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REVIEW 5 major objections 5 minor 27 references

Multi Target Observability

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read One line-of-sight rule decides multi-target observability

desk verdict The main observability theorem is false—per-target zero-output conditions don't couple through the bearing matrix—so the paper's central result doesn't survive scrutiny. read the letter →

arxiv 2507.14765 v1 pith:HJGYGCYA submitted 2025-07-19 eess.SY cs.SY

Debadrita Banerjee , Debjani Mitra , Rajesh Dey , Mudassir Khan , Lalan Kumar This is my paper · ORCID
classification eess.SYcs.SY
keywords multi-targetobservabilitybearing-onlytrackingtrajectoryambiguityDoppler-onlymeasurementshigher-orderdynamicspassivetargetmotionanalysiscondition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that a single observer can uniquely estimate the states of several moving targets only when no two targets ever lie on the same line of sight with the observer, meaning their bearing angles are distinct modulo $\pi$ at every instant. The authors argue this condition is necessary and sufficient regardless of the order of target or observer dynamics, and they derive it through a pseudolinearized observability analysis. They then propose a complementary notion called trajectory ambiguity, in which two targets are indistinguishable if their measurement histories coincide, and they derive necessary and sufficient conditions for such ambiguity under Doppler-only, bearing-only, and combined Doppler-bearing measurements. If correct, the result would reduce a nonlinear, high-dimensional observability problem to a checkable geometric distinctness condition.

What carries the argument

The load-bearing object is the pseudolinearized measurement matrix $C(t)$ together with the block-diagonal state transition matrix $\tilde{\Phi}(t, t_i)$ built from Taylor-polynomial blocks for each target's relative position. The observability argument passes through the matrix $P(t)$ with rows $[\cos\theta_i(t), -\sin\theta_i(t)]$; the authors assert that distinctness of the bearings modulo $\pi$, i.e., rank 2 of $P(t)$ for all $t$, is necessary and sufficient for the zero-output condition to force a zero initial super-state. For Doppler ambiguity, the central object is the time-varying transformation matrix $W(t) = U(t)\left[l' + \frac{b' + c(1 - l')(t - t_i)}{s_j(t)}\right]$, which maps one target's relative position vector into another's; the combined-measurement result states that ambiguity occurs iff $s_j(t)$ is an eigenvector of $W(t)$ with eigenvalue $\alpha'(t)$.

What would settle it

Compute the observability Gramian for two targets that maintain distinct, constant bearings toward a stationary observer, for example two targets moving radially outward along different lines of sight. If the Gramian is singular, Proposition 1 is false, because the range and along-sight velocity of each target are unobservable even though the bearings are distinct modulo $\pi$; equivalently, one can directly inspect equation (19) and seek a nonzero initial state whose polynomial coefficients all vanish without $P(t)$ losing rank.

Watch

Extended reading notes

Core claim

The central claim is Proposition 1: a multi-target system observed by a single passive observer is observable if and only if the bearing angles of the targets are distinct modulo $\pi$ for all $t$ in the observation interval $[t_i, t_f]$, equivalently no target is ever collinear with the observer and another target. The derivation writes each target's relative position as a Taylor polynomial in $(t-t_i)$ and assembles a block-diagonal state transition matrix $\tilde{\Phi}(t, t_i)$; the zero-output observability condition then reduces, in the paper's argument, to a rank condition on the instantaneous matrix $P(t)$ whose rows are $[\cos\theta_i(t), -\sin\theta_i(t)]$. Full rank of $P(t)$ is equivalent to $\theta_j(t) - \theta_i(t) \neq k\pi$ for all integer $k$, i.e., all bearing angles distinct modulo $\pi$. For trajectory ambiguity, the paper states that two targets have identical Doppler measurement histories exactly when $\tilde{s}_i(t) - \tilde{s}_j(t) = (W(t) - I)s_j(t)$ holds for all $t$, with a time-varying transformation matrix $W(t)$ constructed in the proof; with bearing-only measurements, ambiguity requires the analogous relation with a scalar $\alpha'(t)$; and with combined Doppler and bearing measurements, ambiguity occurs precisely when $s_j(t)$ is an eigenvector of $W(t)$ corresponding to the eigenvalue $\alpha'(t)$.

