# Noise Reduction Method of Nanopore Based on Wavelet and Kalman Filter
## 2 The wavelet Method
### Experimental Results  
Design the experiment to obtain the current signals of IgG nanopore: process a solid-state nanopore with a diameter of 13.5 nm on a silicon nitride film with a thickness of 20 nm to construct a direct antibody detection system; secondly, add a certain concentration of IgG antibody (e.g., 0.01875 mg/mL) buffer solution, and apply a bias voltage of 100 mV at both poles of the electrode to induce the IgG antibody to pass through the solid-state nanopore under the action of electrophoretic force. Finally, the patch clamp detection system Axon 200B with high sampling rate (>5 kHz) is used to sample and record the IgG antibody signal.  
According to the smooth discontinuity of nanopore signal, Haar wavelet is selected in this paper, and according to the effect of IgG signal reconstruction, the number of wavelet decomposition layers is 5. Haar wavelet is the simplest one in the family of Dobesian wavelets, which has the advantages of tight support in time domain, orthogonal symmetry, and simple calculation. Haar wavelet is also the only discontinuous Dobessie wavelet, so among all orthogonal wavelet transformations, the Haar wavelet can well resolve and reconstruct the nanopore detection signal in multiple ways by scale transformation and displacement transformation, effectively reducing the noise interference of nanopore detection signal. The IgG antibody nanopore current signal was processed by wavelet decomposition and reconstruction method, hard threshold method, soft threshold method, and modulus maximum method, as shown in Figure 4.  
The above four methods are applied to the nanopore current signal and conclude that since Gaussian white noise exists on the wavelet coefficients of each layer, although the wavelet decomposition and reconstruction method can roughly recover the waveform of the signal, it cannot remove the part of white noise that overlaps with the useful signal, so the noise reduction is not very satisfactory. The Gaussian white noise can be well eliminated by using the hard and soft thresholding methods and the mode maxima method. According to the signal-to-noise ratio (SNR) and root mean square error (RMSE) of the wavelet transform mentioned above [23], as shown in Table 1, the superior performance of the wavelet mode maxima method for noise reduction of nanopore signals is demonstrated. Where \(x_{i}\) refers to the original signal, and \(y_{i}\) refers to the estimated signal after noise reduction through wavelet transform.  
\[SNR=10\log(\frac{\sum_{i=0}^{N-1}x_{i}^{2}}{\sum_{i=0}^{N-1}(x_{i}-y_{i})^{2}}) \tag{6}\]  
\[RMSE=\sqrt{\frac{\sum_{i=0}^{N-1}(x_{i}-y_{i})^{2}}{N}} \tag{7}\]  
Due to the huge amount of nanopore signal data, it is necessary to process the IgG antibody signal by segmentation. Based on the above simulation results, the wavelet mode maxima method is selected for noise reduction, as shown in Figure 5.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Wavelet Decomposition and** & **Hard Threshold** & **Soft Threshold** & **Wavelet Mode** \\  & **Reconstruction Method** & **Method** & **Method** & **Maxima Method** \\ \hline SNR & 31.1687 & 38.1226 & 35.9149 & 41.1490 \\ RMSE & 10.0564 & 4.5159 & 5.8228 & 3.1873 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of SNR and RMSE after processing by four methods.  
Figure 4: Wavelet noise reduction effect: (**a**) original signal; (**b**) wavelet decomposition and reconstruction method; (**c**) hard threshold method; (**d**) soft threshold method; (**e**) wavelet mode maxima method.