## 3 Proposed Technique
### Phase Bias Estimation  
This subsection deals with the estimation issue of the unknown phase bias term, allowing the undesired phase bias to be compensated for. Recalling Equation (5), it is clear that the phase bias can be calculated if the array weight and the satellite steering vector are known. The former is easily obtained from the adaptive array, but the latter is not. Although the satellite direction can be determined from the ephemeris, obtaining the satellite steering vector also requires accurate knowledge of the array manifold and the array attitude. In order to obtain a self-contained phase bias estimate, a phase bias estimation approach is developed below. The estimation of the phase bias is divided into two cases, including the jammer present and absent cases. The ICR in Equation (14) can be used as a judgment condition. If the following condition is found, the jammer is considered to be absent, and vice versa. \[\text{ICR}<T_{\text{J}} \tag{18}\]  
where \(T_{\text{J}}\) is a predefined threshold that can be set to 3 dB. If the jammer is not present, because the satellite signal is much weaker than the noise, \(\mathbf{R}_{xx}\approx\sigma_{n}^{2}\mathbf{I}\), where \(\sigma_{n}^{2}\) is the noise power, and \(\mathbf{I}\) denotes the identity matrix. In this case, it can be considered that no phase bias has been introduced, so the phase bias estimate is \(\hat{\psi}=0\). If the jammer is present, the phase bias is estimated as follows. Suppose that the phase bias is detected at the \(K\)-th correlation epoch, then in the successive correlation epochs of window size \(L\), i.e., \([K,K+L-1]\) correlation epochs, the NCO is tuned using \(\hat{\mathbf{X}}_{k}^{-}\), excluding \(\delta\phi_{k}\) contaminated by the phase bias. Assume that the signal dynamics do not change dramatically and that the phase bias value does not change in these \(L\) epochs, since it is at the millisecond level. Thus, the local carrier replica matches \(s(t)\) well, i.e., \(\delta f\approx 0\) and \(\delta\varphi\approx 0\). Therefore, the discriminator output in Equation (8) can be written as  
\[\delta\tilde{\varphi}_{k}\approx\psi+\xi_{k} \tag{19}\]  
Then, the linear observation function with respect to the unknown phase bias can be written in matrix form as  
\[\mathbf{z}=\mathbf{A}\psi+\mathbf{\Delta} \tag{20}\]  
where \(\mathbf{z}=[\delta\tilde{\varphi}_{K},\delta\tilde{\varphi}_{K+1},\ldots, \delta\tilde{\varphi}_{K+L-1}]^{\text{T}}\) is an \(L\)-dimensional independent observation sequence, \(\mathbf{A}\) is an \(L\)-dimensional all-one observation column vector, and \(\mathbf{\Delta}\) is the \(L\)-dimensional observation error vector. The corresponding error function of Equation (20) can be expressed as  
\[\mathbf{v}=\mathbf{A}\hat{\psi}-\mathbf{z} \tag{21}\]  
where \(\mathbf{v}\) denotes the observation residual vector. Thus, based on the robust estimation theory [20], the estimate can be obtained by the following criterion:  
\[\sum_{i=1}^{L}\rho(v_{i})=\text{min} \tag{22}\]  
where \(v_{i}\) denotes the \(i\)-th element of \(\mathbf{v}\), and \(\rho(\cdot)\) is a bounded convex function used to attenuate the effect of observation outliers. Then the iterative form of the solution can be expressed as  
\[ \begin{cases}\hat{\psi}^{(j+1)}=(\mathbf{A}^{\text{T}}\overline{\mathbf{P}}^{ (j)}\mathbf{A})^{-1}\mathbf{A}^{\text{T}}\overline{\mathbf{P}}^{(j)}\mathbf{ z}\\ \mathbf{v}^{(j+1)}=\mathbf{A}\hat{\psi}^{(j+1)}-\mathbf{z}\end{cases}  \tag{23}\]  
where \(\overline{\mathbf{P}}^{(j)}\) denotes the equivalent weight matrix of the observation vector at the \(j\)-th iteration. In the independent case, \(\overline{\mathbf{P}}\) is a diagonal matrix. The form of the equivalent weight matrix is determined by \(\rho(\cdot)\), and the IGG-III weight function is a good choice, given by [21]:  
\[\overline{P}_{i}=