used as the

initial set to be detected in the current slot. By adaptively exploiting the prior support based on

the corresponding support quality information, the work [27] proposes a prior-information-aided

adaptive subspace pursuit (PIA-ASP) algorithm, which can always outperform the DCS-based

algorithm in [26]. These algorithms include the matrix inversion calculation in each iteration, and

thus suffer high computational complexity in the large-scale problems. Moreover, both the works

[26], [27] assume that the CSI is perfectly known in advance, which is unrealistic in massive

access. By leveraging the system statistics, the authors in [28] propose a sequential AMP (S-

AMP) algorithm to sequentially perform user activity detection and channel estimation in each

frame. Then the work [29] proposes to extract the side information (SI) from the estimation

results in the previous frame to enhance the user activity detection performance in the current

frame based on the AMP framework. Different from [29] which only utilizes the historical

estimation results as SI, our previous work [30] proposes to extract the double-sided information

by considering the estimation from the next frame as well for further exploiting the temporal

correlation.

Compared with these prior works [26]–[30] that exploit the temporal correlation for use activity

detection and channel estimation, this paper differs mainly in the following two aspects. First,

while all the exiting works perform activity detection in a frame-by-frame manner, (i.e., the user

activity is still detected sequentially frame by frame, though the temporal correlation among

adjacent frames has been exploited), this paper performs multi-frame activity detection in a

block-by-block manner. Second, this paper proposes to adaptively learn the parameters of the

system statistics by using the EM technique, while the previous algorithms either do not consider

the system statistics [26], [27] or assume the system statistics are perfectly known [28]–[30].

B. Main Contributions

In this work, we consider the problem of joint user activity detection and channel estimation

in multiple consecutive frames for the temporally-correlated massive access systems. The main

contributions and distinctions of this work are summarized as follows:

1) We propose to perform the user activity detection and channel estimation jointly in multiple

consecutive frames and formulate it as a DCS problem. Based on the probabilistic model,

not only the sparse activity pattern is considered in the problem, but also the statistical

relationships of the activities in these consecutive frames are exploited.

2) In contrast to only making use of the historical estimations in [28], [29], this paper proposes

a HyGAMP-DCS algorithm to fully exploit the temporally-correlated user activities in the

multiple consecutive frames. The HyGAMP-DCS algorithm combines the computationally

efﬁcient GAMP algorithm for channel estimation and the standard MP algorithm for the

activity likelihood update. In particular, the activity likelihood values in each frame are

updated by aggregating both the forward messages from the previous frame and backward

messages from the next frame at each iteration. Numerical results show that HyGAMP-

DCS can achieve superior performance to the existing DCS-based algorithms [26]–[29]

thanks to the bidirectional message propagation.

3) To make the proposed algorithm applicable for the practical system with imperfect system

statistics, this paper incorporates the EM algorithm in HyGAMP-DCS to adaptively learn

the hyperparameters during the estimation procedure. Compared with the traditional EM-

AMP algorithm [31] for the CS problem, the proposed EM-HyGAMP-DCS algorithm

additionally learns the statistical dependencies between the activities in the neighboring

frames. Simulation results demonstrate that EM-HyGAMP-DCS realizes very similar per-

formance to HyGAMP-DCS which requires the perfect system statistics.

4) We provide the performance and complexity analysis of the proposed algorithms. We ﬁrst

introduce the state evolution (SE) to predict the asymptotic performance of HyGAMP-DCS

and EM-HyGAMP-DCS. In particular, we also point out that the SE is quite essential to ﬁnd

the appropriate hyperparameter initialization for EM-HyGAMP-DCS, which is validated by

the numerical results. Then the computational complexity comparison with the state-of-the-

art methods is given to illustrate the computational efﬁciency of our proposed algorithms.

C. Organizations and Notations

The remaining part of this paper is organized as follows. Section II introduces the system model

of temporally-correlated massive access. In Section III, we present the probabilistic model of

the considered system and then introduce the Bayesian inference methods. In Section IV, the

HyGAMP-DCS algorithm is proposed, then the EM algorithm is employed for hyperparameter

update in Section V. Next, we analyze the performance and the computational complexity of

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1MessagePassing-BasedJointUserActivityDetectionandChannelEstimationforTemporally-CorrelatedMassiveAccessWeifengZhu,Member,IEEE,MeixiaTao,Fellow,IEEE,XiaojunYuan,SeniorMember,IEEE,andYunfengGuanAbstractThispaperstudiestheuseractivitydetectionandchannelestimationprobleminatemporally-correlatedmassivea...

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