# Noise Cancellation Method Based on TVF-EMD with Bayesian Parameter Optimization
## 2 Preliminaries  
### Time-Varying Filter Empirical Mode Decomposition  
EMD can decompose a given signal \(x(t)\) into a set of component signals known as IMFs and residual \(r(t)\), as shown in the following equation:  
\[x(t)=\sum_{i=1}^{N}{{{\it im}f_{i}(t)}+r(t)} \tag{1}\]  
where \({{\it im}f_{i}(t)}\) represents the i-th IMF. The screening process of EMD comprises two steps: (1) estimation of the \"local mean\" \(m(t)\) and (2) recursive subtraction of the \(m(t)\) from the input signal until the resulting signal meets the stopping criterion.  
In TVF-EMD, the mono-components are replaced by local narrow-band signals to improve the performance of the EMD method. These signals have properties similar to those of the IMF but provide a meaningful Hilbert spectrum. The local instantaneous bandwidth is used to define local narrow-band signals, which necessitate that the bandwidth be below a given threshold value. This method aims to initially determine the local cut-off frequency and subsequently applies a time-varying filter. The TVF-EMD shifting process is performed using a time-varying filter, which primarily involves three main steps [12]:  
1. Local cutoff frequency rearrangement Step 1: Locate the maximum time consumption of \(x(t)\), expressed as \(u_{i},i=1,2,3\dots\) Step 2: Determine all intermittences, expressed as \(e_{j}\), where \(j=1,2,3\dots\) by setting a threshold value for the rate of change within a certain time span. These breaks should satisfy the following condition: \[\frac{\max(\phi_{bis}^{\prime}(u_{i}:u_{i+1}))-\min(\phi_{bis}^{\prime}(u_{i}: u_{i+1}))}{\min(\phi_{bis}^{\prime}(u_{i}:u_{i+1}))}>\rho\] (2) where \(\rho=0.25\), and \(\phi_{bis}^{\prime}\) is the bisection frequency. Step 3: The condition of the rising edge of \(\phi_{bis}^{\prime}(e_{j}:e_{j+1})\) is \(\phi_{bis}^{\prime}(u_{i+1})-\phi_{bis}^{\prime}(u_{i})>0\), and \(\phi_{bis}^{\prime}(e_{j}:e_{j+1})\) is considered the floor. Similarly, the falling edge is \(\phi_{bis}^{\prime}(u_{i+1})-\phi_{bis}^{\prime}(u_{i})<0\), and \(\phi_{bis}^{\prime}(e_{j+1}:e_{j})\) is considered the floor. The rest of \(\phi_{bis}^{\prime}\) refers to the peak. Step 4: By interpolating between the peaks, the final local cutoff frequency can be obtained. During the local cutoff frequency rearrangement phase, the TVF-EMD addresses the separation and intermittence problems.
2. Shifting process for TVF-EMD Step 1: Address the local cut-off frequency for signal \(x(t)\). Step 2: Filter the input signal \(x(t)\) using the time-varying filter (i.e., B-spline approximation filter) to obtain the local mean. The bandwidth threshold \(\zeta\) determines the separation effect and whether the input signal must be filtered. The B-spline order \(n\) is independent of the cutoff frequency estimate, which determines the attenuation and filtering effect of the TVF. Step 3: Verify that the residual signal satisfies the condition of the stopping criterion indicated below: \[\theta(t)=\frac{B_{Longhlin}(t)}{\varphi_{avg}(t)}\] (3)where \(B_{Loughlin}\) is the Loughlin instantaneous bandwidth, and \(\varphi_{avg}(t)\) is the weighted average instantaneous frequency. The detailed calculation process of TVF-EMD was discussed in previous studies [12; 32].