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Observability Analysis for Fusion of Doppler Measurements in Multistatic Radar Near the Tx-Rx Baseline
Rong Yang, Yaakov Bar-Shalom, Leung Hong Huat Michael
TL;DR
The paper addresses trajectory estimation when overlapping near-baseline ambiguity regions leave Doppler as the only reliable measurement and observability may be marginal or lost. It analyzes observability and combines multiple-initial-point ML initialization with EKF updates; simulations are consistent with the analysis and recover meaningful trajectory information under marginal observability.
Problem
Overlapping near-baseline ambiguity regions make Doppler-only target trajectory estimation highly challenging because trajectory observability may be marginal or completely lost.
Method
The paper analyzes observability under different conditions and uses a multiple-initial-point ML nonlinear estimator for initialization followed by EKF dynamic updates.
Results
Simulation results are consistent with the observability analysis, and position RMSE results indicate meaningful trajectory information can be recovered under marginal observability.
Takeaways & Limitations
The approach supports first-time-seen targets through initiation followed by EKF and targets passing through the ambiguity region using EKF only.
Abstract
from arXiv · showhide
This paper studies multistatic measurement fusion when a target lies within the Tx--Rx (Transmitter-Receiver) baseline ambiguity zone, with particular emphasis on configurations involving two closely spaced stationary Tx--Rx pairs. Such configurations provide overlapping detectable regions and extend the effective detection range compared with sparsely spaced multistatic systems. However, in this region, the accuracy of range and bearing measurements degrades rapidly, and Doppler measurements often remain the only reliable information source. As a result, target trajectory estimation becomes highly challenging, with observability being marginal or even completely lost. To address this problem, the observability of target trajectories is analyzed under various conditions, enabling system designers to assess system performance in advance. A Doppler-only measurement fusion approach is then developed, employing a multiple-initial-point Maximum Likelihood (ML) nonlinear estimator for initial state estimation, followed by dynamic state updates using an Extended Kalman Filter (EKF). Simulation results are presented and shown to be consistent with the observability analysis.
I. INTRODUCTION
The paper examines Doppler-only trajectory estimation when targets lie near overlapping Tx–Rx baseline ambiguity zones, where observability can become marginal or lost. It analyzes observability and develops a fusion approach for closely spaced stationary Tx–Rx pairs.
- Near the Tx–Rx baseline ambiguity zone, trajectory estimation becomes significantly more challenging because observability may be marginal or lost.
- The proposed configuration uses at least two closely spaced Tx–Rx pairs with overlapping ambiguity zones and Doppler-only fusion.Doppler remains the only reliable measurement in the considered region.
- Prior Doppler-only studies generally used widely separated sensors whose non-overlapping ambiguity regions produced more favorable observability.Those settings may also provide bistatic range or bearing information, making them better conditioned than the present scenario.
- Overlapping detectable regions from closely spaced pairs provide a longer effective detection range than sparsely spaced multistatic systems.The geometry enhances received signal strength.
- The paper analyzes observability under different conditions so system designers can assess performance beforehand.It combines this analysis with multiple-initial-point ML initialization and EKF-based dynamic estimation.
II. DOPPLER-ONLY MEASUREMENT FUSION
The measurement-fusion formulation models target motion with a nearly constant-velocity state and uses Doppler measurements from two closely spaced Tx–Rx pairs. Because the measurement models are nonlinear, estimation uses an EKF after defining synchronized or asynchronous measurement cases.
- The target state contains position and velocity components, and the motion model assumes nearly constant velocity.The state is indexed by time and includes x(k), y(k), ẋ(k), and ẏ(k).
- The motion model includes zero-mean white Gaussian process noise with covariance determined by the process-noise variance and state-transition gain.
- The two sensor configurations differ in synchronization: one uses a single synchronized receiver, while the other permits two unsynchronized receivers.These cases correspond to the two Tx–Rx setups shown in Fig. 1.
- The measurement vectors contain Doppler measurements from Tx–Rx pairs 1 and 2, with pair-specific measurements used in the two-receiver case.
- Because the measurement models are nonlinear, the estimator adopts an Extended Kalman Filter to estimate the target state.
III. TRACK INITIATION
Track initiation formulates Doppler-only initial-state estimation as an Iterated Least Squares maximum-likelihood problem. Because the nonlinear estimator is sensitive to initialization, multiple candidate states are sampled, evaluated by residual, and the best candidate is iteratively refined.
- Estimator formulation: The initial-state problem is formulated as an Iterated Least Squares maximum-likelihood estimator applied to an initial batch of measurements.ILS is chosen as the implementation of the Maximum Likelihood Estimator.
- Initialization sensitivity: Different starting points can converge to different optima, making a reliable single initial guess difficult in the Doppler-only scenario.The figure distinguishes a global optimum from multiple local optima.
- Measurement batch: Under the constant-velocity assumption, the batch measurement vector stacks Doppler measurements from both Tx–Rx pairs over the initialization interval.The interval is [0, Tinit], with potentially different start and end indices for the two pairs.
- Measurement model: The nonlinear measurement mapping relates the initial state to the batch measurements, with measurement noise represented by the stacked noise vector.The mapping g(·) is defined under the constant-velocity motion model.
