# An Efficient Algorithm for De-Interleaving Staggered PRI Signals
## 2 Problem Formulation and Proposed Algorithm Description
### Improved Square Sine Wave Interpolation Algorithm
#### 2.3.3 A Threshold Criterion  
In this subsection, a threshold criterion is designed to suppress the interference of sub-PRIs on the frame period estimation and decrease the misidentification of mixed pulse sequences. The detection threshold design process is described in detail below. The SSWI function is shown in Formula (5). The formula shows that the continuous signal generated by the discrete signal through the SSWI can be regarded as a linear superposition of multiple continuous signals. According to the properties of the Fourier transform (FT)  
If  
\[ \begin{cases}x(t)\xleftrightarrow{\varphi}X(j\omega)\\ y(t)\xleftrightarrow{\varphi}Y(j\omega)\end{cases}  \tag{12}\]  
Then  
\[ax(t)+by(t)\xleftrightarrow{\varphi}aX(j\omega)+bY(j\omega) \tag{13}\]  
Therefore, the FT of the SSWI function can be expressed by  
\[\varphi(s(t))=\wp(\sum_{i=1}^{N-1}s_{i}(t))=\sum_{i=1}^{N-1}\wp(s_{i}(t)) \tag{14}\]  
The spectral magnitude threshold of \(s(t)\) is related to two factors. On the one hand, it depends on the spectral magnitude of the SSWI function \(s_{i}(t)\); on the other hand, it depends on the number of the same difference interval. Firstly, compute the FT of \(s_{i}(t)\). Let \(\omega_{0}=\frac{2\pi}{\Delta t_{i}}\), and compute the FT of \(s_{i}(t)\)  
\[ \begin{split}\wp(s_{i}(t))=\int_{x[t]}^{x[t]+\Delta t_{i}}\Delta t _{i}sin(\omega_{0}t)e^{(-j\omega t)}dt\\ =\Delta t_{i}\frac{e^{-j(\omega_{0}+\omega)x[t]}-e^{-j(\omega_{0} +\omega)(x[t]+\Delta t_{i})}}{2(\omega_{0}+\omega)}\\ +\Delta t_{i}\frac{e^{-j(\omega_{0}-\omega)x[t]}-e^{-j(\omega_{0} -\omega)(x[t]+\Delta t_{i})}}{2(\omega_{0}-\omega)}\end{split}  \tag{15}\]  
The spectral amplitude is related to the number of FT points, and the algorithm calculates the number of FT points according to \(s(t)\), not \(s_{i}(t)\). Let the interpolation sample rate be \(F_{S}\), and the amplitude of the one-sided spectrum of \(s_{i}(t)\) is  
\[ \begin{split}|\wp(s_{i}(t))|&=\left|\lim_{\omega \rightarrow\omega_{0}}\frac{e^{-j(\omega_{0}-\omega)x[t]}-e^{-j(\omega_{0}- \omega)(x[t]+\Delta t_{i})}}{2(\omega_{0}-\omega)}\right|\\ &\cdot 2\Delta t_{i}\frac{F_{S}\cdot\Delta t_{i}}{F_{S}\cdot x[N]}\\ &=\Delta t_{i}^{2}/x[N]\end{split}  \tag{16}\]  
The number of the same difference interval is the next concern. When the number of pulses from multiple radiation sources in the observation time is large enough, the difference interval value between adjacent pulses can be set as a random event; that is, the TOA of the pulse is set as the Poisson point. Suppose the observation time is \(t_{N}\), the pulse number is \(N\), the difference interval is \(\Delta t_{i}\), and the probability of \(k\) random points in the difference interval \(\Delta t_{i}\) is  
\[p_{k}(\Delta t_{i})=\frac{(\lambda\Delta t_{i})^{k}}{k!}e^{-\lambda\Delta t_{i} },k=0,1,2,\cdots \tag{17}\]  
where \(\lambda=N/t_{N}\) indicates the average number of events per unit time. The probability that the difference interval between adjacent pulses is \(\Delta t_{i}\) can be derived from the previous expression for \(k=0\)  
\[p_{0}(\Delta t_{i})=e^{-\lambda\Delta t_{i}} \tag{18}\]  
As the histogram is the estimate of the probability distribution function of a random event, the histogram of higher-grade differences will also be in exponential form. The detection threshold of the difference interval \(\Delta t_{i}\) can be expressed as  
\[N_{\tau}=\alpha(N-c)e^{-\lambda\Delta t_{i}} \tag{19}\]  
where \(\alpha\) is a tunable parameter, depending on the supposed maximum percentage of missed pulses. In summary, the detection threshold of the improved SSWI algorithm is  
\[ \begin{split} A_{SW}(\Delta t_{i})&=|\wp(s_{i}(t ))|\cdot N_{\tau}\\ &=\alpha\Delta t_{i}^{2}(N-c)e^{-\lambda\Delta t_{i}}/x[N]\end{split}  \tag{20}\]  
The threshold criterion is stated as follows.