On the Performance of Reconfigurable Intelligent Surface-Aided Cell-Free Massive MIMO Uplink

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On the Performance of Reconfigurable Intelligent
Surface-Aided Cell-Free Massive MIMO Uplink
Manijeh Bashar∗
, Kanapathippillai Cumanan†
, Alister G. Burr†
, Pei Xiao∗
, and Marco Di Renzo‡
∗University of Surrey, UK., †University of York, UK, ‡CNRS and Paris-Saclay University, France
{m.bashar,p.xiao}@surrey.ac.uk, {kanapathippillai.cumanan,alister.burr}@york.ac.uk, marco.direnzo@centralesupelec.fr
Abstract—The uplink of a reconfigurable intelligent sur-
faces (RIS)-aided cell-free massive multiple-input multiple-output
(MIMO) system is analyzed, where the channel state information
(CSI) is estimated using uplink pilots. First, we derive analytical
expressions for the achievable rate of the system with zero
forcing (ZF) receiver, taking into account the effects of pilot
contamination, channel estimation error and the distributed
RISs. The max-min rate optimization problem is considered with
per-user power constraints. To solve this non-convex problem,
we propose to decouple the original optimization problem into
two sub-problems, namely, phase shift design problem and power
allocation problem. The power allocation problem is solved using
a standard geometric programming (GP) whereas a semidefinite
programming (SDP) is utilized to design the phase shifts. More-
over, the Taylor series approximation is used to convert the non-
convex constraints into a convex form. An iterative algorithm is
proposed whereby at each iteration, one of the sub-problems is
solved while the other design variable is fixed. The max-min user
rate of the RIS-aided cell-free massive MIMO system is compared
to that of conventional cell-free massive MIMO. Numerical results
indicate the superiority of the proposed algorithm compared
with a conventional cell-free massive MIMO system. Finally, the
convergence of the proposed algorithm is investigated.
I. INTRODUCTION
Reconfigurable intelligent surfaces (RISs) are capable of
significantly increasing the spectral efficiency, hence contribut-
ing to meet the demanding requirements in fifth generation
(5G) and beyond systems [1], [2]. An RIS with many reflective
elements is designed to beamform and to change the phase
shifts of each element to steer the reflected beam in different
directions. The authors of [3] explain that there are two
main categories of RISs: one based on inexpensive antennas
and another one based on metasurfaces. On the other hand,
cell-free massive multiple-input multiple-output (MIMO) is a
promising technology, where a large number of access points
(APs) are distributed to serve a much smaller number of users
[4], [5]. A cell-free massive MIMO is considered, where the
APs send the received signals and the channel estimates to a
central processing unit (CPU) through fronthaul links [6]–[8].
A practical combination of the aforementioned techniques is
considered to define an RIS-aided cell-free massive MIMO. In
the literature, the effect of RIS in collocated massive MIMO
systems has been investigated [9]. However, in this paper we
exploit distributed massive MIMO.
The work of K. Cumanan and A. G. Burr was supported by H2020-MSCA-
RISE-2015 under grant number 690750.
This work of P. Xiao was supported by the U.K. Engineering and Physical
Sciences Research Council under Grant EP/P008402/2 and EP/R001588/1
The work of M. Di Renzo was supported in part by the European
Commission through the H2020 ARIADNE project under grant agreement
675806.
Figure 1. The uplink of a cell-free massive MIMO system with Ksingle-
antenna users and MAPs. There are TRISs, each equipped with Ltpassive
elements, acting as low-resolution phase shifters. The orange line (named by
gmtlk) demonstrates the cascaded channel from the kth user to the mth AP
through the lth element in the Tth surface.
The contributions of the paper are summarized as follows:
(i) A closed-form expression of the signal-to-interference
plus noise ratio (SINR) of the considered RIS-aided cell-
free massive MIMO system is derived using the zero forcing
(ZF) detector and taking into account the effects of pilot
contamination and channel estimation errors; (ii) A max-min
fairness power control problem is formulated to maximize the
smallest rate of all users under per-user power constraints.
In particular, we propose a novel approach to solve the uplink
max-min rate optimization problem by decoupling the original
problem into two sub-problems, which are solved using an
iterative algorithm. The sub-problems are formulated as a
geometric programming (GP) and a semidefinite programming
(SDP), and both sub-problems are solved at each iteration.
The Taylor series approximation is exploited to transform
the non-convex constraints into tractable convex one; (iii)
Numerical results are provided to validate the convergence of
the proposed scheme.
II. SY ST EM MO DE L
We consider the uplink transmission in a cell-free massive
MIMO system with MAPs and Ksingle-antenna users
randomly distributed in a large area, as shown in Fig. 1.
Each AP has Nantennas. We assume there are TRISs
each equipped with Ltpassive elements [9]. Similar to [9],
we assume that the elements behave as low resolution phase
shifters, attached to the facade of buildings. The authors of
[9] considered a massive MIMO base station (BS) and an RIS
attached to the facade of a building in the line-of-sight (LoS)
of the BS. In this work we assume a large number of APs
distributed in the area. By exploiting the analysis in [10], we

