## 3 Overview of FB Approach for MR
### Statistical Features for MR  
Some types of modulation schemes contain information in the amplitude and phase spectra, which are known as complex signals, and are represented as in-phase and quadrature components. In this context, statistical features are useful for eliminating the background noises caused by channel environments and the presented fluctuation [7; 29]. Statistical features include higher-order statistical (HOS) features, such as moment and cumulants, which are widely used for feature discrimination on ASK, PSK, and QAM modulation types. The main strengths of higher-order statistics are robustness against phase rotation, the capability to remove noises, and reflection of higher-order properties of the signal [45]:  
Moment is a general concept of the probability distribution moment functions. Moment is a method for quantitatively measuring the shape of a function [31]. In 1992, Soliman and Hsue [46] introduced a third-order moment-based modulation classifier for classifying the order of M-PSK; they found that the moment of the phase signal in shared white Gaussian channel is gradually monotonically increasing to M-PSK order. Thus, this feature can be used to classify modulation types. The \(K^{th}\) order moment \(K(r)\) is determined theoretically as [33; 46]  
\[k(r)=\frac{1}{N}\sum_{n=1}^{N}\varphi^{k}(n) \tag{8}\]  
where \(\varphi^{k}(n)\) is the phase for the \(n^{th}\) sample signal received and \(N\) represents the total number of samples. The calculation of different \(k^{th}\) moments of received signal \(r=r[1],r[2],\ldots,r[N]\)[33; 46] can be expressed as  
\[Moment_{xy}(r)=\frac{1}{N}\sum_{n=1}^{N}r^{x}[n]\times r^{y}[n] \tag{9}\]  
where \(x+y=k\) and \(r^{\ast}[n]\) is the complex conjugate of \(r[n]\)[33]. Higher-order cumulants (HOCs) are the second statistical features investigated after the moments. Fourth-order cumulants of the complex signal value are suggested by [47] as the feature for classifying modulation types M-PAM, MPSK, and M-QAM when the carrier phase and frequency offsets are unknown. The HOC equation is obtained from a combination of two or more of higher-order statistical moments (HOMs) [45] as in C\({}_{42}\), which is expressed as  
\[C_{42}=cum[r(n)r(n)r(n)r^{\ast}(n)]=M_{41}-3M_{20}M_{21} \tag{10}\]  
where c42 is the fourth-order moments and cumulants, r[n] is the complex-valued stationary random process, r*[n] is the conjugate of r[n], M20M21 are the second-order moments, and M40 is the second-order moment. HOC features are robust to noise, especially when cumulants higher than the second order are used. HOCs of demodulated signals are approximately equal to those of transmitted signals [48; 49]. The cumulant values can be used for modulation discrimination in two ways. The first method is by creating hierarchical schemes that precisely discriminate the order and type of the digital modulation (M-ASK, M-FSK, M-PSK, M-QAM). The second utilization of cumulants is introduced in [50] by comparing the estimated value to the real value. HOS features are used to classify M-PSK and M-QAM modulation types, which are robust against AWGN [12; 40; 48; 51; 52; 53; 54; 55]. Furthermore, the multipath channel effects can be easily modelled using the HOS features [12; 56; 57; 31], which are robust to frequency, phase offset, and timing errors [52; 53; 58]. In [48; 49], the extracted features from ratio and absolute of HOC are used as a characteristic parameter for classifying between M-FSK and M-PSK, M-ASK, and 16 QAM, and between M-PSK and M-ASK [59]. In [53], the authors used fourth-order cumulants to recognize M-PSK. They divided their algorithm into two steps. In the first step, the fourth-order zero conjugate cumulants of backward difference samples of the receiver signals were used to identify QPSK and OQPSK. In the second step, the fourth-order zero-conjugate cumulants of the received noisy signal and their squares were used to identify QPSK, 8 PSK, and 16 PSK in the presence of frequency and phase offset.