## 3 Nasal Acoustic Signal Processing Methods
### The Nasal Cavity Three-Dimensional Structure Hierarchical Reconstruction Method  
In order to identify the relationship between the input impulse response and the cross-sectional areas at various depths within the nasal cavity, a simplified model of the nasal cavity's structure is established by the nasal cavity three-dimensional structure hierarchical reconstruction method. The process of the nasal cavity three-dimensional structure hierarchical reconstruction method is illustrated in Figure 6.  
In order to facilitate the calculation, the physical structure of the nasal cavity needs to be simplified. The schematic diagram of acoustic wave propagation in a simplified nasal cavity is shown in Figure 7[24]. The physical structure of the nasal cavity is simplified to a physical model with more than \(n\) elastic planes, where planes 0-\(n\) represent the depth of the nasal cavity from shallow to deep. It is assumed that when the pulsed acoustic waves propagate to a space beyond \(n\)-th layers, the intensity of the pulsed acoustic waves has already diminished to zero. To simplify the calculation, the distance between each plane is set to \(x\), measured in centimeters. Each space between two adjacent planes is referred to as a layer. For instance, the area between plane 0 and plane 1 is denoted as layer 1. The signal in each layer can be expressed as a polynomial in the z-transform, \(D_{i}(z)\) is the downward propagating signal in layer \(i\), and \(U_{i}(z)\) is the upward propagating signal in layer \(i\). Specifically, \(D_{1}(z)\) represents the \(z\)-transform of the first-stage incident wave, defined as \(D_{1}(z)=1\), and \(U_{1}(z)\) is the \(z\)-transform of the first-stage reflected wave. Pulsed acoustic waves will be reflected and transmitted on each plane, taking the plane \(i\) as an example, the transmission-reflection pattern on plane \(i\) is shown in Figure 8.  
Taking downward propagation as the positive direction, if the incident wave intensity is 1, the reflectance and transmittance on the plane \(i\) are \(r_{i}\) and \(t_{i}\), respectively. As shown on the left side of Figure 8, the reflectance of the incident wave is \(r_{i}\) and the transmittance is \(t_{i}\). As shown on the right side of Figure 8, the reflectance of the reflected wave is \(-r_{i}\) and the transmittance is \(-t_{i}\). The transmittance and reflectance satisfy Equation (4) [24; 25]:  
Figure 6: The flow diagram of the nasal cavity three-dimensional structure hierarchical reconstruction. \[1=r_{i}^{2}+t_{i}^{2} \tag{4}\]  
Based on the simplified physical properties of the nasal cavity in Figures 7 and 8, the relationship between \(D_{1}(z)\), \(U_{1}(z)\), and \(D_{n+1}(z)\), \(U_{n+1}(z)\) can be represented as follows [24; 25; 26]:  
\[ \begin{bmatrix}D_{n+1}(z)\\ U_{n+1}(z)\end{bmatrix} =\frac{z^{0.5n}}{\sigma_{n}} \begin{bmatrix}P_{n}^{R}(z) &Q_{n}^{R}(z)\\ Q_{n}(z)&P_{n}(z)\end{bmatrix}\begin{bmatrix}D_{1}(z)\\ U_{1}(z)\end{bmatrix}  \tag{5}\]  
\[ \begin{bmatrix}P_{n}^{R}(z)&Q_{n}^{R}(z)\\ Q_{n}(z)&P_{n}(z)\end{bmatrix} =\prod_{i=1}^{n} \begin{bmatrix}z^{-1}&-r_{i}\\ -r_{i}z^{-1}&1\end{bmatrix}  \tag{6}\]  
where \(\sigma_{n}=\prod_{i=1}^{n}t_{i}\), \(t_{i}\) is the transmittance on plane \(i\), \(D_{1}(z)=1\), \(U_{1}(z)\) is the z-transform of the first-stage reflected wave, \(D_{n+1}(z)\) and \(U_{n+1}(z)\) are functions of \(z\) and \(r\). Since \(D_{1}(z)=1\), \(U_{1}(z)\) can be denoted by \(U_{1}(z)=1+x_{1}z^{-1}+x_{2}z^{-2}+...+x_{n}z^{-n}\)[25]. Therefore, according to Equations (4)-(6), the reflectance relationship between the input impulse response sequence and different planes can be calculated [25]:  
\[r_{k}= \begin{cases}0,&k=0\\ x_{1},&k=1\\ \sum_{j=1}^{k-1}P_{j}(z)x_{k+1-j}\Big{[}\prod_{j=1}^{k-1}(1-r_{j}^{2})\Big{]}^ {-1},&k=2,3,...n\end{cases}  \tag{7}\]  
where \(x_{k}\) is the input impulse response intensity value of \(z^{-k}\) when the pulsed acoustic wave is reflected plane by plane from plane \(k\) back to plane \(0\).