# Research on an Ultraviolet Spectral Denoising Algorithm Based on the Improved SVD Method
## 4 Experimentation and Evaluation  
### Denoising of SO\({}_{2}\) UV Spectral Signal Simulation  
We selected the original noiseless differential absorption spectra of 5 uL/L SO\({}_{2}\) gas within the wavelength range of 290 nm to 310 nm. The data had a sampling interval of 0.11 nm and a signal length of 203. We also used simulated signals with added noise at a SNR of 5 dB, as show in Figures 1 and 2. The noisy signal, with a SNR of 5 dB, is transformed into a 101 \(\times\) 103 Hankel matrix. This matrix is then subjected to SVD, resulting in the reconstruction of component signals for each singular value, following the previously described methodology. The resultant component signals are displayed in Figure 3.  
As depicted in Figure 3, the characteristics of the original signal are clearly evident in the singular value component signals. The sum of the first six component signals can be considered the genuine and effective representation of the original signal. A notable change occurs in the seventh component signal. According to the principle of SVD denoising, the order corresponding to the alteration in the singular value, which coincides with the optimal order for singular value noise reduction. In Figure 4, the first-order difference exceeding the predefined threshold of 0.3 serves as the effective order for SVD. It is evident that the order of the first singular value exceeding the preset threshold is 6, consistent with the observation of the singular value component signal transition from the seventh in Figure 3. The first six singular values are retained, and the remaining singular values are set to zero. The noise-reduced Hankel matrix is then reconstructed through the inverse transformation of SVD and condensed into a one-dimensional signal. The signal after denoising is smoothed, and its feature distribution closely aligns with the original signal. The denoising results for the noisy signal with a SNR of 5 dB are presented in Figure 5. The noise reduction algorithm proposed in this study is entirely data-driven and effectively distinguishes between the signal and noise based on the differences in singular value information. This capability enhances the accuracy of selecting the effective order for SVD denoising. Figure 1: Original Spectral Signal. Figure 2: Signal with Noise, Signal-to-Noise Ratio: 5 dB. To evaluate the denoising capability of the optimization algorithm for the UV spectrum of SO\({}_{2}\), we assess the denoising performance of the UV-difference spectrum of SO\({}_{2}\) using the proposed denoising algorithm, the SVDS method, and the wavelet denoising method. For the wavelet denoising method, we choose the \"sym14\" wavelet basis function and set the number of decomposition layers to five [20]. The simulation dataset comprises a total of 30 samples. To gauge the denoising effectiveness, we employ the SNR and RMSE metrics. The denoising results are presented in Figures 6 and 7. Figure 3: Singular Value Component Signals with Noise, SNR: 5 dB. (**a**) 1 st to 4 th singular value component signals; (**b**) 5 th to 8 th singular value component signals. Figure 4: Frequency difference spectrum of SVD signal. Figure 5: The denoising effect of the algorithm in this paper. Figure 6: SNR after denoising. The method presented in this paper demonstrates superior denoising effectiveness at lower SNR. It is shown that the improved SVD noise reduction algorithm proposed in this paper enhances the signal-to-noise ratio (SNR) by 18.02% and 16.86%, and reduces the root-mean-square error (RMSE) by 15.13% and 14.92%, respectively, compared with the SVDS denoising method and wavelet transform denoising method under the condition of low SNR. However, for higher SNR, approximately exceeding 20 dB, the denoising effect becomes comparable to that of the wavelet noise reduction method. The rationale for this approach is rooted in the observation that, once the SNR surpasses 20 dB, the noise content in the signal becomes minimal and closely resembles the original signal.