In many signal processing problems arising in practical applications, we wish to reconstruct an unknown signal from its phaseless measurements with respect to a frame. This inverse problem is known as the phase retrieval problem. For each particular application, the set of relevant measurement frames is determined by the problem at hand, which motivates the study of phase retrieval for structured, application-relevant frames. In this paper, we focus on one class of such frames that appear naturally in diffraction imaging, ptychography, and audio processing, namely, multi-window Gabor frames. We study the question of injectivity of the phase retrieval problem with these measurement frames in the finite-dimensional setup and propose an explicit construction of an infinite family of phase retrievable multi-window Gabor frames. We show that phase retrievability for the constructed frames can be achieved with a much smaller number of phaseless measurements compared to the previous results for this type of measurement frames. Additionally, we show that the sufficient for reconstruction number of phaseless measurements depends on the dimension of the signal space, and not on the ambient dimension of the problem.

翻译：在实际应用中出现的许多信号处理问题中，我们希望从相对于一个框架的无相位测量中重建未知信号。这一逆问题被称为相位恢复问题。对于每个特定应用，相关测量框架的集合由具体问题决定，这促使我们研究结构化、与应用相关的框架的相位恢复。本文聚焦于衍射成像、叠层成像和音频处理中自然出现的一类框架，即多窗口Gabor框架。我们在有限维设置下研究使用这些测量框架的相位恢复问题的单射性，并给出一个无限族可相位恢复的多窗口Gabor框架的显式构造。我们证明，与这类测量框架的先前结果相比，所构造框架的相位恢复性可以用更少的无相位测量实现。此外，我们表明，足以重建的无相位测量数量取决于信号空间的维度，而非问题的环境维度。