## 3 The Scheduling Method Based on Time Shifting Constraints and Pulse Interleaving
### The Description of Proposed Methodology  
After combining the pulse interleaving method with the time pointer scheduling method, it is also necessary to consider the issue of large time shifting. This paper introduces a time shifting constraint mechanism by assigning a maximum allowable time-shift ratio to each task. A task can only be scheduled if its execution time shifting ratio meets this constraint. The constraints of the time shifting-constrained scheduling method, incorporating both the time-pointer and pulse interleaving methods, are as follows: \(t_{ei}\) represents the expected execution time of the \(i\)-th task; \(w_{i}\) represents the time window length of this task; \(t_{0}\) represents the start time of a given scheduling interval; \(t_{end}\) represents the ending time of the scheduling interval; \(t_{si}\) represents the actual execution time of the \(i\)-th task; \(t_{xi}\), \(t_{wi}\), and \(t_{ri}\) are the lengths of the transmit duration, wait interval, and receive duration, respectively; \(N_{exc}\) represents the number of tasks in the radar execution task queue; \(E(t)\) represents the total energy consumption of the radar system at time t; \(E_{th}\) represents the threshold for total energy consumption of the system; \(t_{ck}\) represents the expected execution time of the \(k\)-th task in the deletion task queue, \(w_{k}\) represents its time window length, and \(N_{delete}\) represents the number of tasks in the deletion queue. \[\max(t_{ei}-w_{i},t_{0})\leq t_{si}\leq\min(t_{ei}+w_{i},t_{\text{end}}), \tag{16}\]  
\[\left. \begin{array}{l}\sum_{i=1}^{N_{exc}}\left\{\begin{array}{l}[t_{si},t _{si}+t_{xi}]\cup\\ [t_{si}+t_{xi}+t_{wi},t_{si}+t_{xi}+t_{wi}+t_{ri}]\end{array}\right\}=\varnothing,\end{array} \right. \tag{17}\]  
\[\left. \begin{array}{l}|t_{si}-t_{ei}|/w_{i}<e^{-\alpha*Thr_{i}+\beta},i=1, \ldots,N_{exc},\end{array} \right. \tag{18}\]  
\[\text{E}(t)<E_{th},t\in[t_{0},t_{\text{end}}), \tag{19}\]  
\[t_{ek}+w_{k}<t_{p},k=1,\ldots,N_{\text{delete}}, \tag{20}\]  
Equation (16) states that the actual execution time of each task must be later than the maximum value of the earliest executable time and the start time of the scheduling interval, while the end time of the task must be earlier than the minimum value of the deadline of this task and the end time of the scheduling interval; Equation (17) is designed to prevent time conflicts between scheduled radar tasks. Equation (18) is the time shifting constraint condition, which enforces that a task can only be scheduled if its time shifting rate is below a preset threshold. In this equation, \(Thr_{i}\) represents the threat degree of task \(i\), and \(\alpha\) and \(\beta\) are adjustable parameters used to modify the range of time shifting based on operational requirements or actual conditions. As the priority and threat degree of the target gradually increase, the upper limit of the time shifting for tracking task gradually decreases; Equation (19) represents the energy constraint condition, and Equation (20) is the condition for the task to be dropped. Based on the previous description, the overall flowchart of the algorithm is shown in Figure 7.  
We can summarize the whole processing flow of the method as follows:  
1. Scheduling begins. Initializing the time pointers \(t_{p0}\) and \(t_{p1}\) to point to the start time of the scheduling interval, i.e., \(t_{p0}=t_{p1}=t_{0}\); Initialize counter \(i=0\), system energy \(E_{0}\), and pulse interleaving identifier \(flag\). The variable \(flag\in\)[0,1] is a binary decision variable that takes on the value 1(0) if the current task can be interleaved with the previous task (cannot be interleaved). 2. Let the task request list within the scheduling interval [\(t_{0}\),\(t_{end}\)] be denoted as \(T=\)[\(T_{1}\),\(T_{2}\),...,\(T_{m}\)]. Figure 7: Flowchart of scheduling algorithm. All tasks in \(T\) that satisfy the condition \(t_{e}\)+\(w<t_{p0}\) are moved to the deletion list. Let the total number of such tasks be \(M\), and update the counter by setting \(i=i+M\). 4.