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Generalized Rao Test for Decentralized Detection of an Uncooperative Target

D. Ciuonzo, P. Salvo Rossi, P. Willett

arXiv:1703.03946v1cs.ITmath.IT

TL;DR

Distributed detection is difficult when sensor reports are bandwidth-limited and target parameters are unknown. The paper develops a generalized Rao fusion test for quantized reports and proposes threshold optimization; simulations find performance close to the GLRT with lower computational complexity.

  • Problem

    Unknown target parameters make local likelihood-ratio testing difficult, while one-bit reporting is common under WSN bandwidth and energy constraints.

    Source: §I · p. 1

  • Method

    The paper maximizes a family of Rao statistics over the unknown target position and proposes sensor-threshold optimization for quantized reports sent over imperfect BSC links.

    Source: §I · p. 1, A. Test derivation · p. 2, B. Quantizer Design · p. 3

  • Results

    G-Rao has only marginal detection loss relative to GLRT in the reported location comparisons, performs similarly across the SNR range, and slightly outperforms GLRT at low SNR in Laplace noise.

    Source: §IV · p. 4, §V · p. 5

  • Takeaways & Limitations

    G-Rao is presented as a lower-complexity alternative to GLRT, and zero sensor thresholds are appealing for many noise distributions.

    Source: §VI · p. 5

    The stated model is limited to quantized measurements, zero-mean symmetric unimodal noise, and non-ideal non-identical BSC reporting channels.

    Source: §VI · p. 5

Abstract

from arXiv · show

We tackle distributed detection of a non-cooperative target with a Wireless Sensor Network (WSN). When the target is present, sensors observe an (unknown) deterministic signal with attenuation depending on the distance between the sensor and the (unknown) target positions, embedded in symmetricand unimodal noise. The Fusion Center (FC) receives quantized sensor observations through error-prone Binary Symmetric Channels (BSCs) and is in charge of performing a more-accurate global decision. The resulting problem is a two-sided parameter testing with nuisance parameters (i.e. the target position) present only under the alternative hypothesis. After introducing the Generalized Likelihood Ratio Test (GLRT) for the problem, we develop a novel fusion rule corresponding to a Generalized Rao (G-Rao) test, based on Davies' framework, to reduce the computational complexity. Also, a rationale for threshold-optimization is proposed and confirmed by simulations. Finally, the aforementioned rules are compared in terms of performance and computational complexity.

I. INTRODUCTION

Bandwidth and energy constraints favor one-bit sensor reports, but unknown target parameters often prevent local likelihood-ratio testing. The paper develops a lower-complexity fusion rule for this setting and compares it with the GLRT.

I. INTRODUCTION

  • Unknown target parameters often prevent sensors from computing local likelihood-ratio tests, while optimizing local quantization thresholds can be exponentially complex,.Sensors therefore commonly send bits from simple quantization or suboptimal local rules,,,.Source: §I · p. 1
  • With conditionally independent sensor bits, optimal fusion is a weighted sum whose weights depend on unknown target parameters.Counting-rule and channel-aware alternatives have been proposed –.Source: §I · p. 1
  • For a non-cooperative target, both emitted signal and location are unavailable to the Fusion Center, and the problem is a composite test.A prior GLRT for unknown position and emitted power requires a grid search over both domains.Source: §I · p. 1
  • The paper develops a generalized Rao fusion rule for an unknown deterministic signal and location, then compares detection performance and computational complexity with the GLRT.The setting uses spatially dependent emission, unimodal zero-mean additive noise, one-bit sensor reports, and error-prone Binary Symmetric Channels.Source: §I · pp. 1–2

II. SYSTEM MODEL

The model tests for a target whose unknown deterministic amplitude is attenuated according to sensor–target distance and observed in independent, zero-mean symmetric unimodal noise. Sensors quantize measurements to one bit and report them over noisy Binary Symmetric Channels.

II. SYSTEM MODEL

  • Under H0 sensor k observes noise alone, while under H1 its measurement is θg(xT, xk) plus noise, with unknown signal amplitude θ and target position xT.The target position and sensor position determine the amplitude attenuation function g.Source: §II · p. 2
  • The noise variables are mutually independent, zero-mean, and have symmetric unimodal densities; the target emission is modeled as isotropic with distance-dependent path loss.The framework applies to any suitably defined amplitude attenuation function for the target or event.Source: §II · p. 2
  • Each sensor quantizes its measurement to one bit using threshold τk, and the Fusion Center receives the bit after possible flipping over a Binary Symmetric Channel with known error probability.The model restricts attention to deterministic quantizers; stochastic quantizers are outside the letter's scope.Source: §II · p. 2
  • The noise model includes Gaussian, Laplace, Cauchy, and generalized Gaussian zero-mean distributions.These are given as examples of the assumed noise densities.Source: §II · p. 2
  • Because target position is a nuisance parameter present only under H1, the study seeks a simple decision statistic and optimized sensor thresholds τk.The decision favors H1 when Λ( b̂ ) exceeds threshold γ, and H0 otherwise.Source: §II · p. 2

A. Test derivation

The GLRT requires estimating the unknown signal and target position, with no closed-form maximum-likelihood solution available even for Gaussian noise. The proposed G-Rao test instead maximizes a family of Rao statistics over target position.

