1

Reconfigurable Intelligent Surface for Massive ConnectivityShuhao Xia, Student Member, IEEE, Yuanming Shi, Senior Member, IEEE, Yong Zhou, Member, IEEE, and

Xiaojun Yuan, Senior Member, IEEE,

Abstract—With the rapid development of Internet of Things(IoT), massive machine-type communication has become apromising application scenario, where a large number of devicestransmit sporadically to a base station (BS). Reconfigurableintelligent surface (RIS) has been recently proposed as aninnovative new technology to achieve energy efficiency and cov-erage enhancement by establishing favorable signal propagationenvironments, thereby improving data transmission in massiveconnectivity. Nevertheless, the BS needs to detect active devicesand estimate channels to support data transmission in RIS-assisted massive access systems, which yields unique challenges.This paper shall consider an RIS-assisted uplink IoT networkand aims to solve the RIS-related activity detection and channelestimation problem, where the BS detects the active devices andestimates the separated channels of the RIS-to-device link and theRIS-to-BS link. Due to limited scattering between the RIS andthe BS, we model the RIS-to-BS channel as a sparse channel.As a result, by simultaneously exploiting both the sparsity ofsporadic transmission in massive connectivity and the RIS-to-BSchannels, we formulate the RIS-related activity detection andchannel estimation problem as a sparse matrix factorizationproblem. Furthermore, we develop an approximate messagepassing (AMP) based algorithm to solve the problem based onBayesian inference framework and reduce the computationalcomplexity by approximating the algorithm with the centrallimit theorem and Taylor series arguments. Finally, extensivenumerical experiments are conducted to verify the effectivenessand improvements of the proposed algorithm.

Index Terms—Device activity detection, channel estimation,reconfigurable intelligent surface, approximate message passing,Internet of Things (IoT), massive machine-type communications(mMTC).

I. INTRODUCTION

With the popularity of the Internet of Things (IoT), massiveconnectivity, also known as massive machine-type commu-nication (mMTC), has been regarded as one of the threetypical use cases of the fifth-generation (5G) wireless net-works [1], along with enhanced mobile broadband (eMBB)and ultra-reliable low-latency communication (uRLLC). Amain challenge of mMTC is to support sporadic short-packetcommunications between the base station (BS) and a massivenumber of IoT devices [2]. Grant-free random access that hasthe potential to significantly reduce the signaling overheadand access latency has recently attracted considerable attention[3], where multiple active IoT devices directly transmit theirunique pilot sequences together with the data without the needof obtaining the grant from the BS. With grant-free random

S. Xia, Y. Shi, and Y. Zhou are with the School of Information Scienceand Technology, ShanghaiTech University, Shanghai, 201210, China (E-mail:{xiashh, shiym, zhouyong}@shanghaitech.edu.cn).

X. Yuan is with the Center for Intelligent Networking and Communications,the National Laboratory of Science and Technology on Communications, theUniversity of Electronic Science and Technology of China, Chengdu, 611731,China (E-mail: xjyuan@uestc.edu.cn).

access, the BS generally needs to perform activity detection,channel estimation, and data decoding. Various advancedactivity detection and channel estimation algorithms have beenproposed in the literature. By exploiting the sporadic nature ofthe device activity patterns, several compressed sensing (CS)-based approaches were proposed to solve the joint activity de-tection and channel estimation problem, which is formulated asa sparse signal recovery problem [4]–[6]. Taking into accountthe short-packet transmission, a covariance-based approachwas proposed to jointly detect active devices and decode thedata for massive connectivity [7], [8]. In addition, the authorsin [9] developed a joint design of activity detection, channelestimation, and data decoding, and proposed an approximatedmessage passing (AMP) based algorithm for a trilinear model.

In massive connectivity, the IoT devices are usually locatedin a service dead zone, where the line-of-sight communicationmay not be available. Consequently, the signals received fromthe IoT devices are usually very weak, which makes theaccurate device detection a challenging task for the BS. Onthe other hand, poor channel conditions in the dead zones alsoreduce the link reliability for data transmissions between theactive devices and the BS [10]. To overcome these challenges,a variety of massive access techniques have been recentlyproposed [11]–[18]. In particular, more BSs can be deployed toshorten the communication distance to enhance the coverageand capacity [11], [12], which inevitably leads to higher energyconsumption and deployment costs. Furthermore, the use ofmillimeter wave (mmWave) and terahertz (THz) frequencieswas proposed to enhance the capacity of the IoT networks[16]–[18]. To mitigate the severe propagation loss over dis-tance, mmWave/THz communications are generally combinedwith massive multiple-input-multiple-output (MIMO) [13]–[15], which requires increased hardware and energy cost aswell as signal processing complexity. To address the afore-mentioned issues and limitations, it is exigent to develop noveltechnologies to support massive connectivity with low cost,low complexity, and high energy efficiency.

Reconfigurable intelligent surface (RIS) has recentlyemerged as a promising technology for enhancing the spectralefficiency and energy efficiency in various wireless commu-nication systems [19]–[22]. To be specific, RIS is a man-made surface equipped with a large number of passive andprogrammable reflecting elements integrated with a smartcontroller [23]. RIS plays a similar role as a large-scaleantenna array through performing spatial beamforming, butwith lower hardware and energy cost [24], [25]. By optimizingthe phase shifts based on the instantaneous channel stateinformation (CSI), the signal propagation between the BS andthe IoT devices can be smartly reconfigured to improve thequality of the data transmission. Moreover, the deployment of

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RIS creates more non-line-of-sight links between the BS andthe devices in dead zones via bypassing the obstacle betweenthem [26], [27]. For cell-edge IoT devices, the deployment ofRIS not only helps improve the desired signal power at cell-edge users, but also facilitates the suppression of co-channelinterference from neighboring cells [28]. Hence, RIS has beenregarded as a promising technique to achieve coverage andcapacity enhancement for massive connectivity [10].

In RIS-assisted IoT systems, the aforementioned perfor-mance gains depend on the availability of CSI at the BS.However, acquiring accurate CSI in RIS-assisted wirelessnetworks is much more challenging than that in conventionalwireless networks [29]–[31]. Specifically, the RIS is typicallynot equipped with any radio frequency (RF) chains, and thuslacks the ability to transmit and receive signals. Consequently,the RIS is not able to perform activity detection and channelestimation between the RIS and active devices. Furthermore,the design of passive beamforming at the RIS requires theseparate CSI of the RIS-to-device link and the RIS-to-BSlink [21], [24], [25]. As a result, the BS is responsible forthe tasks of detecting device activity and decoupling thecascaded channels of RIS-to-device and RIS-to-BS links. Inorder to separately estimate the channels for RIS-assistedcommunication systems, various strategies have been recentlyproposed [32]–[36]. From the algorithmic perspective, theauthors in [32] developed a channel estimation algorithmbased on the parallel factor decomposition. The authors in[33] introduced a general two-stage estimation algorithm byformulating the channel estimation problem as bilinear sparsematrix factorization and matrix completion. By exploitingthe channel properties in RIS-assisted communication sys-tems, a series of methods have been developed in [34]–[36].By exploiting the sparsity of the channels, the authors in[34], [35] formulated the channel estimation problem as asparse signal recovery problem. By utilizing the property thatthe RIS-to-BS channel is quasi-static and the RIS-to-devicechannel is fast-varying, the authors in [36] proposed a two-timescale channel estimation framework. Nevertheless, most ofthe existing works on the channel estimation for RIS-assistedcommunication systems [32], [35]–[37] required the allocationof orthogonal pilot sequences to all the devices. However, inmassive connectivity, the length of pilot sequences is usuallymuch smaller than the number of IoT devices. As a result, itis generally impossible to allocate orthogonal pilot sequencesto all IoT devices, which yields unique challenges. Hence,these works cannot be directly applied to RIS-assisted massiveconnectivity systems.

