# A Novel 3D Analytical Scattering Model
## 1 Introduction
Another model for air-to-air communications scenarios is presented in [10], where analytical expressions for time-variant delay Doppler statistics of the channel are derived. A geometry-based study on the delay and Doppler spectrum characterization of A2G/G2A communication channels is presented in [24]. However, both of these studies [10; 24] assume that the scatterers are confined only over a flat 2D surface, and the direction of the air station's (AS) motion is only limited to directly away from and towards each other; whereas considering the propagation of multipath waves in the elevation plane is highly important when realistically modelling A2G/G2A radio propagation environments. Analysis on the angular spread and second order fading statistics of land-mobile radio cellular communication channels is thoroughly studied in the literature, e.g., [25; 26; 27]. These studies for different multipath environments of terrestrial communications investigate the characteristics of the channel in the delay, Doppler and angular domain. A theory of shape factors (SFs) for the quantification of angular spread and second order fading statistics of the channel is proposed in [28]; the SFs include angular spread, angular constriction and the direction of maximum fading. Furthermore, the mathematical relationship of the SFs with the second order fading statistics of the channel is also derived in [28]. Second order fading statistics include the level crossing rate (LCR), average fade duration (AFD), auto-covariance and coherence distance. In [29], these three SFs are redefined on the basis of trigonometric moments and a new quantifier, named the true standard deviation, is proposed. Applications of these SFs in Nakagami-m fading channels are discussed in [30; 31]. A detailed study on second order statistics of the Nakagami-Hoyt fading channel model is given in [32]. Expressions for LCR and AFD are derived. A shadowed Rician model for land mobile satellite channels is proposed in [33], where a detailed analysis on the first and second order statistics of the fading channel is presented. A second order fading statistics-based analysis of channel simulators implementing Nakagami fading is presented in [34]. A detailed analysis on SFs and second order statistics for land-mobile radio cellular communication systems for both geometric models and empirical data is presented in [35; 36; 37]. Despite the established high significance of spatial spread quantification and the second order fading statistics of radio channels, such a study for A2G/G2A communication channels is completely missing in the literature. Nevertheless, there is a scope to develop a realistic 3D analytical model for A2G/G2A radio propagation environments for the characterization of angular spread, the Doppler spectrum and the second order fading statistics of the channel. This paper presents a geometry-based 3D stochastic channel model for A2G/G2A communications. Characterization of the Doppler spectrum, spatial spread and second order fading statistics of the channel are presented. Analytical expressions for joint and marginal probability density functions (PDFs) of Doppler shift, power and AoA are derived. The impact of different physical channel parameters, such as the mobility of AS, the height of AS, the height of GS and the delay of the longest propagation path, on the distribution characteristics of angular spread, Doppler spread and second order statistics is also presented. The rest of the paper is organized as follows: The geometry of the proposed 3D scattering model is presented in Section 2. The derivation of the PDF of the Doppler spectrum and the discussion of the obtained analytical results for the Doppler spectrum are given in Section 3. A detailed analysis on SFs is presented in Section 4. The second order fading statistics are analysed in Section 5. Finally, the paper is concluded in Section 6.