Timing advance and Doppler shift estimation in LEO satellite networks: A recursive Bayesian study

Ashutosh Balakrishnan; Philippe Martins; Pierre Popineau

REVIEW 5 major objections 6 minor 10 references

Timing advance and Doppler shift estimation in LEO satellite networks: A recursive Bayesian study

T0 review · 5 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that one extended Kalman filter over the joint satellite-and-user state can estimate LEO satellite position to within about 1.8 percent mean error and compute timing advance and Doppler shift even when the user moves at…

desk verdict A reasonable EKF state-augmentation for joint LEO satellite-UE tracking, but the Doppler model is physically wrong and the validation can't support the central Doppler claim. read the letter →

arxiv 2506.08739 v1 pith:YKJPIFQJ submitted 2025-06-10 eess.SP

Ashutosh Balakrishnan , Pierre Popineau , Philippe Martins This is my paper

classification eess.SP

keywords LEOsatellitenetworkstimingadvanceDopplershiftextendedKalmanfilterrecursiveBayesianestimationusermobilitynon-terrestrialvisibility

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

The paper argues that timing advance and Doppler shift in a LEO satellite network can be estimated together by modelling the satellite and the ground user as a single joint system and filtering that system with an extended Kalman filter. The satellite motion includes the gravitational acceleration of a central force, the user moves with a constant velocity, and the measurements are slant range and elevation angle. From the filtered state the paper computes timing advance as twice the slant range divided by the speed of light, $\text{TA}(t)=2 d_{lu}(t)/c$, and Doppler shift from the satellite-user relative velocity scaled by the carrier frequency. Simulations show the filter tracking the satellite position with mean percentage error around [1.8166% 0.5595% 0.7725%] even at airplane-speed user motion. If the approach holds, it gives one recursive estimator for the initial-access parameters that LEO networks currently have to compute through separate steps.

What carries the argument

The central object is the joint discrete-time state $x_t = [p_l, v_l, p_u, v_u]$, propagated with gravitational acceleration $a_l = -\mu p_l/\|p_l\|^3$ for the satellite and constant velocity for the user. The argument is carried by the Jacobian $F(t)$, a block matrix with identity and $\Delta t$ blocks plus the gravity-gradient block $A$, which linearizes the nonlinear motion for the extended Kalman filter. Slant range and elevation angle form the measurement model, and the formulas $\text{TA}(t)=2 d_{lu}(t)/c$ and $\Delta f(t)=f_T (v_l-v_u)/c$ translate the filtered state into the two communication parameters. The time-difference-of-arrival relations then turn the same state estimate into a clock-drift estimate when the clocks differ.

What would settle it

Compare the Doppler-versus-elevation curve from a real LEO pass with the paper's simulated curve: physical Doppler must cross zero near the moment of closest approach, whereas the paper's model stays near $f_T \|v_l-v_u\|/c$ and does not drop to zero, so a measured null would settle which formula is right.

Watch

Extended reading notes

Core claim

The paper's central claim is that the position and velocity of a LEO satellite and a mobile user can be concatenated into one state and estimated recursively, with the satellite acceleration linearized through the Jacobian of the inverse-square gravitational field. From that state, the timing advance follows directly as the round-trip light time to the estimated slant range, and the Doppler shift follows as $f_T (v_l-v_u)/c$, the transmitted frequency times the relative velocity divided by the speed of light. The paper reports mean percentage position errors of [1.8166% 0.5595% 0.7725%] for the satellite coordinates in simulation, and shows the estimates staying close to the true values across the satellite visibility window. It further claims that when the satellite and user clocks are not synchronized, the clock drift takes the form $\alpha t + \beta$, so the same filter can separate clock offset from the time difference of arrivals.

Load-bearing premise

The load-bearing assumption is that Doppler shift depends on the total relative speed between satellite and user, not just the speed along the line connecting them; physical Doppler is the line-of-sight component, and that component is zero at closest approach.

