# An Efficient Algorithm for De-Interleaving Staggered PRI Signals
## 2 Problem Formulation and Proposed Algorithm Description
### Sequence Retrieval Algorithm Based on Matched Filter  
Pulse sequence separation is challenging for mixed signals. The cumulative error caused by PRI jitter will lead to pulse separation error. Moreover, the interference between pulse sequences will be serious for multiple pulse sequences arriving simultaneously. A sequence retrieval algorithm is proposed in [20], which solves the cumulative error by the cumulative average method and suppresses the interference between multiple pulse sequences by the clustering method. However, the separation is inaccurate when multiple pulse sequences exist simultaneously. Based on the sequence retrieval algorithm proposed in [20], a class-matched filter algorithm is proposed to solve the interference between multiple pulse sequences in this study, enhancing the radar pulse stream's separation accuracy significantly. The main reason for the mutual interference of multiple pulse sequences is that multiple pulses may meet the search conditions in one search operation. If the correct pulse was not selected for the separation, the periodicity of the remaining pulse sequence would be destroyed. The class-matched filter algorithm is proposed to suppress the mutual interference in this study. By introducing sampling information of the pulse signal, multiple pulses that satisfy the search conditions are filtered. The class-matched filter algorithm is described as follows. The multiple pulses that satisfy the search conditions can be set as  
\[\mathbf{X}(m)=[x_{1}(m),x_{2}(m),\cdots,x_{n}(m)] \tag{21}\]  
where \(x_{i}(m)(i=1,2,\cdots,n)\) indicates the sampled signal of the \(i\)-th pulse that satisfies the search condition. Define \(N_{i}=length(x_{i}(m)),i=1,2,\cdots,n\). The sampled signal of the reference pulse is set to \(x_{0}(m)\), and define class-matched filter operations  
\[\mathbf{R}=x_{0}(m)\bigotimes\mathbf{X}(m)= \begin{pmatrix}x_{0}(m)*x_{1}(N_{1}-m)\\ x_{0}(m)*x_{2}(N_{2}-m)\\ \vdots\\ x_{0}(m)*x_{n}(N_{n}-m)\end{pmatrix} = \begin{pmatrix}\mathbf{r_{1}}\\ \mathbf{r_{2}}\\ \vdots\\ \mathbf{r_{n}}\end{pmatrix}  \tag{22}\]  
The operation \(x_{0}(m)\bigotimes\mathbf{X}(m)\) represents the correlation operation of \(x_{0}(m)\) and each component of \(\mathbf{X}(m)\), respectively. According to the matched filter theory, the correlation operation and the matched filter are equivalent [30]. Assuming that \(x_{0}(m)\) and \(x_{i}(m)\) are pulses emitted by the same radiation source, there will be a peak in \(\mathbf{r_{i}}\); otherwise, there will be no. The peak detection threshold is determined experimentally. Assuming the discrete signal is \(\mathbf{x}\), the estimated PRI is \(q\), the steps of the sequence retrieval algorithm based on matched filter theory are as follows:  
1. Initialization: \(i=1\), \(k_{c}=1\), \(n_{c}=1\), reference pulse \(t_{ref}=x[i];k_{min}\), \(k_{max}\), where \(k_{min}\), \(k_{max}\) and \(k_{c}\), and determine the search scope, determined by the PRI jitter upper limit; \(n_{c}\) is for counting;
2. According to reference pulse \(t_{ref}\) and PRI value \(q\), calculate the search scope: \[ \begin{cases}T_{max}=t_{ref}+q*k_{max}*k_{c}\\ T_{min}=t_{ref}+q*k_{min}*k_{c}\end{cases} \] (23) If \(T_{max}>max(\mathbf{x})*k_{max}\), this round is complete, so go to Step (vi); else, go to Step (iii). 3. Search for the next pulse that meets the condition: \[\mathbf{p_{s}}=find((\mathbf{x}\geq T_{min})\mathbf{\delta}(\mathbf{x}\leq T_{max}))\] (24)where \(find(\bullet)\) is a function used to search for the element number that satisfies the condition, and \(\mathbf{p_{s}}\) indicates the position of elements that meets the condition. If \(length(\mathbf{p_{s}})=0\), it indicates that no pulses meet the search condition; \(k_{c}=k_{c}+1\), so go to Step (ii); if \(length(\mathbf{p_{s}})=1\), it indicates that only one pulse meets the search condition, and \(x[n_{e}]=x[\mathbf{p_{s}}]\), so go to Step (v); if \(length(\mathbf{p_{s}})>1\), it indicates that multiple pulses meet the search condition, so go to Step (iv);
4.