The Radar Blind Spot Made Visible: Tracking Targets That Should Be Invisible

Every radar engineer has met the same quiet nightmare: a target cruising straight along the line between a transmitter and its receiver. It is moving, sometimes fast, but the radar measures nothing. Its Doppler reading — the shift in frequency that reveals motion — is zero. The target has become, for that sensor, invisible. Strip away the range and bearing data, and you are left trying to track an object through a measurement that has gone silent.

This paper from DSO National Laboratories (Singapore) and the University of Connecticut confronts precisely that geometry, and it does so in the hardest possible regime: when Doppler is the only reliable measurement left. The authors, Rong Yang, Yaakov Bar-Shalom, and Leung Hong Huat Michael, study multistatic radar — systems where multiple transmitters and receivers work in concert, reusing existing broadcast signals rather than firing their own — and ask a deceptively simple question: can you still track a target hiding near the baseline, and if so, when exactly do you lose it?

The answer, delivered through a mix of mathematical observability analysis and simulation, is both discouraging and useful. There are target trajectories you will never recover, no matter how clever your filter. But there is a precise, map-like description of exactly which trajectories those are. And for everything else, a carefully constructed estimator can pull a usable track out of measurements that most tracking systems would simply discard.

The Science

To understand the problem, you need to picture how Doppler works in a bistatic radar — one transmitter (Tx) and one receiver (Rx) separated by some distance. When a target moves toward or away from this pair, the frequency of the reflected signal shifts, and that shift is the Doppler measurement. Crucially, the shift is the sum of two contributions: motion relative to the transmitter and motion relative to the receiver (Yang et al., 2026). The formula, for each pair $i$, is

$${\dot{r}}_i(k)={\dot{r}}_i^{\mathrm{T}}(k)+{\dot{r}}_i^{\mathrm{R}}(k)+w_i(k)$$

where the two $\dot{r}$ terms are the projections of target velocity along the lines to the transmitter and receiver, and $w_i$ is white measurement noise.

The trouble arrives near the baseline — the straight line between Tx and Rx. There, the geometry that normally lets a system estimate position collapses. Range and bearing become wildly inaccurate, degrading "rapidly" as the authors put it. But Doppler does something stranger: it sees only the radial component of velocity. A target moving along the baseline moves perpendicular... no, it moves along a line where its distance to both the transmitter and receiver changes in a way that nets out. The bistatic range — the sum of target-to-transmitter and target-to-receiver distances — holds constant. And when that sum is constant, its time derivative, the bistatic range rate $\dot{r}$, is zero. Motion that produces zero Doppler is motion the radar cannot see.

Previous work on Doppler-only tracking largely sidestepped this by assuming sparsely separated transmitters, so that no two pairs ever share an overlapping ambiguity zone. In those configurations the target stays well clear of every near-baseline region, and observability is inherently comfortable. This paper deliberately throws that comfort away. It considers closely spaced Tx–Rx pairs whose ambiguity zones overlap, because such arrangements have a real practical payoff: they extend the effective detection range and improve received signal strength compared with sparse systems. The trade is that the target can sit inside the zone where range and bearing fail simultaneously for multiple pairs, leaving Doppler as the sole witness.

The state to be tracked is a four-vector — position and velocity in the $x$- and $y$-coordinates — evolving under a nearly constant velocity model. The researchers built two sensor configurations, shown in

: one with a single receiver serving two transmitters, the other with a single transmitter feeding two receivers. These mirror real asymmetric deployments where one side of the link already exists as infrastructure.

The tracking pipeline has two stages. First comes initiation: a batch of Doppler measurements from both pairs is fed to a Maximum Likelihood estimator. But the nonlinear measurement model has a pathology — it is riddled with local optima, and which optimum the estimator reaches depends entirely on where it starts (

shows the global optimum as a red dot surrounded by misleading blue ones). Because a good initial guess is nearly impossible to obtain analytically in this geometry, the authors instead sampled thousands of candidate starting states — positions, headings, and speeds drawn uniformly across the detectable zone — ran the residual for each, and kept the best. The second stage hands the winning initial estimate to an Extended Kalman Filter (EKF) that propagates the track forward through time.

What They Found

The core contribution is not the estimator but the map of when estimation is even possible. Observability — whether a target state can be uniquely determined from the measurements — was quantified through a scalar "observability measure," the logarithm of the determinant of the Fisher Information Matrix:

$$O({\mathbf{x}}_0)=\log \left[\det \left(\mathbf{J}({\mathbf{x}}_0{)}^{\prime }\mathbf{J}({\mathbf{x}}_0)\right)\right]$$

A higher value means the measurements are more sensitive to changes in the state, and accurate estimation is easier. When the determinant collapses to zero, observability is lost entirely — the map shows those regions at a floor of $O=-40$.

The researchers swept initial target states across four groups: on the baseline itself, and two kilometers off it, for both sensor configurations, varying position along the range and heading around the full 360 degrees. The results, in

through

, deliver a clean, almost elegant conclusion. A target is unobservable, completely, only when its trajectory runs along a bistatic ellipse associated with some Tx–Rx pair — that is, a curve along which the sum of distances to the transmitter and receiver stays constant, making the Doppler vanish. The baseline itself is just the degenerate case: a bistatic ellipse of zero width.

For targets exactly on the baseline, unobservability strikes at the two headings, 90° and 270°, where motion runs along it. Off the baseline, full loss is never reached, but two bands of low observability appear — again centered on those headings, with branch-like structures at the far ends of the range because only one pair's ellipse can align with the motion at a time (

). Everything else remains observable.

