# 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
In the form of correlation matrix lattice, backward and forward prediction error can be expressed as:  
\[Y=Y_{SI}+Y_{D}+W-Y_{est}, \tag{13}\]5  
\[Y=Y_{SI}+Y_{D}+W-Y_{est}, \tag{13}\]6  
where,  
\[Y=Y_{SI}+Y_{D}+W-Y_{est}, \tag{13}\]7  
With filtered prediction error:  
\[Y=Y_{SI}+Y_{D}+W-Y_{est}, \tag{13}\]8  
\[Y=Y_{SI}+Y_{D}+W-Y_{est}, \tag{13}\]9  
and:  
\[Y=(H_{SI}-H_{est})X_{PA}+H_{D}X_{D}+W, \tag{14}\]0  
\(y(t)\) is the desired output, and \(x_{m}(t,u)\) signal is considered linearly independent. Now the prediction error can be defined as:  
\[Y=(H_{SI}-H_{est})X_{PA}+H_{D}X_{D}+W, \tag{14}\]1  
and in the normalized form:  
\[Y=(H_{SI}-H_{est})X_{PA}+H_{D}X_{D}+W, \tag{14}\]2  
In the GALL filter, the expectation operator is defined with the help of exponential forgetting window \(0<\gamma\leq 1\), as:  
\[E_{\gamma}(x(t))=\left\{ \begin{array}{c}0\ t<0\\ \gamma E_{\gamma}[x(t-1)]+x(t)\ t>0\end{array} \right.\]  
Here, all the terms involved in expectation are dependent upon time and result in a time-variant system. In the GALL filter, correct pole adaption \"\(u\),\" i.e., change in pole direction, the time-variant approach is used in which all the parameters are estimated on a sample-by-sample basis, and for the sign verification of \(c_{m}(u)\)_and_\(c_{m+1}(u)\), the adaption rule is:  
\[u(t+1)=u(t)+\alpha(t)sign[c_{m}(t,u)c_{m+1}(t,u)] \tag{33}\]  
Additionally, \(\alpha(t)\) should be very small, i.e., order of \(10^{-6}\). Any change in \"\(u\)\" will result in a transient state in the GALL filter algorithm, increasing error. GALL filter operates only in the stationary case, so rapid adaption of poles is also worthless, and secondly as \(|u|\ \to 0\), the value of \(\alpha(t)\) must decrease. Coefficients of the filter are adapted to diminish the power of the estimated error. \[e_{m}(t,u)=e_{m-1}(t,u)+h(t,u)*[w_{m}(t,u)b_{m}(t,u)] \tag{34}\]  
and:  
\[w_{m}(t,u)=w_{m-1}(t,u)-\frac{\mu}{{\sigma_{b_{m}}}^{2}(t,u)+\delta}\hat{b}_{m} (t,u)e_{m}(t,u) \tag{35}\]  
Now the updated coefficient equation is defined as:  
\[w(t,u)=w(t-1,u)-\mu(t,u)\sum{}^{-1}(t,u)\hat{b}(t,u)e_{m-1}(t,u) \tag{36}\]  
and hence the residual error can be calculated as follows:  
\[e_{m-1}(t,u)=d(t,u)*\left[w^{T}(t-1,u)b(t,u)\right] \tag{37}\]  
In the next stage, the estimated transmitted signal is used to cancel SI from the received signal to obtain the desired one. Let the estimated SI signal from the GALL filter be \(y_{GALL}(t)\) and error be \(\varepsilon_{GALL}(t)\) now the signal at receiver after analog/RF cancellation by GALL filter can be expressed as:  
\[y_{RF}(t)=(y_{ch}(t)-y_{GALL}(t))+\varepsilon_{GALL}(t) \tag{38}\]  
As can be seen from Figure 6, GALL filter error \"\(\varepsilon_{GALL}(t)\)\" decays at a very rapid pace and approaches to zero within a time of about 50 \(\mu\)s.  
The Kautz filter is an infinite impulse response (IIR) based tapped delay line transversal filter. It can be well-defined by a set of stable poles and a corresponding set of tap-output weights. It offers valuable means for optimization and control. It provides linear in-parameter models and adaptive filtering for both non-fixed and fixed structures and various systems identification and approximation schemes [66]. Kautz filter designing can be seen as two different aspects of the optimizing system model and its equalization: (i) choosing an optimal set of poles of Kautz filter \(\{z_{j}\}\), and (ii) evaluating a corresponding optimal set of Kautz filter tap coefficients or weights \(\{w_{j}\}\). These are the two parameters (poles  
Figure 6: Error analysis of GALL filter. and tap coefficients or weights) that determine the filter structures.