## 2 Methodology
### FFT with A-CFS Method  
The efficacy of the FFT method in noise reduction is significantly enhanced by the choice of an appropriate cutoff frequency. Within the domain of practical signal processing, the FFT is a go-to technique for the removal of noises from signals with a known noise frequency or for the segregation of noisy elements using a fixed cutoff frequency [50; 51; 52; 53]. However, accurately identifying the specific noise frequency is often a complex task. [PERSON] et al. [48] has developed an innovative approach known as the Adaptive Cutoff Frequency Selection (A-CFS) technique, designed to enhance the clarity of time-series data by effectively removing noises. This sophisticated method harnesses the unique spectral properties of noises and employs cross-validation within the FFT framework to pinpoint an optimal cutoff frequency. With the determination of this threshold, the A-CFS can separate and exclude the superfluous high-frequency noises, ensuring that only the significant low-frequency signals are preserved. The specific operation steps are as follows:  
1. Utilize the estimated source signal, denoted as \(s^{\prime}(t)\) from \(\mathbf{\hat{S}}^{\prime}(t)=\left[s_{1}{}^{\prime}(t),s_{2}{}^{\prime}(t), \cdots,s_{n}{}^{\prime}(t)\right]^{T}\) as the input for subsequent processing stages. Establish the maximum signal recognition frequency, referred to as \(m\_f\), which is derived using the formula \(m\_f\leq 0.5\times s\_f\) in accordance with the Nyquist sampling theorem. Here, \(s\_f\) represents the pre-determined signal sampling frequency. Given that signals exceeding the maximum recognition frequency \(m\_f\) are prone to distortion, they are consequently excluded from further analysis to ensure data integrity and accuracy. 2. Acquire the frequency domain representation of the estimated source signal \(s^{\prime}(t)\), denoted as \(f\_s^{\prime}(t)\). It is essential to note that the signal's length in both the temporal and frequency domains remains consistent, ensuring a congruent basis for analysis. 3. Establish the lower frequency boundary, designated as \(low\_f\), and initiate the threshold value \(F\) from this lower limit up to the maximum frequency \(m\_f\). Define the parameters for the search interval as \(\Delta f=m\_f/num\), with \(num\) representing the total count of discrete frequency points within the frequency spectrum. The rationale for establishing a lower frequency threshold is the aim to concentrate on the valuable information, which predominantly resides within a specific lower frequency band. For the subsequent calculations, the value assigned to \(low\_f\) is determined to be \(0.25\times m\_f\), as informed by empirical analysis. 4. Leverage the threshold value \(F_{i}(i=1,2,\cdots num)\) to filter the frequency-domain signal, meticulously eliminating high-frequency noise that surpasses the frequency threshold \(F_{i}\). This targeted filtration is designed to isolate and retain the signal's essence, culminating in the acquisition of a refined, de-noised frequency-domain signal that is liberated from superfluous noise elements. 5. Secure the de-noised time-domain signal, denoted as \(sf^{\prime}(t)\), by employing the inverse FFT method. 6. Calculate the mean value of the estimated source signal \(s^{\prime}(t)\), denoted as \(\widetilde{s}^{\prime}(t)\) Compute the quadratic sum values, expressed as \(E^{2}(i,t)\), through the formula \(E^{2}(i,t)=(sf^{\prime}(t)-\widetilde{s}^{\prime}(t))^{2}\). This calculation serves to quantify the variance of the de-noised signal from its median counterpart across the time-domain samples, providing a measure of the signal's deviation post-de-noising process. 7. Identify the minimal values of \(E^{2}(i,t)\), assuming that the answer comprises \(E^{2}(m,t)\), and select the corresponding \(m\) threshold values \(F_{m}\) as the optimal cutoff frequency. This selection is pivotal for effectively filtering out noise from the time-series data, ensuring the fidelity and reliability of the signal for further analysis. 8.