# On the Distribution of SINR for Cell-Free Massive MIMO Systems

Baolin Chong, Fengqian Guo, Hancheng Lu, and Langtian Qin Baolin Chong,
Fengqian Guo, Hancheng Lu and Langtian Qin are with the Department of
Electronic Engineering and Information Science, University of Science and
Technology of China, Hefei 230027, China. (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>qlt315<EMAIL_ADDRESS>

###### Abstract

Cell-free (CF) massive multiple-input multiple-output (mMIMO) has been
considered as a potential technology for Beyond 5G communication systems.
However, the performance of CF mMIMO systems has not been well studied. Most
existing analytical work on CF mMIMO systems is based on the expected signal-
to-interference-plus-noise ratio (SINR). The statistical characteristics of
the SINR, which is critical for emerging applications that focus on extreme
events, have not been investigated. To address this issue, in this paper, we
attempt to obtain the distribution of SINR in CF mMIMO systems. Considering a
downlink CF mMIMO system with pilot contamination, we first give the closed-
form expression of the SINR. Based on our analytical work on the two
components of the SINR, i.e., desired signal and interference-plus-noise, we
then derive the probability density function and cumulative distribution
function of the SINR under maximum ratio transmission (MRT) and full-pilot
zero-forcing (FZF) precoding, respectively. Subsequently, the closed-form
expressions for two more sophisticated performance metrics, i.e., achievable
rate and outage probability, can be obtained. Finally, we perform Monte Carlo
simulations to validate our analytical work. The results demonstrate the
effectiveness of the derived SINR distribution, achievable rate, and outage
probability.

###### Index Terms:

Cell-free (CF) massive multiple-input multiple-output (mMIMO), Signal-to-
interference-plus-noise ratio (SINR), maximum ratio transmission (MRT), full-
pilot zero-forcing (FZF)

##  I Introduction

Multiple-input multiple-output (MIMO) technology has been extensively deployed
in current communication systems to support diverse service types [1, 2, 3].
However, the arrival of Beyond 5G has introduced a multitude of emerging
applications, such as holographic telepresence, augmented reality, virtual
reality, and the Internet of everything [4], which have imposed more stringent
demands on system performance. The traditional cell-centric MIMO-based
cellular networks have become inadequate to meet the demands of the current
advancements. In traditional cellular networks, where users within a cell are
served by a single base station (BS), users at the cell edge often experience
considerable inter-cell interference caused by signals from neighboring cells.
Hence, inter-cell interference is an important factor limiting the capacity of
cellular networks [5]. Furthermore, users located at different positions
within the cellular network face challenges in receiving uniform service due
to varying distances from the BS.

To overcome aforementioned issues, cell-free (CF) massive MIMO (mMIMO) has
been proposed, which combines the advantages of mMIMO and distributed MIMO
[6]. In a CF mMIMO system, a central processing unit (CPU) is employed to
control multiple access points (APs), enabling the concurrent service of
multiple users on the same time-frequency resources [7, 4]. Particularly, the
distributed deployment of multiple APs eliminates the cell boundaries and
reduces the distance between APs and users [8], with which inconsistent
service quality and large-scale fading effects can be effectively avoided. It
has been proved both spectral efficiency (SE) [9] and energy efficiency (EE)
[10] are substantially improved in the CF mMIMO systems.

The significant technical advantages demonstrated by CF mMIMO have spurred an
increased research focus on its performance analysis. For CF mMIMO systems
under maximum ratio transmission (MRT) precoding, the derivation of expected
signal-to-interference-plus-noise ratio (SINR) was initially undertaken by
authors of [11]. Subsequently, in [12] and [13], the CF mMIMO system’s
performance is investigated under downlink and uplink data transmission
scenarios with different zero-forcing (ZF) precoding and combining schemes.
The performance has also been studied for CF mMIMO systems with other linear
receivers, e.g., minimum mean square error (MMSE) receiver [14]. In [15], the
analytical work was extended to encompass multi-antenna users, coherent
Rayleigh fading channels, and achievable rate, building upon previous
considerations of single-antenna users and Rayleigh fading channels. The
impact of asynchronous reception of signals by users, resulting from the
distributed deployment of APs, was addressed in [16], where a closed-form
expression for SE was derived. Another aspect of performance analysis focused
on the practicality of CF mMIMO systems. In [17], a stochastic geometry
approach was utilized to model the locations of APs in CF mMIMO systems, and
an expression for achievable rates was derived. Furthermore, authors in [18]
and [19] investigated the impact of mobility and oscillator phase noise on
system performance, respectively. In [20], authors considered limited feedback
capacity as well as hardware impairments in both APs and users, which leads to
the derivation of the corresponding expected SINR. Moreover, the performance
of CF mMIMO with other technologies, including low-resolution analog-to-
digital converter (ADC) [21], reconfigurable intelligent surfaces (RIS) [22],
simultaneous wireless information and power transfer (SWIPT) [23], non-
orthogonal multiple access (NOMA) [24], and rate-splitting multiple-access
(RSMA) [25], has also been extensively studied.

Existing analytical work provides a comprehensive and detailed
characterization of the average performance of CF mMIMO systems. However,
capturing the statistical properties of the SINR, which is a critical task in
CF mMIMO systems applied for emerging applications that focus on extreme
events, has not been addressed. The exploration of SINR distribution has been
a prominent research topic in MIMO systems. The SINR distribution for MMSE
MIMO systems was derived by authors of [26], and subsequent advancements by
[27] resulted in the derivation of the exact SINR distribution. Based on the
analyzed SINR distribution, numerous research has delved into the performance
evaluation of MIMO systems, encompassing metrics such as SE and EE [28],
uncoded error and outage probabilities, diversity-multiplexing gain tradeoff,
and coding gain [29]. As MIMO systems evolve into CF mMIMO systems, the
foundational performance analysis concerning SINR distribution continues to be
deficient. Besides, based on the existing research on the expected SINR, only
the lower performance bounds of the CF mMIMO system can be obtained, such as
the lower bound of achievable rates. However, this fails to exactly reflect
the actual system performance. To more precisely characterize the system’s
performance, it becomes imperative to acquire the SINR distribution.
Furthermore, for emerging applications focused on ultra-reliable low-latency
communication, such as intelligent driving, remote healthcare, and tactile
internet, analyzing the system’s performance under extreme conditions becomes
even more crucial, clearly going beyond what can be achieved by relying solely
on the expected SINR. Analyzing the statistical performance of the CF mMIMO
system becomes paramount in such cases. Analyzing the distribution of SINR is
becoming increasingly crucial. While there are already existing studies on the
distribution of signal-to-noise ratio (SNR) in CF mMIMO systems [30, 31], the
impact of interference remains challenging to disregard in many cases.
Therefore, conducting research on the statistical properties of SINR in CF
mMIMO systems is necessary to gain a deeper understanding of their
characteristics.

In this paper, we investigate the statistical characteristics of the SINR in
CF mMIMO systems. It should be noted that such analytical work is non-trivial.
The dense deployment of APs introduces complexities due to the increased
number of signals received by all users. These signals include both target
signals and interference signals originating from multiple APs. Each received
signal represents the accumulation of numerous individual signals, resulting
in intricate interactions among different signals. Furthermore, the reuse of
pilot sequences among different users in the system leads to channel
estimation correlation between them. The correlation introduces coherence in
the precoding vectors and causes the signals transmitted to users employing
the same pilot to exhibit coherence. We overcome the aforementioned challenges
and obtain the distribution of the SINR in CF mMIMO systems. The main
contributions are summarized as follows:

  * •

We consider data transmission in a downlink CF mMIMO system with pilot
contamination. Based on the received signals at the users, we give the
expressions of the SINR when MRT and full-pilot ZF (FZF) precoding schemes are
involved.

  * •

We conduct an analysis of the SINR distribution under MRT and FZF precoding.
Specifically, we begin by decomposing the SINR into two components: the
desired signal (DS) and the interference plus noise (IN). For MRT precoding,
we leverage the Central Limit Theorem (CLT) and random matrix theory to
separately analyze the distributions of DS and IN. This allows us to derive
the PDF and CDF of SINR, taking into account the independence between DS and
IN. In the case of FZF precoding, we directly compute the distribution of DS
and analyze the distribution of IN accordingly. Subsequently, we derive the
overall distribution of SINR based on these individual components.

  * •

In order to compare with the lower bound of the achievable rate, we derive
closed-form expressions for the achievable rate and the outage probability
based on our analytical work on the SINR distribution under the MRT and FZF
precoding, respectively. To validate our derived results, we conduct Monte
Carlo simulations. The simulation results demonstrate the effectiveness of our
analytical work on the distribution of SINR, achievable rate, and outage
probability.

The rest of the paper is organized as follows. Section II gives the downlink
CF mMIMO system with pilot contamination and the expression of SINR. The
distribution of SINR under MRT and FZF precoding for the CF mMIMO system is
obtained in Section III, respectively. Section IV gives closed-form
expressions for the achievable rate and the outage probability under the MRT
and FZF precoding, respectively. Section V presents the simulation results and
analysis. Finally, the conclusion is drawn in Section VI.

_Notations:_ In this paper, vectors, and matrices are denoted by lowercase and
uppercase bold letters, respectively. For a general matrix $\mathbf{A}$,
$\mathbf{A}^{H}$ and $\mathbf{A}^{-1}$represent the Hermitian and inverse of
$\mathbf{A}$, respectively. $\left|\cdot\right|$ and $\left(\cdot\right)^{*}$
denote the modulus and the conjugate of a complex number, respectively.
$\left\|\mathbf{x}\right\|$ denotes $\ell_{2}$ norm of vector $\mathbf{x}$ and
$\mathbb{C}^{x\times y}$ represents the space of $x\times y$ complex number
matrices. $\mathbb{E}\left\\{x\right\\}$ denote the expected value of $x$ and
the calligraphy upper-case letter, such as $\mathcal{K}$, denotes a set.

