# Energy Efficient Constellation for Wireless Connectivity of IoT Devices
## 4 Numerical Analysis and Comments  
The error probabilities computed in Section 3 can be compared to the traditional BPSK signaling. For a fair comparison we set the constant symbol energy of BPSK equal to the averaged symbol energy in Z-BPSK defined in Equation (1), i.e., \(E_{s}^{[BPSK]}=E_{s}^{[Z-BPSK]}=(1-P_{0})E_{1}\). This means that the probability of error of Z-BPSK and BPSK are computed with the same SNR \(\gamma=\frac{E_{s}}{N_{0}}\).  
\begin{table}
\begin{tabular}{c c c} \hline
**Sent** & **Detected** & **Error** \\ \hline \(s_{0}\) & \(s_{0}\) & no error \\ \(s_{0}\) & \(s_{1}\) & false activity \\ \(s_{0}\) & \(s_{2}\) & false activity \\ \(s_{1}\) & \(s_{0}\) & false zero \\ \(s_{1}\) & \(s_{1}\) & no error \\ \(s_{1}\) & \(s_{2}\) & symbol error \\ \(s_{2}\) & \(s_{0}\) & false zero \\ \(s_{2}\) & \(s_{1}\) & symbol error \\ \(s_{2}\) & \(s_{2}\) & no error \\ \hline \end{tabular}
\end{table}
Table 1: Error types.  
Figures 1 and 2 show the three error components \(P_{FA}\), \(P_{FZ}\), \(P_{SE}\) separately, the compound sum \(P_{E}\) for the proposed Z-BPSK signaling as well as the symbol error probability for the legacy BPSK. The graph reports the probabilities versus the signal-to-noise ratio \(\gamma=E_{s}/N_{0}\), where \(N_{0}=2\sigma_{n}^{2}\) is the noise spectral density, for two values of \(P_{0}=\{0.01,0.8\}\). These two values have been selected to represent two extreme situations: when the probability of non-activity (silence) is very low (0.01), and when the probability of non-activity is high (0.8). When the probability of the zero energy symbol is low, the performance present two different behaviors: at low SNRs the zero energy decision region collapses, \(P_{FA}\) is constantly equal to \(P_{0}\), \(P_{FZ}\) drops to zero and the symbol error probability \(P_{SE}\) approaches \(P_{E}\) of legacy BPSK. For high SNRs, the zero symbol is progressively detected and the \(P_{SE}\) outperforms the legacy BPSK.  
A completely different situation is found in Figure 2. For a high probability \(P_{0}\), the zero symbol detection errors still dominate the performance but the constellation performs globally better than traditional BPSK.  
The above comparisons do not take into account the different amount of information the signals are carrying. In a single symbol, Z-BPSK can expose three different states that legacy BPSK cannot deliver. For a more fair comparison, an additional correction has to be made. Let us consider a frame with \(N_{f}\) three-state symbols. If the probability of emission of a zero symbol is \(P_{0}<1/N_{f}\) we can assume optimistically that we will have at most a single zero-energy symbol in the frame. For a legacy BPSK we need to send additional symbols in order to map the eventual occurrence (and position) of the zero energy symbol in the frame. These overhead symbols are \(N_{s}=\log_{2}(N_{f}+1)\). Hence, with a legacy BPSK, we should transmit \(N_{f}+N_{s}\) symbols to deliver \(N_{f}\) three-state symbols with a given \(P_{0}\). These additional symbols are transmitted at the expense of more energy. Therefore, a fair comparison would mean to set:  
\[E_{s}^{[BPSK]}=E_{s}^{[Z-BPSK]}\frac{N_{f}}{N_{f}+N_{s}} \tag{21}\]  
where \(E_{s}^{[BPSK]}\) is the energy per symbol of BPSK and \(E_{s}^{[Z-BPSK]}\) is the energy per symbol of the Z-BPSK. Note that the above computation is biased in favor of BPSK since if there are more than one zero energy symbol in a frame, the number of additional (overhead) symbols is higher.