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Narrowband Internet of Things (NB-IoT) is a recently standardized technology to support machine-type communications (MTC) in Long Term Evolution-Advanced (LTE-A) Pro networks. NB-IoT can enable energy-efficient communication with extended coverage on a narrow bandwidth of 180 kHz for low-cost MTC devices (MTCDs). The main challenge of supporting MTC in LTE-A Pro networks is to provide connectivity to a massive number of MTCDs. To overcome this challenge, in this paper, we propose a power-domain uplink non-orthogonal multiple access (NOMA) scheme for NB-IoT systems. By allowing multiple MTCDs to share the same sub-carrier, NOMA can provide connectivity to more MTCDs than orthogonal multiple access (OMA). We formulate a joint sub-carrier and transmission power allocation problem to maximize the number of MTCDs satisfying the quality of service (QoS) and transmission power …