## 3 Processing Method of Vibration Piezoelectric Signal
### Hardware Damage Condition  
Considering the effect of hardware damage on the transmission of particle vibration piezoelectric signals, specifically, when there is damage to the device, the transmission \(\delta_{i,j}\rightarrow\delta_{i,j+1}\) and \(\delta_{i,j}\to E_{k}\) signals are expressed as follows:  
\[\omega_{\delta_{i,j}\delta_{i,j+1}}\mathbf{\kappa}=\sqrt{P_{\delta_{i,j}}}h_{\delta_ {i,j}\delta_{i,j+1}}(s+\eta_{\delta_{i,j}\delta_{i,j+1}})\mathbf{\kappa}+\mu_{ \delta_{i,j}\delta_{i,j+1}}+\
u_{\delta_{i,j}\delta_{i,j+1}}, \tag{1}\]  
\[\omega_{\delta_{i,j}E_{k}}\mathbf{\delta}=\sqrt{P_{\delta_{i,j}}}h_{\delta_{i,j}E_{ k}}(s+\eta_{\delta_{i,j}E_{k}})+\mu_{\delta_{i,j}E_{k}}+\
u_{\delta_{i,j}E_{k}}, \tag{2}\]  
where \(P_{\delta_{i,j}}\) is the transmit power of the transmitter \(\delta_{i,j}\), h is the link channel coefficient, \(\eta_{\delta_{i,j}\delta_{i,j+1}}\) and \(\eta_{\delta_{i,j}\delta_{i,j+1}}\) represent the noise caused by the damage of the transmitting and receiving equipment, and \(\pi_{\delta_{i,j}\delta_{i,j+1}}\) and \(\mu_{\delta_{i,j}\delta_{i,j+1}}\) represent additive white Gaussian noise (AWGN), which are Gaussian random variables (RVs) with mean zero and variance \(N_{0}\). Considering the effect of additive Gaussian white noise, the signal-processing algorithm of the vibration piezoelectric detection system can be optimized to improve the system's anti-interference ability and monitoring accuracy. Here, \(\gamma_{XY}\) is the channel gain, and the channel is Rayleigh fading. In agriculture, such as in vibrating piezoelectric detection systems used in crop seed drying, signals may be affected by equipment vibration, mechanical interference, or environmental factors, resulting in unstable signal transmission. The Rayleigh fading channel model can help us to understand and predict such signal variations in order to optimize the system design, reduce signal interference and improve the accuracy and reliability of the detection. \(\gamma_{XY}\) follows \(RV\), \(X,Y\in\left\{\delta_{i,j},\delta_{i,j+1},E_{k}\right\}\), and its probability density function (PDF) is shown below:  
\[f_{\gamma_{XY}}(x)=\lambda_{XY}\textit{exp}(-\lambda_{XY}x), \tag{3}\]  
where \(\lambda_{XY}\) represents the parameters of \(\gamma_{XY}\), \(\gamma_{XY}=\left|h_{XY}\right|^{2}\), defined as \(\lambda_{XY}=1/\theta\{\gamma_{XY}\}\), and \(\theta\) is the expectation operator. Considering path loss [18] yields the following:  
\[\lambda_{XY}=d_{XY}^{\beta}, \tag{4}\]  
\(d_{XY}\) is the link distance between \(X\) and \(Y\), and \(\beta\) is the path loss exponent. To simplify the system model, the distortion noise \(\eta_{XY}\)-\(\left(0,\left(\sigma_{XY}^{t}\right)^{2}P_{X}\right)\), \(\mu_{XY}\)-\(\left(0,\left(\sigma_{XY}^{r}\right)^{2}P_{X}\gamma_{XY}\right)\) follows the circular symmetric complex Gaussian distribution, and the damage degree is assumed to be the same as the node architecture to obtain the following:  
\[\left(\sigma_{XY}^{t}\right)^{2}=\sigma_{a}^{2},(\sigma_{XY}^{r})^{2}=\sigma_ {b}^{2}. \tag{5}\]  
Referring to [19], the instantaneous signal-to-noise ratio (SNR) of \(\delta_{i,j}\)\(\rightarrow\)\(\delta_{i,j+1}\) and \(\delta_{i,j}\)\(\rightarrow\)\(\Sigma_{k}\) links can be written as follows:  
\[R_{\delta_{i,j}\delta_{i,j+1}}\mathbf{\kappa}=\frac{P_{\delta_{i,j}}\gamma_{ \delta_{i,j}\delta_{i,j+1}}}{\kappa P_{\delta_{i,j}}\gamma_{\delta_{i,j}\delta_ {i,j+1}}+N_{0}}, \tag{6}\]  
\[R_{\delta_{i,j}E_{k}}\mathbf{\kappa}=\frac{P_{\delta_{i,j}}\gamma_{\delta_{i,j}E_{ k}}}{\kappa P_{\delta_{i,j}}\gamma_{\delta_{i,j}E_{k}}+N_{0}}, \tag{7}\]  
By optimizing the signal-to-noise ratio, the accuracy of the vibrating piezoelectric detection system can be improved, and false and missed detections due to noise interference can be reduced, thus better monitoring the drying process of crop seeds.