RIS and Cell-Free Massive MIMO: A Marriage For Harsh Propagation Environments - arXiv
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RIS and Cell-Free Massive MIMO: A Marriage For Harsh
Propagation Environments
Trinh Van Chien∗ , Hien Quoc Ngo† , Symeon Chatzinotas∗ , Marco Di Renzoξ , and Björn Ottersten∗
∗
Interdisciplinary Centre for Security, Reliability and Trust (SnT), University of Luxembourg, Luxembourg
†
School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast, UK
ξ
Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes, France
Abstract—This paper considers Cell-Free Massive Multiple power amplification. Integrating an RIS into wireless networks
Input Multiple Output (MIMO) systems with the assistance introduces digitally controllable links that scale up with the
of an RIS for enhancing the system performance. Distributed number of engineered scattering elements of the RIS, whose
maximum-ratio combining (MRC) is considered at the access
arXiv:2109.05444v2 [cs.IT] 16 Sep 2021
points (APs). We introduce an aggregated channel estimation estimation is, however, challenged by the lack of digital signal
method that provides sufficient information for data processing. processing units at the RIS. For simplicity, the main attention
The considered system is studied by using asymptotic analysis has so far been concentrated on designing the phase shifts
which lets the number of APs and/or the number of RIS elements with perfect channel state information (CSI) [4], [5] and
grow large. A lower bound for the channel capacity is obtained the references therein. As far as the integration of Cell-Free
for a finite number of APs and engineered scattering elements
of the RIS, and closed-form expression for the uplink ergodic Massive MIMO and RIS is concerned, recent works have
net throughput is formulated. In addition, a simple scheme formulated and solved optimization problems with different
for controlling the configuration of the RIS scattering elements communication objectives under the assumption of perfect
is proposed. Numerical results verify the effectiveness of the (and instantaneous) CSI [6], [7]. Recent results have shown
proposed system design and the benefits of using RISs in Cell- that designs for the phase shifts of the RIS elements based on
Free Massive MIMO systems are quantified.
statistical CSI may be of practical interest and provide good
I. I NTRODUCTION performance [8]. In the depicted context, no prior work has
Cell-Free Massive Multiple Input Multiple Output (MIMO) analyzed an RIS-assisted Cell-Free Massive MIMO system in
has recently been introduced to reduce the intercell interfer- the presence of spatially-correlated channels.
ence of colocated Massive MIMO architectures. This is a
In this work, we consider an RIS-assisted Cell-Free Massive
network deployment where a large number of access points
MIMO under spatially correlated channels. We exploit a chan-
(APs) are located in a given coverage area to serve a small
nel estimation scheme that estimates the aggregated channels
number of users [2]. All the APs collaborate with each
including both the direct and indirect links. We analytically
other via a backhaul network and serve all the users in
show that, even by using a low complexity MRC technique,
the absence of cell boundaries. The system performance is
the non-coherent interference, small-scale fading effects, and
enhanced in Cell-Free Massive MIMO systems because they
additive noise are averaged out when the number of APs
inherit the benefits of the distributed MIMO and network
and RIS elements increases. The received signal includes,
MIMO architectures, but the users are also close to the APs.
hence, only the desired signal and the coherent interference.
When each AP is equipped with a single antenna, maximum-
We derive a closed-form expression of the net throughput
ratio combining (MRC) results in a good net throughput for
for the uplink data transmission. The impact of the array
every user, while ensuring a low computational complexity
gain, coherent joint transmission, channel estimation errors,
and offering a distributed implementation that is convenient
pilot contamination, spatial correlation, and phase shifts of the
for scalability purposes. However, this network deployment
RIS, which determine the system performance, are explicitly
cannot guarantee a good service under harsh propagation
observable in the obtained analytical expressions. With the aid
environments.
of numerical simulations, we verify the effectiveness of the
RIS is an emerging technology that is capable of shaping
proposed channel estimation scheme and the accuracy of the
the radio waves at the electromagnetic level without applying
closed-form expression of the net throughput. The obtained
digital signal processing methods and requiring power am-
numerical results show that the use of RISs enhance the net
plifiers [3]. Each element of the RIS scatters (e.g., reflects)
throughput per user significantly, especially when the direct
the incident signal without using radio frequency chains and
links are blocked with high probability.
