Appl. Sci. 2023, 13, 1872 
60f17 
—1 sin(0sw) 
sum = tar (= A) — Ber 
and sym directly represents the phase error 0e,,. 
We now demonstrate the amplitude and phase error obtained under the influence 
of sky-wave interference for different GSAR and 9cw. Figure 3 presents the amplitude 
of the sum signal on the left (Figure 3a) and the phase error with respect to the ground 
wave on the right side (Figure 3b) for ground-wave-dominated (GSAR = 6 dB) to sky-wave- 
dominated (GSAR = —6 dB) conditions. Starting from the amplitude plot, we can observe 
‘hat, as the GSAR decreases, the amplitude estimate can be greater or smaller than the true 
one. In particular, this depends on the phase difference between the ground-wave and the 
sky-wave and on the GSAR, which influences the resulting interference. When the phase 
difference is zero, the interference is constructive, which means that the signal strength, or 
amplitude, increases, and the reach is the maximum. In contrast, when the signals are out 
of the phase (— 7 or 7), the interference is destructive and the signal strength decreases. 
ta 
27 
-_ 
% 
GSAR 
- -6 dB 
-3 dB 
„.... O0dB 
—— 3dB 
..: 6dB 
; GSAR 
4007 — -6 dB 
-3 dB 
z 2001 0 dB 
; —-:-— 3dB 
=... 6dB 
0) 
Sb 
= 
£ _200° 
zu u =. 
/ Le „un“ 
3 eb 
- mn 
x 
r 
u. 
L 
400 
N 
m‘ 
0 ı 
Bsw [rad] 
(a) 
2 
-L 0 
Bew [rad] 
(b) 
L 
2 
Figure 3. Amplitude (a) and phase error (b) for different GSAR and 60sw. 
Nevertheless, for positioning purposes, the estimated phase or phase error respective 
to the true value is more important than the amplitude itself. By looking at the phase 
error represented in Figure 3b, we can observe that with the decrease in GSAR, the phase 
error increases, in the absolute value sense. If GSAR < 1, the error always increases 
monotonically with the increase in the sky-wave phase. The maximum absolute value error 
is 7 or half the wavelength. In such a case, the sky-wave is stronger than the ground-wave, 
which means that the receiver will finally track the sky-wave, severely corrupting the range 
estimation. This is clearly the worst possible situation. For GSAR > 1, we can observe 
that the phase error has a maximum, in the absolute value sense, between 0 and 7. This 
maximum depends on the value of the GSAR and is given by the following formula 
Ber = tan! ==) 
VGSAR? — 1 
[he reader can find the derivation of Equation (9) in the Appendix A. 
The mitigation of the sky-wave interference poses a real challenge to the usage of the 
MF R-Mode system, and research in this field is ongoing. One of the possible solutions 
would be to redesign the signal waveform modulation in order to have a broader signal 
bandwidth, which would allow the receiver to distinguish between the ground-wave and 
the reflected wave at the receiver side, as suggested in [26]. An alternative approach is to 
design specific antenna patterns in order to attenuate signals arriving with high elevation