  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.
