Performance Bounds and Robust Filtering for LEO Inter-Satellite Synchronization under Cross-Epoch Doppler Coupling

Abstract: Low Earth orbit (LEO) inter-satellite links (ISLs) must achieve joint synchronization and ranging under severe hardware impairments, namely oscillator phase noise, clock drift, and measurement outliers, exacerbated by rapid relative dynamics exceeding 7~km/s. In coherent Doppler processing, the frequency observable depends on the \emph{difference} between consecutive carrier phase states, creating a cross-epoch coupling structure that fundamentally affects estimation-theoretic performance limits. This paper makes three contributions. First, we prove analytically that this cross-epoch Doppler coupling is \emph{necessary} to avoid unbounded carrier phase uncertainty: without it, phase variance grows linearly without bound. Second, we derive a posterior Cramér-Rao bound (PCRB) via the Tichavský recursion that explicitly incorporates the resulting 10$\times$10 block information structure. Third, we propose a hybrid robust filtering framework combining hard gating for impulsive cycle-slip outliers with Huber M-estimation for heavy-tail contamination, using TASD-aware innovation covariance to account for cross-epoch uncertainty in residual normalization. Monte Carlo simulations at Ka-band confirm that the PCRB accurately lower-bounds estimator performance under nominal conditions, while the hybrid method reduces 95th-percentile phase error by 27--93\% compared to standard extended Kalman filtering across different outlier regimes.

• The paper establishes that cross-epoch Doppler coupling is necessary for bounded carrier-phase estimation, reducing phase growth from linear divergence to sub-linear behavior within roughly 10 rad over 500 epochs.
• The paper derives a TASD-aware posterior Cramér–Rao bound using a 10×10 block Fisher information matrix, while showing that a standard EKF is efficient for most states but has a phase efficiency ratio of 2.33.
• The paper introduces hybrid 4σ gating with Huber weighting, reducing 95th-percentile phase error by 93% for impulsive slips and 27% for heavy-tailed Doppler noise without knowing the outlier type in advance.

Motivation and problem setting

Inter-satellite links (ISLs) in low Earth orbit (LEO) mega-constellations must deliver joint time-frequency synchronization and ranging under hardware impairments that are considerably more severe than those encountered in the BDS-3 MEO constellation, where Ka-band dual one-way ranging has demonstrated sub-15 cm SLR residuals. Relative velocities exceeding 7 km/s in LEO amplify three coupled impairments: oscillator phase noise modeled via Allan variance power laws, impulsive carrier-phase cycle slips, and heavy-tailed thermal noise at low carrier-to-noise ratios. The paper's central observation is that in coherent Doppler processing, the frequency observable depends on the difference of consecutive carrier phase states, $\theta_k - \theta_{k-1}$, a structure the authors term time-accumulated signal difference (TASD) coupling. While this differential structure is familiar from GNSS carrier-phase processing, its formal role as a necessary condition for phase observability in ISL synchronization had not previously been established.

System model

The state vector is $$\tilde{\mathbf{x}}_k = [R_k, \dot{R}_k, b_k, u_k, \theta_k]^\top$$, with clock bias and drift expressed in range-equivalent units; this scaling reduces the Fisher information matrix (FIM) dynamic range from $10^{24}$ to approximately $10^6$. Dynamics are linear Gaussian with a block-diagonal integrator transition matrix, random-acceleration range noise ($\sigma_a = 0.1$ m/s$^2$), Allan-variance-parameterized clock noise for an OCXO ($h_0 = 2.2\times10^{-25}$, $h_{-2} = 1.6\times10^{-24}$), and Wiener phase noise with 100 Hz linewidth.

The TASD Doppler measurement in range-rate units is

$$y_D[k] = \dot{R}_k + u_k + \kappa_\theta(\theta_k - \theta_{k-1}) + v_{D,k}, \qquad \kappa_\theta = \frac{c}{2\pi f_c T_{\mathrm{coh}}}$$

which equals 0.0184 m/s/rad at $f_c = 26$ GHz and $$\tilde{\mathbf{x}}_k = [R_k, \dot{R}_k, b_k, u_k, \theta_k]^\top$$0 s. The dependence on $$\tilde{\mathbf{x}}_k = [R_k, \dot{R}_k, b_k, u_k, \theta_k]^\top$$1 creates a binary factor between adjacent epochs. A notable physical ambiguity: a 1 m/s range-rate change is observationally equivalent to a 3.33 ppb clock drift in any single Doppler measurement; separability relies entirely on their differing process-noise statistics.

TASD essentiality and the PCRB

The first analytical contribution is Proposition 1: if $$\tilde{\mathbf{x}}_k = [R_k, \dot{R}_k, b_k, u_k, \theta_k]^\top$$2, the phase receives no information from either Doppler or ToA measurements, and because the phase follows a random walk fully decoupled from the other states, the $$\tilde{\mathbf{x}}_k = [R_k, \dot{R}_k, b_k, u_k, \theta_k]^\top$$3 element of the information matrix obeys a strictly decreasing scalar recursion converging to zero. The posterior phase variance therefore diverges linearly without bound. With TASD active, the cross-epoch term injects information about the phase increment, converting linear divergence into bounded, sub-linear growth within $$\tilde{\mathbf{x}}_k = [R_k, \dot{R}_k, b_k, u_k, \theta_k]^\top$$4 rad over 500 epochs. This is a strong structural claim: cross-epoch Doppler coupling is not merely beneficial but necessary for phase tracking.

