## 5 Noise
### Amplitude Noise and Phase Noise  
To determine the relationship between the introduced voltage noise densities and the power spectral densities (PSD) \(S_{\text{a}}(f)\) and \(S_{\varphi}(f)\) of the random amplitude fluctuations \(\alpha(t)\) and random phase fluctuations \(\varphi(t)\), respectively, Equation (18) is written as  
\[v_{\text{d}}(t)=\tilde{V}_{\text{d}}\cdot\left[1+\underbrace{\hat{\alpha}(f_ {\text{x}})\;\cos(2\pi f_{\text{x}}t)}_{\alpha(t)}\right]\cdot\cos\left(2\pi f _{\text{ex}}t+\underbrace{\hat{\varphi}(f_{\text{x}})\;\cos(2\pi f_{\text{x} }t)}_{\varphi(t)}\right) \tag{42}\]  
in which the measurement signal \(B_{\text{x}}(t)\) and the static phase offset, which are both irrelevant to the noise, are neglected. In this equation, one amplitude noise component \(\hat{\alpha}\) and one phase noise component \(\hat{\varphi}\), both at the frequency \(f_{\text{x}}\), represent other spectral components that can be taken into account by linear superposition. Based on basic trigonometric identities, Equation (42) can be rearranged into  
Figure 6: Noise contributions of the sensor to the overall voltage noise density at the output of the differential input transimpedance amplifier for the exemplary sensor described in Table 1, for feedback capacitances of \(C_{\text{f}}=10\,\text{pF}\), and without any excitation signal. Near the resonance frequency, the total noise is dominated by the thermal mechanical noise \(E_{\text{dm}}\), whereas the thermal electrical noise \(E_{\text{d0}}\) becomes relevant only at some distance from \(f_{\text{res}}\). \[v_{\rm d}(t) =\hat{V}_{\rm d}\cdot\cos(2\pi f_{\rm ex}t) \tag{43}\] \[\quad+\frac{\hat{V}_{\rm d}\,\hat{\rm d}(f_{\rm x})}{2}\cdot\left[ \cos\left(2\pi(f_{\rm ex}-f_{\rm x})t\right)+\cos\left(2\pi(f_{\rm ex}+f_{\rm x })t\right)\right]\] (44) \[\quad-\frac{\hat{V}_{\rm d}\,\hat{\varphi}(f_{\rm x})}{2}\cdot \left[\sin\left(2\pi(f_{\rm ex}-f_{\rm x})t\right)+\sin\left(2\pi(f_{\rm ex}+f _{\rm x})t\right)\right] \tag{45}\]  
revealing the typical structures of an amplitude-modulated signal and a narrow-band small-signal phase-modulated signal, respectively, with a carrier at \(f_{\rm ex}\) and symmetrical sidebands at \(f_{\rm ex}\pm f_{\rm x}\). Following the concept of noise sidebands [50] (p. 243), both carrier-to-noise sideband ratios  
\[\frac{\hat{V}_{\rm d}}{\frac{\hat{V}_{\rm d}\,\hat{\rm d}(f_{\rm x})}{2}}= \frac{\hat{V}_{\rm d}}{\frac{\hat{V}_{\rm d}\,\hat{\varphi}(f_{\rm x})}{2}}= \frac{2}{\hat{\alpha}(f_{\rm x})}=\frac{2}{\hat{\varphi}(f_{\rm x})}=\frac{ \frac{\hat{V}_{\rm d}}{\sqrt{2}}}{\frac{1}{2}\sqrt{E_{\rm d}^{2}(f_{\rm ex}-f _{\rm x})+E_{\rm d}^{2}(f_{\rm ex}+f_{\rm x})}\,\sqrt{\Delta f}} \tag{46}\]  
are equal to the carrier-to-voltage noise ratio where the additional factor \(\sqrt{\Delta f}\) transforms the voltage noise density \(E_{\rm d}\) into an effective voltage noise in the bandwidth \(\Delta f\). Equation (46) directly yields the PSD of random amplitude fluctuations  
\[S_{\rm a}(f_{\rm x})=\left(\frac{\hat{\alpha}(f_{\rm x})}{\sqrt{2}\sqrt{ \Delta f}}\right)^{2}=\frac{E_{\rm d}^{2}(f_{\rm ex}^{\rm AM}-f_{\rm x})+E_{ \rm d}^{2}(f_{\rm ex}^{\rm AM}+f_{\rm x})}{\hat{V}_{\rm d}^{2}}=S_{\rm a0}(f_ {\rm x})+S_{\rm am}(f_{\rm x}) \tag{47}\]  
in units of \(1/\rm Hz\) and the PSD of random phase fluctuations  
\[S_{\varphi}(f_{\rm x})=\left(\frac{\hat{\varphi}(f_{\rm x})}{\sqrt{2}\sqrt{ \Delta f}}\right)^{2}=\frac{E_{\rm d}^{2}(f_{\rm ex}^{\rm PM}-f_{\rm x})+E_{ \rm d}^{2}(f_{\rm ex}^{\rm PM}+f_{\rm x})}{\hat{V}_{\rm d}^{2}}=S_{\varphi 0}(f_{ \rm x})+S_{\rm pm}(f_{\rm x}) \tag{48}\]  
in units of \(\rm rad^{2}/Hz\).