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Stochastic Geometry Methods for Modelling Automotive Radar Interference
Akram Al-Hourani, Robin J. Evans, Sithamparanathan Kandeepan, Bill Moran, Hamid Eltom
TL;DR
Shared-spectrum use and uncoordinated traffic can expose automotive radars to interference that degrades detection. The paper uses stochastic geometry and two vehicle-location models to derive interference and ranging-success characterizations. It reports similar interference statistics for the Poisson and lattice models under practical parameters and presents a method for optimizing random-access duty cycle.
Problem
Shared-spectrum use and uncoordinated traffic can produce radar-to-radar interference that degrades detection in critical road situations.
Source: §I · p. 1Method
The paper applies stochastic geometry to Poisson and Bernoulli-lattice vehicle-location models to characterize interference and estimate radar ranging success.
Source: §I · p. 1, §VII · p. 10Results
Under practical system parameters, Poisson and lattice models produce very similar interference statistics.
Source: §VII · p. 10Takeaways & Limitations
The framework provides tight performance bounds under practical conditions and a method, with a closed-form optimum, for designing random-access duty cycle.
Source: §VII · p. 10The framework does not address time evolution of interference under non-stationary traffic models, which the authors leave for future work.
Source: §VII · p. 10
Abstract
from arXiv · showhide
As the use of automotive radar increases, performance limitations associated with radar-to-radar interference will become more significant. In this paper we employ tools from stochastic geometry to characterize the statistics of radar interference. Specifically, using two different models for vehicle spacial distributions, namely, a Poisson point process and a Bernoulli lattice process, we calculate for each case the interference statistics and obtain analytical expressions for the probability of successful range estimation. Our study shows that the regularity of the geometrical model appears to have limited effect on the interference statistics, and so it is possible to obtain tractable tight bounds for worst case performance. A technique is proposed for designing the duty cycle for random spectrum access which optimizes the total performance. This analytical framework is verified using Monte-Carlo simulations.
I. INTRODUCTION
Expected growth in automotive radar use raises the prospect of interference from shared spectrum and uncoordinated traffic, potentially degrading detection in critical road situations. The paper applies stochastic geometry to model interference and radar performance, and frames its analytic approach against a literature dominated by simulations and simpler analytic scenarios.
I. INTRODUCTION
- Automotive radar deployment is expected to expand, while shared spectrum and uncoordinated traffic create opportunities for radar-to-radar interference.Interference may degrade detection during a critical road situation.Source: §I · p. 1
A. Contribution
- The paper models radar interference with stochastic geometry and develops analytic tools for interference statistics, ranging success probability, and random-access duty-cycle optimization.Its stated contributions also include closed-form expressions and an intuition for convergence between Bernoulli-lattice and Poisson models.Source: §I · p. 1, A. Contribution · pp. 1–2
II. BACKGROUND AND RELATED WORK
- The authors characterize the literature as relying mainly on simulation and empirical approaches, with some analytic studies limited to simple two-vehicle scenarios.They also describe stochastic geometry as a way to model spatial randomness in wireless networks and note its prior use in vehicular ad-hoc networks.Source: §II · p. 2
- The authors report that prior work had not analytically modeled stochastic automotive-radar interference or provided closed-form ranging-success estimates and duty-cycle optimization.They identify this work as the first, to their knowledge, to address those gaps with stochastic-geometry tools.Source: §II · p. 3
III. SYSTEM MODEL
The model treats traffic as a stationary snapshot with homogeneous vehicle density and focuses on long-range radar. It brackets vehicle locations with irregular Poisson and regular lattice models, treating the practical distribution as lying between these extremes.
III. SYSTEM MODEL
- Vehicles are modeled in a stationary traffic snapshot, with geometric statistics treated as stable over a reasonable observation period.The paper notes that these statistics change over the long run, but slowly enough for the snapshot assumption.Source: §III · p. 3
A. Geometrical Model
- The geometric model focuses on long-range radar and treats opposing-lane vehicles beyond a minimum distance as potential interferers.That distance is determined using antenna beamwidth and the separation between the typical vehicle's lane and each opposing lane.Source: A. Geometrical Model · p. 3
- Vehicle locations are represented by two extremes: an uncorrelated Poisson point process and a regular lattice, with practical distributions expected between them.The paper states that under practical road parameters, radar performance is tightly bounded by these two cases; the road segment is modeled with constant average density.Source: A. Geometrical Model · p. 3
1) Poisson Point Process Model:
The paper models vehicle locations as either an independently distributed Poisson point process or a regular lattice, then marks which vehicles interfere. The lattice is translated randomly relative to the typical vehicle.