Load-bearing premise

The argument assumes that if the bearing angles of the targets are distinct modulo $\pi$ at every instant, then the only initial super-state producing identically zero pseudolinearized measurements is the zero vector; this equates a pointwise geometric condition on the measurement matrix with a condition on the whole trajectory over the interval.

Editorial extensions

If this is right

  • A tracking algorithm could pre-check a scenario for observability simply by testing whether any two bearing-angle traces differ by an integer multiple of $\pi$ at any time.
  • The maneuver requirement for a single observer would reduce to breaking collinearity among targets, independent of the order of target or observer dynamics.
  • Doppler-only and combined Doppler-bearing trackers could use the eigenvector condition on $W(t)$ to detect when two targets' measurement histories are exactly compatible and cannot be separated.
  • The NECNDSUF ambiguity conditions give a constructive test: if measured histories satisfy the derived relation, a tracker should declare the two targets indistinguishable and fuse them.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the move from the polynomial zero-output equation (19) to the pointwise rank-2 condition on $P(t)$ skips a coefficient-separation step; because (19) is a polynomial in $(t - t_i)$, its coefficients involve all derivative states, so distinct bearings at each instant may not by themselves rule out every nonzero initial super-state.
  • Editorial extension: classical single-target bearing-only observability requires an observer maneuver (nonzero bearing rate), yet Proposition 1 as stated does not invoke any per-target maneuver condition; this suggests the multi-target claim silently assumes each target is individually observable, and a constant-bearing receding-target counterexample would test that.
  • Testable extension: running an observability Gramian numerical test over randomly generated polynomial trajectories with distinct bearings would reveal whether any nonzero initial super-state yields identically zero output, quantifying how often the geometric condition overstates observability.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. This manuscript addresses observability of multiple targets tracked by a single observer. The authors propose two frameworks. First, for bearing-only measurements, they use the observability Gramian of a pseudo-linearized system and claim that the multi-target system is observable iff all bearing angles are distinct modulo π (Proposition 1), independent of the order of target and observer dynamics (Remark 3). Second, they propose a trajectory-ambiguity framework and derive necessary and sufficient conditions for ambiguity under Doppler-only (Proposition 2), bearing-only (Eq. (26)), and combined Doppler-bearing measurements (Proposition 3), based on a transformation matrix W(t). The paper is entirely analytical and contains no simulations.

Significance. The problem is relevant, and a simple geometric criterion would be practically useful. The manuscript is self-contained in defining observability and attempts direct derivations, which is a strength. However, the central derivations contain load-bearing errors: the bearing-only observability conclusion does not follow from the Gramian condition because the measurement matrix is block diagonal and the zero-output condition decouples per target; the Doppler ambiguity condition is obtained by integrating a scalar range-rate equation into a vector position relation, and it introduces an arbitrary time-varying unitary transformation that makes the claimed necessary condition vacuous. These issues invalidate the main claims and cannot be repaired by local edits.

