Noise reduction in a periodic displacement-measuring heterodyne interferometer using a continuous–discrete Kalman–Bucy filter

Abstract This brief report presents a continuous–discrete Kalman–Bucy filtering (CD-KBF) framework for enhancing the measurement accuracy of nanometre-resolution periodic displacement-measuring heterodyne interferometers, which are affected by environmental noise and thermal drift. Unlike conventional discrete-time approximations, the proposed approach incorporates the exact continuous-time harmonic dynamics of the actuator into the estimation process. This physically informed model enables effective separation of systematic drift from true oscillatory motion without introducing phase delay. Experimental results show that an optimised measurement noise covariance of $$R$$ = 0.001 provides an optimal trade-off between noise suppression and tracking bandwidth. Under this condition, the filter successfully eliminates thermal–mechanical drift while preserving the fidelity of nanometre-scale sinusoidal displacements (~ 1.32 nm), thereby enhancing measurement reliability under non-ideal conditions.