# Positioning Performance Limits of GNSS Meta-Signals and HO-BOC Signals
## 3 GNSS Receiver Signal Processing  
### GNSS ML Baseband Signal Processing and Delay/Phase CRB  
The final goal of a GNSS receiver is to obtain position, velocity, and time (PVT) estimates. Therefore, from am ML perspective and taking into account that the received signal is the superposition of a set of signals related to the same receiver position, the optimal solution is given by the so-called direct position estimation (DPE) [32; 33]. The main disadvantage is that DPE implies solving a high-dimensional minimization problem, which turns into a high computational complexity. Even if DPE approaches are known to provide better position estimates under certain conditions, it has been recently shown that the traditional two-step approach (synchronization + multilateration) is asymptotically optimal [34]. Therefore, in this contribution, we are only interested in the latter, which implies the individual processing of satellite signals thanks to the quasi orthogonality of PRN codes. The goal of the first stage of the receiver is to construct a set of observables for each satellite in view. In our case, this set includes code and phase observables, which are directly linked to the time-delay and phase MLE. Considering the signal model (6), the time-delay and phase MLE are defined as [24]:  
\[\widehat{\tau}=\arg\max_{\tau}\left\{\left|\left(\mathbf{c}\left( \tau\right)^{H}\mathbf{c}\left(\tau\right)\right)^{-1}\mathbf{c}\left(\tau \right)^{H}\mathbf{x}\right|^{2}\right\}, \tag{9}\] \[\widehat{\varphi}\left(\widehat{\tau}\right)=\arg\left\{\left( \mathbf{c}\left(\widehat{\tau}\right)^{H}\mathbf{c}\left(\widehat{\tau} \right)\right)^{-1}\mathbf{c}\left(\widehat{\tau}\right)^{H}\mathbf{x}\right\}, \tag{10}\]  
Figure 4: ACF for different high-order (HO)-BOC signals (i.e., the vertical axis represents the normalized autocorrelation). For comparison the Galileo E5 ACF is also shown. where \(\arg\) is the argument operator, \(\max\) is the maximum operator, \((\cdot)^{H}\) is the Hermitian operator, and \(|\cdot|\) is the Euclidean norm. Then, the phase is estimated as the argument of the cross-ambiguity function evaluated at the time-delay MLE. The study of these MLEs is useful to determine the value of the SNR at the output of the matched filter, which allows reaching the CRB, because such estimators are known to be asymptotically efficient (e.g., in the high SNR regime) for the conditional signal model of interest [35; 36]. It is also fundamental to have the corresponding CRB, which for the time-delay and phase estimation problem of interest was recently derived in a compact closed-form for a generic band-limited signal in [10]. Such a CRB is very convenient because it only depends on the signal sample, and it is summarized in the sequel for completeness. Considering the joint time-delay and phase \(\mathbf{\theta}^{\top}=(\varphi,\tau)\) estimation resorting to Model (6), the CRB is given by [37]:  
\[\mathrm{CRB}_{\tau|\mathbf{\varepsilon}} = \frac{1}{2\mathrm{SNR}_{\mathrm{out}}}\frac{1}{F_{s}^{2}\left( \frac{\xi^{H}\mathbf{v}_{\xi}}{\mathbf{c}^{H}\mathbf{c}}-\mathrm{Im}\left\{ \frac{\xi^{H}\mathbf{A}_{\xi}}{\mathbf{c}^{H}\mathbf{c}}\right\}^{2}\right)} \underset{\text{real signal}}{=}\frac{1}{2\mathrm{SNR}_{\mathrm{out}}}\frac{1} {F_{s}^{2}\frac{\epsilon^{H}\mathbf{v}_{\xi}}{\mathbf{c}^{H}\mathbf{c}}}, \tag{11}\] \[\mathrm{CRB}_{\varphi|\mathbf{\varepsilon}} = \frac{1}{2\mathrm{SNR}_{\mathrm{out}}}\left(1\;+\;\frac{\mathrm{Im }\left\{\frac{\xi^{H}\mathbf{A}_{\xi}}{\mathbf{c}^{H}\mathbf{c}}\right\}^{2}}{ \frac{\epsilon^{H}\mathbf{v}_{\xi}}{\mathbf{c}^{H}\mathbf{c}}-\mathrm{Im} \left\{\frac{\xi^{H}\mathbf{A}_{\xi}}{\mathbf{c}^{H}\mathbf{c}}\right\}^{2}} \right)\underset{\text{real signal}}{=}\frac{1}{2\mathrm{SNR}_{\mathrm{out}}}, \tag{12}\]  
where \(\mathrm{Im}\) represents the imaginary operator, \(\mathrm{SNR}_{\mathrm{out}}=\frac{|\mathbf{a}|^{2}\mathbb{E}}{\left(\sigma_{n} ^{2}/F_{s}\right)}=\frac{|\mathbf{a}|^{2}}{\sigma_{n}^{2}}\mathbf{c}^{H} \mathbf{c}\), and \(\mathbb{E}\) the energy of the signal.