The Discrete Fourier Transform (DFT) is the primary tool of digital signal processing. By defining new variables for iterative transform phase retrieval using a diagonal form of the DFT matrix, the phase retrieval problem appears to decouple in special cases that can increase the computational efficiency of phase retrieval. The current state of the art in increasing the DFT computational efficiency is the Fast Fourier Transform (FFT). In prior art, no acknowledgment of special-purpose computational architectures is given, which is common to some digital signal processing applications.
NASA Goddard developed alternative computational strategies for the DFT that use analysis on geometric manifolds to provide a more general framework for DFT calculations. This technique also delivers a more efficient implementation of the DFT for applications using iterative transform methods, particularly phase retrieval. Diagonalizing the conventional DFT array introduces new variables that facilitate a decoupling of the conventional variables used in iterative transform phase retrieval.
The new DFT method exploits a special computational approach based on analysis of the DFT as a transformation in a complex vector space. Traditional iterative transform techniques can be slow to converge but in this new basis set, the algorithm can decouple to allow a closed form expression in special cases.