# RIS-Aided Cell-Free Massive MIMO: Performance Analysis and Competitiveness

Bayan Al-Nahhas, Mohanad Obeed, Anas Chaaban, and Md. Jahangir Hossain  School
of Engineering, University of British Columbia, Kelowna, Canada.
Email:{bayan.alnahhas, mohanad.obeed, anas.chaaban<EMAIL_ADDRESS>
###### Abstract

In this paper, we consider and study a cell-free massive MIMO (CF-mMIMO)
system aided with reconfigurable intelligent surfaces (RISs), where a large
number of access points (APs) cooperate to serve a smaller number of users
with the help of RIS technology. We consider imperfect channel state
information (CSI), where each AP uses the local channel estimates obtained
from the uplink pilots and applies conjugate beamforming for downlink data
transmission. Additionally, we consider random beamforming at the RIS during
both training and data transmission phases. This allows us to eliminate the
need of estimating each RIS assisted link, which has been proven to be a
challenging task in literature. We then derive a closed-form expression for
the achievable rate and use it to evaluate the system’s performance supported
with numerical results. We show that the RIS provided array gain improves the
system’s coverage, and provides nearly a 2-fold increase in the minimum rate
and a 1.5-fold increase in the per-user throughput. We also use the results to
provide preliminary insights on the number of RISs that need to be used to
replace an AP, while achieving similar performance as a typical CF-mMIMO
system with dense AP deployment.

##  I Introduction

Massive connectivity and demanding coverage requirements in the emerging sixth
generation (6G) wireless networks brought in the need of new and revolutionary
technologies for wireless communication systems. Two technological advances
have recently caught significant attention in literature and are expected to
take an important part in the future 6G wireless networks, namely the cell-
free massive multiple input multiple output (CF-mMIMO) systems and
reconfigurable intelligent surfaces (RISs). CF-mMIMO systems are considered as
a promising technology due to their architecture advantage, which is seen to
reap the benefits of massive MIMO systems and distributed systems together.
CF-mMIMO systems can be viewed as a particular deployment of mMIMO systems,
where the network consists of a large number of APs spread out over a certain
georgraphical area. In CF-mMIMO, the boundaries among cells are removed and
all the APs cooperate to serve all the users, thus the name. The distributed
architecture allows the CF-mMIMO systems to fully exploit macrodiversity and
potentially offers high probability of coverage. The reason is that the users
in CF-mMIMO are close to APs, which improves the systems fairness by enhancing
the service to edge-user [1, 2].
Many recent results on CF-networks show that CF-networks outperfoms the
conventional cellular and small cell networks under various practical settings
[3, 1, 4]. Recent works also studied CF-mMIMO asymptotic performance under
channel hardening and favorable propagation assumptions [5, 6]. Nevertheless,
like many technologies, CF-mMIMO also suffers from drawbacks including
backhauling traffic and infrastracture cost [7].
Recently, the concept of deploying RISs in existing communication systems has
emerged as a cost effective solution. The RIS is envisioned as a planar array
of passive reflecting elements that can independently induce phase-shifts onto
the incident electromagnetic waves for performance enhancement [8, 9]. Several
papers have demonstrated the effectiveness of RISs in envisioning a smart and
interconnected enviroment for future wireless generations [10, 11]. RISs have
also been studied in existing wireless technologies, bringing about names such
as RIS-aided massive MIMO systems, RIS-aided NOMA systems, RIS-aided security
system and RIS-aided UAV communication [12, 13, 14, 15, 16]. All these studies
provide insightful analysis on the improved performance while envisioning
lower cost and higher efficiency than existing systems. Recently, RISs have
also been introduced to CF networks and studied under various setups showing
potential effectiveness [17, 18, 19].

Motivated by the above, this paper considers an RIS-aided CF-mMIMO (RIS-CF-
mMIMO) system and studies the resulting performance enhancement. Specifically,
we study a multi-user downlink scenario in a CF-mMIMO system, wherein
transmission is aided by a number of RISs uniformly distributed in a
geographical area. Unlike previous works on RIS-CF networks, we assume an
imperfect channel state information (CSI) at the AP and derive a closed-form
acheivable rate expression while considering conjugate beamforming at the APs.
We show that in a Rayleigh fading environment and using the direct etimation
(DE) protocol proposed in [12], RISs provide an array gain dependent on the
number of elements, but their phases do not play a significant role in
improving signal-to-interference-plus-noise ratio (SINR) at the user end.
Since a well known advantage of CF-mMIMO systems is providing competitive
coverage performance when compared to small cells, we study the additional
effect of integrating RISs configured with random phase-shifts in further
enhancing coverage. Similar to previous works, we adopt the theoretical
minimum rate and average per user throughput, which accounts for pilot
overhead, as our perfomance criteria. Specifically, we consider a $5\%-20\%$
outage probability among all users for system analysis [3]. We extend the
analysis to study the acheivable sum rate of a CF-mMIMO system with a number
of APs and RISs, and compare it with a typical CF-mMIMO system with a dense AP
deployment. Numerical results show promising preliminary insights on the
potential of deploying RIS in the current CF-mMIMO topology.

The rest of the paper is organized as follows: Sec. II describes the system
model. Sec III explains the channel estimation. Sec. IV investigates the
conjugate beamforming and provides the acheivable rate expression. Sec. V
provides the numerical results and Sec. VI concludes the paper.