##  II System Model

TABLE I: List of Key Notations Symbol |  Description
---|---
$\mathcal{M}$, $\mathcal{K}$ |  Index set of APs and users
$N$ |  Number of antennas each AP
$\mathbf{h}_{mk}$ |  Actual channel between AP $m$ and user $k$
$\hat{\mathbf{h}}_{mk}$ |  Estimated channel between AP $m$ and user $k$
$\bar{\mathbf{h}}_{mk}$ |  Channel estimated error between AP $m$ and user $k$
$\bar{\mathbf{H}}_{m}$ |  Full-rank matrix of channel estimates at AP $m$
$\beta_{mk}$ |  Large scale fading between AP $m$ and user $k$
$c_{mk}$ |  Estimated large scale fading between AP $m$ and user $k$
$l_{p}$ |  Length of pilot sequence
$\mathcal{P}_{k}$ |  Index set of users which use the same pilot sequence with user $k$
$\rho_{p}$, $\rho_{d}$ |  Normalized pilot power and normalized downlink transmission power each AP
$\mathbf{b}_{mk}$ |  Precoding matrix at AP $m$ for user $k$
$\text{DS}_{k}$, $\text{IN}_{k}$ |  DS and IN for user $k$

We consider a downlink CF mMIMO system operating in time-division duplex
(TDD), where a total of $M$ APs, each equipped with $N$ antennas, have the
capability to jointly and coherently serve $K$ users, each equipped with a
single antenna. The set of APs and users are denoted by $\mathcal{M}$ and
$\mathcal{K}$, respectively. All APs are connected to a CPU through backhauls.
The channel vector $\mathbf{h}_{mk}$ between AP $m$ and user $k$ is modeled as

$\mathbf{h}_{mk}=\beta_{mk}^{\frac{1}{2}}\mathbf{g}_{mk},$ (1)

where $\beta_{mk}$ represents the large-scale fading coefficient which is
influenced by path loss and shadowing fading, and
$\mathbf{g}_{mk}\sim\mathcal{CN}(0,\mathbf{I}_{N})$ denotes the small-scale
fading. Table I lists a compilation of the main notations used in this paper.

###  II-A Uplink Training and Channel Estimation

During the uplink training phase, each user simultaneously transmits a pilot
sequence to all APs, with the length of the pilot sequence denoted as $l_{p}$.
Define $i_{k}\in\left\\{1,\dots,l_{p}\right\\}$ as the index of the pilot used
by user $k$, represented by
$\boldsymbol{\phi}_{i_{k}}\in\mathbb{C}^{l_{p}\times 1}$. We assume that
$l_{p}\leq K$, which indicates that some users will share the same pilot
sequence. Consequently, we define $\mathcal{P}_{k}\subset\mathcal{K}$ as the
set of indices, including $k$, corresponding to users assigned the same pilot
as user $k$. Therefore, for any user $k_{1}$ where $k_{1}\neq k$, the
condition $i_{k}=i_{k_{1}}\Leftrightarrow k_{1}\in\mathcal{P}_{k}$ holds true.
The pilot sequences are mutually orthogonal and the inner product of pilot
sequences is given by

$\boldsymbol{\phi}_{i_{k_{1}}}^{H}\boldsymbol{\phi}_{i_{k}}=\left\\{\begin{aligned}
&0,&&k_{1}\notin\mathcal{P}_{k},\\\
&l_{p},&&k_{1}\in\mathcal{P}_{k}.\end{aligned}\right.$ (2)

The pilot signal received by AP $m$, denoted as $\mathbf{Y}_{m}^{p}$, can be
expressed as

$\mathbf{Y}_{m}^{p}=\sqrt{\rho_{p}}\sum_{k\in\mathcal{K}}\mathbf{h}_{mk}\boldsymbol{\phi}_{i_{k}}^{H}+\mathbf{Z}_{m}^{p},$
(3)

where $\rho_{p}$ denotes the normalized pilot power, and
$\mathbf{Z}_{m}^{p}\in\mathbb{C}^{N\times l_{p}}$ corresponds to the additive
Gaussian noise matrix at AP $m$ during the uplink training phase, each element
of which is independent and follows the distribution of $\mathcal{CN}(0,1)$.

To estimate the channel to user $k$, AP $m$ performs a correlation operation
between the received pilot signal and the corresponding pilot sequence
$\boldsymbol{\phi}_{i_{k}}$. Subsequently, it applies the MMSE technique to
obtain the MMSE channel estimate $\hat{\mathbf{h}}_{mk}$ [12], which can be
calculated as

$\hat{\mathbf{h}}_{mk}=\kappa_{mk}\mathbf{Y}_{m}^{p}\boldsymbol{\phi}_{i_{k}},$
(4)

where $\kappa_{mk}$ is defined as

$\kappa_{mk}=\frac{\sqrt{\rho_{p}}\beta_{mk}}{l_{p}\rho_{p}{\textstyle\sum_{k_{1}\in\mathcal{P}_{k}}\beta_{mk_{1}}}+1}.$
(5)

The channel estimation follows the distribution of
$\mathcal{CN}(0,c_{mk}\mathbf{I}_{N})$ with $c_{mk}$ given by [12]

$c_{mk}=\frac{l_{p}\rho_{p}\beta_{mk}^{2}}{l_{p}\rho_{p}{\textstyle\sum_{k_{1}\in\mathcal{P}_{k}}\beta_{mk_{1}}}+1}.$
(6)

Then, denote $\bar{\mathbf{h}}_{mk}=\mathbf{h}_{mk}-\hat{\mathbf{h}}_{mk}$ as
the channel estimation error, which is independent of $\hat{\mathbf{h}}_{mk}$
and follows the distribution of
$\mathcal{CN}(0,(\beta_{mk}-c_{mk})\mathbf{I}_{N})$. When multiple users share
the same pilot sequence, their channel estimates are parallel to each other.
This relationship can be expressed as

$\hat{\mathbf{h}}_{mk}=\frac{\beta_{mk}}{\beta_{mk_{1}}}\hat{\mathbf{h}}_{mk_{1}},\
k_{1}\in\mathcal{P}_{k}.$ (7)

###  II-B Downlink Data Transmission

During downlink data training, all APs transmit signals to all users. The
signal transmitted by AP $m$ is given by

$\mathbf{y}_{m}^{d}=\sqrt{\rho_{d}}\sum_{k\in\mathcal{K}}\sqrt{\eta_{mk}}\mathbf{b}_{mk}q_{k},$
(8)

where $\rho_{d}$ represents the normalized downlink transmission power each AP
and $\eta_{mk}$ denotes power control coefficient from AP $m$ to user $k$. The
symbol $q_{k}$, which satisfies
$\mathbb{E}\left\\{\left|q_{k}\right|^{2}\right\\}=1$, is the symbol intended
for the user $k$ during downlink data transmission and $\mathbf{b}_{mk}$ is
precoding vector.

The precoding vector $\mathbf{b}_{mk}$ for user $k$ at AP $m$ is determined
based on the channel estimation. The matrix of the channel estimates,
$\hat{\mathbf{H}}_{m}=\left[\hat{\mathbf{h}}_{m1},\hat{\mathbf{h}}_{m2},\cdots,\hat{\mathbf{h}}_{mK}\right]\in\mathbb{C}^{N\times
K}$, is rank-deficient due to the pilot sequence length being smaller than the
number of users. To obtain a full-rank matrix of channel estimates, denoted as
$\bar{\mathbf{H}}_{m}\in\mathbb{C}^{N\times l_{p}}$, we define
$\bar{\mathbf{H}}_{m}=\mathbf{Y}_{m}^{p}\boldsymbol{\Phi}$, where
$\boldsymbol{\Phi}=\left[\boldsymbol{\phi}_{1},\dots,\boldsymbol{\phi}_{l_{p}}\right]\in\mathbb{C}^{l_{p}\times
l_{p}}$ is the pilot-book matrix [12]. The channel estimate between AP $m$ and
user $k$ can be expressed in terms of $\bar{\mathbf{H}}_{m}$ as

$\hat{\mathbf{h}}_{mk}=\kappa_{mk}\bar{\mathbf{H}}_{m}\mathbf{\mathbf{e}_{i_{k}}},$
(9)

where $\mathbf{e}_{k}$ represent the $i_{k}$th column of unit matrix
$\mathbf{I}_{l_{p}}$. We consider using MRT and FZF precoding during downlink
data transmission [11, 12], the precoding vector $\mathbf{b}_{mk}$ for user
$k$ can be expressed as follows:

$\mathbf{b}_{mk}=\left\\{\begin{aligned}
&\frac{\bar{\mathbf{H}}_{m}\mathbf{e}_{i_{k}}}{\sqrt{\mathbb{E}\left\\{\left\|\bar{\mathbf{H}}_{m}\mathbf{e}_{i_{k}}\right\|^{2}\right\\}}},&&\text{MRT},\\\
&\frac{\bar{\mathbf{H}}_{m}\left(\bar{\mathbf{H}}_{m}^{H}\bar{\mathbf{H}}_{m}\right)^{-1}\mathbf{e}_{i_{k}}}{\sqrt{\mathbb{E}\left\\{\left\|\bar{\mathbf{H}}_{m}\left[\bar{\mathbf{H}}_{m}^{H}\bar{\mathbf{H}}_{m}\right]^{-1}\mathbf{e}_{i_{k}}\right\|^{2}\right\\}}},&&\text{FZF}.\\\
\end{aligned}\right.$ (10)

Each user receives the signals from all APs and the received signal at user
$k$ is

$r_{k}^{d}=\sqrt{\rho_{d}}\sum_{m\in\mathcal{M}}\sum_{k_{1}\in\mathcal{K}}\sqrt{\eta_{mk_{1}}}\mathbf{h}_{mk}^{H}\mathbf{b}_{mk_{1}}q_{k_{1}}+z_{k},$
(11)

where $z_{k}$ is the noise at user $k$ with the distribution of
$\mathcal{CN}(0,1)$. Then, the SINR of user $k$ during downlink data
transmission is given in (12).

