# In-Band Medium-Frequency R-Mode Signal Quality Estimation +
## 3 Definition of the Signal Quality Indicator  
In principle, to define a signal quality indicator we would need to estimate the amplitude or power of the CWs and noise floor level. Such information could be retrieved from the DFT directly, reducing the complexity of the receiver due to the fact that the DFT is already used to perform the phase estimation. Let us assume we have \(L\) samples of the real received signal \(x\) obtained with sampling frequency \(f_{\text{s}}\). The DFT returns \(L\) complex numbers \(Z_{k}\)  
\[Z_{k}=\sum_{l=0}^{L-1}x_{n}e^{-\frac{2\pi lk}{L}}\qquad k=0,\cdots,L-1. \tag{2}\]  
It is known that the DFT of a real signal has Hermitian symmetry [11]; hence, only the positive frequency component can be considered, which is for \(k=0,...L/2-1\) for even \(L\). The complex number \(Z_{k}\) represents the contribution of the signal with frequency \(kf_{s}/L\), and by selecting \(k\) such that \((kf_{s})/L=f_{\text{CW}}\) we can compute amplitude \(A_{\text{CW}}\) and phase \(\phi_{\text{CW}}\) of the aiding carrier as  
\[A_{\text{CW}}=\frac{|Z_{\text{CW}}|}{L}, \tag{3}\]  
\[\phi_{\text{CW}}=\text{atan}\{\text{imag}(Z_{\text{CW}}),\text{real}(Z_{\text {CW}})\}. \tag{4}\]From the DFT, one can compute the PSD as follows:  
\[\mathbf{P}=\frac{1}{Lf_{S}}\mathbf{Z}\mathbf{Z}^{\star}, \tag{5}\]  
where \(\mathbf{Z}^{\star}\)is the complex conjugate of \(\mathbf{Z}\). In case of AWGN, the noise power is given by [11]  
\[P_{n}=\sigma_{n}^{2}=\frac{N_{0}}{2}f_{s}, \tag{6}\]  
where \(N_{0}/2\) is the noise power density, which can be estimated by using the PSD. Therefore, we can define the carrier to noise density (\(C/N_{0}\))  
\[C/N_{0}=\frac{P_{CW}}{N_{0}}=\frac{A_{CW}^{2}}{2N_{0}}, \tag{7}\]  
by substituting Equation (7) in (1), the bound can be written as a function of the \(C/N_{0}\) as follows:  
\[\sigma_{CRLB}{}^{2}(\phi_{CW})=\frac{2\sigma_{n}^{2}}{A_{CW}^{2}L}=\frac{f_{s} }{2 LC/N_{0}}. \tag{8}\]  
Therefore, under the optimal propagation condition and with an efficient estimator, the receiver performance can be described by Equation (8).  
To estimate the power noise density \(N_{0}\), an average value of the PSD, excluding the sinusoidal tone contribution \(P_{k_{CW}}\), can be calculated as follows:  
\[N_{0}=2\bar{P}=\frac{2}{L}\sum_{k=1}^{\frac{k}{2}-1}P_{k}\qquad\forall k\
eq k _{CW}, \tag{9}\]  
where \(\bar{P}\) is the mean value of the \(P\). Equation (9) holds only if we have a single continuous wave in AWGN. For the R-Mode signal, this is clearly not the case due to the presence of nearby MSK. Moreover, in real applications the full sampling bandwidth might contain different levels of noise per channel and specific channels might be corrupted by nearby interference that does not affect other channels. However, the FFT can be also seen as a bank of filters applied on the target frequencies \((kf_{s})/L\). The bandwidth of each filter depends on \(f_{s}\) and \(L\), and in principle one would need to estimate the noise level within the single filter band. Clearly, this is not possible since the FFT computes the total amount of power entering each filter and cannot distinguish between noise and useful signal component. Therefore, the estimation of the noise floor needs to be carried out locally in the proximity of the target aiding carrier by considering a limited number of samples of \(P_{k}\) around the single tone. Thus, the estimator for the \(C/N_{0}\) becomes  
\[C/N_{0}=\frac{P_{k_{CW}}}{\frac{1}{L_{B}}\left(\sum_{k=k_{CW}-L_{B}}^{k_{CW}+ L_{B}}P_{k}\right)},\qquad\forall l\
eq k_{CW} \tag{10}\]  
where \(L_{B}\) is the number of points considered on the left and right side of the CW. The accuracy of the estimated \(C/N_{0}\) depends on \(L_{B}\), \(f_{s}\), and \(L\) used for the DFT; the power allocated for the MSK and CWs; as well as the noise power and nearby interference. The next section presents some results based on simulations.