# Energy-Aware Control of Error Correction Rate for Solar-Powered Wireless Sensor Networks
## 3 Energy-Aware Reed-Solomon Scheme
### Energy Prediction
#### 3.2.2 Energy Consuming Model  
The energy consumption of a sensor node \(E_{\text{consume}}^{i}\) is generally modeled as follows:  
\[E_{\text{consume}}^{i}=E_{\text{transceiving}}^{i}+E_{\text{system}}^{i}+E_{ \text{sensing}}^{i} \tag{2}\]  
where \(E_{\text{transceiving}}^{i}\) and \(E_{\text{sensing}}^{i}\) are the amounts of energy consumed for data transceiving and data sensing, respectively, and \(E_{\text{system}}^{i}\) is the energy used in the remainder of the system other than data transceiving and sensing. At this time, \(E_{\text{transceiving}}^{i}\) can be expressed as the sum of the energy consumed for data transmission \(E_{\text{Tx}}^{i}\) and that used for data reception \(E_{\text{Rx}}^{i}\). In addition, \(E_{\text{system}}^{i}\) is divided into the FEC-related part and the non-related part as follows:  
\[E_{\text{system}}^{i}=E_{\text{FEC\_sys}}^{i}+E_{\text{general\_sys}}^{i} \tag{3}\]  
where \(E_{\text{FEC\_sys}}^{i}\) refers to the FEC-related energy consumed by the system for encoding and decoding error recovery data, and \(E_{\text{general\_sys}}^{i}\) is the energy used by the system excluding that related to FEC. Therefore, \(E_{\text{consume}}^{i}\) in Equation (2), which represents the total consumed energy of a sensor, can be summarized as follows:  
\[E_{\text{consume}}^{i}=E_{\text{Tx}}^{i}+E_{\text{Rx}}^{i}+E_{\text{FEC\_sys }}^{i}+E_{\text{general\_sys}}^{i}+E_{\text{sensing}}^{i} \tag{4}\]The energy consumption terms we are interested in from Equation (4) are \(E^{i}_{\text{Tx}}\) and \(E^{i}_{\text{FEC\_syx}}\), because these depend on the error recovery rate of FEC (which means the parity length of the RS block). The remaining energy consumption terms are unrelated to this and have almost fixed values. Therefore, from now on, we focus on modeling \(E^{i}_{\text{Tx}}\) and \(E^{i}_{\text{FEC\_sysy}}\) in detail. The energy \(E^{i}_{\text{Tx}}\) consumed for data transmission depends on both the amount of data \(D^{i}_{\text{Tx}}\) to be transmitted and the transmission distance \(d_{i}\). It can be expressed as follows [7]:  
\[E^{i}_{\text{Tx}}=D^{i}_{\text{Tx}}\cdot\beta\cdot(d_{i})^{\alpha} \tag{5}\]  
where \(\beta\) is the energy consumed to transfer one bit of data by one meter distance (\(J/b/m^{\text{a}}\)) and \(\alpha\) is the path loss, which is a natural number with a value of two or higher, depending on the transmission environment. Furthermore, \(D^{i}_{\text{Tx}}\) is the total amount of data that must be delivered by node \(i\), including the amount of data it has sensed (\(D^{i}_{\text{sensing}}\)), the amount of data that is received from neighboring nodes that node \(i\) must relay (\(D^{i}_{\text{relay}}\)), and the parity length resulting from RS encoding (\(D^{i}_{\text{parity}}\)). For the sake of convenience, let the sum of \(D^{i}_{\text{sensing}}\) and \(D^{i}_{\text{relay}}\) be called \(D^{i}_{\text{data}}\). Therefore, \(D^{i}_{\text{Tx}}\) can be expressed as follows:  
\[D^{i}_{\text{Tx}}=(D^{i}_{\text{sensing}}+D^{i}_{\text{relay}})+D^{i}_{\text{ parity}}=(D^{i}_{\text{data}}+D^{i}_{\text{parity}}) \tag{6}\]  
By using Equation (6), Equation (5) can be summarized as follows:  
\[E^{i}_{\text{Tx}}=(D^{i}_{\text{data}}+D^{i}_{\text{parity}})\cdot\beta\cdot( d_{i})^{\alpha} \tag{7}\]  
Moreover, \(E^{i}_{\text{FEC\_sysy}}\) includes the RS encoding and decoding energy and is expressed as follows:  
\[E^{i}_{\text{FEC\_sysy}}=E^{i}_{\text{encoding}}(D^{i}_{\text{data}},D^{i}_{ \text{parity}})+E^{i}_{\text{decoding}}(D^{i}_{\text{relay}},D^{i}_{\text{ parity}}) \tag{8}\]  
\(E^{i}_{\text{encoding}}(A,B)\) and \(E^{i}_{\text{decoding}}(A,B)\) represent the amounts of energy consumed for operation and storage when encoding and decoding, respectively, data of size \(A\) using the parity length \(B\).