# Noise Reduction Method of Nanopore Based on Wavelet and Kalman Filter
## 2 The wavelet Method
### Method Classification  
Based on the characteristics of wavelet transform, it can be applied to nanopore signal noise reduction. The commonly used noise reduction methods include wavelet decomposition and reconstruction method, nonlinear wavelet transform threshold method [11; 12], wavelet mode maxima method, etc. [13]. [PERSON] proposed a multi-resolution analysis method of the signal [14], which decomposes the signal into two parts, low and high frequencies, with the following equation. \[A_{j}(n)=<f(u),\Phi_{2j}(u-2^{j}n)> \tag{1}\]  
\[D_{j}(n)=<f(u),\Psi_{2j}(u-2^{j}n)> \tag{2}\]  
\[A_{j-1}(n)=\sum\limits_{k\in x}h(n-2k)Aj(k)\] \[+\sum\limits_{k\in x}g(n-2k)D_{j}(k) \tag{3}\]  
\(A_{j}\) is the low-frequency scale coefficient obtained as a function of the signal through low-pass filtering [15]. Similarly, \(D_{j}\) is the high-frequency scale coefficient obtained as a function of the signal through band-pass filtering, and the wavelet decomposition principle is shown in Figure 3. The decomposed signal is reconstructed by recurring the approximate discrete signal at higher resolution through the conjugate quadrature filter of both and inserting a zero between every two values of \(A_{j}\) and \(D_{j}\), respectively, and convolving the sum with the corresponding conjugate filter so that the reconstruction of the wavelet signal is completed. Equations (1) and (2) are the process of decomposing the original signal into low- and high-frequency coefficients by passing it through different filtering functions. In Equation (3), the reconstruction of the signal is completed by sieving the high-frequency coefficients. [PERSON] proposed the wavelet threshold method [16], which recovers the sampled data from the original signal into a smoothing function, sets an adaptive threshold, and reduces the noise by setting the coefficient threshold [17]. A property of white noise in wavelet transform is used: the transformation of white noise on any orthogonal basis  
Figure 3: Schematic diagram of wavelet decomposition. is still white noise. Therefore, a finite length signal superimposed with Gaussian white noise is assumed in the method, part of which is a Gaussian white noise sequence, and the wavelet transform is performed on the noise-containing signal, and the absolute value of the wavelet coefficients after wavelet decomposition is quantized by setting the threshold value as a threshold function, and those smaller than the threshold value are set to zero. The threshold method is divided into two methods: hard threshold method and soft threshold method [18; 19]. \[\hat{x}=T_{h}(W,\lambda)=\left\{ \begin{array}{ll}W&|W|\geq\lambda\\ 0&|W|<\lambda\end{array} \right. \tag{4}\]  
\[\hat{x}=T_{h}(W,\lambda)=\left\{ \begin{array}{ll}\text{sgn}(W)(|W|-\lambda)& |W|\geq\lambda\\ 0&|W|<\lambda\end{array} \right. \tag{5}\]  
\(\lambda\) is the threshold value set according to the signal. Wavelet coefficient reconstruction by (4) hard threshold method and (5) soft threshold method to approximate the original signal. Since the amplitude of white noise is invariant in the wavelet transform orthogonal basis, it makes the wavelet coefficients of those noises with smaller amplitude easily discarded by the set threshold, which makes the reconstructed original signal with higher signal-to-noise ratio and better asymptotic properties [20]. The wavelet transform can clearly see the abrupt change point of the signal in the time-frequency domain, and it can describe the characteristics of the local signal. The signal mutation point is the extreme value point of wavelet transform coefficient mode, and the singular value of the signal is related to the change in wavelet coefficient scale, and the wavelet transform has the ability to determine the location and size of the singularity of the signal, so it can reduce the noise by comparing the extreme value. According to the Lipschitz exponent [21], the signal has a singular value at a point in time and its wavelet transform has an extreme value at this point [22].