# Windowed Joint Detection and Decoding with IR-HARQ for Asynchronous SCMA Systems
## 2 System Model
### Asynchronous IR-HARQ SCMA  
We can consider an uplink SCMA system with \(L\) physical resources and each physical resource denoted by \(\mathbf{U}_{i}\), where \(i\in\{1,2,\ldots,L\}\). After mapping, it becomes a codeword \(\mathbf{x}_{i}\) of the SCMA codebooks. If we let \(\mathbf{h}_{i}\) represent the channel vector from the base station to the \(i\)-th physical resource, the received signal \(\mathbf{y}\) can be expressed as  
\[\mathbf{y}=\sqrt{\frac{E_{s}}{L}}\sum_{i=1}^{L}\text{diag}(\mathbf{h}_{i}) \mathbf{x}_{i}+\mathbf{n}. \tag{3}\]The diagram in Figure 1 illustrates the IR-HARQ strategy employed by the asynchronous uplink SCMA system with \(J\) users and \(L\) physical resources. Each user \(j\), \(j\in\{1,2,\ldots,J\}\), encodes a data sequence \(\mathbf{u}_{j}\) of length \(k\) using an RC-LDPC code independently. The result is a codeword \(\mathbf{v}_{j}=\left(v_{j,1},v_{j,2},\ldots,v_{j,N}\right)\) with each \(v_{j,n}\) being either 0 or 1. To simplify, we denote the vector \((v_{j,1},v_{j,2},\ldots,v_{j,N})\) as \(\{v_{j,l}\}_{l=1}^{N}\). The windowed joint detection decoder, boxed in the red line, is utilized at the receiver. Moreover, extrinsic information exchange between \(J\) decoders and the windowed detector is shown.  
This paper discusses an SCMA system that involves 6 users sharing 4 resource elements. Each user has a codebook of size \(M\), denoted by \(\mathcal{X}_{j}=\{\mathbf{a}_{j}^{(i)}\subset\mathbb{C}^{L},1\leq i\leq M\}\). The codebook consists of sparse signal vectors \(\mathbf{a}_{j}^{(i)}=\{a_{j}^{(i)}[\ell]\}_{l=1}^{L}\), where \(n\) non-zero entities are positioned based on a sparse indicator vector \(\mathbf{f}_{j}\). The SCMA encoder is defined as \(\mathbf{v}_{j}=\{v_{j,l}\}_{l=1}^{N}\rightarrow\mathbf{x}_{j}=\{\mathbf{x}_{ j,l}\}_{l=1}^{N}\) with \(\mathbf{x}_{j,l}\subset\mathcal{X}_{j}\). The SCMA system uses an \(L\times J\) indicator matrix \(\mathbf{F}=[\mathbf{f}_{1},\mathbf{f}_{2},\ldots,\mathbf{f}_{J}]\) to assign the vector \(\mathbf{f}_{j}\) to user \(j\). Each \(\mathbf{f}_{j}\) is a binary vector of length \(L\) with row and column weight of \(d_{f}\) and \(n\), respectively. Therefore, each user transmits a coded symbol using \(n=2\) complex symbols. The indicator matrix \(\mathbf{F}\) is given as  
\[\mathbf{F}_{4\times 6}=\left[\begin{array}{cccccc}0&1&1&0&1&0\\ 1&0&1&0&0&1\\ 0&1&0&1&0&1\\ 1&0&0&1&1&0\end{array}\right]. \tag{4}\]  
To learn more about the SCMA encoder, please refer to [7; 26].  
Assuming that all users' signals are perfectly synchronized, we can combine Equations (3) and (4). The received superposed signals at the receiver during the symbol duration \(t\) can be written as  
\[\mathbf{y}_{t}=\sum_{j=1}^{J}\mathrm{diag}(\mathbf{h}_{j,t})\mathbf{x}_{j,t}+ \mathbf{w}_{t}, \tag{5}\]  
where \(\mathbf{h}_{j,t}\) is the channel gain vector and \(\mathbf{w}_{t}\) is the additive complex Gaussian noise vector. In addition, the correlation between each \(\mathbf{h}_{j,t}\) is determined by the coherence time of the channel, or Doppler spread. In situations with high mobility, the channel is rapidly time-varying and experiences multiple channel realizations in each retransmission round.  
This paper considers that each transmitted block has a length of \(N_{0}\) and that \(D_{\max}\) represents the maximum number of transmissions, including the initial transmission and retransmissions. For a user \(j\), \(j\in\{1,2,\ldots,J\}\), the retransmission request will be sent back to the sender if decoding fails. Upon receiving this request, the sender will send  
Figure 1: Block diagram of an asynchronous uplink SCMA system with the RC-LDPC code-based IR-HARQ scheme.  
redundancy bits specific to user \(j\) through the channel. This process will continue until decoding is successful or the maximum number of transmissions is reached.