# Noncoherent Channel Combining for GNSS Signal Tracking with an Adaptive Antenna Array
## 3 Proposed Method
### Noncoherent Channel Combining
Instead of \(\mathbf{b}=[1,0,\ldots,0]^{\mathrm{T}}\), similarly using \(\mathbf{c}_{1}=[1,0,\ldots,0]^{\mathrm{T}}\), \(\mathbf{c}_{2}=[0,1,0,\ldots,0]^{\mathrm{T}}\)\(\ldots\), \(\epsilon_{N}=[0,0,\ldots,0,1]^{\mathrm{T}}\) as the combining vector in Equation (15) yields the \(i\)th (\(i=1,\ldots,N\)) channel output as  
\[\overline{x}_{i}(t) =\epsilon_{i}^{\mathrm{H}}\overline{x}(t) \tag{18}\] \[=\epsilon_{i}^{\mathrm{H}}\mathbf{R}_{xx}^{-1}\mathbf{a}_{0}s(t)+ \epsilon_{i}^{\mathrm{H}}\overline{\mathbf{n}}(t)\] \[=\alpha_{i}\mathrm{e}^{\mathrm{i}\phi_{i}}s(t)+\overline{n}_{i} (t)\]  
where \(\alpha_{i}\) and \(\psi_{i}\) are the attenuation factor and phase bias, respectively, induced by the array for the \(i\)th channel. Note that these variables are different for each channel. The proposed channel combining method is implemented at the correlation level, that is, \(\overline{x}_{i}(t)\) in Equation (18) is correlated with the same local replica signals. Then, the corresponding prompt, early, and late correlator outputs are obtained as  
\[r_{\mathrm{E},i} =\alpha_{i}\sqrt{P}DR(\delta\tau+\Delta\tau)\operatorname{sinc} (\delta f\cdot T)\mathrm{e}^{\mathrm{j}(\pi\delta f\cdot T+\delta\varphi+\psi_ {i})}+n_{\mathrm{E},i} \tag{19}\] \[r_{\mathrm{P},i} =\alpha_{i}\sqrt{P}DR(\delta\tau)\operatorname{sinc}(\delta f \cdot T)\mathrm{e}^{\mathrm{j}(\pi\delta f\cdot T+\delta\varphi+\psi_{i})}+n_{ \mathrm{P},i}\] \[r_{\mathrm{L},i} =\alpha_{i}\sqrt{P}DR(\delta\tau-\Delta\tau)\operatorname{sinc} (\delta f\cdot T)\mathrm{e}^{\mathrm{j}(\pi\delta f\cdot T+\delta\varphi+\psi_ {i})}+n_{\mathrm{L},i}\]  
where \(\delta\tau\), \(\delta f\), and \(\delta\varphi\) are the residual errors of the time delay, Doppler frequency, and carrier phase, respectively, \(\Delta\tau=d_{\mathrm{s}}T_{\mathrm{c}}/2\), where \(d_{\mathrm{s}}\) is the early-minus-late correlator spacing and \(T_{\mathrm{c}}\) is the code chip duration, \(T\) is the coherent integration time, \(R(\cdot)\) is the normalized code auto-correlation function, \(\operatorname{sinc}(x)=\sin(\pi x)/\pi x\), and \(n_{\mathrm{E},i}\), \(n_{\mathrm{P},i}\), and \(n_{\mathrm{L},i}\) are the correlation noise terms. To achieve noncoherent channel combining, a combination approach is performed, and the correlator output after combination can be expressed as  
\[ \begin{cases}|\tilde{r}_{\mathrm{E}}|=\sum_{i=1}^{N}\beta_{i}|r_{\mathrm{E}, i}|\\ |\tilde{r}_{\mathrm{L}}|=\sum_{i=1}^{N}\beta_{i}|r_{\mathrm{L},i}|\end{cases}  \tag{20}\]  
where \(\tilde{r}_{\mathrm{E}}\) and \(\tilde{r}_{\mathrm{L}}\) are the noncoherently combined early and late correlator outputs, respectively, \(|\cdot|\) denotes the absolute operation, and \(\beta_{i}\) is the combination coefficient for the \(i\)th channel. Note that the phase terms of the correlator outputs are eliminated, so only the early and late correlators are combined, while the prompt correlators are not. The optimal combination coefficient has been studied in the joint reception of multiple signal components [17], which can also be applied in this work. The amplitude ratio combination method was found to be better than the equal weight combination method and the power ratio combination method [17]. Based on the amplitude ratio combination method, by assuming that the real and imaginary parts of the correlation noise terms in Equation (19) have unit variance, the combination coefficient can be calculated as  
\[\beta_{i}=\sqrt{2(C/N_{0})_{i}T} \tag{21}\]  
where \((C/N_{0})_{i}\) is the carrier-to-noise ratio (in linear scale) for the \(i\)th channel. With \(r_{\mathrm{P},i}\) in Equation (19), the narrowband-wideband power ratio method [18], which is widely used in GNSS receivers, can be used for \(C/N_{0}\) estimation. Then, the discriminators are used to extract the error signals from the correlator outputs. Typically, the noncoherent early-minus-late envelope code discriminator is used in the code loop. To maintain carrier tracking, similar to the PI algorithm, the reference channel is selected to track the carrier.