Circular Regression in a Dual-Phase Lock-In Amplifier for Coherent Detection of Weak Signal

Lock-in amplification (LIA) is an effective approach for recovery of weak signal buried in noise. Determination of the input signal amplitude in a classical dual-phase LIA is based on incoherent detection which leads to a biased estimation at low signal-to-noise ratio. This article presents, for the first time to our knowledge, a new architecture of LIA involving phase estimation with a linear-circular regression for coherent detection. The proposed phase delay estimate, between the input signal and a reference, is defined as the maximum-likelihood of a set of observations distributed according to a von Mises distribution. In our implementation this maximum is obtained with a Newton Raphson algorithm. We show that the proposed LIA architecture provides an unbiased estimate of the input signal amplitude. Theoretical simulations with synthetic data demonstrate that the classical LIA estimates are biased for SNR of the input signal lower than -20 dB, while the proposed LIA is able to accurately recover the weak signal amplitude. The novel approach is applied to an optical sensor for accurate measurement of NO 2 concentrations at the sub-ppbv level in the atmosphere. Side-by-side intercomparison measurements with a commercial LIA (SR830, Stanford Research Inc., Sunnyvale, CA, USA ) demonstrate that the proposed LIA has an identical performance in terms of measurement accuracy and precision but with simplified hardware architecture.