Editorial extensions

If this is right

When the filter converges, satellite position error stays near the reported mean percentage values and timing advance can be read directly from the estimated slant range.
Timing advance varies with elevation angle, reaching a minimum at maximum elevation and a maximum near the visibility boundary, so a network can predict when to refresh the TA during a pass.
Doppler shift scales with carrier frequency, giving roughly 260 kHz at 10.9 GHz and 680 kHz at 28 GHz in the simulation, so the same state estimate supports frequency precompensation in either band.
The clock-drift expression $\Delta t_{\text{clock drift}} = \alpha t + \beta$ lets the filter estimate user-satellite clock offset from time difference of arrivals without extra hardware.

Reading between the lines

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

A natural extension the paper does not pursue is to replace the full-relative-velocity Doppler expression with its line-of-sight projection, which would let the filter track the Doppler null near closest approach and be checked directly against real satellite beacon measurements.
The same joint-state filter could be extended to multiple visible satellites, allowing the user position to be estimated from several range and elevation measurements instead of assuming a known user trajectory.
The linear clock-drift model suggests a cheap synchronization scheme: estimate $\alpha$ and $\beta$ online from successive TDoA measurements and feed them back to the user's local oscillator.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The useful part of this paper is the state-augmentation: it takes the satellite-only ephemeris EKF from [7] and adds UE states and the gravitational acceleration in the satellite motion model. Constructing the Jacobian for that nonlinear acceleration is correct, and the timing-advance geometry in Eq. (26) is standard but clean. If the paper only claimed the joint EKF for position/velocity estimation, I'd call it a competent, incremental contribution.

But the Doppler half is not sound. Eq. (27) writes Delta_f = f_T (v_l - v_u)/c, using the full relative velocity vector rather than its projection onto the satellite-UE line of sight. A scalar time-derivative is also being equated to a vector difference divided by c, so the equation is internally inconsistent. The physical Doppler is (f_T/c) d(d_lu)/dt, which depends on the radial component and passes through zero near closest approach. Fig. 2(c) is the smoking gun: the reported Doppler stays in narrow bands around 263.36 kHz and 689.17 kHz over 0-40 degrees elevation, matching f_T ||v_rel||/c and not a radial projection. That means the "true" Doppler in the simulation is generated by the wrong law, so the claimed estimation accuracy for Doppler is not established.

Other soft spots are smaller. Eq. (22) is an asserted steady-state Riccati expression with no derivation and unclear notation. The measurement Jacobian is never written out. The clock-drift analysis in Eq. (30) reduces to the definitional identity Delta_tau_hat - Delta_tau = t12*eps2 - t11*eps1 = alpha t + beta, which is not much of a result. Validation is a single in-house simulation with no baseline, no error bars, and no comparison to a standard EKF without UE states, so the reported mean percentage errors are hard to interpret.

I don't think the reader's soundness score is unfair. The Doppler error is load-bearing because Doppler accuracy is half of the advertised contribution. But the paper is not incoherent; the TA part and the joint-state formulation are legitimate and the fix is identifiable: replace the Doppler model with the radial projection and rerun the simulation. If this crosses my desk, I'd send it to review with the expectation of major revision rather than desk-reject, because the topic is timely and the core EKF framework is otherwise usable. For your own work, I wouldn't cite the Doppler results until the model is corrected.

Referee Report

5 major / 6 minor

Summary. The paper proposes an extended Kalman filter (EKF) based recursive Bayesian framework to jointly estimate the position and velocity of a LEO satellite and a mobile user equipment (UE), and then to compute the timing advance (TA) and Doppler shift experienced by the UE. The satellite motion model includes the gravitational acceleration term and its Jacobian, while range and elevation angle are used as measurements. The authors report mean percentage errors of [1.8166% 0.5595% 0.7725%] in satellite position coordinates, show TA variation with time and elevation, and study Doppler shift at 10.9 GHz and 28 GHz. A short clock-drift analysis is also included for unsynchronized satellite and UE clocks.