Figure 1: Sketch of an RIS-CF-mMIMO system.

##  II System Model

As shown in Fig. 1, we consider a system consisting of $M$ single-antenna APs
communicating with $K$ single-antenna users. The system is assisted by $S$
RISs, each composed of $N$ passive reflecting elements installed in the line-
of-sight (LoS) of the APs. In this topology, we assume that the APs, users,
and RISs are uniformly distributed in $D\times D$ $\text{km}^{2}$ geographical
area, where $S,M\gg K$. Furthemore, we consider a time division duplex (TDD)
protocol under which the APs utilize the uplink training phase to obtain the
local channel estimates and the downlink transmission phase to apply conjugate
beamforming and transmit data. Our objective is to study the downlink
acheivable rate at the user end for the proposed IRS-CF-mMIMO system and
derive its closed-form expression for performance analysis. In the following,
we will define channels depicted in Fig. 1 and express the signal model of our
proposed system.

The LoS channel between the $m$’th AP and $s$’th RIS is denoted as
$\mathbf{h}_{1,m}^{s}=\sqrt{\beta_{1,m}^{s}}[{h}_{1,m,1}^{s},\dots,{h}_{1,m,N}^{s}]\in\mathbb{C}^{1\times
N}$, where $\beta_{1,m}^{s}$ is the channel attenuation factor and
${h}_{1,m,n}^{s}$ is the small-scale fading coefficient between the $m$’th AP
and the $n$’th reflecting element in the $s$’th RIS. Also,
$\mathbf{h}_{2,k}^{s}\sim\mathcal{CN}(\mathbf{0},\beta_{2,k}^{s}\mathbf{I}_{N})\in\mathbb{C}^{N\times
1}$ and $h_{d,k}^{m}\sim\mathcal{CN}(0,\beta_{d,k}^{m})\in\mathbb{C}^{1\times
1}$ are the Rayleigh fading channels between user $k$ and the $s$’th RIS, and
user $k$ and the $m$’th AP, respectively, where $\beta_{2,k}^{s}$ and
$\beta_{d,k}^{m}$ are the channel attenuation coefficients. Note that the
subscripts $\\{1,2,d\\}$ denote the links, with $1$ denoting the link between
an AP and an RIS, $2$ denoting the link between the RIS and a user, and $d$
denoting the direct link between the AP and a user. We use block fading model,
where $h_{d,k}^{m}$ and $\mathbf{h}_{2,k}^{s}$ ($1\leq m\leq M,1\leq k\leq
K,1\leq s\leq S$) stay constant during a coherent interval and change
independently between coherent intervals. Additionally, we assume channel
reciprocity, i.e., the channel coefficients for uplink and downlink
transmissions are the same. The response of the RIS is captured in
$\boldsymbol{\Theta}=\text{diag}(\mathbf{v})\in\mathbb{C}^{N\times N}$, where
$\mathbf{v}=[\alpha_{1}\exp(j\theta_{1}),\alpha_{2}\exp(j\theta_{2}),\dots,\alpha_{N}\exp(j\theta_{N})]^{T}\in\mathbb{C}^{N\times
1}$ is the RIS reflect beamforming vector, with $\theta_{n}\in[0,2\pi]$ being
the induced phase-shift and $\alpha_{n}\in[0,1]$ being the given amplitude
reflection coefficient of element $n$. We denote components of $\mathbf{v}$ by
$v_{n}=\alpha_{n}\exp(j\theta_{n})$ in the remainder of this paper.
Given the above definitions, we can now define the total downlink channel at
the user end, i.e., the channel between $m$’th AP to $k$’th user via all RISs,
as
$g_{m,k}=\sum_{s=1}^{S}\mathbf{h}_{1,m}^{s}\boldsymbol{\Theta}\mathbf{h}_{2,k}^{s}+h_{d,k}^{m}$,
which is distributed as:

$\displaystyle g_{m,k}\sim\mathcal{CN}(0,\rho_{m,k}),$ (1)

where
$\rho_{m,k}=\sum_{s=1}^{S}\beta_{2,k}^{s}\mathbf{h}_{1,m}^{s}\boldsymbol{\Theta}\boldsymbol{\Theta}^{H}\mathbf{h}_{1,m}^{s^{H}}+\beta_{d,k}^{m}$.
In the following subsection we define channel estimates under the TDD
protocol.

##  III Channel Estimation

Under the TDD protocol, the AP $m$ exploits the channel reciprocity to
estimate the downlink channels $g_{m,k},k=1\ldots K$ defined in (1). As
mentioned earlier, this paper assumes random phase-shift configuration at each
RIS. Specifically, we assume $\alpha_{n}=1$, $\forall n$, to achieve the
largest array gain, and we choose $\theta_{n}$, $\forall n$ to be uniformly
distributed between $[0,2\pi]$. Under this setting, the APs only need to
acquire the total channel $g_{m,k}$ estimate for each user $k$, without the
need of estimating the individual channels $\mathbf{h}_{2,k}^{s}$, $\forall
s$, $\forall k$, and $h_{d,k}^{m}$, $\forall k$, which is challenging and time
consuming [13, 20].
All users synchronously send pilot sequences
$\boldsymbol{\phi}_{1}\ldots\boldsymbol{\phi}_{K}$ to APs via the direct and
RIS assisted links. The AP antennas then use the uplink training signal to
acquire the channel estimates. Under the assumption that the number of users
is less than the number of the orthogonal pilots, we assign mutually
orthogonal pilot sequences to users, while assuming user mobility of less $10$
km/h and carrier frequency of $1.9$ GHz resulting in negligible pilot
contamination [3]. Let $\tau$ be the length of the coherence interval,
$\tau_{c}$ be the length of uplink training duration per coherence interval,
and $\tau_{d}$ be the length of downlink transmission duration (all in symbol-
durations). It is required that $\tau_{c}<\tau$. Given the pilot sequence for
each user $k$ is of length $\tau_{c}$ symbol-durations,
$\boldsymbol{\phi}_{k}\in\mathbb{C}^{\tau_{c}\times 1},k=1\ldots K$, where
$||\boldsymbol{\phi}_{k}||^{2}=1$, then the $\tau_{c}\times 1$ signal received
at the $m$’th AP is:

