# 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 this approach, CSI is performed using the training period in which a known training sequence or pilot is transmitted over the channel. The required channel matrix is estimated by knowing the transmitted signal and the received antenna's signal. Tx does not directly access CSI; hence, mainly, CSI is utilized at the Rx side, quantized, and fed back to the Tx. Perfect CSI holds when the channel remains static, or the variations are minimal compared to the transmission rate. Separate transmit and receive antennae are used, and the desired signal of infinitesimal strength compared to self-interfering signal is considered for analysis, as shown in Figure 4. The signal at the Rx antenna \"Y\" can be expressed as:\[Y=Y_{SI}+Y_{D}+W, \tag{9}\]  
\[Y=H_{SI}X_{PA}+H_{D}X_{D}+W, \tag{10}\]  
Her,e \(Y_{SI}\) is the SI signal, \(Y_{D}\) is the desired signal, and W is the AWGN noise with mean zero and variance \(\sigma_{W}^{2}\). Channel State Information at Rx (CSIR)  
During the phase of obtaining CSI at the Rx side, the training sequence is transmitted over the channel and channel \(H_{SI}\) is estimated from the knowledge of the received signal \"Y\" and the training sequence \"\(P\)\". Let the training sequence be \(p_{1},p_{2},\ldots\ldots\ldots,p_{T}\), where \(P\) is transmitted over the channel. The statistics of the received signal can be described as:  
\[Y=H_{SI}P+W, \tag{11}\]  
Here, \(Y=[y_{1},y_{2},\ldots\ldots\ldots,\ y_{T}]\), \(P=[p_{1},p_{2},\ldots\ldots\ldots,\ p_{T}]\) and \(W=[w_{1},w_{2},\ldots\ldots,\ w_{T}]\), \"T\" is the length of the training sequence, and \"D\" is the length of the data sequence. The total training interval is defined as \(L=T+D\) and total time spent during training is defined as duty cycle factor \(=\eta=1-\frac{T}{L}\). Now, if each sub-carrier is orthogonal to the other, then the training symbols for N sub-carriers can be represented in matrix form as:  
\[P=\left[ \begin{array}{cccc}p[0]&0&\cdots&0\\ 0&p[1]&0&\cdots&\vdots\\ \vdots&0&\ddots&\ddots&\vdots\\ \vdots&\vdots&&&\vdots\\ 0&\cdots&\cdots&p[N-1]\end{array} \right], \tag{12}\]  
Here, \(E\{p[i]\}=0\) is the mean of \(i\)th sub-carrier and \(Var\{p[i]\}=\sigma_{i}^{2}\) is the variance of \(i\)th sub-carrier, where \(i=0,1,\ldots\ldots\ldots,\ N-1\). The estimators used for channel estimation to obtain CSI at the Rx side are least square (LS) and minimum mean square error (MMSE) channel estimators. Estimated mean square error (MSE) is proportional to the trace of \(P^{HP}\) and the MSE is minimized when \(P^{HP}=TI\), i.e., tr (\(P^{HP}\)) is scaled identity matrix and MSE can also be described as \(MSE=\frac{\sigma_{W}^{2}}{\sigma_{i}^{2}}\). Now, the CSI obtained from the training phase is further utilized for the cancellation of SI. As  
Figure 4: Analog/RF SI cancellation using CSI technique. the channel is assumed to be static, it remains constant for transmission duration. Here in our transceiver model, CSI gained at the receiver side can be used on the Tx side without any overhead of extra circuitry. Once CSI is obtained, SI can be cancelled by a satisfactory amount given by equations below:  
\[Y=Y_{SI}+Y_{D}+W-Y_{est}, \tag{13}\]  
\[Y=(H_{SI}-H_{est})X_{PA}+H_{D}X_{D}+W, \tag{14}\]  
where \(Y_{est}=H_{est}X_{PA}\). Hence after estimation of SI signal using CSI and cancellation of estimated SI signal from the total received signal, the signal left at Rx side can be expressed as:  
\[y_{RF.}(t)=h_{d}(t)*x_{d}(t)+(h_{ch}(t)-h_{est}(t))*x_{PA.}(t)+n_{th}(t), \tag{15}\]  
Blind Estimation deals with the issues of estimating an unknown signal or input at the receiver side. Input that suffered from phase and gained attenuation due to multi-path fading channel. For the proper equalization and estimation of the channel, the receiver does not require any training sequences from the transmitter. Blind estimation primarily relies on higher-order statistics (HOS) and temporal or spatial oversampling.