# Noncoherent Channel Combining for GNSS Signal Tracking with an Adaptive Antenna Array
## 3 Proposed Method
### Comparison with Existing Methods  
From the previous subsection, it is shown that the proposed method can be implemented in a simple logic without any other auxiliary information or sensors. This subsection further compares it with several existing methods. For a fair comparison, in addition to the PI algorithm, two blind algorithms that require signal tracking are considered in this work. Figure 1: Schematic diagram of the proposed method. The first algorithm is called MVDR-like in this paper, which is an extension of the algorithms in [13; 14]. Specifically, each component of \(\overline{\mathbf{x}}(t)\) is individually tracked to estimate the projected satellite steering vector as  
\[\overline{\mathbf{a}}_{0}^{\prime}=[1,\mathbf{e}^{\mathrm{i}\Delta\varphi_{21}}, \ldots,\mathbf{e}^{\mathrm{i}\Delta\varphi_{\mathrm{N1}}}]^{\mathrm{T}} \tag{24}\]  
where \(\Delta\varphi_{n1}=\varphi_{n}-\varphi_{1}\) is the integrated carrier phase difference between the \(n\)th channel and the first channel. The corresponding output signal is obtained as  
\[y_{\text{MVDR-like}}(t)=(\overline{\mathbf{a}}_{0}^{\prime})^{\mathrm{H}} \overline{\mathbf{x}}(t) \tag{25}\]  
The second algorithm is the well-known MMSE algorithm, which is a typical algorithm that uses the reference signal waveform for beamforming. The MMSE weight vector can be formulated as [15]  
\[\mathbf{w}_{\text{MMSE}}=\mathbf{R}_{xx}^{-1}\mathbf{r}_{xd} \tag{26}\]  
where \(\mathbf{r}_{xd}=E[\mathbf{x}(t)d^{*}(t)]\) denotes the cross-correlation vector between the received signal vector \(\mathbf{x}(t)\) and the reference signal \(d(t)\), and \((\cdot)^{*}\) denotes the conjugate operation. Using Equations (26) and (14) in Equation (3) yields  
\[y_{\text{MMSE}}(t)=\mathbf{r}_{xd}^{\mathrm{H}}\overline{\mathbf{x}}(t) \tag{27}\]  
Therefore, the following conclusions can be drawn from Equations (25) and (27):  
1. The MVDR-like and MMSE algorithms can also be expressed as two-stage processing, and the corresponding second-stage combining vectors are \(\mathbf{c}_{\text{MVDR-like}}=\overline{\mathbf{a}}_{0}^{\prime}\) and \(\mathbf{c}_{\text{MMSE}}=\mathbf{r}_{xd}\), by comparing Equations (25) and (27) with Equation (15). 2. For the MVDR-like algorithm, in the ideal case, there are no tracking errors in the integrated carrier phase for each channel, and \(\overline{\mathbf{a}}_{0}^{\prime}=\overline{\mathbf{a}}_{0}\). Therefore, the array gain for the satellite signal is maximized, because in this case, the two vectors \(\mathbf{c}_{\text{MVDR-like}}\) and \(\overline{\mathbf{a}}_{0}\) are parallel, as explained in Equation (17). 3. For the MMSE algorithm, in the ideal case, the reference signal contains no tracking errors, and \(d(t)=s(t)\), therefore, using Equation (1) in the formula of \(r_{xd}\) below Equation (26) obtains \(\mathbf{r}_{xd}=E[s(t)s^{*}(t)]\mathbf{a}_{0}\). This indicates that the MMSE algorithm is equivalent to the MVDR algorithm in the ideal case. As a matter of fact, the practical reference signal for the MMSE algorithm is recovered from the receiver tracking loop, so it is denoted as \(d^{\prime}(t)\) to distinguish. It should be pointed out that the tracking errors are unavoidably involved in \(\overline{\mathbf{a}}_{0}^{\prime}\) and \(d^{\prime}(t)\). Therefore, the performance of the MVDR-like and MMSE algorithms is degraded in two aspects. (1) The tracking errors in \(\overline{\mathbf{a}}_{0}^{\prime}\) and \(d^{\prime}(t)\) reduce the their correlation with \(\overline{\mathbf{a}}_{0}\), thus degrading the algorithm performance, especially under weak signal conditions. (2) The carrier tracking errors will be introduced into the output signal by array weighting, which reduces the carrier tracking precision of these algorithms. Compared to these algorithms, the proposed method is more robust, which can improve code tracking without affecting carrier tracking. Then, the computational complexity associated to different algorithms is investigated.