# Energy Efficient Constellation for Wireless Connectivity of IoT Devices
## 6 Energy Cost  
Most existing wireless mobile networks (including 3G and LTE) employ Discontinuous Reception (DRX) to conserve the power of mobile user devices (MSs). DRX allows an idle MS to power off the radio transceiver for a predefined period (called the DRX cycle) instead of continuously listening to the radio channel. The main idea is to let the MS switch off the radio receiver for several frames if there are not bits to be transmitted. While in sleep mode, the MS listens at the control channel in order to wake up (active mode) if the base station (BS) indicates that there are frames to be transmitted. All these BS-MS control bits (which are not information) are sent by means of additional power. Furthermore, one frame which is partially filled with non-zero symbols has to be transmitted anyway, consuming energy for nothing. Or, the frame is buffered waiting to accumulate non-zero symbols to fill the frame, but this policy does not apply well to delay-sensitive services such as voice and web browsing, or when there are several information sources with constant but different symbol emission rates. Let us focus on a information source and model it with a two-state Markov chain: Talk State (T) and Silence State (S), as depicted in Figure 7.  
Figure 5: Energy capacity of the Z-BPSK and BPSK as a function of the SNR for different values of \(P_{0}\). Figure 6: Energy capacity of the Z-BPSK and BPSK as a function of \(P_{0}\). Given the transition probabilities \(P_{TS}\), from Talk to Silence, and \(P_{ST}\) from Silence to Talk, the probability of Talk \(P_{T}\) and Silence \(P_{S}\) can be derived  
\[P_{T} = \frac{P_{ST}}{P_{ST}+P_{TS}} \tag{35}\] \[P_{S} = 1-P_{T}=\frac{P_{TS}}{P_{ST}+P_{TS}}=P_{0} \tag{36}\]  
The probability \(P_{S}\) is equivalent to the probability of zero-symbol emission that we depicted with \(P_{0}\) in the previous sections. As known, the probability to stay in a state and to leave a state sum up to 1, e.g., \(P_{TT}+P_{TS}=1\). Let us focus now on a traditional frame-based transmission. In order to derive the average energy per transmitted frame (\(E_{f}\)), we have to first calculate the probability of remaining in the silence state for a number of symbols (\(n\)) to fill entirely one frame (\(N_{f}\))  
\[P\{n=N_{f}\}=(1-P_{ST})^{N_{f}-1}P_{ST} \tag{37}\]  
since in a traditional BPSK transmission, if at least one symbol is emitted in a frame, the frame has to be transmitted entirely, using an energy equal to \(N_{f}E_{1}\), where \(E_{1}\) is the energy per non-zero symbol. If the system remains in the silence state for an interval equal to \(N_{f}\) symbols, then that frame is not transmitted and the energy cost is zero, for that frame. Thus, during a silence state, the average energy per frame can be written as  
\[E_{fs}=\sum_{i=1}^{N_{f}-1}P_{ST}(1-P_{ST})^{N_{f}-1}N_{f}E_{1}=N_{f}E_{1}[1-(1 -P_{ST})^{N_{f}-1}] \tag{38}\]  
On the contrary, during a talk state, the average energy per frame is  
\[E_{ft}=P_{T}N_{f}E_{1} \tag{39}\]  
Thus, the overall average energy per frame of the BPSK is  
\[E_{f}^{[BPSK]}=E_{fs}+E_{ft}=N_{f}E_{1}\left(P_{T}+(1-(1-P_{ST})^{N_{f}-1})\right) \tag{40}\]  
Let us now consider the Z-BPSK scheme. In this case the silent state is directly mapped into the symbol constellation (through a symbol \"0\"), and every time the source is in a silent state it simply transmits a zero-energy symbol. Thus, in this case, the average energy per frame can be written as  
\[E_{f}^{[Z-BPSK]}=N_{f}E_{1}\left(1-P_{0}\right)=N_{f}E_{1}\frac{P_{ST}}{P_{ST} +P_{TS}} \tag{41}\]  
Figure 5:0 Markov chain modelling the source emission of zero and non-zero symbols. The average energy-per-frame of a classical signalling method (such as a frame-based BPSK) and the proposed zero-energy mapping method (Z-BPSK) can be compared in order to highlight the cost in terms of energy to transmit a frame of symbols, as a function of the transition probabilities between the talk and silence states.