## 3 Numerical Simulation
However, it is also observed that the NNWSC-based receiver achieves a stable and low BER level when the sub-band spacing factor is approximately \(0.2\), while the conventional receiver requires a sub-band spacing factor of approximately \(0.6\) to achieve a comparable BER performance, highlighting its capability to handle more complex channel environments. Next, we conducted simulations to analyze the effect of roll-off factors on the BER of each sub-band with sub-band spacing factors set to \(0\), \(0.2\), and \(0.7\). The simulation results are shown in Figure 4, illustrating the relationship between the BER of each sub-band and the roll-off factor. Figure 3: **(a)** The relationship between the BER and sub-band spacing factor per sub-band at a roll-off factor of \(0.4\); **(b)** constellation diagrams labeled ‘A’, ‘B’, and ‘C’ depict the constellation at sub-band spacing factors of \(-0.2\), \(0.3\), and \(0.8\), respectively. Constellation diagrams are arranged from left to right as follows: MF + LMS S1, MF + LMS S2, MF + LMS S3, NNWSC S1, NNWSC S2, and NNWSC S3. The simulation results show that, with a sub-band spacing factor of 0.2, the BER of the NNWSC-based receiver across all three sub-bands decreases initially and then stabilizes as the roll-off factor increases. This occurs because increasing the roll-off factor reduces ISI, and once ISI is minimized to a certain level, the BER reaches its lowest point and does not decrease further. The conventional receiver, however, shows a different BER trend across the sub-bands compared to the NNWSC-based receiver. At lower roll-off factors, the spectrum of the pulse-shaping filter is very compact, resulting in a concentrated frequency distribution but longer time-domain signal tails. In this case, a smaller roll-off factor increases ISI, which significantly affects the matched filter in the receiver, impeding effective signal recovery and subsequently reducing the effectiveness of the cascaded LMS  
Figure 4: (**a**) The relationship between the BER and roll-off factor per sub-band at sub-band spacing factors of 0, 0.2, and 0.7; (**b**) constellation diagrams labeled ‘A’, ‘B’, and ‘C’ depict the constellation at a roll-off factor of 0.1, 0.4, and 0.7, respectively. Constellation diagrams are arranged from top to bottom as follows: MF + LMS S1, MF + LMS S2, MF + LMS S3, NNWSC S1, NNWSC S2, and NNWSC S3. equalizer. When the roll-off factor reaches a moderate range, the influence of ISI on the matched filter is reduced, and the combination with the LMS equalizer achieves optimal performance, resulting in the lowest BER. As the roll-off factor continues to increase, while ISI further decreases, the expanded bandwidth and reduced spectral energy concentration make it more challenging for the LMS equalizer to distinguish between signal and noise. Consequently, the LMS equalizer's performance is limited by SNR, causing BER to increase again. For the sub-band spacing factor of 0, the conventional receiver exhibits a similar trend in BER reduction for all sub-bands as the roll-off factor increases. However, the BER performance of the NNWSC-based receiver reaches its minimum level when the roll-off factor is approximately 0.2 but shows a slight increase as the roll-off factor increases further. This behavior can be attributed to the very low sub-band spacing factor value at this point, which results in tightly packed sub-band spectra. As a consequence, the expansion of bandwidth and the reduction in spectral energy concentration introduce certain impacts on the performance of the NNWSC-based receiver. Finally, when the sub-band spacing factor is set to 0.7, the influence of IBI on the performance of both receivers becomes negligible, leading to comparable performance between the two. At higher roll-off factors, the NNWSC is better able to capture linear relationships within the received signal and can achieve a globally optimal result, significantly reducing BER.