\chapter{Cell-Free Massive MIMO System}

\begin{figure}[!]
     \centering
     \includegraphics[width=\textwidth]{figures/fig1_system_model}
     \caption{Uplink transmission in cell-free massive MIMO system.}%\vspace*{-.5em}
     \label{fig:1}
\end{figure}

% 우리는 \cite{2017versus, 2021structured, 2021Hungarian}에서 고려된 모델을 참고하여 시스템모델을 상정하였으며, 우리의 주제에 집중하기 위해 상황의 부수적 요인을 간소화시키는 몇가지 가정을 세팅하였다.
A CF massive MIMO system consisting of $M$ APs and $K$ users ($K\leq M$), all equipped with a single antenna, is considered. Each user is served simultaneously by multiple APs operating with the shared radio resource. The coherence intervals of all channels are thus defined as the product of the coherence time and bandwidth. Individual APs are connected to a central processing unit (CPU) via fronthaul that provides error-free and infinite capacity links for straightforward focus on the PA operation.
%The CPU is responsible for the control for them.

%We assume that the fronthaul connection is perfect so that offer error-free and infinite capacity to the CPU, and channels are constant during the coherence interval. 
% We note that our system model is simplified version of general CF massive MIMO in \cite{2017versus} to concentrate on our main topic, PA optimization. 

This work mainly considers the uplink transmission since there is no need for downlink pilot training \cite{2017versus} and the channel reciprocity allows a straightforward extension to the downlink transmission \cite{2021structured}. In the uplink transmission operating in a time division duplex manner, a coherence interval, with its resource length $N_\mathrm{c}$, is divided into two phases: the uplink pilot training and the uplink payload transmission. During the first training phase of length $N_\mathrm{p}$, users transmit dedicated pilot sequences to APs, and APs attempt to estimate the user channel states. In the second phase of length $N_\mathrm{c}-N_\mathrm{p}$, the uplink signals are detected with those estimates. 

\section{Uplink Training}

Let $ \boldsymbol{\varphi}_{k} \in \mathbb{C}^{N_\mathrm{p} \times 1}$ be the unit norm pilot sequence used by $k$-th user, where $N_\mathrm{p}$ of orthogonal pilot sequences are available. The channel coefficient between the $k$-th user and the $m$-th AP during a coherence interval, denoted as $g_{m k}$, is defined by the small-scale fading $h_{m k}$ and large-scale fading $\beta_{m k}$ of the corresponding user-AP pair as $g_{m k}=\sqrt{\beta_{m k}} h_{m k}$. We assume that the entries of $h_{m k}$ are independent and identically distributed (i.i.d) $\mathcal{N}_{\mathbb{C}}(0,1)$ random variables (RVs), and the channels are constant during the coherence interval. 
%The superscript $(\cdot)^{H}$ means the conjugate-transpose.
The received pilot signal at $m$-th AP is given by
\begin{equation}
    % \mathbf{y}_{m}^{\mathrm{p}}=\sqrt{T } \sum_{k=1}^{K} g_{m k} \boldsymbol{\varphi}_{k}+\mathbf{n}_{m}^{\mathrm{p}},\label{eq:y(SNR)}
    \mathbf{y}_{m}=\sqrt{\omega N_\mathrm{p}} \sum_{k=1}^{K} g_{m k} \boldsymbol{\varphi}_{k}+\mathbf{n}_{m} \, ,\label{eq:y(SNR)}
\end{equation}
where $\omega$ is the normalized transmit signal-to-noise ratio (SNR) of a pilot signal, and $\mathbf{n}_{m}$ is an additive noise vector of the $m$-th AP such that the elements of $\mathbf{n}_{m}$ are i.i.d. $\mathcal{N}_{\mathbb{C}}(0,1)$ RVs.
The estimate of $g_{mk}$ obtained based on a minimum-mean square error is given by 

\begin{equation}
    \hat{g}_{m k}=\frac{\mathbb{E}\left\{\left(\boldsymbol{\varphi}_k^H \mathbf{y}_m\right)^* g_{m k}\right\}}{\mathbb{E}\left\{\left|\boldsymbol{\varphi}_k^H \mathbf{y}_m\right|^2\right\}}\left(\boldsymbol{\varphi}_k^H \mathbf{y}_m\right)=c_{m k}\left(\boldsymbol{\varphi}_k^H \mathbf{y}_m\right),
\end{equation}
where
\begin{equation}
    c_{m k} \triangleq \frac{\sqrt{N_{\mathrm{p}} \omega} \beta_{m k}}{N_{\mathrm{p}} \omega \sum_{k^{\prime}=1}^K \beta_{m k^{\prime}}\left|\boldsymbol{\varphi}_k^H \boldsymbol{\varphi}_{k^{\prime}}\right|^2+1} \ .
\end{equation}
% where the superscript $(\cdot)^{*}$ means the conjugate. 
From the randomness assumptions for corresponding RVs, the mean-square channel estimate, denoted as $\gamma_{m k}$, is given by

