## 1 Introduction  
Frequency-modulated continuous-wave (FMCW) radar is a widely utilized radar system, esteemed for its high measurement accuracy and robust anti-interference capabilities. Its performance is minimally affected by adverse weather conditions such as rain and snow, rendering it highly suitable for applications in military, traffic management, automotive collision avoidance, and liquid level measurement [1]. The FMCW radar generates a beat signal by mixing the transmitted signal with the reflected echo, thereby deriving information on the target's distance and velocity [2]. However, the beat signal acquired by FMCW radar is contaminated by noise, leading to significant frequency errors when using direct FFT. Therefore, the accurate estimation of the beat signal's frequency is a crucial problem in FMCW radar signal processing.  
The purpose of frequency estimation is to estimate the distance or velocity of a target object by using the phase difference or time delay of the beat signal, and its accuracy directly affects the accuracy of the distance and velocity measurements [3]. Due to the non-integer relationship between the sampling frequency and the actual frequency, spectral leakage [4] and the fence effect [5] can result in frequency estimation biases. In practical engineering, FFT-based interpolation is typically employed to enhance the accuracy of frequency estimation. [PERSON] et al. [6] proposed the Rife algorithm, which employs double spectral line interpolation. This algorithm uses the ratio between the maximum spectral line and the adjacent second-largest spectral line to interpolate and estimate the signal, thereby improving the accuracy of frequency estimation. Nevertheless, when the actual frequency is close to the quantized frequency point, the frequency estimation accuracy decreases. [PERSON] et al. [7] proposed an improved Quinn and Rife Automatic Segmentation algorithm, which adaptively selects the optimal estimation method for the frequency interval, thereby improving the stability of the algorithm. Nevertheless, the complexity is not considered. [PERSON] et al. [8] proposed an anticipated Rife interpolation algorithm. This algorithm improved the original single-discriminant method of Rife, enhancing the estimation stability within any frequency range. Nevertheless, the algorithm's noise suppression capability is relatively weak.  
To reduce estimation errors caused by noise and spectral leakage, and to balance both anti-interference performance and complexity, a FMCW beat signal frequency estimation algorithm based on spectral refinement and phase interpolation (SP-Rife) is proposed. This algorithm achieves a more accurate frequency estimation by improving the signal resolution and utilizing phase interpolation. The main steps are as follows:  
1. Spectral Refinement: The post-FFT spectral signal is refined to narrow the frequency search range and enhance frequency resolution, effectively separating the noise signal.
2. Frequency Deviation Factor is Defined: By defining the frequency deviation factor as the ratio of the maximum spectral amplitude to the sum of the maximum and second-largest spectral amplitudes, the relationship of spectral proportions is established. This approach enhances the algorithm's capability to correct frequency deviations.
3. Phase Angle Interpolation: Leveraging the stability of phase angles, the algorithm converts spectral amplitudes into phase angles to determine the frequency deviation factor and interpolation direction, thereby increasing the estimation accuracy.  
The rest of this paper is organized as follows: Firstly, a comprehensive review of existing frequency estimation algorithms is presented in Section 2. Then, the SP-Rife algorithm is proposed in Section 3. Furthermore, the simulation and experimental results are provided to validate the performance of the proposed algorithm in Section 4. Finally, the research work of this paper is summarized in Section 5.