## 5 Basic Concepts for the SMEP Method
### I-bit Position and Generation
Then, in the SMEP method, for integers \(c\geq 1\) and for any integer \(i\geq 1\), \(S_{c,i}\) covers \(I_{0,(i-1)2}{}^{c-1}\bullet I_{0,(i-1)2}{}^{c-1}{}_{+1}\bullet\cdots\bullet I_ {0,i}{}^{2}{}^{c-1}{}_{-1}\)._  
This theorem tells \(S_{c,i}\) is the sensor data selected in \(I_{0,(i-1)2}{}^{c-1}\bullet I_{0,(i-1)2}{}^{c-1}{}_{+1}\bullet\cdots\bullet I_ {0,i}{}^{2}{}^{c-1}{}_{-1}\). By the way, since \(I_{0,k}=[2k,2k+2]\)\({}_{0}\) and \(I_{0,(i-1)2}{}^{c-1}\bullet I_{0,(i-1)2}{}^{c-1}{}_{+1}\bullet\cdots\bullet I_ {0,i}{}^{2}{}^{c-1}{}_{-1}=[2(I-1)2{}^{c-1},\)\(2(i-1)2{}^{c-1}+2]\)\({}_{0}\bullet[2(i-1)2{}^{c-1}+2,\)\(2(i-1)2{}^{c-1}+4]\)\({}_{0}\bullet\cdots\bullet[i2{}^{c}-2),\)\(i2{}^{c}\)\({}_{0}=[(i-1)2{}^{c},i2{}^{c}]\)\({}_{0}\), \(S_{c,i}\) is the sensor data selected in the compression interval \([(i-1)2{}^{c},i2{}^{c}]_{0}\). **Corollary**. : _Let \(S_{c,i}\) be the \(i\)-th position sensor data in a sensor data sequence derived through the \(i\)-th round compression. Then, the position of \(S_{c,i}\) exists in \([(I-1)2{}^{c},i2{}^{c}]_{0}\)._  
As described already before, the positions of the selected data should be necessarily known in order that each of all lost sensor data is restored. The very function that lets the SMEP method know where each sensor data is either selected or lost is I-bit position. Figure 4: A sensor data selection in the merging of a pair of two compression intervals. **Definition 11**. : _I-bit position is defined recursively as a function to generate the bit string about the position of \(S_{c,i}\) in a compression interval according to the following rules:_  
_Rule 1. For any \(c=1\), \(2\), \(\ldots\), \(n\), I-bit position(\(S_{c,0}\)) is not defined._  
_Rule 2. In 1-th round compression, for each \(I_{0,k}\) in \(C_{S^{0}}\) for \(k=0\), \(1\), \(\ldots\), \(|C_{S^{0}}|-1\) and for \(S_{1,i}\) selected from \(I_{0,i-1}=[2i-2\), \(2i]_{0}=S_{0,2i-2}\), \(S_{0,2i-1}\), \(S_{0,2i}\),_  
\[\text{I-bit position}(S_{1,i})=\left\{ \begin{array}{l}0\text{ if }S_{0,2i-1} \text{ is selected as }S_{1,i}\\ 1\text{ if }S_{0,2i}\text{ is selected as }S_{1,i}\end{array} \right.\]  
_Rule 3. In \(c\)-th round compression, for each \(I_{c-1,k}\) in \(C_{S^{c-1}}\) for \(k=0\), \(1\), \(\ldots\), \(|C_{S^{c-1}}|-1\) and for \(S_{c,i}\) selected from \(I_{c-1,i-1}=[2i-2\), \(2i]_{c-1}=S_{c-1,2i-2}\), \(S_{c-1,2i-1}\), \(S_{c-1,2i}\),_  
\[\text{I-bit position}(S_{c,i})=\left\{ \begin{array}{l}0\text{ }\Theta I-bit\text{ position}(S_{c-1,2i-1})\text{ if }S_{c-1,2i-1}\text{ is selected as }S_{c,i}\\ 1\text{ }\Theta I-bit\text{ position}(S_{c-1,2i-1})\text{ if }S_{c-1,2i}\text{ is selected as }S_{c,i}\end{array} \right.\]  
Shortly, we call I-bit position of \(S_{c,i\text{-th}}\) value of _I-bit position(\(S_{c,i}\))_. The I-bit position represents the relative position of a sensor data within a compression interval. In fact, given an _I-bit position(\(S_{c,i}\))_, the real relative position of \(S_{c,i}\) in its interval is _(the value of I-bit position(\(S_{c,i}\)))_ + 1, As compression makes progress, two consecutive compression intervals that two competitive sensor data for selection cover is merged into one compression interval covered by the selected sensor data. The above definition lets us know how the position of a selected sensor data in a merged compression interval is determined by positions of two competitive sensor data. Figure 5 illustrates such a process to generate the I-bit position of a selected sensor data in compression. In the 1 st round compression, \(S_{0,0}\), \(S_{0,2}\), \(S_{0,4}\), \(S_{0,5}\), and \(S_{0,7}\) are selected as \(S_{1,0}\), \(S_{1,1}\), \(S_{1,2}\), \(S_{1,3}\), and \(S_{1,4}\) from compression intervals, \(I_{0,0}\), \(I_{0,1}\), \(I_{0,2}\), and \(I_{0,3}\), respectively. Here, note that \(S_{0,0}\) is unconditionally selected as \(S_{1,0}\) and its I-bit position is undefined according the definition _Rule 1_, and \(S_{0,2}\) is selected as \(S_{1,1}\) in the compression interval \(I_{0,0}\).