# 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  
The online compression of perceptual data to reduce the amount of transmission and storage is one of the means to achieve energy saving in WBAN. Therefore, compression methods need to meet the requirements, such as a fast compression speed, large compression ratio, easy hardware implementation, less resource consumption and high reconstruction accuracy. Common monitoring data compression methods include LZW (Lempel-Ziv-Welch) lossless compression algorithm, wavelet compression, Huffman coding compression, neural network compression, etc., which are based on the characteristics of perceptual data, and achieve data compression by using some means to remove the redundancy of perceptual data. Such compression often needs to design complex compression coding algorithms, which is difficult and time-consuming to implement in an  
Figure 1: TTCF strategy in heterogeneous WBAN. embedded environment. WBAN online data compression expects that the compression process will be as convenient as possible, but the requirements for decompression speed are not strict, which can be completed on the computer offline. In other words, it hopes to find a new algorithm that can separate the complex operation process from the coding. Although the emergence of compressed sensing makes it possible to solve these problems, traditional compressed sensing methods require that the number of columns of the measurement matrix should be equal to the number of rows of the signal when using matrix multiplication, which makes the measurement matrix occupy more memory and have high computational complexity in the practical applications. The STPCS method is a generalized adaptive online compressed sensing technology. It overcomes the limitation of consistent dimension when the measurement matrix is multiplied in traditional compressed sensing methods, and reduces the size of the measurement matrix. It can use the same measurement matrix to measure signals of different sizes, meeting the diversity characteristics of medical data, and can also solve the problem of increased network energy consumption caused by the excessive storage and transmission of sensor data in WBAN. Therefore, for batch medical data with mixed signals, using STPCS technology to reduce the amount of data to be transmitted is a key step to realizing the high-efficiency and energy-saving forwarding model for WBAN. [PERSON] et al. [19] gave an authoritative mathematical model of STPCS as follows:  
\[Y=\Lambda\propto X \tag{1}\]  
in which \(X\in R^{2}\), \(Y\in R^{p\times q}\) and \(\Lambda\in R^{p\times q}\) are sparse signal, observed value and semi-tensor measurement matrix based on perceptual data, respectively, and \(\propto\) denotes a semi-tensor product. Because \(z=lcm(q,z)\), Equation (1) can be converted to the following Equation (2):  
\[Y=(\Lambda\otimes I_{z/q})X \tag{2}\]  
where \(\otimes\) represents tensor product, and Equation (2) can be expressed below:  
\[\left[ \begin{array}{c}y_{1}\\ y_{2}\\ \vdots\\ y_{pz/q}\end{array} \right]=\left[ \begin{matrix}a_{11}&\cdots&a_{1q}\\ \vdots&\ddots&\vdots\\ a_{p1}&\cdots&a_{pq}\end{matrix} \right]\propto\left[ \begin{matrix}x_{1}\\ x_{2}\\ \vdots\\ z_{z}\end{matrix} \right]=\left[ \begin{matrix}a_{11}&\cdots&0&a_{1q}&\cdots&0 \\ \vdots&\ddots&\vdots&\cdots&\vdots&\ddots&\vdots\\ 0&\cdots&a_{11}&&0&\cdots&a_{1q}\\ \vdots&&\ddots&&\vdots&\vdots\\ a_{p1}&\cdots&0&&a_{pq}&\cdots&0\\ \vdots&\ddots&\vdots&\cdots&\vdots&\ddots&\vdots\\ 0&\cdots&a_{p1}&&0&\cdots&a_{pq}\end{matrix} \right]\left[ \begin{matrix}x_{1}\\ x_{2}\\ \vdots\\ x_{z}\end{matrix} \right] \tag{3}\]  
In Equation (3), the row and column number of each small block of matrix \(\Lambda\) is \(z/p\), and there are \(pq\) small blocks in total. The diagonal elements in the small block are \(a_{11}\), \(a_{12}\),..., \(a_{pq}\) and the other elements are 0. The reconstruction method of STPCS is parallel reconstruction method, and the semi-tensor measurement matrix satisfies the properties of cross-correlation, Spark condition and constrained isometry.