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In downlink non-orthogonal multiple access (NOMA) with multiuser clustering, a single beamforming (BF) vector precodes the superimposed signal of strong and weak users. This precoding results in an imperfect inter-cluster interference (ICI) cancellation at weak users since the BF vectors are designed based on strong users' channels. Also, the ICI is highly increased with the number of transmit antennas, causing severe degradation in the performance of weak users and total system. In this paper, first , a new cooperative NOMA is introduced, in which weak user beam-matching (WBM) equalizer is proposed to remove the ICI at weak users by matching weak users' channels with the designed BF vectors. The process of WBM is accomplished with the aid of device-to-device (D2D) channel state information (CSI) sharing between the nearby strong and weak users. Second , based on WBM principle, strong user beam-matching (SBM) equalizer is proposed to eliminate the generated ICI at strong users in the case of limited feedback. Third , a new power allocation strategy is proposed to improve weak users' performance by considering the gained throughput from interference cancellation. Finally , besides the sum-rate, which is adopted as the performance metric by most of the existing NOMA works, the bit error rate (BER) of NOMA users in cooperative NOMA is calculated and compared with those in other schemes. Simulation results show that our cooperative NOMA with the proposed equalizers achieves significant sum-rate and BER improvements over other non-cooperative schemes with both perfect and limited feedback scenarios.