1-D continuous non-minimum phase retrieval using the wavelet transform

The phase retrieval problem arises when a signal must be reconstructed from only the magnitude of its Fourier transform; if the phase information were also available, the signal could simply be synthesized using the inverse Fourier transform. In continuous phase retrieval, most previous solutions rely on discretizing the problem and then employing an iterative algorithm. We avoid this approximation by using wavelet expansions to transform this uncountably infinite problem into a linear system of equations. The wavelet bases permit a solution by incorporating a priori signal information and they provide a structured system of equations which results in a fast algorithm. Our solutions obviate the stagnation problems associated with iterative algorithms, they are computationally simpler and more stable than previous non-iterative algorithms, and they can accommodate noisy Fourier magnitude information. This paper develops our 1-D continuous, non-minimum phase retrieval algorithm and illustrates its effectiveness with numerical examples