## 1 Introduction
However, LI cancellation is implemented at the cost of increased hardware cost and power consumption, particularly as extra-high-resolution analogue-to-digital converters (ADCs) are required to avoid signal saturation before entering digital processing, which may not be desirable in practical system deployments. Several studies have focused on the impact of ADC quantization errors on system performance [16; 17]. In practical communication scenarios, base stations or terminals use various methods, such as least squares (LS) estimation, minimum mean square error (MMSE) estimation, and even deep learning methods [18] to perform channel estimation in order to obtain channel state information (CSI) to select more efficient coding and modulation schemes to improve transmission performance. However, the Doppler shift and noise generated by the pilot signals used in the processing may lead to channel estimation errors. Some research has already studied the impact of channel uncertainty on system performance [19; 20; 21; 22]. Ref. [19] studied the ergodic capacities of both FD and HD modes for two proposed scenarios. By considering channel estimation errors, closed-form expressions and accurate approximations of ergodic capacities for two scenarios were derived, respectively. Ref. [20] investigated the outage performance of FD and HD vehicle-to-vehicle networks over double-Rayleigh channels with channel estimation errors. Ref. [21] studied a coordinated direct and relay NOMA sytem. Closed-form expressions for the outage probabilities and ergodic rates were derived with imperfect channel state information and imperfect SIC. Ref. [22] studied uplink NOMA short-packet communications in the presence of hardware impairments and channel estimation errors with different decoding schemes. Closed-form expressions of the average BLER for the high SNR regime were derived. Recently, the performance of FD C-NOMA systems has been investigated [23; 24; 25; 26]. In [23], the authors analyzed the outage probability and sum rate of the FD NOMA system over Nakagami-\(m\) fading channels. Ref. [24] obtained analytical expressions for the outage probabilities experienced by the users and the system outage probability of FD C-NOMA with an energy-harvesting network under a time-switching relaying protocol. The exact analytical expression of the outage probability for FD C-NOMA with physical-layer network coding was derived in [25]. Ref. [26] studied an FD C-NOMA spectrum-sharing system and derived the analytical expressions for the outage probability and ergodic capacity. Furthermore, some existing studies have considered the impact of various non-ideal factors. In [27], by considering the I/Q imbalance and an imperfect SIC, closed-form and approximate expressions for the outage probability and ergodic sum rate were derived. Ref. [21] investigated exact closed-form expressions for the outage probability and approximate closed-form expressions for the ergodic rates of the downlink users in the presence of an imperfect CSI and imperfect SIC. However, none of these studies considered the impact of ADC resolution and channel estimation errors on system performance. In this paper, we will analyze the system performance under the existence of loopback interference, ADC quantization noise, and channel estimation errors with outage probability and ergodic rate as the main performance metrics. The contributions of this paper are as follows:  
1. We consider a FD C-NOMA wireless communication system that uses an AF relay to assist in signal transmission between the base station and multiple users. The system performance analysis takes into account non-ideal conditions, such as LI caused by FD operation, channel estimation errors, and quantization errors generated by low-resolution ADCs. 2. We derive the theoretical expressions of the ergodic rate and outage probability for an FD C-NOMA multi-user system.