Wideband HAP-MIMO Channels: A 3-D Modeling and Simulation Approach

Abstract

High-altitude platforms (HAPs) are considered as an alternative technology to provide future generation broadband wireless communications services. This paper proposes a three-dimensional (3-D) geometry-based reference model for wideband HAP multiple-input–multiple-output (MIMO) channels. The statistical properties of the channel are analytically studied in terms of the elevation angle of the platform, the antenna arrays configuration, and the angular, the Doppler and the delay spread. Specifically, the space-time-frequency correlation function (STFCF), the space-Doppler power spectrum, and the power space-delay spectrum are derived for a 3-D non-isotropic scattering environment. Finally, a sum-of-sinusoids statistical simulation model for wideband HAP-MIMO channels is proposed. The results show that the simulation model accurately and efficiently reproduces the STFCF of the reference model. The proposed models provide a convenient framework for the characterization, analysis, test, and design of wideband HAP-MIMO communications systems with line-of-sight and non-line-of-sight links.

Appendices

Appendix 1: The SPDS of the NLoS Component

The SDPS of the NLoS component is obtained by calculating the FT of the STCF of the NLoS component in (27)–(30)

$$\begin{aligned} S_{pl,qm}^{NLoS} \left( {\delta _T ,\delta _R ,\nu } \right) =\mathfrak I _{\Delta t} \left\{ {R_{pl,qm}^{NLoS} \left( {\delta _T ,\delta _R ,\Delta t,\Delta f=0} \right) } \right\} . \end{aligned}$$

(58)

Considering that $\Delta f=0,$ the Bessel function $I_0 \left( {\sqrt{w_2^2 +w_3^2 }} \right)$ of (27) can be written as

$$\begin{aligned} I_0 \left( {\sqrt{w_2^2 +w_3^2 }} \right) \mathop =\limits _{\Delta f=0} J_0 \left[ {s_1 \sqrt{\left( {\Delta t+s_2 } \right) ^{2}+s_3^2 }} \right] , \end{aligned}$$

(59)

where

$$\begin{aligned} s_1&= 2\pi f_{R,\max } \cos \left[ {\arctan \left( {H_S /R_S } \right) } \right] , \nonumber \\ s_2&= \frac{q_1 +q_2 -q_3 }{f_{R,\max } \cos \left[ {\arctan \left( {H_S /R_S } \right) } \right] }, \nonumber \\ s_3&= \frac{q_4 +q_5 -q_6 }{f_{R,\max } \cos \left[ {\arctan \left( {H_S /R_S } \right) } \right] }, \nonumber \\ q_1&= \frac{\left( {q-p} \right) \delta _T \sin \theta _T R_S \sin \gamma _R }{\lambda D\cos \beta _T }, \nonumber \\ q_2&= \frac{\left( {m-l} \right) \delta _R \cos \psi _R \cos \left( {\theta _R -\gamma _R } \right) \cos \left[ {\arctan \left( {H_S /R_S } \right) } \right] }{\lambda }, \nonumber \\ q_3&= \frac{jk\cos \left( {\mu -\gamma _R } \right) }{2\pi }, \nonumber \\ q_4&= \frac{\left( {q-p} \right) \delta _T \sin \theta _T R_S \cos \gamma _R }{\lambda D\cos \beta _T }, \nonumber \\ q_5&= \frac{\left( {m-l} \right) \delta _R \cos \psi _R \sin \left( {\theta _R +\gamma _R } \right) \cos \left[ {\arctan \left( {H_S /R_S } \right) } \right] }{\lambda }, \nonumber \\ q_6&= \frac{jk\sin \left( {\mu +\gamma _R } \right) }{2\pi }. \end{aligned}$$

(60)

Using (59) and (60), (58) becomes

$$\begin{aligned}&S_{pl,qm}^{NLoS} \left( {\delta _T ,\delta _R ,v} \right) \nonumber \\&\quad =\int \limits _{H_{S,\min } }^{H_{S,\max } } {\int \limits _{R_{S,\min } }^{R_{S,\max } } {A\int \limits _{-\infty }^\infty {e^{-j2\pi \Delta t\left( {v-f_{T,\max } \cos \gamma _T } \right) }} } } J_0 \left[ {s_1 \sqrt{\left( {\Delta t+s_2 } \right) ^{2}+s_3^2 }} \right] d\Delta tdR_S dH_S ,\nonumber \\ \end{aligned}$$