In this paper, we consider the uplink transmission in an IoTnetwork, where a multi-antenna BS serves a large number ofsingle-antenna IoT devices with the assistance of an RIS. Weadopt the grant-free random access scheme to support massiveconnectivity, where the devices are sporadically active. Tofully unleash the potential of the RIS, the BS is required to de-tect active devices and separately estimate the channels of theRIS-to-device link and the RIS-to-BS link with non-orthogonalpilots, which is referred as a RIS-related activity detectionand channel estimation problem. Our main contributions aresummarized as follows.

• We propose a realistic channel model for the RIS-assisted IoT network with massive connectivity. Takinginto account the physical propagation structure of wire-less channels, we model the channel from the RIS tothe BS follows the geometric distribution. Due to limitedscattering between the RIS and the BS, the number ofspatial paths between them is usually small. Therefore,the RIS-to-BS channel can be represented as a sparsechannel matrix under the virtual angular domain. Onthe other hand, the IoT devices are sporadically activeat any time instant, which results in the sparsity of thedevice transmission pattern. By simultaneously exploitingthe sparsity of both sporadic transmission and the RIS-to-BS channel, we formulate the RIS-related activitydetection and channel estimation problem as a sparsematrix factorization problem.

• As the channel matrices in RIS-assisted IoT networks arehigh dimensional due to the massive number of IoT de-vices and passive elements at the RIS, the computationalcomplexity to infer the active devices and two separatechannels may be prohibitive. To tackle such a high-dimensional inference problem, we develop a unifiedframework based on the Bayesian inference frameworkto jointly detect the active devices and estimate the twoseparate channels. We calculate the posterior mean esti-mators on a factor graph via the canonical sum-productmessage passing algorithm. To further reduce computa-tional complexities, we approximate the messages basedon the AMP framework, which includes central-limit-theorem (CLT) and Taylor-series arguments. Moreover,the sparsity of both device transmission and the RIS-to-BS channels is exploited to enhance the estimationaccuracy.

• We conduct extensive numerical experiments to verify theeffectiveness of our proposed algorithm. Specifically, ouralgorithm outperforms the three-stage algorithm proposedin [38] in terms of both activity detection and channelestimation accuracy. Furthermore, for channel estimation,our proposed algorithm can achieve the similar perfor-mance of Genie-aided MMSE estimator, which assumesall the active devices are known in advance. Finally,our experiments also reveal that massive MIMO cansignificantly improve the estimation accuracy in terms ofboth activity detection and channel estimation for RIS-assisted uplink transmissions.

Organization and Notations

The remainder of this paper is organized as follows. SectionII introduces the system model and the channel models.Section III formulates the RIS-related activity detection andchannel estimation problem and describes the proposed AMP-based algorithm. Section IV presents extensive numericalresults of the proposed algorithm followed by the conclusionsin Section V.

Throughout this paper, the complex number sets is denotedby C. Scalars, vectors and matrices denote regular letters,bold small letters, and bold capital letters, respectively. The

3

imaginary unit is denoted by j ,√−1. We use superscripts

(·)∗, (·)T and (·)H to denote conjugate, transpose and con-jugate transpose. The (i, j)−th entry of X is denoted byxij . I and diag{x} denote the identity matrix and a diagonalmatrix with diagonal entries specified by x. E[·] denotes theexpectation operator and CN (x;µ,Σ) denotes the complexGaussian distributions with mean µ and variance σ2.

II. SYSTEM MODEL

A. RIS-Assisted IoT Networks

In this paper, we consider the uplink transmission in anIoT network depicted in Fig. 1, where a BS equipped withM antennas serves K single-antenna IoT devices. An RISconsisting of N passive reflecting elements is deployed toenhance the communication performance of IoT networks[24]. Each reflecting element of the RIS is able to reflectthe incident signals with desired phase shifts, which can bedynamically adjusted by the RIS controller [28]. Due to thechannel qualities of the direct links between the BS and thedevices are much weak than that of the BS-RIS-device links.Hence, we follow [20], [39]–[41], and assume that the directlinks between the BS and devices are not available.

Active devices

Inactive devices

BS

RIS

Fig. 1: System model.

In this paper, we adopt the grant-free random access schemeto support massive connectivity, where the transmission ofIoT devices is sporadic [42]. Each device is assumed to beactive in each coherence block of length T with probabilityλα. Hence, only a subset of the devices are active within eachtransmission. In particular, the activity of the k-th device isindicated as follows

αk =

{1, if device k is active,0, otherwise, ∀k. (1)

Hence, we have Pr(αk = 1) = λα. Furthermore, we definethe support set of active devices as

A = {k|αk = 1, 1 ≤ k ≤ K}. (2)

For the purpose of detecting active devices and estimatingthe corresponding channels for the RIS-assisted IoT network,the k-th device is assigned to a unique signature sequenceqk ,

[qk,1, . . . , qk,L

]T ∈ CL×1, where the sequence lengthL is typically smaller than coherence length T . We assumea block-fading channel model, where the channels are quasi-static in each coherence block. Namely, all the channels remaininvariant in each coherence block, but vary independently over

different blocks. We define hk ∈ CN×1 and G ∈ CM×N asthe k-th RIS-to-device channel and the RIS-to-BS channel,respectively.

In the l-th time slot, the received signal yl ∈ CM×1 at theBS can be written as

yl =

K∑k=1

GΦhkαkqk,l + nl, (3)

where nl ∼ CN (0, τnI) is the additive white Gaussian noise(AWGN) vector in the l-th time slot, and τn denotes the noisepower. Besides, Φ , diag{φ1, . . . , φN} ∈ CN×N denotesthe phase shift matrix of the RIS, where φn ∈ C denotes thephase shift of the n-th reflecting element. It is assumed that|φn| = 1,∀n = 1, . . . , N and the phase of φn can be flexiblyadjusted within [0, 2π) [25].

Considering all L time slots, the received signals Y =[y1, . . . ,yL] ∈ CM×L at the BS can be rewritten in the matrixform as

Y = GΦHAQ+N , (4)

where H , [h1, . . . ,hK ]T ∈ CK×N is the RIS-to-devicechannel matrix, Q , [q1, . . . , qK ]T ∈ CK×L is the pilotmatrix with qk , [qk,1, . . . , qk,L]

T ∈ CL, N ∈ CL×M is theindependent AWGN and A , diag{α1, . . . , αK} ∈ CK×Kis the activity matrix indicating the activity of devices. Inthis paper, our goal is to detect the device activity αk andestimate the corresponding channel vector hk as well as thechannel matrix G, given the observations Y , the known phaseshift matrix Φ and the known pilot matrix Q in the massiveconnectivity setting.

