Non-Uniqueness Theory in Sampled STFT Phase Retrieval

Abstract

.The reconstruction of a function from its spectrogram (i.e., the absolute value of its short-time Fourier transform (STFT)) arises as a key problem in several important applications, including coherent diffraction imaging and audio processing. It is a classical result that for suitable windows any function can, in principle, be uniquely recovered up to a global phase factor from its spectrogram. However, for most practical applications only discrete samples—typically from a lattice—of the spectrogram are available. This raises the question of whether lattice samples of the spectrogram contain sufficient information for determining a function \(f\in L^2(\mathbb{R}^d)\) up to a global phase factor. In the present paper, we answer this question in the negative by providing general nonidentifiability results which lead to a non-uniqueness theory for the sampled STFT phase retrieval problem. Precisely, given any dimension \(d\) , any window function \(g,\) and any (symplectic or separable) lattice \(\mathcal{L} \subseteq \mathbb{R}^d\) , we construct pairs of functions \(f,h\in L^2(\mathbb{R}^d)\) that do not agree up to a global phase factor, but whose spectrograms agree on \(\mathcal{L}\) . Our techniques are sufficiently flexible to produce counterexamples to unique recoverability under even more stringent assumptions; for example, if the window function is real-valued, the functions \(f,h\) can even be chosen to satisfy \(|f|=|h|\) . Our results thus reveal the non-existence of a critical sampling density in the absence of phase information, a property which is in stark contrast to uniqueness results in time-frequency analysis.Keywordsphase retrievalsamplingtime-frequency analysislatticesymplectic geometrymetaplectic operatordiscretizationMSC codes42A3844A1594A1294A20

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