## 5 Improved Tracking Differentiator (SYSTD)
### Simulation Analysis of SYSTD Performance  
The discrete equation of system S\({}_{2}\) is as follows:  
\[\left\{ \begin{array}{l}x_{1}(k+1)=x_{1}(k)+x_{2}(k)\cdot h+\frac{1}{2}h^{2} \cdot u(k)\\ x_{2}(k+1)=x_{2}(k)+h\cdot u(k)\\ u(k)=R^{2}[-sys(x_{1}(k)-r(k),a_{41},a_{51})-sys(\frac{x_{2}(k)}{R},a_{42},a_ {52})]\end{array} \right. \tag{52}\]  
Among them, \(h\) is the integration step size. Next, conduct the simulation. Analyze the influence of different set values of parameters \(R\), \(a_{41}\), \(a_{51}\), \(a_{42}\), and \(a_{52}\) on SYSTD through the simulation. Since the movement of the MRC in the horizontal direction is a step-by-step movement, the step distance is set to 1 mm, and the survey signal is close to a sine wave. Therefore, in the simulation experiment, white noise and a sine wave are used as the given signals, and their equations are as follows:  
\[r(t)=\sin(t)+\gamma n(t) \tag{53}\]  
Simulation 1: Influence of the time scale R on the tracking characteristics of SYSTD. According to the constraint conditions of Equations (40) and (48), let \(a_{41}=144\), \(a_{51}=16\), \(a_{42}=3\), and \(a_{52}=1\). Take \(R=100\), \(200\), \(300\), and \(500\), respectively, to simulate the tracking of the given signal by SYSTD. The results are shown in Figure 9. Obviously, the value of \(R\) determines the tracking speed of SYSTD. The smaller the value of \(R\), the slower the tracking speed. However, when \(R>300\), the increase in the tracking speed is not obvious, and at the same time, the oscillation amplitude increases. Therefore, the parameter \(R\) has a significant influence on the tracking accuracy of SYSTD. When selecting its value, the balance between the dynamic response and the noise-reduction ability of SYSTD needs to be considered. Simulation 2: Influence of the parameters of \(sys()\) on the tracking characteristics of SYSTD. As can be seen from Section 5.1, when \(a_{4}\) is constant, the parameter \(a_{5}\) directly determines the change in the amplitude of \(sys()\); when \(a_{5}\) is constant, the parameter \(a_{4}\) directly determines the change in the tracking rate of \(sys()\). The parameter settings of  
Figure 9: The tracking results of SYSTD under different values of R.  
SYSTD and the results of following the given signal containing random noise are shown in Figure 10.  
Figure 10: The influence of the _sys(_) parameter on the results of SYSTD tracking. **(a)** The value of \(a_{41}\) changes, while the values of \(a_{51}\), \(a_{42}\), and \(a_{52}\) remain unchanged. **(b)** The value of \(a_{42}\) changes, while the values of \(a_{41}\), \(a_{51}\), and \(a_{52}\) remain unchanged. **(c)** The value of \(a_{51}\) changes, while the values of \(a_{41}\), \(a_{42}\), and \(a_{52}\) remain unchanged. **(d)** The value of \(a_{52}\) changes, while the values of \(a_{41}\), \(a_{42}\), and \(a_{51}\) remain unchanged. Obviously, increasing the parameters \(a_{41}\) and \(a_{42}\) can effectively improve the signal tracking speed, but it will result in a loss of tracking accuracy (Figure 10a,c). Additionally, appropriately increasing \(a_{51}\) and \(a_{52}\) can enhance the denoising ability of SYSTD, but it will lead to a loss of tracking speed (Figure 10b,d). Moreover, when the parameters do not meet the constraint conditions required by Equations (40) and (48), it will cause a large overshoot and seriously reduce the tracking accuracy. To obtain appropriate SYSTD parameters, considering the influence of linear components on the tracking and filtering performance of SYSTD, and, in combination with Simulation 1, first, under the condition of ensuring system stability, roughly select the four parameters of _sys_(), and then adjust R to meet the requirements of tracking speed and accuracy. Then, based on the phenomena in Simulation 2, adjust \(a_{4}\) and \(a_{5}\) to change the amplitude and change rate of _sys_(), and, thus, adjust the tracking and filtering performance of SYSTD. Simulation 3: Performance comparison simulation of TD, MTD, and SYSTD.