Appl. Sci. 2023, 13, 1872 
140f 17 
R-Mode Baltic project through the European Regional Development Fund within the Interreg Baltic 
Sea Region Programme. 
Conflicts of Interest: The authors declare no conflict of interest. 
Abbreviations 
The following abbreviations are used in this manuscript: 
Atmospheric and Ground Delay Factor 
Alternative Position, Navigation and Timing 
German Federal Maritime and Hydrographic Agency 
Continuous Wave 
Differential Global Navigation Satellite System 
German Aerospace Center 
Dilution Of Precision 
Frequency Division Multiple Access 
Fast Fourier transform 
Global Navigation Satellite system 
Global Positioning System 
Ground-wave-to-Sky-wave Amplitude Ratio 
Horizontal Dilution Of Precision 
International Association of Marine Aids to Navigation and Lighthouse Authorities 
International Telecommunication Unit 
Line-Of-Sight 
Medium Frequency 
Minimum Shift Keying 
Open Service Navigation Message Authentication 
Position, Navigation and Timing 
Ranging Mode 
Real-Time Kinematic 
Software-Defined Radio 
Coordinated Universal Time 
Very High Frequency Data Exchange System 
Appendix A 
The section shows the derivation of the formulas used in the paper. As is well-known, 
the sum of two sinusoidal signals with the circular frequency w = 27f is also a sinusoidal 
signal with the same frequency. By using the trigonometric addition formulas, we define 
the two sinusoids, as follows 
sı(t) = Ay cos(wt +81) = A1[cos(wt) cos(81) — sin(wt) sin(91)i 
(A1) 
ınd 
s2(t) = A2cos(wt + 02) = A2[cos(wt) cos(92) — sin(wt) sin(02)]. (A2) 
Then the sum is given as 
53(f) = S1 +52 = cos(wft) [A cos(01) + Az cos(02)] — sin(wt) [Ay sin(91) + Azsin(02)]. (A3) 
Our goal is to find amplitude Az and phase ®; of s3. We define, as follows 
A3 cos(03) = A1cos(01) + A, cos(0;) 
(A4) 
and 
Az sin(83) = A1sin(01) + A» sin(8»). 
a)