## 2 Proposed Scheme and RF Transmission Algorithm  
### Chaos Synchronization Design  
The present research focused on utilizing a chaotic system to encrypt ECG signals, thus enhancing security in telemedicine data transmission. This proposal is based on the master-slave scheme. Firstly, as a master system, consider the following general form of a multi-scroll chaotic system, which can be described as follows:  
\[\dot{x}_{1m} =x_{2m}, \tag{1a}\] \[\dot{x}_{2m} =x_{3m},\] (1b) \[\dot{x}_{3m} =-a_{1}x_{1m}-a_{2}x_{2m}-a_{3}x_{3m}+f(x_{1m},x_{2m},x_{3m}), \tag{1c}\]  
where \(\{a_{1},\ a_{2},\ a_{3}\}\in\mathbb{R}\), and \(f(\cdot)\) is a nonlinear function that generates 1-directional multi-scroll chaotic attractors, and it may be defined by several topologies (e.g., hyperbolic functions, piecewise linear function (PWL), special form of a sine function (SFSF)) [34]. Further, we consider the nonlinear function \(f(\cdot)\) as an additional state, i.e., \(\dot{x}_{4m}=\varphi(t)\). Consequently, system (1) can be represented by the following extended state system:  
\[\dot{x}_{1m} =x_{2m}, \tag{2a}\] \[\dot{x}_{2m} =x_{3m},\] (2b) \[\dot{x}_{3m} =-a_{1}x_{1m}-a_{2}x_{2m}-a_{3}x_{3m}+x_{4m},\] (2c) \[\dot{x}_{4m} =\varphi(t). \tag{2d}\]  
Secondly, the slave system is described by the following ESO:\[\dot{x}_{1s} =x_{2s}+l_{1}(y-x_{1s}) \tag{3a}\] \[\dot{x}_{2s} =x_{3s}+l_{2}(y-x_{1s})\] (3b) \[\dot{x}_{3s} =-a_{1}x_{1s}-a_{2}x_{2s}-a_{3}x_{3s}+x_{4s}+l_{3}(y-x_{1s})\] (3c) \[\dot{x}_{2m} =x_{3m},\]0  
where \(y=x_{1m}\) denotes the measurable output of (1), and \(l_{i}\)\((i=1,2,3,4. )\) represents the observer gains. In this approach, the estimation of all states, including the function \(f(\cdot)\), is performed via the ESO. To address the synchronization problem between (1) and (3), we define the error state as \(e_{i}(t)=x_{im}-x_{is},\ (i=1,2,3,4)\). Thus, the error dynamic system is described as follows:  
\[\dot{x}_{2m} =x_{3m},\]1 \[\dot{x}_{2m} =x_{3m},\]2 (4b) \[\dot{x}_{2m} =x_{3m},\]3 (4c) \[\dot{x}_{2m} =x_{3m},\]4  
The _main objective_ is to present a design technique that can deal with variations in the nonlinear function \(f(x_{1m},x_{2m},x_{3m})\) in system (1) and solve the problem of chaotic synchronization. Here, we analyze the stability of system (4) through Lyapunov criteria. First, let us define the vector error state \(e\triangleq[e_{1},\ldots,e_{4}]^{T}\). Therefore, the error dynamic (4) can be expressed as follows:  
\[\dot{x}_{2m} =x_{3m},\]5  
with  
\[A_{e}\triangleq \begin{bmatrix}-l_{1}&1&0&0\\ -l_{2}&0&1&0\\ -a_{1}-l_{3}&-a_{2}&-a_{3}&1\\ -l_{4}&0&0&0\end{bmatrix} ,\quad B\triangleq \begin{bmatrix}0\\ 0\\ 0\\ 1\end{bmatrix} . \tag{6}\]  
Here, we can select the values of observer gains \(l_{i}\) such that all eigenvalues of matrix \(A_{e}\) have a negative real component. Meanwhile, the following is now considered to hold from this point forward:  
**(A1)**: The solutions of system (1) remain uniformly bounded for all time \(t\in[0,\infty)\). **(A2)**: The scalar function \(\varphi(t)\) in (5) is an unknown function with _a priori_ known upper bound, denoted as \(\varphi^{+}>0\), such that  
\[\big{|}\varphi(t)\big{|}\leq\varphi^{+}. \tag{7}\]  
Now, for the stability analysis of system (5), consider a Lyapunov function defined as \(V(e)=e^{T}Pe\). Here, \(P\in\mathbb{R}^{4\times 4}\) is a positive definite and symmetric matrix.