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In this article, we propose a downlink non-orthogonal 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) relay bridges the connection between the base station and the cell-edge users. Both full-duplex (FD) and half-duplex (HD) protocols are considered for the relay. 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 the ergodic sum rate at high signal-to-noise ratio (SNR) are derived. Numerical results demonstrate that: 1) the FD relaying NOMA system outperforms the HD relaying NOMA system at low SNR, but the situation is exactly the opposite at high SNR; 2) the mutual interference can …