Minimum Length Solution for One-Dimensional Discrete Phase Retrieval Problem Corneliu Rusu Technical University of Cluj-Napoca FETTI, Signal Processing Group Baritiu 26-28, Cluj-Napoca, ROMANIA Email: corneliu.rusu@ieee.org Jaakko Astola Tampere University of Technology TICSP, Signal Processing Laboratory P.O. Box 553, FI-33101, Tampere, FINLAND E-mail: jaakko.astola@tut.fi Abstract—Recently it has been shown that the one-dimensional discrete phase retrieval problem may not have always a causal solution for certain input magnitude data, but it has been proved that the extended form of the one-dimensional discrete phase retrieval problem has always a causal solution within the same conditions. In this work we are looking for the minimum length solution for one-dimensional discrete phase retrieval problem. The Non-uniform Discrete Fourier Transform based approach is introduced and experimental results are also presented. I. PHASE RETRIEVAL PROBLEMS One classical signal recovery task is the reconstruction of a Fourier transform pair from data on either or both domains. There are many examples of reconstruction of Fourier trans- form pair in optics, electrical engineering, quantum physics, and astronomy [1]. Within this class, the problem of phase retrieval is to reconstruct the signal from only magnitude measurements [2]: Problem 1: Given ˜ X ≥ 0, find x such that its Fourier transform X = F{x} satisfies |X| = ˜ X. In practice one deals with sampled data. In case of one- dimensional sequences the Discrete-Time Fourier Transform is used: Problem 2: Given ˜ X(ω k ) ≥ 0, k ∈ K ⊂ Z, find x(n) such that: X(ω)= ∞ ∑ n=-∞ x(n)e -jωn satisfies |X(ω k )| = ˜ X(ω k ) . If we assume that x(n) is causal and of finite length, the Discrete Fourier Transform (DFT) is typically implemented. The main one dimensional discrete phase retrieval (1-D DPhR) problem can be stated as follows: Problem 3: Let ˜ X(k), k =0, 1,...,N - 1 a sequence of nonnegative numbers, which will be called the input magnitude data. A solution of 1-D DPhR problem is a complex signal of length M (M ≤ N ) x(n), n =0, 1,...,N - 1, with x(n)=0 for n = M,M +1,...,N - 1, such that its Fourier transform X(k)= N-1 ∑ n=0 x(n)e -j 2πkn N , k =0, 1,...,N - 1. (1) satisfies |X(k)| = ˜ X(k) (2) for all k =0, 1,...,N - 1. Note that input magnitude data ˜ X(k) correspond to ω k = 2πk N , where k =0, 1,...,N - 1. Also the methods using autocorrelation from circular autocorrelation require M ≤ (N - 1)/2 [3], [4]. One can obtain a solution to 1-D DPhR problem by finding the zeros of z-transform of autocorrelation, Hilbert transform, computation of cepstral coefficients [5]–[7], but the most common approaches are iterative transform algorithms, which alternate between time and frequency domains [4]. Depending on the given input magnitude data ˜ X(k) and the signal length M , the problem may or may not have a solution. Indeed, a solution to the 1-D DPhR problem exists if certain conditions are satisfied by the input magnitude data, namely the corresponding trigonometric polynomial must be nonnegative [3]. Even though we cannot find (or there does not exist) a solution satisfying (2), we can use optimization methods to search that best approximates (2) in some sense. The following least-squares problem or empirical risk minimization [8] is the most well known: Problem 4: Find x(n), a discrete signal of length M (M ≤ N ), such that to minimize min x(n) N-1 ∑ k=0 [ ˜ X 2 (k) -|X(k)| 2 ] 2 (3) where ˜ X 2 (k) are called the measurements, and X(k) are given by (1) for all k =0, 1,...,N - 1. Note that Problem 4 has always solution, but Problem 3 may have solution or not. When Problem 3 has a solution, this verifies also Problem 4. Solutions to both Problem 3 and Problem 4 are subject to ambiguities [9]. Nevertheless, a solution to Problem 4 which is not a solution to Problem 3 gives the magnitudes of the DFT of the solution that are different from the input magnitude data. Unfortunately, small changes in input magnitude data can sometimes provide large changes in the phase of the solution of 1-D DPhR problem [10]. Consequently we may not have always strong arguments that the solution obtained to Problem 4 is indeed 2018 26th European Signal Processing Conference (EUSIPCO) ISBN 978-90-827970-1-5 © EURASIP 2018 475