major comments (5)
  1. [Section 3, Eqs. (18)-(20)] The inference from the zero-output condition to the rank of P(t) is invalid. Condition (18) is equivalent to c_i(t) Φ_i(t,t_i) x_i(t_i) = 0 for every i separately, because C(t) in Eq. (14) is block diagonal. The determinant condition (20) on P(t) involves only pairwise differences of bearing angles and does not enter the per-target zero-output condition. Hence a full-rank P(t) is neither necessary nor sufficient for (18): a nonzero initial state of one target lying on its own bearing line produces identically zero pseudo-measurement regardless of the other targets. Consequently Proposition 1 and Remark 3 are unsupported. In the single-target case (M=1), P(t) has one row and can never be rank 2, so the criterion would declare every single-target bearings-only system unobservable, contradicting the standard maneuver-dependent observability condition in the cited literature [2,3].
  2. [Section 3, Eq. (19)] Equation (19) is one scalar polynomial identity per target, whose coefficients involve all derivative states and the time-varying trigonometric functions cos θ_i(t) and sin θ_i(t). The authors never separate coefficients or analyze the kernel of C(t)Φ(t,t_i); the argument jumps from this polynomial identity to the rank of the instantaneous matrix P(t). This missing coefficient-level analysis is the load-bearing step that would be needed to prove condition (18), and it is absent.
  3. [Section 4.1, Eqs. (23)-(24)] The integration step leading to Eq. (24) is invalid. Equation (23) equates scalar range-rates, and integrating it yields scalar range as a function of time, not the vector identity s_i(t) = l' s_j(t) + b' + c(1-l')(t-t_i). Therefore Eq. (24) does not follow from the Doppler measurement equality, and the transformation W(t) in Eq. (22) is not actually derived from the measurements. This invalidates Proposition 2 and the subsequent ambiguity conditions that rely on W(t).
  4. [Section 4.1, Eq. (22)] The introduction of an arbitrary unitary transformation U(t) makes condition (22) vacuous. Since both unit vectors û_{s_i}(t) and û_{s_j}(t) have unit norm, for any pair of trajectories there exists a unitary transformation mapping one to the other at each time t. Thus Eq. (22) can always be satisfied by choosing U(t) appropriately, so it cannot be a necessary condition for Doppler ambiguity. This is a circularity in the proof of Proposition 2.
  5. [Remark 4, Appendix B, Appendix C] There is an internal contradiction about the status of condition (22). Proposition 2 states that multiple targets have ambiguous Doppler trajectories iff they satisfy (22), but Remark 4 explicitly says that (22) is necessary but not sufficient and lists additional sufficient conditions, and Appendix B derives those additional conditions. A statement cannot be both an iff condition and merely a necessary condition. In addition, Appendix C concludes that the NECNDSUF condition for Doppler-bearing ambiguity is α'(t)=1, which through Eq. (26) reduces to s_i(t)=s_j(t); the claimed eigenvector condition W(t)s_j(t)=α'(t)s_j(t) therefore collapses to a trivial identical-trajectory condition rather than providing a new observability criterion.
minor comments (5)
  1. [Throughout] The notation 'cøs' should be 'cos' (for example in Eqs. (14), (15), and (19)), and the symbol s_i(t) is used both for a vector and for a scalar in Section 4.1, which is confusing.
  2. [Section 3, Eq. (13)] The dimensions are inconsistent: A_i is described as a 2(p_i+1)×(p_i+1) matrix, but each a_i^k is a 2×1 vector, so A_i should be 2×(p_i+1); the dimensions of t_i and of the block-diagonal matrix in the following line should be reconciled.
  3. [Section 3, Eq. (16)] The definition of Φ_i(t,t_i) is not fully specified, and the displayed rows do not clearly match the polynomial coefficient ordering used in Eq. (12), making the transition matrix ambiguous.
  4. [Definition 2] Definition 2 requires equality of outputs for all t ≥ t_i, whereas Definition 1 and the subsequent analysis use a finite interval [t_i,t_f]; this interval mismatch should be reconciled.
  5. [Remark 2] The 3D claim that either bearing or elevation angles should be distinct modulo π for observability is stated without derivation or a formal statement, so it is not verifiable from the present analysis.

Circularity Check

3 steps flagged · score 7.0 of 10

Trajectory-ambiguity NECNDSUF conditions restate their own inputs: Eq. (26) is just the equal-bearing premise rearranged, Doppler W(t) is built from an arbitrary unitary rotation, and the combined Doppler-bearing condition collapses to W=I, α′=1 (identical positions).

  1. self definitional [Section 4.2, Eq. (26)]
    "Let the bearing measurement histories for ith and jth targets be the same, i.e., si(t) = α′(t)sj(t) ∀t ∈ [ti, tf ] for any i ∈ M, j ∈ M, i ̸= j and α′(t) being any scalar constant at time t. Then, the NECNDSUF condition for ambiguity in trajectories between the i-th and the j-th targets is: ˜si(t) − ˜sj(t) = (α′(t) − 1)sj(t) ∀t ∈ [ti, tf ]. (26)"

    The alleged NECNDSUF condition is obtained by substituting the assumed identical-bearing relation s_i(t)=α′(t)s_j(t) into the definition of relative position: ˜s_i−˜s_j = s_i−s_j = (α′−1)s_j. It is therefore an algebraic rearrangement of the premise that the bearing histories are the same, and carries no independent restriction. Since trajectory ambiguity was defined in Definition 3 as identical measurement histories, Eq. (26) merely restates the definition; no dynamics or distinguishability argument is added.

