Optimal phase calibration of nonlinear, delayed sensors

Sensor calibration is a routine task which is often performed under the assumption of linearity and immediate response. The present paper addresses the task of calibrating a statically or zero-memory nonlinear sensor given delayed measurements that can give rise to a multi-valued relationship. A simple, optimal, non-parametric figure-of-merit is proposed to eliminate the delay or phase lag in sensing without the use of parametric models or Fourier transformation. The phase estimation at a selected frequency is not accurate when some nonlinear distortions are present. It is shown that a delayed measurement of a calibration device under periodic oscillations, creates a Lissajous-like curve which encloses an area directly proportional to the delay time. An efficient numerical optimization based on Green's integral, the time-shift of the reference sensor is varied until a non-delayed, single-valued calibration curve is obtained.