# On Self-Interference Cancellation and Non-Idealities Suppression in Full-Duplex Radio Transceivers
## 2 Proposed System Model and Techniques
### Kalman Filter  
The Kalman filter is one of the most significant discoveries of the 20 th century. One of its most promising and great immediate applications is controlling the complex dynamic systems. It propagates the current state of a dynamic system that is extremely important for the statistical analysis of a system. The propagated information includes the statistical properties of random dynamic perturbations and the influence of all the past measurements [70]. The Kalman filter utilizes the recursive structure, and hence helps in real-time execution. It is a statistical technique. It effectively describes the random forms of experimental measurements. The Kalman filter has relatively low complexity and is a discrete-time linear, finite dimension system [71]. The Kalman filter is usually applied to non-linear systems, and it does not require that the system, i.e., dynamic or random systems under analysis, to have static properties. It involves estimating the system's initial state, and the initial state is obtained by detecting it. The Kalman filter is used most solely for two purposes: (i) estimation of desired signal and (ii) estimator's performance analysis [70]. The Kalman filter is habitually applied to real-time applications because of its robustness and ability to deal with unknown signals. In the Kalman filter, mostly dynamic disturbances are modeled as additional unknown parameters or signals [72]. The Kalman filter has fewer computational requirements, it smoothes the noisy data, and estimates the parameters of interest [73]. Our purpose for applying the Kalman filter as a digital cancellation technique at the digital side of the transceiver is to construct an estimator that can accurately estimate the SI signal. This mitigates the residual SI signal before the digital signal processing of the desired signal, as shown in Figure 1. The Kalman filter provides the best estimate of \(x_{n,I}(n)\) and \(x_{n,Q}(n)\) at the Rx side through estimator of the form:  
\[\hat{y}^{\prime}_{n,I}(n)=c_{k,I}(n)x_{n,I}(n)+c_{k,I}(n-1)x_{n,I}(n-1)+\ldots+ c_{k,I}(1)x_{n,I}(1), \tag{42}\]\[\hat{y}^{\prime}{}_{n,Q}(n)=c_{k,Q}(n)x_{n,Q}(n)+c_{k,Q}(n-1)x_{n,Q}(n-1)+\ldots+c_ {k,Q}(1)x_{n,Q}(1), \tag{43}\]  
where \(\hat{y}^{\prime}{}_{n,l}(n)\) and \(\hat{y}^{\prime}{}_{n,Q}(n)\) are the estimated symbols and \(\{c_{k,I}(l)\}\) and \(\{c_{k,Q}(l)\}\), \(l=n,n-1,\ldots,1\) are the coefficients to be adapted to the knowledge of the model, \"\(n\)\" represents the total number of symbols. The Kalman filter involves two successive stages:  
(i) Prediction stage:  
\[\hat{y}^{\prime}{}_{n,I}(n|n-1)=\varnothing_{I}(n-1)\hat{y}^{\prime}{}_{n,I}(n- 1|n-1)+\Psi_{I}(n-1)U_{I}(n-1), \tag{44}\]  
\[\hat{y}^{\prime}{}_{n,Q}(n|n-1)=\varnothing_{Q}(n-1)\hat{y}^{\prime}{}_{n,Q}(n -1|n-1)+\Psi_{Q}(n-1)U_{Q}(n-1), \tag{45}\]  
(ii) Updating stage:  
\[\hat{y}^{\prime}{}_{n,I}(n|n)=\hat{y}^{\prime}{}_{n,I}(n|n-1)+k_{I}(n)\big{(}x _{n,I}(n)-H_{I}(n)\hat{y}^{\prime}{}_{n,I}(n|n-1)\big{)}, \tag{46}\]  
\[\hat{y}^{\prime}{}_{n,Q}(n|n)=\hat{y}^{\prime}{}_{n,Q}(n|n-1)+k_{Q}(n)\big{(}x _{n,Q}(n)-H_{Q}(n)\hat{y}^{\prime}{}_{n,Q}(n|n-1)\big{)}, \tag{47}\]  
\(\hat{y}^{\prime}{}_{n,I}(n)\) and \(\hat{y}^{\prime}{}_{n,Q}(n)\) are updated when \(x_{n,I}(n)\) and \(x_{n,Q}(n)\) become available; the amounts to modify the \(\hat{y}^{\prime}{}_{n,I}(n|n-1)\) and \(\hat{y}^{\prime}{}_{n,Q}(n|n-1)\), i.e., former estimates by a quantity proportional to the innovative part of the new observation. The updating stage of the Kalman filter includes an innovative process which is one of the essential concepts in the Kalman filter technique.