# Difference DV_Distance Localization Algorithm Using Correction Coefficients of Unknown Nodes
## 2 Related Works  
Table 1 presents the description of specific symbols for some improved DV_Distance localization algorithms. In reference [22], [PERSON] introduces both the ratio of actual Euclidean distance to accumulated hop distance and the hops between anchor nodes to design a correction coefficient. However, it still uses the correction model based on anchor nodes, that is, each unknown node receives the corresponding correction coefficient from its nearest anchor node to modify the accumulated hop distance from itself to other anchors. Suppose that, the anchor \(i\) is the nearest anchor node to the unknown node \(k\), and thus the correction equations of [PERSON]'s approach are shown in detail as follows:  
\[r_{i,j}=\frac{D_{i,j}}{d_{i,j}} \tag{1}\]  
\[cv_{i,j}=\frac{r_{i,j}}{h_{i,j}} \tag{2}\]  
\[d_{k,j}^{est}=d_{i,j}-cv_{i,j}\times h_{k,j} \tag{3}\]  
In term of the difference of accumulated hop distance and corresponding actual Euclidean distance between anchor nodes, [PERSON] [23] proposes a difference DV_Distance localization algorithm, and the correction coefficient based on the distance difference between anchor nodes is designed to correct accumulated hop distance from unknown nodes to anchor nodes. However, it has an obvious disadvantage, that is, the accumulated hop distances for unknown nodes on the same path are actually different. The equations for distance correction are as follows:  
\[cv_{i,j}=d_{i,j}-D_{i,j} \tag{4}\]  
\[d_{k,j}^{est}=d_{k,j}-cv_{i,j} \tag{5}\]  
Taking account of the difference of hop number between anchor nodes, [PERSON] [24] evenly assigns the difference of accumulated hop distance and corresponding actual Euclidean distance to each hop, but the problem ignored in this algorithm is that the assigned distance difference for each hop should also be different when the nodes in WSNs are not evenly distributed. The improved distance correction equations of [PERSON]'s method are expressed as:  
\[cv_{i,j}=\frac{d_{i,j}-D_{i,j}}{h_{i,j}} \tag{6}\]  
\[d_{k,j}^{est}=d_{k,j}-cv_{i,j}\times h_{k,j} \tag{7}\]  
On the basis of [PERSON]'s approach, [PERSON] [25] calculated the average distance difference per hop under unit distance after considering the randomness of node distribution, and the specific correction equations for [PERSON]'s method are as follows:  
\[cv_{i,j}=\frac{d_{i,j}-D_{i,j}}{h_{i,j}\times D_{i,j}} \tag{8}\]  
\[d_{k,j}^{est}=\frac{d_{k,j}}{1+cv_{i,j}\times h_{k,j}} \tag{9}\]  
In the above reference [22; 23; 24; 25], the correction model based on anchor nodes is employed. When the accumulated hop distances from unknown nodes to anchors are to be corrected, the unknown nodes will adopt the corresponding correction coefficients from their nearest anchor nodes, which lead to the phenomena that the different unknown nodes may have the same correction coefficient. Therefore, [PERSON] [26] designed a correction model based on unknown nodes in which the unknown nodes use their own correction coefficients, and a dynamic weighted DV_Distance localization algorithm is proposed using the ratio of actual Euclidean distance to accumulated hop distance between anchor nodes. Different anchor nodes are selected to compute their corresponding correction coefficients, and then a dynamic weighting correction approach is designed to obtain the final correction coefficient. The detailed equations of [PERSON]'s method are expressed as follows:  
\[cv_{k,i,j}=\left(\frac{D_{i,j}}{d_{i,j}}\right)^{\frac{h_{k,j}}{h_{k,j}+h_{k,j}}} \tag{10}\]  
\[\omega_{k,i,j}=\frac{\left(\frac{h_{k,j}}{h_{k,j}+h_{k,j}}\right)^{-1}}{\sum \limits_{t=1}^{N}\left(\frac{h_{k,j}}{h_{k,j}+h_{k,t}}\right)^{-1}} \tag{11}\]  
\[d_{k,i}^{est}=\sum\limits_{j=1}^{N}\left(\omega_{k,i,j}\cdot cv_{k,i,j}\right) \times d_{k,i} \tag{12}\]  
We have realized the above algorithms by programming with MATLAB so that we can compare our proposed algorithm with them from lots of aspects. The comparison and analysis will be made after we introduce our algorithm in following sections.