# Two-Tier Cooperation Based High-Reliable and Lightweight Forwarding Strategy in Heterogeneous WBAN
## 3 Two-Tier Collaboration Based High-Reliable and Lightweight Forwarding (TTCF) Mechanism for WBAN
### STPCSE Model in TTCF
Based on the STPCS, this paper proposes a STPCSE model, as shown in Figure 2.  
Figure 2: STPCSE model. Obviously, in the evolution model \(Y=\kappa\Omega_{1}\propto X+\eta\Omega_{2}\) of STPCS, \(\kappa\) and \(\eta\) are variable parameters, \(\Omega_{1}\) is a \(p\)\(\times\)\(q\) chaotic measurement matrix and the compression ratio is \(CR=p/q\). The measurement matrix is generated by the Logistic chaotic system. The chaotic measurement matrix satisfies the constrained isometric property, which ensures the accurate reconstruction of compressed sensing. The auxiliary matrix \(\Omega_{2}\) is generated by the Tent chaotic system. The semi-tensor product breaks the dimension limitation of matrix multiplication and can realize matrix multiplication in the case of \(q\
eq z\), so that the measurement matrix can be used to encrypt signals with different dimensions. The measurement matrix, auxiliary matrix and parameters \(\kappa\) and \(\eta\) in the semi-tensor product contraction matrix evolution model can be flexibly adjusted to meet specific needs. The STPCSE model still conforms to the general form of compressed sensing, and the derivation process is as follows:  
\[Y=\kappa\Omega_{1}\propto X+\eta\Omega_{2}\Rightarrow\frac{Y-\eta\Omega_{2}}{ \kappa}=\kappa\Omega_{1}\propto X\Rightarrow\frac{Y-\eta\Omega_{2}}{\kappa}=( \Omega_{1}\otimes I)X \tag{4}\]  
In Equation (4), let \(Y^{{}^{\prime}}=(Y-\eta\Omega_{2})/\kappa\), \(\Omega_{1}^{{}^{\prime}}=(\Omega_{1}\otimes I)\), and then,  
\[Y^{\prime}=\Omega_{1}^{{}^{\prime}}X \tag{5}\]  
According to the compressed sensing reconstruction algorithm, the signal \(X\) can be reconstructed based on \(Y^{{}^{\prime}}\) and \(\Omega_{1}^{{}^{\prime}}\). Especially, when the parameters \(\kappa=1\) and \(\eta=0\), the model becomes a common STPCS \(Y=\Omega_{1}\propto X\). In Equation (4), it is important to determine the values of \(\Omega_{1}\) and \(\Omega_{2}\), which solution process is as follows. For \(\Omega_{1}\), firstly, the math equation of the Logistic system in the STPCSE model is shown in Equation (6):  
\[x_{q+1}=\lambda x_{q}(1-x_{q}),x_{q}\in(0,1) \tag{6}\]  
where \(\lambda\) (\(\lambda\in\)[3.57, 4]) is the control parameter, \(x(q)\) is the generated Logistic chaotic sequence, \(x_{q}\) is the value of iteration \(q\) and \(x_{q+1}\) is the value of iteration \(q+1\). The initial value of the Logistic system is 0.2 and the parameter is 4. When sampling the Logistic sequence, the initial sampling position is arbitrary. For the convenience of later calculation, the initial sampling value is set to 1. To reduce the correlation of the Logistic series, the sampling interval is set to 4 according to experience. Therefore, the sequence \(n\) can be obtained. According to the transformation \(m=1-2n\), the sequence \(m\) can be obtained. The measurement matrix \(\Omega_{1}\) can be obtained through column priority rearrangement for sequence \(m\), which can be expressed as follows:  
\[\Omega_{1}=\sqrt{\frac{2}{p}} \begin{bmatrix}m_{0}&\cdots&\cdots&m_{p(q-1)}\\ m_{1}&\cdots&\cdots&m_{p(q-1)+1}\\ \vdots&\vdots&\vdots&\vdots\\ m_{p-1}&m_{2p-1}&\cdots&m_{pq-1}\end{bmatrix}  \tag{7}\]  
In Equation (7), the coefficient \(\sqrt{2/p}\) is used for normalization. \(\Omega_{2}\) is the auxiliary matrix of order \(pz/q\)\(\times\)\(r\), which is generated by the Tent chaotic system. The mathematical formula of Tent system is shown in Equation (8). \[ \begin{cases}f_{q+1}=f_{q}/\mu,0<f_{q}<\mu\\ f_{q+1}=(1-f_{q})/(1-\mu),\mu<f_{q}<1\end{cases}  \tag{8}\]  
in which \(\mu\) (\(\mu\in\)(0,1)) is the control parameter, \(f(q)\) is the generated Tent chaotic sequence and \(f_{q}\) and \(f_{q+1}\) are, respectively, the values of iterator \(q\) and iterator \(q+1\). The parameter of the Tent system is 0.3, and the range of its initial value is (0, 1), and this paper sets the initial value of the Tent chaotic system as 0.23.