This work of T. V. Chien, S. Chatzinotas, and B. Ottersten was supported by Notation: Upper and lower bold letters denote matrices and
RISOTTI - Reconfigurable Intelligent Surfaces for Smart Cities under project vectors. The identity matrix of size N × N is denoted by IN .
FNR/C20/IS/14773976/RISOTTI. The work of H. Q. Ngo was supported by
the UK Research and Innovation Future Leaders Fellowships under Grant (·)∗ , (·)T , and (·)H are the complex conjugate, transpose, and
MR/S017666/1. The work of M. Di Renzo was supported in part by the Hermitian transpose. E{·} and Var{·} denote the expectation
European Commission through the H2020 ARIADNE project under grant and variance of a random variable. The circularly symmetric
agreement number 871464 and through the H2020 RISE-6G project under
grant agreement number 101017011. The long version of this paper was Gaussian distribution is denoted by CN (·, ·) and diag(x) is
submitted to the IEEE Trans. Wireless Comm. [1]. the diagonal matrix whose main diagonal is given by x. tr(·)where αm , α̃mk ∈ C are the large-scale channel coefficients.
RIS RIS
controller The matrices in (2) assume that the size of each RIS element
is dH × dV , with dH being the horizontal width and dV
being the vertical height of each RIS element. In partic-
ular, the (l, t)-th element of the spatial correlation matrix
R ∈ CN ×N in (2) is [R]lt = sinc(2kul − ut k/λ), where
CPU λ is the wavelength and sinc(x) = sin(πx)/(πx) is the sinc
Direct link
function. The vector ux , x ∈ {l, t} is given by ux = [0,
Indirect link mod (x − 1, NH )dH , b(x − 1)/NH cdV ]T , where NH and NV
Fig. 1. An RIS-assisted Cell-Free Massive MIMO system where M APs denote the total number of RIS elements in each row and
collaborate with each other to serve K distant users. column, respectively.
is the trace operator. The Euclidean norm of vector x is kxk, B. Uplink Pilot Training Phase
and kXk is the spectral norm of matrix X. Finally, mod(·, ·) is The channels are independently estimated from the τp pilot
the modulus operation and b·c denotes the truncated argument. sequences transmitted by the K users. All the users share the
same τp pilot sequences. In particular, φ k ∈ Cτp with kφφk k2 =
II. S YSTEM M ODEL , C HANNEL E STIMATION , AND RIS
1 is defined as the pilot sequence allocated to the user k. We
P HASE -S HIFT C ONTROL
We consider an RIS-assisted Cell-Free Massive MIMO denote by Pk the set of indices of the users (including the
system, where M APs connected to a central processing user k) that share the same pilot sequence as the user k. The
unit (CPU) serve K users on the same time and frequency pilot sequences are assumed to be mutually orthogonal such
0
resource. All APs and users are equipped with a single that the pilot reuse pattern is φ H
k0 φ k = 1, k ∈ Pk . Otherwise,
H
antenna and they are randomly located in the coverage area. φ k0 φ k = 0. During the pilot training phase, all the K users
The communication is assisted by an RIS that comprises N transmit the pilot sequences to the M APs simultaneously. In
√
engineered scattering elements that can modify the phases of particular, the user k transmits the pilot sequence τpφ k . The
the incident signals. The matrix of phase received training signal at the AP m can be written as
shifts of the RIS is
denoted by Φ = diag [ejθ1 , . . . , ejθN ]T , where θn ∈ [−π, π] K
X √
K
X √
is the phase shift applied by the n-th element of the RIS. ypm = pτp gmkφ k + pτp hH
mΦ zk φ k + wpm , (3)
A. Channel Model k=1 k=1
We assume a quasi-static block fading model where each where p is the normalized signal-to-noise ratio (SNR) of each
coherence interval comprises τc symbols. The APs have pilot symbol, and wpm ∈ Cτp is the additive noise at the
knowledge of only the channel statistics instead of the in- AP m, which is distributed as wpm ∼ CN (0, Iτp ). In order
stantaneous channel realizations. Also, τp symbols (τp < τc ) for the AP m to estimate the desired channels from the user k,
