## 2 A Rapid Output Matching Optimization Method with Dual-Objective Orientation
### Error Function Formulation  
As the S-OID results from compressing the upper space of the P-OID further, which itself is derived by removing a portion of the lower region from the OID, the set of \(Z_{\rm omm}(f_{\rm i})\) over the \(BW_{\rm d}\) may not entirely fall inside the smaller S-OID or simultaneously approach the respective \(Z_{\rm opt,s}(f_{\rm i})\) under a given OMN topology. To optimize the OMN parameters for these two objectives while considering practical constraints, an error function \(Err\) is proposed, which comprises an out-of-domain penalty term and a weighted matching error term, defined in three parts:  
(1) The out-of-domain penalty  
The \(BW_{\rm d}\) is known to consist of the \(BW_{\rm o}\) and a redundant bandwidth of 4 GHz in total. As a hard constraint, all \(Z_{\rm omm}(f_{\rm i})\) within the \(BW_{\rm o}\) must be inside the S-OID; otherwise, the total error (\(E_{\rm tol}\)) is instantly assigned an upper bound of 1 (0 \(\leq E_{\rm tol}\leq\) 1). In contrast, for redundant bands, deviations in \(Z_{\rm omm}(f_{\rm i})\) from the S-OID are allowed, but each outlying impedance point incurs a penalty \(K\) to \(E_{\rm tol}\). The sum of these penalties cannot exceed 10% of \(E_{\rm tol}\) (i.e., 0.1). If there are \(Q\) impedance points under consideration in redundant bands, and \(P\) of them fall outside the S-OID, then \(K=0.1/Q\), yielding a total out-of-domain penalty of \(KP\). (2) The matching error  
The objective is to minimize the distance between \(Z_{\rm omm}(f_{\rm i})\) and \(Z_{\rm opt,s}(f_{\rm i})\), as a smaller distance indicates better impedance matching. The matching error is therefore modeled as the Euclidean distance \(d(f_{\rm i})\) in the complex plane, calculated according to Equation (10), and is normalized by the maximum value among all \(d(f_{\rm i})\). The weighted sum of normalized matching errors is given a coefficient of 0.9 in \(E_{\rm tol}\), maintaining \(E_{\rm tol}\) within the [0, 1] interval. \[d(f_{\rm i})=\left|Z_{\rm omm}(f_{\rm i})-Z_{\rm opt,s}(f_{\rm i})\right| \tag{10}\]  
(3) Weight allocation  
Because the large-signal characteristics of transistors deteriorate as the frequency increases, the matching accuracy at higher frequencies is given greater priority by increasing the matching error weight \(W_{\rm i}\) with respect to \(f_{\rm i}\). In this context, a frequency-dependent weighting scheme is introduced. By evaluating \(P_{\rm 1 dB}\), PAE, and OPG at six equally spaced frequencies over \(BW_{\rm d}\) under the condition that the load of the final-stage transistor is set to \(Z_{\rm opt,s}(f_{\rm i})\), it is found that the PAE is most sensitive to frequency. Let \(\eta(f_{\rm i})\) represent the PAE at \(f_{\rm i}\), with the maximum among these values denoted as \(\eta_{\rm max}\). The definition of \(W_{\rm i}\) is provided in Equation (11), where the adjustment factor \(\beta\) controls the extent of the weight difference between low and high frequencies. Empirically, a recommended range for \(\beta\) is [0.1, 0.3]. The normalized form of the weights, \(w_{\rm i}\), is obtained according to Equation (12). In conjunction with the HB simulation outcomes at the 1 dB compression point, the values of \(w_{\rm i}\) for \(\beta=0.1\) are listed in Table 2.  
\[W_{\rm i}=e^{\beta\left[\eta_{\rm max}-\eta(f_{\rm i})\right]} \tag{11}\]  
\[w_{\rm i}=\frac{W_{\rm i}}{\sum\limits_{i=1}^{N}W_{\rm i}} \tag{12}\]In conclusion, the complete formulation of \(Err\) is given in Equation (13). Lastly, a suitable optimization algorithm, such as particle swarm optimization (PSO), differential evolution, or simulated annealing, is employed to iteratively tune the component values of the OMN so as to minimize \(E_{\text{tol}}\). If \(E_{\text{tol}}\) remains at its initial value \(E_{\text{ini}}\) or consistently equals 1 during the optimization, this implies that the current OMN topology is inadequate and requires modification or increased complexity.