$\displaystyle\gamma_{k}=\frac{\rho_{d}\left|\sum_{m\in\mathcal{M}}\sqrt{\eta_{mk}}\hat{\mathbf{h}}_{mk}^{H}\mathbf{b}_{mk}\right|^{2}}{\rho_{d}\sum_{k_{1}\neq
k}\left|\sum_{m\in\mathcal{M}}\sqrt{\eta_{mk_{1}}}\hat{\mathbf{h}}_{mk}^{H}\mathbf{b}_{mk_{1}}\right|^{2}+\rho_{d}\sum_{k_{1}\in\mathcal{K}}\left|\sum_{m\in\mathcal{M}}\sqrt{\eta_{mk_{1}}}\bar{\mathbf{h}}_{mk}^{H}\mathbf{b}_{mk_{1}}\right|^{2}+z_{k}}.$
(12)

##  III Distribution of SINR for CF mMIMO System

In this section, we derive the distribution of the SINR for the CF mMIMO
system. To begin, we partition the received signal from the user into two
components: the DS and the IN, utilizing the expression of SINR. Subsequently,
we investigate the distribution of DS and IN under the MRT precoding scheme,
employing the CLT and random matrix theory. By leveraging the independence
properties of DS and IN, we obtain the distribution of SINR. Furthermore, we
proceed to change the precoding vector, transitioning to the FZF scheme. We
calculate the value of DS and obtain the distribution of IN. Finally, based on
the aforementioned analysis regarding DS and IN, we derive the distribution of
SINR for the CF mMIMO system.

Based on the expression of SINR for user $k$, we divide the signal that user
$k$ received into DS and IN. Then the expression of SINR $\gamma_{k}$ can be
represented as

$\gamma_{k}=\frac{\text{DS}_{k}}{\text{IN}_{k}}=\frac{\rho_{d}U_{k}^{1}}{\rho_{d}\sum_{k_{1}\neq
k}U_{kk_{1}}^{2}+\rho_{d}\sum_{k_{1}\in\mathcal{K}}U_{kk_{1}}^{3}+z_{k}},$
(13)

where $\text{DS}_{k}$ and $\text{IN}_{k}$ represent the DS and IN for user
$k$. The expression of $\text{DS}_{k}$ and $\text{IN}_{k}$ is given as
follows:

$\displaystyle\text{DS}_{k}$ $\displaystyle=\rho_{d}U_{k}^{1},$ (14)
$\displaystyle\text{IN}_{k}$ $\displaystyle=\rho_{d}\sum_{k_{1}\neq
k}U_{kk_{1}}^{2}+\rho_{d}\sum_{k_{1}\in\mathcal{K}}U_{kk_{1}}^{3}+z_{k},$

The expression of $U_{k}^{1}$, $U_{kk_{1}}^{2}$ and $U_{kk_{1}}^{3}$ are given
by

$\displaystyle
U_{k}^{1}=\left|\sum_{m\in\mathcal{M}}\sqrt{\eta_{mk}}\hat{\mathbf{h}}_{mk}^{H}\mathbf{b}_{mk}\right|^{2},$
(15a) $\displaystyle
U_{kk_{1}}^{2}=\left|\sum_{m\in\mathcal{M}}\sqrt{\eta_{mk_{1}}}\hat{\mathbf{h}}_{mk}^{H}\mathbf{b}_{mk_{1}}\right|^{2},$
(15b) $\displaystyle
U_{kk_{1}}^{3}=\left|\sum_{m\in\mathcal{M}}\sqrt{\eta_{mk_{1}}}\bar{\mathbf{h}}_{mk}^{H}\mathbf{b}_{mk_{1}}\right|^{2}.$
(15c)

For the convenience of the following analysis, we redefine
$U_{k}^{1}=\left|\xi_{mk}\right|^{2}$,
$U_{k}^{2}=\left|\xi_{mkk_{1}}\right|^{2}$ and
$U_{k}^{3}=\left|\psi_{mkk_{1}}\right|^{2}$, where $\xi_{mk}$, $\xi_{mkk_{1}}$
and $\psi_{mkk_{1}}$ is given by
$\xi_{mk}=\sqrt{\eta_{mk}}\hat{\mathbf{h}}_{mk}^{H}\mathbf{b}_{mk}$,
$\xi_{mkk_{1}}=\sqrt{\eta_{mk_{1}}}\hat{\mathbf{h}}_{mk}^{H}\mathbf{b}_{mk_{1}}$
and
$\psi_{mkk_{1}}=\sqrt{\eta_{mk_{1}}}\bar{\mathbf{h}}_{mk}^{H}\mathbf{b}_{mk_{1}}$,
respectively.

###  III-A Maximum Ratio Transmission

In this subsection, we derive the distribution of SINR when MRT precoding is
used. In the presence of multiple users receiving signals from all APs
simultaneously, it becomes evident from equation (12) that all signals are
interconnected, giving rise to the formation of the DS and the IN.
Consequently, directly analyzing the distribution of SINR under MRT precoding
is a challenging task. To overcome this difficulty, we adopt a feasible
approach in the subsequent analysis. Firstly, we scrutinize the expressions
for the distribution of DS and IN individually. Subsequently, by utilizing the
derived expressions for the distributions of DS and IN, we are able to obtain
the distribution of SINR.

When the number of APs $M$ and number of antennas at each AP $N$ are large
enough, the distribution of
$\sum_{m\in\mathcal{M}}\sqrt{\eta_{k}}\mathbf{b}_{mk}^{H}\hat{\mathbf{h}}_{mk}$
is approximated as a Gaussian distribution. Then we can approximate
$U_{k}^{1}$ as a Gamma distribution in the following lemma.

######  Lemma 1.

When MRT precoding is used for downlink data transmission, the distribution of
$U_{k}^{1}$ can be approximated as a Gamma distribution with the shape
parameter $j_{k1}$ and the scale parameter $\chi_{k1}$, i.e., $U_{k}^{1}\sim
Gamma\left(j_{k1},\chi_{k1}\right)$. Therefore, the PDF and cumulative
distribution function (CDF) of $U_{k}^{1}$ is expressed as follows:

$\left\\{\begin{aligned}
&f_{U_{k}^{1}}(x)=\frac{1}{\Gamma(j_{k1})\chi_{k1}^{j_{k1}}}x^{j_{k1}-1}e^{-\frac{x}{\chi_{k1}}},\\\
&F_{U_{k}^{1}}(x)=\frac{1}{\Gamma(j_{k1})}\bar{\gamma}\left(j_{k1},\frac{x}{\chi_{k1}}\right),\end{aligned}\right.$
(16)

where $\Gamma(z)=\int_{0}^{+\infty}t^{z-1}e^{-t}dt$ and
$\bar{\gamma}(a,z)=\int_{0}^{z}t^{a-1}e^{-t}dt$ represent the gamma function
and incomplete gamma function, respectively. The shape parameter $j_{k1}$ and
scale parameter $\chi_{k1}$ are given by

$j_{k1}=\frac{u_{U_{k}^{1}}^{2}}{u_{U_{k}^{1}}^{(2)}-u_{U_{k}^{1}}^{2}},\
\chi_{k1}=\frac{u_{U_{k}^{1}}^{(2)}-u_{U_{k}^{1}}^{2}}{u_{U_{k}^{1}}},$ (17)

where $u_{U_{k}^{1}}=\mathbb{E}\left\\{U_{k}^{1}\right\\}$ and
$u_{U_{k}^{1}}^{(2)}=\mathbb{E}\left\\{\left(U_{k}^{1}\right)^{2}\right\\}$
denote the first and second moments of $U_{k}^{1}$ respectively. The
expressions for $u_{U_{k}^{1}}$ and $u_{U_{k}^{1}}^{(2)}$ are given in (40)
and (41), respectively.

###### Proof.

Please refer to Appendix A. ∎

Since $\text{DS}_{k}=\rho_{d}U_{k}^{1}$, we have
$\mathbb{E}\left\\{\text{DS}_{k}\right\\}=\rho_{d}\mathbb{E}\left\\{U_{k}^{1}\right\\}$
and
$\mathbb{E}\left\\{\text{DS}_{k}^{2}\right\\}=\rho_{d}\mathbb{E}\left\\{\left(U_{k}^{1}\right)^{2}\right\\}$.
Then, $\text{DS}_{k}^{\text{MRT}}$ can be approximated as a Gamma distribution
with shape parameter $j_{k1}$ and scale parameter $\rho_{d}\chi_{k1}$, i.e.,
$\text{DS}_{k}^{\text{MRT}}\sim\text{Gamma}(j_{k1},\rho_{d}\chi_{k1})$, where
$\text{DS}_{k}^{\text{MRT}}$ represents the $\text{DS}_{k}$ with MRT
precoding.

Then we turn to analyze the distribution of $\text{IN}_{k}^{\text{MRT}}$,
where $\text{IN}_{k}^{\text{MRT}}$ represents the $\text{IN}_{k}$ with MRT
precoding. Similarly, the distribution of $\text{IN}_{k}^{\text{MRT}}$ can be
approximated as a Gamma distribution. In the following lemma, we approximate
$\text{IN}_{k}^{\text{MRT}}$ as a Gamma distribution.

######  Lemma 2.

When MRT precoding is used for downlink data transmission, the distribution of
$\text{IN}_{k}^{\text{MRT}}=\rho_{d}\sum_{k_{1}\neq
k}U_{kk_{1}}^{2}+\rho_{d}\sum_{k_{1}\in\mathcal{K}}U_{kk_{1}}^{3}+z_{k}$ can
be approximated as a Gamma distribution with the shape parameter $j_{k2}$ and
the scale parameter $\chi_{k2}$, i.e.,
$\text{IN}_{k}^{\text{MRT}}\sim\text{Gamma}(j_{k2},\chi_{k2})$. The shape
parameter $j_{k2}$ and scale parameter $\chi_{k2}$ are given by

$j_{k2}=\frac{u_{\text{IN}_{k}^{\text{MRT}}}^{2}}{u_{\text{IN}_{k}^{\text{MRT}}}^{(2)}-u_{\text{IN}_{k}^{\text{MRT}}}^{2}},\
\chi_{k2}=\frac{u_{\text{IN}_{k}^{\text{MRT}}}^{(2)}-u_{\text{IN}_{k}^{\text{MRT}}}^{2}}{u_{\text{IN}_{k}^{\text{MRT}}}},$
(18)

where
$u_{\text{IN}_{k}^{\text{MRT}}}=\mathbb{E}\left\\{\text{IN}_{k}^{\text{MRT}}\right\\}$
and
$u_{\text{IN}_{k}^{\text{MRT}}}^{(2)}=\mathbb{E}\left\\{\left(\text{IN}_{k}^{\text{MRT}}\right)^{2}\right\\}$
are first and second moments of $\text{IN}_{k}^{\text{MRT}}$, respectively.
The expressions for $u_{\text{IN}_{k}^{\text{MRT}}}$ and
$u_{\text{IN}_{k}^{\text{MRT}}}^{(2)}$ are given in (50) and (51),
respectively.