$\displaystyle\mathbf{y}_{m}=\sqrt{\tau_{c}p_{c}}\sum_{k=1}^{K}g_{m,k}\boldsymbol{\phi}_{k}+\mathbf{w}_{m},$
(2)

where $p_{c}$ is the normalized transmit signal-to-noise ratio (SNR) of each
pilot symbol and $\mathbf{w}_{m}$ is a vector of additive noise at the $m$’th
AP whose elements are i.i.d. $\mathcal{CN}(0,1)$ random variables (RVs).

Based on the received pilot signal in (2), the $m$’th AP estimates the channel
$g_{m,k},k=1\dots K$, by projecting $\mathbf{y}_{m}$ on
$\boldsymbol{\phi}_{k}^{H}$, such that (s.t.)
$\tilde{\mathbf{y}}_{m}\triangleq\boldsymbol{\phi}_{k}^{H}\mathbf{y}_{m}=\sqrt{\tau_{c}p_{c}}g_{m,k}+\boldsymbol{\phi}_{k}^{H}\mathbf{w}_{m}$.
Therefore, the minimum mean square estimation (MMSE) estimate of $g_{m,k}$
given $\tilde{\mathbf{y}}_{m}$ is defined as:

$\displaystyle\hat{g}_{m,k}=\frac{\sqrt{\tau_{c}p_{c}}\rho_{m,k}\tilde{\mathbf{y}}_{m}}{\tau_{c}p_{c}\rho_{m,k}+1},$
(3)

Let $\tilde{g}_{m,k}=g_{m,k}-\hat{g}_{m,k}$ be the channel estimation error.
Since $\tilde{g}_{m,k}$ and $\hat{g}_{m,k}$ are uncorrelated [21], we can
statistically define the following:

$\displaystyle\hat{g}_{m,k}\sim\mathcal{CN}(0,\gamma_{m,k}),$ (4)

where

$\displaystyle\gamma_{m,k}=\frac{\tau_{c}p_{c}\rho_{m,k}\rho_{m,k}^{*}}{(\tau_{c}p_{c}\rho_{m,k}+1)},$
(5)

and

$\displaystyle\tilde{g}_{m,k}\sim\mathcal{CN}(0,\rho_{m,k}-\gamma_{m,k}).$ (6)

We aim at studying the achievable rate in the proposed RIS-CF-mMIMO system
under this channel estimation protocol. In the following section, we
investigate the conjugate precoding and derive the closed-form expression of
the acheivable rate.

##  IV Downlink Transmission

In the downlink transmission, each AP treats the channel estimates as true
channels and use the conjugate beamforming technique to transmit signals to
$K$ users. Let $s_{k}$, $k=1\ldots K$, where $\mathbb{E}[|s_{k}|^{2}]=1$, be
the symbol intended to user $k$, the transmit signal can then be expressed as
follows:

$\displaystyle
x_{m}=\sqrt{p_{d}}\sum_{k=1}^{K}\eta_{m,k}^{1/2}\hat{g}_{m,k}^{*}s_{k},$ (7)

where $p_{d}$ is the total average power available at any AP, $\eta_{m,k}$,
$m=1\ldots M$, $k=1\ldots K$, are power control coefficients chosen to satisfy
the average power constraints at each AP, $\mathbb{E}[|x_{m}|^{2}]\leq p_{d}$.
With the channel model defined in (1) and its estimate (3), the power
constraints can be rewritten as:

$\displaystyle\sum_{k=1}^{K}\eta_{m,k}\gamma_{m,k}\leq 1,$ (8)

The received downlink signal at user $k$ can be defined as:

$\displaystyle r_{k}$ $\displaystyle=\sum_{m=1}^{M}g_{m,k}x_{m}+n_{k}$ (9)
$\displaystyle=\sqrt{p_{d}}\sum_{m=1}^{M}\sum_{k^{\prime}=1}^{K}\eta_{m,k^{\prime}}^{1/2}g_{m,k}\hat{g}_{m,k^{\prime}}^{*}s_{k^{\prime}}+n_{k}$

where $n_{k}$ represents additive Gaussian noise at the $k$’th user. We assume
that $n_{k}\sim\mathcal{CN}(0,1)$. Then $s_{k}$ will be decoded from $r_{k}$.
Now we define the closed-form expression of the achievable rate. Using similar
analysis as in [22, 23], we assume that each user has the knowledge of signal
statistics but not the channel realizations. Consequently, the received signal
at the user $k$ can be written as:

