# An Efficient Algorithm for De-Interleaving Staggered PRI Signals
## 2 Problem Formulation and Proposed Algorithm Description
### The Principle and Flow of the Proposed Algorithm  
For the de-interleaving of staggered PRI signals, the following issues must be resolved because of the complex electromagnetic environment. 1. The problem of suppressing the interference between the sub-PRIs and the frame period. 2. The problem of extracting the staggered PRI sequence from the mixed pulse sequences. 3. The problem of computing efficiently. An improved algorithm is proposed to solve the problems. The improved square sine wave interpolation (SSWI) algorithm is proposed to estimate the PRI value. The sequence retrieval algorithm incorporating matched filter theory is proposed to separate pulse sequences. Finally, the staggered sequences are extracted by combining the matched filter theory and the intrinsic property of staggered sequences. For the first problem, an improved SSWI algorithm is proposed. The frame period is estimated by calculating multiple difference grade SSWI functions based on cluster. The difference grade of the improved SSWI function is increased step by step. In addition, a threshold criterion is designed for the SSWI algorithm to suppress the misidentification when the difference grade is less than the stagger number. Furthermore, the estimated PRI is the potential PRI. It needs to be verified by the sequence retrieval algorithm, further reducing the frame period misidentification probability. For the second problem, it is challenging to extract the staggered PRI sequence from mixed sequences directly due to the complexity of the staggered PRI sequence. The improved algorithm regards the TOA model of the staggered PRI sequence as multiple pulse sequences with the PRI value as the frame period. Combining the improved SSWI and sequence retrieval algorithms, only one PRI value is estimated, and the corresponding pulse sequence is extracted in each de-interleaving round. The mixed pulse sequence is separated through multiple rounds of operations. When the sequence separation is completed, the class-matched filter algorithm is used to verify whether the pulse sequence with the same PRI comes from the same radar. These sequences are extracted as a staggered PRI sequence if they come from the same radar. A sequence retrieval algorithm incorporating matched filter theory is proposed to enhance the separation accuracy of radar pulse sequences. Figure 1: The TOA models of the mixed pulse sequences: (**a**) Fixed PRI sequence; (**b**) Jitter PRI sequence; (**c**) Staggered PRI sequence, where \(t_{10}\), \(t_{20}\) and \(t_{30}\) represent the initial phase, \(T^{*}\) represents the frame period, S1, S2 and S3 represent the sub-PRIs; (**d**) Mixed pulse sequences; (**e**) Mixed pulse sequences include missing pulses and jamming pulses. For the third problem, the complex electromagnetic environment results in dense pulse signals and a wide PRI range, which seriously affect the real-time performance of the de-interleaving algorithm. An improved SSWI algorithm is proposed in this study, the computational efficiency of which is almost independent of the pulse number and the range of PRI variations, significantly improving the de-interleaving efficiency of mixed signals in complex electromagnetic environments. The flow chart of the proposed algorithm is shown in Figure 2.  
The flow of the proposed algorithm is as follows. 1. Input the discrete signal \(x[n]\), initialize \(c=1\), \(c\) indicates the difference grade. Calculate the improved SSWI function, and execute the fast Fourier transform (FFT) on the interpolation result. Then, determine the estimated PRI values according to the threshold criteria, and go to Step (ii). 2. Use the proposed sequence retrieval algorithm to extract the pulse sequence from the mixed discrete signal \(x[n]\) by the estimated PRI. If sequence search does not succeed, \(c=c+1\). If \(c\) is greater than \(c_{th}\), go to Step (iii); otherwise, go to Step (i). If sequence search succeeds, the residual pulses are set as a new \(x[n]\).