B. Channel Models

1) RIS-to-BS Channels: We assume the BS is equippedwith a uniform linear array (ULA), while the RIS is equippedwith an N1 × N2 uniform rectangular array (URA) withN1N2 = N . By exploiting the physical propagation structureof wireless channel, we consider the RIS-to-BS channel matrixG as a geometric channel [43]. By applying the geometricchannel model [44], the RIS-to-BS channel can be expressedas

G = τG

√MN

P

P∑p=1

κpaB(θp)aHR

(ψp, ωp

), (5)

where P is the total number of spatial paths between the BSand the RIS, κp is the complex-value channel gain of the p-thRIS-to-BS path; τG is the distance-dependent path loss, θp isthe corresponding azimuth angle-of-arrival (AoA) at the BS,ψp (ωp) denote the corresponding azimuth (elevation) angle-of-departure (AoD) at the RIS. We set p = 1 to model theLoS path between the RIS and the BS. In addition, aB andaR are the steering vectors associated with the BS and theRIS antenna geometry, i.e.,

aB(θ) =1√M

[1, e−j

2πρ d sin(θ), · · · , e−j 2π

ρ d(M−1) sin(θ)]T,

(6)aR(ψ, ω) = aR,v(ψ, ω)⊗ aR,h(ψ, ω), (7)

4

where ρ is the carrier wavelength and d is the antennaspacing. Here, aR,v(ψ, ω) and aR,h(ψ, ω) are the horizontaland vertical steering vectors defined asaR,h(ψ, ω)

=1√N1

[1, e−j

2πdρ fh(ψ,ω), · · · , e−j 2πd

ρ (N1−1)fh(ψ,ω)]T,

(8)aR,v(ψ, ω)

=1√N2

[1, ej

2πdρ fv(ψ,ω), · · · , ej 2πd

ρ (N2−1)fv(ψ,ω)]T, (9)

where fh(ψ, ω) = cos(ω) sin(ψ) and fv(ψ, ω) =cos(ω) cos(ψ).

2) RIS-to-Device Channels: For the channels between theRIS and the devices, since the devices are generally sur-rounded by many reflective objects at low elevations in thedense urban environment, there are a rich number of localscatterers at the device side [43]. On the other hand, the densebuildings and other objects are likely to block the LoS linksbetween the RIS and the devices. Based on the above facts,we consider Rayleigh fading channels between the RIS andthe devices, as in [20], [41], , i.e., each entry of hk followsi.i.d complex Gaussian distribution:

p(hnk) = CN (hnk; 0, τh,k), (10)

where τh,k is the distance-dependent path loss for the k-thRIS-to-device link .

C. Problem FormulationBased on the observations Y , the predetermined phase shift

matrix Φ and the known pilot matrix Q, our goal is to detectthe activity of devices {αk} and estimate the correspondingchannel vectors {hk} as well as the channel matrix G in themassive connectivity regime. In this scenario, the sequencelength is typically much smaller than the number of devices,i.e., L � K. Hence, it is impossible to assign the mutuallyorthogonal signature sequences to all devices. Inspired by theprior works [5], [6], we generate the signature sequence qkaccording to i.i.d. complex Gaussian distribution with zeromean and variance 1/L, i.e.,

qk ∼ CN(

0,1

LI

). (11)

Note that each signature sequence is normalized to have unitnorm, i.e., E

[‖qk‖2

]= 1,∀k.

All the reflecting elements of the RIS are switched on andset to have the same phase shift during the period of activitydetection and channel estimation [34]. By setting φn = 1,∀n,(4) can be rewritten as

Y = GHAQ+N . (12)

Recognizing that the RIS-to-BS channels are typicallysparse due to limited scattering between the BS and the RIS,we represent the RIS-to-BS channel matrix G on the virtualangular domain. In the following, by simultaneously exploitingthe sparsity of both sporadic transmission and the RIS-to-BSchannel, we show that the RIS-related channel estimation andactivity detection problem can be formulated as a sparse matrixfactorization problem.

1) Sparsity of Sporadic Transmission: By defining X ,HA, the received signals can be expressed as

Y = GXQ+N . (13)

Due to the sporadic transmission, only a few devices are activeat the same time. Recall that the diagonal entry αk of thematrix A indicates whether the k-th device is active or not,the activity matrix A is diagonal sparse, which further impliesthat the matrix X is a column sparse RIS-to-device channelmatrix. Note that the non-zero columns of X represent thechannel vector between the RIS and the active devices. Hence,we have

xnk =

{hnk, αk = 1,0, αk = 0,

(14)

where xnk is the (n, k)-th entry of matrix X .2) Sparsity of RIS-to-BS Channel Representation: As both

the RIS and the BS are typically installed at high elevations,there are only limited scattering between the BS and the RIS.According to [34], [35], we represent the RIS-to-BS channelG on the virtual angular domain (VAD) to provide a discreteapproximation of the physical channel. Instead of taking AoDsand AoAs from arbitrary physical angles, the VAD repre-sentation parameterizes them with pre-discretized angles withfinite resolutions. Specifically, we employ three pre-discretizedsampling grids ϑ with length M ′, ϕ with length N ′1 and $with length N ′2 to parameterize {θp}1≤p≤P , {ψp}1≤p≤P and{ωp}1≤p≤P , respectively. To well approximate the originalchannel (5), the angular resolutions should be large enough,i.e., M ′ > M , N ′1 > N1 and N ′2 > N2. Following by [45],the RIS-to-BS channel G in (5) can be expressed by

G = ABS(AR)H, (15)

where AR = AR,v ⊗ AR,h. Here, we define AB =[aB(ϑ1), . . . ,aB(ϑM ′)

]∈ CM×M ′

is an over-complete ma-trix with each column representing a steering vector pa-rameterized by the pre-discretized azimuth AoA at BS, andAR,h =

[aR,h(ϕ1), . . . ,aR,h(ϕN ′

1)]∈ CN1×N ′

1 (or AR,v =[aR,v($1), . . . ,aR,v($N ′

2)]∈ CN2×N ′

2 ) is an over-completematrix with each column a steering vector parameterized bypre-discretized azimuth (or elevation) AoD at RIS, respec-tively. In addition, S ∈ CM ′×N ′

is the channel coefficientmatrix in angular domain, where the non-zero (i, j)-th entrycorresponds to the complex gain on the channel consisting ofthe i-th cascaded AoA array steering vector at the BS and thej-th AoD array steering vector at the RIS.

Since there are limited scatters between the BS and the RIS,the number of spatial paths between the BS and the RIS shouldbe small, i.e., P � min{M,N}. Therefore, only a few entriesof S are non-zero, i.e., S is a sparse matrix. By substituting(15) to (12), the system model can be rewritten as

Y = ABSAH

RXQ+N . (16)

With the distorted signals Y , our goal is to recover thesparse coefficient matrix S in angular domain and the sparseRIS-to-device channel matrix X , given the known pilot matrixQ and the predetermined matrices AB and AR. Once weobtain the estimated matrix X , the diagonal entry of activity

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matrix A can be estimated via the group sparsity of X asfollows

αk =

{1, ‖xk‖2 > ε0, ‖xk‖2 ≤ ε 1 ≤ k ≤ K, (17)

where ε is a small positive threshold and xk is the k-th columnof the estimated matrix X [5].

D. Problem Analysis

In terms of the activity detection and channel estimationproblem in the massive connectivity scenario, the RIS-assistedsystem is quite different from the conventional communicationsystems, where the devices are usually served by the BSdirectly. In particular, in the conventional massive connectivityscenario, the BS only needs estimate the channels betweenthe BS and the active devices. Due to the sporadic natureof IoT devices, channel estimation and activity detection canbe achieved in a joint manner. For example, [5] proposed astructured group sparsity estimation approach by exploitingsparsity in the device activity pattern. [42] formulated theproblem as the multiple measurement vector (MMV) problemand solved it via the AMP algorithm. In contrast, in theRIS-assisted scenario, since designing phase-shift matrix Θrequires the knowledge of the RIS-to-BS channel matrix Gand the sparse RIS-to-device channel matrix X separately[24], [25], [46], we need to decouple the RIS-related cascadedchannels GX given the pilot matrix Q and the observationsY . This means that we cannot simply formulate the problemas the MMV problem. Moreover, without RF chains, the RIScannot transmit pilot sequences and process received signals tohelp the BS detect activity and estimate channels [39]. Hence,the BS will bear all the tasks of estimating and detecting.

