## 3 An improved Method Based on WIFFT and DE  
In practice, the envelope current signal is often presented in a trapezoidal envelope form. Figure 1 shows the current signal of an electric locomotive in about 152 s and it has 764,568 data points. Since the gentle exponential envelope and parabolic envelope can be regarded as an oblique envelope, this signal actually contains lots of trapezoidal envelopes. In the trapezoidal envelope, the rise and fall part is the key to the analysis. Therefore, the trapezoidal envelope current signal is an oblique envelope one essentially. Therefore, this paper focuses on the parameter identification of the oblique envelope. The oblique envelope signal model takes this form,  
\[x(t)=\left[(at+b)\cdot\sum_{m=1}^{M}A_{m}\cos(2\pi f_{m}t+\theta_{m})\right]+ B_{0} \tag{6}\]  
In Formula (6), \(f_{\text{m}}\), \(A_{\text{m}}\) and \(\theta_{\text{m}}\) represent the frequency, amplitude and phase of the harmonic signal, respectively. \(a\) is the slope of the oblique envelope, and \(b\) is the intercept. The harmonic parameters can be estimated by Hanning WIFFT algorithm, which is presented in Formula (2). The DC component \(B_{0}\) also corresponds to the DC parameter of FFT and the envelope parameter is given by linear fitting. In order to improve the estimation precision and the iterative efficiency, the pre-estimate value (\(a^{*}\),\(b^{*}\)) obtained by the pretreatment of the envelope points can be used as the initial iterative value. The calculation process of the pre-estimate value is as follows. Figure 1: The current signal of an electric locomotive. (1) Search the envelope points of the signal. Find out the local maximum values and their times. Usually, the interval time between two adjacent local maximum points is about a power frequency cycle. The envelope curve is approximated as \(h(t)\approx p_{1}t+p_{2}\). (2) In order to improve the stability of parameter estimation, the linear fitting method is used to solve \(p_{1}\) and \(p_{2}\). It is effective to eliminate the noise interference in the envelope due to the mean square error minimum criterion is employed. According to Formula (6), \(p_{1}\) and \(p_{2}\) take this form. \[\left\{ \begin{array}{l}p_{1}\approx a\cdot\|\sum\limits_{m}A_{m}\cos(2\pi f_ {m}t+\theta_{m})\|\\ p_{2}\approx b\cdot\|\sum\limits_{m}A_{m}\cos(2\pi f_{m}t+\theta_{m})\|\end{array} \right. \tag{7}\]  
Therefore, the pre-estimate values of envelope parameters \(a^{*}\) and \(b^{*}\) are:  
\[\left\{ \begin{array}{l}a^{*}\approx\frac{p_{1}}{\|\sum\limits_{m}A_{m}\cos(2 \pi f_{m}t+\theta_{m})\|}\\ b^{*}\approx\frac{p_{1}}{\|\sum\limits_{m}A_{m}\cos(2\pi f_{m}t+\theta_{m})\|} \end{array} \right. \tag{8}\]  
Therefore, the pre-estimate values of envelope parameters (\(a^{*}\),\(b^{*}\)) and the harmonic parameters (\(A_{\rm m}\),\(\theta_{\rm m}\)) can be used to form the initial population of DE iteration. The initial population is  
\[\mathbf{P}=[\mathbf{X}_{1},\mathbf{X}_{2},\dots,\mathbf{X}_{M}]^{\rm T} \tag{9}\]  
where \(M\) is the population size and for an arbitrary individual, \(\mathbf{X}=\mathbf{X}_{min}+rand()\cdot(\mathbf{X}_{max}-\mathbf{X}_{min})\), where \(\mathbf{X}_{min}=0.5\)-[\(a^{*}\),\(b^{*}\),\(A_{\rm m}\),\(\theta_{\rm m}\)] and \(\mathbf{X}_{max}=1.5\)-[\(a^{*}\),\(b^{*}\),\(A_{\rm m}\),\(\theta_{\rm m}\)]. Root mean square error (RMSE) is taken as the target function \(f(\mathbf{X})\) of the DE algorithm, and it is defined as  
\[f(\mathbf{X})=RMSE=\sqrt{\frac{\sum\limits_{n}[x(n)-x_{c}(n)]^{2}}{N}} \tag{10}\]  
In Formula (10), \(x(n)\) is the sampling of the signal \(x(t)\), and \(x_{c}(n)\) is the reconstructed sequence by the identified parameters of the envelope signal, and \(N\) is the number of sequence. The smaller RMSE value indicates that the deviation of the actual waveform and the reconstructed signal waveform which is obtained by the combination algorithm is smaller, and the precision of parameter identification is also higher when using this combination method.