# A Cooperative Transmission Scheme in Radio Frequency Energy-Harvesting WBANs
## 4 Cooperation Scheme
### Initialization Phase  
When a sensor transmits packets using limited energy, the probability of the packets being received by the coordinator increases, but the number of uploaded packets decreases when a higher transmission power is used. There is an optimized transmission power that can maximize the energy efficiency of a particular sensor. Typically, sensors that are farther from the coordinator can harvest less energy but need a higher transmission power when uploading packets to ensure a high packet reception ratio. The decision to determine whether a sensor can work as a relay node is based on whether the harvested energy can support the optimized power. The probability of a packet being received successfully is influenced by the signal-to-noise ratio (SNR). For example, in non-coherent FSK, the packet reception ratio can be expressed as  
\[F(\gamma_{ic})=(1-f(\gamma_{ic}))^{L_{i}}=(1-\frac{1}{2}exp^{-\frac{\gamma_{ic }B}{2K}})^{L_{i}}, \tag{2}\]  
where \(f(\gamma_{ic})\) denotes the bit error rate, and \(R\), \(L_{i}\) and \(B\) denote the transmission rate, the packet length in bits, and the noise bandwidth. \(\gamma_{ic}\) is the SNR at the coordinator, which is related to the transmission power \(p_{i}\). \[\gamma_{ic}=\frac{p_{i}|g_{ic}|^{2}}{\sigma^{2}}, \tag{3}\]  
Figure 2: Illustration of the superframe, (**a**) Network, (**b**) Initialize phase, (**c**) Cooperation phase. where \(\sigma^{2}\) is the power of the noise signal and \(g_{ic}\) represents the channel gain. Then the energy efficiency can be expressed as  
\[\mathbb{E}_{i}=\frac{F(\gamma_{ic})R}{p_{i}L_{i}}=\frac{R|g_{ic}|^{2}}{L_{i} \gamma_{ic}\sigma^{2}}(1-\frac{1}{2}exp^{-\frac{\gamma_{ic}R}{2R}})^{L_{i}}. \tag{4}\]  
The maximized energy efficiency problem can be resolved by calculating the derivative of \(\mathbb{E}_{i}\)[27], and the optimized SNR \(\gamma_{ic}^{*}\) can be gained. Corresponding to \(\gamma_{ic}^{*}\), the transmission power required by the sensor is  
\[p_{i}^{*}=\frac{\gamma_{ic}^{*}\sigma^{2}}{|g_{ic}|^{2}}. \tag{5}\]  
When a sensor receives wireless signals broadcast by the coordinator, the received SNR is  
\[\gamma_{ci}=\frac{p_{c}|h_{ci}|^{2}}{\sigma^{2}}, \tag{6}\]  
where \(p_{c}\) and \(h_{ci}\) denote the transmission power of the coordinator and the channel gain. According to the reciprocity of the wireless channel, the receiver and the sender of the same wireless link can derive the same channel state if they observe the channel simultaneously [31]; the channel gains are modeled as \(g_{ic}=h_{ci}=\sqrt{10^{PL(d_{i})}}\). According to Equations (5) and (6), the value of \(p_{i}^{*}\) can be denoted as  
\[p_{i}^{*}=\frac{\gamma_{ic}^{*}}{\gamma_{ci}}p_{c}. \tag{7}\]  
Then, the energy needed to upload a packet using power \(p_{i}^{*}\) is  
\[E_{i}^{*}=\frac{L_{i}}{R}p_{i}^{*}. \tag{8}\]  
A sensor is identified as a potential relay if it can harvest more energy than \(E_{i}^{*}\). The harvested energy of sensor \(i\) during the energy harvesting phase is  
\[E_{i}=\alpha\wp_{c}|h_{ci}|^{2}T_{e}, \tag{9}\]  
where \(\alpha\) and \(\delta\) are the power split ratio and the energy conversion efficiency, and \(T_{e}\) denotes the length of the download period. Then, sensors can be divided into two groups, relay nodes (R = {\(r_{1},r_{2},\ldots,r_{m}\)}, the set of sensors with \(E_{i}>E_{i}^{*}\)) and common nodes (N = {\(s_{1},s_{2},\ldots,s_{n}\)}, the set of sensors with \(E_{i}\leqslant E_{i}^{*}\)). Relay nodes upload packets using power \(p_{i}^{*}\), and the residual energy is used for data relaying. Common nodes upload packets using all the harvested energy. The coordinator needs to maintain some matrices which are used to optimize the relay selection, including metrics such as the correct reception rate when receiving messages from ordinary sensors and relay sensors, the probability that messages sent by ordinary nodes are correctly received by relay nodes, and the length of messages that relay nodes can forward, among others, such as  
\[M_{1}=