# Energy Efficient Constellation for Wireless Connectivity of IoT Devices
## 5 Energy Entropy and Energy Capacity
### Capacity  
The channel capacity \(C_{B}\) of a binary erasure channel (BEC) is well known [30]  
\[C_{B}=1-P_{SE} \tag{27}\]  
where \(P_{SE}\) is the symbol error probability.  
The transition probabilities \(P_{SE},P_{FA},P_{FZ}\) in Equation (17) and the probability of zero symbol emission \(P_{0}\) can be used to derive the capacity \(C_{Z}\) for the Z-BPSK  
\[C_{Z}=\sum_{i=0}^{2}\sum_{j=0}^{2}p(x_{j})p(y_{i}|x_{j})\log\frac{p(y_{i}|x_{j })}{p(y_{i})} \tag{28}\]  
where \(x_{j}\) is the symbol emitted by the source(\(\{s_{0}=0,s_{1}=1,s_{2}=-1\}\) in case of Z-BPSK) and \(y_{i}\) is the symbol received by the destination. After some operations, the capacity can be written as  
\[C_{Z}=P_{FZ}\log\left(\frac{P_{FZ}}{P_{FZ}-3P_{FZ}P_{0}+P_{0}} \right)+P_{SE}(1-P_{0})\log\left(\frac{P_{SE}}{P_{FA}P_{0}+(1-P_{FA})(\frac{1 -P_{0}}{2})}\right)+\\ +(1-P_{FZ}-P_{SE})(1-P_{0})\log\left(\frac{1-P_{FZ}-P_{SE}}{P_{ FA}P_{0}+(1-P_{FA})(\frac{1-P_{0}}{2})}\right)+\\ +2P_{0}P_{FA}\log\left(\frac{P_{FA}}{P_{FA}P_{0}+(1-P_{FA})(\frac {1-P_{0}}{2})}\right)+P_{0}(1-2P_{FZ})\log\left(\frac{1-2P_{FZ}}{P_{FZ}-3P_{FZ} P_{0}+P_{0}}\right) \tag{29}\]  
If the transitions between source and destination have no error, the capacity reaches asymptotically the source entropy, as the classical BPSK  
\[\lim_{P_{SE}\to 0}C_{B}=1 \tag{30}\]  
Figure 3: Entropy per unit of energy of the Z-BPSK (solid line) and classic BPSK (dashed line) as a function of the probability of emission of the zero symbol.  
\[\lim_{P_{\rm SE},P_{FA},P_{FZ}\to 0}C_{Z}=-P_{0}\log P_{0}-(1-P_{0})\log\left(\frac{1-P_{0}}{2}\right) \tag{31}\]  
It is important to note that, unlike \(C_{B}\), the capacity of the Z-BPSK is greater than one for \(0<P_{0}<0.55\). In fact,  
\[\lim_{P_{\rm SE},P_{FA},P_{FZ},P_{0}\to 0}C_{Z} = 1 \tag{32}\] \[\lim_{P_{\rm SE},P_{FA},P_{FZ}\to 0,P_{0}\to 1}C_{Z} = 0 \tag{33}\]  
The capacity of the proposed constellation, i.e., the insertion of a zero-energy symbol, reaches a greater capacity compared to classic BPSK, in error-free channels.  
The energy capacity, as defined in [31], can be thus derived  
\[C_{1}\triangleq\frac{C}{E_{S}}. \tag{34}\]  
We can then compare the bit-per-joule curves for the capacity of the Z-BPSK and the BPSK. The energy per symbol \(E_{s}\) is different for the Z-BPSK and the BPSK, as reported in Equation (25). The capacity of Z-BPSK (\(C_{Z}\)) is shown in Figure 4 as a function of the probability of emission of zero symbol (\(P_{0}\)) and SNR.  
The energy capacity of the Z-BPSK and BPSK are compared in Figure 5 as a function of the SNR for different values of \(P_{0}\), while in Figure 6 the energy capacity is drawn as a function of \(P_{0}\) for a fixed SNR. The SNR (\(E_{s}/N_{0}\)) in Figure 5 is calculated by fixing the energy per symbol \(E_{s}=1\) and varying the noise power \(N_{0}\).  
Figure 4: Capacity of the Z-BPSK as a function the probability of emission zero symbol and SNR.  
The results clearly show how the proposed zero-energy mapping can assure a higher energy efficiency, in terms of higher bit-per-joule, even for low probability of emission of zero symbol.