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.

翻译：模采样技术通过在采样前施加模运算，近期被提出以增强传统模数转换器的动态范围。本文研究了一种测量模型的可辨识性，该模型采用离散傅里叶变换感知矩阵进行测量，随后施加模运算（模-DFT）。首先，我们基于测量次数及原始信号中零元素的索引，推导出模-DFT感知模型唯一可辨识的充要条件。随后，我们对三种特定情况进行了深入分析：测量次数为2的幂、质数以及两倍质数的情况。此外，我们研究了周期带限信号在模采样下的可辨识性，该情形可视为原始信号附加对称性与共轭约束的模-DFT感知模型。我们进一步给出了仅基于单周期内采样点数的充要条件，用于保证周期带限信号在模采样下的唯一可辨识性（尽管存在直流分量的模糊性）。此外，我们证明当过采样因子超过$3(1+1/P)$时，周期带限信号可在直流分量存在模糊性的前提下被唯一辨识，其中$P$表示正频率部分包含基波分量的谐波数量。最后，我们提出了一种通过求解整数线性方程组来估计原始信号的恢复算法，并通过仿真验证了所得结论。