# Comparative Analytical Study of SCMA Detection Methods for PA Nonlinearity Mitigation
## 2 System Model  
In this section, we describe the system model considered. We consider a downlink SCMA scenario, in which the users' bitstreams (considered binary, independent and identically distributed) are grouped and mapped to respective codewords from a codebook \(\left\{\mathbf{x}^{(j)}\in\mathcal{C}^{(j)}\right\}_{j=1}^{J}\), where each codeword, \(\mathbf{x}^{(j)}\in\mathbb{C}^{V}\). Furthermore,the number of codewords in each codebook is denoted by \(\text{Card}[\mathcal{C}^{n}]=M\), with \(M\) denoting the modulation order, and \(\text{Card}[\cdot]\) denoting the number of vectors in a codebook. In this paper, we consider a downlink SCMA system as in ([2] Equation (12.3)), where the users' codewords are overlapped and the superposition, \(\mathbf{x}\), is broadcast through the channel \(\mathbf{h}\). At the receiver, the received vector, \(\mathbf{y}\), is used for MPA-based detection. This is in contrast with the possible uplink scenario presented in ([2] Equation (12.1)) where the users' codewords could arrive asynchronously. For this hypothetical case, there is indeed a possibility of interference between the codewords that could impair their sparsity/algebraic-structure; however, this issue does not arise for downlink SCMA.  
For \(V\) non-interfering resources, the observation at the receiver, \(\mathbf{y}\in\mathbb{C}^{V}\), is given as [2], ([12] Equation (12.3)):  
\[\mathbf{y}=\text{diag}(\mathbf{h})f(\underbrace{\sum_{j=1}^{I}\mathbf{x}^{(j) }}_{\mathbf{x}})+\mathbf{n}, \tag{1}\]  
where \(f(\cdot)\) denotes the PA nonlinearity, \(\mathbf{x}\) denotes the instantaneous superposition of the users' codewords, \(\text{diag}(\cdot)\) is a diagonal matrix that contains elements of \((\cdot)\) in its diagonal, and \(\mathbf{h}\in\mathbb{C}^{V}\) is a vector of channel gains sampled according to a probability density function (PDF) \(p(h)\). The contribution in this work is not constrained by prior statistical assumptions on \(\mathbf{h}\). Furthermore, the complex additive white Gaussian noise (AWGN) vector is given by \(\mathbf{n}=\left[n_{1},n_{2},\cdots,n_{V}\right]^{T}\), with each \(n_{i}\sim p(n)\). Without sacrificing generality, we consider AM-AM Rapp nonlinearity for the PA model, \(f(x)\), which is expressed as follows [18]:\[f(x)=\frac{x}{\left(1+\left|\frac{x}{x_{\text{sat}}}\right|^{2p}\right)^{\frac{1}{2p}}}, \tag{2}\]  
where \(p\) denotes the parameter that controls the severity of the nonlinearity, and \(x_{sat}\) is the PA saturation voltage. It is noted that the RFF-based detectors' performance is not dependent on the nonlinear PA characteristics or their knowledge at the receiver, and existing works show their generalization across different PA characteristics [12].  
The components of the system model are pictorially depicted in Figures 1-3. The transmitter model described mathematically in (1) is shown pictorially in Figure 1. Figure 2 pictorially depicts the overlap of the codewords from each users' dictionary. Finally, the dependence on the user-resources and the variable-nodes is shown by a Tanner graph in Figure 3.  
Figure 1: Depiction of the System Model for SCMA.  
Figure 2: Depiction of the overlapping of codewords for different users.