Gas-insulated switchgear (GIS) is widely used in power systems due to its compact structure and high reliability; however, mechanical defects that develop during long-term operation may trigger partial discharges and pose risks to equipment integrity. To enable monitoring with custom-designed, low-power MEMS vibration sensors operating at low sampling rates, this study proposes an analysis method that combines coherent demodulation with low-frequency feature extraction. We used a 100 Hz fundamental component in simulated signals at multiple sampling rates as the coherent reference. Through coherent demodulation followed by narrowband processing, envelope amplitudes and frequency-correlated amplitude (FCA) features were extracted and used as surrogate indicators for high-frequency defect responses that cannot be directly captured at low sampling rates. Simulation results demonstrated that the method could differentiate between various operating conditions at low sampling rates. This study provides a feasible analysis pathway for low-rate, low-power GIS vibration monitoring and provides implementation guidance for coherent-demodulation-based feature extraction.