On the other hand, for the RIS-assisted communicationsystem, previous works [34], [35], [47] studied the channelestimation problem without considering the massive connec-tivity regime, all active devices are assumed to be knownat the BS. By assuming that the number of active devicesis smaller than the sequence length, i.e., K < L, theyassign mutually orthogonal pilot sequences to all devices [35].However, in the RIS-related massive connectivity regime, thereare a large number of devices in the communication system,which raises unique computational challenges when solvingthe RIS-related channel estimation problem with limited timebudget. In addition, it is infeasible to assign orthogonal pilotsequences to all devices. All these facts impose the criticalchallenges of channel estimation in the RIS-related massiveconnectivity scenario.

To tackle the above challenges brought by the RIS-relatedmassive connectivity system, our previous work [38] formu-lated the RIS-related activity detection and channel estimationproblem as sparse matrix factorization, matrix completion andmultiple measurement vector problem and proposed a three-stage algorithm. However, the proposed algorithm in [38]requires the accuracy guarantee of each stage to ensure theconvergence. To address this issue, this paper further proposesto formulate the RIS-related activity detection and channelestimation problem as a sparse matrix factorization problem

given prior knowledge of the channels. Specifically, we con-currently leverage the channel sparsity of G and the groupsparsity of X . By utilizing the Bayesian inference framework,we propose a unified AMP based framework to efficientlyjointly estimate the sparse channel coefficient matrix S andthe sparse RIS-to-device channel matrix X in (16).

III. AMP-BASED RIS-RELATED ACTIVITY DETECTIONAND CHANNEL ESTIMATION ALGORITHM

In this section, we introduce an AMP-based algorithmicframework to solve the sparse signal recovery problem. Wefirst reformulate the RIS-related activity detection and channelestimation problem as a Bayesian inference problem. Tocompute the optimal minimum mean-squared error (MMSE)estimate of S and X , we derive the posterior probabilities ofS and X conditioned on Y and represent those probabilitieswith a factor graph. Subsequently, we calculate these quantitiesby utilizing sum-product algorithm (SPA) [48]. Furthermore,to implement the SPA in practice, we introduce some approx-imations to the SPA based on the central-limit-theorem (CLT)and Taylor-series arguments.

A. Minimum Mean-squared Error (MMSE) Estimators

In Bayesian inference framework, we treat S and X asrandom variables with known separable probability distribu-tion functions (PDFs) p(S) and p(X). Hence, the Bayesianapproach can exploit prior knowledge to further improve theestimation accuracy. According to [49], the optimal Bayesianestimator is the MMSE estimator. The MMSE estimators ofS and X are given by S = [sm′n′ ] and X = [xnk], where

sm′n′ =

∫sm′n′p

(sm′n′ |Y

)dsm′n′ ,

xnk =

∫xnkp

(xnk|Y

)dxnk.

(18)

The marginal posteriors with respect to sm′n′ and xnk arederived as follows,

p(sm′n′ |Y

)=

∫ ∫p(S,X|Y

)dXd

(S\sm′n′

)and (19)

p(xnk|Y

)=

∫ ∫p(S,X|Y

)dSd

(X\xnk

)(20)

where M\mij denotes the collection of the entries of matrixM excluding the (i, j)-th one. The MMSE for S and Xaccording to (18) are defined as

MMSE(S) ,1

M ′N ′E[‖S − S‖2F

]; (21)

MMSE(X) ,1

M ′N ′E[‖X −X‖2F

]. (22)

According to Bayes’ rule, we arrive at

p(S,X|Y ) =1

p(Y )p(Y |S,X)p(S)p(X)

∝ p(Y |S,X)p(S)p(X), (23)

where p(Y |S,X) is the likelihood function of S and X . Tofacilitate the calculation of the likelihood function p(Y |S,X),we introduce the following two auxiliary variables W =

6

ABSAH

RX and Z = WQ, and the system model (16) isrewritten as

Y = WQ+N = Z +N . (24)

We assume that the likelihood function of Z is element-wise separable. Since the noise N is independent AWGN andzml =

∑Kk=1 wmkqkl, then it can be expressed by

p(Y |Z) =

M∏m=1

L∏l=1

p(yml|wmk,∀k)

=

M∏m=1

L∏l=1

CN (yml;

K∑k=1

wmkqkl, τn). (25)

Similarly, the PDF of the matrix S is assumed to be element-wise separable. Due to the sparsity of the matrix S, we adoptBernoulli-Gaussian distribution as the prior distribution for thematrix S, i.e.,

p(S) =

M ′∏m′=1

N ′∏n′=1

p(sm′n′)

=

M ′∏m′=1

N ′∏n′=1

((1− λs)δ(0) + λsCN (sm′n′ ; 0, τs)

), (26)

where λs is the Bernoulli parameter, τs is the variance of thenonzero entries of S and δ(·) is the Dirac delta function.

Recalling that X = HA, the columns of X share the samesparsity. By separating the activity matrix A and the RIS-to-device channel matrix H , we decompose each entry xnk ofthe matrix X as xnk = αkhnk. To model the sparsity of thematrix X , we model αk as a Bernoulli random variable withPr(αk = 1) = λα. We assume that the PDF of X is assumedto be column-wise independent. Therefore, the prior of the Xcan be written as

p(X) =

N∏n=1

K∏k=1

p(xnk)

=

K∏k=1

N∏n=1

((1− αk)(1− λα)δ(0)

+αkλαCN (hnk; 0, τh,k)). (27)

Recalling the relations among Y ,Z,W ,S, and X in (24),the posterior distribution the posterior distribution of S andX conditioned on the observations Y is given as,

p(S,X|Y ) ∝M∏m=1

L∏l=1

p(yml|wmk,∀k)

×M∏m=1

K∏k=1

p(wmk∣∣ N∑n=1

gmnxnk)

N∏n=1

K∏k=1

p(xnk)

×M∏m=1

N∏n=1

p(gmn|sm′n′ ,∀m′, n′)M ′∏m′=1

N ′∏n′=1

p(sm′n′). (28)

In practice, exact expectations of S and X are generallyprohibitive due to the high-dimensional integrations involvedin the marginalization (see Eq. (19)). However, these quantitiescan be efficiently approximated using loopy belief propagation(LBP) [50].

Fig. 2: The factor graph for a toy-sized problem with dimen-sions M ′ = N = L = 2 and N ′ = M ′ = K = 3.

B. Loopy Belief Propagation

TABLE I: Notations for the factor graph in Fig. 2.

