# Measurement and Analysis of Radar Signals Modulated by the Respiration Movement of Birds
## 2 Method and Materials  
### Theoretical Calculation  
Theoretically, the respiration of a bird could change the physical area and scattering cross-section, and then contribute to the radar signal fluctuation of the bird. Figure 1 demonstrates the structure of an avian respiratory system. References [18; 19] describe the detailed respiration process. In brief, respiration changes the volume by changing the thoracic volume and then changing the cross-section of the bird.  
We utilize the common water sphere model in radar ornithology to analyze signal fluctuations caused by respiration. It was claimed that an approximate model can be obtained by regarding the bird as a water sphere of equivalent mass to the water content because the radar cross-section (RCS) of a bird is predominantly caused by 65% of its water mass [20]. The radius of the \"water\" bird, \(r\), is given by  
\[r=\sqrt[3]{\frac{0.65m}{4}} \tag{1}\]  
where \(m\) is the mass and \(\rho\) is the mean density of the sphere. The RCS, \(\sigma\), of a water bird is given by  
\[\sigma=\pi r^{2} \tag{2}\]  
Considering the periodic respiration movement, the equation should be revised into \(\sigma(t)\), which is given by  
\[\sigma(t)=\pi r^{2}(t) \tag{3}\]  
where \(r(t)\) is the radius function of respiration motion, with the up and down boundary values of \(r_{1}\) and \(r_{2}\). The RCS fluctuation \(\Delta\sigma\), is determined by differences in the water sphere radius.  
\[\Delta\sigma=\left|\pi r_{1}^{2}-\pi r_{2}^{2}\right| \tag{4}\]  
Figure 1: The avian respiratory system with a lateral view of one lung and the water sphere model of a duck.  
The modulation rate is given by the respiration rate of \(\omega_{\sigma}\), which is inversely proportional to the respiratory period of \(T\):  
\[\omega_{\sigma}=\frac{1}{T} \tag{5}\]  
Thus, the radar signals of the duck are modulated by respiration with the periodic respiratory pattern. The original water sphere model mentioned no radar bands, but most of the radar bands used for detecting birds are typical marine radar bands, such as the S-band or X-band. Since radar data of a general bird within these radar bands fall into the resonance region (or the optical region) [15], we used scattering rules (i.e., Mie series solution) in the resonance region (or the optical region) to analyze the radar signals of birds.  
This theoretical simulation indicates that the RCS fluctuation produced by respiration was approximately 1 dB level, much smaller than the 10 dB level. A duck is treated as a two-standard avian target (SAT) object with a 1 kg mass, as defined by the United States Federal Aviation Administration (FAA) [21]. Assuming that the water sphere is 0.65 kg, with a water density of 1 g/cm\({}^{3}\), then the calculated radius of the water duck is approximately 4.65 cm. If the changing radius span is 1 cm, the maximum RCS fluctuation is approximately 1.2 dB, based on the calculation of Equation (4). However, the current measurement of a two-SATs bird has an RCS fluctuation at a 10 dB-level, much larger than the theoretical 1 dB-level fluctuation. If the respiratory movement modulates the 10 dB-level fluctuations, the respiratory radius should have a 10-fold change in its range.