## 4 Overview: Signal Processing Methods
### Signal Feature Extraction Methods  
As per the findings cited in the journal, feature extraction techniques yielding the highest accuracy include methods like Mean Absolute Value (MAV), Waveform Length (WL), Zero Crossing (ZC), Slope Sign Change (SSC), Root Mean Square (RMS), Energy of Wavelet Coefficients (EWLs), and Enhanced Mean Absolute Value (EMAV).  
Mean Absolute Value (MAV) calculates the average magnitude of a signal by averaging the absolute values of its data points. MAV is commonly used in electromyography (EMG) analysis to assess muscle activity by measuring the overall muscle contraction level.  
Figure 17: Discrete Wavelet Transform decomposition, block diagram form, with 4 levels of decomposition.  
\[MAV=\frac{1}{L}\sum_{i=1}^{L}\lvert x_{i}\rvert \tag{3}\]  
Wavelength (WL) quantifies the cumulative length of waveform oscillations by summing the distances between successive peaks or troughs in the EMG signal.  
\[WL=\sum_{i=2}^{L}\lvert x_{i}-x_{i-1}\rvert \tag{4}\]  
Figure 19: Discrete Wavelet Transform decomposition, analysis form, with 4 levels of decomposition.  
Figure 18: Discrete Wavelet Transform decomposition with 6 levels of decomposition.  
Zero Crossing (ZC) refers to the points in a signal where the amplitude changes its sign, where the signal crosses the zero-amplitude axis.  
\[ZC=\sum\limits_{i=1}^{L-1}f(x_{i}) \tag{5}\] \[\textit{where }f(x_{i})=\begin{cases}1,\textit{ if}\{(x_{i}>0\ \&\ x_{i+1}<0)|\ (x_{i}<0\ \&\ x_{i+1}>0)\}\dots\\ &\ \&\ |x_{i}-x_{i+1}|\geq T\\ &0,\textit{ otherwise}\end{cases}\]  
Slope Sign Change (SSC) counts the times the slope of a signal changes within a set interval, aiding in detecting transitions in signal dynamics.  
\[SSC=\sum\limits_{i=2}^{L-1}f(x_{i}) \tag{6}\] \[\textit{where }f(x_{i})=\begin{cases}1,\textit{ if}\{(x_{i}>x_{i-1}\ \&\ x_{i}>x_{i+1})|\ (x_{i}<x_{i-1}\ \&\ x_{i}<x_{i+1})\}\dots\\ &\ \&\ \{(|x_{i}-x_{i+1}|\geq T)|\ (|x_{i}-x_{i-1}|\geq T)\}\\ &0,\textit{ otherwise}\end{cases}\]  
Root Mean Square (RMS) calculates the average power of a signal by taking the square root of the average of the squares of all signal values, offering insight into signal intensity, and finding application in noise analysis, signal characterization, and power estimation.  
\[RMS=\sqrt{\frac{1}{L}\sum\limits_{i=1}^{L}(x_{i})^{2}} \tag{7}\]  
Energy of Wavelet Coefficients (EWLs) quantifies the energy distribution across frequency bands in wavelet-transformed signals, aiding tasks like signal classification and denoising.  
\[\begin{split} EWL=\sum\limits_{i=2}^{L}\left|(x_{i}-x_{i-1})^{p} \right|\\ \textit{where }p=\begin{cases}0.75,\textit{ if }(i>0.2L\ \&\ i<0.8L)\\ &0.50,\textit{ otherwise}\end{cases}\end{split} \tag{8}\]  
Enhanced Mean Absolute Value (EMAV) enhances traditional Mean Absolute Value (MAV) by incorporating weighting factors to emphasize higher amplitude signal components, improving the accuracy of muscle activity representation in EMG signal processing.  
\[\begin{split} EMAV=\frac{1}{L}\sum\limits_{i=1}^{L}\left|(x_{i} )^{p}\right|\\ \textit{where }p=\begin{cases}0.75,\textit{ if }(i\geq 0.2L\ \&\ i\leq 0.8L)\\ &0.50,\textit{ otherwise}\end{cases}\end{split} \tag{9}\]