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 Cram\'{e}r-Rao bound (PCRB) via the Tichavsk\'{y} 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.