Asymptotically efficient GNSS trilateration

Abstract

Localization based on the reception of radio-frequency waveforms is a crucial problem in many civilian or military applications. It is also the main objective of all Global Navigation Satellite System (GNSS). Given delayed and Doppler shifted replicas of the satellites transmitted signals, the most widespread approach consists in a suboptimal two-step procedure. First, estimate the delays and Dopplers from each satellite independently, then estimate the user position and speed thanks to a Least Square (LS) minimization. More accurate and robust techniques, such as a direct Maximum Likelihood (ML) maximization, that exploit the links in between the different channels exist but suffer from an heavy computational burden that prevent their use in real time applications. Two-steps procedures with an appropriate Weighted LS (WLS) minimization are shown to be asymptotically equivalent to the ML procedure. In this paper, we develop a closed-form expression of this WLS asymptotically efficient solution. We show that this simple expression is the sum of two terms. The first one, depending on the pseudo-ranges is the widespread used WLS solution. The second one is a Doppler-aided corrective term that should be taken into account to improve the position estimation when the observation time increases.

Performance Limits of GNSS Code-Based Precise Positioning: GPS, Galileo & Meta-Signals

“…with Br the receiver bandwidth. The PSD of Galileo E5/E6 signals along with the GNSS meta-signal PSD defined by the AltBOC(50,10,5) and AltBOC (35,10,5) are illustrated in Figure 3 (left). Note from these results that the PSD of GNSS meta-signals slightly differ from the PSD of Galileo E5/E6.…”

“…For completeness, we computed the ACF of the meta-signals without intermodulation products, but the difference with the ACF shown in Figure 3 is marginal, with no impact on the delay estimation accuracy. ] because it uses a BPSK(5), and the E5b-I ACF width in [− T c 10 , T c 10 ] because it uses a BPSK (10).…”

“…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.…”

“…As for the Galileo E1 Open Service Signal [24], the Multiplexed BOC (MBOC) modulation is considered, namely a MBOC(6,1,1/11) [25] modulation was selected to transmit the GPS L1C signal. The MBOC(6,1,1/11) is defined in the PSD domain as a mix of a BOC(1,1) and BOC(6,1) modulations, where 10 11 th of the power is associated to the BOC(1,1) modulation and 1 11 th of the power is associated to the BOC(6,1). Note that the BOC modulation is generally denoted BOC(p, q), where p refers to the sub-carrier frequency f sc = p × 1.023 MHz (i.e., sc(t) = sign [sin (2π f sc t)] with sign the sign function) and q to the ranging code frequency f c = q • 1.023 MHz [26].…”

“…For the data component, the PRN sequence of 5115 chips lasts 1 ms, and for the pilot (combination with a secondary code of length 100) 100 ms. This signal is QPSK (5) The ACFs for the different modulations, from the GPS L1 C/A BPSK(1) to the Galileo E5 AltBOC (15,10), are shown in Figure 2.…”