model the channel using a Ricean distribution which includes
a dominant LoS component as well as diffuse scattering. The
total cascaded link between the kth user and the mth AP
through the tth RIS is given by
gRIS, cascaded
mtlk =gmtl exp(jθtl )gtlk ,(1)
where gmtl ∈CN×1denotes the channels from the lth element
in the tthe RIS to the mth AP whereas gtkl is the channel from
the kth user to the lth element in the tth RIS. Moreover, θt=
diag[exp(jθt1),...,exp(jθtLt)] denotes the tth RIS response
where θtl ∈[0,2π]is the phase coefficient of the lth element
in the tth RIS [11]. The uplink direct channel between the
mth AP and the kth user is given by gmk. The direct channel
from the mth AP to the kth user is modeled as
gmk =pζmk rκmk
κmk + 1 ¯
hmk +r1
κmk + 1 ˜
hmk
=√αmk ¯
hmk +pβmk ˜
hmk =¯
gmk +˜
gmk,(2)
where ζmk denotes the large-scale fading coefficient, κmk
is the Ricean K-factor, ¯
hmk and ˜
hmk refer to the LoS and
non-LoS components, respectively. We assume that ˜
hmk ∼
CN(0,IN)and ¯
hmk =e2π
λ(N−1)jsin(ψmk), where ψmk ∈
[−π, π]is the angle of arrival. We consider scenarios where
users are fixed or move slowly, but there are movements of
objects around them. In these scenarios, it is reasonable to
assume that ¯
gmk changes slowly with time, and is known a
priori. In addition, βmk is also assumed to be known a priori.
To model this, we use the following formulation as in [10]:
κmk =PLoS(dmk )
1 + PLoS(dmk ),(3)
where dmk is the distance between the mth AP and the
kth user, PLoS(dmk)is the LoS probability depending on the
distance dmk. For the LoS probability, we use the model from
the 3GPP-UMa as [10]
PLoS(dmk ) = min 18
dmk
,11−e−dmk
63 +e−dmk
63 ,(4)
where dmk is in meters. We use a similar technique to model
the LoS component between the APs and the RISs and the
RISs and the users. The cascaded uplink channel from the kth
user to the mth AP through the lth element in the tth RIS is
given as follows:
gmtlk =pζmtkhmtl htlk
=pζmtk rκmt
κmt + 1 ¯
hmtl +r1
κmt + 1 ˜
hmtl
rκtk
κtk + 1 ¯
htlk +r1
κtk + 1 ˜
htlk
=pζmtkamt atk ¯
hmtl¯
htlk
| {z }
¯
gmktl (known)
+
pζmtk
bmtbtk ˜
hmtl˜
htlk+amtbtk ¯
hmtl˜
htlk+bmtatk ˜
hmtl¯
htlk