  2. self definitional [Section 4.1, Proposition 2 and proof, Eqs. (22) and (25)]
    "W(t) = U(t) h l′ + b′+c(1−l′)(t−ti) sj (t) i ... and the function U(t) is designed to offer a unitary transformation, which ensures that one unit vector is mapped or transformed into another unit vector."

    The proof introduces U(t) as an arbitrary unitary transformation and then defines W(t) in terms of U(t). Consequently, condition (22), ˜s_i−˜s_j=(W−I)s_j, is not an independent property of the two trajectories: W(t) can be chosen, through the free unitary U(t), to map one relative position direction onto the other. The 'necessary condition' is thus satisfied by construction whenever the scalar Doppler identity is imposed; it does not constrain the geometry and cannot serve as a substantive NECNDSUF criterion. The condition is an identity dressed as a derivation.

1 more flagged steps
  1. self definitional [Section 4.3 and Appendix C, Eq. (30)]
    "Thus, the sufficient condition entails: fi,0 = fj,0, W(t) = I and si(t) = rj(t) for all t ∈ [ti, tf ]. ... From Appendix B and (30), the NECNDSUF condition for Doppler bearing tracking is α′(t) = 1"

    The combined-condition derivation equates the two self-definitional relations and then imports the Appendix B sufficient assumptions W(t)=I and s_i(t)=s_j(t). Setting α′=1 in the bearing relation s_i=α′s_j gives s_i=s_j, so the 'NECNDSUF condition' reduces to the statement that the two targets share the same relative position vector—exactly the identical-measurement/identical-trajectory premise already built into Definition 3. No new, externally testable condition is derived; the conclusion is the input assumption restated through W=I and α′=1.

full rationale

The bearing-only observability claim in Proposition 1 is not itself circular: the paper attempts to connect the zero-output Gramian condition to a rank condition on P(t), however unsound that inference may be. The circularity is concentrated in Section 4, where the claimed NECNDSUF trajectory-ambiguity conditions are constructed from their own definitions. In Section 4.2, Eq. (26) is just the equal-bearing premise s_i=α′s_j rearranged. In Section 4.1, the Doppler condition introduces an arbitrary unitary U(t) to define W(t), so Eq. (22) can be made to hold by choice of W rather than by a real constraint on trajectories. In Section 4.3/Appendix C, combining these tautological conditions and imposing Appendix B's sufficient assumptions yields α′=1 and W=I, i.e., identical relative positions, which is the starting assumption of identical measurements. These are self-definitional reductions, not independent criteria. Self-citation is not the issue; the reduction is internal to the paper's own equations. The score reflects that the Section 4 conditions, which are central to the claimed NECNDSUF contribution, reduce by construction to the definitions and assumptions they purport to derive.

Assumptions & free parameters 4 free parameters · 6 assumptions · 2 invented entities

The central claims rest on several unstated modeling choices and ad hoc constructs. The pseudo-linearized measurement, the polynomial motion assumption, and the arbitrary unitary transformation U(t) are introduced without independent justification, and the most important step in the Doppler analysis, converting a scalar range-rate equation into a vector position equation, is mathematically invalid. These ledger entries capture the parts of the derivation the reader did not pay for upstream.