- Multiple-initial-point search: Multiple initial guesses are sampled across detectable-zone positions, headings, and speeds, then the candidate with minimum measurement residual is selected for iterative updating.Candidate speeds are sampled in [smin, smax], where smin is the maximum measured radial Doppler component and smax is the predefined maximum target speed.
IV. OBSERVABILITY ANALYSIS
The paper numerically evaluates observability for selected initial states and sensor configurations, showing that observability is lost or weakened when target motion follows bistatic-ellipse directions. It also finds that observability varies with position and increases with target speed.
- Analysis framework: Observability is quantified numerically using the Jacobian-based observation matrix and a scalar observability measure.For diagonal noise covariance, J(x0)'J(x0) is proportional to the Fisher Information Matrix.
- Evaluation setup: The evaluation uses four groups of initial states across two sensor configurations, including baseline states and states 2 km from the baseline.The tested headings span 0°–359° in 10° increments, with speed fixed at 100 m/s.
- Baseline trajectories: Groups 1 and 3 are unobservable at headings 90° and 270° when the target moves along the Tx1–Rx1 baseline.The associated bistatic range rates remain zero, making the observability matrix singular with det(J'J) = 0.
- Observability condition: A target state is unobservable when its trajectory lies along a bistatic ellipse associated with any Tx–Rx pair; other trajectories remain observable.The Tx–Rx baseline is the zero-width special case of a bistatic ellipse.
- Position and speed effects: Observability is lower near the middle of the x-range for heading 180°, while it increases as target speed increases.The position effect is attributed to similar bistatic range-rate changes between pairs near the middle of the x-range.
V. SIMULATION RESULTS
The simulations evaluate Doppler-only trajectory estimation across two sensor configurations and three starting positions, showing accuracy patterns consistent with observability. Poor observability can make subsequent EKF estimation less accurate than the batch initialization.
- Track initiation: The batch initialization uses 20 measurements, with 10 from each sensor pair, and selects the candidate state with minimum measurement error.Multiple initial states are sampled over predefined ranges before candidate selection.
- Simulation setup: 100 Monte Carlo runs evaluate position RMSE over 30 seconds for two sensor configurations and trajectories starting at 5 km, 25 km, and 45 km.The simulations use a 0.5 s sampling interval and 1 m/s Doppler-error standard deviation for both Tx–Rx pairs.
- Position RMSE: In case (a), the 45 km trajectory has the best position accuracy, the 25 km trajectory the worst, and the 5 km trajectory lies between them.The corresponding position-RMSE curves are shown in Fig. 11.
- Position RMSE: In case (b), the 5 km and 45 km trajectory performances reverse, while the 25 km trajectory remains the worst.The corresponding position-RMSE curves are shown in Fig. 12.
- Observability effects: Under poor observability, batch grid-search initialization can outperform subsequent dynamic estimation because Jacobian ill-conditioning causes numerical problems for ILS and EKF updates.The paper identifies 25 km scenarios, 5 km in case (a), and 45 km in case (b) as examples.
VI. CONCLUSIONS
The paper characterizes when Doppler-only trajectory estimation is observable for closely spaced stationary Tx–Rx pairs and develops a staged estimator for this setting. The simulations agree with the analysis, while future work targets broader trajectories and less Jacobian-sensitive dynamics.
- Observability analysis: A target trajectory is unobservable when it lies along a bistatic ellipse associated with any Tx–Rx pair; other trajectories remain observable.The Tx–Rx baseline is the zero-width special case of a bistatic ellipse.
- Observability analysis: Observability is marginal near the baseline midpoint and improves toward both ends, especially when receivers or transmitters at the ends are spatially separated.Higher target speed also leads to improved observability.
- Estimation approach: The proposed Doppler-only fusion approach uses a multiple-initial-point ML nonlinear estimator for initialization followed by EKF dynamic updates.Simulation results are reported as consistent with the observability analysis.
- Simulation implication: Position RMSE results indicate that meaningful trajectory information can still be recovered under marginal observability conditions.The conclusion is based on the reported simulation study.
- Scope and future work: The approach covers first-time-seen targets through initialization plus EKF and targets passing through the ambiguity region using EKF only.Future work will extend the method to three-dimensional trajectories and alternative dynamic estimators less sensitive to Jacobian ill-conditioning.
APPENDIX A JACOBIAN OF g(x0)
Appendix A derives the Jacobian of the nonlinear measurement function g(x0) with respect to the initial state and defines the relative measurement-time variable.
- Jacobian derivation: The appendix derives the Jacobian of g(x0), omitting the Tx–Rx pair and iteration indices for simplicity.The Jacobian supports the nonlinear estimation procedure.
- State variables: The derivatives are taken with respect to the initial state x0 = [x0 y0 ˙x ˙y]′.These components represent initial position and velocity coordinates.
- Time reference: The variable t denotes the measurement time interval relative to the time associated with x0.This defines the temporal reference used in the derivative expressions.
- Jacobian entries: The displayed derivative expressions provide the Jacobian entries used to characterize sensitivity of the measurement function to the initial state.The appendix presents the partial derivatives of the Doppler-related measurement model.