Two further results sharpen the picture. Observability dips in the middle of the range and improves toward the ends, because near the center both pairs see similar rate changes. And observability rises with target speed: faster targets generate larger Doppler values, and

Observability grows with target speed (off-baseline)

Label	Value
Group 2 (2 km off baseline, case a)	-26 observability O
Group 4 (2 km off baseline, case b)	-24 observability O

Minimum observability measure O reported for target speed 100 m/s vs. 200 m/s at off-baseline states. Observability rises with target speed for both sensor configurations.

shows the minimum observability climbing steadily with speed across both of the studied groups.

The simulations confirmed the analysis with striking fidelity. In case (a), the trajectory starting at 45 km achieved the best accuracy, the 25 km trajectory the worst, and the 5 km trajectory in between — exactly the ranking the observability maps predict. In case (b), the 5 km and 45 km performances reversed, while 25 km remained worst. The 25 km trajectory sits at the low-observability middle of the range in both configurations, and its estimates were the poorest in both.

One finding surprised even the researchers: under poor observability, the brute-force grid-search initial estimate outperformed the refined dynamic tracking that followed it. This is a tale about numerical conditioning. When observability is marginal, the Jacobian matrix — the linear approximation of the measurement model — becomes ill-conditioned, and inverting it, as the iterated least-squares and filter updates require, amplifies tiny numerical errors into catastrophes. The grid search, which evaluates residuals directly without touching that ill-conditioned inverse, dodges the problem entirely.

Under poor observability, initial estimate beats the tracker

Label	Value
Initial batch estimate	25
Dynamic (EKF) estimate	100

Position RMSE (m) for the 25 km trajectory in the low-observability middle region. The grid-search batch estimate holds accuracy while the dynamic filter degrades sharply near 5 s due to the ill-conditioned Jacobian.

shows the position error for the 25 km trajectory climbing sharply around 5 seconds, as the filter stumbles on the same ill-conditioned model; the initial batch estimate, by contrast, holds its accuracy.

Why This Changes Things

To appreciate the significance, consider what these configurations represent. Multistatic radar is attractive precisely because it reuses existing transmitters — television towers, communications links, navigation signals — turning passive receivers into a surveillance network without broadcasting a single Watt. But a sparse network leaves coverage gaps: a target straddles the ambiguous zone around each pair, and the effective detection range shrinks. Bringing the pairs close together extends that range and strengthens the received signal. The catch has always been the blindness near the baseline, where the measurement precision collapses. This paper does something unusual: instead of hoping the problem away, it maps in advance, for any proposed sensor geometry, exactly which regions will be tractable and which will be blind.

That predictive power is the practical gift. The observability measure lets a system designer place transmitters and receivers knowing, before any hardware is built or any target appears, where tracks will be reliable and where they will degrade to marginal or vanish entirely. It converts a continuous, messy engineering trade — "how close is too close?" — into a crisp, precomputable contour map. If a critical corridor of airspace must be monitored, you can now choose the sensor layout so that no plausible target trajectory lies along a bistatic ellipse within it, or accept the limitation and plan staffing and handoff accordingly.

The authors are candid about the costs of the regime they study. The numerical ill-conditioning that handicaps the ILS and EKF under poor observability is not a bug in their code; it is an intrinsic consequence of pursuing a state through an almost-flat likelihood surface. Their honest reporting of this — including the counterintuitive result that a dumb grid search beats a theoretically elegant filter at the margins — is a reminder that in nonlinear estimation, the conditioning of the problem often matters more than the sophistication of the solver. That lesson generalizes well beyond radar, to any system that fuses weak measurements to estimate a hidden state.

Accuracy tracks observability across starting positions

Label	Value
5 km start	30
25 km start	100
45 km start	15

Qualitative position RMSE ranking across the three starting points in case (a). The 25 km trajectory — sitting in the low-observability middle of the range — is by far the worst, matching the observability analysis.

places the accuracy results in context, showing how the position root-mean-square error varies across trajectories and configurations, with the middle-range target performing worst in both.

What's Next

The paper is honest about its scope. The targets move with nearly constant velocity; the real world maneuvers, and the nearly constant velocity model would need to absorb turning and acceleration. The analysis treats stationary transmitters and receivers; in practice, and in the aided single-sensor work the authors cite, one or both sides of the link may move itself — a moving sensor provides multiple perspectives that improve observability, and extending this framework to mobile geometries is a natural next step. Time synchronization is assumed in one configuration; relaxing it introduces unequal sampling that complicates the fusion but mirrors more realistic distributed deployments.

The deepest open question is whether the ill-conditioning that plagued the ILS under marginal observability can be tamed. The authors note their estimator is theoretically sound but practically limited by matrix inversion. Regularized estimation, carefully chosen pseudo-inverses, or full nonlinear optimization that sidesteps the Jacobian altogether could reclaim accuracy in the low-observability bands the analysis now lets us predict. The observability map, having told us where the hard regions are, also tells us where such techniques would be worth the effort.

None of this rescues the truly unobservable case — a target gliding along a bistatic ellipse remains, and will remain, invisible to Doppler. That is not a failure of the method but a fact of physics, and knowing it in advance is itself a form of progress. The value of this work is precisely that it draws the line between the impossible and the merely difficult, and then shows the merely difficult is far more tractable than conventional wisdom assumed. In a domain where the difference between a detected threat and a silent one is measured in seconds, a map of where silence is guaranteed is worth drawing.