Estimation Bounds of Beat Signal in the R-Mode Localization System

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 Cramér-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.

Prediction of Ground Wave Propagation Delay for MF R-Mode

“…The application of the phase-correction scheme compensates the systematic bias significantly (see also the histogram of the Gross Mohrdorf range error in Figure 9). The propagation path with an increased proportion of land after entering the bay causes a larger phase delay as well as a decrease in the signal strength, which directly affects the accuracy of phase estimation [36], explaining the larger standard deviation of the range error after 8:00 GMT. The observed improvement in range level affects the position error as well.…”

Phase Estimation of Single Tones Next to Modulated Signals in the Medium Frequency R-Mode System

“…In our previous publication [18], we had shown that we can assume the Cramér-Rao lower bound (CRB) for the estimation of the phase θn of the n-th subcarrier, in the absence of a modulated signal, to be…”

“…We had shown that the modulated signal affects the accuracy of the phase-estimation of the aiding carriers when the DFT is used as maximum likelihood estimator. We assume, that this is due to the fact that the assumption of white Gaussian noise is not valid any more [17], [18]. However, this estimator is of practical relevance, as it can be implemented in a highly computational effective way [13] to estimate multiple harmonics.…”