# On Self-Interference Cancellation and Non-Idealities Suppression in Full-Duplex Radio Transceivers
## 2 Proposed System Model and Techniques
### Transceiver's Components Linearization
#### 2.1.1 IQ Mixer Linearization  
A blind adaptive feedforward algorithm (BAFF) [48; 49; 50] is implemented for the compensation and estimation of gain and phase mismatch for transmitter chain IQ imbalance. For Rx chain IQ imbalance, a blind adaptive algorithm (BA) [51; 52] is implemented. BAFF algorithm is simple and has low complexity. Both BAFF and BA are blind and do not require any pilot symbols or training elements for estimation and compensation. Because of their blind nature, they can be easily incorporated into any transceiver and any communication standard. It does not involve training of the desired signal; instead, only statistical properties are needed. Both schemes do not have any stability or convergence issues. Figure 2 shows the complete block diagram of the IQ mixer with gain and phase mismatch and IQ mixer linearization technique (BAFF and BA algorithm). Tx: Blind Adaptive Feedforward Algorithm (BAFF)  
For its implementation, three estimators are required to estimate necessary parameters, which involve 1-bit quantization. In the BAFF algorithm, first, we estimate the three parameters \(\phi_{1}\), \(\theta_{2}\), \(\phi_{3}\). These three parameters are further used for the calculation of coefficients \(\gamma_{1},\gamma_{2}\) for the compensation of IQ imbalance, and for calculating new gain and phase \(\hat{g}\)_and_\(\hat{\phi}\). Output after applying BAFF algorithm can be expressed as:  
\[V_{IQ,TX}(t)=\frac{V_{I,TX}(t)+jV_{Q,TX}(t)}{\gamma_{2,TX}} \tag{2}\]  
where,  
\[ \begin{array}{c}V_{I,TX}(t)=g_{TX}\gamma_{2,TX}*U_{I,TX}(t)\\ V_{Q,TX}(t)=\gamma_{1,TX}*U_{I,TX}(t)+U_{Q,TX}(t)\end{array} \]  
IQ Mixer coefficients:  
\[ \begin{array}{c}\gamma_{1,TX}=\frac{\phi_{1,TX}}{\theta_{2,TX}}=g_{TX}sin( \varphi_{TX})\\ \gamma_{2,TX}=\sqrt{\frac{\phi_{3,TX}-\theta_{1,TX}^{2}}{\theta_{2,TX}^{2}}} =g_{TX}cos(\varphi_{TX})\end{array} \]  
Figure 2: IQ mixer imbalance in transceiver and IQ mixer linearization technique (BAFF and BA algorithm). New estimated gain and phase for Tx chain IQ mixer:  
\[\hat{g}_{TX}=\frac{\hat{\sigma}_{3,TX}}{\hat{\sigma}_{2,TX}}\text{ and }\hat{\varphi}_{TX}=\sin^{-1}\left(\frac{\hat{\sigma}_{1,TX}}{\hat{\sigma}_{3,TX}}\right)\]  
For Tx chain IQ mixer:  
\[ \begin{array}{c}\hat{\sigma}_{1,TX}=-E\left\{sgn(U_{I,TX}(t))U_{Q,TX}(t) \right\}\\ \hat{\sigma}_{2,TX}=E\left\{\left|U_{I,TX}(t)\right|\right\}\\ \hat{\sigma}_{3,TX}=E\left\{\left|U_{Q,TX}(t)\right|\right\}\end{array} \]  
Rx: Blind Algorithm (BA)  
As with the Tx chain IQ mixer, asymmetric IQ imbalance is also modeled in the Rx chain. The output of Rx IQ mixer after passing through LPF can be expressed as desired/wanted signal and image/unwanted signal, as shown in Equation (3):  
\[\hat{y}_{IQ,RX}(t)=G_{1,RX}(t)*y_{IQ,RX}(t)+G_{2,RX}(t)*y_{IQ,RX}{}^{*}(t), \tag{3}\]  
where,  
\[ \begin{array}{c}G_{1,RX}(t)=\frac{1+g_{RX}e^{-j\pi_{RX}}}{2}\\ G_{2,RX}(t)=\frac{1-g_{RX}e^{j\pi_{RX}}}{2}=1-G_{1,RX}{}^{*}(t)\end{array} \]  
\(g_{RX}\) and \(\varphi_{RX}\) represents amplitude mismatch and phase mismatch respectively, \(y_{IQ,RX}(t)\) and \(G_{1,RX}(t)\) are the desired/wanted signal components whereas \(y_{IQ,RX}{}^{*}(t)\) and \(G_{2,RX}(t)\) are the image/unwanted signal components. The above Equation (3) can be expressed in terms of orthogonal frequency division multiplexing (OFDM) symbols as non-negative \"\(O_{m}(n)\)\" and mirror \"\(O_{-m}(n)\)\" OFDM sub-carriers with IQ imbalance after ADC as:  
\[\left[ \begin{array}{c}O_{m}(n)\\ O_{-m}(n)\end{array} \right]=\left[ \begin{array}{cc}G_{1,RX}&G_{2,RX}\\ G^{*}{}_{2,RX}&G^{*}{}_{1,RX}\end{array} \right]\left[ \begin{array}{c}\hat {y}_{IQ,RX}(n)\\ \hat{y}^{*}{}_{-IQ,RX}(n)\end{array} \right]\]  
where, \(G=\left[ \begin{array}{cc}G_{1,RX}&G_{2,RX}\\ G^{*}{}_{2,RX}&G^{*}{}_{1,RX}\end{array} \right]\) : is an interference matrix  
\"\(G\)\" can be break into leakage and scaling matrix as:  
\[\hat{G}=\left[ \begin{array}{cc}1&\frac{G_{2,RX}}{G^{*}{}_{1,RX}}\\ \frac{G^{*}{}_{2,RX}}{G_{1,RX}}&1\end{array} \right]\left[