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We analyze the nonlinear Carr\'e 4-steps algorithm including its frequencyresponse, signal-to-noise ratio, and harmonics rejection using linear systemstheory. At first sight the previous statement as well as the title of thispaper seems paradoxical. How can we analyze the 4-step non-linear Carr\'e PhaseShifting Algorithm (PSA) using linear system theory? The short answer is thatthe 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 anon-linear phase-step estimator. Although this fact is well known from thederivation of the Carr\'e algorithm, nobody has properly exploited it. In otherwords, to this day, we do not have explicit mathematical formulae for a) thespectrum, b) the harmonics rejection, and c) the signal-to-noise ratio of thenon-linear Carr\'e algorithm. These are the properties of the Carr\'e's PSAthat we show here with novel and explicit mathematical formulae.

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