# Joint Optimization of Data Freshness and Fidelity for Selection Combining-Based Transmissions
## 6 Numerical Results  
In this section, we firstly perform simulations to validate the correctness of theoretical results. After that, we investigate the effects of system parameters on the overall system freshness and fidelity. The realistic setup is shown in Table 1.  
Figure 3 shows the simulation and analysis results of the average AoI \(\overline{\Delta}(N)\) when the arrival rate of the updates are \(v=4\) and \(v=6\), i.e., in each time slot, there are 4 and 6 updates arriving at the FC respectively. As can be seen intuitively, the average AoI increases while the number of sensor nodes \(N\) grows. This accords with the analysis in Section 4.3.1. It is also validated that the average AoI is overall larger when the arrival rate \(v\) is smaller (cf. Equation (7)). Most importantly, by comparing the curves with the markers we can conclude that our analysis result on average AoI (cf. Lemma 1) is identical to the simulation result in which we invoked 100,000 groups of arrays obeying exponential distribution with parameter \(v\) to calculate the average AoI. Figure 4 shows the simulation and analysis results of the MMSE \(\overline{D}(N)\). The realistic setup in this simulation is introduced as follows. The signal power \(\sigma_{X}^{2}\) is set to 10 dBm (10 mW) and 13 dBm (20 mW) respectively, the channel noise power \(\sigma_{Z}^{2}\) is \(N_{0}B=-173\times 10^{6}\) dBm. For the path loss model, we adopt \(35.5+37.6\lg(d_{k})\), where we assume the distance between the sensors and the FC \(d_{k}\) is 100 m. Then we have the corresponding SNR \(\gamma_{0}\) as 12.5 dB and 15.5 dB respectively. Firstly, it can be seen from the figure that our theoretic analysis on MMSE (Lemma 2) accords with the simulation in which we invoked 100,000 groups of random variables obeying \(|h_{i}|^{2}\sim\exp(1)\) and chose the largest among them as the received signal to further calculate the MMSE. Secondly, it is shown that the MMSE declines with the growth of sensor nodes number \(N\), which also echoes with the analysis in Section 4.3.2. \begin{table}
\begin{tabular}{l l} \hline \hline
**System Parameter** & **Value** \\ \hline Single-sided noise spectral density \(N_{0}\) & \(-173\) dBm\(/\)Hz \\ Available bandwidth \(W_{\max}\) & \(1\) MHz \\ Path loss model \(10\lg(\alpha_{k})\) & \(35.5+37.6\lg(d_{k})\)[27] \\ Maximum transmit power of sensor & \(13\) dBm (20 mW) [27] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters. Figure 3: Simulation and analysis results of the average AoI \(\overline{\Delta}(N)\) w.r.t. the number of sensor nodes \(N\) under different arrival rate \(v\). Figure 5 shows the varying trend of the data freshness-fidelity weighted-sum tradeoff \(f(N)\) with respect to the number of sensor nodes \(N\) under different SNR \(\gamma_{0}\) and weighting factor \(\alpha\). The realistic setup of the system is the same as in Figure 4. In this case, we consider the same arrival rate as \(v=8\) (eight updates arrive at the FC in each time slot). Overall, all curves well demonstrate the monotonicity of the weighted-sum tradeoff which decreases first and goes up then. Comparing circular dotted curve with a circular solid curve which differs in SNR \(\gamma_{0}\), we can conclude that a smaller SNR \(\gamma_{0}\) results in smaller MMSE and so is the tradeoff, whereas the optimal number of sensor nodes \(N^{\star}=4\) remains unchanged. Comparing the circular solid curve with the square solid curve, it is seen that a larger weighting factor \(\alpha\) leads to a smaller optimal number of sensor nodes \(N^{\star}\). This can be ascribed to the fact that a larger weighting factor \(\alpha\) means a larger AoI, hence the curve begins the upward trend earlier. By further reducing the weighting factor \(\alpha\) to 0.45, the optimal number of sensor nodes becomes even larger, see the dotted square curve, reaching \(N^{\star}=12\). Most importantly, it is obvious that our proposed sub-optimal number of sensor nodes \(N^{\star}\) is identical to the optimal \(N^{\star}\).