Significance. If the framework were correct, it would provide a useful single recursive estimator for initial-access parameters in LEO non-terrestrial networks with a mobile UE, an application of current interest. The paper has real strengths: the satellite acceleration model in Eq. (5)-(11) is physically motivated, the TA geometry in Eq. (25)-(26) is standard and correct, and the central position-estimation claim is not circular because no parameters are fitted to the target outputs. The recursive Bayesian formulation and the visibility-aware geometry are appropriate for the problem. However, the Doppler model in Eq. (27) is physically incorrect, and the validation of Doppler accuracy uses exactly this incorrect model. Since accurate Doppler estimation is one of the two headline contributions, the significance of the paper is currently contingent on a substantial correction of the Doppler formulation and a re-run of the corresponding simulations.

major comments (5)

[Section IV-B, Eq. (27)] The Doppler shift formula is physically incorrect and dimensionally inconsistent. The paper writes Δf(t)=f_T (v_l(t)-v_u(t))/c, where v_l and v_u are three-dimensional vectors, so the right-hand side is a vector while the left-hand side is a scalar. The correct non-relativistic Doppler shift is the derivative of the slant range: Δf(t) = (f_T/c) d(d_lu(t))/dt = (f_T/c) [(p_l(t)-p_u(t)) · (v_l(t)-v_u(t))] / ||p_l(t)-p_u(t)||. The chain of equalities in Eq. (27) also treats the time derivative of the slant-range difference as if it were the full relative velocity vector, which is only the radial component. This error is load-bearing because the paper's claimed accurate Doppler estimation rests on this equation.
[Section V, Fig. 2(c)] The simulation confirms that the Doppler model used is the non-physical full-velocity formula rather than the radial projection. At 10.9 GHz the plotted Doppler stays in the narrow band 263.358-263.368 kHz over 0-40 degrees elevation, close to f_T ||v_rel||/c, whereas the true radial Doppler should vary strongly with geometry and pass through zero near closest approach. Consequently, the agreement between estimated and 'true' Doppler in Fig. 2(c) only validates the wrong model; it does not establish that the EKF estimates the physical Doppler shift. The Doppler simulation must be regenerated with the radial Doppler law, and the resulting curves should be shown as a function of elevation and time, including the expected zero crossing.
[Section III-B, Eq. (22)] The steady-state Riccati expression P∞ = (R/H^2)(F + sqrt(F^2 + H^2 Q/R)) is not the algebraic Riccati equation for the EKF. For the matrix-valued F in Eq. (13), the expression is formally undefined, and no scalar-matrix conversion is provided. Since the paper advertises a stability analysis through the Riccati equation, either the correct continuous- or discrete-time algebraic Riccati equation should be stated and solved, or the stability discussion should be removed or substantially revised. This issue does not directly affect the simulation results, but it is presented as a contribution.
[Section II-A / Section III-B] The measurement model h(x_t) is nonlinear (range and elevation in Eqs. (14) and the definition of d_lu), but the EKF update equations (17)-(19) use a matrix H_t without ever providing the Jacobian of h with respect to the state vector. Replacing A_t by F_t in the linear Kalman equations does not supply the measurement Jacobian. Without H_t, the EKF implementation used to produce the results is not fully specified, and the claim that the framework is a standard EKF cannot be verified from the manuscript alone. The measurement Jacobian should be derived and reported.
[Section IV-B, Eqs. (28)-(30)] The clock-drift analysis is a definitional identity rather than an estimation method. Equation (30) shows that Δτ̂_1 - Δτ_1 = t_12 ε_2 - t_11 ε_1 = α t + β, which is just algebra from the definitions in Eq. (28). The paper does not provide a procedure to estimate ε_1 and ε_2, nor does it connect the EKF output to these quantities. The statement that the proposed framework 'computes the clock drift' is therefore unsupported. This is an advertised contribution in the abstract and Section I-C, even though it is not central to the TA estimation.

