On the Identifiability from Modulo Measurements under DFT Sensing Matrix

Modulo sampling (MS) has been recently introduced to enhance the dynamic range of conventional ADCs by applying a modulo operator before sampling. This paper examines the identifiability of a measurement model where measurements are taken using a discrete Fourier transform (DFT) sensing matrix, followed by a modulo operator (modulo-DFT). Firstly, we derive a necessary and sufficient condition for the unique identification of the modulo-DFT sensing model based on the number of measurements and the indices of zero elements in the original signal. Then, we conduct a deeper analysis of three specific cases: when the number of measurements is a power of $2$, a prime number, and twice a prime number. Additionally, we investigate the identifiability of periodic bandlimited (PBL) signals under MS, which can be considered as the modulo-DFT sensing model with additional symmetric and conjugate constraints on the original signal. We also provide a necessary and sufficient condition based solely on the number of samples in one period for the unique identification of the PBL signal under MS, though with an ambiguity in the direct current (DC) component. Furthermore, we show that when the oversampling factor exceeds $3(1+1/P)$, the PBL signal can be uniquely identified with an ambiguity in the DC component, where $P$ is the number of harmonics, including the fundamental component, in the positive frequency part. Finally, we also present a recovery algorithm that estimates the original signal by solving integer linear equations, and we conduct simulations to validate our conclusions.