# Performance Assessment of the Medium Frequency R-Mode Baltic Testbed at Sea near Rostock
## 3 Mf Signal Impairments
### Sky-Wave Self-Interference  
MF radio waves can travel either as ground-waves, by following the earth's surface, or as a sky-wave reflection between the earth and the ionosphere [18]. While the ground-wave is characterized by the propagation phenomena, which attenuate and delay the signal, as described in Section 3.1, the sky-wave can be observed as a multipath effect which generates self-interference on the receiver side, degrading the ranging accuracy. This effect depends on several factors, such as the frequency of the signal itself, the ionospheric reflection height, the angle of the incidence, the geographical region of interest and solar activity [13]. Radio waves in the 300 kHz frequency range are significantly absorbed by the D-layer of the ionosphere during the daytime. Therefore, the sky-wave interference effect is negligible, since its field strength is far smaller than the ground-wave in the service area of the radio beacon. In contrast, at night, the D-layer tends to disappear. The signal is therefore no longer attenuated and hits the upper ionospheric E-layer, which is capable of reflecting signals below 10 MHz. In a simplified scenario, we can assume a one-ray reflection model, where the sky-wave sum-up to the ground-wave signal at the receiver side. By neglecting the influence of the noise, we can define the real value ground-wave signal, as follows  
\[s_{\text{GW}}=A_{\text{GW}}(t)\cos(2\pi ft+\theta_{\text{GW}}) \tag{2}\]  
where \(A_{\text{GW}}\) is the signal amplitude, \(f\) is the frequency of a particular CW and \(\theta_{\text{GW}}\) is the phase shift due to the propagation path. Similarly, the sky-wave can be given as follows  
\[s_{\text{SW}}=A_{\text{SW}}(t)\cos(2\pi ft+\theta_{\text{SW}}) \tag{3}\]  
where now \(A_{\text{SW}}\) represents the sky-wave amplitude, and \(\theta_{\text{SW}}\) is the phase shift due to the sky-wave propagation path, which clearly differs from the ground-wave one. The sum of the two signals can be expressed as follows  
\[s_{\text{SUM}}=s_{\text{GW}}+s_{\text{SW}}=A_{\text{SUM}}(t)\cos(2\pi ft+ \theta_{\text{SUM}}) \tag{4}\]  
where  
\[A_{\text{SUM}}=\sqrt{A_{\text{GW}}^{2}+A_{\text{SW}}^{2}+2A_{\text{GW}}A_{ \text{SW}}\cos(\theta_{\text{GW}}-\theta_{\text{SW}})} \tag{5}\]  
and  
\[\tan(\theta_{\text{SUM}})=\frac{A_{\text{GW}}\sin(\theta_{\text{GW}})+A_{ \text{SW}}\sin(\theta_{\text{SW}})}{A_{\text{GW}}\cos(\theta_{\text{GW}})+A_{ \text{SW}}\cos(\theta_{\text{SW}})}. \tag{6}\]  
The derivation of Equations (4)-(6) is given in the Appendix A. We attempt to explain the impact of a sky-wave on the phase error and amplitude in a simulated scenario. We assume \(A_{\text{GW}}=1\), \(\theta_{\text{GW}}=0\) for simplicity and we define the ground-wave to the sky-wave amplitude ratio (GSAR) as follows  
\[\text{GSAR}=\frac{A_{\text{GW}}}{A_{\text{SW}}}. \tag{7}\]  
With these assumptions, Equation (6) becomes  
Figure 2: Größ Mohrdorf AGDF calculated for the southern Baltic Sea in radians. \[\theta_{\text{SUM}}=\tan^{-1}\left(\frac{\sin(\theta_{\text{SW}})}{\text{GSAR}+ \cos(\theta_{\text{SW}})}\right)=\theta_{err} \tag{8}\]  
and \(\theta_{\text{SUM}}\) directly represents the phase error \(\theta_{err}\). We now demonstrate the amplitude and phase error obtained under the influence of sky-wave interference for different GSAR and \(\theta_{\text{GW}}\). 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 that, 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.