Adaptive-additive algorithm

In the studies of Fourier optics, sound synthesis, stellar interferometry, optical tweezers, and diffractive optical elements (DOEs) it is often important to know the spatial frequency phase of an observed wave source. In order to reconstruct this phase the Adaptive-Additive Algorithm (or AA algorithm), which derives from a group of adaptive (input-output) algorithms, can be used. The AA algorithm is an iterative algorithm that utilizes the Fourier Transform to calculate an unknown part of a propagating wave, normally the spatial frequency phase (k space). This can be done when given the phase’s known counterparts, usually an observed amplitude (position space) and an assumed starting amplitude (k space). To find the correct phase the algorithm uses error conversion, or the error between the desired and the theoretical intensities.

ϕnk respectively. Fourier transform the wave in k-space to x space.

I0f, where

ε against the convergence requirements. If the requirements are not met then mix the transformed amplitude A¯nf=[aAf+(1−a)Anf],

where a is mixing ratio and

A¯nk and ϕnk. Increase loop by one a=0 then Röbel, Axel (2006), "Adaptive Additive Modeling With Continuous Parameter Trajectories", IEEE Transactions on Audio, Speech, and Language Processing, 14 (4): 1440–1453, doi:10.1109/TSA.2005.858529, S2CID 73476.