## 2 Analysis of Signal Coherent Characteristics  
As we all know, the premise of emitter identification is to classify the received pulses from the same radar emitter into one class by some criteria. In view of the stable phase relationship between the signals transmitted from the coherent radar, the purpose of analyzing the signal's coherent characteristics is to find out the relationship of the initial phase among the received pulse train, so as to use this feature to distinguish the coherent and noncoherent signals for REID. In this section, the received signal model is given. Then, the phase relationship between the received pulse trains is analyzed to validate the feasibility of using the coherent feature as the recognition feature. The relationship between the reconnaissance equipment and the target radar in a radar reconnaissance system is shown in Figure 1; this figure is only a sketch map between the radar emitter and the receiving equipment. The key point is to derive the relationship between the initial phases of the pulse train from the same radar emitter. Therefore, multipath and other jammings in the actual environment are not listed in this sketch map. In Figure 1, \(r(t)\) denotes the relative distance between the receiving equipment and the target radar. Supposing \(x(t)\) is the signal transmitted from the target radar and \(s(t)\) is the received signal, the relationship between \(x(t)\) and \(s(t)\) can be shown, as below. \[\left\{ \begin{array}{l}s(t)=ux(t-\Delta)+v(t)\\ \Delta\approx r(t)/c\end{array} \right. \tag{1}\]  
where \(u\) denotes the attenuation coefficient of signal amplitude, \(c\) is the signal transmission speed, \(v(t)\) is the noise. \(x(t)\) is defined as the transmitting signal from the radar emitter; it can be expressed as follows:  
\[x(t)=A\exp[j(2\pi f_{c}t+\phi_{0})] \tag{2}\]  
where \(A\), \(f_{c}\), and \(\phi_{0}\) denote the amplitude, frequency, and initial phase of the transmitting signal. Without considering signal attenuation, the received signal \(s(t)\) can be expressed as follows:  
\[s(t)=A\exp\{j[2\pi f_{c}(t-r(t)/c)+\phi_{0}]\}+v(t) \tag{3}\]  
When \(t=t_{0}\), expanding \(r(t)\) according to [PERSON]'s formula,  
\[r(t)=r(t_{0})+r^{\prime}(t_{0})(t-t_{0})+\frac{1}{2!}r^{\prime\prime}(t_{0}) (t-t_{0})^{2}+\cdots+\frac{1}{n!}r^{(n)}(t_{0})(t-t_{0})^{n}+\cdots \tag{4}\]  
Substitute \(r(t)\) into \(s(t)\), then the received signal \(s(t)\) can also be expressed as below. \[s(t)=A\cdot\exp\{j\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\  
Then, using the second-order approximation to describe \(r(t)\), \(\varphi(t)\) can be further simplified as below.  
\[\varphi(t)\approx\varphi_{0}+2\pi(f_{c}+f_{d})(t-t_{0})+\pi f_{d}{}^{\prime}(t_{ 0})(t-t_{0})^{2} \tag{8}\]  
When \(t_{0}=0\), it is assumed that the Doppler change rate (\(f_{d}{}^{\prime}(t_{0})\)) can be ignored in a very short observation time, so the phase function \(\varphi(t)\) can be expressed as below. \[\varphi(t)=\varphi_{0}+2\pi(f_{c}+f_{d})t \tag{9}\]  
According to Formula (9), the received pulse train can be expressed as follows. \[s(t)=\sum_{p=1}^{N}A_{p}\text{-rect}\bigg{(}\frac{t-T_{p}}{\tau}\bigg{)}\exp \big{\{}\big{[}2\pi\big{(}f_{cp}+f_{d}\big{)}t+\varphi_{0}\big{]}\big{\}}+ \upsilon(t) \tag{10}\]  
where \(A_{p}\), \(T_{p}\), \(f_{cp}\) represent the PA, time of arrival (TOA), RF of the \(p\)th pulse in a pulse train, \(\text{rect}(\cdot)\) denotes the rectangular window function, \(N\) is the number of the received pulses. After passing through the mixer, \(s\) (\(t\)) can be expressed as below. \[s(t)=\sum_{p=1}^{N}A_{p}\text{-rect}\bigg{(}\frac{t-T_{p}}{\tau}\bigg{)}\exp \big{\{}\big{[}2\pi f_{p}t+\varphi_{0}\big{]}\big{\}}+\upsilon(t) \tag{11}\]  
where \(f_{p}\) denotes the center frequency (CF) of the \(p\)th pulse in a pulse train, \(f_{p}=f_{cp}+f_{d}-f_{L}\), (\(f_{d}<<f_{cp}\)), \(f_{L}\) denotes the local frequency.