|{z }
˜
gmktl (unknown)
=¯
gmktl +˜
gmktl,(5)
where √ζmtk is the large-scale fading coefficient, κmt and
κmt are the Ricean K-factors, ¯
hmtl and ˜
hmtl refer to the
LoS and non-LoS components, respectively. We assume that
˜
hmtl ∼ CN(0,IN)and ¯
hmtl =e2π
λ(N−1)jsin(ψmt), where
ψmt ∈[−π, π]is the angle of arrival. Moreover, κtk is the
Ricean K-factor, ¯
htlk and ˜
htlk refer to the LoS and non-
LoS components, respectively. In addition, we have amt =
qκmt
κmt+1 ,bmt =q1
κmt+1 ,atk =qκtk
κtk+1 , and btk =
q1
κtk+1 . We assume that ˜
hmtl ∼ CN (0,1) and ¯
hmtl =ejψtk ,
where ψtk is the angle of arrival. Note that using the assump-
tion that the LoS component changes slowly and is known a
priori, the first term in (5), i.e., ¯
gmktl, is known. The Ricean
K–factors and large–scale fading coefficients vary depending
on the locations of users and APs. Also, we have
βmtk=En|[˜
gmtlk]n|2
o=ζmtkb2
mtb2
tk +a2
mtb2
tk +b2
mta2
tk,(6)
and αmtk =En|[¯
gmtlk]n|2o=ζmtk a2
mta2
tk.
A. Uplink Channel Estimation
We use the time-division duplex (TDD) transmission pro-
tocol, where each AP estimates the channels of all users.
Given the fact that the RISs are passive [9], the APs need
to estimate all the channels. Throughout the uplink training
phase, the users transmit uplink pilot symbols. Similar to [9],
the channel estimation time is divided into PT
t=1 Lt+ 1 sub-
phases. During the first sub-phase, all elements at all RISs
are OFF and the APs estimate the direct channel gmk for
all users. During the (isub−p+ 1)th sub-phase, where in each
sub-phase isub−p= 11, . . ., 1L1,21, . . ., 2L2, . . ., T 1, . . ., T LT
only one element of one RIS is ON to assist all APs to estimate
gmtlk,∀k , m, for all users, while all other RIS elements are
OFF.
1) Step 1: In the first phase, all elements are OFF and each
AP estimates the direct channels between itself and all users.
In order to estimate the channel coefficients in the uplink, the
APs employ a minimum mean-square error (MMSE) estimator.
During the training phase, all Kusers simultaneously transmit
their pilot sequences of length τsymbols to the APs. Let
√τφ
φ
φk∈Cτ×1, where kφ
φ
φkk2= 1, be the pilot sequence
assigned to the kth user. Then, the received signal at the mth
AP is given by
Yp
m=√τpp
K
X
k=1
gmkφ
φ
φH
k+Wp
m,(7)
where the matrix Wp
m∈CN×τis the noise whose elements
are i.i.d. CN(0,1). Moreover, pp=¯pp
pnis the normalized
transmit signal-to-noise ratio (SNR) of each pilot symbol,
where ¯ppand pnare to the transmit pilot power and the noise
power, respectively. The APs exploit the pilot sequence φ
φ
φkto
correlate the received signal with the pilot sequence as follows
[4]:
ˇ
yp
m,k=Yp
mφ
φ
φk=√τppgmk+√τ pp
K
X
k06=k
gmk0φ
φ
φH
k0φ
φ
φk+˙
wp
mk,(8)
where ˙
wp
mk ,Wp
mφ
φ
φH
k. The linear MMSE estimate of gmk is

