Hall sensors are commonly used to collect the rotor position information of the Permanent magnet synchronous motor (PMSM). Due to the low resolution of Hall sensors, only six discrete accurate positions can be returned after the motor rotor completes one revolution. However, the vector control system requires real-time access to the motor rotor position. Therefore, a novel method for rotor position estimation using the Luenberger observer with a variable bandwidth treatment is proposed. The method first transforms the signal returned by the Hall sensor into a vector and extracts the fundamental component of the transformed orthogonal signal through Fourier analysis. Then, a Luenberger observer based on the mechanical motion equation of the PMSM is established. In order to improve the position observation performance of the Luenberger observer under a broad range of speed and loaded conditions, the bandwidth of the observer is dynamically adjusted based on the detection of the motor operation status of the control system. The effectiveness and rationality of the proposed method are validated through simulation and experimentation.