## 4 Overview: Signal Processing Methods
### Discrete Wavelet Transform  
_Biorthogonal 3.3_  
The Fourier transform stands as a powerful tool for data analysis, yet it lacks efficiency in representing abrupt changes. This limitation arises because the Fourier transform represents data as a sum of infinite sinusoids, which do not inherently consider time or spatial localization. Representing the signals in the time-frequency domain proves beneficial, particularly for non-stationary signals. In wavelet analysis, two primary transforms exist: the discrete wavelet transform, and the continuous wavelet transform. These transforms diverge in terms of how the wavelets are scaled and shifted. The DWT was selected due to its explicit utility in analyzing bioelectrical signals. As a novel signal processing tool, the DWT technique offers advantages in decomposing signals and images into different frequency bands, enabling efficient analysis while preserving crucial information. This method, characterized by its dual filter bank structure, stands out for its ability to maintain a balance between time and frequency localization, allowing for enhanced feature extraction and denoising capabilities across various domains. By comprehensively delving into the intricacies of this transformative method, the research aims to harness its potential in diverse applications, spanning from biomedical signal processing to image compression, with a focus on optimizing its functionalities for real-world scenarios. The DWT filters the signal using both a low-pass filter and a high-pass filter, followed by down sampling. This separation of frequency components results in the high-pass components known as  
Figure 14: PSD analysis of EMG signals captured from the wrist after the application of the 60 Hz Notch filter. detail coefficients and the low-pass components known as approximate coefficients. The filtering process occurs iteratively and can be conceptualized as involving multiple filter banks. The formula for the DWT that involves multiple levels of decomposition can be split into approximate (low-frequency components) (1) and detail (high-frequency components) (2) components. Figure 16: PSD analysis post filtering of EMG signals captured from the forearm. Figure 15: PSD analysis post filtering of EMG signals captured from the wrist. \[Approximate\to y[n]=\sum_{k=-\infty}^{\infty}x[k]\cdot g[2n-k] \tag{1}\]  
where \(g\) represents the high-pass filter coefficients. \[Detail\to y[n]=\sum_{k=-\infty}^{\infty}x[k]\cdot h[2n-k] \tag{2}\]  
where \(g\) represents the high-pass filter coefficients. The research initially utilized 6 levels of DWT decomposition. However, considering EMG's higher frequency range, it was determined that employing 4 levels of DWT decomposition would yield superior results. Figure 17 illustrates the process of 4-level DWT decomposition. In the reconstruction phase of wavelet signal processing using Biorthogonal 3.3 DWT, the approximation and detail coefficients obtained during the decomposition phase are utilized. These coefficients represent the different frequency components of the original signal at various levels of resolution. Biorthogonal wavelets provide a balanced reconstruction process by utilizing synthesis filters that complement the analysis filters employed during decomposition. Subsequently, through an inverse wavelet transform, the approximation and detail coefficients are merged to reconstruct the original signal. Biorthogonal 3.3 DWT offers an accurate reconstruction, preserving essential signal characteristics while effectively compressing the data. This phase plays a crucial role in recovering the original signal from its decomposed components, ensuring fidelity and reliability in signal reconstruction. An illustration of wavelet decomposition using DWT at the sixth level is shown in Figure 18, while Figure 19 depicts wavelet decomposition at the fourth level. It is evident that over-decomposition occurs at levels 5 and 6 of the detail coefficients.