## 3 Overview of FB Approach for MR
### Spectral Features for MR
\begin{table}
\begin{tabular}{p{142.3 pt} p{142.3 pt}} \hline
**Features** & **Mathematical Equation** \\ \hline
**Maximum value of PSD \(\gamma_{max}\) of the normalized centered instantaneous amplitude** & \(\gamma_{\max}=\frac{\max\left|DTT(A_{in}(i))^{2}\right|}{N_{s}}\), where DFT is the discrete Fourier transform of the modulated signal, \(N_{s}\) is the sample number, \(A_{cn}=\frac{A_{s}}{A_{s}}-1\), \(A_{i}\) is the \(i^{th}\) instantaneous amplitude and \(\mu_{A}\) is the sample mean \\ \hline
**Standard deviation of the absolute values of the centered nonlinear components of instantaneous phase \(\sigma_{ap}\)** & \(\sigma_{ap}=\sqrt{\frac{1}{N_{c}}\left(\sum\limits_{A_{s}(i)>A_{s}}\varphi_{NL} ^{2}(i)\right)-\frac{1}{N_{c}}\left(\sum\limits_{A_{s}(i)>A_{s}}\varphi_{NL} (i)\right)^{2}}\), where \(N_{c}\) is the number of sample(s) in \(\left|\zeta_{NL}\right|\) for \(A_{n}\) (\(i\)) \(>A_{t}\), where \(A_{s}\) is the threshold value of \(A_{n}\) (\(i\)) when the filter provides the minimum amplitude of the signal sample due to high noise sensitivity and \(\phi_{NL}\) (\(i\)) is the nonlinear component of the \(i^{th}\) instantaneous phase of the sample. \\
**Standard deviation of the absolute value of the normalized centered instantaneous amplitude in the nonweak segment of signal \(\sigma_{a}\)** & \(\sigma_{a}=\sqrt{\frac{1}{N_{c}}\left(\sum\limits_{A_{s}(i)>A_{s}}\varphi_{NL} ^{2}(i)\right)-\frac{1}{N_{c}}\left(\sum\limits_{A_{s}(i)>A_{s}}\varphi_{NL} (i)\right)^{2}}\), where all parameters are similar to \(\sigma_{ap}\) but differs in the absence of the absolute operator in the nonlinear component of the instantaneous phase. \\
**Standard deviation of the absolute value of the normalized centered instantaneous amplitude of signal segment \(\sigma_{aa}\)** & \(\sigma_{aa}=\sqrt{\frac{1}{N_{c}}\left(\sum\limits_{i=1}^{N}A_{cn}^{2}(i) \right)-\frac{1}{N_{c}}\left(\sum\limits_{i=1}^{N}A_{cn}^{2}(i)\right)^{2}}\), where \(A_{cn}\) is the normalized and centered instantaneous amplitude of the incoming signal at the time instant. \\
**Standard deviation of the absolute value of the normalized centered instantaneous frequency of signal segment \(\sigma_{df}\)** & \(\sigma_{df}=\sqrt{\frac{1}{N_{c}}\left(\sum\limits_{A_{s}(i)>A_{s}}f_{N}^{2}(i )\right)-\frac{1}{N_{c}}\left(\sum\limits_{A_{s}(i)>A_{s}}\left|f_{N}(i)\right| \right)^{2}}\), where \(f_{N}\) is the normalized frequency. \\
**Kurtosis of the normalized centered instantaneous frequency \(\mu_{42}^{a}\)** & \(\mu_{42}^{a}=\frac{E\left[A_{cn}^{a}[i]\right]}{\left[E\left[A_{cn}^{a}[i] \right]\right]^{2}}\), where \(A_{cn}\) is the normalized and centered instantaneous amplitude of the incoming signal at the time instant. \\ \hline
**Kurtosis of the normalized centered instantaneous amplitude \(\mu_{42}^{a}\)** & \(\mu_{42}^{a}=\frac{E\left[A_{cn}^{a}[i]\right]}{\left[E\left[A_{cn}^{a}[i] \right]\right]^{2}}\), where \(f_{N}\) is the normalized normalized frequency. \\
**Kurtosis of the normalized centered instantaneous frequency \(\mu_{42}^{f}\)** & \(\mu_{42}^{f}=\frac{E\left[A_{cn}^{a}[i]\right]}{\left[E\left[A_{cn}^{a}[i] \right]\right]^{2}}\). \\ \hline \end{tabular}
\end{table} 
Table 1: Spectral features used for modulation classification [33]. phase. \(\sigma_{ap}\) is used to discriminate the order of PSK [4; 44], and it distinguishes among ASK2, ASK4, PSK2, and PSK4, Therefore, the gap mainly depends on the absolute phase information to discriminate the modulation formats. \(\sigma_{dp}\) uses the direct phase information to distinguish the modulation types (containing the direct information phase) from those that do not have the direct information phase, such as distinguishing among ASK2, ASK4, and PSK [35; 43]. Although these features are simple to extract, they are sensitive to noise and can produce estimation errors. Furthermore, the extraction of instantaneous information fully depends on thresholds that should be set normally in advance.