in each coherence interval are dedicated to the channel es- the received training signal in (3) is projected on φ H
k as
timation and the remaining (τc − τp ) symbols are the data √
ypmk = φ H pτp gmk + hH
transmission. k ypm = m Φ zk
X √
pτp gmk0 + h Φ zk0 + wpmk , (4)
H
The following notation is used: gmk is the channel between + m
the user k and the AP m, which is the direct link [9]; k0 ∈Pk \{k}
hm ∈ CN is the channel between the AP m and the RIS;
and zk ∈ CN is the channel between the RIS and the where wpmk = φH k wpm ∼ CN (0, 1). We emphasize that
user k. In this paper, we consider a realistic channel model by the co-existence of the direct and indirect channels due to
taking into account the spatial correlation among the scattering the presence of the RIS results in a complicated channel
elements of the RIS, which is due to their sub-wavelength estimation process. In particular, the cascaded channel in
size, sub-wavelength inter-distance, and geometric layout. In (4) results in a nontrivial procedure to apply the minimum
an isotropic propagation environment, in particular, gmk , hm , mean-square error (MMSE) estimation method, as reported in
and zk can be modeled as follows previous works, for processing the projected signals [2], [11].
Based on the specific signal structure in (4), we denote the
gmk ∼ CN (0, βmk ), hm ∼ CN (0, Rm ), zk ∼ CN (0, R e k ), channel between the AP m and the user k through the RIS
(1) as
where βmk is the large-scale fading coefficient; Rm ∈ CN ×N umk = gmk + hH m Φ zk , (5)
and Re k ∈ CN ×N are the covariance matrices. The covariance
which is referred to as the aggregated channel that comprises
matrices in (1) correspond to a general model, which can be
the direct and indirect link between the user k and the AP m.
further particularized for application to typical RIS designs
By capitalizing on the definition of the aggregated channel
and propagation environments. A correlation model that is
in (5), the required channels can be estimated in an effective
applicable to isotropic scattering with uniformly distributed
manner even in the presence of the RIS. In particular, the
multipath components in the half-space in front of the RIS
aggregated channel in (5) is given by the product of weighted
was recently reported in [10], whose covariance matrices are
complex Gaussian and spatially correlated random variables,
Rm = αm dH dV R and R
e k = α̃k dH dV R, (2) as given in (1). Conditioned on the phase shifts, we employ thelinear MMSE method for estimating umk at the AP. Despite phase shift matrix Φ of the RIS so as to minimize the total
the complex structure of the RIS-assisted channels, Lemma 1 NMSE obtained from all the users and all the APs as follows
provides analytical expressions of the estimated channels.
M X
X K
Lemma 1. By assuming that the AP m employs the linear minimize NMSEmk
MMSE estimation method based on the observation in (4), {θn }
m=1 k=1
(11)
the estimate of the aggregate channel umk is formulated as subject to − π ≤ θn ≤ π, ∀n.
∗
umk }ypmk /E{|ypmk |2 } = cmk ypmk , (6)
ûmk = E{ypmk The optimal phase shifts solution to problem (11) is obtained
∗ by exploiting the statistical CSI that include the large-scale
where cmk = E{ypmk umk }/E{|ypmk |2 } has the following
fading coefficients and the covariance matrices. Problem (11)
closed-form expression
is a fractional program, whose globally-optimal solution is
√
pτp βmk + tr Φ H RmΦ R ek not simple to be obtained for an RIS with a large number of
cmk = P . (7) independently tunable elements. Nonetheless, in the special
pτp k0 ∈Pk βmk0 + tr Φ H RmΦ R e k0 + 1
network setup where the direct links from the APs to the users
The estimated channel in (6) has zero mean and variance γmk are weak enough to be negligible with respect to the RIS-
equal to assisted links, the optimal solution to problem (11) is available
√ in a closed-form expression as summarized in Corollary 1.
γmk = E{|ûmk |2 } = pτp βmk + tr Φ H RmΦ R
e k cmk .
