## 3 Proposed Signal Demodulation Method  
In order to facilitate frequency parameter settings, this paper proposes a new signal demodulation method, which is shown in Figure 4. The main advantage of using an SFF to replace a BPF and an HPF in the traditional signal demodulation process is that the center frequency of an SFF can be adaptively adjusted according to the estimated rotational speed. Therefore, the frequency parameter setting is convenient while maintaining accurate rotor position estimation and good dynamic performance. It can be seen from the above flow chart that the response current \(i_{\text{afk}}\) in the ABC coordinate system can be detected. Then, the stator sampling current \(i_{\text{afk}}\) is transformed by [PERSON] to obtain the current \(i_{\text{a}\beta}\) in the \(\alpha\beta\) coordinate system. The current \(i_{\text{a}\beta}\) contains signal components with different frequencies and mainly includes the fundamental frequency feedback current \(i_{\text{a}\beta\prime}\), the high-frequency response current \(i_{\text{a}\beta\prime}\), and the high harmonic current \(i_{\text{a}\beta\prime}\) due to the switching device. The other sub-harmonic components are not considered because of their small content. The current \(i_{\text{a}\beta}\) can be expressed as follows:  
\[i_{\text{a}\beta}=i_{\text{a}\beta\prime}+i_{\text{a}\beta\prime}+i_{\text{a }\beta\prime} \tag{13}\]  
Figure 4: Flow chart of the signal demodulation. To extract rotor position information from high-frequency signals, the different frequency components of the signal should be separated. Transforming \(i_{\alpha\beta}\) using synchronous coordinate transformation to a coordinate system rotated synchronously by \(\omega_{h}\), the three components \(i_{\alpha\beta}\), \(i_{\alpha\beta h}\) and \(i_{\alpha\beta z}\) in the above equation become:  
\[i_{\alpha\beta f}=\mathrm{e}^{\mathrm{i}(-\omega_{h}t)}i_{\alpha\beta f}=I_{ \alpha\beta f}\mathrm{e}^{\mathrm{i}[(\omega_{e}-\omega_{h})t]} \tag{14}\]  
\[i_{\alpha\beta i2}=\mathrm{e}^{\mathrm{i}(-\omega_{h}t)}i_{\alpha\beta h}=I_{ ph}+I_{ht}\mathrm{e}^{\mathrm{i}2(\omega_{e}-\omega_{h})t]} \tag{15}\]  
\[i_{\alpha\beta c}=\mathrm{e}^{\mathrm{i}(-\omega_{h}t)}i_{\alpha\beta z}=I_{ \alpha\beta z}\mathrm{e}^{\mathrm{i}[(\omega_{p}-\omega_{h})t]} \tag{16}\]  
where \(\omega_{p}\) is the frequency of the harmonic components generated by the switching device, which is much higher than the frequency of the injection voltage. According to Equations (14)-(16), the frequency of \(I_{\alpha\beta}\mathrm{e}^{\mathrm{i}2(\omega_{e}-\omega_{h})t]}\) is obviously different from that of the other items \(I_{\alpha\beta f}\mathrm{e}^{\mathrm{i}[(\omega_{e}-\omega_{h})t]}\), \(I_{ph}\) and \(I_{\alpha\beta z}\mathrm{e}^{\mathrm{i}[(\omega_{p}-\omega_{h})t]}\). Therefore, a traditional method uses a BPF to extract the \(I_{ht}\mathrm{e}^{\mathrm{i}2(\omega_{e}-\omega_{h})t]}\) component. The upper and lower cut-off frequencies of the bandpass filter are fixed, but the frequency of the \(I_{ht}\mathrm{e}^{\mathrm{i}2(\omega_{e}-\omega_{h})t]}\) component will alter with the change in rotor speed. If the passband of the BPF is set too wide, the filtering effect will be poor. Otherwise, the useful signal will be filtered out. In order to achieve an ideal filtering effect, a BPF can be designed to change the center frequency adaptively with the speed Information. An SFF can adaptively change its center frequency with the motor speed, which can be used as a real-time variable center frequency BPF. The SFF transfer function used in this paper is shown in Equation (17).