## 7 Numerical Results and Discussion
### Results and Discussion  
For QPSK-OFDM signals, the CM reduction performance was investigated for original unmodified OFDM, SLM, SCF, ICF, ANN, and the proposed SNN-CF schemes in terms of the complementary cumulative distribution function, \(\mathcal{F}(\Omega)=Pr\{\Omega>\Omega_{0}\}\), for different cubic metric threshold (\(\Omega_{0}\)) levels ranging from 2.5 to 6 dB as shown in Figure 5(a). The proposed SNN-CF scheme outperforms the best of the three conventional approaches and shows more than a 0.5 dB performance improvement compared to ANN at CCDF levels as low as \(10^{-3}\). Similarly, Figure 5(b) shows that the proposed SNN-CF scheme maintains its superior CM reduction performance for the higher order 16-QAM-OFDM case. This is accomplished by scanning the neighborhood of SCF during the training stage to find an SNN-CF network with the highest possible performance. The proposed SNN-CF exhibits lower OoB power leakage than ANN for both QPSK and 16-QAM modulations, as shown in Figure 7. This figure plots the power spectral density (PSD) curves of OFDM signals as a function of normalized frequency. Only original OFDM is included as a baseline for comparison since SLM has identical power distribution as the original OFDM and conventional ICF and SCF are both assumed ideal, rendering their PSD performance matching with the original unprocessed OFDM. In the neighborhood of the in-band, both ANN and the proposed SNN-CF show similar leakage for QPSK-OFDM as shown in Figure 6(a). However, the OoB spectrum radiation caused by the proposed SNN-CF scheme is rapidly diminished compared to ANN, facilitating more relaxed guard-band requirements between different operators. For 16-QAM-OFDM, OoB power leakage of the proposed SNN-CF scheme is consistently lower than ANN, as shown in Figure 6(b). This is mainly because the proposed SNN-CF is able to learn the amplitude variations of SCF more accurately than ANN. Figure 6: CM reduction performance of the proposed SNN-CF vs. ANN [12], selected mapping [4], simplified-CF [8], iterative-CF [5], and unmodified OFDM for (a) QPSK-OFDM, and (b) 16-QAM-OFDM. The BER performance of all QPSK-based schemes are compared in Figure 8a for \(E_{b}/N_{0}\) ranging from 0 to 9 dB. As expected, SLM preserves the signal integrity and results in the lowest possible BER which is identical to the BER performance of original OFDM. SCF and ICF come next and are closely followed by ANN and the proposed SNN-CF scheme. For 16-QAM-OFDM transmissions, the proposed SNN-CF scheme remains remarkably close to conventional SCF in terms of BER performance, as shown in Figure 8b. Although ANN maintains its CM reduction performance, it starts to exhibit a deteriorated BER performance resulting in about a 3 dB performance gap from conventional SCF at the \(10^{-3}\) BER level. In contrast, the proposed SNN-CF scheme remains within approximately 0.3 dB from conventional SCF at the same BER level. This is facilitated by the more accurate representation of SNN-CF to SCF signals. Figure 9a plots the BER performance of the proposed SNN-CF scheme vs. ANN, SLM, SCF, ICF, and raw OFDM signals for 16-QAM-based OFDM transmissions deploying a non-linear solid state HPA with the signal envelope amplitude, \(E\), undergoing an amplitude-to-amplitude (AM/AM) conversion accomplished by the transfer function [22]:  
\[f(E)=\frac{gE}{\left(1+\left(g\frac{E}{E_{0}}\right)^{2R}\right)^{1/2R}}, \tag{10}\]  
Figure 8: BER performance of the proposed SNN-CF vs. ANN [12], selected mapping [4], simplified-CF [8], iterative-CF [5], and unmodified OFDM for (**a**) QPSK-OFDM; and (**b**) 16-QAM-OFDM. Figure 7: Power leakage performance of the proposed SNN-CF vs. ANN [12] and unmodified OFDM in terms of normalized PSD vs. normalized frequency for (**a**) QPSK-OFDM; and (**b**) 16-QAM-OFDM modulation. where \(g\) is the small-signal gain, \(E_{0}\) is the output limiting parameter, and \(R\) is a parameter that controls the transition rate from the linear operation region to the limiting operation region.