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

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.