# Interference of Spread-Spectrum EMI and Digital Data Links under Narrowband Resonant Coupling
## 4 Discussion  
Based on the results presented in Section 3, the effects of SS-modulation under resonant EMI coupling are now investigated and discussed. For this purpose, a bandpass channel with bandwidth \(B\) ranging from \(f_{B_{\text{min}}}=4B\) to \(f_{B_{\text{max}}}=5B\) is considered and an aggressor generating periodic or random SS-modulated EMI with a fixed total power 20 dB below the total signal power, concentrated in a spectral line at frequency \(f_{0}\) falling in correspondence of the central frequency of the channel \(f_{\text{C}}=\frac{f_{B_{\text{min}}}+f_{B_{\text{max}}}}{2}=f_{0}=4.5B\) is considered, as depicted in Figure 4. The signal-to-background noise ratio of the channel is assumed to be 47 dB so that the background thermal noise is negligible compared to EMI. Moreover, it is assumed that the EMI source is coupled to the victim receiver by a resonant coupling function \(G(f)\) in (3), with resonant frequency \(f_{\text{R}}\) and quality factor \(Q\), where the scaling constant \(G_{0}\) is chosen so that to have unity peak amplitude  
\[\max_{f}|G(f)|=1.\]  
The channel capacity is evaluated accordingly as in Section 3 and is plotted in Figure 4 for \(Q=100\) versus the resonant frequency \(f_{\text{R}}\) and for different values of the SS EMI bandwidth \(W\) normalized with respect to the channel bandwidth \(B\), for \(W/B\) ranging from 0 (no SS) up to 25%. Based on (2), depending on the EMI harmonic order \(k\), the SS-modulation depth can be expressed in terms of the normalized bandwidth \(W/B\) as  
\[\delta=\frac{W}{kf_{0}}=\frac{W}{B}\frac{1}{4.5\cdot k}. \tag{14}\]The same analysis is repeated in Figure 5 for different values of the quality factor \(Q\) ranging from 1 to 1,000,000 to discuss the effects of SS-modulations on signal capacity under different resonant coupling quality factors. For \(Q=1\), the resonant behaviour is weak and the situation is similar to that considered in [37] for wideband EMI coupling (In this study, just the case of EMI bandwidth fully included in the communication channel bandwidth is considered. For extremely narrow-bandwidth communications, EMI coupling can be always considered nearly uniform over the channel bandwidth even in the presence of resonant coupling and the considerations presented in [37] can be directly extended) and an increasing EMI-induced channel capacity loss, weakly dependent on \(f_{\mathrm{R}}\), is observed for an increased SS spreading bandwdith \(W/B\), i.e., for an increased modulation depth \(\delta\), which is consistent with [37]. For an increased quality factor \(Q\), the channel capacity loss decreases since the EMI components far from the resonant frequency are partially filtered. Moreover, the capacity loss is more sensitive to the resonant frequency \(f_{\mathrm{R}}\) and expectedly reaches a minimum when \(f_{\mathrm{R}}\) is close to the EMI central frequency, i.e., for \(f_{\mathrm{C}}\simeq f_{\mathrm{R}}\). Even in this case, however, for \(Q\) up to 10,000, the EMI-induced capacity loss gets worse and worse by increasing the spreading bandwidth \(W/B\) of SS-modulations. A beneficial effect of SS-modulations, which could be possibly expected in view of the reduced SS-EMI coupled power under resonant conditions, as highlighted in Figure 1, is observed only for extremely high \(Q\) values exceeding 10,000, where the peak capacity loss decreases for an increasing spreading bandwidth \(W/B\). Even in these cases, for SS-modulations, a capacity impairment is experienced over a wider bandwidth compared to the case with lower \(W/B\) or no SS-modulation. Such extremely high values of the quality factor, however, are not realistic for EMI coupling. Figure 4: Normalized capacity versus resonant frequency \(f_{\mathrm{R}}/B\) for a communication channel with bandwidth \(B=(f_{1},f_{2})\) in the presence of EMI.