(8) Corollary 1. If the direct links are weak enough to be negli-
Also, the channel estimation error emk = umk − ûmk and gible and the RIS-assisted channels are spatially correlated as
the channel estimate ûmk are uncorrelated. The channel formulated in (2), the optimal maximizer of the optimization
estimation error has zero mean and variance equal to problem in (11) is θ1 = . . . = θN , i.e., the equal phase shift
E |emk |2 = βmk + tr Φ H RmΦ R
e k − γmk . (9) design is optimal.
Proof. The proof follows by analyzing the objective function
Proof. It is similar to the proof in [12], and is obtained by
of problem (11) with respect to the phase-shift elements. The
applying similar analytical steps to the received signal in (4)
detailed proof is available in the journal version [1].
and by taking into account the structure of the RIS-assisted
channel and the spatial correlation matrices in (1). Corollary 1 provides a simple but effective option to de-
Lemma 1 shows that, by assuming Φ fixed, the aggregated sign the phase shifts of the RIS while ensuring the optimal
channel in (5) can be estimated without increasing the pilot estimation of the aggregated channels according to the sum-
training overhead, as compared to a conventional Cell-Free NMSE minimization criterion, provided that the direct link are
Massive MIMO system. The obtained channel estimate in completely blocked and the spatial correlation model in (2)
(6) unveils the relation ûmk0 = ccmk 0
ûmk if the user k 0 holds true. Therefore, an efficient channel estimation protocol
mk
uses the same pilot sequence as the user k. Because of pilot can be designed even in the presence of an RIS with a large
contamination, it may be difficult to distinguish the signals of number of engineered scattering elements. The numerical
these two users. In the following, the analytical expression of results in Section IV show that the phase shift design in
the channel estimates in Lemma 1 are employed for signal Corollary 1 offers good gains in terms of net throughput even
detection in the uplink data transmission. They are used also if the direct links are not negligible.
to optimize the phase shifts of the RIS in order to minimize III. U PLINK DATA T RANSMISSION AND P ERFORMANCE
the channel estimation error and to evaluate the corresponding A NALYSIS W ITH MR C OMBINING
ergodic net throughput. In this section, we introduce a procedure to detect the
C. RIS Phase-Shift Control uplink transmitted signals and derive an asymptotic closed-
Channel estimation is a critical aspect in Cell-Free Massive form expression of the ergodic net throughput.
MIMO. As discussed in previous text, in many scenarios, A. Uplink Data Transmission Phase
non-orthogonal pilots have to be used. This causes pilot In the uplink, all the K users transmit their data to the
contamination, which may reduce the system performance M APs simultaneously. Specifically, the user k transmits a
significantly. In this section, we design an RIS-assisted phase modulated symbol sk with E{|sk |2 } = 1. This symbol is
√
shift control scheme that is aimed to improve the quality of weighted by a power control factor ηk , 0 ≤ ηk ≤ 1. Then,
channel estimation. To this end, we introduce the normalized the received baseband signal, yum ∈ C, at the AP m is
mean square error (NMSE) of the channel estimate of the K
user k at the AP m as follows √ X√
ym = ρ ηk umk sk + wm , (12)
2 2
NMSEmk = E{|emk | }/E{|umk | } k=1
where ρ is the normalized uplink SNR of each data symbol
Φ H Rm Φ R
pτp βmk + tr(Φ e k) (10)
=1− P . and wm is the normalized additive noise with wm ∼ CN (0, 1).
pτp k0 ∈Pk βmk0 + tr(Φ H
Φ Rm Φ Rk 0 ) + 1
e
For data detection, the MRC method is used at the CPU, i.e.,
where the last equality is obtained from (9). We optimize the ûmk , ∀m, k, in (6) is employed to detect the data transmittedby the user k. In mathematical terms, the corresponding inserting (16) into the decision variable in (14), we obtain
decision statistic is the following deterministic value
M K M
√ X X√ X 1 P
rk = ρ ηk û∗mk umk0 sk0 + û∗mk wm . (13) rk −−−−→
M M →∞
m=1 k0 =1 m=1 M
1 X X√
ηk0 pτp ρu cmk βmk0 +tr Φ H RmΦ R
Based on the observation rk , the uplink ergodic net throughput e k0 sk0 ,
M
of the user k is analyzed in the next subsection. k0 ∈Pk m=1
(17)
B. Asymptotic Analysis
Since the number of APs, M , and the number of tunable because Tk2 /M → 0 and Tk3 /M → 0 as M → ∞.
