# In-Band Medium-Frequency R-Mode Signal Quality Estimation +
## 4 Simulation  
To evaluate if Equation (10) can be an appropriate measure of the received signal quality, we perform Monte Carlo simulations. The following signal cases are considered and compared:Case 1: Single tone with AWGN
* Case 2: Single channel R-Mode signal (MSK + CWs) in AWGN  
It is important to specify that the simulated MSK bit sequence is generated such that bits are random and evenly distributed. This is also quite close to the real MSK bit distribution. For both cases, we assume the reference CW to be located at 300.225 kHz and the channel center frequency to be 300 kHz. Therefore, the spectrum looks like the one represented in Figure 1. We also assume that 50% of the power is allocated to the MSK, whereas 25% is allocated to each CW. We can fix the sampling rate to 1 MS/s, a common value that is used in our receiver, and fix \(L_{\mathrm{B}}=25\). If the number of samples for the FFT is \(L=1\) MS, the resolution of the FFT is 1 Hz and the bandwidth on which the noise density is calculated is approximately 50 Hz. This means that the noise (and interference of the MSK) is evaluated within the transmitter channel around the target CW. A total number of 1000 runs are simulated for each case and for each value of \(C/N_{0}\). Figure 2 presents the mean value (a) and root mean squared error (RMSE) (b) of the estimate \(C/N_{0}\) in dB-Hz for different values of the true \(C/N_{0}\) used in the simulation. It appears clear that for the case of a single tone in AWGN, the red squared line, the estimated mean value is matching the reference, which is the true value (blue line). For values below 7 dB-Hz, the curve starts to diverge due to the fact that noise power becomes too strong. It can be seen that, by increasing the value of \(N_{0}\), the estimated (\(C/N_{0}\)) tends to 0 as the amplitude of the CW becomes hidden in the noise. This behavior is similar for the R-Mode signal case represented by the green line. On the other side, for values of \(C/N_{0}>42\) dB-Hz the estimated mean value is diverging from the reference line and tends to a constant value. This is due to the fact the noise floor is estimated over a 50 Hz bandwidth by using the side bins of the DFT, which are affected by the presence of the MSK side-lobes, similar to a self interference effect. As the noise decreases, the presence of the MSK remains constant, hence the estimated \(C/N_{0}\) level will also stay constant and differ from the true value. Regarding the RMSE of the estimate, it can be seen that it increases as the \(C/N_{0}\) level decreases, as expected. Therefore, for \(C/N_{0}<42\) dB-Hz the results for the two cases are similar, whereas for \(C/N_{0}>42\) dB-Hz there is a concrete difference. While the RMSE approaches a constant value for the single tone case, the RMSE increases considerably for the R-Mode case due to the bias of the \(C/N_{0}\) estimation. Figure 3 presents the accuracy of the phase estimation in meters obtained in simulation for the two specific cases. In particular, the RMSE of the phase estimation is given for the  
Figure 2: Mean value of the estimated \(C/N_{0}\) for the two cases compared with the reference line (a) and its RMSE (b) in dB-Hz. single tone case (red) and for the R-Mode signal (green). Additionally, the CRLB for the single tone (blue) and the predicted accuracy (black) obtained by using the estimated values of \(C/N_{0}\) for the case 2 are also presented. Good agreement between the curves is visible for \(C/N_{0}\) between 7 and 42 dB-Hz. Below 7 dB-Hz, the signal is strongly compromised by the noise. The estimated RMSE approaches the theoretical limit of \(\lambda\sqrt{(1/12)}\) (represented by the horizontal gray dotted line) with the \(\lambda\) wavelength of the signal, deriving from the fact that the distribution of the observed phase becomes uniform at \([0,2\pi)\).