## 4 Simulation Analysis
### Estimated Performance at Different Frequencies  
In this subsection, the performance of the SP-Rife algorithm is evaluated across various frequencies. The quantization frequency \(f_{0}=19\) Hz is set, \(0.05\Delta f\) is the quantization interval, and 11 quantization frequency points \(f_{i}\) are taken between \(\left(f_{0}\sim f_{0}+\Delta f/2\right)\). The SNR is \(-4\) dB.  
The results comparing the SP-Rife algorithm with other Rife algorithms are illustrated in Figures 2 and 3. The Rife algorithm exhibits significant estimation errors near quantized frequencies. Despite improvements in the ASIQ-Rife algorithm aimed at reducing estimation errors near quantized frequencies, its fundamental spectral interpolation method has not been altered. Consequently, the estimation accuracy remains consistent within the optimal range. The PAI-Rife algorithm changes the criteria for determining the correction direction and the interpolation method, resulting in a more accurate estimation of the frequency deviation factor, but there are still fluctuations in errors near quantized frequencies. In contrast, SP-Rife achieves the most stable frequency estimation performance by increasing the frequency resolution through spectral refinement.  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Parameters** & **Value** \\ \hline Monte Carlo times & 10,000 \\ Sampling points (N) & 1024 \\ Quantization frequency (\(f_{0}\)) & 19 Hz \\ Sampling frequency (\(f_{s}\)) & 1024 Hz \\ Frequency resolution (\(\Delta f\)) & 1 Hz \\ Spectrum refinement range & 18 Hz: 0.01 Hz: 20 Hz \\ SNR & \(-20\) dB: 2 dB: 20 dB \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation parameters.  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Algorithm** & \begin{tabular}{l} **Complex** \\ **Multiplications** \\ \end{tabular} & **Complex Additions** & \
\begin{tabular}{l} **Computation** \\ **Points** \\ \end{tabular} &
\begin{tabular}{l} **Increased** \\ **Complexity** \\ \end{tabular} \\ \hline Rife & \(C_{M}\) & \(C_{A}\) & 15,360.0 & 0\% \\ ASIQ-Rife & \(C_{M}+8N\) & \(C_{A}+5.3(N-1)\) & 28,973.9 & 88.6\% \\ PAI-Rife & \(C_{M}+1.2N\) & \(C_{A}+0.8(N-1)\) & 17,407.2 & 13.3\% \\ SP-Rife & \(C_{M}+2\cdot 200\) & \(C_{A}+2\cdot(200-1)\) & 16,158.0 & 5.2\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Complexity of different Rife algorithms (\(N=1024\)).  
Based on the statistical analysis of the standard deviation, the mean standard deviation of the Rife algorithm across 11 frequency points is approximately 0.068. For the ASIQ-Rife algorithm, the mean standard deviation is about 0.045, while the PAI-Rife algorithm has a mean standard deviation of approximately 0.018. In contrast, the SP-Rife algorithm exhibits a mean standard deviation of around 0.013. This indicates a significant improvement in frequency estimation accuracy with the SP-Rife algorithm, achieving an 80.9% reduction compared to the Rife algorithm and a 27.8% reduction compared to the PAI-Rife algorithm. The SP-Rife algorithm consistently delivers a superior frequency estimation accuracy across various frequency points relative to other algorithms.  
Figure 3: The standard deviation of SP–Rife and other algorithms at different frequencies.  
Figure 2: The mean of SP–Rife and other algorithms at different frequencies.