The second contribution derives a posterior Cramér-Rao bound (PCRB) via the Tichavský recursion, explicitly incorporating the resulting $$\tilde{\mathbf{x}}_k = [R_k, \dot{R}_k, b_k, u_k, \theta_k]^\top$$5 block FIM over the joint state $$\tilde{\mathbf{x}}_k = [R_k, \dot{R}_k, b_k, u_k, \theta_k]^\top$$6. With $$\tilde{\mathbf{x}}_k = [R_k, \dot{R}_k, b_k, u_k, \theta_k]^\top$$7 m/s, the dominant off-diagonal entry $$\tilde{\mathbf{x}}_k = [R_k, \dot{R}_k, b_k, u_k, \theta_k]^\top$$8 couples phase information into the range-rate and clock-drift dimensions. Under nominal Gaussian noise over 500 Monte Carlo trials, the bound is validated with zero violation rate across all steady-state epochs. An important caveat stated by the authors: under outlier-corrupted Doppler, the Gaussian-assumed PCRB is no longer a strict lower bound.

Efficiency analysis isolates the EKF's deficiency

A sharp numerical contrast emerges under nominal conditions. The EKF achieves efficiency ratios $$\tilde{\mathbf{x}}_k = [R_k, \dot{R}_k, b_k, u_k, \theta_k]^\top$$9 for range, clock bias, range rate, and clock drift (e.g., $10^{24}$0, RMSE 0.714 m/s against a $10^{24}$1 of 0.710), but only $10^{24}$2 for phase (RMSE 49.5 rad versus a bound of 21.2 rad). Because the standard EKF updates only $10^{24}$3 and discards the information about $10^{24}$4 carried by the cross-epoch Doppler residual, the gap is attributable specifically to the TASD structure rather than generic filter nonlinearity. The authors note that fixed-lag smoothing could plausibly close part of this gap, but do not quantify the achievable gain. They also observe that since ToA observes only $10^{24}$5, the marginal bounds on $10^{24}$6 and $10^{24}$7 individually remain near the prior level—a single-link gauge ambiguity inherent to the scenario.

Hybrid robust filtering

The third contribution is a hybrid update combining hard gating ($10^{24}$8) for sparse extreme outliers with Huber M-estimation ($10^{24}$9) for moderate contamination, both operating on residuals normalized by a TASD-aware innovation covariance that adds the previous-epoch term $10^6$0 absent from single-epoch EKF implementations. The authors show that the omitted cross terms simplify to an expression whose leading component is strictly negative, so the adopted covariance conservatively overestimates $10^6$1 by roughly 40% at steady state, making thresholds more permissive and preserving stability—an approximation acknowledged rather than eliminated.

The design rationale is quantitative: at a $10^6$2 impulsive outlier, Huber weighting alone inflates effective noise by only $10^6$3, leaving an effective residual near $10^6$4 that still dominates the update—hence Huber alone cannot handle cycle slips, while gating alone discards useful moderate outliers. Threshold sensitivity is modest: varying both parameters by $10^6$5 changes p95 phase error by less than 5%, and the false rejection rate at $10^6$6 under Gaussian noise is 0.006%.

Simulation results

With 500 trials over 100 epochs, two outlier mechanisms corrupting Doppler only (ToA remains Gaussian):

Regime	EKF	Gating	Huber	Hybrid
Impulsive slips (5%, $10^6$7)	1406 rad	97 rad	771 rad	98 rad
Heavy-tail (15%, $10^6$8 mixture)	191 rad	142 rad	184 rad	139 rad

The Hybrid reduces 95th-percentile phase error by 93% under impulsive slips and 27% under heavy-tail contamination relative to the EKF, achieving lowest or near-lowest p95 in both regimes without prior knowledge of the outlier type. Single-trial trajectories confirm the mechanism: the EKF absorbs the first cycle slip at $10^6$9 and sustains a permanent $\sigma_a = 0.1$0 rad bias, whereas the Hybrid tracks phase within 1–2× the $\sigma_a = 0.1$1 floor between outlier events.

Limitations and open questions

The paper concedes several restrictions explicitly. The analysis assumes a single link with time-invariant parameters; extension to constellation-scale distributed PCRBs remains open. The Gaussian-assumed PCRB loses its strict lower-bound status under outlier corruption, so tail-performance claims rest on simulation rather than theory. The innovation covariance omits cross terms justified only by a conservative-sign argument, and the efficiency gap $\sigma_a = 0.1$2 suggests—but does not demonstrate—that fixed-lag smoothing would improve phase estimation. All results are simulation-based at Ka-band with specific oscillator and dynamics parameters; no experimental ISL data are used.

Conclusion

This work establishes that cross-epoch Doppler coupling is a necessary condition for bounded carrier-phase estimation in LEO ISL synchronization, provides a validated TASD-aware PCRB through an explicit $\sigma_a = 0.1$3 block information structure, and demonstrates a hybrid gating-plus-Huber filter that cuts p95 phase error by 27–93% across outlier regimes. The efficiency contrast ($\sigma_a = 0.1$4 versus $\sigma_a = 0.1$5) cleanly attributes EKF suboptimality to mishandling of the cross-epoch structure, making the framework a concrete benchmark for future synchronization estimators on LEO constellations (2603.11280).