$\displaystyle
r_{k}=\mathcal{D}_{k}s_{k}+\mathcal{B}_{k}s_{k}+\sum_{k^{\prime}\neq
k}^{K}\mathcal{U}_{kk^{\prime}}s_{k^{\prime}}+n_{k},$ (10)

where
$\mathcal{D}_{k}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sqrt{p_{d}}\mathbb{E}\left[{\sum}_{m=1}^{M}\eta_{m,k}^{1/2}g_{m,k}\hat{g}_{m,k}^{*}\right]$
is the deterministic value of the strength of the desired signal,
$\mathcal{B}_{k}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sqrt{p_{d}}\left(\sum_{m=1}^{M}\eta_{m,k}^{1/2}g_{m,k}\hat{g}_{m,k}^{*}-\mathbb{E}\left[{\sum}_{m=1}^{M}\eta_{m,k}^{1/2}g_{m,k}\hat{g}_{m,k}^{*}\right]\right)$
is the beamforming gain uncertainty, and
$U_{kk^{\prime}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sqrt{p_{d}}\sum_{m=1}^{M}\eta_{m,k^{\prime}}^{1/2}g_{m,k}\hat{g}_{m,k^{\prime}}^{*}$
is the interefence due to user $k^{\prime}$.

It can be shown, through straightforward calculation, that the effective
noise,
$E_{k}=\left(\sqrt{p_{d}}\sum_{m=1}^{M}\sum_{k^{\prime}=1}^{K}\eta_{m,k^{\prime}}^{1/2}g_{m,k}\hat{g}_{m,k^{\prime}}^{*}s_{k^{\prime}}+n_{k}\right)-\mathcal{D}_{k}s_{k}$,
and the desired signal are uncorrelated. Knowing that uncorrelated Gaussian
noise represents the worst case, we obtain the following achievable rate of
the $k$’th user for the RIS-CF-mMIMO system:

$\displaystyle
R_{k}=\log_{2}\left(1+\frac{|\mathcal{D}_{k}|^{2}}{\mathbb{V}\text{ar}\\{E_{k}\\}}\right).$
(11)

In the following theorem, we provide the closed-form expression of (11), for a
finite number of APs.

###### Theorem 1

The acheivable rate of transmission from the APs to the $k$’th user via all
$S$ RISs is given by (12) at the top of next page.

$\displaystyle{R}_{k}=\log_{2}\left(1+\frac{p_{d}\left(\sum_{m}^{M}\eta_{m,k}^{1/2}\gamma_{m,k}\right)^{2}}{p_{d}\sum_{k^{\prime}=1}^{K}\sum_{m=1}^{M}\eta_{m,k^{\prime}}\gamma_{m,k^{\prime}}\rho_{m,k}+1}\right)$
(12)

###### Proof:

The proof is provided in Appendix A. ∎

From the expression in (12), we can notice the following. The closed-form
expression is a function of $\rho_{m,k}$ and $\gamma_{m,k}$, which are defined
under a Rayleigh fading environment, while considering the estimation protocol
proposed in Sec. III. Under the RIS response considered in Sec. III, where the
amplitude $\alpha_{n}=1$, and $\theta_{n}$ is random, $\forall n$, we can
define
$\rho_{m,k}=\sum_{s=1}^{S}\beta_{2,k}^{s}\mathbf{h}_{1,m}^{s}\mathbf{h}_{1,m}^{s^{H}}+\beta_{d,k}^{m}$,
since $\boldsymbol{\Theta}\boldsymbol{\Theta}^{H}=I_{N}$. Therefore, we can
notice that under the considered channel fading and estimation method, both
$\rho_{m,k}$ and $\gamma_{m,k}$ are independent of RIS phase-shifts
$\theta_{n}$, $\forall n$. If an estimation protocol, such as ON-OFF protocol
proposed in [12], were used, the RIS phase-shifts $\theta_{n}$ would appear in
terms related to the direct channel estimation error. In such case, it is
important to consider phase optimization. In our setting, however, we consider
a sub-optimal scenario where the RISs are randomly configured. Therefore, it
is reasonable to consider the derived acheivable rate as a lower bound to the
RIS-assisted cell free performance metric.

##  V Numerical Results

In this section, we quantitively study the performance of RIS-CF-mMIMO and
compare it to that of CF-mMIMO system. Specifically, we look into a CF-mMIMO
system with $M$ APs and $K$ users uniformly distributed in a $D\times D$
$\text{km}^{2}$ geographical area. Under a similar setting, i.e., same number
of uniformly distributed APs and user density, we additionally deploy $S$
uniformly distributed RISs, where each is equipped with $N$ reflecting
elements to simulate the RIS-CF-mMIMO system.