\begin{equation}
    %    \gamma_{m k} \triangleq \mathbb{E}\left\{\left|\hat{g}_{m k}\right|^{2}\right\}=\frac{\beta_{m k}^2}{\sum_{k^{\prime}=1}^{K} \beta_{m k^{\prime}}\left|\boldsymbol{\varphi}_{k}^{H} \boldsymbol{\varphi}_{k^{\prime}}\right|^{2}+1/T}\ .\label{eq:mean-sqr channel est}
    \gamma_{m k} \triangleq \mathbb{E}\left\{\left|\hat{g}_{m k}\right|^{2}\right\}=\frac{N_\mathrm{p}\beta_{m k}^2}{N_\mathrm{p}\sum_{k^{\prime}=1}^{K} \beta_{m k^{\prime}}\left|\boldsymbol{\varphi}_{k}^{H} \boldsymbol{\varphi}_{k^{\prime}}\right|^{2}+1/\omega} \ .\label{eq:mean-sqr channel est}
\end{equation}
%If $T \geq K$, then we can choose pairwisely orthogonal pilot sequences $\boldsymbol{\varphi}_{1}, \boldsymbol{\varphi}_{2}, \cdots, \boldsymbol{\varphi}_{K}$ such that $\left|\boldsymbol{\varphi}_{k}^{H} \boldsymbol{\varphi}_{k^{\prime}}\right|^{2}=0$ if $k^{\prime} \neq k$. 
The orthogonal PA leads to $\left|\boldsymbol{\varphi}_{k}^{H} \boldsymbol{\varphi}_{k^{\prime}}\right|^{2}=0$ if $k^{\prime} \neq k$. The number of users $K$ generally exceeds the number of orthogonal pilot resources $N_\mathrm{p}$, and the same pilot sequences are necessarily reused in the assignment\cite{2017versus, 2018coded}. 
The quality of the channel estimate $\hat{g}_{m k}$ is degraded by the signal transmitted from a pilot-sharing user $k^\prime$ with $\boldsymbol{\varphi}_{k^{\prime}}=\boldsymbol{\varphi}_{k}$. This induces the pilot contamination. Fig. \ref{fig:1}(a) illustrates a simple example. Two pilot resources are assigned to three users. While AP 1 estimates the channel for user 1, pilot signals from users 2 and 3 interfere due to the pilot reuse.
% In the uplink payload data transmission, the received signal at the $m$-th AP is given by
% \begin{equation}
% y_{m}^{\mathrm{u}}= \sum_{k=1}^{K} g_{m k} s_{k}+n_{m}^{\mathrm{u}},
% \end{equation}
% where $s_{k}$ is data symbol of $k$-th user which satisfy $\mathbb{E}\big\{\left|s_{k}\right|^{2}\big\}=1$.
% $n_{m}^{\mathrm{u}} \sim \mathcal{N}_{\mathbb{C}}(0,1)$ is additive noise at the $m$-th AP.
% The received signal $y_{m}^{\mathrm{u}}$ and its channel estimate $\hat{g}_{m k}$ are processed into a quantity $\hat{g}_{m k}^{*} y_{m}^{\mathrm{u}}$ by $m$-th AP, and it is transmitted to the CPU through fronthaul link for detection of the symbol $s_{k}$.

\section{Uplink Payload Data Transmission}

In uplink payload transmission from users in Fig. \ref{fig:1}(b), all $K$ users simultaneously send their data to the APs, and each AP detects the signals from channel estimates and sends them to the CPU. According to an analysis in \cite{2017versus}, the received signal collected at the CPU can be decomposed into the desired signal (DS) component, the beamforming uncertainty (BU) component, the user interference (UI) component, and noise. Their average powers of $k$-th user are given, respectively, by
\begin{subequations}
\begin{align}
    \mathrm{DS}_{k} =&  ~\omega \eta_{k}\left(\sum\limits_{m=1}^{M} \gamma_{m k}\right)^{2},\\
    \mathrm{BU}_{k} =&  ~\omega\sum_{k^{\prime}=1}^{K} \eta_{k^{\prime}} \sum_{m=1}^{M} \gamma_{m k} \beta_{m k^{\prime}}  \ ,\\
    \mathrm{UI}_{k k^{\prime}} =&
         ~\omega\eta_{k^{\prime}}\left(\sum\limits_{m=1}^{M} \gamma_{m k} \frac{\beta_{m k^{\prime}}}{\beta_{m k}}\right)^{2} \left|\boldsymbol{\varphi}_{k}^{H} \boldsymbol{\varphi}_{k^{\prime}}\right|^{2},\label{eq:contaminationfactor}
\end{align}
\end{subequations}
where $\sqrt{\eta_{k}}$ is a power control coefficient of the $k$-th user, i.e., $\eta_{k} \in (0, 1]$. 
We assume that the normalized uplink transmit SNR is equal to that of pilot signals, $\omega$.
Note that \eqref{eq:contaminationfactor} corresponds to the degradation effect of the pilot reuse on an achievable rate of $k$-th user in case of $k\neq k^{\prime}$. The signal to interference and noise ratio (SINR) of $k$-th user and the corresponding achievable rate are given, respectively, by
\begin{subequations}
\begin{align}\label{eq:SINR}
    \mathrm{SINR}_{k} &=
    \frac{\mathrm{DS}_{k}}{\mathrm{BU}_{k}+\sum\limits_{k^{\prime} \neq k}^{K} \mathrm{UI}_{k k^{\prime}}+\sum\limits_{m=1}^{M} \gamma_{m k}}\ ,\\
    R_{k} &=\left(1-\frac{N_\mathrm{p}}{N_\mathrm{c}}\right) \log _{2}\left(1+\mathrm{SINR}_{k}\right) \, . \label{eq:achivableRate}
\end{align}
\end{subequations}
%\section{PLACEHOLDER}

The network throughput maximization is the main purpose of this work. In the next chapter, the PA optimization task is formulated with the achievable rate for each user.