## 3 Experiment Results
### PAM Modulation  
In Figure 2a,b, the DESN symbols exhibit a clearer eye pattern compared to the RAW symbols. The Q factor for DESN symbols visibly surpasses that of the RAW symbols, indicating an optimal decision point of the PAM symbols. This observation implies that the BER could be improved through DeepESN processing. Figure 3a shows the BER curves for PAM4 signal processed by different equalization methods: RAW for raw signal (no signal processing), RLS for RLS equalization, and DESN for DeepESN equalization. With 250-Msamples/s signal generation/acquisition, the PAM4 signal oversampling factors are five samples per symbol (SPS) for 100 Mbps, 4 SPS for 125 Mbps, and 3 SPS for 167 Mbps. To observe the improvement of the system's transmission performance, BER curves without equalization (RAW signals) and with RLS are plotted alongside BER curves processed by DeepESN (DeepESN signals). It can be observed that, in all cases, the BER curves reached the error-free limit as the received power gradually increased. The BER curves without equalization and with RLS (RLS signals) experienced over 3.89 dB of power penalty when the data rate increased from 100 Mbps to 167 Mbps. As the received power increased from \(-27.5\) dBm to \(-20\) dBm, the RLS processed signals consistently performed better than the RAW signals. The RLS processed signal's BER curves reached the error-free limit at received powers (\(P\)) of \(-25.2\) dBm, \(-22.5\) dBm, and \(-21.4\) dBm for 100 Mbps, 125 Mbps, and 167 Mbps, respectively. Due to the system's nonlinear frequency response (lower response for higher frequency signals), the power penalty in RLS signals followed the trend observed in most communication systems. Figure 3a reveals a trend in BER curves: lower-data-rate signals reach its error floor more readily than those with higher data rates when they use the same equalization conditions. Furthermore, given a fixed data rate, DeepESN signals reach their error floor prior to both RAW and RLS signals. Additionally, this observation suggests that DeepESN signals exhibit better performance than RAW and RLS signals as they require less power to achieve error-free transmission. In contrast, DeepESN signals exhibited an exceptional performance that overcomes hardware limitations and provides improved BER for higher-data-rate signals. The BER curves of DeepESN-processed signals reached \(10^{-3}\) when \(P=-21\) dBm for 100 Mbps, \(P=-21.8\) dBm for 125 Mbps, and \(P=-22.5\) dBm for 167 Mbps. This trend shows that  
Figure 2: The eye diagrams for the 167 Mbps PAM signal (**a**) and the signal with DeepESN processing (**b**). (**c**) is the receiving waveform of the 100 Mbps OFDM signal (RAW: without equalization, DESN: with DeepESN processing, and Tx: transmitted signal). Comparing to the Tx signal, the mean square error (MSE) of RAW and DESN signals were 0.0095 and 0.0042, respectively. the DeepESN-processed signals performed significantly better at higher transmission rates. Higher-data-rate signals achieved better performance despite being limited by the hardware's bandwidth response. This phenomenon is also evidenced in the power-saving plot at the right of Figure 3a. This plot illustrates the difference in received power (power saving) when the BER curves reach the error-free limit under different data rates and equalization conditions. Compared to the RAW signal at 167 Mbps, all other cases achieve a certain power saving, up to 4.92 dBm, to attain error-free transmission. For the RAW and RLS signals, their power saving decreases with the growth of the data rate. In contrast, the power saving of DeepESN  
Figure 3: BER curves for PAM (**a**) and OFDM (**b**) modulations with different transmission speeds (“M for Mbps) and signal-processing methods (RAW for no equalization, RLS for recursive least squares filter, and DESN for DeepESN processing). Additionally, the power saving plot to reach the error-free limit for both PAM and OFDM is displayed.