elements of the RIS, N , can be large, we analyze the perfor- The result in (17) unveils that, for a fixed N , the channels
mance of two case studies: (i) N is fixed and M is large; become asymptotically orthogonal. In particular, the small
and (ii) both N and M are large. The asymptotic analysis is scale fading, the non-coherent interference, and the additive
conditioned upon a given setup of the CSI. To this end, the noise vanish. The only residual impairment is the pilot con-
uplink weighted signal in (13) is split into three terms based tamination caused by the users that employ the same pilot
on the pilot reuse set Pk , as follows sequence. Due to pilot contamination, the system performance
M
cannot be improved by adding more APs if MRC is used.
√ X X √ The contributions of both the direct and RIS-assisted in-
rk = ρ ηk0 û∗mk umk0 sk0 +
direct channels appear explicitly in (17) through βmk and
k0 ∈Pk m=1
| {z } Φ H Rm Φ R
tr(Φ e k0 ), respectively.
Tk1
(14) 2) Case II: Both N and M are large, i.e., N → ∞ and
M M
√ X X √ X M → ∞. We first need some assumptions on the covariance
ρ ηk0 û∗mk umk0 sk0 + û∗mk wm , matrices Rm and R e k , as summarized as follows.
k0 ∈P
/ k m=1 m=1
Assumption 1. For m = 1, . . . , M and k = 1, . . . , K, the
| {z } | {z }
Tk2 Tk3
covariance matrices Rm and Re k are assumed to fulfill the
where Tk1 accounts for the signals received from all the users following properties
in Pk , and Tk2 accounts for the mutual interference from the
1
users that are assigned orthogonal pilot sequences. The impact lim sup kRm k2 < ∞, lim inf tr(Rm ) > 0, (18)
of the additive noise obtained after applying MR combining is N N
N
given by Tk3 . From (4)-(6), we obtain the following identity e k k2 < ∞, lim inf 1 tr(R
lim sup kR e k ) > 0. (19)
N N N
M M
X √ X X √ The assumptions in (18) and (19) imply that the largest
ηk0 û∗mk umk0 = ηk0 pτp cmk umk0 u∗mk00
m=1 k00 ∈P k \{k0 } m=1 singular value and the sum of the eigenvalues (counted with
M M their mutiplicity) of the N × N covariance matrices that
X √ X √ ∗ characterize the spatial correlation among the channels of the
+ ηk0 pτp cmk |umk0 |2 + ηk0 cmk umk0 wpmk ,
m=1 m=1 RIS elements are finite and positive. Dividing both sides of
(15) (15) by M N and applying Tchebyshev’s theorem, we obtain
1) Case I: N is fixed and M is large, i.e., M → ∞. In 1 X√
M
P
this case, we divide both sides of (15) by M and exploits ηk0 û∗mk umk0 −−−−→
M N m=1 M →∞
Tchebyshev’s theorem [13]1 to obtain N →∞
M
1 X √
ηk0 pτp ρu cmk tr Φ H RmΦ R
M e k0 . (20)
1 X√ P MN
ηk0 û∗mk umk0 −−−−→ m=1
M m=1 M →∞
M We observe that Φ H Rm Φ Re k0 is similar to
1 X√ 1/2 H e 1/2
Rk0 Φ RmΦ Rk0 , which is a positive semi-definite
ηk0 pτp cmk βmk0 + tr Φ H RmΦ R
e k0 , (16) e
M m=1 matrix.3 Because similar matrices
have the same eigenvalues,
it follows that tr Φ H RmΦ R
e k0 > 0. Based on Assumption 1,
P
where −→ denotes the convergence in probability.2 Note that we obtain the following inequalities
the second and third terms in (15) converge to zero. By
1 (a) 1
tr Φ H RmΦ R
e k0 ≤ kΦ
Φk2 tr RmΦ R e k0
1 Let X1 , . . . , Xn be independent random variables such that E{Xi } = N N (21)
x̄i and Var{Xi } ≤ c < ∞. Then, Tchebyshev’s theorem states (b) 1 (c) 1 e
1 Pn P 1 P = e k 0 Rm ≤
tr Φ R kRk0 k2 tr(Rm ),
n n0 =1 Xn0 −−−−→ n n→∞ n0 x̄n0 . N N
2A sequence {Xn } of random variables converges in probability to the
random variable X if, for all > 0, it holds that limn→∞ Pr(|Xn − X| > 3 Two matrices A and B of size N × N are similar if there exists an
) = 0, where Pr(·) denotes the probability of an event. invertible N × N matrix U such that B = U−1 AU.
