## 5 The Random Grouping Selection Algorithm for the JSCC-IA Scheme in the Asymmetric Networks
### The Complexity Analysis  
In this subsection, the computational complexity of the proposed RGS algorithm is analyzed and compared with the BFS algorithm. The number of complex multiplications is employed as the measurement of the complexity. As shown in Algorithm 2, the computation complexity mainly comes from the Max-SINR algorithm in step (2) and the SINR calculation in step (3). Detail analysis on each step is provided as follows:  
1. From the description of the Max-SINR method in Algorithm 1, for the \(k\)-th user's interference-plus-noise covariance matrices \(\mathbf{J}_{k,l}\), we can firstly calculate the terms \(\mathbf{H}_{k}\mathbf{V}_{j}\) with \(MNd\) complex multiplications and then compute the product of \(\mathbf{H}_{k}\mathbf{V}_{j}\) and \((\mathbf{H}_{k}\mathbf{V}_{j})^{\mathrm{H}}\) which takes \(N^{2}d\) complex multiplications. Therefore, the number of complex multiplications required by calculating the interference-plus-noise covariance matrix of each data stream can be expressed as \((K_{0}-1)(NMd+N^{2}d)+(d-1)(NM+N^{2})\). As there are \(Kd\) data streams in total, the sum complexity for calculating all the interference-plus-noise covariance matrices is \(Kd[(K_{0}-1)(NMd+N^{2}d)+(d-1)(NM+N^{2})]\). 2. For the decoding vector \(\mathbf{U}_{k,l}\), the calculation of \(\mathbf{J}_{k,l}^{-1}\) requires \((N-1)!(N-2)N^{2}+N! (N-1)\) complex multiplications and one complex division. The multiplication of \(\mathbf{J}_{k,l}^{-1}\), \(\mathbf{H}_{kk}\) and \(\mathbf{V}_{k,l}\) takes \(MN^{2}+NM\) complex multiplications and the normalization needs \(N\) complex multiplications and \(N\) complex divisions. Therefore, it requires \((N-1)!(N-2)N^{2}+N! (N-1)+MN^{2}+NM+N\) complex multiplications and \(N\) complex divisions to attain each decoding vector. As there are \(Kd\) data streams in total, the sum complexity for calculating all the decoding vectors is \(Kd[(N-1)!(N-2)N^{2}+N! (N-1)+MN^{2}+NM+N]\) complex multiplications and \(KNd\) complex divisions. 3. The complexity for calculating \(\mathbf{\tilde{J}}_{k,l}\) is similar as that of \(\mathbf{J}_{k,l}\), and the total number of complex multiplications is \(Kd[(K_{0}-1)(NMd+M^{2}d)+(d-1)(NM+M^{2})]\). 4. The complexity for calculating \(\mathbf{V}_{k}\) is similar as that of \(\mathbf{U}_{k}\), and the total number of complex multiplications is \(Kd[(M-1)!(M-2)M^{2}+M!(M-1)+NM^{2}+NM+M]\). 5. For the JSCC-IA scheme with identical propagation delay, the calculation of the SINR for each data stream requires \(Kd[(MN+M)+3]\) complex multiplications and one real division according to Equation (33). For the JSCC-IA scheme with different propagation delays, the calculation of SINR requires \((2K-1)d[(MN+M)+3]\) complex multiplications and one real division according to Equation (60). As there are \(Kd\) data streams in total, the overall computational complexity for all the data streams is \(K^{2}d^{2}[(MN+M)+3]\) and \(K(2K-1)d^{2}[(MN+M)+3]\) for the identical and different propagation delays cases, respectively. In all, the total numbers of complex multiplications per iteration for the RGS algorithm with the identical and different propagation delays can be respectively summarized as follows:\[ \begin{split} c_{1}=& Kd[(K_{0}-1)(NMd+N^{2}d)+(d-1)(NM+N^{2} )]+\\ & Kd[(N-1)!(N-2)N^{2}+N!(N-1)+MN^{2}+NM+N]+\\ & Kd[(K_{0}-1)(NMd+M^{2}d)+(d-1)(NM+M^{2})]+\\ & Kd[(M-1)!(M-2)M^{2}+M!(M-1)+M^{2}N+NM+M]+\\ & K^{2}d^{2}[(MN+M)+3]\\ & c_{2}=& Kd[(K_{0}-1)(NMd+N^{2}d)+(d-1)(NM+N^{2})]+\\ & Kd[(N-1)!(N-2)N^{2}+N!(N-1)+MN^{2}+NM+N]+\\ & Kd[(K_{0}-1)(NMd+M^{2}d)+(d-1)(NM+M^{2})]+\\ & Kd[(M-1)!(M-2)M^{2}+M!(M-1)+M^{2}N+NM+M]+\\ & K(2K-1)d^{2}[(MN+M)+3].\end{split}