Notation Factor node Distributionpyml|wmk p(yml|wmk,∀k) CN (yml;

∑Kk=1 wmkqkl, τn)

pwmk|gmn,xnk p(wmk|gmn, xnk, ∀n) δ(wmk −∑Nn=1 gmnxnk)

pxnk p(xnk)(1− αk)(1− λα)δ(0)+αkλαCN (hnk; 0, τh,k)

pgmn|sm′n′ p(gmn|sm′n′ , ∀m′, n′) δ(gmn −∑M′

m′=1

∑N′

n′=1aB,mm′sm′n′aR,nn′ )

psm′n′ p(sm′n′ )(1− λs)δ(0)+λsCN (sm′n′ ; 0, τs)

We construct a factor graph shown in Fig. 2 to representthe PDFs in (28). The random variables and the factors ofthe posterior probabilities in (28) are represented by variablenodes appearing as while circles and factor nodes appearingas black squares, respectively. The notations of the factornodes are summarized in Table I. To simplify the notations,we omit the subscripts of the variable nodes in Fig. 2.By applying the traditional message passing algorithm, theestimators in (21) can be approximately computed. In high-dimensional inference problems, exact implementation of theSPA is impractical, which motivates approximations of theSPA. In the sequel, we will first derive the messages forthe SPA, and then approximate these messages. As we shallsee, the approximations are primarily based on central-limit-theorem (CLT) and Taylor-series arguments.

TABLE II: Notations of means and variances for the messages.

Message Mean Variance∆tl←mk(wmk) wmk,l(t) vwmk,l(t)

∆tk←mn(gmn) gmn,k(t) vgmn,k(t)

∆tm←nk(xnk) xnk,m(t) vxnk,m(t)

∆tmn←m′n′ (sm′n′ ) sm′n′,mn(t) vs

m′n′,mn(t)

∆twmk

(wmk) wmk(t) vwmk(t)∆tgmn

(gmn) gmn(t) vgmn(t)∆tsm′n′ (sm′n′ ) sm′n′ (t) vs

m′n′ (t)

∆txnk

(xnk) xnk(t) vxnk(t)

7

C. Sum-Product Algorithm

To employ the SPA, we define the following notationsof messages: ∆t

ij→mn(xmn) denotes the message from thefactor node fij to the variable node xmn in the t-th iteration,∆tij→mn(xmn) denotes the message from the variable node

xmn to the factor node fij , and ∆tx(x) denotes the marginal

message computed at variable node x.By applying the SPA to the factor graph in Fig. 2 and

following the procedures in [48], we obtain the followingupdate rules of the messages:

1) Messages between variable nodes {wmk} and factornodes {p(yml|wmk,∀k)} are given by:

∆tl→mk(wmk) ∝

∫p(yml|wmk,∀k

)∏j 6=k

∆tl←mj(wmj)dwmj ,

(29)

∆t+1l←mk(wmk) ∝ Ptwmk(wmk)

∏j 6=l

∆tj→mk(wmk). (30)

Here, the auxiliary distribution Ptwmk(wmk) denotes the mes-sages from the variable nodes {gmn} and {xnk}, which isdefined as

Ptwmk(wmk) ∝∫p(wmk|gmn, xnk,∀n

)×

N∏n=1

∆tk←mn(gmn)dgmn∆t

m←nk(xnk)dxnk. (31)

2) Messages between variable nodes {gmn} and factornodes {p(wmk|gmn, xnk,∀n)} are given by:

∆tk→mn(gmn) ∝

∫ L∏l=1

∆tl→mk(wmk)p(yml|wmk,∀k)dyml

×N∏n=1

∆tm←nk(xnk)dxnk

∏j 6=n

∆tk←mj(gmj)dgmj , (32)

∆t+1k←mn(gmn) ∝ Ptgmn(gmn)

∏j 6=k

∆tj→mn(gmn). (33)

Here, the auxiliary distribution Ptgmn(gmn) denotes the mes-sages from the variable nodes {sm′n′}, which is defined as

Ptgmn(gmn) ∝∫p(gmn|sm′n′ ,∀m′, n′)

×M ′∏m′=1

N ′∏n′=1

∆tmn←m′n′(sm′n′)dsm′n′ . (34)

3) Messages between variable nodes {sm′n′} and factornodes {p(gmn|sm′n′ ,∀m′, n′)} are given by:

∆tmn→m′n′(sm′n′) ∝

∫p(gmn|sm′n′ ,∀m′, n′)

×K∏k=1

∆tk→mn(gmn)dgmn

∏(i,j)6=(m′,n′)

∆imn←ij(sij)dsij ,

(35)

∆t+1mn←m′n′(sm′n′) ∝ p(sm′n′)

∏(i,j)6=(m,n)

∆tij→m′n′(sm′n′).

(36)

4) Messages between variable nodes {xnk} and factornodes {p(wmk|gmn, xnk,∀n)} are given by:

∆tm→nk(xnk) ∝

∫ L∏l=1

∆tl→mk(wmk)p(yml|wmk,∀k)dyml

×N∏n=1

∆tk→mn(gmn)dgmn

∏j 6=n

∆tm←jk(xjk)dxjk, (37)

∆t+1k←nk(xnk) ∝ p(xnk)

∏j 6=m

∆tj→nk(xnk). (38)

5) Marginal messages at variable nodes are given by:

∆t+1wmk

(wmk) ∝ Ptwmk(wmk)

L∏l=1

∆tl→mk(wmk), (39)

∆t+1gmn(gmn) ∝ Ptgmn(gmn)

K∏k=1

∆tk→mn(gmn), (40)

∆t+1sm′n′ (sm′n′) ∝ p(sm′n′)

M∏m=1

N∏n=1

∆tmn→m′n′(sm′n′). (41)

∆t+1xnk

(xnk) ∝ p(xnk)

M∏m=1

∆tm→nk(xnk). (42)

D. Approximated Message Passing for SPA

Due to the high-dimensional integrations, the messages in(29)–(41) are generally computationally intractable. Hence,we approximate the the SPA updates (29)–(41) based onthe central-limit-theorem (CLT) and the Taylor-series argu-ments that are almost exact in the large-system limit, i.e.,M,M ′, N,N ′,K, L→∞ with the fixed ratios M/K, M ′/K,N/K, N ′/K, and L/K, which is widely adopted in [51], [52].Specifically, we will neglect terms that vanish relative to othersas K → ∞, which is reasonable in massive connectivity.We first outline the main steps of the approximation methodas follows, and the details of derivations can be referred toAppendix A.1. We first adopt a second-order Taylor expansion to approx-

imate Ptwmk as a Gaussian distribution. Due to the CLTarguments, we also approximate

∏j 6=k ∆t

j→mk(wmk) asa Gaussian distribution. According to (29), ∆t+1

l←mk(wmk)is thus characterized as a Gaussian distributions with thetractable mean and variance.

2. We find that ∆tl←mk(wmk) differs from ∆t

wmk(wmk) in

only one term ∆tl→mk(wmk) that will vanish in the large-

system limit. As a result, ∆twmk

(wmk) also becomes aGaussian distribution. Then we use the same mean wmk(t+1) and variance vwmk(t+ 1) to characterize these two mes-sages. Consequently, the closed-loop updating formulas forwmk and vwmk can be obtained. As we shall see, the high-dimensional integration can be so that the computationalcomplexity is significantly reduced.

