In this paper, we establish a comprehensive uplink system model for in-band and guard-band Narrowband Internet of Things (NB-IoT) with arbitrary sample duration in the NB-IoT device. The mathematical expressions of received LTE and NB-IoT signals are derived. Moreover, the close-form interference power on the LTE signal from the adjacent NB-IoT signal is given analytically. The result shows that the sample duration of NB-IoT device has significant impact on its desired signal and on the interference to the LTE user equipment (UE). Numerical results show that the analytical expressions match the simulated ones perfectly, which verifies the effectiveness of proposed system model and derivations. The work in this paper provides a valid guidance for NB-IoT system deployment and co-existence analysis.