Unlimited sampling was recently introduced to deal with the clipping or saturation of measurements where a modulo operator is applied before sampling. In this paper, we investigate the identifiability of the model where measurements are acquired under a discrete Fourier transform (DFT) sensing matrix first followed by a modulo operator (modulo-DFT). Firstly, based on the theorems of cyclotomic polynomials, we derive a sufficient condition for uniquely identifying the original signal in modulo-DFT. Additionally, for periodic bandlimited signals (PBSs) under unlimited sampling which can be viewed as a special case of modulo-DFT, the necessary and sufficient condition for the unique recovery of the original signal are provided. Moreover, we show that when the oversampling factor exceeds $3(1+1/P)$, PBS is always identifiable from the modulo samples, where $P$ is the number of harmonics including the fundamental component in the positive frequency part.

翻译：无限采样最近被引入以处理测量中的削波或饱和问题，其方法是在采样前应用模数算子。本文研究了在离散傅里叶变换（DFT）感知矩阵之后紧跟模数算子（模数-DFT）的测量模型的可辨识性。首先，基于分圆多项式定理，我们推导出在模数-DFT中唯一识别原始信号的充分条件。此外，对于可视为模数-DFT特例的无限采样下的周期性带限信号（PBS），给出了原始信号唯一恢复的充要条件。进一步地，我们证明当过采样因子超过$3(1+1/P)$时，PBS总能从模数样本中可辨识，其中$P$是正频率部分中包括基波在内的谐波数量。