# Interference of Spread-Spectrum EMI and Digital Data Links under Narrowband Resonant Coupling
## 3 EMI-Induced Channel Capacity Loss  
Under the hypotheses introduced in the previous Section, the capacity \(C\) of a communication channel in the presence of EMI, for a single EMI harmonic overlapping completely or in part with the channel bandwith \(B\), as depicted in Figure 3 can be calculated as  
\[C=C_{1}+C_{2} \tag{4}\]  
where \(C_{1}\) is the capacity of sub-channel where the signal bandwidth overlaps with the EMI bandwidth, i.e., over the bandwidth  
\[W^{\star}=B\bigcap W=[W^{\star}_{\text{min}},W^{\star}_{\text{max}}]=\left[ \max\left(f_{B_{\text{min}}},f_{W_{\text{min}}}\right),\min\left(f_{B_{\text{ max}}},f_{W_{\text{max}}}\right)\right]. \tag{5}\]and can be expressed as  
\[C_{1} = \int_{W^{*}}\log_{2}\left(1+\frac{S_{\mathrm{S}}(f)}{S_{\mathrm{N}} (f)+|G(f)|^{2}S_{\mathrm{EM}}(f)}\right)\mathrm{d}f \tag{6}\]  
where:  
\[S_{\mathrm{S}}(f)=\frac{P_{\mathrm{S}}}{B}\Pi_{B}(f-f_{\mathrm{B,min}})\]  
is the signal p.s.d.,  
\[S_{\mathrm{N}}(f)=N_{0}\]  
is the background noise p.s.d.,  
\[S_{\mathrm{EM}}(f)=\frac{P_{\mathrm{EMI}}}{W}\Pi_{\mathrm{W}}(f-f_{\mathrm{W, min}})\]  
is the EMI p.s.d. and \(G(f)\) is the resonant coupling transfer function defined in (3). By introducing in (6) the expressions of \(S_{\mathrm{S}}(f)\), \(S_{\mathrm{N}}(f)\) and \(G(f)\) derived above, the capacity of the sub-channel \(C_{1}\) affected by EMI can be explicitly evaluated as  
\[C_{1} = \int_{B\bigcap W}\log_{2}\left(1+\frac{S_{\mathrm{S}}(f)\left[1+ \left(\frac{1}{Q^{2}}-2\right)\frac{f_{2}^{2}}{f_{\mathrm{R}}^{2}}+\frac{f_{ \mathrm{R}}^{4}}{f_{\mathrm{R}}}\right]}{S_{\mathrm{N}}(f)+\left[S_{\mathrm{N }}(f)\left(\frac{1}{Q^{2}}-2\right)+\frac{G_{\mathrm{N}}^{2}}{Q^{2}}S_{ \mathrm{EM}}(f)\right]\frac{f_{2}^{2}}{f_{\mathrm{R}}^{2}}+S_{\mathrm{N}}(f) \frac{f_{\mathrm{R}}^{4}}{f_{\mathrm{R}}}}\right)\mathrm{d}f \tag{7}\] \[= \int_{B\bigcap W}\log_{2}\left(1+\frac{P_{\mathrm{S}}}{BN_{0}} \frac{1+\left(\frac{1}{Q^{2}}-2\right)\frac{f_{2}^{2}}{f_{\mathrm{R}}^{2}}+ \frac{f_{\mathrm{R}}^{4}}{f_{\mathrm{R}}}}{1+\left(\frac{1}{Q^{2}}-2+G_{0} \frac{P_{\mathrm{EMI}}}{WN_{0}}\right)\frac{f_{2}^{2}}{f_{\mathrm{R}}^{2}}+ \frac{f_{\mathrm{R}}^{4}}{f_{\mathrm{R}}^{4}}}\right)\mathrm{d}f\] \[= \int_{B\bigcap W}\log_{2}\left(1+\alpha\right)+\log_{2}\left( \frac{1+a_{1}\frac{f_{2}^{2}}{f_{\mathrm{R}}^{2}}+\frac{f_{\mathrm{R}}^{4}}{ f_{\mathrm{R}}^{2}}}{1+a_{2}\frac{f_{\mathrm{R}}^{2}}{f_{\mathrm{R}}^{2}}+ \frac{f_{\mathrm{R}}^{4}}{f_{\mathrm{R}}^{4}}}\right)\mathrm{d}f\]  
where  
\[a_{1}=\frac{\frac{1}{Q^{2}}-2+\alpha\left(\beta+1\right)}{1+\alpha},\]  
\[W^{\star}=B\bigcap W=[W^{\star}_{\text{min}},W^{\star}_{\text{max}}]=\left[ \max\left(f_{B_{\text{min}}},f_{W_{\text{min}}}\right),\min\left(f_{B_{\text{ max}}},f_{W_{\text{max}}}\right)\right]. \tag{5}\]0  
and where \(\alpha\) is the signal-to-noise ratio as in (1) and \(\beta=P_{\mathrm{EMI}}/P_{\mathrm{S}}\) is the overall EMI-to-signal power ratio. Since the integral in (7) can be calculated analytically in closed form in terms of:  
\[W^{\star}=B\bigcap W=[W^{\star}_{\text{min}},W^{\star}_{\text{max}}]=\left[ \max\left(f_{B_{\text{min}}},f_{W_{\text{min}}}\right),\min\left(f_{B_{\text{ max}}},f_{W_{\text{max}}}\right)\right]. \tag{5}\]1 \[W^{\star}=B\bigcap W=[W^{\star}_{\text{min}},W^{\star}_{\text{max}}]=\left[ \max\left(f_{B_{\text{min}}},f_{W_{\text{min}}}\right),\min\left(f_{B_{\text{ max}}},f_{W_{\text{max}}}\right)\right]. \tag{5}\]2  
the capacity \(C_{1}\) can be finally expressed as  
\[W^{\star}=B\bigcap W=[W^{\star}_{\text{min}},W^{\star}_{\text{max}}]=\left[ \max\left(f_{B_{\text{min}}},f_{W_{\text{min}}}\right),\min\left(f_{B_{\text{ max}}},f_{W_{\text{max}}}\right)\right]. \tag{5}\]3  
where \(C_{0}\) is the capacity of the EMI-free channel.