# A Light Signaling Approach to Node Grouping for Massive MIMO IoT Networks
## 5 Optimization
### Formulation  
We can now formulate the optimization problem, which we will refer to as Maximization of Minimum Angular Difference (MMAD) as follows. \[\max \quad B \tag{1}\] \[\quad\sum_{g\in\mathcal{G}}u_{kg}=1, k\in\mathcal{K}\] (2) \[\quad\sum_{k\in\mathcal{K}}u_{kg}\leq P, g\in\mathcal{G}\] (3) \[\quad\sum_{p\in\mathcal{P}}Y_{kg}^{p}=u_{kg}, k\in\mathcal{K}, g\in\mathcal{G}\] (4) \[\quad\sum_{k\in\mathcal{K}}Y_{kg}^{p}\leq 1, p\in\mathcal{P}, g\in\mathcal{G}\] (5) \[\quad\sum_{k\in\mathcal{K}}Y_{kg}^{p}\leq\sum_{k\in\mathcal{K}}Y_ {kg}^{p+1}, p\in\mathcal{P}\setminus\{P\}, g\in\mathcal{G}\] (6) \[\quad t_{g}^{p}=\sum_{k\in\mathcal{K}}\theta(k)Y_{kg}^{p}, p\in\mathcal{P}, g\in\mathcal{G}\] (7) \[\quad t_{g}^{p}\leq t_{g}^{d}+2\pi\left(1-\sum_{k\in\mathcal{K}}Y_ {kg}^{p}\right), p, q\in\mathcal{P}, p<q, g\in\mathcal{G}\] (8) \[\quad s_{g}^{p}\leq\pi\sum_{k\in\mathcal{K}}\sigma(k)Y_{kg}^{p} \in\mathcal{P}, g\in\mathcal{G}\] (9) \[\quad T_{g}\geq t_{g}^{p}, p\in\mathcal{P}, g\in\mathcal{G}\] (10) \[\quad\sum_{g\in\mathcal{G}}u_{kg}=1, k\in\mathcal{K}\]0 (11) \[\quad\sum_{g\in\mathcal{G}}u_{kg}=1, k\in\mathcal{K}\]1 (12) \[\quad\sum_{g\in\mathcal{G}}u_{kg}=1, k\in\mathcal{K}\]2 (13) \[\quad\sum_{g\in\mathcal{G}}u_{kg}=1, k\in\mathcal{K}\]3 (14) \[\quad\sum_{g\in\mathcal{G}}u_{kg}=1, k\in\mathcal{K}\]4 (15) \[\quad\sum_{g\in\mathcal{G}}u_{kg}=1, k\in\mathcal{K}\]5 (16) \[\quad\sum_{g\in\mathcal{G}}u_{kg}=1, k\in\mathcal{K}\]6 (17) \[\quad\sum_{g\in\mathcal{G}}u_{kg}=1, k\in\mathcal{K}\]7 (18) \[\quad\sum_{g\in\mathcal{G}}u_{kg}=1, k\in\mathcal{K}\]8  
The objective function, (1), maximizes the minimum angular difference, adjusted for the angular spectrum spread, between any two nodes in any one group, based on their symbolic placement around a circle. We assume that performance, in terms of achievable SINR, will in general improve for any given node the greater the angle between it and its closest neighbors that are scheduled in the same group. However, this relationship is not straightforward for channels with multipath components, since nodes of which the dominant directions lie close to one another may nonetheless have little correlation between their channels if they have significant power in multipath components with different angles of arrival. The minimum angular difference that is maximized therefore takes the angular spectrum spread of the nodes' channels into account, via the constraints that will be explained below. The first constraint, (2), ensures that each node is placed in exactly one group, and constraint (3) then ensures that the number of nodes placed in each group does not exceed the number of available pilots, \(P\). The decision variables \(Y_{kg}^{p}\) are used to keep track of which group each node is placed in, along with which order it is placed in. Hence, \(Y_{kg}^{p}\) should be set to 1 if node \(k\in\mathcal{K}\) is placed in group \(g\in\mathcal{G}\) at position \(p\leq P\), and 0 otherwise. This is enforced using constraints (4) and (5). Constraint (4) requires that exactly one position in group \(g\in\mathcal{G}\) be assigned to node \(k\in\mathcal{K}\), if the node is placed in that group (\(u_{kg}=1\)), and that no positions in the group are assigned to it if it was not placed in that group (\(u_{kg}=0\)). Constraint (5) ensures that at most one node is assigned to each position in each group, i.e., a strict ordering is imposed in which two or more nodes cannot occupy the same position. Finally, constraint (6) prevents gaps in the node ordering for each group, that is, no position in the group may be filled until all the preceding positions are filled. The next three constraints determine the angles and angular tolerances of each node in each group. Constraint (7) sets the angle of each node position in each group to be the dominant direction of the node assigned to that position, through the decision variable \(t_{g}^{p}\).