## 3 Proposed Sensing Waveform
### Waveform Solution Based on the Simulated Annealing Algorithm  
For the optimization problem (9), the solution is NP-hard. However, we do not need to find the optimal solution. We only need to find a solution that satisfies the constraints and meets the requirements of the sensing scenario. Therefore, we can use a suboptimal solution as a substitute for the optimal one. Traditional solving algorithms include simulated annealing [33], genetic algorithms, ant colony algorithms, and so on. For simulated annealing, it is easy to consider each sensing OFDM symbol as a particle inside an object. As the temperature decreases, the particles tend to stabilize. Eventually, the internal energy reaches a minimum, resulting in an optimal solution. In theory, the other algorithms mentioned above are also feasible, but encoding the solving process of the optimization problem (9) into the corresponding physical processes of other algorithms is difficult. Therefore, we use the simulated annealing algorithm to solve the optimization problem (9). A summary of the simulated annealing algorithm is shown in Algorithm 1. Step 1: Initialize the temperature of the system as \(T_{temp}=T_{max}\), generate a \(\mathbf{U}\) randomly, according to (9), and obtain the sensing waveform \(\mathbf{S}_{t}\) according to (5). Step 2: Calculate the number of fluctuation symbols \(N_{r}\) and the fluctuation range based on \(T_{temp}\). Select \(N_{r}\) non-zero elements in \(\mathbf{U}\) to exchange with zero elements in the fluctuation range and judge whether (9) is satisfied. If not, repeat Step 2, and if satisfied, generate a new \(\mathbf{U}^{\prime}\). Then, obtain a new sensing waveform \(\mathbf{S}^{\prime}_{t}\) according to (5). Step 3: Calculate the cost functions \(F(\mathbf{U})\), \(F(\mathbf{U}^{\prime})\) according to (6)-(10) and \(\Delta F=F(\mathbf{U}^{\prime})-F(\mathbf{U})\). Step 4: If \(\Delta F\leq 0\), the new solution \(\mathbf{U}=\mathbf{U}^{\prime}\) and \(\mathbf{S}_{t}=\mathbf{S}^{\prime}_{t}\) is obtained. If \(\Delta F>0\), the new solution is received when \(e^{-\Delta F/(T_{k}T_{temp})}>\delta\), where \(\delta\) takes a uniform distribution of [0, 1]. If the value of \(\Delta F\) is smaller and the value of \(T_{temp}\) is larger, the probability of the new solution being accepted is higher. Step 5: Reduce system temperature \(T_{temp}=\beta T_{temp}\), where \(\beta\) represents the temperature fading factor. Step 6: Determine if the current temperature of the system \(T_{temp}\) is less than the minimum temperature \(T_{min}\). If \(T_{temp}<T_{min}\), the algorithm terminates and outputs the optimal sensing waveform \(\mathbf{S}_{t}\). Otherwise, repeat steps 2 to 5 until \(T_{temp}<T_{min}\). ```
Initialization:  Generate a \(\mathbf{U}\) randomly according to (10) to obtain \(\mathbf{S}_{t}\); \(T_{temp}=T_{max}\); Repeat:  Generate a new \(\mathbf{U}^{\prime}\) according to (9), \(\mathbf{U}\) and \(T_{temp}\) to  obtain \(\mathbf{S}^{\prime}_{t}\); \(\Delta F=F(\mathbf{U}^{\prime})-F(\mathbf{U})\); If\(\Delta F\leq 0\) Then \(\mathbf{U}=\mathbf{U}^{\prime}\), \(\mathbf{S}_{t}=\mathbf{S}^{\prime}_{t}\); Else Accept \(\mathbf{U}^{\prime}\) and \(\mathbf{S}^{\prime}_{t}\) with a probability \(e^{-\Delta F/(T_{k}T_{temp})}\); \(T_{temp}=\beta T_{temp}\); Until\(T_{temp}<T_{min}\); Output:\(\mathbf{S}_{t}\);
```  
**Algorithm 1** The simulated annealing algorithm. After the above algorithm optimization, the designed sensing waveform is \(\mathbf{S}_{t}\). Considering a point target, the received signal \(y_{l}(t)\) of the \(l\)th sensing symbol of the sensing waveform can be described as  
\[ \begin{split} y_{l}(t)=&\sigma s^{\prime}_{n_{l}}(t- \tau)e^{2\pi f_{0}(t-\tau)}e^{2\pi f_{d_{l}}n_{l}T_{k}}+c_{l}(t)+n_{l}(t)\\ =&\sum_{m=0}^{N_{c}-1}\sigma e^{2\pi f_{0}(t-\tau)}e ^{2\pi\tau m\Delta f(t-\tau-n_{l}T_{s})}\\ & e^{2\pi f_{d_{l}}n_{l}T_{s}}\text{rect}\left[\frac{t-\tau-n_{l} T_{s}}{T_{s}}\right]+c_{l}(t)+n_{l}(t),\end{split}  \tag{12}\]  
where \(\tau\) is the round-trip propagation time of the target.