## 5 The Random Grouping Selection Algorithm for the JSCC-IA Scheme in the Asymmetric Networks
### The Random Grouping Selection Algorithm  
In the JSCC-IA scheme with \(G\) groups and \(K_{0}\) users per group, the number of all possible clustering combinations is \(D=C_{G\kappa_{0}}^{K_{0}}C_{(G-1)K_{0}}^{K_{0}}...C_{2K_{0}}^{K_{0}}\) and the set that consists all the possible combinations is \(\mathrm{S}=\{P_{1},\,P_{2},...,\,P_{D}\}\). Therefore, the optimal combination under the criterion of optimal BER can be expressed as:  
\[P_{\mathrm{opt-BER}}=\arg\ \min_{P_{\mathrm{in}}}\left\{\overline{\mathrm{BER}}_{ \mathrm{JSCC}},m=1,2,...,D\right\} \tag{74}\]  
where \(\overline{\mathrm{BER}}_{\mathrm{JSCC}}\) represents \(\overline{\mathrm{BER}}_{\mathrm{JSCC1}}\) or \(\overline{\mathrm{BER}}_{\mathrm{JSCC2}}\) for the JSCC-IA systems with identical or different propagation delays, respectively. Although the objective function in Equation (74) can achieve the optimal overall error performance, the computational complexity of the theoretical BER by Equation (42) or Equation (65) increases exponentially with the number of users, which hinders the application of the grouping selection strategy. Therefore, we introduce the minimal received SINR as the alternative objective function, _i.e._,  
\[P_{\mathrm{opt-SINR}}=\arg\ \max_{P_{\mathrm{in}}}\left\{\min_{k,l}\left\{ \mathrm{SINR}_{\mathrm{JSCC}}(k,l),k=1,2,...K,l=1,2,...,d\right\},m=1,2,...,D\right\} \tag{75}\]where \(\mathtt{SNR}_{\mathtt{RCC}}\) represents \(\mathtt{SINR}_{\mathtt{RCC1}}\) or \(\mathtt{SINR}_{\mathtt{SCC2}}\) for the JSCC-IA systems with identical or different propagation delays, respectively. Since the received SINR has a significant impact on the error performance, the minimal received SINR can serve as the measurement of the QoS, and the grouping combination that has the highest minimal SINR is chosen as the optimal solution in Equation (75). Moreover, the computational complexity of calculating the received SINR is much lower than that of the BER. Therefore, the proposed objective function can serve as a good measurement to evaluate the performance of different grouping combinations with low computational complexity. When the number of users is small, brute-force searching (BFS) can be employed to select the optimal grouping combination. However, the number of candidates increases dramatically with the network scale, and searching the optimal clustering combination is a NP-hard problem. Therefore, it is impossible to enumerate all the possible combinations in the case of massive users due to the limited computational time in the practical systems. For the high price of searching for the optimal combination, we can turn to the suboptimal solutions by the RGS algorithm with much lower computational complexity and good performance. As shown in Algorithm 2, the RGS algorithm provides a grouping combination which can achieve the preset objective minimal received SINR \(\xi\) or returns the one that has the highest minimal SINR within the \(\Theta\) tested combinations. In the small size networks, \(\Theta\) can be set as \(D\) so that all the available candidates can be tested, and the optimal grouping combination can be attained. However, in the large scale networks, \(D\) is such a huge number that \(\Theta\) should be much smaller than \(D\). Although the solution in the case of \(\Theta<D\) is not optimal, the RGS JSCC-IA scheme can improve the error performance significantly without too much additional computational complexity. ```
1. Set the maximal number of iterations as \(\Theta\) and the objective minimal received SINR as \(\xi\). The variable \(i\) is used to represent the number of iterations with initial value of 1. Select \(\Theta\) grouping combinations randomly from S as \(S^{\prime}\!=\!\{P^{\prime}_{1},P^{\prime}_{2},\ldots P^{\prime}_{\Theta}\}\). Begin iteration. 2. Select the grouping combination \(P^{\prime}_{i}\) from \(S^{\prime}\), and calculate the precoding and decoding matrices of each group according to the Max-SINR algorithm as \(\mathrm{W}_{i}=\{\{\mathbf{V}_{i}\},\{\mathbf{U}_{i}\}\}\). 3.