# Frequency Response Extension Method of MET Vector Hydrophone Based on Dynamic Feedback Network
## 3 Materials and Methods
### Calculation of Compensation Method for Bandwidth Sensitivity  
The key to the adaptive compensation of bandwidth sensitivity is to calculate the sensitivity compensation parameter \(G\) to meet the system's expectations for bandwidth sensitivity.  
The system's expectation for bandwidth and sensitivity is defined based on two critical requirements: (1) maintaining a flat frequency response within the operational bandwidth; (2) achieving target sensitivity levels compatible with subsequent signal processing stages. In this paper, the operational bandwidth is extended beyond its original range while balancing sensitivity suppression and application-specific considerations. Specifically, the expected low-frequency cutoff is set to 1 Hz, and the expected closed-loop sensitivity is set to \(-\)180 dB.  
First, in the closed-loop state, turn off bandwidth sensitivity compensation, run the excitation coil self-test, and obtain the closed-loop transfer function \(H_{\text{cl}}\) of the current system.  
The closed-loop transfer function \(H_{\text{cl}}\) is as shown in Equation (26):  
\[H_{\text{cl}}=\frac{W_{\text{s}}W_{\text{f}}}{1+W_{\text{s}}W_{\text{f}}W_{ \text{a}}W_{\text{fb}}} \tag{26}\]  
where \(W_{\text{s}}\), \(W_{\text{f}}\), \(W_{\text{a}}\), and \(W_{\text{fb}}\) are the transfer functions of the sensor, preamplifier, power amplifier, and feedback system, respectively.  
First, we must have the feedback system \(W_{\text{fb}}\), which is set to \(W_{\text{fb0}}\) after the low-frequency cutoff frequency compensation is completed.  
Next, to compensate for bandwidth sensitivity, we introduce the sensitivity compensation parameter \(G\), which introduces gain adjustment in the closed-loop transfer function. The new closed-loop transfer function is shown in Equation (27):  
\[H_{\text{cl}}^{\prime}=G\cdot H_{\text{cl}} \tag{27}\]  
The desired bandwidth sensitivity is \(S_{\text{desired}}\), so it needs to satisfy Equation (28), as follows:  
\[H_{\text{cl}}^{\prime}(\omega_{\text{p}})=S_{\text{desired}} \tag{28}\]  
where \(\omega_{\text{p}}\) is a frequency point in the bandwidth, usually selected at the center frequency of the bandwidth.  
Using the measured \(H_{\text{cl}}(\omega_{\text{p}})\), the sensitivity compensation parameter \(G\) can be calculated, as shown in Equation (29):  
\[G=\frac{S_{\text{desired}}}{H_{\text{cl}}(\omega_{\text{p}})} \tag{29}\]  
Apply \(G\) to the system, i.e., update the sensitivity compensation parameter online and turn on bandwidth sensitivity compensation.  
It should be noted that, in order to ensure the stability of the system, the range of \(G\) should meet certain conditions. Generally speaking, \(G\) should be a positive real number and it should not be too large or too small, so as to avoid excessive amplification or attenuation of the system.  
Through the above method, adaptive compensation of the bandwidth sensitivity can be achieved, ensuring that the system maintains stable sensitivity performance during long-term operation.  
In the digital signal processor (DSP), the sensitivity compensation parameter \(G\) can be implemented by adjusting the gain coefficient of the digital filter. Specifically, an adjustable gain module can be introduced into the signal path, with the gain set to \(G\).  
The implementation of the gain module can be completed with a multiplication operation, i.e., for the input signal \(x(t)\), the output signal \(y(t)\) is as shown in Equation (30):  
\[y(t)=G\cdot x(t) \tag{30}\]  
Via a real-time adjustment of the value of \(G\), online compensation and adaptive adjustment of bandwidth sensitivity can be achieved.  
The above is the calculation of the compensation method for bandwidth sensitivity. By calculating the sensitivity compensation parameter \(G\), adaptive compensation of the system's bandwidth sensitivity can be achieved.