## 3 Nasal Acoustic Signal Processing Methods
### Nasal Cavity Acoustic Signal Preprocessing Method
#### 3.1.1 Separation of Incident and Reflected Waves  
The propagation of acoustic waves in a short-source acoustic tube is shown in Figure 5. Initially, the pulsed acoustic waves are generated by the pulse generator at the bottom of the short-source acoustic tube. It propagates upwards through the tube and continues towards the test object after being captured by the microphone for the first time. Upon the pulsed acoustic waves' contact with the surface of the test object, an acoustic wave reflection phenomenon occurs, and some reflected waves return to the short-source acoustic tube, being captured for a second time by the microphone. Subsequently, these waves propagate back down the short-source acoustic tube, undergo reflection at the bottom, and serve as secondary pulsed acoustic waves for the second round of detection of the test object. Due to energy consumption during this cyclic process, the intensity of captured pulsed acoustic waves gradually weakens to zero after multiple rounds of repeated detection.  
Based on the above analysis, it can be concluded that, except for the first captured pulsed acoustic wave, the intensity of each of the two adjacent captured pulsed acoustic waves should be proportional. The \((2n-1)\)st captured pulsed acoustic wave is called the \(n\)-stage incident wave \((n=1,2,...)\), and the \(2n\)th captured pulsed acoustic wave is called the \(n\)-stage reflected wave \((n=1,2,...)\). Since the first-stage incident wave serves as the original pulsed acoustic wave generated by the pulse generator, the amplitude of the first-stage reflected wave represents the maximum energy, which is beneficial to the enhancement of detection accuracy. Therefore, the separated first-stage incident wave and first-stage reflected wave are used as the incident wave \(P_{i}(t)\) and reflected wave \(P_{r}(t)\) to detect the three-dimensional structure of the nasal cavity.  
Figure 4: Nasal cavity acoustic signal preprocessing flow diagram.  
Figure 5: Acoustic wave propagation in a short-source acoustic tube.  
After separating the incident and reflected waves, the input impulse response \(x(t)\) is calculated using the incident wave \(P_{i}(t)\) and the reflected wave \(P_{r}(t)\), and the relationship between the three in the time domain is [16]  
\[P_{r}(t)=\int_{0}^{t}x(t-\tau)P_{i}(\tau)d\tau \tag{1}\]  
where \(\tau\) is the duration of the incident wave. The time domain convolution of the input impulse response and the incident wave is the reflected wave; then, in the frequency domain, the input impulse response is the ratio of the reflected wave to the incident wave:  
\[x(\omega)=\frac{P_{r}(\omega)}{P_{i}(\omega)} \tag{2}\]  
where \(\omega\) is the discrete angular frequency, \(P_{i}(\omega)\) is the incident wave in the frequency domain, and \(P_{r}(\omega)\) is the reflected wave in the frequency domain. \(x(\omega)\) is the input impulse response in the frequency domain, and the input impulse response sequence \(x(t)\) in the time domain is calculated by the inverse fast Fourier transform.  
When using the reflected and incident waves in the frequency domain to calculate the input impulse response, the division operation in the calculation amplifies the influence of the DC offset component and high-frequency noise, which leads to a large error in the final results of the nasal cavity three-dimensional structure reconstruction and fails to meet the accuracy of practical applications.