minor comments (6)

[Section II-A, Eq. (12)] The text states that the UE is assumed to move with constant acceleration, but Eq. (12) shows v_u_i(t+1)=v_u_i(t), i.e., constant velocity. Please reconcile the wording with the model.
[Section III-A, Eq. (15)] The initial condition is written as x0 ~ N(0, P_{1|0}); this notation is confusing because the initial mean is typically nonzero and the initial covariance is P_0, not P_{1|0}. Please clarify.
[Section V, Fig. 2(c)] The vertical axis appears to list Doppler values in kHz with values such as 263.358, but the units label says 'KHz'; please use consistent SI notation and verify the scaling.
[Section V, Fig. 2(a)] The mean percentage error is reported as [1.8166% 0.5595% 0.7725%], but the definition of 'percentage error' per coordinate is not stated. Please specify how the percentage is computed relative to the true coordinate magnitude or another reference.
[Section IV-B, Eq. (27)] The notation t11 and t12 is introduced without clear definition of the two time instants; the TDoA expression should be stated with explicit times, for example d_lu(t+Δt)-d_lu(t), to avoid ambiguity.
[General] There are several typographical issues, including 'Algebraic Riccatti' instead of 'Riccati', and a missing space in 'Constellation Technologies & Operations' in the author affiliation. These do not affect the technical content.

Circularity Check

1 steps flagged · score 2.0 of 10

No load-bearing circularity: the EKF position/TA derivation is self-contained; a minor definitional identity appears in the clock-drift sidebar and the Doppler truth model is internally self-referential.

self definitional [Section IV-B, Eqs. (28)-(30)]
"Taking difference of the erroneous TDoA and the ideal TDoA, we get the corresponding clock drift between the two time intervals which is given as ∆tclock drift =∆ˆτ1 − ∆τ1 = (t12ǫ2 − t11ǫ1)=tα + β. Hence, it is inferred that ∆tclock drift is of the form αt+β, with the corresponding rate of change of clock drift being, d/dt(∆tclock drift )=α."

The quantity called 'clock drift' is defined as the difference between the erroneous TDoA and the ideal TDoA. Substituting (28) makes this difference identically t12·ε2 − t11·ε1, which is linear in time by algebraic construction. The 'inference' that the drift has the form αt+β therefore follows immediately from the definitions, not from the EKF or from any measurement. This is a self-definitional step, but it is a side remark in the Doppler/clock discussion and is not load-bearing for the central EKF position, TA, or Doppler estimates.

full rationale

The paper's central derivation is a standard EKF over a satellite-UE state vector. The motion model (6)-(13) is written from gravitational acceleration (5), the Jacobian is computed explicitly, and the measurement model (range and elevation) is a deterministic function of the state. No parameter is fitted to the reported TA, Doppler, or position errors; the reported mean percentage error is a measured discrepancy between the EKF state estimate and the simulated trajectory. The clock-drift conclusion (30) is the one genuinely definitional step: 'clock drift' is introduced as the difference between the erroneous and ideal TDoA, so its expression as t12·ε2 − t11·ε1 and hence as αt+β follows immediately from the definitions in (28) rather than from estimation. Because this step is a side remark and not used to generate the TA or Doppler estimates, it does not infect the central claim. The Doppler law in Eq. (27) uses the full relative velocity vector instead of its line-of-sight projection, and Fig. 2(c) is consistent with that same total-velocity model; this is a physical-modeling or correctness risk, not a circularity, since the estimator does not fit Doppler to itself—the Doppler output is a deterministic function of the state estimate. There are no self-citations and no imported uniqueness theorem. Overall the derivation is self-contained; the only self-referential element is the minor definitional clock-drift identity, warranting a low score.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central derivation uses only standard filtering assumptions and physical constants; no new physical entities are introduced. The main caveats are the hand-chosen noise covariances with no sensitivity analysis, the truncated gravity model, and the unverified constant-velocity UE model.