3. Similarly, we can show that ∆t+1gmn(gmn), ∆t+1

xnk(xnk) and∏

m,n ∆tmn→m′n′(sm′n′) can be approximated as Gaussian

distributions with corresponding means and variances aswell. Taking the prior information (26)–(27) and the mes-sages (41)–(42) into account, we can obtain the closed-loop updating formulas for gmn(t), vgmn(t), xnk(t), vxnk(t),

8

Algorithm 1: The proposed algorithm.Input: Y ;AB ;AR;Q; τn; τs; τh,k;λs;λα; εOutput: S; X; AInitialize:∀m′, n′: Sample sm′n′(1) according to p(sm′n′);∀n, k: Sample xnk(1) according to p(xnk);∀m,n:gmn(1) =

∑M ′

m′=1

∑N ′

n′=1 aB,mm′ sm′n′,mn(1)aR,n′n;

∀m, k: wmk(1) =∑Nn=1 gmn(1)xnk(1);

for i = 1, 2, · · · , Imax do∀m, k: update wmk(t) and vwmk(t) via (51c)–(51f),(52), and (58);∀m,n: update gmn(t) and vgmn(t) via (56),

(55a)–(55b), and (57);∀m′, n′: update sm′n′(t) and vsm′n′(t) via (64c)–(64b) and (65);∀n, k: update xnk(t) and vxnk(t) via (64c)–(64f)and (66);

endActivity matrix A = diag{αk} is determined by

αk =

{1, ‖xk‖2 > ε0, ‖xk‖2 ≤ ε 1 ≤ k ≤ K.

and sm′n′(t), vsm′n′(t). The notations of the messages aresummarized in Table II.

The whole procedure is summarized in Algorithm 1, and thealgorithm will continue until meeting a convergence condition,i.e., the maximum number of iterations Imax is reached.

Remark 1: The computational complexity of each step isshown in Table III. The complexity of the proposed algo-rithm in each iteration is O(MK) + O(MN) + O(NK) +O(MNM ′N ′). Note that, with the fixed ratios M/K, M ′/K,N/K, N ′/K, and L/K, the overall complexity of the pro-posed algorithm can be simplified as O(IK4). This poly-nomial scaling of the complexity with respect to problemdimensions gives encouragement that our algorithm shouldefficiently handle large-scale problems. In contrast, the exactMMSE estimators of X and S in (18) involving integrationswith respect to X and G, therefore the corresponding com-putational complexity grows exponentially with K2.

Remark 2: Although this paper models the channels asin Section II-B, the proposed algorithm still needs the ex-act parameters that determine these distributions, i.e., τn.In general, these parameters are usually unobtainable. Totackle this problem, we can parameterize the priors in theproposed algorithm and exploit the expectation-maximization(EM) based approach to tune the parameter. Due to the spacelimitation, we will not go into the details of the method in thispaper, which can be referred to [53].

TABLE III: Computational complexities of each step in Algo-rithm 1.

Step ComplexityUpdating wmk(t) and vwmk(t) O(MK)Updating gmn(t) and vgmn(t) O(MN)Updating xnk(t) and vxnk(t) O(NK)Updating sm′n′ (t) and vs

m′n′ (t) O(MNM ′N ′)

Fig. 3: Horizontal locations of the BS, RIS and devices.

IV. NUMERICAL EXPERIMENTS

A. Simulation Setting

In this section, we conduct several numerical experimentsto verify the effectiveness of the proposed algorithm. Thereare totally K = 1000 devices, where 80 of them are active,i.e., λa = 0.08. Under a three dimensional (3D) Cartesiancoordinate system, we consider that the BS is equipped with auniform linear array (ULA) aligned with z-axis; while the RISis equipped with a uniform rectangular array (URA) parallelto the x − z plane. For illustration, we assume that both theBS and the RIS are located at the same altitude above thedevices by zR m. In addition, we assume that all devices arerandomly and uniformly located in a circular coverage areaof radius R = 50 meters (m) with a center whose locationis (Ox, Oy, 0). The locations of the BS, the RIS and thek-th device are set as (0, 0, zR), (xR, yR, zR) and (Ox +∆kx, Oy + ∆k

y , 0), respectively, whose horizontal projectionsare illustrated in Fig. 3. The 3D distances for the RIS-to-BSlink and the k-th RIS-to-device link can be obtained as dG =√x2R + y2R m and dk =

√(Ox + ∆k

x)2 + (Oy + ∆ky)2 + z2R

m, respectively. For all numerical experiments, we set zR = 10m, and the horizontal distances from the projection of the RISon the y-axis and x-axis to the RIS as xR = 5 m and yR = 100m.

We consider the distance-dependent path loss for all chan-nel, which is modeled as

τ = τ0

(d

d0

)−µ, (43)

where τ0 = −30 dB denotes the path loss at the referencedistance d0 = 1 m followed by [21]; µ denotes the path lossexponent. In our setting, τG and τh,k are denoted as the pathloss of the RIS-to-BS link and the k-th RIS-to-device link,respectively; the path loss exponents for the corresponding

9

links are set as µG = 2.2 and µh,k = 2.5, respectively.According to (10), the RIS-to-device channel matrix H ismodeled by Rayleigh fading with hnk ∼ CN (0, τh,k),∀k, n.On the other hand, we generate the RIS-to-BS channel matrixG by (5) with 10 clusters of paths and 5 subpaths per cluster.We draw the central azimuth AoA at the BS of each clusteruniformly over [−π/2, π/2]; draw the central azimuth (orelevation) AoD at the RIS of each cluster uniformly over[−π, π] (or [−π/2, π/2]); and draw each subpath with a π/12angular spread. Moreover, every complex-value channel gainκp is drawn from CN (0, 1). In our experiments, we set the pre-discretized sampling grids ϑ, ϕ and $ to be uniform samplinggrids over their corresponding domain, and the length of thesampling grids are set to have a fixed ratio to the antennadimensions, i.e., M ′/M = N ′1/N1 = N ′2/N2 = 2. Unlessspecifically mentioned, we set other system parameters asλa = 0.08,M = 40, N1 = N2 = 7(N = 49) and Imax = 2000for the proposed algorithm. All the numerical results areaveraged over 50 independent channel realizations.

In the following, we provide numerical experiments toevaluate the performance of the proposed algorithm from theaspects of activity detection and channel estimation.

B. Simulation for Activity Detection

1) Performance Metric: We evaluate the algorithms withthe following metric. We define the probabilities of falsealarm and missed detection to evaluate the performance ofthe proposed algorithm for activity detection. In particular,the probability of false alarm, pF is defined as the probabilitythat a device is active but the detector declares the device isinactive, and the probability of missed detection, pM is definedas the probability that a device is inactive but the detectordeclares the device is active.

2) Baselines: In addition to the proposed algorithm, weintroduce the following two algorithms as baselines for com-parison.• AMP-MMV: Instead of estimating the matrices G andX , we estimate the cascaded channel matrix W = GX ,and then detect the activity matrix A. This problem isformulated as a Multiple Measurement Vector (MMV)problem which can be efficiently solved by the AMP-MMV algorithm [54]. Since we do not estimate thematrices G and X separately, the solution of AMP-MMV should be the performance lower bound for theprobabilities of false alarm and missed detection.

• Three-stage [38]: By designing the phase shift matrixΦ as a sparse matrix, the problem can be formulated assparse matrix factorization, matrix completion and MMVthree stages. The solution of activity detection can beachieved in the third stage.

