# Interference of Spread-Spectrum EMI and Digital Data Links under Narrowband Resonant Coupling
## 4 Discussion
To better highlight the effect of SS-modulations, the worst case channel capacity and the average EMI-induced capacity loss over different values of \(f_{\text{R}}\) ranging from \(2B\) to \(7B\) are reported in Figures 6 and 7, respectively, versus \(W/B\). It can be clearly observed that the worst case capacity monotonically decreases with \(W/B\) for practical values of \(Q\) below 10,000 and shows an increasing behavior only for extremely high values of \(Q\) and for quite large \(W/B\) ratios, according to the previous discussion. By contrast, the average channel capacity has found to be monotonically decreasing with \(W/B\) for any value of \(Q\). Figure 5: Normalized capacity versus resonant frequency \(f_{\text{R}}/B\) for the channel in Figure 3 for different values of the resonant coupling quality factor \(Q\) under constant peak coupling, reported in each plot for different SS-modulation depth \(\delta\) ranging from 0 (no SS) to 30%. To further investigate the effect of SS-modulation under resonant coupling, the analysis described above has been repeated choosing the scaling constant \(G_{0}\) in (3) so that the coupling function \(G(f)\) has unitary energy, i.e., imposing  
\[\int_{-\infty}^{+\infty}|G(f)|^{2}\mathrm{d}f=1.\]  
In analogy with Figures 6 and 7, the average and the worst case capacity loss over different values of \(f_{\mathrm{R}}\) are reported in Figure 8 versus the SS bandwidth \(W\) normalized with respect to the channel bandwidth. In this case, a monotonic decrease of both the average and of the worst case channel capacity is observed for increased \(W/B\) for all the considered values of the quality factor \(Q\). Under the hypotheses and limitations considered in the proposed analysis, i.e., that SS EMI can be described as a band-limited Gaussian random process, which is coupled to a wideband AWGN channel by resonant coupling described by the transfer function (3), it can be observed that the application of SS-modulations on periodic interfering signals gives rise to a larger EMI-induced capacity loss under the same EMI total power compared to non-SS-modulated EMI for almost all practical resonant EMI coupling conditions. In other words, the results on the impact of SS-modulations on EMI-induced capacity loss studied in [37] for wideband EMI coupling can be extended to practical resonant coupling conditions according to the proposed analysis. Figure 6: Worst case normalized capacity loss over different resonant frequencies \(f_{\mathrm{R}}\) versus SS-modulation depth \(\delta\) ranging from 0 (no SS) to 30% for different values of the resonant coupling quality factor \(Q\) under constant peak coupling. A detail of the top figure is reported in the bottom figure. As observed in [37], such a larger capacity loss is not always related to an increased bit error rate (BER) in communication channels operating at sub-capacity bit rates, by the way it is expected to give rise to a worse and worse BER degradation as far as the data rate approaches the Shannon capacity limit, as in real world communication channels employing advanced channel coding schemes (e.g., Turbo coding). Under this perspective, the impact of SS-modulations on the BER of a 4-QAM digital link featuring Turbo coding will be considered as a test case in what follows. Figure 7: Average normalized capacity for resonant frequencies \(f_{\text{R}}\) ranging from \(2B\) to \(7B\) versus SS-modulation depth \(\delta\) ranging from 0 (no SS) to 30% for different values of the resonant coupling quality factor \(Q\) under constant peak coupling. A detail of the top figure is reported in the bottom figure.