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Joint Channel Estimation and Signal Recovery in
RIS-Assisted Multi-User MISO Communications
Li Wei∗, Chongwen Huang†, George C. Alexandropoulos‡, Zhaohui Yang§, Chau Yuen∗, and Zhaoyang Zhang†
∗Singapore University of Technology and Design, 487372 Singapore
†Department of Information and Electronic Engineering, Zhejiang University
‡Department of Informatics and Telecommunications, National and Kapodistrian University of Athens, Greece
§Centre for Telecommunications Research, Department of Engineering, King’s College London, WC2R 2LS, UK
Abstract—Reconfigurable Intelligent Surfaces (RISs) have been
recently considered as an energy-efficient solution for future
wireless networks. Their dynamic and low-power configuration
enables coverage extension, massive connectivity, and low-latency
communications. Channel estimation and signal recovery in RIS-
based systems are among the most critical technical challenges,
due to the large number of unknown variables referring to the
RIS unit elements and the transmitted signals. In this paper,
we focus on the downlink of a RIS-assisted multi-user Multiple
Input Single Output (MISO) communication system and present
a joint channel estimation and signal recovery scheme based on
the PARAllel FACtor (PARAFAC) decomposition. This decompo-
sition unfolds the cascaded channel model and facilitates signal
recovery using the Bilinear Generalized Approximate Message
Passing (BiG-AMP) algorithm. The proposed method includes
an alternating least squares algorithm to iteratively estimate the
equivalent matrix, which consists of the transmitted signals and
the channels between the base station and RIS, as well as the
channels between the RIS and the multiple users. Our selective
simulation results show that the proposed scheme outperforms a
benchmark scheme that uses genie-aided information knowledge.
We also provide insights on the impact of different RIS parameter
settings on the proposed scheme.
Index Terms—Reconfigurable intelligent surfaces, PARAFAC,
message passing, MISO, channel estimation, signal recovery.
I. INTRODUCTION
Reconfigurable Intelligent Surfaces (RISs) are lately con-
sidered as a candidate technology for beyond fifth Gener-
ation (5G) wireless communication due to their potentially
significant benefits in low powered, energy-efficient, high-
speed, massive-connectivity, and low-latency communications
[1]–[9]. In its general form, a RIS is a reconfigurable planar
metasurface composed of a large number of hardware-efficient
passive reflecting elements [1], [10], [11]. Each unit element
can alter the phase of the incoming signal without requiring a
dedicated power amplifier, as for example needed in conven-
tional amplify-and forward relaying systems [1], [3], [11].
The energy efficiency potential of RIS in the downlink
of outdoor multi-user Multiple Input Single Output (MISO)
communications was analyzed in [3], while [12] focused on
an indoor scenario to illustrate the potential of RIS-based
indoor positioning. It was shown in [13], [14] that the latter
potential is large even in the case of RISs with finite resolution
unit elements and statistical channel knowledge. Recently, a
novel passive beamforming and information transfer technique
was proposed in [15] to enhance primary communication, as
well as a two-step approach at the receiver to retrieve the
information from both the transmitter and RIS. RIS-assisted
communication in the millimeter wave [16] and THz [17]
bands was also lately investigated. However, the joint channel
estimation and signal recovery problem in RIS-assisted sys-
tems has not been studied before. Most of the existing research
works in RIS-assisted multi-user communications only focus
on channel estimation.
The recent works [18] and [19] presented compressive sens-
ing and deep learning approaches for recovering the involved
channels and designing the RIS phase matrix. However, the
deep learning approaches require extensive training during
offline training phases, and the compressive sensing framework
assumes that RIS has some active elements, specifically, a fully
digital or hybrid analog and digital architecture attached at a
RIS portion. The latter architectures increase the RIS hardware
complexity and power consumption. A low-power receive
radio frequency chain for channel estimation was considered
in [5], [20], which also requires additional energy consumption
compared to passive RISs. Very recently, [21] and [22] adopted
PARAllel FACtor (PARAFAC) decomposition to estimate all
involved channels, however, the signal recovery probem was
not considered. In addition, some generalized approximate
message passing algorithms are proposed for joint estimation
and signal recovery, such as bilinear generalized approx-
imate message passing (BiG-AMP) [23], bilinear adaptive
generalized vector approximate message passing [24], and
generalized sparse Bayesian learning algorithm [25]. In [26],
the framework of expectation propagation was employed to
perform joint channel estimation and decoding in massive
multiple-input multiple-output systems with orthogonal fre-
quency division multiplexing.