###### Proof.

Please refer to Appendix B. ∎

Based on the above analysis, we have obtained the distributions of the two
components of the SINR: DS and IN. By leveraging the independence property
between these two components, we can obtain the distribution of SINR in the CF
mMIMO system which is shown in the following theorem.

######  Theorem 1.

In the CF mMIMO system with pilot contamination, when MRT precoding is used
for downlink data transmission, the PDF and CDF of the SINR for user $k$,
$\forall k\in\mathcal{K}$ can be expressed as follows:

$\left\\{\begin{aligned}
f_{\gamma_{k}}^{\text{MRT}}(x)=&\frac{\Gamma(j_{k1}+j_{k2})x^{j_{k1}-1}(\frac{1}{\chi_{k2}}+\frac{x}{\rho_{d}\chi_{k1}})^{-j_{k1}-j_{k2}}}{\Gamma(j_{k1})\Gamma(j_{k2})\left(\rho_{d}\chi_{k1}\right)^{j_{k1}}\chi_{k2}^{j_{k2}}},\\\
F_{\gamma_{k}}^{\text{MRT}}(x)=&\frac{\Gamma(j_{k1}+j_{k2})\chi_{k2}^{j_{k1}}x^{j_{k1}}}{j_{k1}\Gamma(j_{k1})\Gamma(j_{k2})\left(\rho_{d}\chi_{k1}\right)^{j_{k1}}}\\\
&\times
H(j_{k1},j_{k1}+j_{k2},j_{k1}+1,-\frac{\chi_{k2}x}{\rho_{d}\chi_{k1}}),\end{aligned}\right.$
(19)

where $j_{k1}$, $\chi_{k1}$, $j_{k2}$, and $\chi_{k2}$ are given in Lemma 1
and Lemma 2, respectively, $H(a,b,c)$ is the hypergeometric function [32].

###### Proof.

Please refer to Appendix C. ∎

###  III-B Full-Pilot Zero-Forcing

Different from MRT precoding, FZF precoding can suppress inter-user
interference. The FZF precoding vector used by AP $m$ towards user $k$ is
given in (10), and the denominator of the precoding vector is given in closed
form by [12]

$\displaystyle\mathbb{E}\left\\{\left\|\bar{\mathbf{H}}_{m}\left[\bar{\mathbf{H}}_{m}^{H}\bar{\mathbf{H}}_{m}\right]^{-1}\mathbf{e}_{i_{k}}\right\|^{2}\right\\}$
$\displaystyle=\mathbb{E}\left\\{\left[\left(\bar{\mathbf{H}}_{m}^{H}\bar{\mathbf{H}}_{m}\right)^{-1}\right]_{i_{k}i_{k}}\right\\}$
(20) $\displaystyle\overset{(a)}{=}\frac{\kappa_{mk}^{2}}{(N-l_{p})c_{mk}},$

where $(a)$ is obtained based on [33, Lemma 2.10], for a $l_{p}\times l_{p}$
central complex Wishart matrix with $M$ degrees of freedom satisfying $M\geq
l_{p}+1$. Interference between users using different pilot sequences is
suppressed and the product between $\hat{\mathbf{h}}_{mk}^{H}$ and
$\mathbf{b}_{mk^{\prime}}$ can be calculated as follows [12]:

$\displaystyle\alpha_{mkk_{1}}=\hat{\mathbf{h}}_{mk}^{H}\mathbf{b}_{mk_{1}}$
(21)
$\displaystyle=(\kappa_{mk}\bar{\mathbf{H}}_{m}\mathbf{e}_{i_{k}})^{H}\bar{\mathbf{H}}_{m}\left(\bar{\mathbf{H}}_{m}^{H}\bar{\mathbf{H}}_{m}\right)^{-1}\mathbf{e}_{i_{k_{1}}}\sqrt{\frac{(N-l_{p})c_{mk}}{\kappa_{mk}^{2}}}$
$\displaystyle=\left\\{\begin{aligned} &0,&&k_{1}\notin\mathcal{P}_{k},\\\
&\sqrt{(N-l_{p})c_{mk}},&&k_{1}\in\mathcal{P}_{k}.\end{aligned}\right.$

Then we can obtain the value of $\text{DS}_{k}$ based on (21) as follows:

$\displaystyle\text{DS}_{k}^{\text{FZF}}$
$\displaystyle=\rho_{d}\left({\textstyle\sum_{m\in\mathcal{M}}}\sqrt{\eta_{mk}}\alpha_{mkk}\right)^{2}$
(22)
$\displaystyle=\rho_{d}\left({\textstyle\sum_{m\in\mathcal{M}}}\sqrt{\eta_{mk}(N-l_{p})c_{mk}}\right)^{2},$

where $\text{DS}_{k}^{\text{FZF}}$ represents the $\text{DS}_{k}$ with FZF
precoding.

Similar to the analysis under MRT precoding, we approximate the distribution
of $\text{IN}_{k}^{\text{FZF}}$ under FZF precoding as a Gamma distribution in
the following lemma.

######  Lemma 3.

When FZF precoding is used for downlink data transmission, the distribution of
$\text{IN}_{k}^{\text{FZF}}=\rho_{d}\sum_{k_{1}\neq
k}U_{kk_{1}}^{2}+\rho_{d}\sum_{k_{1}\in\mathcal{K}}U_{kk_{1}}^{3}+z_{k}$ can
be approximated as a Gamma distribution with the shape parameter $j_{k2}$ and
the scale parameter $\chi_{k2}$, i.e., $\text{IN}_{k}^{\text{FZF}}\sim
Gamma(j_{k2},\chi_{k2})$. The shape parameter $j_{k2}$ and scale parameter
$\chi_{k2}$ are given by

$j_{k2}=\frac{u_{\text{IN}_{k}^{\text{FZF}}}^{2}}{u_{\text{IN}_{k}^{\text{FZF}}}^{(2)}-u_{\text{IN}_{k}^{\text{FZF}}}^{2}},\
\chi_{k2}=\frac{u_{\text{IN}_{k}^{\text{FZF}}}^{(2)}-u_{\text{IN}_{k}^{\text{FZF}}}^{2}}{u_{\text{IN}_{k}^{\text{FZF}}}},$
(23)

where
$u_{\text{IN}_{k}^{\text{FZF}}}=\mathbb{E}\left\\{\text{IN}_{k}^{\text{FZF}}\right\\}$
and
$u_{\text{IN}_{k}^{\text{FZF}}}^{(2)}=\mathbb{E}\left\\{\left(\text{IN}_{k}^{\text{FZF}}\right)^{2}\right\\}$
are first and second moments of $\text{IN}_{k}^{\text{FZF}}$, respectively.
The expressions for $u_{\text{IN}_{k}^{\text{FZF}}}$ and
$u_{\text{IN}_{k}^{\text{FZF}}}^{(2)}$ are given in (26) and (27),
respectively.

###### Proof.

Define $U_{k}^{2}=\rho_{d}\sum_{k_{1}\neq k}U_{kk_{1}}^{2}$, similar to (22),
the value of $U_{k}^{2}$ can be expressed as follows:

$U_{k}^{2}=\rho_{d}\sum_{k_{1}\neq
k}\left(\sum_{m\in\mathcal{M}}\sqrt{\eta_{mk_{1}}}\alpha_{mkk_{1}}\right)^{2}.$
(24)

Similar to (48), the second and fourth moment of $\psi^{d}_{mkk_{1}}$ can be
calculated as follows:

$\displaystyle\mathbb{E}\left\\{\left|\psi_{mkk_{1}}\right|^{2}\right\\}=\eta_{mk_{1}}\left(\beta_{mk}-c_{mk}\right)$
(25)
$\displaystyle\mathbb{E}\left\\{\left|\psi_{mkk_{1}}\right|^{4}\right\\}=\eta_{mk_{1}}^{2}\frac{2\left(N+1\right)}{N}\left(\beta_{mk}-c_{mk}\right)^{2}$

Then we can obtain the first moment and second moment of $U_{kk_{1}}^{3}$
under FZF precoding, i.e., $\mathbb{E}\left\\{U_{kk_{1}}^{3}\right\\}$,
$\mathbb{E}\left\\{\left(U_{kk_{1}}^{3}\right)^{2}\right\\}$, which is similar
to (49) in the proof of Lemma 2. Then first moment of
$\text{IN}_{k}^{\text{FZF}}$ can be expressed as follows:

$u_{\text{IN}_{k}^{\text{FZF}}}=U_{k}^{2}+\rho_{d}\sum_{k_{1}\in\mathcal{K}}\mathbb{E}\left\\{U_{kk_{1}}^{3}\right\\}+1.$
(26)

The second moments of $\text{IN}_{k}^{\text{FZF}}$ can be calculated as
follows:

$\displaystyle u_{\text{IN}_{k}^{\text{FZF}}}^{(2)}$ (27)
$\displaystyle=\rho_{d}^{2}\sum_{k_{1}\in\mathcal{K}}\mathbb{E}\left\\{\left(U_{kk_{1}}^{3}\right)^{2}\right\\}+2\rho_{d}\left(U_{k}^{2}+1\right)\sum_{k_{1}\in\mathcal{K}}\mathbb{E}\left\\{U_{kk_{1}}^{3}\right\\}$
$\displaystyle+\rho_{d}^{2}\sum_{k_{1}\in\mathcal{K}}\sum_{k_{2}\neq
k_{1}}\mathbb{E}\left\\{U_{kk_{1}}^{3}\right\\}\mathbb{E}\left\\{U_{kk_{2}}^{3}\right\\}+\left(U_{k}^{2}\right)^{2}+2U_{k}^{2}+2.$

∎

We calculate the value of $\text{DS}_{k}^{\text{FZF}}$ and obtain the
distribution of $\text{IN}_{k}^{\text{FZF}}$ above. Then the distribution of
SINR under the FZF precoding can be obtained as shown in the following
theorem.