where (a) is obtained by an inequality on the trace of the with δmk0 = βmk0 + tr Φ H RmΦ R
e k0 , cmk given in (7), and
product of matrices; (b) follows because kΦ Φk2 = 1; and (c) γmk given in (8).
is obtained from Assumption 1. Based on Assumption 1, in
Proof. The main idea of proof is to average out the ran-
addition, the last inequality in (14) is bounded by a positive
domness by using the use-and-then-forget capacity bounding
constant. From (20) and (21), therefore, the decision variable
technique and fundamental properties of Massive MIMO. The
in (14) can be formulated as
detailed proof is available in the journal version [1].
1 P
rk −−−−→ By direct inspection of the SINR in (24), the numerator
MN M →∞
N →∞ increases with the square of the sum of the variances of the
M channel estimates, γmk , ∀m, thanks to the joint coherent trans-
1 X X √
ηk0 pτp ρu cmk tr Φ H RmΦ R
e k0 sk0 . (22) mission. On the other hand, the first term in the denominator
MN
k0 ∈Pk m=1 represents the power of the interference. Due to the limited
The expression obtained in (22) reveals that, as M, N → ∞, and finite number of orthogonal pilot sequences being used, it
the post-processed signal at the CPU consists of the desired represents the impact of pilot contamination. The last term is
signal of the intended user k and the interference from the the additive noise. The SINR in (24) is a multivariate function
other users in Pk . Compared with (17), we observe that of the matrix of phase shifts of the RIS and of the channel
(22) is independent of the direct links and depends only on statistics, i.e., the channel covariance matrices. Compared with
the RIS-assisted indirect links. This highlights the potentially conventional Cell- Free Massive MIMO systems, the strength
promising contribution of an RIS, in the limiting regime of the desired signal increases thanks to the assistance of
M, N → ∞, for enhancing the system performance. an RIS. However, the coherent and non-coherent interference
become more severe as well, due to the need of estimating
C. Uplink Ergodic Net Throughput Analysis with a Finite
both the direct and indirect links in the presence of an RIS.
Number of APs and Phase Shifts
We now focus our attention on the practical setup in which IV. N UMERICAL R ESULTS
M and N are both finite. By utilizing the user-and-then forget We consider a geographic area of size 1 km2 that is
channel capacity bounding method [1], the uplink ergodic net wrapped around at the edges. The locations of 100 APs
throughput of the user k can be computed in a closed-form and 10 users are given in terms of (x, y) coordinates. To
expression for (23) as given in Theorem 1. simulate a harsh communication environment, the APs are uni-
formly distributed in the sub-region x, y ∈ [−0.75, −0.5] km,
Theorem 1. If the CPU utilizes the MRC method, a lower while the users are uniformly distributed in the sub-region
bound closed-form expression for the uplink net throughput x, y ∈ [0.375, 0.75] km. An RIS with N = 900 is located
of the user k is given as follows at the origin, i.e., (x, y) = (0, 0). Each coherence interval
comprises τc = 200 symbols and τp = 5 orthonormal pilot
Rk = Bν (1 − τp /τc ) log2 (1 + SINRk ) , [Mbps], (23)
sequences. The large-scale fading coefficients αm and α̃mk
where B is the system bandwidth measured in MHz and are generated according to the three-slope propagation model
0 ≤ ν ≤ 1 is the portion of each coherence interval that is in [2]. The large-scale fading coefficient βmk is formulated as
dedicated to the uplink data transmission. The effective uplink βmk = β̄mk amk , where β̄mk is generated by the three-slope
signal-to-noise-plus-interference ratio (SINR) is propagation model in [2]. The binary variables amk accounts
!2 for the probability that the direct links are unblocked, and it is
M
X defined as amk = 1 with probability p̃. Otherwise, amk = 0
SINRk = ρηk γmk (MIk + NOk ), (24)
with probability 1 − p̃, where p̃ ∈ [0, 1] is the probability
m=1
that the direct link is not blocked.The covariance matrices
where MIk is the mutual interference and the noise denoted are generated according to the spatial correlation model in (2)
by NOk are, respectively, given by with dH = dV = λ/4. The pilot power is 100 mW and ν = 1.
