Application of complex-valued FXLMS adaptive filter to Fourier basis control of adaptive optics

In this paper, the Filtered-X Least Mean Square (FXLMS) adaptive filter with bias integration technique is applied to an adaptive optics system where the Discrete Fourier Transform is used to project the measured phase onto the Fourier basis for modal control. The control law is applied in the complex-valued coefficient space and the FXLMS algorithm is modified accordingly for the complex-valued control. Numerical analysis is conducted for a feedback loop of a single Fourier mode in the presence of a disturbance representing a frozen flow atmospheric turbulence. The performance is compared with a Kalman estimator based control law proposed in the literature called Predictive Fourier Control (PFC). The proposed method demonstrated a similar performance for a stationary disturbance and improved performance for a slowly drifting disturbance. Whereas the performance of the PFC is very sensitive to the accuracy of the identification of the disturbance, the proposed method does not require such an explicit identification and produces minimum error for the given disturbance.