## Appendix E Power Spectral Density
The following definitions are convenient:  
\[A_{\phi} \equiv\Big{(}i\Omega+\tau_{\mathrm{e}}^{-1}\Big{)}\bigg{(}i \Omega\mathrm{K}_{\mathrm{c}}+\frac{a_{g}G_{0}}{i\Omega+\tau_{\mathrm{e}}^{-1}} S_{0}+\zeta_{\mathrm{s}}\Big{)} \tag{112a}\] \[2A_{S} \equiv i\Omega\mathrm{K}_{\mathrm{s}}\frac{i\Omega+\tau_{\mathrm{e}}^{-1}} {S_{0}}-\alpha_{\mathrm{H}}a_{g}G_{0}\] (112b) \[2A_{N} \equiv\frac{\alpha_{\mathrm{H}}A_{\phi}+2S_{0}A_{S}}{i\Omega+ \tau_{\mathrm{e}}^{-1}}a_{g}\] (112c) \[B_{\phi} \equiv\mathrm{K}_{\mathrm{c}}A_{\phi}-\alpha_{\mathrm{H}}a_{g}G_{ 0}S_{0}\kappa_{g}+i\Omega\mathrm{K}_{\mathrm{s}}^{2}\Big{(}i\Omega+\tau_{ \mathrm{e}}^{-1}\Big{)}. \tag{112d}\]  
First, \(N_{\mathrm{0p}}\) is extracted from Equation (21c):  
\[N_{\mathrm{0p}}=\ \frac{\widehat{F}_{N}-G_{0}S_{\mathrm{0p}}}{i\Omega+\tau_{ \mathrm{e}}^{-1}}. \tag{113}\]Next, replacing Equation (A37) into Equation (21a) yields the expression for \(S_{0\text{p}}\):  
\[(i\Omega\mathsf{K}_{c}+\zeta_{ S})S_{0\text{p}} =a_{g}\frac{\widehat{F}_{N}-G_{0}S_{0\text{p}}}{i\Omega+\tau_{\text {e}}^{-1}}S_{0}-2i\Omega S_{0}\mathsf{K}_{s}\phi_{0\text{p}}+\widehat{F}_{S}\] \[\Leftrightarrow\frac{A_{\phi}}{i\Omega+\tau_{\text{e}}^{-1}}S_{0 \text{p}} =a_{g}S_{0}\frac{\widehat{F}_{N}}{i\Omega+\tau_{\text{e}}^{-1}}-2i \Omega S_{0}\mathsf{K}_{s}\phi_{0\text{p}}+\widehat{F}_{S}\] (A38) \[\Leftrightarrow S_{0\text{p}} \stackrel{{\eqref{eq:S00}}}{{=}}\frac{a_{g}S_{0} \widehat{F}_{N}-2 iS_{0}\Omega\mathsf{K}_{s}\big{(}i\Omega+\tau_{\text{e}}^{-1 }\big{)}\phi_{0\text{p}}+\big{(}i\Omega+\tau_{\text{e}}^{-1}\big{)}\widehat{F }_{S}}{A_{\phi}}.\]  
Finally, inserting Equation (A37) and (A38) into Equation (21b) and grouping the terms with \(\phi_{0\text{p}}\), \(\widehat{F}_{S}\), \(\widehat{F}_{\phi}\) and \(\widehat{F}_{N}\) yields:  
\[2i\Omega B_{\phi}\phi_{0\text{p}} \stackrel{{\eqref{eq:S00}}}{{=}}\left[-\alpha_{ {}_{\text{H}}}a_{g}G_{0}S_{0}+\alpha_{{}_{\text{H}}}A_{\phi}+\,i\Omega \mathsf{K}_{s}\Big{(}i\Omega+\tau_{\text{e}}^{-1}\Big{)}\right]\frac{a_{g}}{i \Omega+\tau_{\text{e}}^{-1}}\,\widehat{F}_{N}+2A_{S}\,\widehat{F}_{S}+2\,A_{ \phi}\widehat{F}_{\phi}\] (A39) \[\stackrel{{\eqref{eq:S00}}}{{=}}\left(\alpha_{ {}_{\text{H}}}A_{\phi}+2S_{0}A_{S}\right)\frac{a_{g}}{i\Omega+\tau_{\text{e}} ^{-1}}\,\widehat{F}_{N}+A_{S}\,\widehat{F}_{S}+2\,A_{\phi}\widehat{F}_{\phi}\] \[2A_{S} \equiv i\Omega\mathrm{K}_{\mathrm{s}}\frac{i\Omega+\tau_{\mathrm{e}}^{-1}} {S_{0}}-\alpha_{\mathrm{H}}a_{g}G_{0}\]0  
