Kalman Filtering

for GNSS Receiver Tracking Loops for Scintillating Signals

Scintillation

amplitude fading
- degrades CN0
phase fluctuations
- induces cycle slips and/or loss-of-lock

Why Kalman Filter?

minimize mean-squared phase error
can adapt over gradual changes in signal characteristics

Disadvantage

degrades or diverges under abrupt changes in signal characteristics

Steps to Kalman Filter

Observation Matrix
System Model
Noise Model (Covariance Matrix)
Filter Output

A = M\vec{x}

A = M ​ x ​ ⃗ ​ ​

\text{signal}_k\star \text{ref}_k = ADC(n T_s + \tau_k)C(n T_s+\hat{\tau_k}) e^{j\left[2\pi(f_{d_k} - \hat{f_{d_k}})n T_s + \phi_k - \hat{\phi}_k\right]} + \eta(nT_s)

s i g n a l ​ k ​ ​ ⋆ r e f ​ k ​ ​ = A D C (n T ​ s ​ ​ + τ ​ k ​ ​) C (n T ​ s ​ ​ + ​ τ ​ k ​ ​ ​^​ ​) e ​ j [2 π (f ​ d ​ k ​ ​ ​ ​ - ​ f ​ d ​ k ​ ​ ​ ​ ​^​ ​) n T ​ s ​ ​ + ϕ ​ k ​ ​ - ​ ϕ ​^​ ​ ​ k ​ ​] ​ ​ + η (n T ​ s ​ ​)

correlation output at block

k^{th}

k ​ t h ​ ​

\approx ADR(\Delta\tau) N\text{sinc}(2\pi\Delta f_d T_I)\exp(j2\pi \Delta f_{d_k} \frac{T_I - T_s}{2})\exp(j\Delta\phi_k)

\approx A D R (Δ τ) N s i n c (2 π Δ f ​ d ​ ​ T ​ I ​ ​) exp (j 2 π Δ f ​ d ​ k ​ ​ ​ ​ \frac{​ T ​ I ​ ​ - T ​ s ​ ​ ​ ​}{​ 2 ​}) exp (j Δ ϕ ​ k ​ ​)

\eta(n T_s)

η (n T ​ s ​ ​)

for fine acq., add and subtract adjacent overlapping blocks to cancel noise? investigate, but probs no b/c mult by signal means diff noise

is zero-mean white noise with variance

\sigma_n^2 = P_n = \frac{P_s}{10^{\frac{C/N_0}{10}}}

σ ​ n ​ 2 ​ ​ = P ​ n ​ ​ = \frac{​ P ​ s ​ ​ ​ ​}{​ 1 0 ​ \frac{​ C / N ​ 0 ​ ​ ​ ​}{​ 1 0 ​} ​ ​ ​}

Kalman Filtering

Scintillation

Why Kalman Filter?

Steps to Kalman Filter

The End