\chapter{Numerical Results}\label{Results}

% \begin{figure}
%     \centering
%     \includegraphics[width=\columnwidth]{figures/result.pdf}
%     \caption{Simulation results} % in case resource per user lamda is 1.23 and 1.75 (a) cumulative distribution of sum rate with N=80 (b) sum rate versus the number of users
%     \label{fig:results}
% \end{figure}

% 파라미터 셋업+비교스킴, 비교 내용 + 결과

\section{Experimental Setup}
This section presents numerical results to test the performance of the developed algorithm. A CF massive MIMO configuration is considered so that $M$ APs and $K$ users are equipped with a single antenna and randomly distributed over a $500\mathrm{m}\times500\mathrm{m}$ square area. The simulation region is wrapped around in order to mitigate the boundary effects \cite{2017versus}. 
The heights of APs and users are set as $10$m and $1.5$m, respectively. The communication bandwidth of 20$\mathrm{MHz}$ and carrier frequency of $2\mathrm{GHz}$ are used. The normalized transmit SNR is set to $\omega=\bar{\omega}/\omega_{\text{noise}}$, with $\bar{\omega}=100 \mathrm{mW}$ and the noise power $\omega_{\text{noise}}=-92 \mathrm{dBm}.$ 
The 3GPP Urban Microcell path loss model is used to set large-scale propagation conditions, where the shadow fading model is set to $z_{m k}\sim \mathcal{N}(1,\sigma_{\mathrm{sh}}^2)$ with $\sigma_{\mathrm{sh}}=4\mathrm{dB}$ \cite[Table B.1.2.1-1]{3GPP.36.814}.
% {\color{red} The path loss and shadow fading models described in \cite{2017versus, 2022topological} are used to set large-scale propagation condition.} 
% Without shadow fading "correlation"$.
The length of the coherence interval $N_\mathrm{c}$ is taken as 200.
We assume that the channels are constant during the coherence interval, and all users transmit with full power so that $\eta_k = 1$.

%\textcolor{orange}{(K-means, topologic? 추가)}
The proposed algorithm is compared with three existing PA schemes based on greedy algorithm \cite{2017versus}, K-means clustering \cite{2021structured}, and Hungarian algorithm \cite{2021Hungarian}.
The greedy strategy repetitively identifies the pilot-user pair with the least user throughput and assigns a new pilot sequence to the identified user while fixing all other PA configurations. In K-means clustering based PA, all users are grouped by $N_\mathrm{p}$ members, then pilots are assigned orthogonally to users in each cluster. At this time, $\lceil K/N_\mathrm{p} \rceil$ centroids for the deployment of the APs are used for user clustering, and the centroids have been trained through the K-means clustering algorithm. On the other hand, Hungarian algorithm-based strategy execute the following tasks sequentially for each user: select $N_\mathrm{p}-1$ users with the highest proximity and conduct the bipartite matching for PA while the assignment of other users is fixed. 
%%% fig(a)와 (b) 각각 내용을 문단별로 설명하자. K, T 규모,e.g. a CF massive MIMO configuration is considered so that $M=100$ APs and users ranging $15\leq K \leq 75$, 및 genetic 등 설명
%The proposed algorithm is evaluated based on the initial setup policy described in Section \ref{section:formulation}. 
In each simulation, $N_\mathrm{p}$ users are chosen at random and orthogonal pilots are preassigned to them so that PA is needed for the other $K-N_\mathrm{p}$ users.
Note that pre-assignment for some users poses a risk of performance degradation instead of improving the PA complexity.
For other benchmarks, we set up those to show full performance without pre-assignment.
We evaluate the additional gain for SP framework with respect to Hungarian algorithm, K-means clustering, and greedy algorithm, respectively. The corresponding percentages are calculated based on the performance of each benchmark. 
% 우리는 우리가 제안하는 알고리즘이 상술한 initial setup policy을 바탕으로 구동되었을 시의 성능을 검증하였다. 일부 유저의 파일럿 할당을 고정하는 전략은 PA의 복잡도를 개선하는 대신에 성능 하락을 감수한다는 점을 유의하라. 다른 벤치마크의 경우 별도의 pre-assignment 없이 기존에 보고된 성능을 보이도록 설정하였고 이를 우리의 제안과 비교하였다.

\section{Results and Discussions}

% \begin{figure}[!t]
%     \centering
%     \includegraphics[width=\columnwidth]{figures/fig4_results}
%     \caption{Simulation results in case $M=100$ and $K/N_{\mathrm{p}}=3$. (a) CDF of system throughput where $K=45$. (b) Average system throughput versus number of users.}
%     \label{fig:results}
% \end{figure}

\begin{figure}[]
    \centering
    \includegraphics[width=.7\textwidth]{figures/fig5-1_results_CDF.eps}
    \caption{Cumulative distribution function of system throughput where $K=45, M=100, N_{\mathrm{p}}=15$.}
    \label{fig:5-1}
\end{figure}

Fig. \ref{fig:5-1} illustrates the cumulative distribution function of the system throughput in uplink with $M=100$, $K=45$, and $N_\mathrm{p}=15$. 
The proposed algorithm, SP, outperforms other benchmarks constantly over the simulation, thus indicating that the pilot contamination is effectively mitigated. 
The average system throughput obtained by SP is about 744$~\mathrm{Mbps}$, which corresponds to the improvement over compared schemes by $2\%$, $5\%$, and $10.3\%$ respectively.
%The SP framework enhances the average system throughput by $2\%$, $5\%$, and $10.3\%$ with respect to Hungarian algorithm, K-means clustering, and greedy algorithm, respectively. %In particular, only $39\%$, $26\%$, and $13\%$ can meet the median performance of the SP framework, respectively.
% This indicates that SP-based solution is more successful at searching all solutions of feasible PA configuration spaces than the existing schemes.
%This indicates that the SP framework can explore optimal PA solutions over existing performance even under pre-assignment conditions. %In addition, it also shows performance advantages in addition to reinforcing the sustainability.
In addition, a popular metric for PA is the network throughput for the worst $5\%$ PA configurations \cite{2021structured, 2021Hungarian}.
SP obtains the value of 657$~\mathrm{Mbps}$, and this amounts to $3\%$, $6\%$, and $13\%$ improvement over others, respectively.
Thus, the population of low throughput users is decreased by the proposed method.

\begin{figure}[]
    \centering
    \includegraphics[width=.7\textwidth]{figures/fig5-2_results_avg.eps}
    \caption{Average system throughput versus number of users where $K\in\{15,30,45,60,75\}, M=100, N_{\mathrm{p}}=15$.}
    \label{fig:5-2}
\end{figure}
Fig. \ref{fig:5-2} depicts the average system throughput where $M=100$ and $K\in\{15, 30, 45, 60, 75\}$. 
The number of users per orthogonal pilot resources, $K/N_\mathrm{p}$, is fixed to $3$.
For large user populations, the system throughput growth under all PA strategies attenuates due to the additional beamforming uncertainty and user interference from the network congestion. 
At the largest population of $K=75$, the SP framework results about 907$~\mathrm{Mbps}$ then outperforms Hungarian algorithm, K-means clustering, and greedy strategies, respectively, by $1\%$, $4\%$, and $9\%$. This shows the robustness to the network congestion with respect to benchmarks. 

The SP approach is suited to optimize corresponding tasks for network throughput maximization from the advantage achieved by organizing the PA task from a group-wise matching framework. This allows the achievable rate to be directly utilized in network management. Through the developed algorithm, the PA operation can be configured to explore the solution space of the PA efficiently.