# Performance Limits of GNSS Code-based Precise Positioning: GPS, Galileo & Meta-Signals
## 1 Introduction  
Synchronization is a key first stage in many applications, e.g., radar, sonar, communications, or navigation, to name a few [1; 2; 3; 4; 5]. This typically implies the estimation of both propagation delay and Doppler shift (or higher order Doppler terms in high dynamics scenarios), in order to identify, localize, and/or track radiating sources. Several approaches and estimators are available in the literature to perform such synchronization, for instance, standard Global Navigation Satellite Systems (GNSS) receiver architectures rely on a scalar (i.e., different satellite signals are processed with independent channels) acquisition and tracking approach. In this case, the former provides a coarse point estimate of the synchronization parameters (identify/acquire visible satellites), and the latter keeps track of their time-varying evolution [6]. These two stages, i.e., acquisition and tracking, can be seen as particular instances of the Maximum Likelihood (ML) solution. The optimal positioning solution is the so-called Direct Position Estimation (DPE) [7; 8], that is, the direct MLposition estimation from the sum of signals to all visible satellites. Indeed, the set of time-delays and Doppler shifts to each individual satellite are related to the same receiver position; thus, not exploiting them together is suboptimal because we are not taking into account the geometry constraint. However, DPE is not useful in practice because it requires solving a high-dimensional optimization problem. Therefore, thanks to the quasi-orthogonality of GNSS signals, the standard solution is a two-step (synchronization + multilateration) positioning approach: (i) first, individual time-delay estimates to each visible satellite are used to build a set of independent pseudoranges; and (ii) then these pseudoranges are used used to solve a multilateration problem to estimate the user position, typically via a weighted least-squares (WLS) [9]. 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 is asymptotically optimal [10]. Then, the ultimate achievable standard point positioning (SPP) GNSS receiver performance is directly driven by the achievable time-delay estimation performance. By ultimate performance, we refer to the asymptotic region of the time-delay ML estimator. In that perspective, we do not assume external errors such as ionospheric/tropospheric delays, orbital or satellite clock errors, or environment-specific effects such as multipath [9]. For a comprehensive analysis on the specific impact of such external errors and different types of corrections, refer to [11]. The asymptotic region is characterized considering only the thermal noise and a high signal-to-noise (SNR) ratio to ensure the efficient behavior of the ML estimator. Moreover, these external errors are the same whatever the processing, so that we do not need to consider them with an aim of comparing SPP and Precise Point Positioning (PPP) schemes. Notice that the standard way to obtain precise positioning solutions is to exploit the signal phase information. Indeed, this measurement is linked to the wavelength which is much smaller than the baseband signal resolution (i.e., for a legacy Global Positioning System (GPS) L1 C/A signal, the wavelength is approximately 19 cm while the baseband signal resolution is 300 m). Unfortunately, exploiting this phase information implies solving a much more complicated problem, mainly because the carrier phase measurement is ambiguous (i.e., unknown number of cycles inside the baseband signal resolution), rather than such ambiguity resolution being the bottleneck [9] (Chapters 21 and 23). Two main approaches are available in the literature: (i) differential techniques such as Real-Time Kinematics (RTK) [9] (Chapter 26); and (ii) PPP techniques [9] (Chapter 25).