# Mechanisms of Noise Transmission in Velocity Broad-Band Seismometers: Modeling and Analysis
## 3 Methodology
### Noise Transfer Model in Forward Path
#### 3.1.2 Spectral Characteristics Analysis  
From Equation (23), it is evident that the noise transfer function of the suspension pendulum is solely determined by the input mechanical constants. The gain of this noise transfer function, in decibels, remains constant across the entire frequency spectrum and is expressed as follows:  
\[|H_{n_{m}}^{d_{i}}(\omega)|=-20\log(K_{m})\text{ dB}. \tag{26}\]  
Increasing \(K_{m}\) can reduce the contribution of mechanical noise to the total noise.  
Similarly, from Equation (25), it becomes clear that the frequency characteristics of the noise transfer function for the total electronic noise in the forward path are influenced by \(K_{m}\), \(H_{a}^{d}(s)\), \(K_{d}^{c}\), and \(K_{c}^{v}\). Increasing these parameters can reduce the contribution of forward path noise to the total noise. This relationship is further expressed as follows:  
\[\left|H_{n_{cv}}^{a_{i}}(s)\right|=20\log\Biggl{(}\frac{1}{K_{m}\left|H_{a}^{d} (s)\right|K_{d}^{c}K_{c}^{v}}\Biggr{)}\text{dB}. \tag{27}\]  
From Equation (27), it can be observed that the spectral characteristics of the noise transfer function for the total electronic noise in the forward path are primarily governed by the transfer function \(H_{a}^{d}(s)\), since \(K_{m}\), \(K_{d}^{c}\), and \(K_{c}^{v}\) are constants.  
The transfer function \(H_{a}^{d}(s)\) acts as a low-pass filter with a cutoff frequency \(\omega_{0}\) (typically near \(2\pi\) rad/s). Below the cutoff frequency, the amplitude of the transfer function is approximately \(\frac{1}{\omega_{0}^{2}}\), and the product of \(K_{m}K_{d}^{c}K_{c}^{v}\) is significantly larger than 1. Therefore, the noise transfer function for the total electronic noise in the forward path exhibits noise suppression for frequencies \(\omega<\omega_{0}\), with a fixed gain in decibels given by the following:  
\[\left|H_{n_{cv}}^{a_{i}}(\omega<\omega_{0})\right|=20\log\biggl{(}\frac{{ \omega_{0}}^{2}}{K_{m}K_{d}^{c}K_{c}^{v}}\biggr{)}\text{ dB}. \tag{28}\]  
For \(\omega>\omega_{0}\), the amplitude-frequency response of the transfer function follows an asymptotic behavior, increasing at a rate of approximately 40 dB/dec. As the frequency increases beyond a certain point, the noise suppression effect gradually diminishes, leading to noise amplification. The frequency at which the gain crosses the 0 dB threshold, denoted as \(\omega_{cv,z}\), is given by:  
\[\omega_{cv,z}=\sqrt{-2\zeta^{2}{\omega_{0}}^{2}-{\omega_{0}}^{2}+\sqrt{4\zeta ^{4}-4\zeta^{2}+{K_{m}}^{2}{K_{d}^{c}}^{2}{K_{c}}^{2}}}. \tag{29}\]  
It can be observed that the angular frequency \(\omega_{cv,z}\) at the 0 dB point of the transfer function is jointly determined by the parameters \(\zeta\), \(\omega_{0}\), \(K_{m}\), \(K_{d}^{c}\), and \(K_{c}^{v}\). It is inversely proportional to \(\omega_{0}\) and directly proportional to \(K_{m}\), \(K_{d}^{c}\), and \(K_{c}^{v}\). The relationship with \(\zeta\) is more complex: within a smaller range, \(\omega_{cv,z}\) decreases as \(\zeta\) increases, whereas for larger values of \(\zeta\), \(\omega_{cv,z}\) tends to stabilize or slightly increase. This nonlinear behavior reflects the combined effects of damping and system gain parameters on the angular frequency at the 0 dB point.