## 2 Related Works
They first estimated the signal parameters through a linear regression model, and then constructed a frequency estimation algorithm using the standard gradient descent approach. The method ensured the global exponential convergence of the frequency estimation and had a certain degree of robustness. However, the method relied on the persistently exciting (PE) assumption of the signal, and its performance could be affected when the noise level was high or the signal did not meet the PE condition. [PERSON] et al. [17] introduced a frequency estimation method based on the amplitude ratio of two DFT samples from zero-filled signals. This method employed the analytical expression of the DFT signal after zero-filling, defining the amplitude ratio function between the two DFT samples nearest to the actual frequency. Utilizing the monotonicity of this ratio function, the residual frequency was estimated using either the Newton-Raphson method or a leveling search. While this method markedly improved the estimation accuracy, it also increased the complexity of signal processing and the overall algorithm. To tackle the issue of incorrect interpolation direction in estimation algorithms that rely on spectral value ratios, [PERSON] et al. [18] proposed a frequency estimation method that uses the peak of the sidelobe. It was not always possible to determine the peak of the first sidelobe from the DFT spectrum due to the fence effect. The method zero-filled the signal and then performed a Fourier transform to accurately locate the peaks of the main flap and the corresponding peaks of the first para flap. However, the method was significantly affected by noise. It was suitable for signals with high signal-to-noise ratios, and a sufficiently large number of samples were required to ensure that the method contributed to the identification of the peak. [PERSON] et al. [19] proposed frequency estimation in the interpolated discrete Fourier transforms with a generalized maximum sidelobe attenuation window, defining the Fourier analysis concerning the generalized MSD (GMSD) window. By double-zero padding, the frequency values were estimated using a three-point interpolation algorithm, which provided a greater flexibility for noisy signals but had specific requirements on the signal-to-noise ratio of the processed signals. [PERSON] et al. [20] introduced a phase angle interpolation-based improved Rife frequency estimation algorithm called PAI-Rife, which utilized frequency shifting techniques. First, they redefined the existing interpolation method. Then, frequencies outside the frequency shift threshold range were moved to the optimal estimation space. This proposed algorithm reduced the impact of the quantization frequency distribution on estimation accuracy and improved the frequency estimation precision. However, the algorithm's noise resistance was limited, being effective only within the range of \(-10\) dB to \(10\) dB, making it unsuitable for harsher environments. Based on the frequency bias for the corrected frequency estimation, [PERSON] et al. [21] proposed the Golden-MacLeod (GM) algorithm. By improving the MacLeod algorithm and employing an iterative method to obtain the frequency bias, followed by the application of the golden ratio algorithm for frequency estimation, this method significantly reduced estimation errors and enhanced stability. However, the iterative process was influenced by the number of sample points, which led to an increased computational load. An analysis of the discussed algorithms indicates that the performance of frequency estimation methods is predominantly limited by the trade-off between accuracy and efficiency, especially under low signal-to-noise ratio conditions. The key research objective is to reconcile the reduced computational complexity with the improved accuracy. This paper tackles this issue by combining phase spectrum refinement and phase angle interpolation with the Rife algorithm, aiming to optimize the overall performance of frequency estimation.