Injectivity Conditions for STFT Phase Retrieval on $${\mathbb {Z}}$$, $${\mathbb {Z}}_d$$ and $${{\mathbb {R}}}^d$$

Abstract

We study the phase retrieval problem for the short-time Fourier transform on the groups $${\mathbb {Z}}$$ Z , $${\mathbb {Z}}_d$$ Z d and $${{\mathbb {R}}}^d$$ R d . As is well-known, phase retrieval is possible once the windows’s ambiguity function vanishes nowhere. However, there are only few results for windows that fail to meet this condition. The goal of this paper is to establish new and complete characterizations for phase retrieval with more general windows and compare them to existing results. For a fixed window, our uniqueness conditions usually only depend on the signal’s support and are therefore easily comprehensible. In the discrete settings, we also provide examples which show that a non-vanishing ambiguity function is not necessary for a window to do phase retrieval.