## 2 System Model
### NOMA in Small-cells  
This section of the paper provides the system model for downlink NOMA in hybrid-access small-cells using ideal and nonideal SIC conditions. The section starts with the basic NOMA model and then moves toward the ideal and nonideal SIC part. One important thing to remember is that once the MU becomes an SU, both uplink and downlink are provided by the serving SBS. In NOMA, the user equipments (UEs) of a superposition-coded signal are considered as paired users. In the existing literature, the maximum number of paired UEs is not yet defined, so for now, it is supposed that the total number of \(N\) UEs can share a common link of the same SBS using NOMA. Let us assume that there are \([1\ \ldots\ B]\) small-cells underlaid on a macrocell and the \(b\)th SBS transmits a superposition-coded signal \(S\) for the \(k\) paired SUs, given as:  
\[S=\sum_{k=1}^{N}s_{b,k}\sqrt{P_{b,k}}\, \tag{1}\]  
where \(k\in\{1,2,3,\ldots,N\}\), \(s_{b,k}\) is the signal of the \(k\)th UE, and \(P_{b,k}\) is the total transmission power for the \(k\) paired UEs. Considering NOMA in small-cells, as aforementioned in Section 1, in user pairing, there are two kinds of UEs, i.e., high gain (small-cell center users (SCUs)) and low gain (small-cell edge users (SEUs)). The ascending value of index \(k\) is assigned in descending order of the UE channel gain, i.e., \(k=1\) is the highest-gain user, categorized as SCU, and \(k=N\) is the lowest-gain user, categorized as SEU. The received superposition-coded signal at the \(k\)th UE in the \(b\)th SBS is given as:  
\[y_{b,k}=h_{b,k}\times S+n+I. \tag{2}\]  
Using (1) and (2) gives:  
\[y_{b,k}=h_{b,k}\times\left(\sum_{k=1}^{N}s_{b,k}\sqrt{P_{b,k}}\right)+n+I, \tag{3}\] \begin{table}
\begin{tabular}{c l} \hline \hline
**Notations** & **Description** \\ \hline \(P_{SBS}\) & Serving SBS transmission power \\ \hline \(n\) & Channel Noise \\ \hline \(\sigma^{2}\) & Variance \\ \hline \(I\) & Inter-cell Interference \\ \hline \(h_{b,k}\) & Transmission channel response from the \(b\)th SBS to the \(k\)th UE \\ \hline \(H_{i}\) & WHT matrix with \(i=2^{l}\) length code \((l\in\mathbb{Z}^{+})\) \\ \hline \(H_{q}\) & Complex channel matrix between the \(q\)th small-cell and UE \\ \hline \(\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{ \mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{\mathsf{ \mathsf{       }}}}}}}}}}}}}}}}\) \\ \hline \(W_{q}\) & Precoding weight matrix of the \(q\)th small-cell \\ \hline \(W_{IRC}\) & The IRC receiver weight matrix, containing covariance matrix and the estimated channel matrix \\ \hline \(W_{ideal\ IRC}\) & Ideal IRC weight matrix at UE-\(m\) served by \(b\)th SBS \\ \hline \(G_{bm}\) & Composite channel of the \(b\)th small-cell \\ \hline \(R_{yy}\) & The covariance matrix respectively \\ \hline \(\mathbf{BW}\) & Bandwidth \\ \hline \(Q\) & Modulation order of the transmitted signal \\ \hline \(T\) & Throughput \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cont. where \(h_{b,\ k}\) gives the impulse response of the transmission channel from the \(b\)th SBS to the \(k\)th UE; we assume in this work a Rayleigh fading channel model, \(n\) represents the additive white Gaussian noise (AWGN) with zero mean, and \(\sigma^{2}\), and \(I\) represent the ICI that is created at the SEU receiver from a nearby MU. Here, \(P_{k}\) is the transmission power associated with the \(k\)th UE in the pair given as:  
\[P_{b,k}=\frac{\alpha_{b,k}\times P_{SBS}}{N_{SB}}, \tag{4}\]  
where \(\alpha_{k}\) is the ratio of power assignment to the \(k\)th UE, \(P_{SBS}\) represents the serving SBS transmission power, and the number of sub-bands in the system is given by \(N_{SB}\). In general, \(\alpha_{k}\) can take any theoretical value [18]. Now, as mentioned before, to extract the desired signal information from the received superposition-coded signal of the \(k\) paired UEs, a UE-receiver must perform SIC. To do this correctly, it is very important that the UE receiver knows the optimum order of decoding within its paired UEs.