# In-Band Medium-Frequency R-Mode Signal Quality Estimation +
## 2 The MF R-Mode Signal  
The system shares the medium with a frequency division multiple access (FDMA) scheme in which a bandwidth of 500 Hz in the European area is allocated to each transmitter. Different geographical regions might have slightly different allocation strategies that are not considered in this paper. The transmitted MF R-Mode signal is composed of a minimum shift keying (MSK) component, which contains the D(differential)GPS information, and two aiding carriers indicated as \(CW_{1}\) and \(CW_{2}\) placed, respectively, at \(f_{MSK}-225\,\mathrm{Hz}\) and \(f_{MSK}+225\,\mathrm{Hz}\), where \(f_{MSK}\) is the center frequency of the channel and carrier frequency of the MSK signal. Figure 1 presents the power spectral density (PSD) of a simulated MF R-Mode signal for one channel with a center frequency of \(300\,\mathrm{kHz}\) (blue line). The different signal components are indicated, and the theoretical PSD of the MSK signal is also presented (red dashed line) to better distinguish them.  
The MF R-Mode receiver exploits the aiding carriers to perform ranging by computing their phase. By knowing that at the transmitter side at each full second the phase is zero,  
Figure 1: Simulated MF R-Mode signal PSD (blue curve). The red dashed line represents the theoretical spectrum of a MSK signal.  
the observed phase can be directly translated to a distance measurement. The tracking stage is used to compute the evolution of the phase over time to describe the receiver movement and clock drift. Due to the signals' wavelength of about 1 km and an approximate service coverage of 300 km, the phase observations are affected by ambiguities. Such ambiguity can be initially solved by performing a calibration procedure by using GNSS for few seconds [4] or by using an apriori knowledge of the location of the receiver. As an alternative, the phase obtained by mixing the of two aiding carriers, referred to as the beat signal phase, can be used to solve the ambiguity since the resulting wavelength is about 666 km. As discussed, the first step of the receiver is the computation of the CW phase. To do that, the discrete Fourier transform (DFT) or its fast implementation, also known as fast Fourier transform (FFT), is used due to the efficiency of the estimation process and low computational complexity. Additionally, the FFT provides the estimates for all the channels in view in one shot. To bound the performance of the phase estimation process, usually the Cramer-Rao Lower Bound (CRLB) is adopted, which describes the optimal achievable performance of an unbiased estimator [9]. Each estimator can be compared with such a bound, and if the estimator attains the bound the estimator is said to be efficient. The CRLB for the phase estimation of the CWs in the MF R-Mode signal is given in [10], where the author describes the assumptions and validity criteria in detail. Under additive white Gaussian noise (AWGN) and by assuming that the bit sequence of the MSK signal is uniformly distributed, the bound for the phase estimate is  
\[\sigma_{CRLB}{}^{2}(\phi_{\text{CW}})=\frac{2\sigma_{n}{}^{2}}{A_{\text{CW}}^ {2}L}=\frac{\sigma_{n}{}^{2}}{P_{\text{CW}}L}, \tag{1}\]  
where \(\sigma_{n}{}^{2}\) is the noise power, \(A_{\text{CW}}\) is the amplitude of the sinusoidal tone, and \(L\) is the number of samples. The formula can also be expressed by using the sinusoidal tone power since \(P_{\text{CW}}=(A_{\text{CW}}^{2})/2\). By using Equation (1), the optimal achievable performance can be predicted. The challenge now is to properly estimate the noise power and the amplitude of CW to finally obtain the prediction that can be used to monitor the received signal quality under optimal propagation condition.