## 2 Conventional SB_DSICED3_W/OF Scheme  
### System Model  
Let us consider a short-packet transmission model between two users, A and B, which are equipped with two antennas for simultaneously transmitting and receiving information in FD modes, as shown in Figure 1. At both User A and B's transceivers, the channel coding scheme is based on 5G QC-LDPC codes for both uplink and downlink short-packet transmissions [34]. We assumed that the intended channel gains between two users and the SI channel gains at User A itself are denoted by \(\mathbf{h}_{BA}\) and \(\mathbf{h}_{AA}\), respectively, which are i.i.d. complex Gaussian random variables with \(\mathcal{CN}(0,1)\). In FD transmissions, the SI channel normally consists of two components: Line-of-Sight (LoS) and Non Line-of-Sight (NLoS). By using passive cancellation and analogue cancellation techniques, the LoS component can be cancelled significantly while the reflections remain [22]. Therefore, in the digital domain, the SI channel can be modelled as a Rayleigh fading distribution. In FD transmissions, the channel estimation and decoding performance will be the same for both Users A and B because of the symmetric characteristic. Without loss of generality, we considered that the receiver side of User A and its received signal in the time domain can be given by:  
\[y(t)=y_{SI}(t)+y_{BA}(t)+w(t), \tag{1}\]  
where \(y_{SI}(t)\) is the self-interference component consisting of the SI signal \(\mathbf{x}^{\prime}_{SI}\) (the transmitted signal at User A is denoted by \(\mathbf{x}^{\prime}_{SI}\) rather than \(\mathbf{x}^{\prime}_{A}\) in order to distinguish the SI and the intended signal) propagating through the SI channel \(\mathbf{h}_{AA}\) and \(y_{BA}(t)\) is the received component consisting of the intended signal \(\mathbf{x}^{\prime}_{B}\) passing through the propagation channel \(\mathbf{h}_{BA}\). Therefore, Equation (1) can be expressed as:  
\[y(t)=(\mathbf{h}_{AA}*\mathbf{x}^{\prime}_{SI})(t)+(\mathbf{h}_{ BA}*\mathbf{x}^{\prime}_{B})(t)+w(t). \tag{2}\]  
The operator (\(*\)) denotes the convolution, and \(w(t)\) is the complex Gaussian noise with \(\mathcal{CN}(0,\sigma_{w}^{2})\). Then, the received summation signal is passed to the Analogue-to-Digital Converter (ADC) process to be converted to the discrete time domain signal, \(y[n]\). Here, in order to overcome the effects of residual quantisation noise error, the bit resolution and voltage dynamic range of Digital-to-Analogue Converter (DAC)/ADC devices should be chosen carefully and highly enough, which was implemented in [35], or the oversampling method should be applied if a low-bit resolution of ADC devices is used [36]. In this paper, we further assumed that the impacts of DAC/ADC, other hardware impairments on the SI  
Figure 1: SISO FD transmission with the SB_DSICED3_W/OF process. cancellation, and the problem of the synchronisation process were not considered (which is outside of the scope of this study, but essential in practice). After that, the DSIC process is applied to obtain the estimated SI channel \(\hat{\mathbf{h}}_{AA}\) by using an adaptive filter with the Recursive Least-Squares (RLS) algorithm, and its forgetting factor \(\lambda\) should be chosen between 0.9 and 1 [37]. The RLS algorithm was chosen because it has a faster convergence and better performance in the DSIC process compared to other algorithms such as the Least Mean Squares (LMS) or Normalised Least Mean Squares (NLMS) [38; 39; 40]. A reference signal \(x_{SI}^{\prime}[n]\) from the transmitter side of User A can be used to cancel the SI component to obtain:  
\[\tilde{y}[n]=y[n]-\hat{y}_{SI}[n]=y[n]-(\hat{\mathbf{h}}_{AA}*\mathbf{x^{ \prime}}_{SI})[n]. \tag{3}\]  
Then, an equaliser using the RLS algorithm is applied with the assistance of pilot symbols, which are integrated into the information message modulated sequence in the encoder process at the transmitting side of User B, in order to estimate the intended channel and obtain the equalised signal.