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Deinterleaving of Discrete Renewal Process Mixtures with Application to Electronic Support Measures

Jean Pinsolle, Olivier Goudet, Cyrille Enderli, Sylvain Lamprier, Jin-Kao Hao

TL;DR

A new deinterleaving method for mixtures of discrete renewal Markov chains based on the maximization of a penalized likelihood score, which competes favorably with state-of-the-art methods on simulated warfare datasets.

Abstract

In this paper, we propose a new deinterleaving method for mixtures of discrete renewal Markov chains. This method relies on the maximization of a penalized likelihood score. It exploits all available information about both the sequence of the different symbols and their arrival times. A theoretical analysis is carried out to prove that minimizing this score allows to recover the true partition of symbols in the large sample limit, under mild conditions on the component processes. This theoretical analysis is then validated by experiments on synthetic data. Finally, the method is applied to deinterleave pulse trains received from different emitters in a RESM (Radar Electronic Support Measurements) context and we show that the proposed method competes favorably with state-of-the-art methods on simulated warfare datasets.

Deinterleaving of Discrete Renewal Process Mixtures with Application to Electronic Support Measures

TL;DR

A new deinterleaving method for mixtures of discrete renewal Markov chains based on the maximization of a penalized likelihood score, which competes favorably with state-of-the-art methods on simulated warfare datasets.

Abstract

In this paper, we propose a new deinterleaving method for mixtures of discrete renewal Markov chains. This method relies on the maximization of a penalized likelihood score. It exploits all available information about both the sequence of the different symbols and their arrival times. A theoretical analysis is carried out to prove that minimizing this score allows to recover the true partition of symbols in the large sample limit, under mild conditions on the component processes. This theoretical analysis is then validated by experiments on synthetic data. Finally, the method is applied to deinterleave pulse trains received from different emitters in a RESM (Radar Electronic Support Measurements) context and we show that the proposed method competes favorably with state-of-the-art methods on simulated warfare datasets.
Paper Structure (28 sections, 8 theorems, 48 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 28 sections, 8 theorems, 48 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mixture of renewal processes
  3. Markov renewal chains
  4. Deinterleaving Scheme
  5. Semi-Markov Chain Estimation
  6. Global model estimation and deinterleaving scheme
  7. Solving the Combinatorial Problem in the Space of Partitions
  8. First experimental analysis
  9. Synthetic dataset generation
  10. Empirical verification of the consistency of the score
  11. Experimental validation
  12. State-of-the-art algorithms
  13. Parameter settings
  14. Experiments on synthetic datasets
  15. Electronic Warfare experiments
  16. ...and 13 more sections

Key Result

Theorem 1

Given a generative model $P = G_{\Pi}(\{P^e\}_{e \in E(\Pi)})$, corresponding to a partition $\Pi$ of the alphabet $\mathcal{A}$, under assumptions $(\mathcal{P})$ and $(\mathcal{Q})$, if $P =G_{\Pi'}(\{P^e\}_{e \in E(\Pi')})$ for some partition $\Pi'$, then $\Pi = \Pi'$ must hold.

Figures (5)

  • Figure 1: Display of data generated for a scenario of size 200, with 5 different symbols coming from 3 different emitters. The y-axis represents the symbols, the x-axis their arrival times. The ground truth is $\Pi_{truth} = \{a,e\} \cup \{b,d\} \cup \{c\}$: emitter 0 in blue emits symbols $a$ and $e$, emitter 1 in red emits symbols $b$ and $d$, emitter 2 in green emits symbol $c$.
  • Figure 2: Average success rate of the proposed deinterleaving scheme when an exhaustive search in the partition space is performed and displayed for different numbers of symbols and different sequence sizes $n$. The horizontal line represents the $99\%$ threshold of correctly deinterlaced sequences.
  • Figure 3: Comparison of the deinterleaving algorithms on synthetic data of size $n=500$, 2000 and 5000. Tests were made for 5, 10, 20 and 50 symbols. Each box represents the distribution of the V-measure scores obtained by the compared algorithms and computed for 100 independent scenarios.
  • Figure 4: For both plots the x-axis is the time of arrival of observed data. The y-axis in the first plot shows the frequency values as they are observed. The second plot shows the result of the preprocessing step where the y-axis represents the symbols when clustering pulses based on their frequency. Ground truth is given: each color represents one emitter. There are four emitters. Here we can see emitter 1 (in blue) emits only symbol $d$, while emitter 4 (red) emits 10 symbols $a,b,c,e,f,h,i,j,k,p$.
  • Figure 5: Comparison of deinterleaving algorithm on simulated PDWs. One box represents the distribution of the V-measure scores of an algorithm on the 50 test scenarios.

Theorems & Definitions (16)

  • Theorem 1
  • Proposition 1
  • Theorem 2
  • proof
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • ...and 6 more