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In this paper, we propose a downlink nonorthogonal multiple access (NOMA) based coordinated direct and relay system with one cell-center user and multiple cell-edge users, where a decode-and-forward (DF) and full-duplex (FD) relay is used to assist the communication between the base station and the cell-edge users. We assume that the performance of the cell-edge users is subjected to the relay, and the cancellation of the mutual interference between the relay and cell-center user is imperfect. Both the exact analytical expression of outage probability and an approximate expression of ergodic sum rate at high singal-to-noise (SNR) are derived. Numerical results demonstrate that the proposed FD relaying NOMA system outperforms the half-duplex relaying NOMA system at low SNR.