# GPU Accelerated PIC and SIC for OFDM-NOMA
## 2 System Model
At iteration \(k\), the time domain OFDM signal for the \(k^{\text{th}}\) user \(\hat{s}_{k}(t)\) becomes, assuming perfect cancellation at each previous iteration and perfect phase/amplitude adjustment during regeneration [11],  
\[\hat{s}_{k}(t)=\sqrt{P}\alpha_{i}s_{k}(t)+\sum_{i=k+1}^{K}\sqrt{P}\alpha_{i}s_ {i}(t)+n(t). \tag{3}\]  
The signal-to-interference and noise ratio (SINR) of the \(k^{\text{th}}\) user per subcarrier can be written as:  
\[\text{SINR}_{k}=\frac{P|\alpha_{k}|^{2}}{\sigma_{n}^{2}+\sum_{i=k+1}^{K}P| \alpha_{i}|^{2}}. \tag{4}\]  
Then, the bit error rate (BER) for the \(k^{\text{th}}\) user per subcarrier with the SIC can be written for binary transmission as \(\text{BER}_{k}=Q(\sqrt{2\text{SINR}_{k}})\) where \(Q(. )\) is the standard Q-function [12]. In this work, the channel is assumed to be frequency flat, so that the SNR and BER per subcarrier values become the overall values for that particular user \(k\). In the PIC receiver (see Figure 2), interference cancellation occurs in two stages. In the first stage, the information signal of each user is decoded collectively [13]. For example, in order to decode the \(k^{\text{th}}\) user's signal, the other users' signals are decoded and regenerated in parallel, then their sum is subtracted from the received signal. The stripped signal includes the signal for the \(k^{\text{th}}\) user only so that, in the second stage, it can be decoded using OFDM demodulation. Once the set of regenerated OFDM signals is obtained in the first stage, the signals of each user can easily be obtained. For PIC, the time domain OFDM signal for the \(k^{\text{th}}\) user \(\hat{s}_{k}(t)\) can be written as:  
\[\hat{s}_{k}(t)=\sum_{i=1}^{K}\sqrt{P}\alpha_{i}s_{i}(t)-\sum_{i=1,i\
eq k}^{K }\sqrt{P}\alpha_{i}\hat{s}_{i}(t)+n(t). \tag{5}\]  
Figure 1: Block diagram of a typical successive interference cancellation (SIC). Derivation of closed-form SINR for PIC is mathematically intractable since it depends on the decoding errors in the first stage. The BER for the \(k^{\text{th}}\) user per subcarrier for a binary transmission can still be expressed as [7; 14; 15]:  
\[BER_{k} = Q\left\{\left[\frac{1}{2(P_{k}/\sigma_{n}^{2})}\left(\frac{1-(\frac {K-1}{3})^{s+1}}{1-(\frac{K-1}{3})}\right)+\frac{1}{3^{s+1}}\right.\right. \tag{6}\] \[\left.\times\left(\frac{(K-1)^{s-1}-(-1)^{s+1}}{K}\left(\frac{ \sum_{i=1,i\
eq k}^{K}P_{i}}{P_{k}}+1\right)+(-1)^{s+1}\right)\right]^{-1/2}\right\}\]  
where \(s\) is the number of stages in the PIC receiver (which is two in our case). Note that Equation (6) is obtained based on a similar approach as in multi-stage receivers in code division multiple access (CDMA) with unit processing gain [7]. In CDMA systems, however, each user has signature codes that help to distinguish them. In power domain NOMA systems, on the other hand, these unique codes may not be present, and they can fail to support large number of users since the interference grows stronger as the number of users increases. Code domain NOMA can be considered in order to have comparable reliability for both SIC and PIC; however, comparison of computation performances will be analogous to the ones in the power domain [16; 17]. In both techniques, any decoding error that occurs in the intermediate stages propagates to the other stages of the receiver. Furthermore, both receivers rely on accurate power allocation among the users to ensure successful interference cancellation. These will have an impact on the reliability of the receivers (e.g., bit error rate performances); however, they will not change the computational complexity. In this work, without loss of generality, equal power allocation among the users and power domain NOMA is considered.