1) Poisson Point Process Model:
- PPP vehicle locations are independent, and random thinning with retention probability ξ models the chance an opposing vehicle uses the same resources.After excluding vehicles within δo, the interferers form a PPP with intensity ξλ.Source: 1) Poisson Point Process Model · pp. 3–4
2) Lattice Model:
- The lattice has fixed spacing δ, but a single uniform random variable U translates all approaching vehicles relative to the typical vehicle.A Bernoulli mark selects the interfering subset of lattice vehicles.Source: 2) Lattice Model · p. 4
B. Channel Model
The channel model accounts for propagation loss and stochastic effects on both the reflected ranging signal and interfering signals. The aggregated interference is modeled from approaching vehicles, especially those traveling in the opposite direction.
B. Channel Model
- Propagation loss is modeled with a general exponent α to account for atmospheric attenuation and absorption beyond inverse-square decay.For the ranging signal, millimeter-wave absorption motivates an exponent slightly above 2 in practical cases.Source: B. Channel Model · p. 4, 1) The Ranging Signal · p. 4
1) The Ranging Signal:
- The reflected ranging signal follows a modified radar equation using target range, antenna parameters, transmit power, and target radar cross-section.The model uses γ1 and γ2 parameters tied to operating frequency and the propagation exponent.Source: 1) The Ranging Signal · p. 4
2) The Interfering Signals:
- Interfering signals use the same α-law decay while incorporating stochastic channel fading and clutter in an i.i.d. channel variable.The power from an interferer depends on its Euclidean distance from the radar.Source: 2) The Interfering Signals · pp. 4–5
IV. RADAR INTERFERENCE
- With front-mounted directive antennas, the model treats approaching opposite-direction vehicles as the main interferers and characterizes their aggregate power through a point process.The interference statistics are described by the mean, standard deviation, and cumulative distribution function.Source: §IV · p. 5
A. Interference Mean
The mean interference is the same for the Poisson and translated Bernoulli lattice models under the stated lane-spacing simplification. It scales with effective interferer density and is sensitive to antenna beamwidth.
A. Interference Mean
- For the PPP mean interference, Campbell’s theorem sums over interferers; averaging uses channel and geometric expectations with effective intensity λI = ξλ, independent i.i.d. propagation channels, and unit average channel gain E[g] = 1.Source: A. Interference Mean · p. 5
- Mean interference is linearly proportional to ξλ, while for α ≈ 2 it is inversely proportional to δo, so a narrower beam reduces interference.Source: A. Interference Mean · p. 5
- The PPP and translated Bernoulli lattice give the same mean interference when lane spacing is ignored, and their plotted means match.The comparison uses Ln = 0 and the numerical parameters in Table I.Source: A. Interference Mean · pp. 5–6, Figure 4 · p. 6
B. Interference Distribution
The paper derives interference distributions from characteristic or Laplace transforms and obtains a tractable Lévy form under simplifying assumptions. The PPP and lattice CDFs are reported to be similar, with the PPP analysis consistent with Monte Carlo simulations.
B. Interference Distribution
- For the PPP, the characteristic function is formed using the homogeneous PPP generating functional over active interferers beyond δo, then inverted to obtain the CDF.With multiple lanes, the CDF can be obtained numerically using the Gil-Pelaez inversion theorem.Source: B. Interference Distribution · p. 6
- With δo = 0, α ≈ 2, ignored lane spacing, and no channel fading, the PPP interference follows a Lévy distribution, and its CDF agrees with Monte Carlo simulations.The worst-case setup assumes γ1Po remains constant as antenna gain changes, adjusting Po to compensate for wider beamwidth.Source: B. Interference Distribution · p. 6, Figure 5 · p. 7
- The Bernoulli lattice CDF is obtained by inverting its Laplace transform, and its distribution is reported to be very similar to the PPP CDF in Monte Carlo comparisons.The lattice calculation averages over the independent Bernoulli marks.Source: B. Interference Distribution · p. 7
C. The Convergence of Lattice to Poisson
The section explains how a Bernoulli-thinned lattice converges in distribution to a Poisson point process as lattice spacing decreases while intensity remains constant. The paper also reports similar interference CDFs for the two models.
