# On Self-Interference Cancellation and Non-Idealities Suppression in Full-Duplex Radio Transceivers
## 2 Proposed System Model and Techniques
### SI Cancellation Techniques
#### 2.2.1 Analog/RF SI Cancellation Techniques
Many HOS-based blind approaches for blind estimation exist, i.e., Laguerre filter, Kautz filter, and gradient adaptive Laguerre lattice (GALL) filter, etc. [62]. As stated above, the Rx has no prior knowledge about the signal being transmitted over the channel. Due to the multi-path channel and reflection from the nearby environment, the Rx receives multiple copies of the transmitted signal, i.e., delayed, attenuated, and phase-shifted copies of the transmitted signal. To estimate and segregate the transmitted signal from the Rx side, two filters are used for blind estimation: (i) Kautz filter and (ii) GALL filter. After the estimation, segregation, and recovering of the SI signal at the Rx side, it is subtracted from the total received signal. This can be expressed as  
\[Y=(Y_{SI}+Y_{D}+W)-Y_{Kautz}, \tag{16}\]  
\[Y=(Y_{SI}+Y_{D}+W)-Y_{GALL}, \tag{17}\]  
Here, \(Y_{SI}\) denotes SI signal, \(Y_{D}\) denotes the desired signal, \(W\) indicates additive white Gaussian noise (AWGN) with mean zero and variance \(\sigma_{W}^{2}\) and \(Y_{GALL}\), and \(Y_{Kautz}\) denotes the estimated, segregated, and recovered SI signal. A GALL filter is a filter in which a gradient-based approach is used for pole adaption [63]. Pole adaption using a gradient-based method is a straightforward and less complicated technique. Hence, this makes GALL filter implementation easy and straightforward. A GALL filter is used because it wraps time in filter coefficients, i.e., requires less time to calculate and obtain filter coefficients. It also has a lower delay, reduced-order modeling, and requires a smaller number of coefficients and fewer computational operations [64]. It has a fast convergence response and can attain stability in less time. The GALL filter contains an additional free parameter, i.e., a free variable for the independence of memory and the number of adaptive coefficients. Thus, the GALL filter can easily model an extended response filter by using relatively few parameters. Both perform well in steady-state and under tracking conditions [65]. Here, in this paper, for pole adaption, a sign-based gradient algorithm is used. The transfer function of \(n\)th order Laguerre filter:  
\[LF_{n}(z,u)=L(z,u)\sum_{m=0}^{n}w_{n,m}A^{m}(z,u) \tag{18}\]Laguerre filters have an all-pass filter \(A^{m}(z,u)\) with a low pass pre-processing filter stage \(L(z,u)\). \(w_{n,i}\) shows adaptive coefficients of the filter. The Laguerre filter's correlation matrix can be expressed as:  
\[x_{m}(t,u)=L(q,u)A^{m}(q,u)x(t) \tag{19}\]  
Here, \(x_{m}(t,u)\) represents a Toeplitz matrix. For a lattice description of the Laguerre filter, consider a lattice predictor that transforms the filtered reference signal \"\(v(t,u)\)\" into orthogonal filter backward prediction errors \"\(b_{m}(t,u)\)\", \(0<m<M-1\), where M is the order of lattice predictor and \(\hat{h}(t,u)\) will be the estimated secondary path. The filtered reference signal \"\(v(t,u)\)\" can be expressed as:  
\[v(t,u)=x(t,u)*\hat{h}(t,u) \tag{20}\]  
Lattice backward and forward prediction error orthogonalization can be obtained as per Figure 5 and can be defined as:  
\[\hat{f}_{m+1}(t,u)=\hat{f}_{m}(t-1,u)-\kappa_{m}(u-1)\hat{b}_{m}(t-1,u) \tag{21}\]  
\[Y=Y_{SI}+Y_{D}+W-Y_{est}, \tag{13}\]0  
The assumption for orthogonality:  
\[Y=Y_{SI}+Y_{D}+W-Y_{est}, \tag{13}\]1  
\[Y=Y_{SI}+Y_{D}+W-Y_{est}, \tag{13}\]2 Analog/RF SI cancellation using GALL filter. The adjustment of \(\kappa_{m}(u)\) is carried out as follows by the recursive formula:  
\[Y=Y_{SI}+Y_{D}+W-Y_{est}, \tag{13}\]3  
where \(\overline{\mu}\) is a step size parameter and \(\delta\) is a regularization constant, and the power of both forward and backward prediction error of the \(m\)th stage can be defined as:  
\[Y=Y_{SI}+Y_{D}+W-Y_{est}, \tag{13}\]4  
where \(\beta\) is the smoothing parameter.