## 3 Proposed Mitigation Architecture for an RF Front-End
### Mitigation Architecture
\begin{array}{cccc}z_{2}(1)&z_{2}(2)&...&z_{2}(N)\end{array}  \right]^{T}\) and the notation \(H\) is the transposed-conjugate operator. Once the searching range of the best delay is determined, the optimization problem becomes:  
\[\underset{\omega,\tau}{\operatorname{argmin}}E(\omega,\tau) \tag{25}\]  
Equation (25) reveals that for each delay \(\tau\) within the searching range, \(E(\omega,\tau)\) reaches its minimum value after optimization. Among these values, the minimal one is chosen as the optimum solution and its corresponding \(\tau\) and \(\omega\) are taken as the best delay and the coefficients of nonlinear behavior kernels. As \(\omega\) in the objective function belonging to the complex number field, applying deviation directly to \(E(\omega,\tau)\) is infeasible. Let:  
\[\omega_{k}=\rho_{k}e^{j\varphi_{k}},d=U^{H}Z_{2}(\tau)=\left[ \begin{array}{ ccccc}d_{1}&d_{2}&...&d_{M}\end{array} \right]^{T},R=U^{H}U=\left[ \begin{array}{ ccccc}r_{11}&r_{12}&...&r_{1M}\\ r_{21}&r_{22}&...&r_{2M}\\...&...&...&...\\ r_{M1}&r_{M2}&...&r_{MM}\end{array} \right] \tag{26}\]  
where \(\rho_{k}\) and \(\varphi_{k}\) represent the amplitude and phase of \(\omega_{k}\), respectively. From Equation (24), we obtain:  
\[E(\omega,\tau)=\omega^{H}R\omega-\omega^{H}d-d^{H}\omega+Z_{2}^{H}(\tau)Z_{2} (\tau) \tag{27}\]  
Taking the derivative of \(E(\omega,\tau)\) with respect to \(\rho_{k}\) and \(\varphi_{k}\):  
\[\frac{\partial E}{\partial\rho_{k}}=e^{-j\varphi_{k}}\sum_{i=1}^{M}r_{ki} \omega_{i}+e^{i\varphi_{k}}\sum_{i=1}^{M}\omega_{i}^{*}r_{ik}-d_{k}^{*}e^{j \varphi_{k}}-e^{-j\varphi_{k}}d_{k}=0 \tag{28}\]  
\[\frac{\partial E}{\partial\varphi_{k}}=-j\rho_{k}e^{-j\varphi_{k}}\sum_{i=1}^{ M}r_{ki}\omega_{i}+j\rho_{k}e^{j\varphi_{k}}\sum_{i=1}^{M}\omega_{i}^{*}r_{ik}-jd_{k}^{*} \rho_{k}e^{j\varphi_{k}}+j\rho_{k}e^{-j\varphi_{k}}d_{k}=0 \tag{29}\]  
Let: \(\varphi=\left[ \begin{array}{cccc}e^{j\varphi_{1}}&&\\ &e^{j\varphi_{2}}&&\\ &...&\\ &&e^{j\varphi_{M}}\end{array} \right]\), \(\rho=\left[ \begin{array}{cccc}\rho_{1}&&\\ &\rho_{2}&&\\ &&...&\\ &&&\rho_{M}\end{array} \right]\), \(\frac{\partial E}{\partial\rho}=\left[\frac{\partial E}{\partial\rho_{1}}\frac{ \partial E}{\partial\rho_{2}}\)... \(\frac{\partial E}{\partial\rho_{M}}\right]^{T}\), \(\frac{\partial E}{\partial\varphi}=\left[\frac{\partial E}{\partial\varphi_{1}} \frac{\partial E}{\partial\rho_{2}}\)... \(\frac{\partial E}{\partial\rho_{M}}\right]^{T}\), \(\frac{\partial E}{\partial\varphi}=\left[\frac{\partial E}{\partial\varphi_{1}} \frac{\partial E}{\partial\rho_{2}}\)... \(\frac{\partial E}{\partial\rho_{M}}\right]^{T}\), \(\frac{\partial E}{\partial\varphi}=\left[\frac{\partial E}{\partial\varphi_{1}} \frac{\partial E}{\partial\varphi_{2}}\)... \(\frac{\partial E}{\partial\rho_{M}}\right]^{T}\), \(\frac{\partial E}{\partial\varphi}=\left[\frac{\partial E}{\partial\varphi_{1}} \frac{\partial E}{\partial\varphi_{2}}\)... \(\frac{\partial E}{\partial\rho_{M}}\right]^{T}\) and substituting them into Equations (28) and (29):  
\[\frac{\partial E}{\partial\rho}=\varphi^{*}R\omega+\varphi R^{*}\omega^{*}- \varphi d^{*}-\varphi^{*}d=0 \tag{30}\]  
\[\frac{\partial E}{\partial\varphi}=-j\rho\varphi^{*}R\omega+j\rho\varphi R^{*} \omega^{*}-j\rho\varphi d^{*}+j\rho\varphi^{*}d=0 \tag{31}\]  
Combining Equations (30) and (31), the result is \(R\omega=d\), i.e.,  
\[U^{H}U\omega=U^{H}Z_{2}(\tau) \tag{32}\]