## 4 Dual-Channel Signal Enhancement Processing Methodology  
To achieve more stable demodulated vibration signals with enhanced data quality, this paper proposes a dual-wavelength synchronous detection scheme. Leveraging the low correlation in coupling efficiency between two signals with significantly different wavelengths, we combine the velocity signals through a channel-weighting approach. The specific process is illustrated in Figure 7. The dual-channel signals undergo in-phase and quadrature (IQ) demodulation to yield the envelopes of both signals, \(Env_{1}\) and \(Env_{2}\)[15], as well as the laser phases, Phase 1 and Phase 2, induced by target vibrations. The dual-channel weights, \(w_{1}\) and \(w_{2}\), and the velocities, \(v_{1}\) and \(v_{2}\), are then derived. Finally, the combined velocity, \(V\), is obtained through the weighted combination.  
Figure 6: The statistical variation relationship between coupling efficiency and wavelength.  
The velocity values obtained through dual-channel demodulation can be regarded as a combination of the true velocity and noise [16]. Based on Equation (3), the relationship between the effective value of velocity noise, \(v_{n1},v_{n2}\), and the envelope can be expressed as follows:  
\[v_{n1}=\frac{a}{Env_{1}};v_{n2}=\frac{a}{Env_{2}}, \tag{17}\]  
In the equation, \(a\) represents the conversion factor between the envelope and the effective value of velocity noise. To ensure \(V\) is an unbiased estimate of the true velocity, the following condition must be satisfied:  
\[\left\{\begin{aligned} w_{1}+w_{2}=1\\ V=w_{1}v_{1}+w_{2}v_{2}\end{aligned}\right.^{\prime}, \tag{18}\]  
The combined velocity noise, \(V_{n}\), can be expressed as follows:  
\[V_{n}=\sqrt{{w_{1}}^{2}{v_{n1}}^{2}+\left(1-{w_{1}}\right)^{2}{v_{n2}}^{2}}, \tag{19}\]  
To minimize the combined noise, the derivative of \(V_{n}\) with respect to \(w_{1}\) must equal zero. The final solutions for \(w_{1}\) and \(w_{2}\) are as follows:  
\[\left\{\begin{aligned} w_{1}=\frac{Env_{2}}{Env_{2}+Env_{2}}^{2} \\ w_{2}=\frac{Env_{2}}{Env_{1}+Env_{2}}^{2}\end{aligned}\right., \tag{20}\]  
The final combined velocity noise is expressed as follows:  
\[V_{n}=\frac{a}{\sqrt{Env_{1}}^{2}+Env_{2}}, \tag{21}\]  
In the above equation, it can be concluded that the combined velocity noise is lower than the velocity noise obtained from any individual channel demodulation. As illustrated in Figure 6, the effective velocity noise corresponding to the envelope of Channel 1 is used as a baseline (indicated by a dashed line). Figure 8 compares the relationship between the combined velocity noise and the single-channel velocity noise as the amplitude envelope ratio between Channel 2 and Channel 1 varies.  
In Figure 8, it can be observed that the combined velocity noise curve lies below the individual velocity noise curves of Channel 1 and Channel 2, confirming that the combined velocity noise is lower than that of any single-channel demodulation. When there is a significant amplitude difference between Channel 1 and Channel 2, the combination primarily functions as a channel-gating mechanism, suppressing spike noise and enhancing the stability of vibration measurement. Conversely, when the amplitudes of Channel 1 and Channel 2 are similar, the noise can be minimized to \(1/\sqrt{2}\) of its original value, effectively improving the instrument's detection resolution.  
Figure 7: Flow chart of dual-channel signal reinforcement.