## 3 System Implementation
### Active Self-Interference Cancel
#### 3.2.1. Canceler Noise Analysis  
The canceler output noise is added directly at the input of the receiver and must be kept low to achieve low overall NF. Both AMP\({}_{1}\) and AMP\({}_{\text{Q}}\) add noise proportionally based on their transconductance, with AMP\({}_{\text{Q}}\) contributing relatively less noise thanks to the capacitive (noiseless) degeneration. The SI signal at the receiver input determines the cancelling condition set by the magnitude (G\({}_{\text{m}}\)) and phase (\(\theta\)) of the canceler transconductance (i.e., the gain settings of AMP\({}_{1}\) and AMP\({}_{\text{Q}}\)). The noise current injected at the input of the LNTA from the canceller, neglecting the feedback resistors, is given as follows:  
\[\text{i}_{\text{n,canc}}^{2}=8\text{KT}\gamma\left(\sigma\text{G}_{\text{mI}}+ \text{G}_{\text{mQ}}\frac{\omega_{\text{o}}}{\omega_{\text{p}}}\right), \tag{9}\]  
where \(\sigma=\frac{\gamma+\text{gm},\text{R}_{\text{c}}}{1+\text{gm},\text{R}_{ \text{c}}}\sigma=\frac{\gamma+\text{gm},\text{R}_{\text{c}}}{1+\text{gm}, \text{R}_{\text{c}}}\) (\(\sigma\approx 1\) for \(\gamma\) close to 1). The noise factor expression can be simplified further as a function of the magnitude (G\({}_{\text{m}}\)) and phase (\(\theta\)) in relation to the self-interference signal as expressed in (10).  
\[\text{i}_{\text{n,canc}}^{2}=8\text{KT}\gamma\text{G}_{\text{m}}\text{f}( \theta), \tag{10}\]  
where \(\text{f}(\theta)=\left(\cos\theta+\sin\theta\frac{\omega_{\text{o}}}{\omega_{ \text{p}}}\right)\) is the excess noise factor accounting for the different contributions from AMP\({}_{1}\) and AMP\({}_{\text{Q}}\). When AMP\({}_{\text{Q}}\) is OFF f(\(\theta)\approx 1\) and. f(\(\theta)\ll 1\) when AMP\({}_{1}\) is off, where  
Figure 8: Canceler constellation: (**a**) before compensation; (**b**) after compensation.  
Figure 7: Quadrature error compensation.  
\(\omega_{\rm p}\gg\omega_{\rm o}\). The canceler noise factor referred to its input is thus calculated as \({\rm F_{\rm canc}}=1+\frac{\rm i^{2}_{\rm canc}}{4{\rm KTR_{\rm s}}G_{\rm m}^{2}}\). The noise current of the canceler referred to the input of the receiver is expressed by: \({\rm i^{2}_{\rm canc,ref}}=8{\rm KT}({\rm G_{m}}+\frac{1}{{\rm R_{H}}}+\frac{1} {{\rm R_{\rm FQ}}})f(\theta)/[(\frac{2{\rm n}}{{\rm R_{\rm LNA}}})^{2}]\), where \({\rm n}\) is the LNA input transformer turn ratio, \({\rm R_{\rm LNA}}\) is the differential LNA input impedance and \({\rm R_{\rm FI}}\), \({\rm R_{\rm FQ}}\) are the cross coupled feedback resistances for AMPI and AMPQ respectively. For \({\rm n}=0.7\), the canceler \({\rm Gm}=20\) mS and \({\rm Rs}=50\)\(\Omega\), the maximum noise contribution to the system when AMPI is fully ON, for \({\rm f(\theta)}=1\) is a factor of 0.58. The entire noise factor of the system is therefore calculated as \({\rm F_{\rm TOT}}={\rm F_{\rm RX}}+{\rm G^{2}}({\rm F_{\rm canc}}-1)\), where \({\rm G}\) is the canceler gain.  
Figure 9 shows the simulated and calculated noise figure of the canceler and LNA only where the calculated noise is expressed as: \({\rm NF_{\rm LNA+Canc}}=10\log_{10}[{\rm F_{\rm LNA}}+{\rm G^{2}}({\rm F_{\rm canc }}-1)]\), where \({\rm F_{\rm LNA}}=1.44\) at 2 GHz from Figure 6b in Section 3.1.