## 3 Overview of FB Approach for MR
### Spectral Features for MR  
Spectral-based features are frequency-based features that help in correlating physical attributes to perceptual attributes. Each feature is selected because of two reasons, namely, the mathematical calculation, which proves the goodness of the feature criteria, and the reported usefulness of the feature in previous research [29]. Spectral-based features of a signal was presented for the first time by Azzouz and Nandi at the end of the 20 th century [32; 33]. The parameters of instantaneous amplitude, phase, and frequency (which are presented as \(a_{n},\phi_{NL}\) and \(f_{N}\), respectively) were used for signal classification. This process is an intuitive way of identifying the modulation class of incoming signals. FSK signals are defined by the fixed instantaneous amplitude. ASK signals are classified on the basis of their amplitude variations, and the PSK information is in their phase. They can be obtained in many ways, and a common method used is Hilbert transform [34]. Instantaneous amplitude A(t), phase \(\omega\)(t), and frequency \(f_{N}\) can be obtained using Equations (5)-(7), respectively, [8; 35]:  
\[A(t)=\sqrt{r^{2}(t)+\hat{r}^{2}(t)} \tag{5}\]  
\[\varphi(t)=\tan^{-1}\frac{\hat{r}(t)}{r(t)} \tag{6}\]  
\[f_{N}=\frac{1}{2\pi}\frac{d_{quaw}(t)}{dt} \tag{7}\]  
where r(t) is the real valued modulated signal and \(\hat{\text{r}}\)(t) is its Hilbert transform. Several spectral features are used for digital modulation identification, which are proposed in the literature and are tabulated in Table 1[33]. Previous studies have presented several FB approaches on time and frequency domains. Table 1 presents the specific purpose for each feature. The maximum power of spectral density (PSD) depends on the information conveyed through a signal envelope, such as M-ASK and M-QAM for digital modulation, and the \(\gamma_{\max}\) value must be nonzero. When the modulated signal has a constant amplitude, such as M-FSK and M-PSK, the \(\gamma_{\max}\) value must be zero. Therefore, \(\gamma_{\max}\) can be used to discriminate between M-ASK and M-QAM in addition to constant amplitude M-FSK and M-PSK digital modulation schemes [4; 31; 33; 35; 36; 37; 38; 39]. The average value of \(\gamma_{\max}\) is used to discriminate among continuous phase FSK, Gaussian FSK, and Gaussian minimum shift keying (GMSK) [40]. In addition, several studies have used these features to discriminate the order of modulation schemes, such as M-ASK and M-FSK [41]. Features \(\sigma_{aa}\) and \(\sigma_{a}\) are used to detect the variant of modulated signal amplitude. These features are similar to \(\gamma_{\max}\). Therefore, these features are used to classify the order of M-ASK and M-QAM [42]. In [35; 38; 43], \(\sigma_{aa}\) is used to discriminate between the orders of ASK. \(\sigma_{a}\) feature is helpful for recognizing ASK from PSK [31]. Features \(\mu_{42}^{a}\) and \(\mu_{42}^{f}\) represent the distribution of amplitude and frequency, respectively. \(\mu_{42}^{a}\) is used to separate FSK/PSK from ASK/QAM, whereas \(\mu_{42}^{f}\) is used to separate FSK from PSK [8; 36]. Feature \(\sigma_{df}\) highlights the variation in signal frequency; therefore, it can be used to recognize the order in FSK [7; 35; 42]. Features \(\sigma_{ap}\) and \(\sigma_{dp}\) characterize the variations in a signal's instantaneous