## 4 Simulation of Parameter Identification for Envelope Current Signal
### Simulation of Oblique Envelope Signal with 3 Times Harmonic
#### 4.1.2 The Analysis of WIFFT and Envelope Parameters Estimation  
The previous step, Hanning WIFFT, is employed to calculate the harmonic parameters. In addition, the signal envelope curve is used to estimate the envelope parameters \(a\) and \(b\). The two steps are separated in this analysis. Therefore, the frequency and phase precision of harmonic parameter identification is consistent with the previous analysis. The process of envelope parameter estimation is as follows. The coefficients \(p_{1}\) and \(p_{2}\) are given by linear fitting of the envelope, and the envelope parameters \(a^{*}\) and \(b^{*}\) are given by Formula (8). For the simulation model, the harmonic amplitude parameter in the Formula (8) can be a set value which is able to investigate the performance of the fitting algorithm. Meanwhile, in the analysis of the actual measured signal, the amplitude parameter should be the estimated values computed by WIFFT algorithm. In addition, the following formula  
\[(a\cdot nT_{\text{s}}+b)\cdot\sum\limits_{m=1}^{3}A_{m}\cos(2\pi\text{m}f_{0} \cdot nT_{\text{s}}+\theta_{m})=\frac{(a\cdot nT_{\text{s}}+b)}{k}\cdot\sum \limits_{m=1}^{3}k\cdot A_{m}\cos(2\pi\text{m}f_{0}\cdot nT_{\text{s}}+\theta _{m}),\forall k\
eq 0 \tag{12}\]  
indicates that the envelope parameters are correlated with the amplitude of harmonics. Hence, if the amplitude parameter has an error (the error of amplitude is about 10% in this example), the envelope parameters \(a\) and \(b\), calculated by the Formula (8), still have errors even if the fitting process is zero-error. The envelope parameters identification of this example is shown in Table 2.  
Similarly, the reconstructed signal based on the estimated harmonic parameters and envelope parameters is shown in Figure 4.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & **Valuation (\(A_{\text{m}}\) Is** & **Relative Error (\%)** & **Valuation (\(A_{\text{m}}\) Is the** & **Relative Error (\%)** \\  & **the Set Value)** & **Estimated Value)** & **Relative Error (\%)** \\ \hline \(a^{*}\) & 0.7161 & 43.22 & 0.6510 & 30.2041 \\ \(b^{*}\) & 0.9139 & 8.6111 & 0.8308 & 16.9166 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Identification and errors of envelope parameters of the oblique envelope signal.  
Figure 4: Simulation results of WIFFT and envelope parameters estimation method: (**a**) Comparison of original signal and reconstructed signal; (**b**) Absolute error of original signal and reconstructed signal.  
It can be seen from Figure 4 that the reconstructed signal was able to reflect the change of envelope since it considers the envelope parameters \(a\) and \(b\) in this analysis model. However, according to Table 2, the estimation precision of envelope parameters is poor. The errors of \(a\) and \(b\) are 8.6% and 43.2%, respectively, even if the harmonic amplitudes are the set values. In addition, RMSE is 3.5218, which is better than the previous analysis. Therefore, the relative independent method of parameter identification is still insufficient, and it cannot be used in envelope signal analysis.