K X
X M M
X X The power control coefficients are ηk = 1, ∀k. Without loss
MIk = ρu η γmk δ
k0 mk0 + pτp ρu ηk0 c2mk of generality, in particular, the N phase shifts in Φ are all set
k0 =1 m=1 k0 ∈P k m=1 equal to π/4, except in Fig. 2(c) with different phase shifts.
K
X M X
X X M Three system configurations are considered for comparison:
Θ2mk0 ) + pτp ρu
× tr(Θ ηk0 cmk cm0 k × i) RIS-Assisted Cell-Free Massive MIMO: This is the pro-
k0 =1 k00 ∈Pk m=1 m0 =1 posed system model where the direct links are unblocked
!2
X M
X with probability p̃. It is denoted by “RIS-CellFree”.
Θmk0 Θ m0 k00 ) + pτp ρu
tr(Θ ηk 0 cmk δmk0 , ii) Conventional Cell-Free Massive MIMO: This is the same
k0 ∈Pk \{k} m=1 as the previous model with the only exception that the
(25) RIS is not deployed. It is denoted by “CellFree”.
M
X iii) Cell-Free Massive MIMO without the direct links: This
NOk = γmk , (26) is the worst case study in which the direct links are
m=1 blocked with unit probability and the uplink transmission(a) (b) (c)
Fig. 2. The sum net throughput [Mbps]: (a) Average sum net throughput versus the unblocked probability of the direct links; (b) CDF of the sum net
throughput with the unblocked probability of the direct links p̃ = 0.2; (c) Sum net throughput as a function of the phase shift setup (equal or random) and
the unblocked probability of the direct links p̃ = 0.2.
is ensured only through the RIS. This setup is denoted closed-form expression of the ergodic net throughput for the
by “RIS-CellFree-NoLOS”. uplink data transmission phase has been proposed. Based
In Fig. 2(a), we illustrate the sum net throughput as a function on them, the performance of RIS-assisted Cell-Free Massive
of the probability p̃.P
In particular, the average sum net through- MIMO has been analyzed as a function of the fading spatial
K correlation and the blocking probability of the direct AP-user
put is defined as k=1 E{Rk }. Cell-Free Massive MIMO
provides the worst performance if the blocking probability links. The numerical results have shown that the presence of
is large (p̃ is small). If the direct links are unreliable, as an RIS is very useful if the AP-user links are mostly unreliable
expected, the net throughput offered by Cell-Free Massive with high probability.
MIMO tends to zero if p̃ → 0. In addition, the proposed R EFERENCES
RIS-assisted Cell-Free Massive MIMO setup offers the best [1] T. V. Chien, H. Q. Ngo, S. Chatzinotas, M. Di Renzo, and B. Otternsten,
net throughput, since it can overcome the unreliability of the “Reconfigurable intelligent surface-assisted Cell-Free Massive MIMO
systems over spatially-correlated channels,” IEEE Trans. Wireless Com-
direct links. An RIS is particularly useful if p̃ is small since mun., 2021, submitted for publication.