######  Theorem 2.

In the CF mMIMO system with pilot contamination, when FZF precoding is used
for downlink data transmission, the PDF and CDF of the SINR for user $k$,
$\forall k\in\mathcal{K}$ can be expressed as follows:

$\displaystyle
f_{\gamma_{k}}^{\text{FZF}}(x)=\frac{1}{\Gamma(j_{k2})\chi_{k2}^{j_{k2}}x}\left(\frac{\text{DS}_{k}^{\text{FZF}}}{x}\right)^{j_{k2}}e^{-\frac{\text{DS}_{k}^{\text{FZF}}}{x\chi_{k2}}},$
(28) $\displaystyle
F_{\gamma_{k}}^{\text{FZF}}(x)=1-\frac{1}{\Gamma(j_{k2})}\bar{\gamma}\left(j_{k2},\frac{\text{DS}_{k}^{\text{FZF}}}{x\chi_{k2}}\right),$

where $\text{DS}_{k}^{\text{FZF}}$ is given in (22), $j_{k2}$, and $\chi_{k2}$
are given in Lemma 3.

###### Proof.

The CDF of $\gamma_{k}$ when using the FZF precoder can be calculated as
follows:

$\displaystyle F_{\gamma_{k}}^{\text{FZF}}(x)$
$\displaystyle=P\left\\{\frac{\text{DS}_{k}^{\text{FZF}}}{\text{IN}_{k}^{\text{FZF}}}\leq
x\right\\}=P\left\\{\text{IN}_{k}^{\text{FZF}}\geq\frac{\text{DS}_{k}^{\text{FZF}}}{x}\right\\}$
(29)
$\displaystyle=1-F_{\text{IN}_{k}^{\text{FZF}}}\left(\frac{\text{DS}_{k}^{\text{FZF}}}{x}\right),$

where $F_{\text{IN}_{k}^{\text{FZF}}}\left(\cdot\right)$ represents the CDF of
$\text{IN}_{k}^{\text{FZF}}$.

Based on (29), the PDF of $\gamma_{k}$ when using FZF precoder can be
calculated as follows:

$\displaystyle f_{\gamma_{k}}^{\text{FZF}}(x)$
$\displaystyle=\frac{\mathrm{d}F_{\gamma_{k}}^{\text{FZF}}(x)}{\mathrm{d}x}=-\frac{\mathrm{d}F_{\text{IN}_{k}}^{\text{FZF}}(\frac{\text{DS}_{k}^{\text{FZF}}}{x})}{\mathrm{d}x}$
(30)
$\displaystyle=\frac{\text{DS}_{k}^{\text{FZF}}}{x^{2}}f_{\text{IN}_{k}^{\text{FZF}}}\left(\frac{\text{DS}_{k}^{\text{FZF}}}{x}\right),$

where $f_{\text{IN}_{k}^{\text{FZF}}}\left(\cdot\right)$ represents the PDF of
$\text{IN}_{k}^{\text{FZF}}$. ∎

##  IV Performance Analysis

The lower bound of achievable rate for users in the CF mMIMO system has been
investigated [12]. However, to provide a more comprehensive characterization
of the system’s performance, this section focuses on deriving the achievable
rate of users under both MRT and FZF precoding. This analysis is based on the
previously obtained distribution of SINR, allowing for a more accurate
assessment of system performance. Besides, we also derive the outage
probability of the CF mMIMO system.

###  IV-A Maximum Rate Transmission

When MRT precoding is used in the CF mMIMO system, then the lower bound of
achievable downlink rate is given in (33) [12]. The achievable rate in the CF
mMIMO system with MRT precoding employed is given in the following lemma.

######  Lemma 4.

In the CF mMIMO system with pilot contamination, when MRT precoding is
employed for downlink data transmission, the achievable rate for user $k$,
$k\in\mathcal{K}$, is given in (35), where
$H^{P}\left(\left\\{\cdot\right\\},\left\\{\cdot\right\\},\left\\{\cdot\right\\}\right)$
represents the generalized hypergeometric function [34]. The values of
$j_{k1}$ and $\chi_{k1}$ are obtained from Lemma 1, while $j_{k2}$ and
$\chi_{k2}$ are derived from Lemma 2.

###### Proof.

When MRT precoding is employed for downlink data transmission, the PDF of SINR
for user $k$, $k\in\mathcal{K}$, is given in Theorem 1. Then the achievable
rate for user $k$ can be calculated directly by
$R_{k}^{\text{MRT}}=\int_{0}^{\infty}\log_{2}(1+x)f_{\gamma_{k}}^{\text{MRT}}(x)dx$.
∎

Consider the encoding rate of user $k$ is $r_{k}^{\text{MRT}}$ in the CF mMIMO
system unser MRT precoding. An outage event for user $k$ when the SINR cant
support the target rate $r_{k}^{\text{MRT}}$. The outage probability of user
$k$ in the CF mMIMO system is given by

$\displaystyle P_{\text{out},k}^{\text{MRT}}\left(r_{k}^{\text{MRT}}\right)$
$\displaystyle=P\left(\log_{2}\left(1+\gamma_{k}\leq
r_{k}^{\text{MRT}}\right)\right)$ (31)
$\displaystyle=F_{\gamma_{k}}^{\text{MRT}}\left(2^{r_{k}^{\text{MRT}}}-1\right).$

###  IV-B Full-Pilot Zero-Forcing

When FZF precoding is used in the CF mMIMO system, then the lower bound of
achievable downlink rate is given (34) [12].

The achievable rate in the CF mMIMO system with FZF precoding employed is
given in the following lemma.

######  Lemma 5.

In the CF mMIMO system with pilot contamination, when FZF precoding is
employed for downlink data transmission, the achievable rate for user $k$,
$k\in\mathcal{K}$, is given in (36), where $\text{DS}_{k}^{\text{FZF}}$ is
given in (22), while $j_{k2}$ and $\chi_{k2}$ are obtained from Lemma 3.

###### Proof.

The proof is similar to the proof of Lemma 4, which is omitted for simplicity.
∎

Consider the encoding rate of user $k$ is $r_{k}^{\text{MRT}}$ in the CF mMIMO
system under FZF precoding. The outage probability of user $k$ is given by

$\displaystyle
P_{\text{out},k}^{\text{MRT}}\left(r_{k}^{\text{MRT}}\right)=F_{\gamma_{k}}^{\text{FZF}}\left(2^{r_{k}^{\text{FZF}}}-1\right).$
(32)

$\bar{R}_{k}^{\text{MRT}}=\log_{2}\left(1+\frac{N\rho_{d}\left(\sum_{m\in\mathcal{M}}\sqrt{\eta_{mk}c_{mk}}\right)^{2}}{N\rho_{d}\sum_{k_{1}\in\mathcal{P}_{k}\setminus\left\\{k\right\\}}\left(\sum_{m\in\mathcal{M}}\sqrt{\eta_{mk}c_{mk}}\right)^{2}+\rho_{d}\sum_{m\in\mathcal{M}}\sum_{k}^{\mathcal{K}}\eta_{mk}\beta_{mk}+1}\right).$
(33)

$\displaystyle\bar{R}_{k}^{\text{FZF}}=\log_{2}\left(1+\frac{\left(N-l_{p}\right)\rho_{d}\left(\sum_{m\in\mathcal{M}}\sqrt{\eta_{mk}c_{mk}}\right)^{2}}{\left(N-l_{p}\right)\rho_{d}\sum_{k_{1}\in\mathcal{P}_{k}\setminus\left\\{k\right\\}}\left(\sum_{m\in\mathcal{M}}\sqrt{\eta_{mk}c_{mk}}\right)^{2}+\rho_{d}\sum_{m\in\mathcal{M}}\sum_{k}^{\mathcal{K}}\eta_{mk}\beta_{mk}+1}\right).$
(34)

$\displaystyle R_{k}^{\text{MRT}}=$
$\displaystyle\frac{\pi\csc(j_{k2}\pi)}{\Gamma(j_{k1})\Gamma(j_{k2})\ln
2}\left(\frac{\left(\rho_{d}\chi_{k1}\right)^{j_{k2}}\Gamma(j_{k1}+j_{k2})}{\chi_{k2}^{j_{k2}}\Gamma(1+j_{k2})}H(j_{k2},j_{k1}+j_{k2},1+j_{k2},\frac{\rho_{d}\chi_{k1}}{\chi_{k2}})\right.$
(35)
$\displaystyle\left.-\frac{\rho_{d}\chi_{k1}\Gamma(1+j_{k1})}{\chi_{k2}\Gamma(2-j_{k2})}H^{P}\left(\left\\{1,1,1+j_{k1}\right\\},\left\\{2,2-j_{k2}\right\\},\frac{\rho_{d}\chi_{k1}}{\chi_{k2}}\right)\right).$

$\displaystyle R_{k}^{\text{FZF}}=$
$\displaystyle\frac{\left(-1\right)^{-j_{k2}}\pi\csc(j_{k2}\pi)}{\Gamma(j_{k2})\ln
2}\bar{\gamma}\left(-\frac{\text{DS}_{k}^{\text{FZF}}}{\chi_{k2}},j_{k2}\right)+\frac{\text{DS}_{k}^{\text{FZF}}\Gamma(-1+j_{k2})}{\chi_{k2}\Gamma(j_{k2})\ln
2}H^{P}\left(\left\\{1,1\right\\},\left\\{2,2-j_{k2}\right\\},\frac{\text{DS}_{k}^{\text{FZF}}}{\chi_{k2}}\right).$
(36)

##  V Simulation Results

In this section, we validate our derived results by conducting Monte Carlo
simulations across various scenarios and subsequently perform corresponding
performance analysis.