3) Simulation Results: Fig. 4a investigates the impacts ofsequence length L on pM with the fixed pF = 0.1. It canbe shown that pM obtained by the proposed algorithm arevery close to the ones obtained by the AMP-MMV algorithm,while the proposed algorithm estimates separately the matrixG and X . Moreover, the proposed algorithm outperformsthe three-stage algorithm [38], and the gap between the two

100 120 140 160 180 200L

10−4

10−3

10−2

10−1

100

p M

Proposed algorithm

AMP-MMV

Three-stage

(a)

0 10 20 30 40 50SNR (dB)

10−6

10−5

10−4

10−3

10−2

10−1

100

p M

M = 40

M = 50

M = 60

(b)

Fig. 4: The performance of pM with the fixed pF = 0.1.

algorithms increases when L increases. Until the sequencelength L ≥ 130, the probabilities of the three-stage algorithmbegin to decrease. The reason is that the estimated error inthe first two stages will greatly affect the performance ofactivity detection. When the estimated error in the first twostages is relatively large, the third stage cannot detect activityaccurately. In contrast, our algorithm provides a unifyingframework instead of three-stage framework, and then is ableto iteratively update all factor nodes in each message passingiteration, which avoids the problem arising in the three-stagealgorithm. Fig. 4b shows the performance of activity detectionas SNR varies for different number of antennas M at theBS. By comparing the cases when M = 40, M = 50 andM = 60 with the fixed sequence length L = 130 and thefixed pF = 0.1, we observe that under the our proposedframework, pM decreases significantly as the number of BSantennas increases.

C. Simulation for Channel Estimation

1) Performance Metric: The performance of the proposedalgorithm for channel estimation is evaluated by the normal-ized MSEs (NMSEs) of the RIS-to-BS channels G and theaverage NMSEs of the active RIS-to-device channels {hk},which are defined by

NMSE of G ,‖G− G‖2F‖G‖2F

, (44)

Average NMSE of {hk} ,1

|A|∑k∈A

‖hk − hk‖22‖hk‖22

. (45)

2) Baselines: We also consider two baseline algorithmsfor comparison. The three-stage algorithm [38] has beenintroduced in the previous subsection. Here, we introduceGenie-aided MMSE estimator where all the active devicesare assumed to be known at the BS in advance. Hence, theperformance of Genie-aided MMSE estimator is referred asthe bound of the NMSE performance for channel estimation.

3) Simulation Results: Fig. 5a and Fig. 5b show the effectsof the sequence length on the NMSE of G and the averageNMSE of {hk}, respectively. We observe that the performanceof the proposed algorithm can reach the bound (Genie-aidedMMSE estimator) in both figures, namely, the proposed algo-rithm can correctly identified the active devices. In addition,we can see that the proposed algorithm achieves much lowerNMSE of both G and {hk} than the three-stage algorithm

10

120 140 160 180 200 220L

−50

−40

−30

−20

−10

0

Average

NMSEof

hk

Proposed algorithm

Three-stage

Genie-aided MMSE

(a)

120 140 160 180 200 220L

−45

−40

−35

−30

−25

−20

−15

NMSEof

G

Proposed algorithm

Three-stage

Genie-aided MMSE

(b)

15 20 25 30 35 40 45SNR (dB)

−35

−30

−25

−20

−15

−10

−5

Average

NMSEof

hk

M = 40

M = 50

M = 60

(c)

15 20 25 30 35 40 45SNR (dB)

−35

−30

−25

−20

−15

−10NMSEof

GM = 40

M = 50

M = 60

(d)

Fig. 5: The performance of channel estimation.

over all sequence lengths, since the three-stage algorithm islimited to the correlation issue. The main advantage of theproposed algorithm is to simultaneously exploit the sparsityof the RIS-to-BS channel G and the sporadic transmission inmassive connectivity.

On the other hand, we also investigate the channel estima-tion performance as SNR varies the cases when M = 40,M = 50 and M = 60 with the fixed sequence lengthL = 130. We find that the NMSEs of the proposed algorithmdecrease as SNR increases overall. Specifically, Fig 5c andFig. 5d show that increasing the number of BS antennas Mbrings significant improvement, i.e., for M = 60, the proposedalgorithm approaches best performance in estimating G andH compared to other two cases. In other words, deployingmassive antennas at the BS can significantly improve theestimation performance.

Finally, we demonstrate the performance limits of the pro-posed algorithm in terms of some critical systems parameters(i.e., M , N and L). We declare the successful recovery ofG and {hk} if their NMSE are both less than −30dB. Weperform 30 trials for each system setting to average and drawthe phase transition diagrams in term of the success rate in Fig.6. It illustrates that two sharp phase-transition curves separatesuccess and failure regions. Based on the phase-transitioncurves, we can set appropriate system parameters for theproposed algorithm to efficiently support massive connectivityin RIS-assisted systems. For example, the minimal sequencelength can be determined to accurately detect the activedevices and estimate the channels.

V. CONCLUSION

In this paper, we studied the RIS-related activity detectionand channel estimation problem in the RIS-assisted massiveIoT network. We first modeled the RIS-to-BS channel as asparse channel due to limited scattering between the RIS andthe BS. Consequently, we formulated the problem as a sparse

25 30 35 40 45

50

45

40

35

30 0

0.2

0.4

0.6

0.8

1

su

cce

ss r

ate

(a)

140 150 160 170 180 190 200

50

45

40

35

30 0

0.2

0.4

0.6

0.8

1

su

cce

ss r

ate

(b)

Fig. 6: The phase transitions of channel estimation.

matrix factorization problem by simultaneously exploiting thesparsity of both sporadic transmission in massive connectivityand the RIS-to-BS channels. Based on Bayesian inferenceframework, we further proposed an AMP-based algorithm tojointly estimate the detect active devices and two separatedchannels of the RIS-to-BS link and the RIS-to-device link.The computational complexity is further reduced by utilizingthe central limit theorem and Taylor series arguments. Fur-thermore, we also conducted several numerical experiments toconfirm the effectiveness and improvements of the proposedalgorithm for the RIS-related activity detection and channelestimation problem.

APPENDIXDETAILS OF DERIVATIONS

A. Approximated messages of (29)-(31)

Applying the Fourier inversion theorem and a second-order Taylor expansion to (31), we can approximate thatPtwmk(wmk) ≈ CN (wmk; pmk(t), vpmk(t)), where

vpmk(t) =

N∑n=1

(∣∣gmn,k(t)∣∣2 vxnk,m(t)

+vgmn,k(t)∣∣xnk,m(t)

∣∣2 + vgmn,k(t)vxnk,m(t)), (46a)

pmk(t) =

N∑n=1

gmn,k(t)xnk,m(t). (46b)

The details can be referred to [55, Eqs. (51)-(53)].To approximate the message ∆t

l→mk(wmk), we definezml,k ,

∑j 6=k wmjqjl ∼

∏j 6=k ∆t

l←mj(wmj). Accord-ing to the CLT, we treat zml,k as a Gaussian randomvariable with mean zml(t) − wmk,l(t)qkl and variancevzml(t) − vwmk,l(t)|qkl|

2, where zml(t) ,∑Kk=1 wmk,l(t)qkl

and vzml(t) ,∑Kk=1 v

wmk,l(t)|qkl|

2. Then we can approximatethe message (29) as∆tl→mk(wmk)

∝∫

dzml,kCN (yml; zml,k + wmkqkl, τn)

× CN(zml,k; zml(t)− wmk,l(t)qkl, vzml(t)− vwmk,l(t)|qkl|2

)= CN

(yml; zml(t) + (wmk − wmk,l(t))qkl,

τn + vzml(t) + (|wmk|2 − vwmk,l(t))|qkl|2). (47)

11

By exploiting the arguments in [56, Eqs. (A.6)–(A.16)], weobtain the following approximations

∆t+1l→mk(wmk) = CN

(wmk; wmk,l(t+ 1), vwmk,l(t+ 1)

),

(48)

∆t+1wmk

(wmk) = CN(wmk; wmk(t+ 1), vwmk(t+ 1)

),(49)

