\chapter{Introduction}

Cell-free (CF) massive multiple-input multiple-output (MIMO) technology has been spotlighted as one of the most promising architectures for future wireless networks \cite{2017versus, 2020prospective, 2022survey, 2021structured}. In the CF MIMO transmission scenario, massive-scale of access points (APs) are spatially distributed across the coverage to jointly serve users with shared radio resources as illustrated in Fig. \ref{fig:1.1}. 

\section{Motivation}

\begin{figure}[!]
     \centering
     \includegraphics[width=.7\textwidth]{figures/fig1-1_cellfree}
     \caption{Cell-free massive MIMO system.}%\vspace*{-.5em}
     \label{fig:1.1}
\end{figure}

The quality-of-service with improved user fairness can be enhanced by the macro-diversity of CF massive MIMO techniques \cite{2017precoding}. %, while it also encompasses several virtues of other structures such as virtual MIMO\cite{2013virtual}, network MIMO\cite{2011network}, collocated massive MIMO\cite{2016fundamentals}, and small-cell network\cite{2013holistic}. % which are previously investigated,
For the full potential of this concept, the qualified channel state information (CSI) from channel estimation based on pilot training is of utmost importance \cite{2017massive}. % In order to achieve the capabilities
However, the elimination of cell boundaries incurs a main trade-off challenge in the pilot usage (see Fig. \ref{fig:1.2}); The pilot reuse is unavoidable due to insufficient sequence diversity for orthogonal pilot assignment (PA) over the coverage \cite{2019ubiquitous}. Non-orthogonal PA among pilot-sharing users is subject to the pilot contamination, degrading the channel estimation accuracy and the subsequent system performance \cite{2017massive}. A proper strategy is essential for alleviation of the contamination and improvement in network throughput. 

Despite the urgent need for the optimal assignment of the pilot sequence, a well-structured PA framework remains yet partially addressed, and the associated combinatorial challenges are found computationally demanding \cite{2022topological}. Furthermore, the resulting throughput gain to a group of pilot-sharing users is hard to address. A variety of PA frameworks have been investigated for this challenge. A simple approach sticks to a greedy policy that begins with a random assignment and repeated modification of the worst case user for gradual throughput performance improvement \cite{2017versus}.
An iterative user selection is developed to assign orthogonal pilots via bipartite matching implemented using Hungarian algorithms \cite{2021Hungarian}. 
In \cite{2021WGF}, a weighted graphic framework have been proposed that the contamination factors defined between two UEs are used to assignment. 
The usage of a variant of the hierarchical agglomerative clustering algorithm has been suggested in \cite{2022clusteringPA}.
There have also been attempts to apply methods such as tabu-search \cite{2020tabu} or graph coloring \cite{2020color} to PA in CF massive MIMO. 
Furthermore, a topological interference management is considered in sparse connectivity cases \cite{2022topological}. Although a structured PA framework is also proposed using $K$-means clustering, geographic information is used instead of throughput measurements \cite{2021structured}. 

\begin{figure}[!]
     \centering
     \includegraphics[width=.7\textwidth]{figures/fig1-2_pilotreuse}
     \caption{Pilot reuse problem in CF massive MIMO.}%\vspace*{-.5em}
     \label{fig:1.2}
\end{figure}

% Although there have been many investigations on PA, the inherent difficulties of this combination problem have not been overcome, and naturally the gap between achievable performance and current solution remains as ever.

\section{Contributions}

This work develops a new optimization framework for the CF massive MIMO PA task.
While existing works require a completely new PA on every coherence interval, we consider some succession of previous assignments for sustainable PA. 
In this regard, the maximization of the network utility is conducted under the pre-assigned states for some users. % while maintaining the last PA for some users whose channel condition is preserved compared to the last coherence interval.
%On every coherence interval that requires a new PA, the PA is conducted to maximize the network utility under the consideration of the pre-assigned users. 
% The pre-assigned users, for sustainable PA, maintain the previous assignment results from the last coherence interval.
% To handle this operation, we directly deal with feasible groups of users to which the same pilot is assigned so that the pilot contamination and throughput can be calculated for each group.
The resulting optimization task is formulated into a special class of quadratic assignment problems \cite{koopmans1957assignment}, which fall in a class of {\em NP}-hard problems \cite{sahni1976QAPisNPhard}.
The PA challenge entails group-wise matching where fixing an assignment of a group enables or disables assignments of many other groups that share the same users. The PA solution space is thus separated into interacting {\em clusters} of feasible configurations, thereby rendering the PA search quickly demanding as the system scales up.

To tackle this, a novel approach is developed by borrowing the principles of survey propagation (SP) that has originally been derived to address the interaction relationship among multiple particles from statistical physics \cite{2005survey, 2016beyound}. Based on this formalism, multiple states of pilot to user-group association can be efficiently described and solved using a distributed algorithm that obtains an efficient solution. % In this regard, a new algorithm is constructed that inherits the philosophy of two message-passing algorithms, affinity propagation\cite{2007affinity} for max-sum optimization, and survey propagation\cite{2005survey, 2016beyound} for constraint satisfaction.
% \textbf{SP}We inspired to the survey propagation that quantization.....
Thus, the proposed distributed algorithm inspired by survey propagation accomplishes a comprehensive search over the clusters of PA solution space without brute-force enumeration.

\section{Thesis Outline and Notations}

The remainder of this thesis is organized as follows. The system model of CF massive MIMO and the main metric are first introduced.
In chapter 3, the corresponding PA problem is formulated in combinatorial way.
Our proposed framework is provided in chapter 4. The design principles and derivations of SP algorithm are also described in this chapter.
The numerical results in chapter 5 demonstrate that the network throughput is consistently achieved higher than existing techniques. Finally, the major conclusion is suggested in chapter 6.

Throughout this thesis, a boldface lowercase letter, $x$, denote column vectors.
The superscripts $*$ and $\mathrm{H}$ denote conjugate and conjugate transpose, respectively. 
The expected value of $\mathbf{x}$ is denoted as $\mathbb{E}\{\mathbf{x}\}$. %, and $\|\mathbf{x}\|$ denotes the Euclidean norm of $\mathrm{x}$. 
$\mathcal{N}_{\mathbb{C}}(\mathbf{0}, \mathbf{R})$ denotes the multivariate circularly symmetric complex Gaussian distribution with correlation matrix $\mathbf{R}$. Other mathmetical notations in some expressions are described in the first part where they first appear.