# Noncoherent Channel Combining for GNSS Signal Tracking with an Adaptive Antenna Array
## 3 Proposed Method
### Noncoherent Channel Combining  
This subsection first unifies the conventional anti-jamming methods using the concept of channel combining. Then, the noncoherent channel combining is proposed. Referring to Equation (3), and using Equations (5) and (7), the conventional anti-jamming processing methods can be written in the unified form as  
\[y(t)=\mathbf{c}^{\mathrm{H}}\overline{\mathbf{x}}(t) \tag{15}\]  
where \(\mathbf{c}\) is a combining vector that combines different channels of \(\overline{\mathbf{x}}(t)\), and it varies for different algorithms. For ease of expression, the real scalars in the denominator of the weight vectors in Equations (5) and (7) are ignored hereafter. Thus, different combining vectors can be obtained as \(\mathbf{c}_{\mathrm{MVDR}}=\mathbf{a}_{0}\), and \(\mathbf{c}_{\mathrm{PI}}=\mathbf{b}\). Based on Equations (14) and (15), the conventional anti-jamming methods can be regarded as two-stage processing. The common first-stage processing is interference suppression described in Equation (14). Different algorithms differ in the second-stage combining processing, as shown in Equation (15). To compare different algorithms in the second stage, using Equations (1), (13), and (14) in Equation (15) yields  
\[ \begin{split} y(t)&=\mathbf{c}^{\mathrm{H}}\mathbf{ R}_{xx}^{-1}\mathbf{a}_{0}s(t)+\mathbf{c}^{\mathrm{H}}\mathbf{R}_{xx}^{-1} \mathbf{n}(t)\\ &=\mathbf{c}^{\mathrm{H}}\overline{\mathbf{a}}_{0}s(t)+\mathbf{c} ^{\mathrm{H}}\overline{\mathbf{n}}(t)\\ &=\frac{1}{\sigma_{\mathrm{n}}^{2}}\mathbf{c}^{\mathrm{H}} \mathbf{\tilde{a}}_{0}s(t)+\frac{1}{\sigma_{\mathrm{n}}^{2}}\mathbf{c}^{ \mathrm{H}}\mathbf{\tilde{n}}(t)\end{split}  \tag{16}\]  
where \(\mathbf{\tilde{a}}_{0}=\mathbf{U}_{\mathrm{I}}^{\perp}\mathbf{a}_{0}\), and \(\mathbf{\tilde{n}}(t)=\mathbf{U}_{\mathrm{I}}^{\perp}\mathbf{n}(t)\). It is known that each element of \(\mathbf{\tilde{n}}(t)\) still has the variance of \(\sigma_{\mathrm{n}}^{2}\)[16]. In addition, \(\|\mathbf{\tilde{a}}_{0}\|=\|\mathbf{a}_{0}\|=\sqrt{N}\) since \(\mathbf{U}_{\mathrm{I}}\) is formed by orthonormal vectors, where \(\|\cdot\|\) denotes the \(\ell_{2}\)-norm of a vector. The ratio of the output SNR to the input SNR can be used to determine the gain (in logarithmic scale) provided by the array in the satellite direction, which is defined as  
\[ \begin{split} G&=10\lg\biggl{(}\frac{|\mathbf{c} ^{\mathrm{H}}\mathbf{\tilde{a}}_{0}|^{2}}{\|\mathbf{c}\|^{2}}\biggr{)}=10\lg \Bigl{(}\|\mathbf{\tilde{a}}_{0}\|^{2}|\cos\beta|^{2}\Bigr{)}\\ &=10\lg N+20\lg|\cos\beta|\end{split}  \tag{17}\]  
where \(\beta\) is the angle between vectors \(\mathbf{c}\) and \(\mathbf{\tilde{a}}_{0}\) (or \(\mathbf{\overline{a}}_{0}\)). If \(G<0\), the satellite signal will be attenuated, while if \(G>0\), the satellite signal can be enhanced. Specifically, if the two vectors are parallel, then the gain can reach its maximum value \(10\lg N\). Considering different combining vectors, the following results can be obtained:  
1. For the PI algorithm, \(\mathbf{c}_{\mathrm{PI}}=\mathbf{b}\) means that it selects the reference channel in \(\overline{\mathbf{x}}(t)\) to output. That is, it is performed without using channel combining. It usually causes signal attenuation since it discards the useful signal power in other channels. 2. For the MVDR algorithm, \(\mathbf{c}_{\mathrm{MVDR}}=\mathbf{a}_{0}\), thus \(\mathbf{c}_{\mathrm{MVDR}}^{\mathrm{H}}\mathbf{R}_{xx}^{-1}\mathbf{a}_{0}\) is real, which means \(\mathbf{c}_{\mathrm{MVDR}}\) and \(\mathbf{R}_{xx}^{-1}\mathbf{a}_{0}\) are linearly correlated. Therefore, the satellite signals in different channels are coherently accumulated, which is called coherent channel combining. In order to appropriately utilize the useful signal power from different channels to achieve better signal tracking with a simple implementation, unlike coherent channel combining (which requires \(\mathbf{a}_{0}\)), the noncoherent channel combining method is proposed below.