Appl. Sci. 2023, 13, 1872 
30f17 
The MF radio beacon system is based on the Frequency Division Multiple Access 
(FDMA) scheme. Thus, a channel of 500 Hz is allocated for each terrestrial beacon in 
the marine frequency band between 283.5 kHz and 315 kHz. These values are valid in 
European countries and might differ in other regions, where a different frequency allocation 
strategy has been selected. An important characteristic of the radio waves in the MF band 
is that they propagate as a ground-wave. This means that the signals are not limited to the 
ıine-of-sight (LOS) propagation, as it is for GNSS or VHF signals, but they propagate far 
beyond the horizon following the curvature of the earth. Therefore, the R-Mode positioning 
only provides 2D solutions and does not provide any height information. The transmitted 
signal includes a Minimum Shift Keying (MSK)-modulated signal, which represents the 
‘'egacy signal carrying the D(ifferential) GNSS information, plus two additional pilot signals, 
each referred to as a Continuous Wave (CW). Such sinusoidal signals are placed in the 
fourth zero crossing of the MSK spectrum to minimize interference on the legacy signal 
and, to be precise, they are spaced by 225 Hz on the left and right of the carrier frequency. 
Additionally, the signals are synchronized to the R-Mode System Time in order to have a 
zero-phase crossing to the full seconds, such that synchronization is achieved among all 
the transmitters. The R-Mode System Time is the reference time for all R-Mode system 
components. It can be aligned to the GPS time, as it is conducted by the GPS-stabilized 
rubidium clocks at the moment but also other options are possible, such as the time and 
frequency transfer using fiber optic cable. The research on cost-efficient synchronization 
strategies is ongoing [15]. 
The fundamental idea to obtain pseudorange measurements at the receiver side is 
to exploit phase estimates of the two CW and to track the evolution of these phases as 
the vessel moves. This technique is similar to the one used in the GNSS receivers, which 
exploit carrier phase observations to obtain high-accuracy positioning solutions [16]. The 
pseudorange p;,(f) can then be represented as follows 
pjk(t) = [Nie(£) + je (8) + BI + OEL Ajı (1) 
where j is the station index and k = 1,2 is an index which distinguishes between the two 
CW. A is the wavelength of the CW. N represents the integer number of the full wavelengths 
between the transmitter and the receiver, referred to as the ambiguity in the GNSS domain, 
and @; x represents the fractional part of the wavelength. Finally, % represents the error 
caused by the clock at the receiver and transmitter side j, while Pix includes the errors 
introduced by the different impairments encountered by the signal, as will be explained in 
Section 3. 
A Software-Defined Radio (SDR) receiver has been developed by the DLR, as described 
in [14]. The receiver setup is composed of an Ettus X310 connected to an E-field antenna 
hrough an amplifier and a filter. These components represent the front-end of the receiver, 
which provides as an output the discretized digital raw samples to a computer. A C++ 
and Python-based receiver implementation performs digital signal processing and phase 
and position estimation on the computer. As explained, the phase measurements are 
characterized by the presence of the ambiguity, which needs to be estimated. Due to the 
fact that the transmitters and receiver are not calibrated, the current implementation of 
che receiver solves the ambiguity by using a calibration process. The mechanism is based 
on the knowledge of the accurate location provided by the GNSS receiver at the time of 
calibration. A 30 s averaging time window allows one to calibrate the measurements and 
solve the ambiguities for all the available signals [14]. Afterwards, the tracking is in charge 
of computing the accumulated phase measurements. With the calibrated measurements, 
the receiver can estimate the latitude and longitude in the WGS-84 coordinate frame and 
the receiver time offset by using an iterative least squares algorithm. Due to the signal 
propagation, the transmitter-receiver distance is modelled by the Vincenty’s formula [17], 
which gives the distance between two points on the WGS-84 ellipsoid, assuming that the