## 3 The Scheduling Method Based on Time Shifting Constraints and Pulse Interleaving  
### Pulse Interleaving Technology  
To further enhance the utilization of time resources, pulse interleaving technology is introduced. Pulse interleaving uses the wait interval of a task (during which the radar  
Figure 2: The mapping relationship from the task request list to the actual dwell tasks. The left side of the figure represents the task request list, where each row corresponds to a task and contains its associated information, including priority, dwell time, and other parameters. The right side illustrates the mapping of this information onto the time axis. The time window \(w\) defines the allowable range within which a task can be executed relative to its expected execution time. Specifically, \(t_{e}-w\) is the earliest possible execution time for the task, while \(t_{e}+w\) is the latest permissible execution time (i.e., the deadline). is idle) to execute the transmit and/or receive dwell of another task. For tracking tasks, the length of the task waiting period can be calculated based on the distance of the target. However, for search tasks, in the absence of prior information, it is not possible to calculate the waiting period between the transmission and reception of echoes due to the lack of distance information of the target. Therefore, pulse interleaving is not applicable to search tasks, which instead immediately enter the reception phase after transmission. Let the task request list based on pulse interleaving be \(T=[T_{1},T_{2},\ldots,T_{N}]\), where \(T_{i}\) represents the \(i\)-th task in the list \(T\). The task model is defined as follows:  
\[T_{i}=\{P,t_{e},w,t_{s},t_{x},t_{w},t_{r},p_{x}\}, \tag{2}\]  
where \(t_{x}\), \(t_{w}\), and \(t_{r}\) are denoted as the length of the transmit dwell time, the wait interval, and the receive dwell time, respectively. In general, the power consumed by receiving a signal is significantly less than the power consumed by transmitting a signal. Therefore, when considering energy constraints in this paper, we ignore the power consumption of receiving signals and only consider the power consumption of transmitting signals \(p_{x}\). The dwell task model based on pulse interleaving is shown in Figure 3.  
Based on the model shown in Figure 3, two types of pulse interleaving methods are presented, as illustrated in Figure 4. Let the previously executed task be \(T_{i-1}\), and the currently executing task is \(T_{i}\). As can be seen, there are two modes of pulse interleaving. One is that only the transmit interval of a task can be interleaved with the wait interval of another task, which is termed partial interleaving. The other is that the entire duration of a task can fi  
Figure 4: Two types of pulse interleaving methods. (**a**) Partial interleaving. The transmission of task \(T_{i}\) is initiated during the waiting period of task \(T_{i-1}\). After \(T_{i-1}\) completes its reception phase, \(T_{i}\) continues with its own waiting and reception phases. This interleaving approach allows tasks to share idle periods effectively. (**b**) Fully interleaving. The entire execution process of task \(T_{i}\) (including transmission, wait, and reception) is fully embedded within the wait interval of the previous task \(T_{i-1}\). This requires that the dwell time of task \(T_{i}\) must be shorter than the wait interval of \(T_{i-1}\). Figure 3: Dwell task model based on pulse interleaving. The task is divided into transmit, wait, and receive phases, where only the transmit phase consumes power and is considered in the energy-constrained scheduling model. of another task, which is termed fully interleaving. Therefore, if two tasks are eligible for pulse interleaving, their transmission and reception periods must not overlap. The time constraints that must be met to implement the two types of pulse interleaving mentioned above are as follows:  
For the mode in Figure 4a:  
\[ \begin{cases}t_{w1}\geq t_{x2},t_{w2}\geq t_{r1},\\ t_{x2}+t_{w2}\geq t_{w1}+t_{r1}\end{cases}