## 4 Experimental Design
### Data Pre-Processing
Wavelet transform denoising is fast and suitable for detection systems with high real-time requirements. Since wavelet transform permits adjustment in time-domain resolution, it can be optimized by selecting  
Figure 3: Changes before and after interpolation of 30 subcarriers (**a**) Before interpolation algorithm processing; (**b**) After interpolation algorithm processing. suitable parameters based on the user's signal to obtain the best denoising effect. It is also convenient to extract the gesture information afterwards. The threshold-based wavelet transform denoising model is expressed below:  
\[s(n)=f(n)+\sigma e(n) \tag{6}\]  
where \(s(n)\) is the data interposed with high frequency noise, \(f(n)\) is the required data after removing high frequency noise, which is the signal to be filtered out in this study, \(e(n)\) is the Gaussian white noise, and \(\sigma\) is the noise intensity. As for threshold selection, this study adopted the wavelet transform based on heuristic thresholding for denoising. Figure 5 illustrates the flow of wavelet denoising. The noise reduction process was executed using the following three steps:  
First, wavelet transform was performed on \(s(n)\) to obtain wavelet transform coefficients \(\omega(n)\). Second, threshold r was selected and combined with the wavelet coefficients obtained after processing, thus making the reconstructed signal smoother by taking a threshold to avoid intermittent points. Third, the scale coefficients were reconstructed to complete the noise removal and obtain the reconstructed signal \(f(n)\). 1. Wavelet decomposition;
2. The two most commonly used wavelet bases are Daubechies (\(db\)) and Symlets (\(sym\)) wavelet systems. Although the \(db\) wavelet system has an excellent effect of signal smoothness after denoising, its division in the frequency domain leads to a weaker support in the time domain, thus increasing computation time and worsening real-time performance. Meanwhile, the \(sym\) Wavelets system has better regularity and is comparable to the \(db\) wavelet system in terms of continuity and filter length. Its better symmetry can reduce the phase distortion caused by the decomposition and reconstruction of the signal to a certain extent. Hence, the \(sym\) wavelet was selected in this study to carry out the wavelet decomposition of the signal in three layers. The \(sym\) wavelet function is presented below:  
\[SW_{(\omega)}=\frac{1}{\sqrt{2}}\sum_{k=0}^{2N-1}h_{k}e^{-jk\omega} \tag{7}\]  
where \(\omega\) is the wavelet coefficient. 3. Threshold processing;  
Figure 4: Data packet after outlier algorithm processing. Figure 5: Wavelet denoising process. The four thresholds are unbiased likelihood estimation, fixed threshold, heuristic threshold, and extreme value threshold. Heuristic threshold is a combination of unbiased and fixed thresholds based on the algorithm of unbiased likelihood estimation. Let \(m\) be the sum of squares of wavelet coefficients and n be the number of wavelet coefficients. When \(m=\sum\limits_{i=1}^{n}\omega_{i}^{2}\)  
, let \(\alpha=\text{m}-\text{n}/\text{n}\), \(\beta=(log_{2}n)^{1.5}\sqrt{n}\), and the heuristic threshold is:  
\[\text{T}= \begin{cases}c,&\alpha<\beta;\\ d,&\alpha\geq\beta;\end{cases}  \tag{8}\]  
where \(c\) is the unbiased threshold, and \(d\) is the fixed threshold. Figure 6 (below) portrays the comparison of before and after the wavelet denoising. As can be seen from the comparison above, after the wavelet filter for denoising, the perturbations in the data were successfully processed, and we obtained clearer data on the trend of gesture changes. The next step will be the selection of subcarriers. Subcarrier selection. After the above pre-processing steps, the data were clear from high-frequency noise and outliers. Since the data had 30 subcarriers, but due to the different sensitivity of each subcarrier to environmental changes, the amplitude of the extracted subcarrier amplitudes differed.