## 3 Minimum Nonlinearity PL Design Guidelines
### Extension of the IMPM to Account for Nonlinearities
\right] \tag{8}\]  
From the expressions above, we can compute all the unknown wave amplitudes from the known input amplitudes \(\mathbf{a_{A}}\):  
\[\mathbf{a_{B}} = \mathbf{\Gamma_{B}}\mathbf{b_{B}} \tag{9}\] \[\mathbf{b_{A}} = \left\{\mathbf{S_{AA}}+\mathbf{S_{AB}}\mathbf{\left(\Gamma_{B}}^ {-1}-\mathbf{S_{BB}}\right)^{-1}\mathbf{S_{BA}}\right\}\mathbf{a_{A}}\] (10) \[\mathbf{b_{B}} = \mathbf{\left(I-S_{BB}\ \Gamma_{B}\right)^{-1}}\mathbf{S_{BA}\ a_{A}}. \tag{11}\]  
Now, the aggregated radiated field for a particular excitation and load condition can be calculated as follows from both \(\mathbf{a_{A}}\) and \(\mathbf{a_{B}}\):  
\[\bar{E}_{tot}(\theta,\phi)=\sum_{n\in pts}a_{n}\bar{E}_{n}(\theta,\phi). \tag{12}\]  
A number of quantities of interest can now be obtained from these wave amplitudes:  
\[P_{in} = \|\mathbf{a_{A}}\|^{2}-\|\mathbf{b_{A}}\|^{2} \tag{13}\] \[P_{sw} = \|\mathbf{b_{B}}\|^{2}-\|\mathbf{a_{B}}\|^{2}\] (14) \[P_{ant} = \|\mathbf{a_{A}}\|^{2}+\|\mathbf{a_{B}}\|^{2}-\|\mathbf{b_{A}}\|^ {2}-\|\mathbf{b_{B}}\|^{2}\] (15) \[\eta_{tot} = \frac{\eta_{ant}P_{ant}}{P_{in}}, \tag{16}\]  
where \(P_{in}\) is the net input power to the reconfigurable antenna system, \(P_{sw}\) is the power dissipated by the switches, \(P_{ant}\) is the net input power to the elementary radiator (the passive/linear portion of the system), and \(\eta_{tot},\ \ \eta_{ant}\) are the efficiencies for the full reconfigurable antenna system and the passive antenna element, respectively. Under a pure tone excitation, and considering that the antenna building materials and surrounding environment have a linear behavior for all operating circumstances, the only source of nonlinear distortion are the internal port loads and their frequency conversion characteristics. Although the presented formulation relates only one input frequency to the same output frequency, it can be extended to account for harmonics induced by the nonlinear loads by properly adapting the vectors and matrices involved in the calculations. In particular, if \(N_{H}\) is the number of harmonics to consider, the vector quantities in the presented formulation can be expressed as composed by sub-vectors of \(N_{H}\) terms each, forming a longer extended vector of size \(N_{p}N_{H}\). This implies also the extension of the involved matrices to include the relationships at the harmonic frequencies, therefore each matrix entry will be a sub-matrix of size \(N_{H}\times N_{H}\), totaling a size of \((N_{p}N_{H})\times(N_{p}N_{H})\) for a full \(N_{p}\)-ports matrix. In this case, \(\mathbf{S}\), which involves the linear antenna portion, will be a matrix composed by diagonal matrices, that is, for any pair of ports \((p,q)\), and harmonics \((m,n)\), \(S_{pq}(m,n)=0\), \(\forall m\
eq n\). On the other hand, given the non-linearity of the terminating loads (switches), \(\mathbf{\Gamma_{B}}\) will be a block diagonal matrix with each block representing a given switch, and all blocks being fully populated; therefore the reflection coefficient of a single load \(\Gamma^{l}\) must provide a representation for harmonic frequencies conversion, taking the form  
\[\Gamma^{l}= \begin{bmatrix}\Gamma^{l}(1,1)&\dots&\Gamma^{l}(1,N_{H})\\ \vdots&\ddots&\vdots\\ \Gamma^{l}(N_{H},1)&\dots&\Gamma^{l}(N_{H},N_{H}).\end{bmatrix}  \tag{17}\]  
From this expression, the convenience of the X-parameter formulation for the loads modelling results evident as it embeds the required frequency conversion as a function of the input power. As an additional remark, it must be noted that when input power is low enough, the cross-frequency components are negligible and consequently the \(\Gamma^{l}\) reduce to diagonal matrices, thus the harmonics calculated are only affected by signals in their same frequency, and the harmonic-extended system is reduced to a linear relationship.