In this paper, motivated by the PARAFAC decomposition
based channel estimation method [21], [22], we present an
efficient joint channel estimation and signal recovery technique
for the downlink of RIS-aided multi-user MISO systems.
Based on the PARAFAC decomposition, the proposed chan-
nel estimation method includes an Alternating Least Squares
(ALS) algorithm to iteratively estimate two unkown matrices
[27] [28]. With the estimated matrix, the signal and another
unknown channel can be estimated simultaneously using BiG-
AMP [23]. In such way, we can estimate all involved channels
and recover the transmitted signals simultaneously. Represen-
arXiv:2011.13116v1 [cs.IT] 26 Nov 2020
tative simulation results validate the accuracy of the proposed
scheme and its superiority over benchmark techniques.
Notation: Fonts a,a, and Arepresent scalars, vectors, and
matrices, respectively. AT,AH,A−1,A†, and kAkFdenote
transpose, Hermitian (conjugate transpose), inverse, pseudo-
inverse, and Frobenius norm of A, respectively. amn is the
(m, n)-th entry of A.|·| and (·)∗denote the modulus and
conjugation, respectively, while ◦represents the Khatri-Rao
matrix product. Finally, notation diag(a)represents a diagonal
matrix with the entries of aon its main diagonal.
II. SY ST EM A ND SI GNA L MOD EL S
In this section, we first describe the system and signal
models for the considered downlink RIS-assisted multi-user
MISO communication system, and then present the considered
PARAFAC decomposition for the end-to-end wireless commu-
nication channel.
A. System Model
We consider the downlink communication between a Base
Station (BS) equipped with Mantenna elements and Ksingle-
antenna mobile users. We assume that this communication is
realized via a discrete-element RIS deployed on the facade of
a building existing in the vicinity of the BS side, as illustrated
in Fig.1. The RIS is comprised of Nunit cells of equal
small size, each made from metamaterials capable of adjusting
their reflection coefficients. The direct signal path between
the BS and the mobile users is neglected due to unfavorable
propagation conditions, such as large obstacles. The received
discrete-time signals at all Kmobile users for Tconsecutive
time slots using the p-th RIS phase configuration (out of the
Pavailable in total, hence, p= 1,2, . . . , P) can be compactly
expressed with Yp∈CK×Tgiven by
Yp,HrDp(Φ)HsX+Wp,(1)
where Dp(Φ),diag(Φp,:)with Φp,:representing the p-th
row of the P×Ncomplex-valued matrix Φ. Each row of Φ
includes the phase configurations for the RIS unit elements,
which are usually chosen from low resolution discrete sets
[13]. Hs∈CN×Mand Hr∈CK×Ndenote, respectively,
the channel matrices between RIS and BS, and between all
users and RIS. Additionally, X∈CM×Tincludes the BS
transmitted signal within Ttime slots (it must hold T≥M
for channel estimation), and Wp∈CK×Tis the Additive
White Gaussian Noise (AWGN) matrix having zero mean and
unit variance elements.
B. Decomposition of the Received Training Signal
The channel matrices Hsand Hrin (1) are in general
unknown and need to be estimated. We hereinafter assume that
these matrices have independent and identically distributed
complex Gaussian entries; the entries between the matrices
are also assumed independent. Our objective in this work
is to continuously track Hs,Hr, and Xunder the known
observation signals Yand few training pilots. To achieve this
goal, we propose the following two-stage method. Firstly, we
Figure 1. The considered RIS-based multi-user MISO system consisting of
aM-antenna base station simultaneously serving in the downlink Ksingle-
antenna mobile users.
estimate the channel Hrand the equivalent matrix He,
HsX∈CN×Tusing in (1) and the PARAFAC decomposition,
according to which the intended channels are represented using
tensors [29], [30]. Then, the channel Hsand the transmitted
signal Xare simultaneously estimated using the BiG-AMP
algorithm [23]. Thus, each (k, t)-th entry of Zp, which is
the noiseless version of Yp, with k= 1,2, . . . , K and
t= 1,2, . . . , T is obtained as
[Zp]kt =
N
X
n=1
[Hr]kn[He]nt [Φ]pn,(2)
where [He]nt,[Hr]kn , and [Φ]pn denote the (n, t)-th entry of
He, the (k, n)-th entry of Hr, and the (p, n)-th entry of Φ,
respectively, with n= 1,2, . . . , N.