C. The Convergence of Lattice to Poisson
- As lattice spacing δ tends to zero while intensity λ remains fixed, the Bernoulli-thinned lattice converges in distribution to a Poisson point process.The argument uses independent count measures that converge to Poisson counts over disjoint Borel subsets.Source: C. The Convergence of Lattice to Poisson · p. 7
- The geometric PPP and Bernoulli-lattice models have very similar interference CDFs, as validated by Monte Carlo simulations.The CDFs are computed using the inversion theorem for PPP and inverse Laplace transform for the lattice model.Source: Figure 6 · p. 7
- The two-dimensional illustration shows the Bernoulli lattice approaching a Poisson point process as δ vanishes while preserving intensity λ = ξδ^2.The lattice is thinned while its spacing decreases.Source: Figure 7 · p. 8
V. RADAR PERFORMANCE
The section defines successful ranging through an SINR threshold and derives ranging-success probabilities from the interference statistics. In the worst-case, interference-limited scenario, the paper gives a simpler expression and reports that success probability is independent of common transmit power and antenna characteristics.
V. RADAR PERFORMANCE
- Successful ranging and detection occur when SINR exceeds threshold T, and the success probability can be calculated from the interference distribution.The interference may be derived from the PPP or Bernoulli-lattice model.Source: §V · p. 8
- The worst-case ranging-success probability is plotted against ranging distance and compared with Monte Carlo simulations.The worst-case plot uses ξ = 1/100.Source: Figure 8 · p. 8, §V · p. 8
- In the worst-case interference-limited scenario, ranging success is independent of common transmit power P_o and antenna characteristic γ_1.The paper compares the closed-form probability with Monte Carlo simulations and uses it as a lower performance bound for the general case.Source: §V · p. 8
A. Radar Performance Optimization
The section optimizes random spectrum access by maximizing spatial success probability, balancing successful detections against spectrum use. It extends the design to stochastic target range by averaging over the distance to a chosen nearest vehicle.
A. Radar Performance Optimization
- Higher spectrum-collision probability ξ lowers radar success, while reducing ξ also lowers spectrum utilization by limiting concurrent access.The proposed duty-cycle design seeks an optimum that improves overall performance for users.Source: A. Radar Performance Optimization · p. 8
- A finite optimum interferer density exists for each target range, maximizing spatial success probability β.The paper obtains the optimum numerically using z_o ≈ 0.532, then derives the optimum transmission probability.Source: A. Radar Performance Optimization · p. 9, Figure 10 · p. 9
- For stochastic range, the duty cycle is optimized by choosing a target number n of nearest vehicles and averaging over contact distance R.The nearest-vehicle distance distribution depends on linear vehicle density λ.Source: A. Radar Performance Optimization · p. 9
- The optimum transmission duty cycle drops as measured detection range increases.The plotted optimum varies with the target nth neighbour and vehicle density.Source: A. Radar Performance Optimization · p. 9, Figure 10 · p. 9
- Detecting farther vehicles decreases spectral efficiency, whereas increased vehicle density improves spectrum utilization.The duty-cycle comparison is presented for different vehicle densities and target-neighbour values.Source: A. Radar Performance Optimization · p. 9
VI. SIMULATION PROCEDURE
The simulations deploy vehicles over a 10,000 m road segment and repeat the scenario at least 5,000 times. They estimate interference statistics and, for ranging-performance plots, count SINR-threshold exceedances across sampled distances.
VI. SIMULATION PROCEDURE
- The duty-cycle figure plots optimum transmission probability against target nth neighbour for different vehicle densities.Source: Figure 11 · p. 10
- Each simulation deploys vehicles over 10,000 m and sums interference at the origin; the scenario runs at least 5,000 times.Interference statistics are used for Figs. 4, 5, and 6.Source: §VI · p. 10
- For ranging-probability plots, each run calculates SINR and counts cases exceeding threshold T, then repeats across ranging-distance samples.The simulated success probability is obtained as the ratio of successful runs to total runs.Source: §VI · p. 10
VII. CONCLUSION
The paper develops stochastic-geometry models for automotive radar interference using Poisson and Bernoulli lattice vehicle distributions, finding similar interference statistics under practical parameters. It also derives target-detection success probabilities and an optimization method for random-spectrum-access duty cycle.
VII. CONCLUSION
- Poisson and Bernoulli lattice models produce very similar interference statistics under practical system parameters.The paper also gives an intuitive explanation for their convergence.Source: §VII · p. 10
- The framework estimates target-detection success probability from vehicle density, transmit power, antenna beamwidth, lane separation, and random-spectrum-access duty cycle.Source: §VII · p. 10
- A duty-cycle optimization methodology provides a closed-form expression for the optimum random-spectrum-access value.Future work will incorporate interference evolution under non-stationary traffic models.Source: §VII · p. 10