## 2 System Model and Problem Formulation  
### System Model  
Consider a large-scale uplink multi-cell multi-user network, where \(M\) single-antenna base stations (BSs) are collected by the set \(\mathcal{M}=\{1,2,\ldots,M\}\) are deployed at the center of \(M\) cells, respectively. Assume that there are \(N\) users collecting by the set \(\mathcal{L}_{m}=\{l_{m,1},l_{m,2},\ldots,l_{m,N}\}\) in each cell \(m\), where \(l_{m,n}\) denotes the index of the \(n\)-th user in the \(m\)-th BS. The total users of the considered system are collected by the set \(\mathcal{K}=\{1,2,\ldots,K\}\), where \(K=MN\). The total bandwidth of the considered system is denoted as \(W\) and is divided into three widths, which are collected by the set \(\mathcal{B}=\{B_{i}\}=\{15\text{kHz},30\text{kHz},60\text{kHz}\}\), where \(i\in\{1,2,3\}\)[40]. Let \(\mathcal{X}_{i}=\{1,2,\ldots,X_{i}\}\) denote the set of the sub-bands of the width \(i\) of the bandwidth, where \(X_{i}\) is the total number of allocated bandwidth of width \(i\).  
Since users in different cells would occupy the same frequency band when transmitting their uplink signals, there exists interference between these users. This interference is called co-channel interference [41]. In this paper, each cell occupies the same frequency band and serves the same number of users \(N\). For each user \(l_{m,n}\), some users in the neighboring cells can cause co-channel interference. In other words, users in the same cell can use different frequency band sub-carriers, and, thus, each user is subject to co-channel interference from users in other cells. Let \(\mathcal{M}^{\prime}=\{l_{m^{\prime},n}|m^{\prime}\in\mathcal{M},m^{\prime}\
eq m\}\) denote the set of interfering users. Thus, these users from different cells belonging to the set \(\mathcal{M}^{\prime}\) will interfere with user \(l_{m,n}\). The channel gain between user \(l_{m^{\prime},n}\) and BS \(m\) at the slot \(t\) is represented by the following:  
\(g(d_{l_{m^{\prime},n},m})=h_{l_{m^{\prime},n}}[\beta(d_{l_{m^{\prime},n^{\prime },m}})]^{\frac{1}{2}}\), where \(\beta(d_{l_{m^{\prime},n^{\prime},m}})=10^{\frac{P_{l_{m^{\prime},n^{\prime}} +\rho_{n}^{2}\beta}}{10}}\) is the large scale fading corresponding to the distance \(d_{l_{m^{\prime},n},m}\) between user \(l_{m^{\prime},n}\), BS \(m\), \(PL_{l_{m^{\prime},n}^{\prime}}\) is the path loss of user \(l_{m^{\prime},n}\), \(\sigma_{\beta}\) is the standard deviation of shadow fading, \(z_{\beta}\sim\mathcal{N}(0,1)\) is a Gaussian random variable, and \(h_{l_{m^{\prime},n}}\sim\mathcal{CN}(0,\sigma_{h}^{2})\) is the small-scale fading with variance \(\sigma_{h}^{2}\). Then, the power of co-channel interference on user \(l_{m,n}\) is expressed as follows:  
\[I_{l_{m,n}}=\sum_{m^{\prime}\in\mathcal{M}^{\prime}}g\left(d_{l_{m^{\prime},n ^{\prime},m}},t\right)p_{l_{m^{\prime},n}}, \tag{1}\]  
where \(p_{l_{m^{\prime},n}}\) denotes the transmit power for user \(l_{m^{\prime},n}\).  
The signal \(y_{l_{m,  
complex Gaussian noise. As a result, the received signal-to-interference-plus-noise ratio (SINR) at BS \(m\) of user \(l_{m,n}\) is given by  
\[\xi_{l_{m,n}}\big{(}\mathbf{p_{l_{m,n}}},B_{i,l_{m,n}}\big{)}=\frac{p_{l_{m,n} }\delta(d_{l_{m,n},m},t)}{\sigma_{l_{m,n}}^{2}+l_{l_{m,n}}}, \tag{3}\]  
where \(\sigma_{l_{m,n}}^{2}=n_{f}B_{i,l_{m,n}}\) indicates the variance of the Gaussian white noise, and \(n_{f}\) is the power spectral density of noise. \(\mathbf{p_{l_{m,n}}}\) is the power vector that includes the power of user \(l_{m,n}\) and its interfering users, and \(B_{i,l_{m,n}}\) is the \(i\)-th width of the bandwidth allocated to the user \(l_{m,n}\). Then, by considering the normalized rate [42], the achievable throughput of user \(l_{m,n}\) at BS \(m\) is  
\[TH_{l_{m,n}}=\log_{2}\big{(}1+\xi_{l_{m,n}}\big{(}\mathbf{p_{l_{m,n}}},B_{i,l_{ m,n}}\big{)}\big{)}. \tag{4}\]