Resorting to the PARAFAC decomposition [31]–[34], each
matrix Zpin (2) out of the Pin total can be represented using
three matrix forms. These matrices form the horizontal, lateral,
and frontal slices of the tensor composed in (2). The unfolded
forms of the mode-1, mode-2, and mode-3 of Zp’s are given
in [21, eqs. (6)–(8)]. In the sequel, we present the proposed
joint channel estimation and signal recovery scheme for the
simultaneous estimation of the channel matrices Hrand Hs,
and the recovery of the transmitted signals X.
III. PROP OS E JOI NT CH AN NE L ESTIMATION
AN D SIG NAL RE COVERY
The proposed joint channel estimation and signal recov-
ery scheme, which is graphically summarized in Fig.2, is
composed of two main blocks. The first block deals with the
estimation of the channels Hsand He. Capitalizing on the
latter estimation, the second block simultaneously estimates
the channel Hrand recovers the transmitted signal X. The
details of the considered algorithms for these two blocks,
as well as the treatment of their possible ambiguities, are
presented in following subsections.
A. The First Block
In this block, the ALS-based channel estimation algorithm
of [21] is adopted. Specifically, the channels Hrand Heare
obtained in an iterative manner by alternatively minimizing
conditional Least Square (LS) criteria using the tensor Yfor
the received signal and the noiseless unfolded forms for all P
Figure 2. The proposed framework for the joint channel estimation and signal recovery in RIS-assisted multi-user MISO systems.
matrices Zp(similar to [21, eqs. (6)–(8)]). As shown in Fig.2
(a), two intermediate matrices A1=HeT◦Φ∈CP T ×Nand
A2=Φ◦Hr∈CKP ×Nare iteratively updated based on
the estimates for Hrand Heat each iteration. At the i-th
algorithmic iteration, the i-th estimation for Hr, denoted by
b
Hr
(i), is obtained from the minimization of the following LS
objective function:
Jb
Hr
(i)=
Y1−b
A(i−1)
1b
Hr
(i)T
2
F
,(3)
where b
A(i−1)
1=b
He
(i−1) ◦Φ. The solution of (3) is given by
b
Hr
(i)T
=b
A(i−1)
1†
Y1.(4)
Similarly, for the Heestimation, we use the mode-2 un-
folded form and formulate the following LS objective function
for the i-th estimation for He, which is denoted by b
He
(i):
Jb
He
(i)=
Y2−b
A(i)
2He
2
F,(5)
where Y2∈CKP ×Tis the mode-2 matrix-stacked form of
tensor Y, and b
A(i)
2=Φ◦b
Hr
(i). The solution of (5) is easily
obtained as b
He
(i)=b
A(i)
2†
Y2.(6)
The latter estimation method is summarized in Algorithm 1.
B. The Second Block
In this block, we jointly estimate Hsand Xgiven the esti-
mated matrix He=HsXand using the BiG-AMP algorithm
[23]. First, we derive the following posterior distribution:
pX,Hs|He(X,Hs|He)
=pHe|X,Hs(He|X,Hs)pX(X)pHs(Hs)/pHe(He)
∝pHe|Z(He|HsX)pX(X)pHs(Hs)
="N
Y
n=1
T
Y
t=1
phe
nt|znt he
nt |
M
X
k=1
hs
nkxk t!#
×"M
Y
m=1
T
Y
t=1
pxmt (xmt)#"N
Y
n=1
M
Y
m=1
phs
nm (hs
nm)#
(7)
Algorithm 1 Estimation of Channel Hrand He[21]
Input: A feasible Φ, > 0, and the number of maximum
algorithmic iterations Imax.
1: Initialization: Initialize with a random feasible phase ma-
trix Φand b
He
(0) obtained from the Nnon-zero eigenvalues
of YH
2Y2, and set algorithmic iteration i= 1.
2: for i= 1,2, . . . , Imax do
3: Obtain b
Hr
(i)using b
Hr
(i)=b
A(i−1)
1†
Y1T
.