ˆ
gmk=cmk
√τppgmk +√τ pp
K
X
k06=k
gmk0φ
φ
φH
k0φ
φ
φk+˙
wp
mk

,(9)
where the expectations are taken over small-scale fading and
noise and cmk =√τ ppβmk
τ ppPK
k0=1 βmk0|φ
φ
φH
k0φ
φ
φk|2+1 [4]. The estimated
channels in (9) are used by the APs to design the receiver
filter coefficients and to determine the power allocations at
the users to maximize the minimum rate of the users. The
power on the nth component of gmk is defined as γmk =
√τppβmkcmk. In this work, we investigate the cases of both
random and orthogonal pilot assignments in cell-free massive
MIMO. The term “orthogonal pilots” refers to the case where
unique orthogonal pilots are assigned to all users. In the case
of orthogonal pilots, the length of pilots is τ≥K. For the
case with “random pilot assignment”, we have τ < K. Hence,
similar to the scheme in [4], each user is randomly assigned
a pilot sequence from a set of orthogonal sequences of length
τ(< K).
2) Step tl + 1:In step tl + 1, the lth element at the tth RIS
is ON, while all the other elements are OFF. The received pilot
at the mth AP is given by
Yp,tot
mtl =√τpp
K
X
k=1
gmtlkφ
φ
φH
k+√τpp
K
X
k=1
gmkφ
φ
φH
k+Wp,tot
mtl .(10)
By using (7) and (10), we have
Yp
mtl =Yp,tot
mtl −Yp
m=√τpp
K
X
k=1
gmtlkφ
φ
φH
k+Wp
mtl,(11)
The linear MMSE estimate of gmtlk is
ˆ
gmtlk=cmtk
√τppgmtlk +√τ pp
K
X
k06=k
gmtlk0φ
φ
φH
k0φ
φ
φk+˙
wp
mtlk
,(12)
where the expectations are taken over small-scale fading and
noise and cmtk =√τ ppβmtk
τ ppPK
k0=1 βmtk0|φ
φ
φH
k0φ
φ
φk|2+1 . The power on the
nth component of gmtlk is defined as γmtk =√τ ppβmtk cmtk.
B. Uplink Transmission
In this subsection, we consider the uplink data transmission,
where all users send their signals to the APs. The transmitted
signal from the kth user is given by
xk=√ρ qksk,(13)
where sk(E{|sk|2}= 1) and qkare the transmitted symbol
and the transmit power, respectively. Moreover, ρdenotes the
normalized uplink SNR. The N×1received signal at the mth
AP is given by
ym=√ρ
K
X
k=1
T+1
X
t=1
Lt
X
l=1
gmktlej θtl √qksk+nm,(14)
where nm∼ CN(0,IN)is the noise vector at the mth AP.
The mth AP forwards the estimate of the channel coefficients
and the received signals to the CPU through error-free and
perfect fronthaul links. The received signal for the kth user
after using the ZF detector at the CPU is given by
rk=
M
X
m=1
T+1
X
t=1
Lt
X
l=1
ˆ
uH
mktlym
=√ρqk
M
X
m=1
T+1
X
t=1
Lt
X
l=1
ˆ
uH
mktl
T+1
X
t0=1
Lt0
X
l0=1
ˆ
gmk0t0l0ejθt0l0sk
+√ρ
K
X
k06=k
√qk0
M
X
m=1
T+1
X
t=1
Lt
X
l=1
ˆ
uH
mktl
T+1
X
t0=1
Lt0
X
l0=1
ˆ
gmk0t0l0ejθt0l0sk0
+√ρ
K
X
k0=1
√qk0
M
X
m=1
T+1
X
t=1
Lt
X
l=1
ˆ
uH
mktl
T+1
X
t0=1
Lt0
X
l0=1
˜
gmk0t0l0ejθt0l0sk0
+
M
X
m=1
ˆ
uH
mknm,(15)
where ˜
gmktl is the channel estimation error, and ˆ
U=
ˆ
Gˆ
GHˆ
G−1
, where ˆ
U= [ˆ
u1,...,ˆ
uK]and ˆ
uk=
[ˆ
uT
1k,...,ˆ
uT
Mk ]T. The SINR of the kth user in the RIS-aided
cell-free massive MIMO system is given by (16) (defined at
the top of the next page), where θtl is the phase shift of
the lth element in the tth surface, ˆ
fmk =
T+1
P
t=1
Lt
P
l=1
ˆ
umktl and
ˆ
gmk =hˆ
gT
mk11,...,ˆ
gT
mkT LT+1 iT
, where ˆ
gmktl ∈CN×1.
We consider Tsurfaces, each equipped with Ltelements.
Moreover, ˆ
gmk(T+1)LT+1 denotes the direct path from the mth
AP to the kth user. Note that in (16), we have
M
X
m=1
ˆ
fH
mk
T+1
X
t0=1
Lt0
X
l0=1
ˆ
gmkt0l0
ejθt0l0=
T+1
X
t0=1
Lt0
X
l0=1
wkl0t0ejθt0l0
=vwk,(17)
where v= [v1,...,vT+1]and vt=
[exp (jθt1),...,exp (jθtLt)] denotes the response
of the lth element in the tth RIS, and we set
vT+1 = [1,...,1]. In addition, wk∈C(T+1)Lt×1,
where wkl0t0=PM
m=1 ˆ
fH
mk ˆ
gmkt0l0, and ∆k=wkwH
k, where
∆k0=wk0wH
k0. Finally, we obtain the achievable rate as
ratek= log2(1 + SINRk).(18)
III. PROP OS ED MA X-MI N RATE SCHEME
In this section, we formulate the max-min rate optimization
problem in the RIS-aided cell-free massive MIMO system,
where the minimum uplink rate of all users is maximized
while satisfying the per-user power constraints. The max-min
rate optimization problem can be formulated as the following
optimization framework:
P1: max
qk,vmin
k=1,...,K ratek
s.t. |vtl|= 1,∀t, l,
0≤qk≤p(k)
max,∀k,
(19a)
(19b)
(19c)