###  V-A Simulation Setting

In our simulations, we consider a system of randomly distributed APs and users
within a rectangular area measuring $1km\times 1km$. The large-scale fading
coefficient, which accounts for both path loss and shadowing effects, is
denoted as $\beta_{mn}=PL_{mn}+z_{mn}$, where $PL_{mn}$ represents the path
loss component, while $z_{mn}\sim\mathcal{CN}(0,\delta_{sh}^{2})$ represents
the shadowing component following a complex Gaussian distribution with a mean
of zero and a variance of $\delta_{sh}^{2}$. To characterize the path loss, we
employ a three-slope model proposed in [11] and adopt the same parameter
settings as described in that paper. In addition, we set the pilot power to 20
dBm and the downlink transmit power to 23 dBm. The system bandwidth is 2 MHz
and the power of Gaussian noise is -174 dB/Hz. The pilots are sequentially
assigned to users, i.e., $i_{k}=\text{rem}\left(k,l_{p}\right)$, where
$\text{rem}\left(,\right)$ represents the remainder operation. We employ the
heuristic power allocation scheme introduced in [12], and the power allocation
coefficients are computed using the following equation:

$\eta_{mk}=\frac{c_{mk}}{{\textstyle\sum_{k\in\mathcal{K}}c_{mk}}},\forall
m,\forall k.$ (37)

For the specific configuration details regarding the number of APs, users, the
number of antennas per AP, and the pilot length in the system, we will provide
them during the corresponding experiments.

###  V-B CDF of SINR

Figure 1: CDF of DS under MRT precoding with different number of antennas each
AP. System parameters: $M$ = 120, $K$ = 20, $l_{p}$ = 10. Figure 2: CDF of IN
under MRT precoding with different number of antennas each AP. System
parameters: $M$ = 120, $K$ = 20, $l_{p}$ = 10.

Once MRT precoding is implemented in CF mMIMO systems, the CDF of DS and IN,
concerning the variation in the number of antennas at the AP, are illustrated
in Fig. 1 and Fig. 2, respectively. As anticipated, both DS and IN exhibit a
corresponding increase with the augmentation of antenna quantity. This
phenomenon can be attributed to the fact that a higher number of antennas
enables the AP to transmit a greater number of signals, consequently leading
to an enhancement in signal strength. Furthermore, the results derived from
our theoretical analysis align closely with the outcomes obtained through
Monte Carlo simulations. This congruence serves as evidence supporting the
accuracy of our conclusions.

Figure 3: CDF of IN under FZF precoding with different number of antennas each
AP. System parameters: $M$ = 120, $K$ = 20, $l_{p}$ = 10.

In Fig. 3, we investigate the impact of the variation in the number of
antennas at each AP on the CDF of IN when FZF precoding is deployed. As the
number of antennas increases at each AP, interference becomes more severe,
resulting in a proportional increase of IN. Moreover, the graph illustrates a
close alignment between the distribution derived from our theoretical analysis
and the distribution obtained through Monte Carlo simulations, providing
strong evidence that our conclusions are accurate.

Figure 4: CDF of SINR under FZF precoding with different number of users.
System parameters: $M$ = 120, $N$ = 11, $l_{p}$ = 10. Figure 5: CDF of SINR
under MRT precoding with different number of users. System parameters: $M$ =
120, $N$ = 2, $l_{p}$ = 10. Figure 6: CDF of SINR under FZF precoding with
different number of antennas each AP. System parameters: $M$ = 120, $K$ = 20,
$l_{p}$ = 10. Figure 7: CDF of SINR under MRT precoding with different number
of antennas each AP. System parameters: $M$ = 120, $K$ = 20, $l_{p}$ = 10.

We compare the CDF of the SINR for varying numbers of users under FZF
precoding, as illustrated in Fig. 4. It is evident that as the number of users
increases, the derived results closely converge with the Monte Carlo
simulation results, eventually reaching a consistent agreement. This
convergence is attributed to our approximation of the IN based on the CLT. As
the number of users grows, this approximation becomes increasingly accurate
and approaches the true results. Furthermore, with an increasing number of
users, the SINR correspondingly decreases. Despite the unchanged length of the
pilot sequence, more users share the same pilot, resulting in significant
pilot contamination and subsequently causing severe interference, ultimately
leading to a lower SINR.

In Fig. 5, we present the CDF of the SINR under the deployment of MRT
precoding, showcasing its variation with respect to the number of users. As
the number of users increases, the derived distribution of SINR exhibits a
growing consistency with the results obtained from Monte Carlo simulations.
This increasing alignment can be attributed to the heightened applicability of
the CLT as the number of users expands. Furthermore, similar to the
observation in the case of FZF precoding, the SINR decreases with an
increasing number of users. The reason for this degradation in SINR is the
same. Additionally, under MRT precoding, users who do not employ the same
pilot sequence also introduce interference with each other. Consequently,
employing FZF precoding in CF mMIMO systems generally yields higher SINR,
indicating superior performance compared to MRT precoding.

We turn our attention to comparing the impact of varying the number of
antennas on each AP under FZF precoding on the CDF of the SINR in CF mMIMO
systems, as depicted in Fig. 6. It is evident that as the number of antennas
increases, the derived results align more closely with the results obtained
from system analysis. This convergence can be attributed to the increased
number of constituent signals in the DS and IN due to the growing number of
antennas, thereby enhancing the accuracy of the approximation based on the
CLT. Furthermore, the increase in the number of antennas leads to an increase
in SINR. From (22), it can be observed that DS increases with the growing
number of antennas. While, according to (24), the $U_{kk_{1}}^{2}$ component
of IN also increases with the number of users, (25) indicates that
$U_{kk_{1}}^{3}$ remains unchanged with an increasing number of antennas.
Hence, an increase in the number of antennas results in an improvement in
SINR.

Fig. 7 illustrates the distribution of SINR under MRT precoding for different
numbers of antennas. Similar to the scenario with FZF precoding, it is
observed that as the number of antennas increases, the derived CDF of SINR
closely aligns with the results obtained from Monte Carlo simulations.
Furthermore, an increase in the number of antennas leads to an improvement in
SINR. As MRT precoding only mitigates the influence of noise, while FZF
precoding suppresses the effects of interference, FZF precoding generally
achieves better performance.

###  V-C Achievable Rate and Outage Probability Analysis

Figure 8: Achievable rate under FZF precoding with different number of
antennas each AP. System parameters: $M$ = 120, $l_{p}$ = 10. Figure 9:
Achievable rate under MRT precoding with different number of antennas each AP.
System parameters: $M$ = 120, $l_{p}$ = 10.

We have investigated the impact of varying the number of antennas on each AP
and the number of users on achievable rates under both FZF precoding and MRT
precoding, as depicted in Fig. 8 and Fig. 9, respectively. It can be observed
that the results derived from our analysis align with the results obtained
from Monte Carlo simulations. As the number of users decreases and the number
of antennas increases, the achievable rates also increase. This is because the
reduction in the number of users and the increase in the number of antennas
result in higher SINR, thereby leading to an increase in achievable rates.
Besides, it can be observed that our derived results are higher than the lower
bound with different scenarios. Under MRT precoding, our derived results are
significantly higher than the lower bound. However, under FZF precoding, our
derived results are only slightly higher than the lower bound. This is
because, with FZF precoding, both desired signals and interference can be
directly computed, and the uncertainty in SINR mainly comes from channel
estimation errors. Therefore, compared to MRT precoding, FZF precoding yields
a more stable SINR, resulting in actual achievable rates that are closer to
the lower bound.

Figure 10: Outage probability under FZF precoding with different number of
antennas each AP. System parameters: $M$ = 120, $l_{p}$ = 10. Figure 11:
Outage probability under MRT precoding with different number of antennas each
AP. System parameters: $M$ = 120, $l_{p}$ = 10.

We conducted separate investigations on the impact of different coding rates
and the number of antennas on each AP for outage probability under FZF
precoding and MRT precoding, as depicted in Fig. 10 and Fig. 11, respectively.
As anticipated, the outage probability increases continuously with higher
coding rates and a reduced number of antennas on the APs. Moreover, it is
noteworthy that our derived results closely align with the results obtained
from Monte Carlo simulations across various scenarios.

##  VI Conclusion

In this paper, we provided the PDF and CDF of the SINR in CF mMIMO systems
considering both MRT and FZF precoding schemes. Moreover, we have performed a
comprehensive performance analysis based on these results. Specifically, we
modeled a CF mMIMO system with pilot contamination and derived the expression
for the SINR. Then, by considering the independence of the signals, we divided
the SINR into two components, i.e., DS and IN. Under MRT precoding, we derived
the distributions of DS and IN using CLT and random matrix theory, enabling
the analysis of the distribution of the SINR. Under FZF precoding, we directly
computed the DS and derived the distribution of IN, which allowed us to obtain
the PDF and CDF of the SINR. Based on the aforementioned analysis, we derived
expressions for the achievable rate and the outage probability in CF mMIMO
systems under MRT and FZF precoding, respectively. Finally, simulation results
demonstrated the accuracy of our derivations.

Currently, research on the statistical characteristics of SINR in the CF mMIMO
system is still in its preliminary stage, and further investigation is
required. Firstly, when deploying other precoding schemes, such as local ZF or
MMSE precoding, in CF mMIMO systems, the corresponding SINR distributions also
need to be analyzed. Secondly, in a more realistic CF mMIMO system, factors
such as backhaul constraints and hardware impairments must be considered to
evaluate their impact on the SINR distribution and derive appropriate models.
Lastly, it would be a meaningful endeavor to design new power allocation
schemes, pilot allocation schemes, and other system design schemes based on
the SE and EE derived from our current analysis of the statistical
characteristics of SINR in CF mMIMO systems. These endeavors aim to better
harness the potential of CF mMIMO systems.

##  Appendix A Proof of Lemma 1

When MRT precoding is used for downlink data transmission, the precoding
vector can be expressed as

$\mathbf{b}_{mk}=\frac{\bar{\mathbf{H}}_{m}\mathbf{e}_{i_{k}}}{\sqrt{\mathbb{E}\left\\{\left\|\bar{\mathbf{H}}_{m}\mathbf{e}_{i_{k}}\right\|^{2}\right\\}}}=\frac{\hat{\mathbf{h}}_{mk}}{\sqrt{Nc_{mk}}}.$
(38)

Estimated channel $\hat{\mathbf{h}}_{mk}$ follows the distribution of
$\mathcal{CN}(0,c_{mk}\mathbf{I}_{N})$, then
$\xi_{mk}=\sqrt{\frac{\eta_{mk}}{Nc_{mk}}}\hat{\mathbf{h}}_{mk}^{H}\hat{\mathbf{h}}_{mk}$
follows Gamma distribution with shape parameter $N$ and scale parameter
$\sqrt{\frac{\eta_{mk}c_{mk}}{N}}$, i.e., $\xi_{mk}\sim
Gamma(N,\sqrt{\frac{\eta_{mk}c_{mk}}{N}})$.

