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.

Figures (5)

• Figure 1: LEO inter-satellite link synchronization scenario. Two satellites exchange ranging signals subject to clock drift, phase noise, and measurement outliers. The TASD Doppler measurement couples consecutive carrier phase states $\theta_{k-1}$ and $\theta_k$.
• Figure 2: TASD information structure and phase observability. (a) Minimum eigenvalue of $\mathbf{J}_k$: with TASD ($\kappa_\theta \neq 0$), $\lambda_{\min}$ reaches a process-noise-limited floor; without TASD, it decays toward zero. (b) Phase $\sqrt{[\mathbf{P}_k]_{55}}$ over 500 epochs: TASD yields sub-linear growth within $\mathcal{O}(10)$ rad, while $\kappa_\theta = 0$ diverges, confirming Proposition \ref{['prop:tasd']}.
• Figure 3: PCRB validation under nominal Gaussian noise ($k \geq 20$, 500 trials). (a) Phase $\theta$: empirical MSE exceeds the PCRB at all steady-state epochs with efficiency ratio $\eta_\theta = 2.33$. The gap reflects the EKF's inability to jointly update $\theta_{k-1}$ and $\theta_k$ from the cross-epoch Doppler measurement. (b) Range rate $\dot{R}$: the EKF achieves $\eta_{\dot{R}} = 1.01$, confirming near-optimal performance for states without cross-epoch coupling. The contrast between (a) and (b) isolates the TASD structure as the source of the efficiency gap.
• Figure 4: Robust performance comparison (500 trials). (a) Phase error cumulative distribution function (CDF) under impulsive slips (5%, $300\sigma$): the Hybrid method achieves p95 = 98 rad, matching Gating (97 rad) and reducing the EKF baseline (1406 rad) by 93%. (b) p95 phase error across both outlier regimes: solid bars represent impulsive slips, hatched bars represent heavy-tail contamination. The Hybrid achieves the lowest or near-lowest p95 in both scenarios without prior knowledge of the outlier type.
• Figure 5: A representative single-trial phase error trajectory under impulsive outliers. Red shading marks Doppler outlier epochs. The standard EKF diverges after the first cycle-slip event at $k \approx 3$ and does not recover. The Hybrid method rejects the outliers via hard gating and tracks $\theta$ within $1$--$2\times$ the $\sqrt{\mathrm{PCRB}}$ floor between outlier events.

Theorems & Definitions (2)

• Proposition 1: TASD Essentiality
• proof