## 3 Theoretical Analysis
### Adaptive Order Algorithm
When the order \(k\) is less than eight, the system is in a state of over-denoising, where noise signals are filtered out along with a large number of noiseless signals. As \(k\) increases from two to eight, the filtered noiseless signal decreases rapidly and the \(SNR\) increases rapidly. When \(k=8\), \(SNR\) reaches its maximum and SVD has optimal denoising effect. At this point, the noiseless signal is maximally retained, whereas the noisy signal is maximally filtered out. As \(k\) increases further, the system enters the state of under-denoising, as the filtered noise signal decreases and the \(SNR\) gradually drops. Figure 2b presents the curve of \(ESNR\) obtained by Equation (6). For \(k\) of less than eight, the system is in a state of over-denoising. When \(k\) increases from two to eight, the over-denoising state diminishes and \(ESNR\) increases rapidly. When \(k>8\), the system enters the state of under-denoising, while \(ESNR\) continues to increase gently from this point onwards. This can be explained by Equation (6), where the noise signal not filtered in the under-denoising state is retained in the signal after denoising, \(S^{\prime}(i)\). This leads to a gentle increase of \(S^{\prime}(i)\) and a gentle decrease of \(|X(i)-S^{\prime}(i)|\) with an increasing \(k\), thus resulting in a gentle increase in \(ESNR\). Figure 2: Curves of \(SNR\) against order values and the slopes. (**a**) Curve of accurate SNR. (**b**) Curve of ESNR. (**c**) Change in slope of SNR curve. (**d**) Change in slope of ESNR curve. From Figure 2a,b, both the curves of accurate \(SNR\) and \(ESNR\) are divided into two apparent stages, with the turning points in both being \(k=8\). Moreover, it can be deduced from Figure 2a that \(k=8\) is the optimal order for SVD denoising. Hence, in practical applications, this optimal order can be determined by finding the state turning point of the \(ESNR\) curve. Figure 2c depicts the change in slope of the curve in Figure 2a. It can be observed that before point \(k=8\), the accurate \(SNR\) curve has a positive curve, \(SNR\) increases rapidly, and SVD's denoising capability is continuously enhanced. From \(k=8\) onwards, however, the slope is less than or equal to zero, and \(SNR\) gradually declines, with the denoising capability of SVD constantly weakened. Similarly, Figure 2d depicts the change in slope corresponding to the curve in Figure 2b. It can be seen that before \(k=8\), the \(ESNR\) curve has a positive slope and increases rapidly. After \(k=8\), the slope becomes positive and close to zero, and \(ESNR\) increases gently. Hence, it can be inferred that the change in slope of the accurate \(SNR\) curve demonstrates a consistent pattern with that of the \(ESNR\) curve, which further corroborates that the change in the \(ESNR\) can provide a basis for the selection of effective order \(k\), and the state turning point of the \(ESNR\) can be found based on the slope of the \(ESNR\) curve. Therefore, the following equation can be established according to the characteristics of the slope of the \(ESNR\) curve:  
\[\left\{ \begin{array}{l}f(k)=\frac{1}{2}[ESNR(k+2)-ESNR(k)],\frac{k}{2}\in N ^{+}\\ k_{opt}=arg[f(k)<0.1]\end{array} \right., \tag{8}\]  
where \(f(k)\) represents the slope of the \(ESNR\) curve and \(k_{opt}\) denotes the solution of \(f(k)\) when inequality \(f(k)<0.1\) holds for the first time, which is essentially the optimal order for SVD denoising. In short, when applying the SVD denoising algorithm, the optimal order \(k\) of SVD denoising can be determined adaptively according to Equation (8), thereby avoiding the influence of different noise intensities on the SVD denoising performance and achieving higher adaptability and robustness.