Then the first moment to fourth moment of $\xi_{mk}$ can be expressed as
follows:

$\displaystyle\mathbb{E}\left\\{\xi_{mk}\right\\}=\sqrt{N\eta_{mk}c_{mk}},$
(39a)
$\displaystyle\mathbb{E}\left\\{\xi_{mk}^{2}\right\\}=(N+1)\eta_{mk}c_{mk},$
(39b)
$\displaystyle\mathbb{E}\left\\{\xi_{mk}^{3}\right\\}=\frac{(N+1)(N+2)}{\sqrt{N}}\eta_{mk}^{\frac{3}{2}}c_{mk}^{\frac{3}{2}},$
(39c)
$\displaystyle\mathbb{E}\left\\{\xi_{mk}^{4}\right\\}=\frac{(N+1)(N+2)(N+3)}{N}\eta_{mk}^{2}c_{mk}^{2}.$
(39d)

The first moment of the
$U_{k}^{1}=\left|\sum_{m\in\mathcal{M}}\xi_{mk}\right|^{2}$ can be expressed
as follows:

$\displaystyle u_{U_{k}^{1}}$
$\displaystyle=\mathbb{E}\left\\{U_{k}^{1}\right\\}$ (40)
$\displaystyle=\mathbb{E}\left\\{\left|\sum_{m\in\mathcal{M}}\xi_{mk}\right|^{2}\right\\}\overset{(a)}{=}\mathbb{E}\left\\{\left(\sum_{m\in\mathcal{M}}\xi_{mk}\right)^{2}\right\\}$
$\displaystyle\overset{(b)}{=}\sum_{m\in\mathcal{M}}\mathbb{E}\left\\{\xi_{mk}^{2}\right\\}+\sum_{m\in\mathcal{M}}\sum_{m_{1}\neq
m}\mathbb{E}\left\\{\xi_{mk}\right\\}\mathbb{E}\left\\{\xi_{m_{1}k}\right\\},$

where $(a)$ follows the fact that $\xi_{mk}$ is real variable, and $(b)$ is
obtained based on the independence between $\xi_{mk}$ and $\xi_{m^{\prime}k}$
when $m\neq m^{\prime}$. By inserting (39a) and (39b) into (40), we can obtain
the first moment of the $U_{k}^{1}$.

Similarly, the second moment of $U_{k}^{1}$ is given in (41). The second
moment of $U_{k}^{1}$ is obtained by using (39a)-(39d) and (41).

$\displaystyle u_{U_{k}^{1}}^{(2)}$
$\displaystyle=\mathbb{E}\left\\{\left(U_{k}^{1}\right)^{2}\right\\}=\mathbb{E}\left\\{\left|\sum_{m\in\mathcal{M}}\xi_{mk}\right|^{4}\right\\}=\mathbb{E}\left\\{\left(\sum_{m\in\mathcal{M}}\xi_{mk}\right)^{4}\right\\}=\mathbb{E}\left\\{\left(\sum_{m\in\mathcal{M}}\xi_{mk}^{2}+\sum_{m\in\mathcal{M}}\sum_{m1\neq
m}\xi_{mk}\xi_{m1k}\right)^{2}\right\\}$ (41)
$\displaystyle=\sum_{m\in\mathcal{M}}\mathbb{E}\left\\{\xi_{mk}^{4}\right\\}+\sum_{m\in\mathcal{M}}\sum_{m_{1}\neq
m}\sum_{m_{2}\neq
m,m_{1}}6\mathbb{E}\left\\{\xi_{mk}^{2}\right\\}\mathbb{E}\left\\{\xi_{m_{1}k}\right\\}\mathbb{E}\left\\{\xi_{m_{2}k}\right\\}+\sum_{m\in\mathcal{M}}\sum_{m_{1}\neq
m}4\mathbb{E}\left\\{\xi_{mk}^{3}\right\\}\mathbb{E}\left\\{\xi_{m_{1}k}\right\\}$
$\displaystyle\quad+\sum_{m\in\mathcal{M}}\sum_{m_{1}\neq
m}3\mathbb{E}\left\\{\xi_{mk}^{2}\right\\}\mathbb{E}\left\\{\xi_{m_{1}k}^{2}\right\\}+\sum_{m\in\mathcal{M}}\sum_{m_{1}\neq
m}\sum_{m_{2}\neq m,m_{1}}\sum_{m_{3}\neq
m,m_{1},m_{2}}\mathbb{E}\left\\{\xi_{mk}\right\\}\mathbb{E}\left\\{\xi_{m_{1}k}\right\\}\mathbb{E}\left\\{\xi_{m_{2}k}\right\\}\mathbb{E}\left\\{\xi_{m_{3}k}\right\\}.$

##  Appendix B Proof of Lemma 2

According to (7), users using the same pilot sequence have the parallel
estimated channels. Thus the first and second moments of $U_{kk_{1}}^{2}$ can
be obtained similarly to $U_{k}^{1}$ when user $k$ and user $k_{1}$ use the
same pilot sequence. The first to fourth moment of $\xi_{mkk_{1}}$ can be
calculated as follows:

$\displaystyle\mathbb{E}\left\\{\xi_{mkk_{1}}\right\\}=\sqrt{N\eta_{mk_{1}}c_{mk}},$
(42a)
$\displaystyle\mathbb{E}\left\\{\xi_{mkk_{1}}^{2}\right\\}=(N+1)\eta_{mk_{1}}c_{mk},$
(42b)
$\displaystyle\mathbb{E}\left\\{\xi_{mkk_{1}}^{3}\right\\}=\frac{(N+1)(N+2)}{\sqrt{N}}\eta_{mk_{1}}^{\frac{3}{2}}c_{mk}^{\frac{3}{2}},$
(42c)
$\displaystyle\mathbb{E}\left\\{\xi_{mkk_{1}}^{4}\right\\}=\frac{(N+1)(N+2)(N+3)}{N}\eta_{mk_{1}}^{2}c_{mk}^{2}.$
(42d)

Then the first moment of $U_{kk_{1}}^{2}$ can be expressed as follows:

$\displaystyle\mathbb{E}\left\\{U_{kk_{1}}^{2}\right\\}=$ (43)
$\displaystyle\sum_{m\in\mathcal{M}}\mathbb{E}\left\\{\xi_{mkk_{1}}^{2}\right\\}+\sum_{m\in\mathcal{M}}\sum_{m_{1}\neq
m}\mathbb{E}\left\\{\xi_{mkk_{1}}\right\\}\mathbb{E}\left\\{\xi_{m_{1}kk_{1}}\right\\}$

The second moment of $U_{kk_{1}}^{2}$ is given in (44). The first and second
moments of $U_{kk_{1}}^{2}$ can be obtained by using (42), (43) and (44).

$\displaystyle\mathbb{E}\left\\{\left(U_{kk_{1}}^{2}\right)^{2}\right\\}$
$\displaystyle=\sum_{m\in\mathcal{M}}\mathbb{E}\left\\{\xi_{mkk_{1}}^{4}\right\\}+\sum_{m\in\mathcal{M}}\sum_{m_{1}\neq
m}\sum_{m_{2}\neq
m,m_{1}}6\mathbb{E}\left\\{\xi_{mkk_{1}}^{2}\right\\}\mathbb{E}\left\\{\xi_{m_{1}kk_{1}}\right\\}\mathbb{E}\left\\{\xi_{m_{2}kk_{1}}\right\\}$
(44) $\displaystyle\quad+\sum_{m\in\mathcal{M}}\sum_{m_{1}\neq
m}4\mathbb{E}\left\\{\xi_{mkk_{1}}^{3}\right\\}\mathbb{E}\left\\{\xi_{m_{1}kk_{1}}\right\\}+\sum_{m\in\mathcal{M}}\sum_{m_{1}\neq
m}3\mathbb{E}\left\\{\xi_{mkk_{1}}^{2}\right\\}\mathbb{E}\left\\{\xi_{m_{1}kk_{1}}^{2}\right\\}$
$\displaystyle\quad+\sum_{m\in\mathcal{M}}\sum_{m_{1}\neq m}\sum_{m_{2}\neq
m,m_{1}}\sum_{m_{3}\neq
m,m_{1},m_{2}}\mathbb{E}\left\\{\xi_{mkk_{1}}\right\\}\mathbb{E}\left\\{\xi_{m_{1}kk_{1}}\right\\}\mathbb{E}\left\\{\xi_{m_{2}kk_{1}}\right\\}\mathbb{E}\left\\{\xi_{m_{3}kk_{1}}\right\\}.$

When user $k$ and user $k_{1}$ use different pilot sequences, i.e.,
$k_{1}\notin\mathcal{P}_{k}$, the second and fourth moment of $\xi_{mkk_{1}}$
can be expressed as follows:

$\displaystyle\mathbb{E}\left\\{\xi_{mkk_{1}}^{*}\xi_{mkk_{1}}\right\\}=\eta_{mk_{1}}c_{mk},$
(45a)
$\displaystyle\mathbb{E}\left\\{\left(\xi_{mkk_{1}}^{*}\xi_{mkk_{1}}\right)^{2}\right\\}=\frac{2\left(N+1\right)}{N}\eta_{mk_{1}}^{2}c_{mk}^{2},$
(45b)

The first moment of $U_{kk_{1}}^{2}$ when $k_{1}\notin\mathcal{P}_{k}$ can be
expressed as follows:

$\displaystyle\mathbb{E}\left\\{U_{kk_{1}}^{2}\right\\}=\mathbb{E}\left\\{\left(\sum_{m\in\mathcal{M}}\xi_{mkk_{1}}^{*}\right)\left(\sum_{m\in\mathcal{M}}\xi_{mkk_{1}}\right)\right\\}$
(46)
$\displaystyle=\sum_{m\in\mathcal{M}}\mathbb{E}\left\\{\xi_{mkk_{1}}^{*}\xi_{mkk_{1}}\right\\}+\sum_{m\in\mathcal{M}}\sum_{m_{1}\neq
m}\mathbb{E}\left\\{\xi_{mkk_{1}}^{*}\xi_{m_{1}kk_{1}}\right\\}$
$\displaystyle\overset{(a)}{=}\sum_{m\in\mathcal{M}}\mathbb{E}\left\\{\xi_{mkk_{1}}^{*}\xi_{mkk_{1}}\right\\}$

where $(a)$ is obtained based on the fact that the exception of
$\xi_{mkk_{1}}$ is zero when user $k$ and user $k_{1}$ use different pilot
sequences and the independence between $\xi_{mkk_{1}}$ and $\xi_{m_{1}kk_{1}}$
when $m\neq m_{1}$. Then the second moment of $U_{kk_{1}}^{2}$ when user $k$
and user $k_{1}$ use different pilot sequences is given in (47).

