On the stability and sensitivity of multidimensional signal reconstruction from Fourier transform magnitude

In this paper, we deal with the problem of retrieving a finite-extent signal from the magnitude of its Fourier transform. We consider the discrete phase retrieval problem as a special case of a more general problem which consists of recovering a real-valued signal x from the magnitude of the output of a linear distortion: |Hx| (j), j=1 ....,n. An important result concerning the conditioning of this problem will be obtained for this general setting by means of algebraic-geometric techniques. In particular, the problems of the existence of a solution for phase retrieval, conditioning of the problem and stability of the (essentially) unique solution will be addressed.