## 3 Receiver Design for NOMA
### Conventional NOMA  
In conventional NOMA environment, there are two different scenarios [3;7]: (1) a reference user (typically closer to the base station) that presents received power level lower than that of interfering users, corresponding to the scenario where the SIC is effective as it performs the cancellation of the higher power users (typically users further from the base station); (2) a reference user with received power level higher than that of interfering users (typically further from the base station), disabling the use of the SIC (the SIC is only applicable to cancel higher power interfering users), where the lower power users represent some level of residual interference (user 1 depicted in Figure 3). It should be referred that the SIC focuses on received power levels, which depends on several factors, such as near-far problem, fading and power control. Nevertheless, the most dominant factor is the near-far problem.  
Let us assume that there are a total of \(U\) users sharing the spectrum. The received signal at the \(r\)th receiving antenna and \(n\)th data symbol, is the cumulative sum of the \(\{U;(u=1\ldots U)\}\) signals that share the spectrum (NOMA signals):  
\[y_{n}^{(r)}(t)=\sum_{u=1}^{U}y_{u,n}^{(r)}(t) \tag{6}\]  
Considering only NOMA users, (1) can be re-written as:  
\[\mathbf{Y}_{u,k}=\left[Y_{u,k}^{(1)},\ldots,Y_{u,k}^{(R)}\right]^{T}=\mathbf{ H}_{u,k}\mathbf{X}_{u,k}+\mathbf{N}_{u,k} \tag{7}\]  
As depicted in Figure 4, the SIC detector [18] subtracts the interference estimated for each interfering user by their descending order of received power levels. It detects, regenerates and cancels the signal of each interfering user with power levels higher than those of the reference user. Due to its descending received power order of cancellation, all weaker users benefit from the fact that stronger users are cancelled first. Furthermore, it leads to a better acquisition and more accurate detection of higher power signals and so,  
Figure 3: Illustration of NOMA and Cooperative NOMA.  
take advantage of users with different powers. The signal \(y^{\prime}_{n}\) at the output of the SIC, for the \(n\)th data symbol, after the cancellation of the higher power users \(U^{\prime}\), comes:  
\[y^{\prime}_{n}=y_{u,n}-\sum_{i=1}^{U^{\prime}}\hat{y}_{i,n} \tag{8}\]  
where \(\hat{y}_{i,n}\) stands for the estimate of the received signal of the \(i\)th interfering user and \(n\)th data symbol.  
In case the reference user presents higher power, the SIC is not able to cancel interference and the lower power users represent residual interference. Note that the SIC introduces a delay in the signal corresponding to one symbol period for each cancellation stage, i.e., for each cancelled user. Assuming, e.g., that the reference user is the third more powerful user, this means that the two more powerful users need to be cancelled, introducing a delay of two symbol periods in the signal detection. However, these delays can be reduced if a fast processing is employed (in general, we can have a single symbol delay (to receive the full block) plus the processing delay for each iteration, with the later dependent on the signal processing approach)