## 2 LFM Signal Detection Based on FrFT
### FrFT Detection Method for LFM Signal  
The LFM signal [37]\(s(t)\) can be expressed as  
\[s(t)=A\exp\Big{(}\mathrm{j}\tau\mu t^{2}+\mathrm{j}2\pi f_{0}t+\varphi\Big{)},-T /2\leq t\leq T/2, \tag{8}\]  
where \(A\) is the signal amplitude, \(\mu\) is the chirp rate, \(f_{0}\) is the starting frequency, \(\varphi\) is the initial phase, \(T\) is the time width of LFM signal.  
Without loss of generalization, both the starting frequency \(f_{0}\) and the initial phase \(\varphi\) in (8) are set to 0, and (8) is reduced to  
\[s(t)=A\exp\Big{(}\mathrm{j}\tau\mu t^{2}\Big{)}. \tag{9}\]  
Substituting (9) into (1), we obtain the FrFT of the signal \(s(t)\):  
\[\begin{split} S_{p}(u)&=\{F^{p}[s(t)]\}(u)\\ &=A\sqrt{1-\mathrm{j}\cot\alpha}\exp\Big{(}\mathrm{j}\tau u^{2} \cot\alpha\Big{)}\int_{-\infty}^{+\infty}\exp\Big{(}\mathrm{j}\tau\mu t^{2} \Big{)}\exp\Big{(}\mathrm{j}\tau t^{2}\cot\alpha-\mathrm{j}2\pi tu\csc\alpha \Big{)}dt.\end{split} \tag{10}\]  
For an infinite time width LFM signal, when \(\alpha=\mathrm{arccot}(-\mu)\), Equation (10) can be simplified as [38]  
\[\begin{split} S_{p}(u)&=\{F^{p}[s(t)]\}(u)\\ &=A\sqrt{1-\mathrm{j}\cot\alpha}\exp\Big{(}\mathrm{j}\tau u^{2} \cot\alpha\Big{)}\delta(u\csc\alpha),\end{split} \tag{11}\]  
where \(\delta\) is Kronecker delta function. Equation (10) is a pulse function of \(u\).  
For a determined time width \([-T/2,T/2]\) signal, when \(\alpha=\mathrm{arccot}(-\mu)\), Equation (10) can be simplified as [6]  
\[\begin{split} S_{p}(u)&=\{F^{p}[x(t)]\}(u)\\ &=A\sqrt{1-\mathrm{j}\cot\alpha}\exp\Big{(}\mathrm{j}\tau u^{2} \cot\alpha\Big{)}T\sin[\pi T\csc(\alpha)u].\end{split} \tag{12}\]  
When \(T\to+\infty\), \(T\sin[\pi T\csc(\alpha)u]=\delta(u\csc\alpha)\), it means that (12) is equivalent to (11). The result shows that the finite-length LFM signal in the FrFT domain follows the sinc function distribution under the condition that the rotation angle is \(\alpha_{opt}=\mathrm{arccot}(-\mu)\), which is called the optimal rotation angle. This indicates that the LFM signal has its energy concentrated in the optimal order fractional Fourier domain, where the optimal order \(p_{opt}=2\alpha_{opt}/\pi\). We can use this property to detect LFM signal in the optimal order domain. To this end, the objective is to find the optimal order \(p_{opt}\), i.e.,  
\[\begin{split} p_{opt}=&\underset{p}{\arg\max}\big{|} S_{p}(u)\big{|}^{2}\quad\forall u,\\ &\text{subject to}\quad 0\leq p\leq 2\end{split} \tag{13}\]  
which is a non-convex problem, we can employ the brutal searching method to find the optimal solution. However, it will cost heavy computations. In this paper, we adopt an existing coarse-to-fine scanning method [39], as shown in Algorithm 1.  
```
1: Initialization: setting the transform order range \(\mathcal{P}\in[0,2]\); setting the scanning space \(s_{s}=0.1\); setting the initial iteration number \(Num=0\);
2:while\(I>Num\)do
3: Compute \(p^{\prime}=\max_{p^{\prime}\in\mathcal{P}}\left\{\max\lvert S_{p}(u)\rvert^{2}\right\}\);
4: Update: \(\mathcal{P}=[\max(0,p^{\prime}-s_{s}/2),\min(p^{\prime}+s_{s}/2,2)];s_{s}=s_{s} /10\); \(Num=Num+1\);
5:return result Output: The optimal transform order \(p_{opt}=p^{\prime}\).
```  
**Algorithm 1** The optimal transform order corresponding to the maximum value in FrFT domain