## 3 The Scheduling Method Based on Time Shifting Constraints and Pulse Interleaving
### Pulse Interleaving Technology
, \tag{3}\]  
for the mode in Figure 4b:  
\[t_{w1}\geq t_{x2}+t_{w2}+t_{r2}. \tag{4}\]  
The task scheduling based on pulse interleaving increases the continuous operation time of the radar, leading to higher energy consumption and a rise in the system's temperature. When the temperature rises beyond the limit range that the radar system can withstand, it can cause damage to the radar system. Therefore, energy constraints must also be considered, as shown in Figure 5. \(t_{p0}\) is the end time of the launch period for task \(T_{i-1}\), \(E_{0}\) is the total energy consumption of the system at the end of task \(T_{i-1}\) launch, and \(E_{t}\) is the total energy consumption of the system at the end of task \(T_{i}\) launch. If the energy consumption of the system at time \(t_{0}\) is \(E\) and the radar does not perform any tasks in the following time, then at time \(t_{0}+\Delta T\), the system energy becomes the following:  
\[E=E_{0}e^{-\Delta t/\tau}, \tag{5}\]  
In (5), \(\tau\) is the look-back period, which characterizes the heat dissipation performance of the radar. If task \(T_{i}\) is executed at time \(t_{p0}\), the total energy consumption of the system at the end of the mission launch period is as follows:  
\[E_{t}=E_{0}e^{-t_{x2}/\tau}+p_{x2}\tau\Big{(}1-e^{-t_{x2}/\tau}\Big{)}. \tag{6}\]  
Let \(E_{th}\) be denoted as the endurable maximal instantaneous energy consumption; then, the energy constraint is expressed as follows:  
\[E_{t}<E_{th}. \tag{7}\]  
Figure 5: Energy constraints of pulse interleaving. \(E_{0}\) is the system’s energy consumption at the end of the transmission period of \(T_{i-1}\), and \(E_{t}\) is the system’s energy consumption after the transmission period of \(T_{i}\) when executed in an interleaved manner. Therefore, to enable the interleaved execution of \(T_{i}\) and \(T_{i-1}\), it is necessary to guarantee that the system’s energy consumption at the end of \(T_{i-1}\)’s transmission period remains below a predefined threshold. In summary, two tasks can be interleaved only if they satisfy either (3) or (4), and also meet the energy constraint in (7), as illustrated in Figure 6.  
When two tasks are successfully interleaved, two new idle periods, \(o_{1}\) and \(o_{2}\), are created. To simplify the scheduling process, the two interleaved tasks are merged into a composite task, and \(\max\{o_{1},o_{2}\}\) is considered the wait interval for the new task. After forming the new task, its parameters are updated as follows:  
1. If the interleaving method shown in Figure 4a is adopted, the parameters of the new task are updated as follows: when \(o_{1}>o_{2}\): \[ \begin{cases}t_{x1}=t_{x1}+t_{x2},t_{w1}=o_{1}\\ t_{r1}=t_{r1}+o_{2}+t_{r2}\end{cases} ,\] (8) \[ \begin{cases}t_{p0}=t_{p0}+t_{x2},E_{0}=E_{t}\\ t_{p1}=t_{p1}+o_{2}+t_{r2}\end{cases} .\] (9) when \(o_{1}\leq o_{2}\): \[ \begin{cases}t_{x1}=t_{x1}+t_{x2}+t_{r1}+o_{1}\\ t_{w1}=o_{2},t_{r1}=t_{r2}\end{cases} ,\] (10) \[ \begin{cases}t_{p0}=t_{p0}+t_{x2}+o_{1}+t_{r1}\\ E_{0}=E_{t}e^{-(o_{1}+t_{x1})/\tau}\\ t_{p1}=t_{p1}+o_{2}+t_{r2}\end{cases} .\] (11) 2. If the interleaving method shown in Figure 4b is adopted, the parameters of the new task are updated as follows: when \(o_{1}>o_{2}\): \[ \begin{cases}t_{x1}=t_{x1}+t_{x2},t_{w1}=o_{1}\\ t_{r1}=t_{r1}+o_{2}+t_{r2}\end{cases} ,\] (12) \[ \begin{cases}t_{p0}=t_{p0}+t_{x2},t_{p1}=t_{p1}\\ E_{0}=E_{t}e^{-(o_{1}+t_{r1}+o_{2}+t_{r2})/\tau}\end{cases} .\] (13) when \(o_{1}\leq o_{2}\): \[ \begin{cases}t_{x1}=t_{x1}+t_{x2}+t_{r1}+o_{1}\\ t_{w1}=o_{2},t_{r1}=t_{r1}\end{cases} ,\] (14)  
Figure 6: Parameter updates for the pulse interleaving method. The left part of the figure represents the result after partial interleaving, while the right part shows the result after fully interleaving. \(t_{p0}\) denotes the time point at which the transmission of the task ends and serves as the baseline of the system’s energy state before interleaving. \(t_{p1}\) represents the time point at which the reception of the task ends.