## 3 Theoretical Analysis
### Algorithm Flow
### Signal Denoising

To begin with, four groups of signals with noise are denoised under different orders \(k\) as described in Sections 3.1 and 3.2. Their _ESNR_ curves after denoising are obtained according to Equations (5) and (6) and are plotted in Figure 5. The variations of the slopes of the _ESNR_ curves are shown in Figure 6.

Figure 4: Waveforms of the four groups of noisy signals and the noiseless signal. Figure 5: _ESNR_ curves of four groups of noisy signals. As can be seen from Figure 5, the \(ESNR\) curves of the four groups of noisy signals are divided into two apparent stages, and the order of turning points of the two stages is the effective order. From Figure 6, the slopes of the \(ESNR\) curves are approximately zero when the order is greater than or equal to eight. According to the principle of the adaptive order determination algorithm and Equation (8), the effective orders of the four groups of noisy signals are all selected as eight. On the other hand, the noiseless signal \(y(t)\) is composed of four frequency components, suggesting that the optimal order of SVD denoising is eight. This example highlights the adaptive order algorithm's ability to find the optimal order of SVD denoising under different noise intensities and thus its strong adaptability. Figure 7 presents the waveforms of the four groups of denoised signals and the noiseless signal. Noticeably, the denoised signal maximally resembles its noiseless counterpart under different noise intensities with no apparent signal distortion. This indicates that the denoising is effective and stable. Figure 6: Slopes of \(ESNR\) curves of four groups of noisy signals. Figure 7: Waveforms of the four groups of signals after denoising and the noiseless signal. Next, the proposed denoising method is compared with the SVD mean method, the SVD mutation method and the wavelet soft thresholding method. According to Equations (4) and (5), adopting _SNR_ and _RMSE_ as the evaluation metrics and with reference to Equations (4) and (5), their denoising performance is evaluated in Figures 8 and 9.

method fails. The SVD mutation method is on par with the proposed method on the 30 dB signal, where the \(SNR\) is improved by about 20 dB. Nevertheless, when the noise in the signal gradually increases, the singular value representing the noise becomes larger. This is when the mutation between the singular value representing the noise and the noiseless signals is weakened, and the denoising performance deteriorates. For the signals of 5 dB to 20 dB, its resultant \(SNR\) improvement is 4 dB lower than that of the proposed method. Compared with the three considered methods, the denoising performance of the proposed method is less affected by noise, and the selected order \(k\) is more effective. The proposed method increases the \(SNR\)s of the four groups of noisy signals by about 20 dB on average and results in the minimum \(RMSE\), manifesting strong adaptability and robustness. ### Detecting Harmonic and Interharmonic Signals

The Prony algorithm is highly sensitive to noise and its fitting accuracy can be easily compromised. It can have many false components in its fitting results, and the amplitude of certain false components may be higher than that of non-false components, making it difficult to get rid of the spurious components. These drawbacks lead to low reliability of the Prony algorithm in practical application. In this section, the Prony algorithm and the proposed method are applied, respectively, to detect harmonics and interharmonics of the four groups of noisy signals.