4: Obtain b
He
(i)using b
He
(i)=b
A(i)
2†
Y2.
5: Until k
b
Hr
(i)−
b
Hr
(i−1)k2
F
k
b
Hr
(i)k2
F
≤, or i=Imax.
6: end for
Output: b
Hr
(i)and b
He
(i)that are the estimations of Hrand
He, respectively.
This joint distribution of Hsand Xgiven Hecan be rep-
resented with the factor graph in Fig. 2 (b). In this graph,
the factor nodes are represented with black boxes, while
the variable nodes are represented with white circles. The
estimated b
Hefrom the first block is treated as the parameters
of the pHe|Zfactor nodes. Applying the BiG-AMP algorithm,
the channel Hsand signal Xcan be estimated, as shown in
Algorithm 2.
In Algorithm 2, Steps 3and 4compute the ‘plug in’ estimate
of b
Heand its corresponding variances. Based on this estimate,
the Onsager correction is applied in Steps 5and 6. Afterwards,
Steps 7and 8calculate the marginal posterior means and
variances, respectively, of b
He. Considering that b
Heis the
output of the first stage (i.e., the output of Algorithm 1),
we assume the noise to be zero, thus the scaled residual
and corresponding variances are computed in Steps 9and
10 without the AWGN term. In the sequel, steps 11 and 12
compute the means and variances of the corrupted version of
the estimation for X, and then, Steps 13 and 14 calculate the
means and variances of the AWGN-corrupted observation of
the estimated channel Hs. Finally, based on the priors of X
and Hs, the posteriors of Xand Hsare computed in Steps
15 until 18, as follows:
pxmt|rmt (xmt |brmt(i); νr
mt(i))
∝pxmt (xmt)N(xmt ;brmt(i), νr
mt(i)) (8)
and phs
nm|qnm (hs
nm |bqnm(i); νq
nm(i))
∝phs
nm (hs
nm)N(hs
nm;bqnm (i), νq
nm(i)) (9)
The posterior probability (8) is used to compute the vari-
ances and means of xmt ∀m, t in Step 15 and 16. In fact,
the probability (8) can be interpreted as the exact poste-
rior probability of each xmt given the observation rmt un-
der the prior model pxmt (xmt)and the likelihood model
prmt|xmt (brmt(i)|xmt ;νr
mt(i)) = N(brmt(i); xmt , νr
mt(i)).
The latter model is assumed in each i-th iteration of the BiG-
AMP algorithm. Similarly, the posterior probability (9) is used
to compute the variances and means of hs
nm in Steps 15 and
16. It can be interpreted as the posterior probability of hs
nm
given the observation qnm under the prior model phs
nm (hs
nm)
and the likelihood model pqnm|hs
nm (bqnm(i)|hs
nm;νq
nm(i)) =
N(bqmt(i); hs
nm, ν q
nm(i)).
C. Ambiguity Removing
The proposed joint estimation and signal recovery scheme
involves ambiguities in the two blocks in Fig. 2. In the
first block, the iterative estimation of Heand Hr, using the
ALS approach included in Algorithm 1, encounters a scaling
ambiguity from the convergence point, which can be resolved
with adequate normalization [30], [35]. To tackle this issue,
the first column of Hrhas been normalized.
In the second block, phase and scaling ambiguities exist in
the estimation of Hsand Xusing Algorithm 2. To remove the
latter ambiguity, we assume initial pilot signal transmission in
X. For the phase ambiguity, we adopt the method in [36]. In
particular, the transmitted signal is designed to be a full-rank
matrix. It is generated from the Bernoulli Gaussian distribution
with sparsity β(βis the portion of nonzero elements in
transmitted signal), and sparsity pattern known to the receivers.
With the latter settings, the ambiguities can be removed from
the proposed scheme.
D. Computational complexity
In Algorithm 1, the computational complexities for the
estimations of Hsand Heare O(N3+ 4N2KP −N KP +
2NMKP −NM)and O(N3+ 4N2MP −NMP +
2NMKP −NK), respectively. Thus, using the feasibility
condition in the proposed PARAFAC-based structure [21], the
total computational complexity for this algorithm becomes
O(2N3+ 4N M KP ). The complexity of Algorithm 2 is
dominated by the matrix multiplications per algorithmic it-
eration. In Steps 3up to 5, as well as from Steps 11 until
14, each computation requires NMT multiplications, where
the computations within Steps 6and 10 are of the order
O(NM +M T ). Thus, Algorithm 2 requires O(NMT +
NM +M T )mathematical operations per iteration. Putting
all above together, the total computational complexity of the
Algorithm 2 The BiG-AMP Algorithm [23]
Input: b
He, > 0, and the number of maximum algorithmic
iterations Imax.
