# 6G Goal-Oriented Communications: How to Coexist with Legacy Systems?
## 5 Numerical Evaluation  
In this section, we evaluate the performance of the proposed GO optimization strategy in Algorithm 1 in the proposed coexistence network setting. To be able to compare the performance with fixed PER and \(P_{d}\), we consider the same scenario of the previous evaluation (Figure 7). As a first result, in Figure 8, we show the trade-off between goal effectiveness and goal cost, for a fixed goal-effectiveness threshold \(\mathcal{E}_{g}=82\%\), two different values of total system bandwidth (Figure 7(a) with \(W=1\) GHz and Figure 7(b) with \(W=500\) MHz) and for different goal value thresholds (the same as the previous figure). Moreover, let us recall that our solution hinges on the problem approximation that replaces the first indicator function in (30) with its expectation computed on a validation set. Therefore, in Figure 7(a),b, we compare this solution with the one involving a _genie_ that uses a posteriori knowledge on the resulting entropy at the output of the classifier. Obviously, this solution cannot be implemented in practice, but it represent our benchmark, as it approaches the optimum of (19) as the trade-off parameter \(\Omega\) increases. Finally, we compare our goal-oriented optimization method with a strategy that splits the bandwidth across the two systems, thus resulting in zero co-channel interference leaked from one system to another, however, at the cost of less radio bandwidth available for each of the two systems. In this case, only the PER of the GO user is selected dynamically. This can also be considered as a goal-oriented communication KPIs selection, but the interference to and from a DO system is not managed, as the system is assumed to not be affected by interference. Note that the bandwidthsplitting is empirically selected to obtain the target level of goal effectiveness to compare the results in terms of goal cost. In other words, \(UE_{g}\) is allocated the minimum amount of bandwidth needed to achieve the target goal effectiveness, while the remaining portion of the bandwidth is allocated to \(UE_{d}\). In the curves of Figure 8a, the hyper-parameter \(\Omega\) increases from right to left. The curves with the same colors are obtained with the same system parameters, but they represent the three different strategies: _(i)_ GO optimization (i.e., our proposed strategy) with solid lines, _(ii)_ genie-aided GO optimization with dashed lines and (iii) bandwidth splitting scenario with the pentagrams. First, we can notice how our solution is able to always guarantee the goal-effectiveness constraint, with the goal cost decreasing towards the point in which the constraint is exactly attained (left-hand side of the plot). Moreover, as the goal value threshold (\(\Theta_{\text{th}}\)) increases, the minimum achieved goal cost increases, as expected (see, e.g., yellow curve vs. blue curve). In particular, for the case with larger bandwidth (Figure 8a), the effect is more visible, but still limited, through the approximation, while the genie-aided solution is able to keep the cost close to optimal even when increasing \(\Theta_{\text{th}}\). However, as \(\Theta_{\text{th}}\) decreases, the approximated solution becomes closer to the genie solution. Notably, _both non-orthogonal spectrum sharing_ solutions outperform the one with bandwidth splitting, with considerable gain for the case of limited total available bandwidth (up to 50%--Figure 8a). This suggests that the power of a GO system optimization is even more relevant under communication resource scarcity, as also pointed out in the introduction of this work. Still, it should be noted the large difference between the GO communication KPIs selection and the legacy one, which is the purpose of the following performance evaluation. As a second result, Figure 9 builds on the results of Figure 7 to compare our optimization method with the fixed PER and \(P_{d}\) setting (i.e., the non-optimized case). In particular, let us focus on Figure 9a, i.e., the case with \(\Theta_{\text{th}}=-0.8\).