free parameters (4)
  • l' = f_j0/f_i0
    Ratio of the two targets' radiated tonals; an unknown target parameter that the Doppler ambiguity relation depends on. It is not estimated from data, but it introduces an extra degree of freedom into the derived condition.
  • b' = s_i0 - l' s_j0
    Initial relative position difference; a free vector constant in the Doppler ambiguity condition. It is not determined by the measurements, and its role in W(t) is unclear because b' is a vector divided by a vector.
  • alpha'(t)
    Time-varying scalar relating s_i(t) and s_j(t) in the bearing ambiguity condition; it is introduced ad hoc and can absorb any range ratio along a common bearing.
  • U(t)
    Arbitrary unitary transformation introduced to map unit vectors in the Doppler ambiguity proof. Since U(t) is unconstrained, it makes the necessary condition (22) satisfiable for almost any pair of trajectories.
assumptions (6)
  • domain assumption The system is noise-free, linear time-varying, and target and observer motions are polynomial with constant course (Section 2, eq (1)-(2)).
    The analysis assumes away measurement noise and restricts trajectories to polynomials, which is a strong modeling simplification not justified for general target motion.
  • domain assumption The pseudo-linearized measurement z_i(t)=x_i(t)cos(theta_i(t))-y_i(t)sin(theta_i(t)) can serve as the measurement (Section 3, after eq (13)).
    This quantity is zero on the true trajectory and is not the measured bearing; replacing the nonlinear bearing measurement with this pseudo-measurement without a rigorous equivalence argument is asserted.
  • standard math Observability is characterized by Definition 1, i.e., zero output over [ti,tf] implies zero initial state (Section 2, Definition 1).
    This is a standard zero-input observability definition for linear time-varying systems, but it is applied to a pseudo-measurement that is identically zero for the true trajectory, which complicates its interpretation.
  • domain assumption The state transition matrix Phi_i(t,ti) has the block polynomial form of equation (16), so all derivative states evolve deterministically with no coupling between targets.
    The higher-order dynamics are assumed to be exactly polynomial and independent per target; the analysis does not derive this from a physical motion model.
  • ad hoc to paper Integrating the scalar Doppler range-rate equation (23) yields the vector position relation (24).
    The Doppler frequency equation constrains the derivative of scalar range, not the full position vector. Treating the integrated result as a vector equality is mathematically invalid and is a load-bearing step in Proposition 2.
  • ad hoc to paper An arbitrary unitary transformation U(t) can relate the unit vectors of two targets without loss of generality (Section 4.1 proof).
    No physical constraint is given for U(t); it is introduced to construct W(t), which makes the necessary condition tautological.
invented entities (2)
  • W(t) transformation matrix
    purpose: Expresses the relation between two target position vectors in the Doppler ambiguity condition (22).
    W(t) is constructed using the arbitrary unitary transformation U(t) and a ratio of radiated tonals; it has no independent physical meaning or empirical handle.
  • U(t) arbitrary unitary transformation
    purpose: Maps the unit vector of target j to that of target i in the Doppler ambiguity proof.
    U(t) is unconstrained and can be chosen to make any pair of trajectories satisfy the proposed necessary condition, so it carries no falsifiable content.

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Cite this review

Pith. "Pith review of Multi Target Observability." pith.science (2026). https://pith.science/paper/HJGYGCYA

@misc{pith2026250714765,
  author       = {Pith},
  title        = {Pith review of: Multi Target Observability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HJGYGCYA}},
  note         = {Machine review of arXiv:2507.14765}
}
read the original abstract

In this paper, we mainly focus on the problem of multi-target observability, focusing on the unique state estimation criteria for multiple targets. We derive the condition which is necessary as well as sufficient for observability using bearing angles with multiple higher-order dynamics observed by a single observer. We then establish an alternative notion of observability by analyzing ambiguous target trajectories and deriving the condition which is NECNDSUF (Nec. and Suff.) for multi-target observability, considering three types of measurements: Doppler-only, bearing-only, and combined Doppler and bearing measurements, which offers insights that can improve target distinguishability, trajectory reconstruction, and overall tracking accuracy.

Figures

Figures reproduced from arXiv: 2507.14765 by the authors.

Figure 1
Figure 1. Bearing only tracking target moving from C at time t to D at time t+ 1 and the observer moving from A at time t to B at time t + 1) given by, θi(t) = tan−1  xi(t) yi(t)  . (10) xi(t) ∈ R 1×2 for i-th target be defined as, xi(t) = xi(t) yi(t) [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Multi-Target Scenario 4 Observability through Trajectory Ambiguity Another notion of observability in the multi-target scenario is to ensure that the trajectory of one target is not a compatible trajectory (i.e., the trajectory with identical measurement) of any other target. In such a case, shrinking the compatible trajectory of one target (which is the true trajectory of another target) to its true trajectory to e… view at source ↗

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