1: Initialization: Initialize ˆsnt (0) = 0, and choose
νx
mt(1),bxmt (1), νh
nm(1),b
hs
nm(1), n = 1, . . . , N, t =
1, . . . , T, m = 1, . . . , M .
2: for i= 1,2, . . . , Imax do
3: ∀n, t : ¯νp
nt(i) = PN
n=1 b
hs
nm(i)
2νx
mt(i) +
νh
nm(i)|bxmt (i)|2
4: ∀n, t : ¯pnt(i) = PN
n=1 b
hs
nm(i)bxmt (i)
5: ∀n, t :νp
nt(i) = ¯νp
nt(i) + PN
n=1 νh
nm(i)νx
mt(i)
6: bpnt(i) = ¯pnt(i)−bsnt(t−1)¯νp
nt(i)
7: ∀n, t :νz
nt(i) = var {znt |pnt =bpnt(i); νp
nt(i)}
8: ∀n, t :bznt(i) = E {znt |pnt =bpnt(i); νp
nt(i)}
9: ∀n, t :νs
nt(i) = (1 −νz
nt(i)/νp
nt(i)) /νp
nt(i)
10: ∀n, t :bsnt (i)=(bznt(i)−bpnt (i)) /νp
nt(i)
11: ∀m, t :νr
mt(i) = PM
m=1 b
hs
nm(i)
2νs
nt(i)−1
12:
∀m, t :brmt(i) = bxmt (i) 1−νr
mt(i)
M
X
m=1
νh
nm(i)νs
nt(i)
!
+νr
mt(i)
M
X
m=1 b
hs∗
nm(i)bsnt(i)
13: ∀n, m :νq
nm(i) = PL
l=1 |bxmt(i)|2νs
nt(i)−1
14:
∀n, m :bqnm (i) = b
hs
nm(i) 1−νq
nm(i)
L
X
l=1
νx
mt(i)νs
nt(i)
!
+νq
nm(i)
L
X
l=1 bx∗
mt(i)bsnt(i)
15: νx
mt(i+ 1) = var {xmt |rmt =brmt(i); νr
mt(i)},∀m, t
16: bxmt(i+ 1) = E {xmt |rmt =brmt(i); νr
mt(i)},∀m, t
17: νh
nm(i+1) = var {hs
nm |qnm =bqnm(i); νq
nm(i)},∀n, m
18: b
hs
nm(i+1) = E {hs
nm |qnm =bqnm(i); νq
nm(i)},∀n, m
19: Until Pm,l |¯pnt(i)−¯pnt(i−1)|2≤Pm,l |¯pnt (i)|2.
20: end for
Output: [b
Hs]nm =b
hs
nm and [b
X]mt =bxmt, respectively.
proposed joint channel estimation and signal recovery scheme
is O(2N3+ 4NMKP +NMT +NM +M T ).
IV. PERFORMANCE EVALUATION RESULTS
In this section, we present computer simulation results for
the performance evaluation of the proposed joint channel
estimation and signal recovery scheme. We have particularly
simulated the Normalized Mean Squared Error (NMSE) of
the proposed scheme using the metrics kHr−b
Hrk2kHrk−2,
kHs−b
Hsk2kHsk−2, and kX−b
Xk2kXk−2. The scaling
ambiguity in our two-stage estimation has been removed by
normalizing the first column of Hr, such that it has all one
elements. For Φ, we have considered the discrete Fourier
transform matrix, which satisfies ΦHΦ=INand has been
suggested as a good choice for ALS [30]. All NMSE curves
were obtained after averaging over 500 independent Monte
0 5 10 15 20 25 30
10-6
10-5
10-4
10-3
10-2
10-1
20
1.7
1.8
1.9
2
2.1 10-4
Figure 3. NMSE performance comparisons between the proposed algorithm,
LSKRF [22], and genie-aided LS estimation versus the SNR in dB for K=
32,N= 16,M= 12,N= 16,T= 100,P= 16, and β= 0.2.