$\displaystyle\mathbb{E}\left\\{\left(U_{kk_{1}}^{2}\right)^{2}\right\\}$
$\displaystyle=\mathbb{E}\left\\{\left|\sum_{m\in\mathcal{M}}\xi_{mkk_{1}}\right|^{4}\right\\}=\mathbb{E}\left\\{\left(\sum_{m\in\mathcal{M}}\xi_{mkk_{1}}^{*}\xi_{mkk_{1}}+\sum_{m\in\mathcal{M}}\sum_{m_{1}\neq
m}\xi_{mkk_{1}}^{*}\xi_{m_{1}kk_{1}}\right)^{2}\right\\}$ (47)
$\displaystyle=\sum_{m\in\mathcal{M}}\mathbb{E}\left\\{\left(\xi_{mkk_{1}}^{*}\xi_{mkk_{1}}\right)^{2}\right\\}+2\sum_{m\in\mathcal{M}}\sum_{m_{1}\neq
m}\mathbb{E}\left\\{\xi_{mkk_{1}}^{*}\xi_{mkk_{1}}\right\\}\mathbb{E}\left\\{\xi_{m_{1}kk_{1}}^{*}\xi_{m_{1}kk_{1}}\right\\}.$

By using (45), (46) and (47), we can obtain the first and second moment of
$U_{kk_{1}}^{2}$ when $k_{1}\notin\mathcal{P}_{k}$.

We calculated the first and second moments of $U_{kk_{1}}^{3}$ using a similar
way with $U_{kk_{1}}^{2}$. The second and fourth moment of $\psi_{mkk_{1}}$
can be calculated as follows:

$\displaystyle\mathbb{E}\left\\{\psi_{mkk_{1}}^{*}\psi_{mkk_{1}}\right\\}=\eta_{mk_{1}}\left(\beta_{mk}-c_{mk}\right),$
(48)
$\displaystyle\mathbb{E}\left\\{\left(\psi_{mkk_{1}}^{*}\psi_{mkk_{1}}\right)^{2}\right\\}=\frac{2\left(N+1\right)}{N}\eta_{mk_{1}}^{2}\left(\beta_{mk}-c_{mk}\right)^{2},$

Then the first and second moment of $U_{kk_{1}}^{3}$ can be expressed as
follow:

$\displaystyle\mathbb{E}\left\\{U_{kk_{1}}^{3}\right\\}=\sum_{m\in\mathcal{M}}\mathbb{E}\left\\{\psi_{mkk_{1}}^{*}\psi_{mkk_{1}}\right\\},$
(49)
$\displaystyle\mathbb{E}\left\\{\left(U_{kk_{1}}^{3}\right)^{2}\right\\}=\sum_{m\in\mathcal{M}}\mathbb{E}\left\\{\left(\psi_{mkk_{1}}^{*}\psi_{mkk_{1}}\right)^{2}\right\\}$
$\displaystyle+2\sum_{m\in\mathcal{M}}\sum_{m_{1}\neq
m}\mathbb{E}\left\\{\psi_{mkk_{1}}^{*}\psi_{mkk_{1}}\right\\}\mathbb{E}\left\\{\psi_{m_{1}kk_{1}}^{*}\psi_{m_{1}kk_{1}}\right\\}.$

The first and second moment of $U_{kk_{1}}^{3}$ can be obtained by using (48)
and (49). The first order of $\text{IN}_{k}^{\text{FZF}}$ can be expressed as
follows:

$u_{\text{IN}_{k}^{\text{FZF}}}=\rho_{d}\sum_{k_{1}\neq
k}\mathbb{E}\left\\{U_{kk_{1}}^{2}\right\\}+\rho_{d}\sum_{k_{1}\in\mathcal{K}}\mathbb{E}\left\\{U_{kk_{1}}^{3}\right\\}+1.$
(50)

The second moments of $\text{IN}_{k}^{\text{FZF}}$ can be expressed as

$\displaystyle
u_{\text{IN}_{k}^{\text{FZF}}}^{(2)}=\mathbb{E}\left\\{\left(\rho_{d}\sum_{k_{1}\neq
k}U_{kk_{1}}^{2}+\rho_{d}\sum_{k_{1}\in\mathcal{K}}U_{kk_{1}}^{3}+z_{k}\right)^{2}\right\\}$
(51) $\displaystyle=\rho_{d}^{2}\left(\sum_{k_{1}\neq
k}\mathbb{E}\left\\{\left(U_{kk_{1}}^{2}\right)^{2}\right\\}+\sum_{k_{1}\neq
k}\sum_{k_{2}\neq
k,k_{1}}\mathbb{E}\left\\{U_{kk_{1}}^{2}\right\\}\mathbb{E}\left\\{U_{kk_{2}}^{2}\right\\}+\sum_{k_{1}\in\mathcal{K}}\mathbb{E}\left\\{\left(U_{kk_{1}}^{3}\right)^{2}\right\\}+2\right.$
$\displaystyle\left.+\sum_{k_{1}\in\mathcal{K}}\sum_{k_{2}\neq
k_{1}}\mathbb{E}\left\\{U_{kk_{1}}^{3}\right\\}\mathbb{E}\left\\{U_{kk_{2}}^{3}\right\\}+2\left(\sum_{k_{1}\neq
k}\mathbb{E}\left\\{U_{kk_{1}}^{2}\right\\}\right)\left(\sum_{k_{1}\in\mathcal{K}}\mathbb{E}\left\\{U_{kk_{1}}^{3}\right\\}\right)\right)+2\rho_{d}\left(\sum_{k_{1}\neq
k}\mathbb{E}\left\\{U_{kk_{1}}^{2}\right\\}+\sum_{k_{1}\in\mathcal{K}}\mathbb{E}\left\\{U_{kk_{1}}^{3}\right\\}\right).$

Then the first and second moments of $\text{IN}_{k}^{\text{FZF}}$ can be
obtained.

##  Appendix C Proof of Theorem 1

In the CF mMIMO system with MRT precoding, the distribution of
$\text{DS}_{k}^{\text{MRT}}$ can be approximated as a Gamma distribution with
shape parameter $j_{k1}$ and scale parameter $\rho_{d}\chi_{k1}$ for user $k$,
and $\text{IN}_{k}^{\text{MRT}}$ can be seemed as Gamma distribution with
shape parameter $j_{k1}$ and scale parameter $\chi_{k2}$. Since the
$\rho_{d}U_{k1}$ and $\text{IN}_{k}^{\text{MRT}}$ are independent with each
other. The CDF of $\gamma_{k}$ can be calculated as follows:

$\displaystyle P\left\\{\gamma_{k}\leq x\right\\}$
$\displaystyle=P\left\\{\frac{\text{DS}_{k}^{\text{MRT}}}{\text{IN}_{k}^{\text{MRT}}}\leq
x\right\\}=P\left\\{\text{DS}_{k}^{\text{MRT}}\leq\text{IN}_{k}^{\text{MRT}}x\right\\}$
(52)
$\displaystyle=\int_{0}^{\infty}F_{\text{DS}_{k}^{\text{MRT}}}(xx_{1})f_{\text{IN}_{k}^{\text{MRT}}}(x_{1})dx_{1}.$

where $F_{\text{DS}_{k}^{\text{MRT}}}(\cdot)$ denotes the CDF of
$\text{DS}_{k}^{\text{MRT}}$ and $f_{\text{IN}_{k}^{\text{MRT}}}(\cdot)$
represnets the PDF of $\text{IN}_{k}^{\text{MRT}}$. Then the PDF of
$\gamma_{k}$ can be calculated based on (52), which is given in (53)

Based on the PDF of $\gamma_{k}$, the CDF of $\gamma_{k}$ can be calculated
using
$F_{\gamma_{k}}^{\text{MRT}}(x)=\int_{0}^{x}f_{\gamma_{k}}^{\text{MRT}}(x_{1})dx_{1}$
directly.

$\displaystyle f_{\gamma_{k}}^{\text{MRT}}(x)$
$\displaystyle=\frac{\mathrm{d}P\left\\{\gamma_{k}\leq
x\right\\}}{\mathrm{d}x}=\int_{0}^{\infty}x_{1}f_{\text{DS}_{k}^{\text{MRT}}}(xx_{1})f_{\text{IN}_{k}^{\text{MRT}}}(x_{1})dx_{1}.$
(53)
$\displaystyle=\int_{0}^{\infty}x_{1}\frac{1}{\Gamma(j_{k1})\left(\rho_{d}\chi_{k1}\right)^{j_{k1}}}\left(xx_{1}\right)^{\left(j_{k1}-1\right)}e^{-\frac{xx_{1}}{\chi_{k1}}}\frac{1}{\Gamma(j_{k2})\left(\chi_{k2}\right)^{j_{k2}}}\left(x_{1}\right)^{\left(j_{k2}-1\right)}e^{-\frac{x_{1}}{\chi_{k2}}}dx_{1}$
$\displaystyle=\frac{\Gamma(j_{k1}+j_{k2})}{\Gamma(j_{k1})\Gamma(j_{k2})\left(\rho_{d}\chi_{k1}\right)^{j_{k1}}\chi_{k2}^{j_{k2}}}x^{j_{k1}-1}(\frac{1}{\chi_{k2}}+\frac{x}{\rho_{d}\chi_{k1}})^{-j_{k1}-j_{k2}}.$

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