A Dual-Clock Stability Feature-Based Noise-Adaptive Clock Steering Approach for Single-Satellite Time Reference Generation

. 2026 Sep 8;26(18):5699. doi: 10.3390/s26185699

Algorithm 1. Variational Bayesian Kalman filter for atomic clocks

Clock State Prediction:

Predict state:

{\hat{x}}^{-} (t) = ϕ (δ) \hat{x} (t - δ)

Predict MMSE matrix:

C^{-} (t) = ϕ (δ) C (t - δ) ϕ {(δ)}^{T} + Q (δ)

Adaptive Measurement Noise Estimation:
Predict noise parameters:

α^{-} (t) = ρ * [α (t - 1) + 1]

β^{-} (t) = ρ * β (t - 1)

Estimate measurement noise variance:

R (t) = β (t - 1) / α (t - 1)

Update noise parameters:

α (t) = α^{-} (t)

β (t) = β^{-} (t - 1) + {(y (t) - H \hat{x} (t - 1))}^{2} + H C (t) H^{T}

Clock State Estimation Update:
Update MMSE matrix:

C (t) = C^{-} (t) - C^{-} (t) H^{T} {(H C^{-} (t) H^{T} + R (t))}^{- 1} H C^{-} (t)

Update Kalman gain:

K (t) = C^{-} (t) H^{T} {(H C^{-} (t) H^{T} + R (t))}^{- 1}

Upstate clock state estimation:

\hat{x} (t) = {\hat{x}}^{-} (t) + K (t) [y (t) - H {\hat{x}}^{-} (t)]