## 3 Spatial and Temporal Correlation Characteristics
By applying the equality \(\mathbb{E}\{e^{jtx}\}=\frac{e^{jtx}-e^{jna}}{jt(b-a)}\), where \(x\sim\mathrm{U}(a,b)\), we have  
\[R_{qp,\tilde{q}\tilde{p}}(t,\Delta t)=\mathbb{E}\{H_{qp}(t,f)H_{\tilde{q} \tilde{p}}^{*}(t-\Delta t,f)\} \tag{60}\]0 \[R_{qp,\tilde{q}\tilde{p}}(t,\Delta t)=\mathbb{E}\{H_{qp}(t,f)H_{\tilde{q} \tilde{p}}^{*}(t-\Delta t,f)\} \tag{60}\]1 \[R_{qp,\tilde{q}\tilde{p}}(t,\Delta t)=\mathbb{E}\{H_{qp}(t,f)H_{\tilde{q} \tilde{p}}^{*}(t-\Delta t,f)\} \tag{60}\]2 \[R_{qp,\tilde{q}\tilde{p}}(t,\Delta t)=\mathbb{E}\{H_{qp}(t,f)H_{\tilde{q} \tilde{p}}^{*}(t-\Delta t,f)\} \tag{60}\]3 \[R_{qp,\tilde{q}\tilde{p}}(t,\Delta t)=\mathbb{E}\{H_{qp}(t,f)H_{\tilde{q} \tilde{p}}^{*}(t-\Delta t,f)\} \tag{60}\]4  
Based on the definition of the \(\mathrm{sinc}\) function, i.e., \(\mathrm{sinc}(x)=\frac{\sin(\pi x)}{\pi x}\), the spatial-temporal CCF of the \(\mathrm{LoS}\) component is further written as  
\[R_{qp,\tilde{q}\tilde{p}}(t,\Delta t)=\mathbb{E}\{H_{qp}(t,f)H_{\tilde{q} \tilde{p}}^{*}(t-\Delta t,f)\} \tag{60}\]5 \[R_{qp,\tilde{q}\tilde{p}}(t,\Delta t)=\mathbb{E}\{H_{qp}(t,f)H_{\tilde{q} \tilde{p}}^{*}(t-\Delta t,f)\} \tag{60}\]6 \[R_{qp,\tilde{q}\tilde{p}}(t,\Delta t)=\mathbb{E}\{H_{qp}(t,f)H_{\tilde{q} \tilde{p}}^{*}(t-\Delta t,f)\} \tag{60}\]7 \[R_{qp,\tilde{q}\tilde{p}}(t,\Delta t)=\mathbb{E}\{H_{qp}(t,f)H_{\tilde{q} \tilde{p}}^{*}(t-\Delta t,f)\} \tag{60}\]8 \[R_{qp,\tilde{q}\tilde{p}}(t,\Delta t)=\mathbb{E}\{H_{qp}(t,f)H_{\tilde{q} \tilde{p}}^{*}(t-\Delta t,f)\} \tag{60}\]9  
Employing a similar procedure, the spatial-temporal CCF of the NLoS component can be expressed as  
\[R_{qp,\tilde{q}\tilde{p}}(t,\Delta t)=R_{qp,\tilde{q}\tilde{p}}^{\mathrm{LoS} }(t,\Delta t)+R_{qp,\tilde{q}\tilde{p}}^{\mathrm{NLoS}}(t,\Delta t). \tag{61}\]0 \[R_{qp,\tilde{q}\tilde{p}}(t,\Delta t)=R_{qp,\tilde{q}\tilde{p}}^{\mathrm{LoS} }(t,\Delta t)+R_{qp,\tilde{q}\tilde{p}}^{\mathrm{NLoS}}(t,\Delta t). \tag{61}\]1 \[R_{qp,\tilde{q}\tilde{p}}(t,\Delta t)=R_{qp,\tilde{q}\tilde{p}}^{\mathrm{LoS} }(t,\Delta t)+R_{qp,\tilde{q}\tilde{p}}^{\mathrm{NLoS}}(t,\Delta t). \tag{61}\]2 \[R_{qp,\tilde{q}\tilde{p}}(t,\Delta t)=R_{qp,\tilde{q}\tilde{p}}^{\mathrm{LoS} }(t,\Delta t)+R_{qp,\tilde{q}\tilde{p}}^{\mathrm{NLoS}}(t,\Delta t). \tag{61}\]3 \[R_{qp,\tilde{q}\tilde{p}}(t,\Delta t)=R_{qp,\tilde{q}\tilde{p}}^{\mathrm{LoS} }(t,\Delta t)+R_{qp,\tilde{q}\tilde{p}}^{\mathrm{NLoS}}(t,\Delta t). \tag{61}\]4  
The spatial-temporal CCF plays a key role in channel characteristic analysis since it is the foundation for deriving many other channel statistics. For example, by setting \(\bar{p}=p\) and \(\bar{q}=q\), (61) reduces to the temporal autocorrelation function (ACF), i.e., \(R_{qp}(t,\Delta t)\). By setting \(\Delta t=0\), (61) reduces to the spatial CCF, i.e., \(R_{qp,\bar{p}\bar{q}}(t)\). By setting \(a_{tm}=a_{rm}=0\), (61) reduces to the spatial-temporal CCF, neglecting the UAV vibrations. Additionally, the Doppler PSD depicts how the channel power is distributed across the Doppler frequency and can be calculated as the Fourier transform of the temporal ACF in terms of \(\Delta t\), i.e.,  
\[R_{qp,\tilde{q}\tilde{p}}(t,\Delta t)=R_{qp,\tilde{q}\tilde{p}}^{\mathrm{LoS} }(t,\Delta t)+R_{qp,\tilde{q}\tilde{p}}^{\mathrm{NLoS}}(t,\Delta t). \tag{61}\]5In addition, the coherence time is the duration over which the channel stays almost constant and can be defined as the minimum time interval in which the normalized temporal ACF is below a certain threshold \(c\), expressed as [46]  
\[R_{qp,\tilde{q}\tilde{p}}(t,\Delta t)=R_{qp,\tilde{q}\tilde{p}}^{\mathrm{LoS} }(t,\Delta t)+R_{qp,\tilde{q}\tilde{p}}^{\mathrm{NLoS}}(t,\Delta t). \tag{61}\]6