free parameters (3)

Process noise variance Q (per-coordinate sigma^2_iQ) = 10^-4 km/s^2 as stated
Chosen as a simulation input; the notation confuses variance and standard deviation and no sensitivity analysis is reported.
Measurement noise variance R (per-coordinate sigma^2_iR) = 0.1 km
Chosen for the range and elevation measurement model; no calibration method or sensitivity study is given.
Initial covariance P_{1|0} = not stated
Initial state x0 and its covariance are assumed known from ephemeris; the paper does not quantify initialization error or its effect on TA and Doppler estimates.

assumptions (5)

domain assumption Process and measurement noises are Gaussian, zero-mean, independent, and the system is Markov (Eqs. (3)-(4)).
This underlies the KF/EKF MMSE derivation and is not validated against real satellite-UE channel statistics.
domain assumption Satellite acceleration is exactly the two-body central gravity term -mu*p_l/||p_l||^3 (Eq. (5)); J2, drag, solar radiation pressure, and third-body effects are neglected.
At 375 km altitude these perturbations are non-negligible over long passes; the paper gives no error bound for the truncation.
domain assumption The Earth is modeled as a sphere of radius Re for geometry and coordinate conversion (Section II).
This simplifies the slant range and angle equations, and the geodetic-to-geocentric conversion is exact only for a sphere.
domain assumption The UE moves at constant velocity in the state model (Eq. (12) sets v_u_i(t+1)=v_u_i(t)).
The simulation uses one constant airplane speed; the text claims support for time-varying UE motion but does not demonstrate it.
domain assumption The initial satellite state from the known ephemeris is accurate enough for the filter to converge (Section II, Section V).
No analysis of initialization error or filter divergence is provided.

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Cite this review

Pith. "Pith review of Timing advance and Doppler shift estimation in LEO satellite networks: A recursive Bayesian study." pith.science (2026). https://pith.science/paper/YKJPIFQJ

@misc{pith2026250608739,
  author       = {Pith},
  title        = {Pith review of: Timing advance and Doppler shift estimation in LEO satellite networks: A recursive Bayesian study},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YKJPIFQJ}},
  note         = {Machine review of arXiv:2506.08739}
}

read the original abstract

Low earth orbit (LEO) satellite based non-terrestrial networks are a key theme of the upcoming 6G networks. These space networks are proposed to be used for high-mobility use-cases like airplanes and vehicles. The initial access process between a base station (BS) and a user equipment (UE) involves timing advance (TA) value computation at the BS, requiring precise BS location information at the UE. It becomes more challenging in LEO satellite networks due to the fast moving LEO satellites and large pathloss, in addition to the mobile UE. This paper aims to compute the TA and Doppler shift experienced at the UE by modeling the joint system dynamics in a LEO satellite-mobile UE network through an extended Kalman filter (EKF) based recursive Bayesian framework. The framework accurately models the joint system dynamics by considering the LEO satellite acceleration. It constructs the Jacobian to linearize the inherent non-linearities present in the motion. Probabilistic insights regarding the state-update and propagation are also provided. The analytical framework factors in the limited satellite visibility at the UE and the satellite-UE geometry w.r.t. the earth center. The proposed framework is also useful when the satellite and UE clocks are not in sync, with the corresponding clock drift a function of the measured time difference of arrivals. Our results showcase the efficacy and robustness of the proposed EKF framework to estimate the TA and Doppler shift, even at very high UE speeds. The work is expected to be extremely useful in realizing LEO satellite based non-terrestrial networks.

Figures

Figures reproduced from arXiv: 2506.08739 by the authors.

**Figure 2.** Illustrating (a) estimated and measured coordina [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** (a) Illustrating variation of TA and radial distan [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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Reference graph

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