## 3 Adaptive Carrier-to-Noise Ratio (C/N\({}_{0}\)) Estimation Method
The amplitude Kalman filter is a single-state filter. The discretized system model and measurement equation are as follows:  
\[\left\{ \begin{array}{l}X_{k}=X_{k-1}+w_{k-1}\\ Z_{k}=X_{k}+v_{k}\end{array} \right. \tag{20}\]  
\[X_{k}=A^{2}+2\sigma_{n}^{2} \tag{21}\]  
where \(A\) is the effective amplitude of the signal; \(Z_{k}\) is the filter measurement, the value of which is equal to \(\overline{A}\) in Equation (14); and \(w\) and \(v\) are the system noise and the measurement noise, respectively. The state transition matrix \(\Phi_{k,k-1}\) and the measurement matrix \(H_{k}\) are both \(1\). The variance of the measurement noise is:  
\[\sigma_{v,k}^{2}=E\left\{\left[\overline{A}_{k}-E\left(\overline{A}_{k} \right)\right]^{2}\right\}\approx 4\delta_{n,k}^{2}\left(\overline{A}_{k}- \delta_{n,k}^{2}\right) \tag{22}\]  
The main reason for the sudden change of the C/N\({}_{0}\) due to signal occlusion is the sudden change in the signal power, namely, the signal amplitude (filter state) has a large jump. After the filter reaches a steady state, the classical Kalman filter (see Table 1) state covariance matrix (\(P_{k}\)) tends to a fixed value. The Kalman filter gain approaches zero. The time update effect is strengthened, and the measurement update function is weakened [19]. This causes the state estimate of the filter to deviate substantially from the true value of the catastrophic state. To solve the divergence problem of CKF, the strong tracking Kalman filter introduces a time-varying fading factor into the calculation of the state-one-step prediction mean square error matrix, thereby improving the performance in tracking the state. The covariance propagation of the amplitude Kalman filter that is based on the STKF is as follows:  
\[P_{k/k-1}=\lambda_{k}\Phi_{k,k-1}P_{k-1}\Phi_{k,k-1}^{T}+Q_{k-1}=\lambda_{k}P _{k-1}+Q_{k-1} \tag{23}\]  
where \(\lambda_{k}\geq 1\) is a time-varying fading factor. Figure 2 depicts the time-varying fading factor calculation process. In Figure 2, \(0<\kappa\leq 1\) is the forgetting factor and \(L\geq 1\) is weakening factor. \begin{table}
\begin{tabular}{l l} \hline \hline \multirow{3}{*}{**System propagation**} & 1. State prediction \(\bar{X}_{k/k-1}=\Phi_{k,k-1}\bar{X}_{k-1}\) \\  & 2. Covariance propagation \\  & \(\mathbf{P}_{k/k-1}=\Phi_{k,k-1}P_{k-1}\Phi_{k-1}^{T}+Q_{k-1}\) \\  & 3. Kalman gain \(\mathbf{K}_{k}=\mathbf{P}_{k/k-1}\mathbf{H}_{k}^{T}\left(\mathbf{H}_{k}P_{k/k-1}\mathbf{H}_{k}^{T }+\mathbf{R}_{k}\right)^{-1}\) \\
**Measurement update** & 4. State update \(\bar{X}_{k}=\bar{X}_{k/k-1}+K_{k}\left(\mathbf{Z}_{k}-\mathbf{H}_{k}\bar{X}_{k/k-1}\right)\) \\  & 5. Covariance update \(P_{k}=(\mathbf{I}-\mathbf{K}_{k}\mathbf{H}_{k})P_{k/k-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Classical Kalman filter (CKF) process [19]. In the amplitude Kalman filter, the noise variance is estimated via Equation (22). In the case of low \(\zeta\)/\(N_{0}\), \(\widetilde{A}\) may be smaller than the noise variance \(\delta_{n}^{2}\). Calculating the noise variance using Equation (22) will result in a negative value, thereby causing the filter to be abnormal. To avoid a negative value of the measurement noise variance, a measurement noise adaptive method that is based on the Allan variance is used to realize the online estimation of the noise variance in real time. The measurement variance estimation algorithm that is based on the Allan variance is [20]:  
\[ \begin{array}{l}\hat{R}_{k}=\frac{1}{2(k-1)}\sum\limits_{i=2}^{k}{(Z_{i}-Z_{i -1})^{2}}\\ \quad=\frac{1}{2(k-1)}\sum\limits_{i=2}^{k-1}{(Z_{i}-Z_{i-1})^{2}}+{(Z_{k}-Z_{ k-1})^{2}}\\ \quad=\left(1-\frac{1}{k-1}\right)\hat{R}_{k-1}+\frac{1}{2(k-1)}{(Z_{k}-Z_{k-1 })^{2}}\end{array}  \tag{24}\]  
where \(k=2,3,4,\cdots\) and the initial value \(\hat{R}_{1}\) is calculated via Equation (22).