in this case the direct links are not able to support a high [2] H. Q. Ngo, A. Ashikhmin, H. Yang, E. G. Larsson, and T. L. Marzetta,
throughput. Fig. 2(b) compares the three considered systems “Cell-free massive MIMO versus small cells,” IEEE Trans. Wireless
PK Commun., vol. 16, no. 3, pp. 1834–1850, 2017.
in terms of sum net throughput (defined as k=1 Rk ) when [3] T. A. Le, T. Van Chien, and M. Di Renzo, “Robust probabilistic-
p̃ = 0.2. We observe the net advantage of the proposed constrained optimization for IRS-aided MISO communication systems,”
RIS-assisted Cell-Free Massive MIMO system. The worst- IEEE Wireless Commun. Lett., vol. 10, no. 1, pp. 1–5, 2021.
[4] M.-M. Zhao, Q. Wu, M.-J. Zhao, and R. Zhang, “Intelligent reflecting
case RIS-assisted Cell-Free Massive MIMO system setup (i.e., surface enhanced wireless network: Two-timescale beamforming opti-
p̃ = 0) outperforms the Cell-Free Massive MIMO setup mization,” IEEE Trans. Wireless Commun., 2020.
in the absence of an RIS. Fig. 2(c) focuses on the RIS- [5] N. S. Perović, L.-N. Tran, M. Di Renzo, and M. F. Flanagan, “Achiev-
able rate optimization for MIMO systems with reconfigurable intelligent
asissted Cell-Free Massive MIMO setup, since it provides the surfaces,” IEEE Trans. Wireless Commun., vol. 20, no. 6, pp. 3865 –
best performance. We compare the sum net throughput as a 3882, 2021.
function of the phase shifts of the RISs (random and uniform [6] T. Zhou, K. Xu, X. Xia, W. Xie, and J. Xu, “Achievable rate optimization
for aerial intelligent reflecting surface-aided Cell-Free Massive MIMO
phase shifts according to Corollary 1 in the presence of system,” IEEE Access, 2020.
spatially-correlated and spatially-independent fading channels [7] Z. Zhang and L. Dai, “Capacity improvement in wideband recon-
according to (2). In the presence of spatial correlation, we con- figurable intelligent surface-aided cell-free network,” in Proc. IEEE
e k = α̃k dH dV IN , ∀m, k. If SPAWC, 2020, pp. 1–5.
sider Rm = αm dH dV IN and R [8] T. Van Chien, A. K. Papazafeiropoulos, L. T. Tu, R. Chopra,
the spatial correlation is not considered, there is no significant S. Chatzinotas, and B. Ottersten, “Outage probability analysis of IRS-
difference between the random and uniform phase shifts setup. assisted systems under spatially correlated channels,” IEEE Wireless
Commun. Lett., 2021.
In the presence of spatial correlation, on the other hand, the [9] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless
proposed uniform phase shift design, which is obtained from network via joint active and passive beamforming,” IEEE Trans. Wire-
Corollary 1, provides a much better throughput. This result less Commun., vol. 18, no. 11, pp. 5394–5409, 2019.
[10] E. Björnson and L. Sanguinetti, “Rayleigh fading modeling and channel
highlights the relevance of using even simple optimization hardening for reconfigurable intelligent surfaces,” IEEE Wireless Com-
designs for RIS-assisted communications in the presence of mun. Lett., vol. 10, no. 4, pp. 830 – 834, 2021.
spatial correlation. [11] T. Van Chien, E. Björnson, and E. G. Larsson, “Joint power allocation
and load balancing optimization for energy-efficient cell-free Massive
V. C ONCLUSION MIMO networks,” IEEE Trans. Wireless Commun., vol. 19, no. 10, pp.
We have considered an RIS-assisted Cell-Free Massive 6798–6812, 2020.
[12] S. Kay, Fundamentals of Statistical Signal Processing: Estimation
MIMO system and have introduced an efficient channel es- Theory. Prentice Hall, 1993.
timation scheme to overcome the high channel estimation [13] H. Cramér, Random variables and probability distributions. Cambridge
overhead. An optimal design for the phase shifts of the University Press, 2004, vol. 36.
RIS that minimizes the channel estimation error has been
introduced and has been used for system analysis. Also, aYou can also read
