Generalized phase-shifting algorithms: error analysis and minimization of noise propagation

Abstract

Phase shifting is a technique for phase retrieval that requires a series of intensity measurements with certain phase steps. The purpose of the present work is threefold: first we present a new method for generating general phase-shifting algorithms with arbitrarily spaced phase steps. Second, we study the conditions for which the phase-retrieval error due to phase-shift miscalibration can be minimized. Third, we study the phase extraction from interferograms with additive random noise, and deduce the conditions to be satisfied for minimizing the phase-retrieval error. Algorithms with unevenly spaced phase steps are discussed under linear phase-shift errors and additive Gaussian noise, and simulations are presented.

Signal detection algorithms for interferometric sensors with harmonic phase modulation: miscalibration of modulation parameters

“…Secondly, the distortions of the measured phase analyzed in this article do not arise in the interference scheme or in the blocks of signal formation and registration, but in the process of phase demodulation by the eqs. ( 9) or (10) due to the fact that these expressions were derived under the assumption Δψ = Δθ = 0. Given these circumstances, it is appropriate to use numeric simulation for verification of the obtained analytical results, including the direct calculation of the interference signal samples and their demodulation according to the eqs.…”

“…Given these circumstances, it is appropriate to use numeric simulation for verification of the obtained analytical results, including the direct calculation of the interference signal samples and their demodulation according to the eqs. ( 9) and (10).…”

“…Since this analysis does not include noise, in order to make a direct numerical calculation of the error Δφ for fixed parameters of the signal φ0, ψm, Δψ and Δθ, it is sufficient to calculate 4 (for the algorithm "OLS-4") or 5 (for the 4+1 algorithm) samples of the form (8), then calculate φr according to eqs. ( 9) or (10), and then find the difference Δφ = φr -φ0. Such relatively simple calculations not only allow us to check the correctness of the eqs.…”

“…This issue has already been raised in a number of works dedicated to interference signal demodulation. In [7][8][9][10][11][12][13][14][15][16] the influence of this step deviation on the resulted phase φr was analyzed, and also special equations, allowing to reduce the resultant error were developed for the case of linear auxiliary modulation with step 2π/N. As for harmonic auxiliary modulation, the error caused by amplitude and initial phase miscalibration for a specific demodulation algorithm with fixed parameters, minimizing the impact of additive noises is investigated in [17].…”

Signal detection algorithms for interferometric sensors with harmonic phase modulation: distortion analysis and suppression

“…Initially, interferometric systems utilized analog modulation and demodulation methods [1,[8][9][10], although, some of them are used nowadays. However, most state of the art interferometric systems utilize digital demodulation techniques, allowing more flexibility to algorithm design [2,3,7,[11][12][13][14]. Such techniques generally imply some demodulation interval, formed by Q signal samples.…”