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Statistical Properties of Target Localization Using Passive Radar Systems

Mats Viberg, Daniele Gerosa, Tomas McKelvey, Thomas Eriksson

TL;DR

The paper analyzes the statistical properties of the Extended Cancellation Algorithm (ECA) for target localization in passive radar with multiple receiver nodes, each equipped with a Reference Channel (RC) and a Surveillance Channel (SC). It treats ECA as an approximate maximum-likelihood estimator, derives a compact Cramér-Rao bound (CRB) for the target state that includes an explicit penalty term for RC-noise, and provides a clear sufficient condition under which RC-noise effects are negligible. The authors establish consistency under mild unambiguity conditions and derive the asymptotic covariance as the sum of the ideal CRB and a RC-noise correction, supported by extensive simulations in bi-static and multi-static scenarios. The work yields practical guidelines for system design, including frame-based processing and range-migration handling, and demonstrates how RC-SNR and batch processing choices impact localization accuracy and efficiency.

Abstract

Passive Radar Systems have received tremendous attention during the past few decades, due to their low cost and ability to remain covert during operation. Such systems do not transmit any energy themselves, but rely on a so-called Illuminator-of-Opportunity (IO), for example a commercial TV station. A network of Receiving Nodes (RN) receive the direct signal as well as reflections from possible targets. The RNs transmit information to a Central Node (CN), that performs the final target detection, localization and tracking. A large number of methods and algorithms for target detection and localization have been proposed in the literature. In the present contribution, the focus is on the seminal Extended Cancelation Algorithm (ECA), in which each RN estimates target parameters after canceling interference from the direct-path as well as clutter from unwanted stationary objects. This is done by exploiting a separate Reference Channel (RC), which captures the IO signal without interference apart from receiver noise. We derive the statistical properties of the ECA parameter estimates under the assumption of a high Signal-to-Noise Ratio (SNR), and we give a sufficient condition for the SNR in the RC to enable statistically efficient estimates. The theoretical results are corroborated through computer simulations, which show that the theory agrees well with empirical results above a certain SNR threshold. The results can be used to predict the performance of passive radar systems in given scenarios, which is useful for feasibility studies as well as system design.

Statistical Properties of Target Localization Using Passive Radar Systems

TL;DR

The paper analyzes the statistical properties of the Extended Cancellation Algorithm (ECA) for target localization in passive radar with multiple receiver nodes, each equipped with a Reference Channel (RC) and a Surveillance Channel (SC). It treats ECA as an approximate maximum-likelihood estimator, derives a compact Cramér-Rao bound (CRB) for the target state that includes an explicit penalty term for RC-noise, and provides a clear sufficient condition under which RC-noise effects are negligible. The authors establish consistency under mild unambiguity conditions and derive the asymptotic covariance as the sum of the ideal CRB and a RC-noise correction, supported by extensive simulations in bi-static and multi-static scenarios. The work yields practical guidelines for system design, including frame-based processing and range-migration handling, and demonstrates how RC-SNR and batch processing choices impact localization accuracy and efficiency.

Abstract

Passive Radar Systems have received tremendous attention during the past few decades, due to their low cost and ability to remain covert during operation. Such systems do not transmit any energy themselves, but rely on a so-called Illuminator-of-Opportunity (IO), for example a commercial TV station. A network of Receiving Nodes (RN) receive the direct signal as well as reflections from possible targets. The RNs transmit information to a Central Node (CN), that performs the final target detection, localization and tracking. A large number of methods and algorithms for target detection and localization have been proposed in the literature. In the present contribution, the focus is on the seminal Extended Cancelation Algorithm (ECA), in which each RN estimates target parameters after canceling interference from the direct-path as well as clutter from unwanted stationary objects. This is done by exploiting a separate Reference Channel (RC), which captures the IO signal without interference apart from receiver noise. We derive the statistical properties of the ECA parameter estimates under the assumption of a high Signal-to-Noise Ratio (SNR), and we give a sufficient condition for the SNR in the RC to enable statistically efficient estimates. The theoretical results are corroborated through computer simulations, which show that the theory agrees well with empirical results above a certain SNR threshold. The results can be used to predict the performance of passive radar systems in given scenarios, which is useful for feasibility studies as well as system design.
Paper Structure (22 sections, 3 theorems, 80 equations, 5 figures)

This paper contains 22 sections, 3 theorems, 80 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Symbols and notation
  3. Problem description
  4. Data model
  5. Parameter Estimation and Target Localization
  6. Interference Cancellation and Delay-Doppler Estimation
  7. Target Localization
  8. Frame-Based Processing
  9. Including Range Migration
  10. Performance Analysis
  11. Consistency
  12. Asymptotic Covariance Matrix
  13. Statistical Efficiency
  14. Numerical Examples
  15. Baseline validation: single bi-static pair
  16. ...and 7 more sections

Key Result

Theorem 1

Let $s(t)$ be a circularly symmetric complex-valued bandlimited wide-sense stationary Gaussian stochastic process, with zero mean and spectral density $\mathcal{S}_s (f) = 1$ supported in $[-B/2,B/2]$, with $B > 0$. Let $0 \le t_1 < t_2 < \dots < t_N$, $N \ge 3$, be some fixed real sampling times, a and the random field Then, $\hbox{\bf v}$ and $\hbox{\bf w}(\tau,\omega)$ are linearly independent

Figures (5)

  • Figure 1: General Passive Radar Geometry
  • Figure 2: Empirical and theoretical MSE for estimating time-delay (left plots) and Doppler frequency (right plots) versus SC SNR (top) and RC SNR (bottom) respectively. The RC SNR was kept constant to $75 \text{ dB}$ in \ref{['fig:tauSCsweep']}-\ref{['fig:omegaSCsweep']} while the SC SNR was kept constant to $15 \text{ dB}$ in \ref{['fig:tauRCsweep']}-\ref{['fig:omegaRCsweep']}.
  • Figure 3: Empirical and theoretical RMSE of $\tau$ (a) and $\omega$ (b) for consecutive and non-consecutive batches respectively.
  • Figure 4: Monte-Carlo simulation versus theory for localization with $3$ RNs. Plots for $y$ and $v_y$ are qualitatively identical and thus omitted. SC SNR is an average over the different RNs.
  • Figure 5: Target tracking illustration.

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • proof
  • Proposition C.1: 6.11 in azais_wschebor_2009
  • proof : Proof of Theorem \ref{['thm:rf_unambiguous']}