A. Test derivation

  • The GLRT maximizes the likelihood over unknown signal amplitude and target position, but their maximum-likelihood estimates lack a closed form even for Gaussian noise.Implementation typically uses a grid over both parameters.Source: A. Test derivation · p. 2
  • Davies’ approach handles target position present only under H1 by maximizing over position a family of Rao statistics computed as if each position were known.This generalized Rao, or G-Rao, test avoids estimating the nuisance position under H0.Source: A. Test derivation · p. 2
  • The G-Rao implementation requires a grid only over target position, since it does not require the GLRT’s estimate of signal amplitude.This is the stated rationale for its reduced implementation complexity.Source: A. Test derivation · p. 2
  • The Rao statistic is expanded using independence of sensor measurements and reporting channels, with its likelihood terms expressed through the received bits and their probabilities.The resulting statistic depends on sensor thresholds τk through its bit-related terms and associated quantities.Source: A. Test derivation · p. 3
  • Both ΛR and ΛG depend on sensor thresholds, which can be optimized to improve performance.The dependence arises because the bit-related quantities and ψk,0 depend on τk.Source: A. Test derivation · p. 3

B. Quantizer Design

Thresholds are chosen to maximize the non-centrality parameter, a criterion that separates into per-sensor optimizations independent of target position and supports a simpler threshold-optimized G-Rao test.

B. Quantizer Design

  • Larger λQ(xT) predicts better position-clairvoyant GLRT and Rao performance at xT, motivating its use as the threshold-design objective.Because performance expressions for Davies-based tests are unavailable, the modified rationale is confirmed by simulations.Source: B. Quantizer Design · p. 3
  • The threshold optimization decouples into K independent per-sensor problems that do not depend on the unknown target position.A direct joint optimization would otherwise produce thresholds dependent on xT and therefore impractical.Source: B. Quantizer Design · p. 3
  • For Gaussian, Laplace, Cauchy, and generalized-normal noise with 0 ≤ ϵ ≤ 2, τk = 0 maximizes ψk,0; it is also a good suboptimal choice, especially with noisy reporting channels.Source: B. Quantizer Design · p. 3
  • The threshold-optimized G-Rao expression avoids the GLRT’s joint optimization over target position and signal amplitude.Source: B. Quantizer Design · p. 3

C. Computational Complexity

The G-Rao implementation searches target-position bins rather than jointly searching position and amplitude, yielding a reported reduction in computational complexity relative to the GLRT.

C. Computational Complexity

IV. GAUSSIAN NOISE ANALYSIS

The Gaussian-noise analysis compares G-Rao and GLRT detection performance and validates the zero-threshold choice. Across the tested SNRs and reporting-channel conditions, their detection performance is very similar, while τ=0 is nearly optimal in the low-SNR regime.

IV. GAUSSIAN NOISE ANALYSIS

  • The evaluation models target detection in the unit-square surveillance area using 49 sensors on a regular square grid, Gaussian noise, and power-law attenuation with η=0.2 and α=4.Initially, reporting channels are ideal BSCs, with Pe,k=0 for every sensor.Source: §IV · p. 4
  • The GLRT search discretizes target position and signal amplitude, whereas G-Rao discretizes only target position.The experiment samples the surveillance area on a regular 2-D grid.Source: §IV · p. 4
  • At low SNR, a common threshold τ = 0 is nearly optimal for sensors observing a target located uniformly at random within A, with PF0 = 0.01.Source: §IV · p. 4
  • Across target locations at SNR = 5 dB, G-Rao incurs only marginal detection loss versus GLRT, with both showing lower detection at surveillance-area boundaries.The authors attribute the boundary pattern to regular displacement of the WSN within the area.Source: Figure 2 · p. 4, §IV · p. 4
  • G-Rao and GLRT perform very similarly across the tested SNR range, false-alarm rates, and reporting-channel qualities.Figure 3 compares PD0 versus SNR for PF0 ∈ {0.05, 0.01} and Pe ∈ {0, 0.1}.Source: Figure 3 · p. 4, §IV · p. 4

V. LAPLACE NOISE ANALYSIS

Under Laplace noise, the paper finds that zero sensor thresholds remain nearly optimal at low SNR, while G-Rao performs similarly to GLRT across target locations and SNRs.

V. LAPLACE NOISE ANALYSIS

  • The Laplace-noise simulations use unit-variance noise and initially assume ideal reporting channels.The noise is modeled as Laplace with zero mean and sensor-specific scale parameters.Source: §V · p. 4
  • At low SNR, τ = 0 is nearly optimal because the numerically optimal threshold depends on the unknown signal polarity.The threshold comparison uses a common sensor threshold and PF0 = 0.01.Source: §V · p. 5, Figure 4 · p. 5
  • At SNR = 5 dB, G-Rao incurs only marginal detection-probability loss relative to GLRT across target locations.The comparison uses PF0 = 0.01 and ideal reporting channels.Source: §V · p. 5, Figure 5 · p. 5
  • G-Rao and GLRT perform very similarly across the SNR range and reporting-channel qualities, with G-Rao slightly outperforming GLRT at low SNR.Figure 6 compares detection probability at PF0 ∈ {0.05, 0.01} and Pe ∈ {0, 0.1}.Source: §V · p. 5, Figure 6 · p. 5

VI. CONCLUSIONS

The conclusion presents G-Rao as a lower-complexity alternative to GLRT for decentralized detection with unknown target signal and location, and reports practically the same simulated performance.

VI. CONCLUSIONS

  • G-Rao avoids GLRT’s grid search over signal amplitude and target location while achieving practically the same performance in the simulated cases.The stated model includes quantized measurements, zero-mean symmetric unimodal noise, and non-ideal, non-identical BSCs.Source: §VI · p. 5
  • The G-Rao statistic maximizes a family of Rao decision statistics over target location, a nuisance parameter present only under H1.Source: §VI · p. 5
  • The paper proposes zero sensor thresholds as an appealing choice for many noise PDFs and uses them to optimize G-Rao and GLRT performance.Source: §VI · p. 5
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