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Distribution Bounds on the Conditional ROC in a Poisson Field of Interferers and Clutters

Gourab Ghatak

TL;DR

This work addresses radar ROC variability when interferers and clutter form a Poisson field by elevating ROC analysis to meta-distributions of the conditional false-alarm and conditional detection probabilities, $P_{ m FA_ ilde{Φ}}$ and $P_{ m D_ ilde{Φ}}$. It develops a higher-order Campbell–Mecke framework to obtain the first two moments of $P_{ m FA_ ilde{Φ}}$, derives Cantelli-type tail bounds, and introduces a beta-distribution approximation for the CFA meta-distribution, with extensions to the CD probability. Key contributions include closed-form expressions for the CFA mean and variance, tight tail bounds, and a beta-approximation that enables percentile evaluations, along with CD bounds under both stochastic and deterministic signal-power models. The results advocate designing radar thresholds and processing algorithms based on percentile guarantees rather than solely on mean false-alarm or detection probabilities, enabling robust performance in high-density interference and clutter scenarios.

Abstract

We present a novel analytical framework to characterize the distribution of the conditional receiver operating characteristic (ROC) in radar systems operating within a realization of a Poisson field of interferers and clutters. While conventional stochastic geometry based studies focus on the distribution of signal to interference and noise ratio (SINR), they fail to capture the statistical variations in detection and false-alarm performance across different network realizations. By leveraging higher-order versions of the Campbell-Mecke theorem and tools from stochastic geometry, we derive closed-form expressions for the mean and variance of the conditional false-alarm probability, and provide tight upper bounds using Cantelli's inequality. Additionally, we present a beta distribution approximation to capture the meta-distribution of the noise and interference power, enabling fine-grained performance evaluation. The results are extended to analyze the conditional detection probability, albeit with simpler bounds. Our approach reveals a new approach to radar design and robust ROC selection, including percentile-level guarantees, which are essential for emerging high-reliability applications. The insights derived here advocate for designing radar detection thresholds and signal processing algorithms based not merely on mean false-alarm or detection probabilities, but on tail behavior and percentile guarantees.

Distribution Bounds on the Conditional ROC in a Poisson Field of Interferers and Clutters

TL;DR

This work addresses radar ROC variability when interferers and clutter form a Poisson field by elevating ROC analysis to meta-distributions of the conditional false-alarm and conditional detection probabilities, and . It develops a higher-order Campbell–Mecke framework to obtain the first two moments of , derives Cantelli-type tail bounds, and introduces a beta-distribution approximation for the CFA meta-distribution, with extensions to the CD probability. Key contributions include closed-form expressions for the CFA mean and variance, tight tail bounds, and a beta-approximation that enables percentile evaluations, along with CD bounds under both stochastic and deterministic signal-power models. The results advocate designing radar thresholds and processing algorithms based on percentile guarantees rather than solely on mean false-alarm or detection probabilities, enabling robust performance in high-density interference and clutter scenarios.

Abstract

We present a novel analytical framework to characterize the distribution of the conditional receiver operating characteristic (ROC) in radar systems operating within a realization of a Poisson field of interferers and clutters. While conventional stochastic geometry based studies focus on the distribution of signal to interference and noise ratio (SINR), they fail to capture the statistical variations in detection and false-alarm performance across different network realizations. By leveraging higher-order versions of the Campbell-Mecke theorem and tools from stochastic geometry, we derive closed-form expressions for the mean and variance of the conditional false-alarm probability, and provide tight upper bounds using Cantelli's inequality. Additionally, we present a beta distribution approximation to capture the meta-distribution of the noise and interference power, enabling fine-grained performance evaluation. The results are extended to analyze the conditional detection probability, albeit with simpler bounds. Our approach reveals a new approach to radar design and robust ROC selection, including percentile-level guarantees, which are essential for emerging high-reliability applications. The insights derived here advocate for designing radar detection thresholds and signal processing algorithms based not merely on mean false-alarm or detection probabilities, but on tail behavior and percentile guarantees.

Paper Structure

This paper contains 12 sections, 6 theorems, 22 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Context and Related Work
  3. Limitations of Current Studies
  4. Contributions, Organization, and Notation
  5. System Model and Performance Metrics
  6. Unified Model for Clutter Returns and Interference
  7. Metrics of Interest
  8. Distribution Bound for the CFA Probability
  9. Approximation using Beta Functions
  10. Distribution Bound for the CD Probability
  11. Numerical Results and System Design Insights
  12. Conclusions and Discussion

Key Result

lemma 1

chiu2013stochastic Let $\Phi$ be a simple and locally finite point process on $\mathbb{R}^d$, and let $f: \mathbb{R}^d \times \mathbb{R}^d \times \mathcal{N} \to [0, \infty)$ be a measurable function, where $\mathcal{N}$ denotes the space of locally finite counting measures on $\mathbb{R}^d$. Then t where $\rho^{(2)}(x, y)$ denotes the second-order product density of $\Phi$. The function $f$ may d

Figures (3)

  • Figure 1: Illustration of the insights from fine-grained (right) analysis as compared to the standard ROC (left).
  • Figure 2: Empirical distributions of the CFA and CD probability along with the corresponding bounds and the beta-approximation for the distribution of the CFA probability.
  • Figure 3: Illustration of a sample ROC empirically obtained while the bound based analysis reveals the operating confidence region.

Theorems & Definitions (9)

  • Remark 1
  • Definition 1
  • lemma 1
  • Theorem 1
  • Theorem 2
  • Remark 2
  • Corollary 1
  • Corollary 2
  • lemma 2