The large scale fading described in channel definition in (1) is a function of
pathloss and shadow fading defined as follows [1]:

$\displaystyle\beta^{t}_{l,w}=\mathcal{P}^{t}_{l,w}\cdot
10^{\frac{\sigma_{sh}z^{t}_{l,w}}{10}}$ (13)

where $t\in[s,m]$, $l\in[1,2,d]$, and $w\in[m,k]$. The pathloss parameter
$\mathcal{P}^{t}_{w,l}$ represents the pathloss between the $t$’th and $w$’th
nodes via the $l$’th link, and $10^{\frac{\sigma_{sh}z^{t}_{l,w}}{10}}$
represents shadow fading with standard deviation $\sigma_{sh}$, and
$z^{t}_{l,w}\sim\mathcal{N}(0,1)$. Let us denote the distance between the
$t$’th node and the $w$’th node as $d_{t,w}$, where $t,w\in[m,s,k]$ for
$m=1\ldots M,s=1\ldots S$ and $k=1\ldots K$. We use a three-slope model for
the path loss [1]: In the case when $d_{t,w}>d_{1}$, we employ the Hata-
COST231 propagation model, where the path loss exponent equals $3.5$ for
${h}_{d,k}^{m}$, $2.8$ for $\mathbf{h}_{2,k}^{s}$ and $2.0$ for
$\mathbf{h}_{1,m}^{s}$[20, 12]. Otherwise, we set it equals to $2$ if
$d_{1}<d_{t,w}\leq d_{0}$, and equals $0$ if $d_{t,w}\leq d_{0}$. Note that
when $d_{t,w}\leq d_{1}$ there is no shadowing [1].

In all our results, we use the parameters defined in Table I, unless otherwise
noted. The quantities $\bar{p}_{d}$ and $\bar{p}_{c}$ in the table are the
transmit powers for the transmit data and pilot symbol, respectively. The
corresponding normalized transmit SNRs $p_{d}$ and $p_{c}$ can be computed by
dividing these powers by the noise power, where the noise power is given by:
noise power = bandwidth $\times k_{B}\times T_{0}\times$ noise figure (W),
$k_{B}=1.381\times 10^{-23}$ (Joule per Kelvin) is the Boltzmann constant, and
$T_{0}=290$ (Kelvin) is the noise temperature.

TABLE I: System Parameters Parameter | Value
---|---
carrier frequency | 1.9 GHz
bandwidth | 20 MHz
noise figure | 9 dB
AP height | 15 m
RIS height | 18 m
user-device height | 1.65 m
$\sigma_{sh}$ | 8 dB
$\bar{p}_{d}$, $\bar{p}_{c}$ | 200 mW [3]
$d_{0}$, $d_{1}$ | 10 m, 50 m
$\tau$ (coherence period) | 200 samples

In Fig. 2, we plot the closed-form expression of acheivable sum rate using
(12), s.t.:

$\displaystyle R_{sum}=(\frac{1-\tau_{c}/\tau}{2})\sum_{k=1}^{K}R_{k},$ (14)

versus the varying number of APs $M$, along with the Monte-Carlo simulated
curve. For the purpose of our simulation, we assume half the data samples are
spent for downlink transmission. Specifically, we compare the sum rate under
the achievable rate in (12) with the rate in following expression:

$\displaystyle\tilde{R}_{k}=\mathbb{E}\left[\log_{2}\left(1+\frac{p_{d}|\sum_{m=1}^{M}\eta_{m,k}^{1/2}g_{m,k}\hat{g}^{*}_{m,k}|^{2}}{p_{d}\sum_{k^{\prime}\neq
k}^{K}|\sum_{m=1}^{M}\eta_{m,k^{\prime}}^{1/2}g_{m,k}\hat{g}^{*}_{m,k^{\prime}}|^{2}+1}\right)\right]$
(15)

which is the achievable rate under imperfect CSI at the users. Fig. 2 shows
the comparison between (12), which assumes that the users only know the
channel statistics, and (15), which assumes knowledge of the realizations. The
small gap depicted in the plot proves the tightness of the expressions at a
practical system dimension.

$50$$75$$100$$125$$150$$175$$200$$1.2$$1.8$$2.4$Number of APs ($M$)Average sum
rate (bps/Hz)User knows the effective channel gainUser only knows the channel
statistics  Figure 2: Validation of the closed form expression against the
Monte-Carlo (MC) values, under imperfect CSI. Here we assume $D=1\times
1\text{km}^{2}$, $\beta_{l,w}^{t}=1$, $K=40,S=30$, and $N=10$.

We fix the number of reflecting elements per RIS in the following simulation
figures to $N=30$. In Fig. 3, we compare the minimum rate, i.e.,
$\text{min}_{k}$ $R_{k}$, achieved for both RIS-CF-mMIMO and CF-mMIMO, to
demonstrate the coverage enhancement under RIS deployment in the CF-mMIMO
system. Specifically, the figure shows the cumulative distribution (CDF) of
the minimum (min) rate for a CF-mMIMO with and without RIS deployment, under
three different scenarios. We use the closed-form rate expression defined in
(12) for RIS-CF-mMIMO and the closed-form rate expression defined in Theorem 1
in [3] for CF-mMIMO system. For all scenarios depicted in Fig. 3, we consider
$M=100$ and $S=80$. Specifically, the blue curves examine the performance
under $K=45$ users uniformly distributed in a $2\times 2$ km2 georaphical
area, i.e., $D=2$, green curves are under $K=90$ users uniformly distributed
in a $4\times 4$ km2, and the red curves are under $K=45$ users uniformly
distributed in a $4\times 4$ km2 area.

$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0$$0.2$$0.4$$0.6$$0.8$$1$min Rate
(bps/Hz)CDFRIS-A-CF, $45$ users in $2\times 2\text{km}^{2}$CF, $45$ users in
$2\times 2\text{km}^{2}$RIS-A-CF, $90$ users in $4\times 4\text{km}^{2}$CF,
$90$ users in $4\times 4\text{km}^{2}$20%-outage5%-outage Figure 3: The
minimum rate for RIS-CF-mMIMO and CF-mMIMO under different $K$ and $D$
settings. Here, $M=100$ and $S=80$ and red curves are for $K=45$ and $D=4$.

