The 1/f noise in micro hemispherical resonator gyroscopes (mHRGs) induces bias instability (BI);however, theoretical understanding of 1/f noise propagation in force-to-rebalance (FTR) systems remains inadequate. For the first time, we have investigated the effect of 1/f noise attached to the amplitude of the AC signal on the instability of the output of the mHRG under constant angular velocity input in the FTR system, and propose a new model for the propagation of 1/f noise. We first demonstrate the fundamental dependence of 1/f noise Allan variance on low-frequency spectral components under large correlation time regimes. Through rigorous mathematical analysis, we prove that Allan variance is predominantly governed by low-frequency characteristics, with high-frequency components becoming negligible. We subsequently identify and characterize the flat amplitude-frequency response of FTR transfer functions within the lowfrequency domain, establishing their critical role in noise amplification. Analytical expressions are derived for this flat gain characteristic, which directly governs propagation coefficients mapping various 1/f noise sources to gyroscope bias output. We further develop quantitative models describing regulatory mechanisms by which system parameters control noise propagation. The influence of drive mode amplitude, frequency splitting, and proportional-integral controller coefficients on low-frequency gain is systematically analyzed, revealing distinct regulatory effects on five identified noise sources: external angular velocity fluctuations, anchor voltage variations,
mechanical parameter instabilities, demodulation artifacts, and feed-through interference.
Experimental validation through FPGA-based implementation confirms theoretical predictions,
providing a physics-based foundation for systematic noise optimization.