Abstract
The R-Mode system is a terrestrial navigation system currently under development, which exploits existing means of medium frequency radio transmission. The positioning and timing performance depends on the estimation of the signals’ phase offset, from which the ranging information is derived. For an analogous problem such as the single-tone phase estimation, the Cramer-Rao bound (CRB) describes the minimal achievable performance in the mean squared error sense. For R-Mode, the problem involves the estimation of the phase offset for a beat signal, which can be described as the difference of phase estimation for the two aiding carriers next to the signal. This estimates are not statistically independent for finite observation, as we show in this paper. The effect becomes stronger for short observation times, which are important for a near real time application. In this contribution, we are interested in phase offset estimation for the signal models relevant to R-Mode: a beat signal and a beat signal combined with an MSK signal. A closed-form lower CRB is proposed for the aforementioned signal models phase estimation, as well as a generalization of the bound for the phase-difference estimation. Based on this derivation, optimized bit sequences are shown to improve performance of the estimates. The validity of the proposal is verified based on a simulation setup. Measurements acquired during a measurement campaign serve to further justify the usefulness of the bound. Some possible applications of such a bound are R-Mode coverage prediction and the associated phase estimators’ performance.
Highlights
Global navigation satellite systems (GNSS) are the backbone for today’s position, navigation and timing (PNT) information
We show that the estimates of the tones for the described R-Mode signal are not statistically independent for finite observation, so they cannot be considered as separate sinusoids as in [18]
Based on our derivation of the bound, we show optimized bit patterns that nearly satisfy statistical independence of R-Mode signal components, which is a prerequisite for an optimized navigation message, that minimizes the variance of CW phase estimation
Summary
Global navigation satellite systems (GNSS) are the backbone for today’s position, navigation and timing (PNT) information. The resulting signal has a frequency of 450 Hz and the phase offset at the full second is derived by a simple subtraction of the estimates of both single tones. We are interested in deriving a deterministic CramérRao bound (CRB) for the amplitude and phase of a beat signal. This information is a precondition for the estimation if resolving the ambiguities of the single-tones in a certain region is possible and improves the choice of estimation parameter in the receiver. Based on our derivation of the bound, we show optimized bit patterns that nearly satisfy statistical independence of R-Mode signal components, which is a prerequisite for an optimized navigation message, that minimizes the variance of CW phase estimation. In the asymptotic regimen case for which the lower bound is attained by an estimator, the estimation of each signal part is statistically independent
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