There are three important takeaways which we can draw from Fig. 3. First,
looking at the blue curves (both dashed and solid) we can see that by
deploying RISs, specifically $S=80$ and $N=30$ in this example, under a 5%
outage constraint, the minimum rate can be increased by a factor of $\approx$
$1.82$ as compared to CF-mMIMO without RIS. Second, an RIS-CF-mMIMO system
serving double the number of users (green solid curve), $K=90$ in a $4\times
4$ km2 area (same user density but half AP density), can achieve an equivalent
minimum rate as that of a CF-mMIMO serving $K=45$ in $2\times 2$ km2 area
(blue dashed curve) under the same outage probability of 20%. This implies
that with the same number of AP deployment, we can double the network’s
service (serve more users) over a wider area. Again, note that our perfomance
metric used, defined in (12), is a lower bound to the RIS-assisted
performance. Therefore, the results in this work could be seen as a lower
bound on the performance improvement that can be realized using an RIS in a
CF-mMIMO system. Finally, the solid red curve looks at a scenario where an
RIS-CF-mMIMO system serves $K=45$ users uniformly distributed in a $4\times 4$
km2 area and compares it with the CF-mMIMO system (red dashed line) serving
users under a similar setting. Note that in this scenario, we have only spread
the $45$ users over a wider area. This increases the average distance between
the APs and users, which weakens AP-user link due to the induced pathloss
values. The results show that the minimum rate is doubled by deploying RISs
into the system under a 20% outage constraint, and by a factor of about $1.87$
under a 5% outage constraint. This shows that by deploying enough RISs, we can
achieve better service for users suffering from weak AP-user links, such as in
rural areas with low population.

Next, we study the per-user throughput performance. Specifically, we consider
the following per-user net throughput expression, which takes into account the
channel estimation overhead [1]:

$\displaystyle S_{k}=B\frac{1-\frac{\tau_{c}}{\tau}}{2}R_{k},$ (16)

where $B$ is the bandwidth. Fig. 4 plots the average per-user throughput for
RIS-CF-mMIMO (solid curves) and CF-mMIMO (dashed curves), under $K=45$ users
(red curves) and $K=65$ users (green curves) uniformly distributed in a
$2\times 2$ $\text{km}^{2}$ area. From the red curves we can see that the
average per-user throughput can be increased by almost a factor of $1.42$ in
an RIS-CF-mMIMO system under a 5% outage constraint. Additionally, as shown
from the solid green curve, increasing the number of users to $65$ while
deploying RISs, specifically $S=100$ and $N=30$ in this example, we can
achieve the same throughput as that of CF-mMIMO with fewer users served per
unit area, with an outage of 13%.

$2$$4$$6$$8$$10$$12$$14$$16$$18$$0$$0.2$$0.4$$0.6$$0.8$$1$Average per-user
throughput (Mbits/s)CDFCF, $45$ users in a $2\times 2\text{km}^{2}$RIS-A-CF,
$45$ users in a $2\times 2\text{km}^{2}$CF, $65$ users in a $2\times
2\text{km}^{2}$RIS-A-CF, $65$ users in a $2\times
2\text{km}^{2}$13%-outage5%-outage Figure 4: The average per-user throughput
for RIS-CF-mMIMO and CF-mMIMO under different $K$ settings. Here, $M=100$,
$S=80$ and $D=2$.

Finally, we look into a primitive approach of studying the effect of deploying
RISs in CF-mMIMO on the sum-rate performance. Specifically, Fig. 5 plots the
average sum rate curve of CF-mMIMO over varying number of $M$ APs, while
fixing the number of APs in the RIS-CF-mMIMO to $M=70$, and varying the number
of RISs in $S\in\\{80,200\\}$. Fig. 5 shows that an RIS-CF-mMIMO system with
$M=70$ APs and $S=80$ RISs can perform as well as a CF-mMIMO with $M=92$ APs,
thus saving up to $22$ APs for an equivalent average sum rate performance.
Additionally, an RIS-CF-mMIMO system with $M=70$ APs and $S=200$ RISs can
perform as well as a CF-mMIMO with $M=107$ APs. Thus, again we can save up to
$37$ APs by integrating low cost RISs into CF-mMIMO systems. Though we cannot
draw a clear relationship between the number of RISs or reflecting elements
per RIS that are needed per saved AP, we can still see a promising initial
insight on resource saving when deploying RISs into a CF-mMIMO system. Further
study is required to compute the potential number of RISs needed to deploy in
CF-mMIMO to save an AP.

$80$$90$$100$$110$$120$$130$$140$$9$$10$$11$$12$M APsAverage sum rate
(Bps/Hz)CF-mMIMORIS-A-CF-mMIMO, M=70, N=30, S=80RIS-A-CF-mMIMO, M=70, N=30,
S=200 Figure 5: The average sum rate comparison for RIS-CF-mMIMO and CF-mMIMO
under different number of RISs $S$ deployment.

