## 7 Numerical Results and Discussion  
### Simulation Setup  
The parameters of the simulation environment are configured as in Table 2. We assume an OFDM system utilizing \(N=256\) sub-carriers loaded with QPSK or 16-QAM data. An oversampling factor \(\eta=4\) was chosen for the accurate tracking of the time-domain signal envelope. The RCM of the reference signal is simulated at 1.52 dB and the empirical factor \(x\) is fixed at the value of 1.56 for all simulated schemes. The clipping ratio defining the maximum allowable amplitude of the time-domain OFDM signal processed by ICF and SCF schemes is set to 6 dB. Three ICF iterations are used and the equivalency factor used in SCF is set to the same value. The signal's standard deviation parameter \(\sigma\) is set to \(1/2\) for the proper estimation of the frequency-domain scaling coefficient \(\rho\). The block size representing the total number of alternative sequences in SLM is set to 16. The feed-forward multi-layer perceptron NN architecture model is adopted in both ANN and the proposed SNN-CF scheme due to its simplicity and robustness in accurately implementing various mappings in the processing of wireless communication signals. ANN uses two NN modules with two hidden layers in each module. The first layer has two neurons and the second layer has one neuron. The activation function used in both hidden layers of ANN is the triangular function defined as \(f(x)\triangleq 1-|x|,|x|\leqslant 1\), and \(f(x)\triangleq 0\) otherwise. A design target \(E_{b}/N_{0}|_{t}\) is defined at the input of the PAPR reduction where the achievable BER is confined to a limiting threshold \(\mathcal{E}_{t}\). The proposed SNN-CF scheme deploys a single NN module which has two hidden layers. Both layers have two neurons \begin{table}
\begin{tabular}{c c} \hline \hline
**Scheme** & **Complexity** \\ \hline SLM [4] & \(\mathcal{O}(U(1.5 Nlog_{2}N+N+1))\) \\ \hline ICF [5] & \(\mathcal{O}(LN(3 log_{2}N+1))\) \\ \hline SCF [8] & \(\mathcal{O}(N(3 log_{2}N+4))\) \\ \hline ANN [12] & \(\mathcal{O}(20N)\) \\ \hline Proposed SNN-CF & \(\mathcal{O}(24N)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Complexity requirements of the investigated schemes. each. The activation function used to transfer the weighted input plus bias from the input terminal of a neuron to its output terminal is the hyperbolic tan-sigmoid transfer function defined as \(g(x)\triangleq 2/(1+e^{-2x})-1\ \forall\ x\in\mathbb{R}\). Both ANN and the proposed SNN-CF scheme use the _purelin_ activation function, \(h(x)\triangleq x\ \forall\ x\in\mathbb{R}\), for the output layer's neurons. The design threshold \(\Omega_{0d}\) used in Algorithm 2 for SNN-CF network selection is arbitrary and is set to 3 dB for both QPSK and 16 QAM cases. In addition, the maximum allowable BER, \(\mathcal{E}_{t}\), is set to \(3\times 10^{-4}\) and the design target \(E_{b}/N_{0}|_{t}\) is set to 8 dB for QPSK signals. For the 16-QAM case, \(\mathcal{E}_{t}\) is set to \(4\times 10^{-3}\) and target \(E_{b}/N_{0}|_{t}\) is set to 10 dB. The Levenberg-Marquardt algorithm [21] is used to train ANN to emulate the time-domain OFDM signal produced by conventional SCF when fed by the original unmodified OFDM signal at its input. The training epochs defining the number of training iterations