L∏l=1

∆tl→mk(wmk) = CN (wmk; emk(t), vemk(t)), (50)

wherewmk,l(t+ 1) ≈ wmk(t+ 1)− vwmk(t+ 1)qklγml(t), (51a)vwmk,l(t+ 1) ≈ vwmk(t+ 1), (51b)

vwmk(t+ 1) =vpmk(t)vemk(t)

vpmk(t) + vemk(t), (51c)

wmk(t+ 1) =vpmk(t)emk(t) + pmk(t)vemk(t)

vpmk(t) + vemk(t), (51d)

vemk(t) =

L∑l=1

vγml(t)|qkl|2

−1 , (51e)

emk(t) = wmk(t) + vemk(t)

L∑l=1

q∗klγmt(t). (51f)

In the above equations (51), we define the following variables

vγml(t) =(vβml(t) + τn

)−1, (52a)

γml(t) = vγml(t)(yml − βml(t)

), (52b)

vβml(t) =

K∑k=1

vwmk(t)|qkl|2 , (52c)

βml(t) =

K∑k=1

wmk(t)qkl − vβml(t)γml(i− 1), . (52d)

B. Approximated messages for (32)–(33) and (40)

Substituting (55) into (32) and (37), we then arrive at∆tk→mn(gmn) ∝∫CN

N∑n=1

gmnxnk; emk(t), vemk(t)

p(yml|wmk,∀k)dyml

×N∏n=1

∆tm←nk(xnk)dxnk

∏j 6=n

∆tk←mj(gmj)dgmj , (53a)

∆tm→nk(xnk) ∝∫CN

N∑n=1

gmnxnk; emk(t), vemk(t)

p(yml|wmk,∀k)dyml

×N∏n=1

∆tk→mn(gmn)dgmn

∏j 6=n

∆tm←jk(xjk)dxjk. (53b)

Note that the forms of (53) have the same pattern with of[53, Eq. (13)]. Hence, we use the same arguments in [53, Sec.II-D–Sec. II-E], and approximate that

∆t+1xnk

(xnk) ≈ p(xnk)CN (xnk; bnk(t), vbnk(t)), (54a)K∏k=1

∆tk→mn(gmn) = CN (gmn; cmn(t), vcmn(t)), (54b)

where

vbnk(t) =

M∑m=1

∣∣wmn(t)∣∣2 vomk(t)

−1 , (55a)

bnk(t) =

1− vbnk(t)

M∑m=1

vgmn(t)vomk(t)

gnk(t)

+ vbnk(t)

M∑m=1

w∗mn(t)omk(t), (55b)

vcmn(t) =

K∑k=1

∣∣gnk(t)∣∣2 vomk(t)

−1 , (55c)

cmn(t) =

1− vcmn(t)

K∑k=1

vxnk(t)vomk(t)

wmn(t)

+ vcmn(t)

K∑k=1

g∗nk(t)omk(t). (55d)

In the above equations (55), we define the following auxiliaryvariables

vomk(t) =vpmk(t)− vwmk(t)

(vpmk(t))2, (56a)

omk(t) =wml(t)− pml(t)

vpmk(t). (56b)

Substituting (27) into (54a), we can obtain the mean andvariance of ∆t+1

xnkas follows

xnk(t+ 1) =

∫xnk∆t+1

xnk(xnk)dxnk, (57a)

vxnk(t+ 1) =

∫x2nk∆t+1

xnk(xnk)dxnk −

∣∣xnk(t+ 1)∣∣2 .(57b)

By using the same arguments in [53, Sec. II-F], we can rewrite(46) asvpmk(t) =

N∑n=1

(∣∣gmn(t)

∣∣2 vxnk(t) + vgmn(t)∣∣xnk(t)

∣∣2 + vgmn(t)vxnk(t)),

(58a)

pmk(t) =

N∑n=1

xmn(t)gnk(t)

−omk(i− 1)

N∑n=1

(∣∣xmk(t)∣∣2 vxnk(t) + vgmn(t)

∣∣gnk(t)∣∣2) .(58b)

12

C. Approximated messages for (34)–(40) and (41)Substituting (54b) into (35), we obtain∆tmn→m′n′(sm′n′) ∝∫CN

M ′∑m′=1

N ′∑n′=1

aB,mm′sm′n′aR,n′n; cmn(t), vcmn(t)

×

∏(i,j)6=(m′,n′)

∆imn←ij(sij)dsij . (59)

To simplify, we define gmn,m′n′ ,∑(i,j)6=(m′,n′) aB,misijaR,jn. According to the CLT, we

obtain that gmn,m′n′ becomes a Gaussian random variablewith mean gmn(t) − aB,mm′ sm′n′,mn(t)aR,n′n and variancevgmn(t)− aB,mm′vsm′n′,mn(t)aR,n′n, where

gmn(t) =

M ′∑m′=1

N ′∑n′=1

aB,mm′ sm′n′,mn(t)aR,n′n, (60a)

vgmn(t) =

M ′∑m′=1

N ′∑n′=1

∣∣aB,mm′∣∣2 vsm′n′,mn(t)

∣∣aR,n′n

∣∣2 . (60b)

Substituting (60) into (59), we obtain that∆tmn→m′n′(sm′n′)

∝ CN(aB,mm′sm′n′aR,n′n + gmn,m′n′ ; cmn(t), vcmn(t)

)∝ CN

(gmn(t)− aB,mm′ sm′n′,mnaR,n′n; . (61)

Using the same arguments in [56, Eqs. (A.6)–(A.16)], wearrive atM ′∏m′=1

N ′∏n′=1

∆tmn←m′n′(sm′n′) = CN

(gml; gmn(t), vg

mn(t)),

(62)M∏m=1

N ′∏n′=1

∆tmn→m′n′(sm′n′) = CN

(sm′n′ ; dm′n′(t), vdm′n′(t)

),

(63)

where

vgmn(t) =

M ′∑m′=1

N ′∑n′=1

∣∣aB,mm′∣∣2 vsm′n′(t)

∣∣aR,n′n

∣∣2 , (64a)

gmn(t) =

M ′∑m′=1

N ′∑n′=1

aB,mm′ sm′n′(t)aR,n′n − vgmn(t)αmn(i− 1),

(64b)

vαmn(t) =(vgmn(t) + vcmn(t)

)−1, (64c)

αmn(t) = vαmn(t)(cmn(t)− gmn(t)

), (64d)

vdm′n′(t) =

M∑m=1

N ′∑n=1

∣∣aB,mm′∣∣2 vαmn(t)

∣∣aR,n′n

∣∣2−1 ,(64e)

dm′n′(t) = sm′n′

+ vdm′n′(t)

M∑m=1

N∑n=1

|aR,B,mm′ |2αmn(t)|aR,n′n|2.

(64f)

Substituting (62) into (34) along with (54b), we obtain

vgmn(t+ 1) =vgmn(t)vcmn(t)

vgmn(t) + vcmn(t)

, (65a)

gmn(t+ 1) =vgmn(t)cmn(t) + vcmn(t)gmn(t) + vcmn(t)

vgmn(t) + vcmn(t)

.

(65b)

Substituting (63) into (41), we obtain that

∆t+1sm′n′ (sm′n′) ∝ p(sm′n′)CN

(sm′n′ ; dm′n′ , vdm′n′

)(66a)

sm′n′(t+ 1) =

∫sm′n′∆t+1

sm′n′ (sm′n′)dsm′n′ , (66b)

vsm′n′(t+ 1) =∫s2m′n′∆t+1

sm′n′ (sm′n′)dsm′n′ −∣∣sm′n′(t+ 1)

∣∣2 .(66c)

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