# Opportunities and Challenges for Error Control Schemes for Wireless Sensor Networks: A Review
## 4 Review on Energy Consumption and Efficiency Model
### Single Hop Network  
We introduce a network including single hop and multi-hop in Section 3.2. In this section, we will review on energy consumption and efficiency model for single hop WSN. An energy consumption model based on a radio model considering distance between sensor nodes and number of transmission bits is introduced in [60]. The proposed energy consumption model does not consider encoding and decoding energy consumption. Therefore, the model does not reflect all aspects of energy consumption involved in WSN. Another energy consumption model based on the radio model and incorporating encoding and decoding energy consumption is introduced in [61; 62]. However, this model does not consider the distance between sensor nodes and number of transmission bits during energy consumption modeling. A communication link of a single hop wireless link between two nodes is illustrated in Figure 11[63; 64; 65]. The energy consumption by sensor node for sending \(l\) bits over a single hop wireless link of distance \(d\) incorporating encoding and decoding energy consumption can be expressed as in Equation (10) [63; 64]:  
\[e(l,d)=\left(e_{t}(l,d)+P_{lst}T_{lst}+e_{Enc}\right)+\left(e_{r}(l)+P_{rst}T_{ rst}+e_{Dec}\right) \tag{10}\]  
where \(e_{t}(l,d)\) is the energy spent by the transmitter electronics, \(P_{lst}T_{lst}\) is the start-up energy of the transmitter (\(e_{st}\)), \(e_{Enc}\) is the encoding energy cost per frame, \(e_{r}(l)\) is the energy spent by the receiver electronics, \(P_{rst}T_{rst}\) is the start-up energy of the receiver (\(e_{sr}\)) and \(e_{Dec}\) is the decoding energy per frame. Equation (10) is valid for both coded and uncoded systems. In an uncoded system, there is no encoding and decoding energy cost and the number of transmission bits is less than the coded system. The concept of energy efficiency (\(\eta\)) for a single hop wireless link is introduced in [36] with a packet format as illustrated in Figure 12, with length of preamble \(\alpha\), payload \(n\) and trailer \(\tau\). The energy efficiency captures both throughput (\(\eta_{e}\)) and reliability factors of a WSN link and is defined as in Equation (11):  
Figure 11: A single hop wireless link. Figure 10: Radio model [60]. \[\eta=\eta_{e}r \tag{11}\]  
where \(r\) is the Packet Acceptance Rate (PAR) as a reliability factor. The packet acceptance rate is \((1-PER)\) and Packet Error Rate \((PER)\) of a communication link depends on application of error control coding, distance between nodes and quality of channel. Let us consider a channel with error probability of BER of \(p\), the reliabilities of an uncoded and coded system with \(t\) correction capability are given by Equations (12) and (13), respectively [36]:  
\[r=(1-p)^{n+\alpha} \tag{12}\]  
\[r=\sum_{j=0}^{t}\binom{\alpha+n+\tau}{j}p^{j}(1-p)^{n-j} \tag{13}\]  
The energy throughput is based on packet format, energy cost associated with transmitter, receiver and RF transmission. The length of a preamble and a trailer can be considered fixed for a single standard. However, the length of payload can be varied with a single standard according to the specific application. The total payload is equal to information bits for an uncoded system, whereas the total payload is higher than information bits when FEC is used. The total payload for coded system depends on the code rate (refer to Equation (2)). The energy consumption per bit i.e., \(e_{b}\) can be expressed as in Equation (14) [64]:  
\[e_{b}=e_{tx}+e_{rx}+\frac{e_{Dec}+e_{Enc}}{n} \tag{14}\]  
If we assume that encoding energy is negligible, which is a good assumption for many WSN applications as we discussed in Section 2.2, Equation (14) is reduced to Equation (15):  
\[e_{b}=e_{tx}+e_{rx}+\frac{e_{Dec}}{n} \tag{15}\]  
where \(e_{tx}\) and \(e_{rx}\) are the transmitter energy consumption per bit and receiver energy consumption per bit, respectively.