One of the unique characteristics of a ground target is its micro-motion, which can be used for target classification and identification. In this study, methods for vibrating ground target detection and feature extraction of the one-stationary bistatic frequency-modulated continuous-wave Synthetic Aperture Radar (SAR) are studied. The Displaced Phase Center Antenna (DPCA) technique is adopted to suppress the ground clutter, allowing the ground-vibrating targets to be detected. Analysis of the received signal indicates that the DPCA processing results in a slow time-varying envelope, known as the Slow Time Envelope (STE). The STE has a direct effect on the micro-Doppler time-frequency curve, which therefore cannot be obtained unbroken. Furthermore, vibrating features are extracted by utilizing their relationship with the STE term. Finally, some simulations are provided to validate the theoretical derivation and effectiveness of the proposed extraction method.