With Equation (A39) it is possible to calculate an expression for Equation (22):  
\[2A_{S} \equiv i\Omega\mathrm{K}_{\mathrm{s}}\frac{i\Omega+\tau_{\mathrm{e}}^{-1}} {S_{0}}-\alpha_{\mathrm{H}}a_{g}G_{0}\]1 (A40) \[2A_{S} \equiv i\Omega\mathrm{K}_{\mathrm{s}}\frac{i\Omega+\tau_{\mathrm{e}}^{-1}} {S_{0}}-\alpha_{\mathrm{H}}a_{g}G_{0}\]2 \[2A_{S} \equiv i\Omega\mathrm{K}_{\mathrm{s}}\frac{i\Omega+\tau_{\mathrm{e}}^{-1}} {S_{0}}-\alpha_{\mathrm{H}}a_{g}G_{0}\]3  
It can be seen from Equation (A36) that the coefficients \(A_{i}\) are independent of time, and assuming an ergodic process they can be taken out of the average in Equation (A40), obtaining:  
\[2A_{S} \equiv i\Omega\mathrm{K}_{\mathrm{s}}\frac{i\Omega+\tau_{\mathrm{e}}^{-1}} {S_{0}}-\alpha_{\mathrm{H}}a_{g}G_{0}\]4 (A41) \[2A_{S} \equiv i\Omega\mathrm{K}_{\mathrm{s}}\frac{i\Omega+\tau_{\mathrm{e}}^{-1}} {S_{0}}-\alpha_{\mathrm{H}}a_{g}G_{0}\]5 \[2A_{S} \equiv i\Omega\mathrm{K}_{\mathrm{s}}\frac{i\Omega+\tau_{\mathrm{e}}^{-1}} {S_{0}}-\alpha_{\mathrm{H}}a_{g}G_{0}\]6 \[2A_{S} \equiv i\Omega\mathrm{K}_{\mathrm{s}}\frac{i\Omega+\tau_{\mathrm{e}}^{-1}} {S_{0}}-\alpha_{\mathrm{H}}a_{g}G_{0}\]7 \[2A_{S} \equiv i\Omega\mathrm{K}_{\mathrm{s}}\frac{i\Omega+\tau_{\mathrm{e}}^{-1}} {S_{0}}-\alpha_{\mathrm{H}}a_{g}G_{0}\]8 \[2A_{S} \equiv i\Omega\mathrm{K}_{\mathrm{s}}\frac{i\Omega+\tau_{\mathrm{e}}^{-1}} {S_{0}}-\alpha_{\mathrm{H}}a_{g}G_{0}\]9 (A42)Furthermore, using the following definitions:  
\[2A_{N} \equiv\frac{\alpha_{\mathrm{H}}A_{\phi}+2S_{0}A_{S}}{i\Omega+ \tau_{\mathrm{e}}^{-1}}a_{g}\]0 \[2A_{N} \equiv\frac{\alpha_{\mathrm{H}}A_{\phi}+2S_{0}A_{S}}{i\Omega+ \tau_{\mathrm{e}}^{-1}}a_{g}\]1 (113b) \[2A_{N} \equiv\frac{\alpha_{\mathrm{H}}A_{\phi}+2S_{0}A_{S}}{i\Omega+ \tau_{\mathrm{e}}^{-1}}a_{g}\]2 (113c) the expression for \[2A_{N} \equiv\frac{\alpha_{\mathrm{H}}A_{\phi}+2S_{0}A_{S}}{i\Omega+ \tau_{\mathrm{e}}^{-1}}a_{g}\]3 from Equation ( 113d ) can be rewritten as: \[2A_{N} \equiv\frac{\alpha_{\mathrm{H}}A_{\phi}+2S_{0}A_{S}}{i\Omega+ \tau_{\mathrm{e}}^{-1}}a_{g}\]4 (114a) \[2A_{N} \equiv\frac{\alpha_{\mathrm{H}}A_{\phi}+2S_{0}A_{S}}{i\Omega+ \tau_{\mathrm{e}}^{-1}}a_{g}\]5 \[2A_{N} \equiv\frac{\alpha_{\mathrm{H}}A_{\phi}+2S_{0}A_{S}}{i\Omega+ \tau_{\mathrm{e}}^{-1}}a_{g}\]6 \[2A_{N} \equiv\frac{\alpha_{\mathrm{H}}A_{\phi}+2S_{0}A_{S}}{i\Omega+ \tau_{\mathrm{e}}^{-1}}a_{g}\]7 (114b) with which the expression of the FN PSD becomes: \[2A_{N} \equiv\frac{\alpha_{\mathrm{H}}A_{\phi}+2S_{0}A_{S}}{i\Omega+ \tau_{\mathrm{e}}^{-1}}a_{g}\]8