## 3 PAI-Rife Algorithm
### PAI-Rife Algorithm
The comparison results are shown in Figures 4 and 5. Since the estimation error of the PAI-Rife algorithm and these two algorithms at different frequencies is small, the mean is relatively close. We conclude from the partially enlarged image of Figure 4 that the mean value calculated by PAI-Rife is closer to the actual frequency. With the change of the quantization frequency points, the standard deviation of the PAI-Rife algorithm is the smallest, and the estimation performance is the most stable. At the same time, since the phase angle interpolation is used, its frequency deviation correction ability is improved, and the estimation accuracy of different frequencies is better than the other two algorithms. Figure 3: Standard deviation of PAI\(-\)Rife and other Rife algorithms at different frequencies. Figure 2: Mean of PAI\(-\)Rife and other Rife algorithms at different frequencies. ### Estimated Performance at Different SNRs

The previous subsection analyzed the estimation performance of several algorithms for different frequencies. This subsection will investigate the interference of noise on frequency estimation. The quantization frequency point \(f_{i}=19.2\) Hz is set. The comparison results with other Rife algorithms are shown in Figures 6 and 7. We can conclude that since the ASIQ-Rife, An-Rife, and CQRE algorithms are improved in different ways, the estimation accuracy at low SNR is improved. Our proposed PAI-Rife algorithm changes the traditional spectral interpolation method to the phase angle interpolation method. It used the stability of the phase angle to reduce the influence of noise on the signal. Therefore, the PAI-Rife algorithm's mean of frequency estimation is the most stable in the environment with different SNRs. Figure 4: Mean of PAI\(-\)Rife and other frequency domain estimation algorithms at different frequencies. Figure 5: Standard deviation of PAI\(-\)Rife and other frequency domain estimation algorithms at different frequencies. It is calculated that the mean value of the standard deviation of the Rife algorithm is about 0.063. The CQRE algorithm performed the best among the three modified Rife algorithms, with a mean value of the standard deviation of about 0.02. The mean value of the standard deviation of the PAI-Rife algorithm is about 0.012. Compared with the CQRE algorithm, the anti-noise performance of the PAI-Rife algorithm is improved by 40%. The anti-noise performance is better than Rife, ASIQ-Rife, An-Rife, and CQRE algorithms. Especially in a low SNR environment, the improvement is more significant. The standard deviation is relatively closer to the CRLB. The comparison results between the PAI-Rife and other frequency-domain estimation algorithms under different SNRs are obtained from Figures 8 and 9. Due to the increased number of iterative computations, the length of the signal estimated by the HAQSE algorithm is limited. As the signal length increased, the estimation accuracy of the HAQSE algorithm gradually decreased. Therefore, in the low SNR environment, the estimation performance of the HAQSE algorithm is the worst. The PAI-Rife algorithm estimated a more precise frequency deviation factor through the proportional relationship between the phase angles. The estimated frequency value is closer to the actual frequency, which is not affected by the length of the signal. Figure 6: Mean of PAI\(-\)Rife and other Rife algorithms at different SNRs. Figure 7: Standard deviation of PAI\(-\)Rife and other Rife algorithms at different SNRs. ### Correction Direction Misjudgment Rate

There are two reasons for the error in frequency estimation: one is that the estimate of the frequency deviation factor is not accurate enough, and the other is that the correction direction is misjudged. In this subsection, the influence of the frequency deviation factor on the frequency estimation is analyzed, and the judgment results of the correction direction will be compared.