SINRk=qkv∆kvH
PK
k06=kqk0v∆k0vH+PK
k0=1 qk0PM
m=1 

ˆ
fmk


2PT+1
t0=1 Lt0(βmk0t0−γmk0t0) + 1
ρPM
m=1 

ˆ
fmk


2.(16)
P4: max
$,v$
s.t. qkTr [∆kV]
K
P
k06=k
qk0Tr [∆k0V]+
K
P
k0=1
qk0
M
P
m=1 

ˆ
fmk


2T+1
P
t0=1
Lt0(βmk0t0−γmk0t0)+ 1
ρ
M
P
m=1

ˆ
fmk


2≥$, ∀k,
Diag [V] = 1,V=VH,V0,rank [V]=1.
(22a)
(22b)
(22c)
where p(k)
max is the maximum transmit power available at user
k. It is not possible to jointly solve Problem P1in terms of
vand qk. Therefore, this problem cannot be directly solved
through existing convex optimization software. To tackle this
non-convexity issue, we decouple the original Problem P1
into two sub-problems: phase shift design problem and the
power allocation problem. To obtain a solution, these sub-
problems are alternately solved as explained in the following
subsections.
A. Power Allocation
In this subsection, we solve the power allocation problem
for a given set of fixed receiver filter coefficients which can be
formulated as the following max-min optimization problem:
P2: max
qk
min
k=1,...,K SINRk
s.t. 0≤qk≤p(k)
max.
(20a)
(20b)
Proposition 1. P2can be formulated into a standard GP.
B. Phase Shift Design
In this subsection, we solve the phase shift design problem
to maximize the uplink rate of each user for a given set of
transmit power allocations at all users. We define the phase
shift optimization problem as follows:
P3: max
vmin
k=1,...,K SINRk
s.t. |vtl|= 1,∀t, l.
(21a)
(21b)
The phase shift design Problem P3can be rewritten by
introducing a new slack variable as P4(defined at the top
of this page), where V=vHv, and we have v∆kvH=
Tr ∆kvHv=Tr [∆kV]. Next, the phase shift design
Problem P3can be rewritten by introducing a new slack
variable as
P5: max
$,v,ηk
$
s.t. qkTr [∆kV]
ηk≥$, ∀k,
K
X
k06=k
qk0Tr [∆k0V]+
K
X
k0=1
qk0
M
X
m=1||ˆ
fmk||2
T+1
X
t0=1
Lt0(βmk0t0−γmk0t0)+1
ρ
M
X
m=1||ˆ
fmk||2≤ηk,∀k,
Diag [V] = 1,V=VH,V0,rank [V] = 1,
(23a)
(23b)
(23c)
(23d)
Algorithm 1 Proposed algorithm to solve Problem P1
1. Initialize q(0) = [q(0)
1, q(0)
2, . . . , q(0)
K],n= 1
2. Repeat steps 3-5 until SINR(n)
k−SINR(n−1)
k
SINR(n−1)
k≤, ∀k
3. Determine the optimal phase shifts V(i)through solving
Problem P7for a given q(n−1),
4. Compute q(n)through solving Problem P2for a given V(n)
5. n=n+ 1
which is equivalent to the following problem:
P6: max
$,v,ηk
$
s.t. qkTr [∆kV]≥ηk$, ∀k,
K
X
k06=k
qk0Tr [∆k0V]+
K
X
k0=1
qk0
M
X
m=1||ˆ
fmk||2
T+1
X
t0=1
Lt0(βmk0t0−γmk0t0)+1
ρ
M
X
m=1||ˆ
fmk||2≤ηk,∀k,
Diag [V] = 1,V=VH,V0,rank [V]=1.
(24a)
(24b)
(24c)
(24d)
Using the second order Taylor series approximation, we have:
ηk$≈$(i−1)
η(i−1)
k+η(i−1)
k$−$(i−1)+$(i−1)
ηk−η(i−1)
k.(25)
Hence, we have
P7: max
$,v,ηk
$
s.t. qkTr [∆k
V]≥$(i−1)η(i−1)
k+η(i−1)
k
$−$(i−1)
+$(i−1) ηk−η(i−1)
k,∀k,
K
X
k06=k
qk0Tr [∆k0V]+
K
X
k0=1
qk0
M
X
m=1||ˆ
fmk||2
T+1
X
t0=1
Lt0(βmk0t0−γmk0t0)+1
ρ
M
X
m=1||ˆ
fmk||2≤ηk,∀k,
Diag [V] = 1,V=VH,V0,rank [V]=1.
(26a)
(26b)
(26c)
(26d)
The rank one constraint, i.e., rank [V] = 1, is not convex.
However, it is easy to show that Problem P7is a standard SDP
without the non-convex constraint rank[V]=1. To handle the
non-convexity, we define the relaxed Problem P8by removing
the rank-one constraint as follows:

