## 3 Proposed Technique
### Phase Bias Detection  
This subsection first gives the reason for the phase bias detection, then the detection method is presented. The dynamic behavior of the NCO can be characterized by a state vector \(\mathbf{X}_{k}=[\varphi_{k},\omega_{k},\dot{\omega}_{k}]^{\text{T}}\), with the corresponding parameters being initial carrier phase \(\varphi_{k}\), angular Doppler frequency \(\omega_{k}\), and angular Doppler frequency rate of change \(\dot{\omega}_{k}\) at the \(k\)-th time epoch, and \([\cdot]^{\text{T}}\) denotes the transpose operation. The NCO can be tuned by the prediction model, and the locally predicted NCO is obtained as  
\[\hat{\mathbf{X}}_{k}^{-}=\mathbf{F}\hat{\mathbf{X}}_{k-1}+\mathbf{B}\mathbf{u}_ {k-1} \tag{11}\]  
where \(\mathbf{B}=[T,0,0]^{\text{T}}\), \(\mathbf{u}_{k}=f_{\text{IF}}\), and \(\mathbf{F}\) has the form  
\[\mathbf{F}= \begin{bmatrix}1&T&T^{2}/2\\ 0&1&T\\ 0&0&1\end{bmatrix}  \tag{12}\]  
The discriminator output in Equation (8) is used to adjust the NCO prediction as  
\[\hat{\mathbf{X}}_{k}=\hat{\mathbf{X}}_{k}^{-}+\mathbf{K}_{k}\delta\tilde{ \varphi}_{k} \tag{13}\]  
where \(\mathbf{K}_{k}\) denotes the loop gain, which can be determined by the conventional tracking loop design or the Kalman filter approach [18]. In this work, the latter is used to obtain reasonable tracking performance. When the discriminator output is contaminated by the phase bias, as seen in Equation (8), it will therefore misadjust the NCO through Equation (13). If the induced phase bias can be detected, the negative effect can be cut off immediately, so two indicators for phase bias detection are designed below. Monitoring the change of interference conditions is an effective method for phase bias detection. Therefore, the interference cancellation ratio (ICR) [19] can be adopted, which is approximately calculated as  
\[\text{ICR}=\frac{P_{\text{in}}}{P_{\text{out}}} \tag{14}\]  
where \(P_{\text{in}}\) and \(P_{\text{out}}\) are the array input and output power, respectively. It should be pointed out that the interference is assumed to be stable and far enough away from the target receiver. That is, it is assumed that the jamming power does not vary over the receiver operating time. The ICR can then be used to monitor changes as the jammer appears or disappears. As a complement, the phase of weight (POW) can also reflect the change in array-induced phase bias, which is defined by the array weight as  
\[\text{POW}=\angle\left[\sum_{n=1}^{N}w_{n}^{*}\right] \tag{15}\]  
where \(\angle\) denotes the phase of a complex scalar, \(w_{n}\) is the \(n\)-th element of the array weight, and \((\cdot)^{*}\) denotes the complex conjugate. The POW actually represents a specific phase bias when the GNSS signal arrives perpendicular to the array plane, since, in this case, the satellite steering vector is an all-one column. The POW can then be used to monitor changes as the interference direction varies. Note that both the ICR and the POW are obtained from the adaptive array, which is not affected by the receiver itself. In order to locate the time when the phase bias appears, the first-order differential method can be easily applied to these two indicators to obtain  
\[d_{\text{ICR}}=|\text{ICR}(k)-\text{ICR}(k-1)|>T_{\text{I}} \tag{16}\]  
\[d_{\text{POW}}=|\text{POW}(k)-\text{POW}(k-1)|>T_{\text{P}} \tag{17}\]  
where \(d_{\text{ICR}}\) and \(d_{\text{POW}}\) denote the detection statistics, with \(T_{\text{I}}\) and \(T_{\text{P}}\) being the corresponding predefined thresholds, which are empirically set to 3 dB and 8 degrees, respectively. \(k\) and \(k-1\) are the current and previous time epochs, respectively. Further discussions about the detection proceed as follows. The detection thresholds are chosen based on experience because it is not easy to determine the specific value based on the detection probability for a given false alarm probability.