# A Chaotic Compressive Sensing Based Data Transmission Method for Sensors within BBNs
## 5 Feasibility Analysis  
This section contains two parts. First, we demonstrate the feasibility of the proposed method theoretically. Because the reconstruction process of full-authorized receivers is rather similar to the reconstruction process of traditional compressive sensing, in this section, we only discuss the feasibility of the reconstruction of restricted-authorized receivers. In the second part, we conduct experiments to verify that the proposed method is practically feasible. To explicitly explain the processes of critical information concealment and retrieval, we expand Equation (9) in the following way, and assume that \(s_{p}\) is the element that involves critical information. \[\left[ \begin{array}{c}y_{e1}\\ y_{e2}\\ y_{e3}\\ \vdots\\ y_{eM}\end{array} \right]=\left(\left[ \begin{array}{ccccc}a_{11}&a_{12}&a_{ 13}&\cdots&a_{1N}\\ a_{21}&a_{22}&a_{23}&\cdots&a_{2N}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ a_{M1}&a_{M2}&a_{M3}&\cdots&a_{MN}\end{array} \right]+\right.\]  
\[\left[ \begin{array}{ccccc}0&0&\cdots&-z_{1}*a_{1p}&\cdots&0\\ 0&0&\cdots&-z_{2}*a_{2p}&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&-z_{M}*a_{Mp}&\cdots&0\end{array} \right]\right)\left[ \begin{array} []{c}s_{1}\\ s_{2}\\ \vdots\\ s_{p}\\ \vdots\\ s_{N}\end{array} \right] \tag{19}\]  
\[=\left[ \begin{array}{ccccc}a_{11}&a_{12}&\cdots&(1-z_{1})*a_{1p}&\cdots&a_{ 1N}\\ a_{21}&a_{22}&\cdots&(1-z_{2})*a_{2p}&\cdots&a_{2N}\\ \vdots&\vdots&\ddots&\vdots&\ddots&\vdots\\ a_{M1}&a_{M2}&\cdots&(1-z_{M})*a_{Mp}&\cdots&a_{MN}\end{array} \right]\left[ \begin{array}{c}s_{1}\\ s_{2}\\ \vdots\\ s_{p}\\ \vdots\\ s_{N}\end{array} \right]\]  
Let \(w_{i}=1-z_{i}=w+\Delta_{i},i=1,2,3\cdots,M\), then Equation (19) can be rewritten as follows:  
\[\left[ \begin{array}{c}y_{e1}\\ y_{e2}\\ y_{e3}\\ \vdots\\ y_{eM}\end{array} \right]=\left[ \begin{array}{ccccc}a_{11}&a_{12}&a_{13}& \cdots&a_{1N}\\ a_{21}&a_{22}&a_{23}&\cdots&a_{2N}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ a_{M1}&a_{M2}&a_{M3}&\cdots&a_{MN}\end{array} \right]\left[ \begin{array}{c}s_ {1}\\ s_{2}\\ \vdots\\ ws_{p}\\ \vdots\\ s_{N}\end{array} \right]+\]  
\[\left[ \begin{array}{ccccc}0&0&\cdots&\frac{\Delta_{1}}{w}*a_{1p}&\cdots&0\\ 0&0&\cdots&\frac{\Delta_{2}}{w}*a_{2p}&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\frac{\Delta_{M}}{w}*a_{Mp}&\cdots&0\end{array} \right]\left[ \begin{array}{c}s_{1}\\ s_{2}\\ \vdots\\ ws_{p}\\ \vdots\\ s_{N}\end{array} \right] \tag{20}\]To further simplify the discussion, we assume that all \(\Delta_{i}=0,i=1,2,3\cdots,M\), so we get,  
\[y_{e}=As^{{}^{\prime}} \tag{21}\]  
\[\text{where }s^{{}^{\prime}}=\left[ \begin{array}{c}s_{1}\\ s_{2}\\ \vdots\\ ws_{p}\\ \vdots\\ s_{N}\end{array} \right],\,s^{{}^{\prime}}\in\mathbb{R}^{N\times 1}\text{. When restricted-authorized receivers use the OMP algorithm with the input \(y_{e}\) and \(A\), they obtain the signal \(s^{\prime}\) as a result. For the normal case, if at least one \(\Delta_{i}\)\(\
eq\)0, then Equation (21) could be rewritten as,  
\[y_{e}=(A+\Delta A)s^{{}^{\prime}} \tag{22}\]  
\[\text{where }\Delta A=\left[ \begin{array}{cccccc}0&0&\cdots&\frac{\Delta_{i}}{ \text{P}}*a_{1p}&\cdots&0\\ 0&0&\cdots&\frac{\Delta_{i}}{\text{P}}*a_{2p}&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\frac{\Delta_{M}}{\text{P}}*a_{Mp}&\cdots&0\end{array} \right],\, \text{and, as explained in \@@cite[cite]{[\@@bibref{}{Paschke2012}{}{}]}},\, \text{no reduction in the reconstruction can be achieved when the noise added to the measurement matrix is not arbitrarily large. Clearly, the boundedness characteristic of chaotic systems is an additional prerequisite of the successful reconstruction of signal \(s^{\prime}\). We conduct a substantial number of experiments to verify the feasibility of the proposed method from a practical perspective.