## 3 Overview of FB Approach for MR
### Transform Features for MR  
The spectral characteristics of an FSK signal is proven to be recognized from PSK and QAM signals and to split various M-ary FSK schemes [61]. Spectrum analysis-based classification is a scheme that is used to identify MFSK modulation schemes by using fast Fourier transformer classifier (FFTC) and DFT via fast Fourier transform (FFT) [62]. In [61; 63], the features were extracted from the FFT and then compared with the threshold values, which were used to discriminate the types of M-FSK. This technique can be used in real time and is useful in reducing the input noise and decimating the input signal to the automatic modulation classification (AMC) [64]. WT provides an environment for analyzing signals under different frequencies and different resolutions. Hence, particular features of signals are only revealed in this domain or cannot be extracted using other forms of transform [30]. WT can be categorized into continuous wavelet transform (CWT) and discrete wavelet transform (DWT). CWT is used to overcome the resolution problem, and short-time Fourier transform can be alternatively used [65]. CWT of received \(S\) is defined as the integral of \(S(t)\) times the conjugate transpose of wavelet function \(\Psi_{a,x}(t)\) over a time duration, as follows:  
\[CWT(a,\tau)=\frac{1}{\sqrt{a}}\int_{-\infty}^{\infty}S(t)\psi_{a,\tau}^{*} \Big{(}\frac{t-\tau}{a}\Big{)}dt \tag{11}\]  
where \(\Psi^{*}\) is its complex conjugate, \(a\
eq 0\) is the scale, and \(\tau\) is the translation variable. The wavelet function has been extensively discussed in the literature, such as by Morlet, Haar, and Shannon, and the most applied wavelet within the modulation field is Haar wavelet because of its simplicity and computation convenience. Haar used a simple rectangular waveform as the reference wavelet for a classifier to decompose the received modulating signal [66]. Most studies on the field of wavelet domain-based features have focused on the histogram computation of CWT and/or DWT wavelet coefficient of demodulated signals. The number of peaks in histograms is used for identifying different types of digital modulation [67; 68; 69; 70]. The histogram peaks in WT magnitude, mean, and variance of normalized histogram are utilized for identifying the type of digital modulation [71]. In [72], the author analyzed the CWT instantaneous features (mean, variance, and central moment values) for recognizing M-ASK, M-PSK, M-PSK, M-QAM, OOK, and MSK. In [73], the histogram peaks in WT magnitude, mean, and HOM of normalized histogram were adopted as features for digital modulation classification. The wavelet variation coefficient difference and similarity measurement functions from statistics were used as the features to classify MASK, MPSK, and MQAM [74]. In practical scenarios, modulating signal power exhibits uncertainty. Therefore, an optimal solution that covers the uncertainty should be determined. In 1994, Gardner firstly introduced cyclostationary analysis for the received signal to investigate the diversity in the spectrum appearance of all modulation types [75]. Random signals are classified as cyclostationary when the HOMs of their envelope are periodic [33]. Each modulation type has different periodic spectrum patterns [45]. A common signal \(x(t)\) of a sinusoidal function is considered to be a cyclostationary feature or second-order periodicity when its cyclic autocorrelation function can be expressed as  
\[R_{x}^{\alpha}(\tau)=\underset{T\to 0}{lim}\frac{1}{T}\int_{-T/2}^{T/2}X(t+ \frac{\tau}{2})\times X^{*}\bigg{(}t-\frac{\tau}{2}\bigg{)}e^{-j2\pi t} \tag{12}\]  
where \(R_{x}^{\alpha}(\tau)\) is a Fourier series coefficient called cyclic autocorrelation function, \(X(t)\) is the signal itself, \(X^{*}(t)\) is its conjugate and \(T\) is the time interval. The spectral correlation function can be defined as the cyclic spectrum.