##  VI Conclusion

This paper studied the performance of an RIS-CF-mMIMO under imperfect CSI. By
deriving a closed-form expression of the acheivable rate, we show that under
Rayleigh fading and a DE channel estimation protocol, the system’s performance
is independent of the RIS phase configuration. We further compare the
performance of a RIS-CF-mMIMO system with the well known CF-mMIMO system. The
results show that RISs deployment can improve the current CF-mMIMO topology in
terms of system’s coverage, nearly doubling the minimum rate relative to a CF-
mMIMO system. Additionally, RIS technology can lead to a significant resource
saving, where a number of low cost RISs may be deployed instead of a number of
APs while achieving equivalent sum rate performance. A more accurate and
qualitative comparison between the potential number of RIS reflecting
elements/RIS surfaces needed to replace one AP under Rician fading channels
and optimal RIS configuration will be studied as a future extension of this
work.

###  -A Proof of Theorem 1

To derive the closed-form expression for the acheivable rate given in (12), we
need to compute $\mathcal{D}_{k}$ and $\mathbb{V}\text{ar}\\{E_{k}\\}$. Recall
that
$E_{k}=\left(\sqrt{p_{d}}\sum_{m=1}^{M}\sum_{k^{\prime}=1}^{K}\eta_{m,k^{\prime}}^{1/2}g_{m,k}\hat{g}_{m,k^{\prime}}^{*}s_{k^{\prime}}+n_{k}\right)-\mathcal{D}_{k}s_{k}$.
It can be shown that $\mathcal{B}_{k}$, $\mathcal{U}_{kk^{\prime}}$ and
$n_{k}$ are uncorrelated, therefore we can define
$\mathbb{V}\text{ar}\\{E_{k}\\}=\mathbb{E}[|\mathcal{B}_{k}|^{2}]+\sum_{k^{\prime}\neq
k}^{K}\mathbb{E}[|\mathcal{U}_{kk^{\prime}}|^{2}]+1$.
Let $\mathcal{E}=g_{m,k}-\hat{g}_{m,k}$ be the channel estimation error. Then

$\displaystyle\mathcal{D}_{k}$
$\displaystyle=\sqrt{p_{d}}\mathbb{E}\left[{\sum}_{m=1}^{M}\eta_{m,k}^{1/2}g_{m,k}\hat{g}_{m,k}^{*}\right]$
$\displaystyle=\sqrt{p_{d}}\mathbb{E}\left[{\sum}_{m=1}^{M}\eta_{m,k}^{1/2}(\mathcal{E}+\hat{g}_{m,k})\hat{g}_{m,k}^{*}\right].$

Owing to the properties of MMSE estimation, $\mathcal{E}$ and $\hat{g}_{m,k}$
are independent, thus

$\displaystyle\mathcal{D}_{k}$
$\displaystyle=\sqrt{p_{d}}{\sum}_{m=1}^{M}\eta_{m,k}^{1/2}\gamma_{m,k}.$ (17)

Next, we compute $\mathbb{E}[|\mathcal{B}_{k}|^{2}]$. Recall that the variance
of sum of independent RVs is equal to sum of the variances, therefore,

$\displaystyle\mathbb{E}[|\mathcal{B}_{k}|^{2}]$
$\displaystyle=p_{d}\sum_{m=1}^{M}\eta_{m,k}\mathbb{E}\left[|g_{m,k}\hat{g}_{m,k}^{*}-\mathbb{E}\left[g_{m,k}\hat{g}_{m,k}^{*}\right]|^{2}\right]$
$\displaystyle=$ $\displaystyle
p_{d}\sum_{m=1}^{M}\eta_{m,k}\left(\mathbb{E}\left[|g_{m,k}\hat{g}_{m,k}^{*}|^{2}\right]-|\mathbb{E}\left[g_{m,k}\hat{g}_{m,k}^{*}\right]|^{2}\right)$
$\displaystyle=$ $\displaystyle
p_{d}\sum_{m=1}^{M}\eta_{m,k}\left(\mathbb{E}\left[\left|\mathcal{E}\hat{g}_{m,k}^{*}+|\hat{g}_{m,k}|^{2}\right|^{2}\right]-\gamma_{m,k}^{2}\right).$

Following similar argument in the proof of Theorem 1 provided in [1], it can
be shown that the above can be simplified to:

$\displaystyle\mathbb{E}[|\mathcal{B}_{k}|^{2}]=p_{d}\sum_{m=1}^{M}\eta_{m,k}\gamma_{m,k}\rho_{m,k}.$
(18)

Finally, to compute $\mathbb{E}[|\mathcal{U}_{kk^{\prime}}|^{2}]$, note that

$\displaystyle\mathbb{E}[|\mathcal{U}_{kk^{\prime}}|^{2}]=p_{d}\mathbb{E}\left[|\sum_{m=1}^{M}\eta_{m,k^{\prime}}^{1/2}g_{m,k}\hat{g}_{m,k^{\prime}}^{*}|^{2}\right].$
(19)

Recall that ($g_{m^{\prime},k}$, $g_{m,k}$), and
($g_{m,k^{\prime}}$,$g_{m,k}$), for $m^{\prime}\neq m$ and $k^{\prime}\neq k$,
are independent. By expanding the expression inside the expectation and
following a similar proof as that of Theorem 1 in [1], while considering our
setting, i.e., orthogonal pilots, (omitted here due to lack of space) we can
express (19) as:

$\displaystyle\mathbb{E}[|\mathcal{U}_{kk^{\prime}}|^{2}]=p_{d}\sum_{m=1}^{M}\eta_{m,k^{\prime}}\rho_{m,k}\gamma_{m,k^{\prime}}.$
(20)

Plugging (20) and (18) in the definition $\mathbb{V}\text{ar}\\{E_{k}\\}$, and
using the definition and (17) back in (12) completes the proof.

