## 2 Materials and Methods  
### Background of Energy Operators  
The Teager-Kaiser energy operator (TKEO) is a signal processing technique that was proposed by [PERSON] to calculate the instantaneous \"energy\" of nonlinear signals based on a conversation with [PERSON], and it has important applications in the processing of various nonlinear signals, such as speech signals [39; 40], physiological signals [23; 30; 41], and mechanical vibration and noise signals [42; 43; 44; 45]. The most commonly used discrete domain TKEO formula is obtained by converting the continuous domain to the discrete domain through the forward difference method:  
\[\Psi_{d}[x(n)]=x^{2}(n)-x(n+1)x(n-1) \tag{1}\]  
where \(x(n)\) denotes the data value at the \(n\)th sampling point, and \(n+1\) and \(n-1\) represent the pre- and post-sampling points adjacent to the \(n\)th sample, respectively. Equation (1) shows that the TKEO can monitor the \"energy\" of the signal in a very narrow window of three sampling points, with simple operations and no division, making it particularly suitable for processing zero-crossing signals. The TKEO is actually a special case under a broader operator paradigm. [PERSON] et al. generalized the TKEO into a matrix framework and interpreted the general framework through the determinant of a Toplitz matrix containing the signal and its derivatives. Using this, we can obtain the determinant of Equation (1), as follows [46]:  
\[\Psi_{d}[x(n)]=\begin{pmatrix}x[n]&x[n-1]\\ x[n+1]&x[n]\end{pmatrix} \tag{2}\]  
The determinant is time-invariant for a signal with constant frequency. If such a matrix is generalized to an \(M\times M\) Toeplitz matrix by adding delayed \(x[n]\) up to \(x[n\pm(\mathrm{M}-1)]\), the determinant is also time-invariant but for signals with multi frequency [46; 47]:  
\[\Psi_{M}[x(n)]=\begin{pmatrix}x[n]&x[n-(M-1)]\\ x[n+(M-1)]&x[n]\end{pmatrix} \tag{3}\]  
Different \(M\) values can enhance different frequency components [47], which not only makes the algorithm 'observe' more signal information to enhance the robustness but also improves the signal conditioning performance of the energy operators.  
In fact, converting a discrete signal into a determinant form can be understood as slicing the signal:  
\[X_{n}=[x[n]\;x[n-m]\;\ldots\;x[n-(d-1)m]]^{T} \tag{4}\]  
where \(m\) and \(d\) are integer time-lag values, and \((\cdot)^{T}\) denotes the transpose operator. The sliced signal of Equation (4) can still be matrised by the Toeplitz matrix [38; 46]:  
\[X_{n}=\left(\begin{array}{llll}x[n]&x[n+s]&\ldots&x[n+(d-1)s]\\ x[n-m]&x[n-m+s]&\ldots&x[n-m+(d-1)s]\\ \vdots&\vdots&\ddots&\vdots\\ x[n-(d-1)m]&x[n-(d-1)m+s]&\ldots&x[n-(d-1)(m-s)]\end{array}\right) \tag{5}\]where \(s\
eq 0\) is a time-lag value, as are \(m\) and \(d\). Important factors for maintaining the efficiency of energy operators are the length of their corresponding index windows and the symmetry of their time alignment [44]. Therefore, we set \(s=m=1\), \(d-1=\) k and converted Equation (5) into a more intuitive format of temporal symmetry, as follows:  
\[X_{n}=\left(\begin{array}{cccc}x[n]&x[n+1]&\ldots&x[n+\mathrm{k}]\\ x[n-1]&x[n]&\ldots&x[n+\mathrm{k}-1]\\ \vdots&\vdots&\ddots&\vdots\\ x[n-\mathrm{k}]&x[n-\mathrm{k}+1]&\ldots&x[n]\end{array}\right) \tag{6}\]  
Equations (3) and (6) address two key issues of EONND. Firstly, Equation (6) converts 1-D discrete signal data into the form of 2-D Toeplitz matrices, providing the mathematical basis for the combination of energy operator methods with neural networks. Secondly, Equation (3) provides a computational method for obtaining energy operators from 2-D matrices data. It is worth mentioning that Equation (6) can be regarded as a matrix form of a multi-resolution Teager energy operator (MTEO) [48; 49].