Carlo channel realizations. We have used = 10−5and
Imax = 15 in all NMSE performance curves.
The performance evaluation of the proposed scheme versus
the Signal-to-Noise Ratio (SNR) is given in Figs.3 and 4.
Considering that the joint estimation and recovery in RIS-
assisted system has not been studied before, we adopt the
channel estimation methods: i) LS Khatri-Rao factorization
(LSKRF) [22]; and ii) conventional LS estimation, for the
estimation part in the benchmark schemes. In Fig.3, using
the setting K= 32,N= 16,M= 12,N= 16,T= 100,
P= 16, and β= 0.2, it is shown that proposed scheme
achieves similar performance with the one that adopts LSKRF
estimation. However, our scheme is more robust to parameter
setting changes compared to the one with LSKRF estimation.
As depicted in Fig.4, the parameter setting is changed to
K= 32,N= 20,M= 12,N= 16,T= 100,P= 16,
and β= 0.2. It is observed that the estimation performance
of the LSKRF scheme gets dramatically deteriorated with
slight changes of N, witnessing its increased sensitivity in
the parameter setting. More details for solely the channel
estimation comparison between our scheme and LSKRF can
be found in [21]. In addition, the illustrated abnormality in the
estimation of the signal Xhappens due to its high sparsity
and pilots. It is shown in Figs.3 and 4 that the proposed
scheme has only about 2.5dB performance gap from the LS
scheme that assumes perfect channel knowledge. This behavior
substantiates the robustness and favorable performance of our
proposed scheme.
In Fig.5, we have used the simulation parameters K= 16,
N= 32,T= 100,P= 16,β= 0.2, and various values
of Mto illustrate the NMSE performance of the proposed
scheme as a function of the SNR. It is evident that there
exists an increasing performance loss when Mincreases. In
those cases, the numbers of unknown variables for estimation
increase, which result in performance loss. The impact of the
sparsity level βin the NMSE performance of the proposed
scheme is investigated in Fig.6 as a function of the SNR
for K= 20,N= 32,M= 16,T= 100, and P= 16.
It is shown that there exists a tradeoff between the NMSE
0 5 10 15 20 25 30
10-6
10-5
10-4
10-3
10-2
10-1
20
1.2
1.4
1.6
1.8 10-4
Figure 4. NMSE performance comparisons between the proposed algorithm,
LSKRF [22], and genie-aided LS estimation versus the SNR in dB for K=
32,N= 20,M= 12,N= 16,T= 100,P= 16, and β= 0.2.
0 5 10 15 20 25 30
10-5
10-4
10-3
10-2
10-1
Figure 5. NMSE performance of the proposed algorithm versus the SNR in
dB for K= 16,N= 32,T= 100,P= 16,β= 0.2, and various values
Mfor the BS antennas.
performance and β. The scheme for β= 0.5suffers the worst
performance. This happens because more non-zero variables
need to be estimated with increasing β, which makes the BiG-
AMP algorithm to performs worse. It can be also seen that the
performance for β= 0.1is worse than for β= 0.3(of around
3dB), even that less unknown variables are involved. This
behavior accounts for the possible sparsity pattern collision.
Actually, when multiple users are located close to each other,
recovery failures happen [37].
V. CONCLUSION
In this paper, we presented a joint channel estimation and
signal recovery scheme for RIS-assisted multi-user MISO
communication systems, which capitalizes on the PARAFAC
decomposition of the received signal model. All involved
channels are estimated and the transmitted signal is recovered
using a two-stage approach. In the first stage, the channel
between RIS and the users and another equivalent channel
are estimated using ALS. Then, in the second stage using the
equivalent channel estimation, the channel between the BS and
RIS and the transmitted signals are recovered via the BiG-
AMP algorithm. Our simulation results showcased that the
proposed joint channel estimation and signal recovery scheme
0 5 10 15 20 25 30
10-6
10-5
10-4
10-3
10-2
10-1
Figure 6. NMSE performance of the proposed algorithm versus the SNR in
dB for K= 20,N= 32,M= 16,T= 100,P= 16, and various values
βfor the sparsity parameter.
outperforms various benchmarks. In addition, we observed that
the number of BS antennas Mand the sparsity level βexert
a significant effect on our proposed algorithm.
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