## 4 Simulation Results and Analysis
### Analysis of Algorithm Complexity  
We will split the algorithm into two parts for analysis: initial threshold calculation and follow-up iterations. To simplify the discussion, we will focus on comparing and analyzing the parts where the three algorithms differ. Initial Threshold: Given that the signal data length is \(N\), FCME uses the Heapsort algorithm [21] for sorting, with an average time complexity of \(Nlog_{2}N\). The proposed algorithm divides the signal data of length \(N\) into \(p\) segments, then further divides a segment of length \(N/p\) into \(n\) segments. By incorporating the initial threshold complexity estimation method from [PERSON] et al. 's differential amplitude-based FCME algorithm [19], the initial threshold complexity of the proposed algorithm is \(N/p+N+p+n+2\). Follow-up Iterations: The CME algorithm compares the amplitudes of \(N\) frequencies with the initial threshold one by one for classification, resulting in a complexity of \(N\). The FCME algorithm compares the amplitudes of the remaining \(0.9N\) frequencies with the initial threshold one by one for classification, resulting in a complexity of \(0.9N\). Similarly, the proposed algorithm compares the mean values of the \(p-1+n\) non-uniformly segmented data segments with the initial threshold, followed by a final comparison of \(N\) frequency  
Figure 8: Average false detection rate of multi-tone interference. amplitudes with the final threshold. Thus, the iterative computation complexity of the proposed algorithm is \(N+p+n-1\). By adding the complexities of the initial threshold computation and the follow-up iterations, the total complexity of each algorithm is obtained. The complexity of the CME algorithm is \(O(N)\), the complexity of the FCME algorithm is \(O(Nlog_{2}N+0.9N)\), while the complexity of the proposed algorithm is \(O(N/p+2N+2p+2n+1)\). Table 1 presents a comparison of the computational complexity and simulation time among the CME algorithm, the FCME algorithm, and the proposed algorithm. The simulation time is the cumulative time taken to perform 5000 Monte Carlo simulations for single-tone interference, multi-tone interference, and partial-band interference detection on a computer equipped with an Intel Core i5-10210U 4-core processor (Intel Corporation, Santa Clara, CA, USA) using MATLAB R2023b. It can be seen that, compared to the CME algorithm, the proposed algorithm is slightly slower in detecting single-tone and multi-tone interference, but slightly faster in \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Algorithm**} & \multirow{2}{*}{**Complexity**} & \multicolumn{2}{c}{**Simulation Time for Different Types of Interference (s)**} \\ \cline{3-5}  & & **Single-Tone** & **Multi-Tone** & **Partial-Band** \\  & & **Interference** & **Interference** & **Interference (10\%)** \\ \hline CME & \(O(N)\) & 2.307 & 4.658 & 14.566 \\ \hline FCME & \(O(Nlog_{2}N+0.9N)\) & 21.436 & 23.382 & 25.921 \\ \hline Proposed Algorithm & \(O(N/p+2N+2p+2n+1)\) & 3.406 & 5.342 & 13.316 \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of algorithm complexity between FCME and oroposed algorithm. Figure 9: Average false detection rate of partial-band interference (RBW = 10%). detecting partial-band interference. In contrast, compared to the FCME algorithm, the proposed algorithm improves the detection speed by nearly seven times for single-tone interference, four to five times for multi-tone interference, and nearly two times for partial-band interference. Moreover, as the number of interference frequencies increases, the detection time for both algorithms also increases. This is because a greater number of interference frequencies leads to a more dispersed interference signal energy, making it more challenging to determine the optimal segmentation threshold between interference and noise. Consequently, more iterations are required, resulting in longer detection times.