P8: max
$,v,ηk
$
s.t. qkTr [∆k
V]≥$(i−1)η(i−1)
k+η(i−1)
k
$−$(i−1)+$(i−1) ηk−η(i−1)
k,∀k,
K
X
k06=k
qk0Tr [∆k0V]+
K
X
k0=1
qk0
M
X
m=1||ˆ
fmk||2
T+1
X
t0=1
Lt0(βmk0t0−γmk0t0)+1
ρ
M
X
m=1||ˆ
fmk||2≤ηk,∀k,
Diag [V] = 1,V=VH,V0.
(27a)
(27b)
(27c)
(27d)
The relaxed Problem P8is a SDP and can be solved via the
existing convex optimization software. However, note that the
relaxed Problem P8may not yield a rank one solution. This
implies that the SDP relaxation is not tight [12]. Therefore,
we exploit the additional steps to construct a rank-one solution
from the optimal non-rank-one solution to the relaxed Problem
P8. We first employ the eigenvalue decomposition as follows
V= ΥΣΥH.Next, to obtain the sub-optimal solution to
Problem P7, we generate the following vector:
¯
v= ΥΣ1
2r,(28)
where r∈CPT
t=1 Lt×1.
Remark 1. With independently generated Gaussian random
vector r, the objective value of Problem P7is approximated
as the maximum one obtained by the best ¯
vamong all r’s
[12].
Finally, the sub-optimal solution vto Problem P7is recov-
ered as follows:
v=exp jarg  ¯
v
¯
vT L .(29)
It has been shown that such a semidefinite relaxation (SDR)
approach followed by sufficiently large number of randomiza-
tions of r(i.e., Nrelax
randomizations) guarantees an π
4-approximation
of the optimal solution of Problem P7[12]. We propose
an algorithm to iteratively solve sub-Problems P2and P7.
Note that Problem P8is solved iteratively. In problem P8,
we set i= 1, . . . , NTaylor, where NTaylor is a pre-determined
value referring to the total number of iterations in Problem
P7. To have a set of initial feasible parameters, we need to
formulate a problem to examine the feasibility. For simplicity,
here we use VIni =1, where 1refers to a matrix with size
(PT
t=1 Lt+ 1) ×(PT
t=1 Lt+ 1) with all one elements and
refers to the case with θtl = 0,∀t, l. Finally, Problem P1
is efficiently solved through the existing convex optimization
software. Based on sub-Problems P2and P7, an iterative
algorithm has been developed by alternately solving both sub-
problems at each iteration, which is summarized in Algorithm
1.
IV. NUMERICAL RES ULT S AN D DISCUSSION
In this section, we provide numerical results to validate
the performance of the proposed max-min rate scheme with
different parameters. A cell-free massive MIMO system with
MAPs and Ksingle-antenna users is considered in a D×D
simulation area, where all APs, users and RISs are uniformly
located at random points. In the following subsections, we
define the simulation parameters and then present the corre-
sponding simulation results.
A. Large-Scale Fading Model
The channel coefficients between users and APs are mod-
eled in (2), where the coefficient ζmk is given by ζmk =
PLmk10 σsh zmk
10 ,where PLmk is the path-loss from the kth
user to the mth AP and 10σshzmk
10 , denotes the shadow fading
with standard deviation σsh = 8 dB, and zmk ∼ N(0,1). In
the simulation, a three-slope COST Hata model for the path-
loss is given by [4]
PLmk=


