# Resource-Efficient Multicast URLLC Service in 5G Systems
## 3 Analyses of Possible Solutions and Their Adaptation to mURLLC
### Precoder Selection  
For each transmission, the gNB constructs a special matrix called a precoder that determines how signals are generated from different gNB antennas. A proper precoder can significantly boost the performance of the M-MIMO system. The corresponding problem for one-antenna UEs is formulated as follows. For each UE \(i\) and RB \(j\), the gNB has estimations of (i) the channel matrix \(H_{i,j}\) of size \(1\times M\) that is updated based on periodic SRS, and (ii) the interference plus noise power \(\sigma_{j}^{2}\) in RB \(j\), which is obtained based on the channel quality indicator reports provided by the UEs. For each RB \(j\) used for the DL transmission, the gNB constructs a precoder \(W_{j}\), which is a matrix of size \(M\times 1\). The Signal to Interference plus Noise Ratio (SINR) for UE \(i\) in RB \(j\) is estimated as:  
\[SINR_{i,j}(W_{j})=\frac{|H_{i,j}W_{j}|^{2}}{\sigma_{j}^{2}}. \tag{1}\]  
Since the UE with the worst SINR limits multicast transmission parameters, the precoder selection problem for the set \(U\) of UEs is stated as follows:  
\[ \begin{split}&\max_{W_{j}}\left[\min_{i\in U}SINR_{i,j}(W_{j}) \right],\\ &\text{s.t.: }\left\|W_{j}\right\|_{F}^{2}\leq P_{TX},\end{split}  \tag{2}\]  
where \(P_{TX}\) is the transmission power allocated for a single RB. In the literature, authors often consider a similar problem statement [37]:  
\[ \begin{split}&\min_{W_{j}}\|W_{j}\|_{F}^{2},\\ &\text{s.t.: }SINR_{i,j}(W_{j})\geq\gamma,\quad\forall i\in U, \end{split}  \tag{3}\]  
where \(\gamma\) is the SINR constraint. Problems (2) and (3) are similar in the sense that the solution to Problem (2) gives the solution to Problem (3) with a proper scaling, which depends on constraints \(P_{TX}\) and \(\gamma\). Both optimization problems are non-convex and proven to be NP-hard in case \(M\leq N\)[37]. Let us classify and analyze numerous approaches to the multicast precoding problem. One of the approaches is to reformulate Problem (3) as follows (see [37]):  
\[\min_{X_{j}}trace(X_{j}), \tag{4a}\] \[\text{s.t.: }trace(X_{j}Q_{i,j})\geq\gamma\sigma_{j}^{2},\quad \forall i\in U,\] (4b) \[X_{j}\succeq 0,\] (4c) \[rank(X_{j})=1, \tag{4d}\]where \(X_{j}=W_{j}W_{j}^{H}\), \(Q_{i,j}=H_{i,j}^{H}H_{i,j}\), \(trace(X)\) and \(rank(X)\) denote the trace and rank of matrix \(X\), respectively. Note that the objective function (4a) and constraints (4b) and (4c) are convex, while only the constraint (4d) is non-convex. By relaxing the constraint (4d), we obtain the Semi-Definite Relaxation (SDR) (4a), (4b), (4c) of the optimization problem, Problem (4). That SDR can be solved with state-of-the-art convex solvers. However, the obtained SDR solution gives the solution to Problems (3) and (4) only if the matrix \(X_{j}\) has rank equal to one. Otherwise, it only provides a lower bound on the objective function for Problem (3) and a corresponding upper bound for the Problem (2). Thus, we use that upper-bound solution to evaluate the performance of other precoder algorithms. Since the usage of convex solvers results in high computational complexity inappropriate for mURLLC applications with tight latency requirements, below we consider different approaches proposed in the literature that have lower computational complexity. [PERSON] et al. [35] propose the application of the Successive Convex Approximations (SCA) method to the Problem (3) and use the Alternating Direction Method of Multipliers (ADMM) on each SCA iteration, which is a relatively low-complexity state-of-the-art method for convex problems. Unfortunately, the resulting algorithm, which we refer to as SCA-ADMM, still has high complexity that depends on the convergence threshold as we show in Section 4. Other approaches considered in the literature are based on different heuristics. For example, [PERSON] at al. [33] derived a closed-form solution for multicast Problem (3) with only two UEs, and presented an FF-C2 (Full Featured Combine-2) algorithm that performs a full search over all possible pairs of UEs.