## 2 System Model
### NOMA in Small-cells
This order is determined through the channel-gain of the \(k\) paired UEs in a way that only the high-gain users (SCUs) perform SIC to remove the signal information of the low-gain users (SEUs), as shown in Figure 3. Let us consider a high-channel-gain user \(m\), in the \(k\) paired UEs, connected to the \(b\)th SBS. Then, UE-\(m\) must perform SIC to extract its signal by removing the signal information of the low-gain UEs within its pairing. The received signal of UE-\(m\) in the \(b\)th SBS is given as:  
\[y_{b,m}=\underbrace{h_{b,m}\times s_{b,m}\sqrt{P_{b,m}}}_{Wanted\ Signal}+ \underbrace{h_{b,m}\times\left(\sum_{k=m+1}^{N}s_{b,k}\sqrt{P_{b,k}}\right)} _{Unwanted\ Signal:}+\underbrace{n}_{noise}+\underbrace{I}_{ICI} \tag{5}\]  
where, the \(ICI\) presented by \(I\) consists of two kinds of interference signals, i.e., cross-tier interference and co-tier interference signals. In the co-channel environment, cross-tier interferences occur when the UE of an SBS and a nearby UE of the overlaying macrocell base station (MBS) are allocated the same resource block (RB); similarly, co-tier interferences occur when UEs of neighboring SBSs reuse the same RB. Thus, if there are \([1\ \ldots\ M]\) resource blocks in an MBS and the underlaid small-cells reuse the same RBs, then the ICI created at the UE-\(m\) of the serving \(b\)th SBS is given as:  
\[I=\underbrace{\sum_{a=1}^{M}\lambda_{a,m}h_{a,m}s_{a}\sqrt{P_{a}}}_{Cross-tier \ Interference}+\underbrace{\sum_{b^{\prime}\
eq b}\lambda_{b,b^{\prime}}h_{b^{ \prime},m}^{a}s_{b^{\prime},m}^{a}\sqrt{P_{b^{\prime}}}}_{Co-tier\ Interference}, \tag{6}\]  
where, \(s_{a}\) and \(s_{b^{\prime},m}^{a}\) are transmitted symbols from the MBS to its MU-\(a\) and from SBS-\(b^{\prime}\) to its SU, respectively; \(h_{a,m}\) and \(h_{b^{\prime},m}\) are channel responses from MBS and from SBS-\(b^{\prime}\) to UE-\(m\) of the serving SBS, respectively; \(P_{a}\) and \(P_{b^{\prime}}\) are transmission powers of the MBS and SBS-\(b^{\prime}\), respectively; \(\lambda_{a,m}\) is the RB allocation indicator for cross-tier interference; \(\lambda_{a,m}=1\) when the same RB is being occupied by both SBS and the interfering UE of the MBS; otherwise, \(\lambda_{a,m}=0\). Similarly, \(\lambda_{b,b^{\prime}}\) is the RB allocation indicator for co-tier interference, \(\lambda_{b,b^{\prime}}=1\) if both serving SBS-\(b\) and interfering SBS-\(b^{\prime}\) use the same RB; otherwise, \(\lambda_{b,b^{\prime}}=0\). Replacing \(I\) in (5) gives:  
\[y_{b,m}=\underbrace{h_{b,m}\times s_{b,m}\sqrt{P_{b,m}}}_{Wanted\ Signal}+ \underbrace{h_{m}\times\left(\sum_{k=m+1}^{N}s_{b,k}\sqrt{P_{b,k}}\right)}_{ Unwanted\ Signal:}+\underbrace{\sum_{a=1}^{M}\lambda_{a,m}h_{a,m}s_{a}\sqrt{P_{a}}}_{ Cross-tier\ Interference}+\underbrace{\sum_{V\
eq b}\lambda_{b,b^{\prime}}h_{b^{ \prime},m}^{a}s_{b^{\prime},m}^{a}\sqrt{P_{b^{\prime}}}}_{Co-tier\ Interference}. \tag{7}\]To simplify the problem and as per the existing literature for the NOMA receiver, let us consider a two-user case. Thus, for the two-UE case, the received signal at UE-1 is given by using (5) as:  
\[y_{b,1}=\underbrace{h_{b,1}\times s_{b,1}\sqrt{P_{b,1}}}_{Wanted\ Signal}+ \underbrace{h_{b,1}\times s_{b,2}\sqrt{P_{b,2}}}_{Unwanted\ Signal}+ \underbrace{n}_{noise}+\underbrace{I}_{ICI}, \tag{8}\]  
where, \(s_{b,1}\), \(P_{b,1}\), and \(h_{b,1}\) are the signal-information, power, and channel-response, respectively, of the high-gain user UE-1, and \(s_{b,2}\) and \(P_{b,2}\) are signal information and power, respectively, of the low-gain user UE-2. For further investigation of the received signal at the receiver, two cases are considered, i.e., NOMA with perfect and imperfect SIC.