## 4 Numerical Analysis of the Thermal Performance
### Experiment Data Processing and Analysis  
Factors such as manufacturing error, deformation error, and electromagnetic interference (EMI) can be major error sources for the temperature measurement. Therefore, proper data post-processing is necessary for identify actual information from noisy observations. Proposed by [PERSON] et al. [24], the technique of empirical mode decomposition (EMD) separates an original one-dimensional time domain signal into a number of oscillatory components named as intrinsic mode functions (_IMF_s). Furthermore, EMD also employs Hilbert transform to these _IMF_s such that meaningful information of instantaneous frequency and instantaneous amplitude can be revealed [25]. Based on the principles of EMD, a new type of mean filter is proposed to obtained smooth trend of the measurement and suppresses interfering noises.  
Let \(s(n)\) be the original one-dimension discrete series. EMD can be regarded as a mathematical tool characterized by the following representation [26]:  
\[s(n)=\sum_{k=1}^{K}IMF_{k}(n)+r_{K+1}(n) \tag{18}\]  
where \(IMF_{k}(n)\) is the \(k\)-th oscillatory mode (or _IMF_) of the signal, and \(r_{K+1}(n)\) is the residual trend (usually a component in the form of low-order polynomials).  
Affected by actual environmental noises and the uncertainty of measurements, noisy contents were incorporated as interfering fluctuations occurred around the actual dynamic temperature curve in the measurement, shown in Figure 13a. The noisy fluctuations can be considered as high-frequency noises embedding in the low-frequency trend signal. Therefore, the original discrete signal \(s(n)\) can be mathematically modeled as:  
\[s(n)=\sum_{k=1}^{K}r_{k}(n)+LFS \tag{19}\]  
where \(r_{k}(n)\) is the \(k\)-th noise mode of the signal and \(LFS\) is the low-frequency component.  
The procedure of EMD can be briefly summarized as searching the extremes of the data, connecting the entire local maximum via cubic spline curves as an upper envelope curve and a lower envelope curve [27; 28]. As presented in Figure 13b, the enclosed area within the upper and the lower envelopes, marked by double dot dash lines, cover all the date samples between them. The instantaneous mean values can be evaluated as the trend line (Figure 13c). According to EMD, on condition that the resultant trend line does not satisfy the requirement, additional iterations named as the sifting process stated above is further conducted.  
Figure 12: Photo of the experimental setup.  
After performing post-processing, introduced as above, to the measured experimental data, temperatures of the measured points are presented in Figure 14. In Figure 14, the solid line is the first measure point nearest to the tool tip and the double dot dash line denotes the temperatures of the measure point of the greatest distance from the tool tip. The dimension of the horizontal axis indicates the time with a unit of second, while the dimension of the vertical axis indicates the temperature of the measure point with the unit of degrees Celsius. In Figure 14a, original curves demonstrating the temperatures of the specified points are displayed, from which it is remarkable that the high-frequency noises are so great that it is unable to distinguish the realistic temperatures of the points. In Figure 14b, the filtered data is shown. It is evident that the high-frequency noises have been substantially suppressed.  
Figure 14: Filter result. **(a)** Measured temperature signal; **(b)** filtered temperature signal.  
Figure 13: Filter principles of empirical mode decomposition (EMD). **(a)** The original data; **(b)** the envelope process; **(c)** the residue.