−L−35 log10(dmk ), dmk > d1,
−L−15log10(d1)−20 log10(dmk ), d0<dmk≤d1,
−L−15 log10(d1)−20 log10
(d0), dmk ≤d0,
(30)
and Lis defined in [10]. It is assumed that that ¯ppand
¯ρdenote the pilot sequence and the uplink data powers,
respectively, where pp=¯pp
pnand ρ=¯ρ
pn. In simulations,
we set ¯pp= 200 mW and ¯ρ= 200 mW. Similar to [4],
we assume that the simulation area is wrapped around at the
edges which can simulate an area without boundaries. Hence,
the square simulation area has eight neighbours. We evaluate
the average rate of the system over 100 random realizations of
the locations of APs, users and shadow fading. We model the
large-scale fading coefficient between the mth AP, the tth RIS
and the kth user as ζmtk. Using the analysis for far-field in
[13], the path-loss for the cascaded link is modeled as follows:
ζmtk =ζmtζtk ,(31)
where ζmt and ζtk the path-loss from the AP to the tth
RIS and the path-loss from the tth RIS to the kth user,
respectively. Note that we have ζmt =PLmt10
σsh zmt
10 ,and
ζtk =PLtk10
σsh ztk
10 ,where the path-loss PLmt is given by
PLmt=



−L−22 log10(dmt ), dmt > d1,
−L−15log10(d1)−20 log10(dmt), d0<dmt≤d1
,
−L−15 log10(d1)−20 log10(d0), dmt ≤d0.
(32)
Moreover, we have
PLtk=


−L−35 log10(dtk ), dtk > d1,
−L−15log10(d1)−20 log10(dtk ), d0<dtk≤d1
,
−L−15 log10(d1)−20 log10(d0), dtk ≤d0,
(33)
where dmt and dtk denote the distances from the AP to the
tth RIS and from the tth RIS to the kth user, respectively.
B. Simulation Results
In this subsection, we investigate the effect of the max-min
rate problem on the system performance. First, we consider an
RIS-based cell-free massive MIMO system with 1 RIS (T=
1), each equipped with L= 20 elements, 30 APs (M= 20),
each equipped with 4 antennas (N= 4), and 20 users (K=
20), who are randomly distributed over the area of size D×D
km2, where D= 0.4km. Moreover, we consider τ= 10 as

0.2 0.4 0.6 0.8 1 1.2
Min-user uplink rate (bits/s/Hz)
0
0.2
0.4
0.6
0.8
1
Cumulatice distribution
Without RIS
Ref. [4]
Proposed
Algorithm 1
Figure 2. The cumulative distribution of the per-user uplink rate, with M=
20,N= 4,K= 30, and τ= 10.
0.1 0.2 0.3 0.4 0.5 0.6
Min-user uplink rate (bits/s/Hz)
0
0.2
0.4
0.6
0.8
1
Cumulatice distribution
Without RIS
Ref. [4]
Proposed
Algorithm 1
Figure 3. The cumulative distribution of the per-user uplink rate, with M=
20,N= 4,K= 40, and τ= 10.
12345
Number of iterations
0.55
0.6
0.65
0.7
0.75
0.8
Min-user uplink rate (bits/s/Hz)
Channel 1
Channel 2
Figure 4. The convergence of the proposed max-min SINR approach for
M= 20,N= 4,K= 30, and τ= 10.
the length of pilot sequences. Fig. 2 compares the cumulative
distribution of the achievable uplink rates for our proposed
algorithm with the system without the RISs, i.e., the scheme
in [4]. The figure shows that adding the RISs can significantly
increase the performance of the system. In Fig. 3, we compare
the performance of the proposed max-min rate approach with
that of the scheme in [4] for the case with 1 RIS (T= 1), each
equipped with L= 20 elements, M= 20, each equipped with
1 antennas (N= 4), and serving 10 users (K= 40), D= 0.4
km and τ= 10 as the length of the pilot sequence.
Fig. 3 shows the superiority of the proposed iterative
algorithm over the system without the RISs as in [4]. Fig.
4 investigates the convergence of the proposed max-min rate
algorithm for a set of different channel realizations. The figure
demonstrates that the proposed algorithm converges after a few
iterations.
V. CONCLUSIONS
We have investigated the performance of the RIS-aided cell-
free massive MIMO system relying with the ZF receiver. A
closed-form expression was derived for the achievable rate of
the system. Next, a novel optimization scheme was developed
through formulating the max-min optimization problems as
standard GP and SDP, which efficiently solve the max-min rate
optimization problem. The simulation results demonstrated the
effectiveness of the proposed scheme in terms of maximis-
ing the minimum rate of the users compared with existing
schemes.
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