## References

  * [1] H. Q. Ngo, A. Ashikhmin, H. Yang, E. G. Larsson, and T. L. Marzetta, “Cell-free massive mimo versus small cells,” _IEEE Transactions on Wireless Communications_ , vol. 16, no. 3, pp. 1834–1850, 2017.
  * [2] J. Zhang, S. Chen, Y. Lin, J. Zheng, B. Ai, and L. Hanzo, “Cell-free massive mimo: A new next-generation paradigm,” _IEEE Access_ , vol. 7, pp. 99 878–99 888, 2019.
  * [3] E. Nayebi, A. Ashikhmin, T. L. Marzetta, and H. Yang, “Cell-free massive mimo systems,” in _2015 49th Asilomar Conference on Signals, Systems and Computers_ , 2015, pp. 695–699.
  * [4] H. Q. Ngo, A. Ashikhmin, H. Yang, E. G. Larsson, and T. L. Marzetta, “Cell-free massive mimo: Uniformly great service for everyone,” 2015.
  * [5] Z. Chen and E. Bjoernson, “Can we rely on channel hardening in cell-free massive mimo?” in _2017 IEEE Globecom Workshops (GC Wkshps)_ , 2017, pp. 1–6.
  * [6] Z. Chen and E. Björnson, “Channel hardening and favorable propagation in cell-free massive mimo with stochastic geometry,” 2018.
  * [7] A. A. I. Ibrahim, A. Ashikhmin, T. L. Marzetta, and D. J. Love, “Cell-free massive mimo systems utilizing multi-antenna access points,” in _2017 51st Asilomar Conference on Signals, Systems, and Computers_ , 2017, pp. 1517–1521.
  * [8] C. Huang, Member, IEEE, R. Mo, C. Yuen, and S. Member, “Reconfigurable intelligent surface assisted multiuser miso systems exploiting deep reinforcement learning,” 2020.
  * [9] M. D. Renzo, M. Debbah, D.-T. Phan-Huy, A. Zappone, M.-S. Alouini, C. Yuen, V. Sciancalepore, G. C. Alexandropoulos, J. Hoydis, H. Gacanin, J. de Rosny, A. Bounceu, G. Lerosey, and M. Fink, “Smart radio environments empowered by ai reconfigurable meta-surfaces: An idea whose time has come,” 2019.
  * [10] S. Gong, X. Lu, D. T. Hoang, D. Niyato, L. Shu, D. I. Kim, and Y. C. Liang, “Towards smart wireless communications via intelligent reflecting surfaces: A contemporary survey,” _IEEE Communications Surveys Tutorials_ , pp. 1–1, 2020.
  * [11] C. Pan, H. Ren, K. Wang, M. Elkashlan, M. Chen, M. D. Renzo, Y. Hao, J. Wang, A. L. Swindlehurst, X. You, and L. Hanzo, “Reconfigurable intelligent surface for 6g and beyond: Motivations, principles, applications, and research directions,” 2020.
  * [12] B. Al-Nahhas, Q.-U.-A. Nadeem, and A. Chaaban, “Intelligent reflecting surface assisted miso downlink: Channel estimation and asymptotic analysis,” 2020.
  * [13] Q.-U.-A. Nadeem, A. Kammoun, A. Chaaban, M. Debbah, and M.-S. Alouini, “Intelligent reflecting surface assisted wireless communication: Modeling and channel estimation,” 2019.
  * [14] Z. Chu, W. Hao, P. Xiao, and J. Shi, “Intelligent reflecting surface aided multi-antenna secure transmission,” _IEEE Wireless Communications Letters_ , vol. 9, no. 1, pp. 108–112, 2020.
  * [15] X. Yue and Y. Liu, “Performance analysis of intelligent reflecting surface assisted noma networks,” 2020.
  * [16] D. Ma, M. Ding, and M. Hassan, “Enhancing cellular communications for uavs via intelligent reflective surface,” 2020.
  * [17] S. Huang, Y. Ye, M. Xiao, H. V. Poor, and M. Skoglund, “Decentralized beamforming design for intelligent reflecting surface-enhanced cell-free networks,” 2020.
  * [18] Z. Zhang and L. Dai, “Capacity improvement in wideband reconfigurable intelligent surface-aided cell-free network,” 2020.
  * [19] Q. N. Le, V.-D. Nguyen, and O. A. Dobre, “Energy efficiency maximization in ris-aided cell-free network with limited backhaul,” 2020.
  * [20] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming,” _IEEE Transactions on Wireless Communications_ , vol. 18, no. 11, pp. 5394–5409, 2019.
  * [21] J. Hoydis, S. ten Brink, and M. Debbah, “Massive mimo in the ul/dl of cellular networks: How many antennas do we need?” _IEEE Journal on Selected Areas in Communications_ , vol. 31, no. 2, pp. 160–171, 2013.
  * [22] B. Hassibi and B. M. Hochwald, “How much training is needed in multiple-antenna wireless links?” _IEEE Transactions on Information Theory_ , vol. 49, no. 4, pp. 951–963, 2003.
  * [23] H. Yang and T. L. Marzetta, “Capacity performance of multicell large-scale antenna systems,” in _2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton)_ , 2013, pp. 668–675.