Full-duplex systems are expected to double the spectral efficiency compared to conventional half-duplex systems if the self-interference signal can be significantly mitigated. Digital cancellation is one of the lowest complexity self-interference cancellation techniques in full-duplex systems. However, its mitigation capability is very limited, mainly due to transmitter and receiver circuit's impairments. In this paper, we propose a novel digital self-interference cancellation technique for full-duplex systems. The proposed technique is shown to significantly mitigate the self-interference signal as well as the associated transmitter and receiver impairments. In the proposed technique, an auxiliary receiver chain is used to obtain a digital-domain copy of the transmitted Radio Frequency (RF) self-interference signal. The self-interference copy is then used in the digital-domain to cancel out both the self-interference signal and the associated impairments. Furthermore, to alleviate the receiver phase noise effect, a common oscillator is shared between the auxiliary and ordinary receiver chains. A thorough analytical and numerical analysis for the effect of the transmitter and receiver impairments on the cancellation capability of the proposed technique is presented. Finally, the overall performance is numerically investigated showing that using the proposed technique, the self-interference signal could be mitigated to ~3dB higher than the receiver noise floor, which results in up to 76% rate improvement compared to conventional half-duplex systems at 20dBm transmit power values.