## 4 The Joint Spatial-Code Clustered IA Scheme
### The Mechanism of the JSCC-IA Scheme  
The JSCC-IA scheme is depicted in Figure 2 for the (\(M\times N\), \(d\))\({}^{K}\) interference channel where the number of users exceeds the IA feasibility constraint, _i.e._, \(K>(M+N)/d-1\). The basic idea of the JSCC-IA scheme is to partition the users into \(G\) groups, with \(K_{0}\) users in each group that satisfies:  
\[K_{0}\leq(M+N)/d-1 \tag{22}\]  
Therefore, the precoding and decoding matrices can be designed to mitigate the intra-group interference just as the conventional IA. The key point of the JSCC-IA scheme is that the data within each group is modulated by a distinct PN code to suppress the inter-group interference according to the idea of spread spectrum. Therefore, the inter-group and intra-group interference can be respectively aligned in the code and spatial domains, and thus the networks can accommodate many more users to communicate simultaneously via the same frequency band.  
Without loss of generality, we assign the \(k\)-th user to the [\(k\)/\(K_{0}\)]-th group, and the \(j\)-th group is assigned with the PN sequence \(C_{j}(n)\). We will employ \(\underline{k}\) to represent [\(k\)/\(K_{0}\)] henceforth to avoid cumbersome notation. The PN sequence \(C_{j}(n)\) can be given by:  
\[C_{j}(n)=\begin{cases}\pm 1,\ 0\leq n\leq L-1\\ 0,\ \ \ \ \text{else}\end{cases} \tag{23}\]  
where \(L\) is the length of the PN sequence. Then the baseband waveform associated with \(C_{j}(n)\) can be expressed as:  
\[c_{j}(t)=A\sum_{n=0}^{L-1}C_{j}(n)\operatorname{rect}_{T_{c}}(t-nT_{c}),0\leq t \leq LT_{c} \tag{24}\]  
Figure 2: The JSCC-IA scheme with \(G\) groups and \(K_{0}\) users in each group.  
where \(\varDelta\) and \(T_{c}\) represent the amplitude of the transmitted signal and the duration of one PN chip, respectively. The relationship of \(T_{b}\), \(T_{c}\) and \(L\) can be expressed as:  
\[T_{b}=LT_{c} \tag{25}\]  
Therefore, the waveform function for the JSCC-IA system can be given by:  
\[g_{k}(t)=c_{\underline{k}}(t) \tag{26}\]  
And the bit energy \(E_{b}\) of the JSCC-IA scheme can be calculated as:  
\[E_{b}=\int_{0}^{T_{b}}[m_{k,j}g_{k}(t)]^{2}dt=\int_{0}^{T_{b}}[g_{k}(t)]^{2}dt= \int_{0}^{T_{b}}[c_{\underline{k}}(t)]^{2}dt=A^{2}T_{b}=LA^{2}T_{c} \tag{27}\]  
The most significant difference between the proposed JSCC-IA and the conventional IA systems lies in the receiver, and the demodulator of the JSCC-IA scheme can suppress the inter-group interference by correlating the received signal with the local PN waveform. To facilitate the analysis in the next subsections, we define the correlation between two PN sequences as:  
\[\rho_{ij}(I)=\sum_{n=0}^{L-1}C_{i}(n)C_{j}(n-I) \tag{28}\]  
According to the properties of PN sequences, the autocorrelation value \(\rho_{ii}(0)\), which equals to \(L\), is significantly larger than the cross-correlation value \(\rho_{ij}(0)\) with \(i\
eq j\). Therefore, the correlation of the PN codes can not only provide extra gains to the desired signals within each cluster, but also mitigate the interference from the unintended groups. There are various types of PN sequences such as m-sequence, M-sequence and Gold sequence. We employ the Gold sequence in the JSCC-IA system due to its advantage of low cross-correlation value, easy implementation, large number of available sequences and so on. Further discussion on the properties of different PN sequences is beyond the scope of this paper, and the readers can refer to [34] for more details.