Linear analysis of the 4-step Carr\'e phase shifting algorithm: spectrum, signal-to-noise ratio, and harmonics response

TL;DR

This paper presents a novel linear systems analysis of the nonlinear 4-step Carré phase shifting algorithm, deriving explicit formulas for its spectrum, harmonics rejection, and signal-to-noise ratio by decomposing it into linear and nonlinear components.

Contribution

It introduces explicit mathematical formulas for the spectrum, harmonics rejection, and SNR of the Carré algorithm by exploiting its decomposition into linear and nonlinear parts.

Findings

Derived explicit formulas for spectrum, SNR, and harmonics rejection.

Demonstrated the linear decomposition approach for analyzing nonlinear algorithms.

Provided new insights into the properties of the Carré phase shifting algorithm.

Abstract

We analyze the nonlinear Carr\'e 4-steps algorithm including its frequency response, signal-to-noise ratio, and harmonics rejection using linear systems theory. At first sight the previous statement as well as the title of this paper seems paradoxical. How can we analyze the 4-step non-linear Carr\'e Phase Shifting Algorithm (PSA) using linear system theory? The short answer is that the non-linear Carr\'e algorithm may be decomposed into two building blocks. The first block is a tunable linear 4-step PSA, and the second one is a non-linear phase-step estimator. Although this fact is well known from the derivation of the Carr\'e algorithm, nobody has properly exploited it. In other words, to this day, we do not have explicit mathematical formulae for a) the spectrum, b) the harmonics rejection, and c) the signal-to-noise ratio of the non-linear Carr\'e algorithm. These are the properties of the Carr\'e's PSA that we show here with novel and explicit mathematical formulae.