(61)

where

$$\begin{aligned} A&= \frac{e^{j2\pi \left( {\frac{\left( {q-p} \right) \delta _T \cos \theta _T }{\lambda \cos \beta _T }+\frac{\left( {m-l} \right) \delta _R \sin \psi _R \sin \left[ {\arctan \left( {H_S /R_S } \right) } \right] }{\lambda }} \right) }e^{\left[ {-\frac{1}{2\sigma ^{2}}\hbox {ln}^{2}\left( {\frac{H_S }{H_{S,\mathrm{mean}} }} \right) } \right] }}{H_S \cosh ^{2}\left( {aR_S } \right) \sigma \sqrt{2\pi }I_0 \left( k \right) \sqrt{\left( {K_{pl} +1} \right) \left( {K_{qm} +1} \right) }} \nonumber \\&\times \frac{a}{\left[ {F_{H_S } \left( {H_{S,\max } } \right) -F_{H_S } \left( {H_{S,\min } } \right) } \right] \left[ {\tanh \left( {aR_{S,\max } } \right) -\tanh \left( {aR_{S,\min } } \right) } \right] }. \end{aligned}$$

(62)

It is well known that $e^{\pm jx}=\cos x\pm j\sin x,$ the integral of the product of an odd function, i.e., sin $x$, and an even function, i.e., cos $x$ and $J_{0}(x)$, from $-\infty$ to $\infty$ is equal to zero, the product of two even functions is an even function, and the integral of an even function from $-\infty$ to $\infty$ is twice the integral from 0 to $\infty$. Under these considerations and using the equality$\int _0^\infty {J_0 \left( {a\sqrt{x^{2}+z^{2}}} \right) \cos \left( {bx} \right) dx=\cos \left( {z\sqrt{a^{2}-b^{2}}} \right) /\sqrt{a^{2}-b^{2}} }$ [27, eq. 6.677-3], (61) becomes

$$\begin{aligned}&S_{pl,qm}^{NLoS} \left( {\delta _T ,\delta _R ,\nu } \right) \nonumber \\&\quad =\int \limits _{H_{S,\min } }^{H_{S,\max } } {\int \limits _{R_{S,\min } }^{R_{S,\max } } {\frac{Ae^{j2\pi s_2 \left( {\nu -f_{T,\max } \cos \gamma _T } \right) }}{\pi f_{R,\max } \cos \left[ {\arctan \left( {H_S /R_S } \right) } \right] \sqrt{1-\left( {\frac{\nu -f_{T,\max } \cos \gamma _T }{f_{R,\max } \cos \left[ {\arctan \left( {H_S /R_S } \right) } \right] }} \right) ^{2}}}} } \nonumber \\&\quad \times \cos \left[ {2\pi s_3 f_{R,\max } \cos \left[ {\arctan \left( {H_S /R_S } \right) } \right] \sqrt{1-\left( {\frac{\nu -f_{T,\max } \cos \gamma _T }{f_{R,\max } \cos \left[ {\arctan \left( {H_S /R_S } \right) } \right] }} \right) ^{2}}} \right] dR_S dH_S.\nonumber \\ \end{aligned}$$

(63)

The double integral in (61) has to be evaluated numerically, since there is no closed-form solution. One observes that (61) is applicable for the range

$$\begin{aligned}&0\le \left| {\nu -f_{T,\max } \cos \gamma _T } \right| \le f_{R,\max } \cos \left[ {\arctan \left( {H_S /R_S } \right) } \right] \nonumber \\&\qquad \qquad \Rightarrow 0\le \left| {\nu -f_{T,\max } \cos \gamma _T } \right| \le \nu _{\min } , \end{aligned}$$

(64)

where

$$\begin{aligned} \nu _{\min }&= \mathop {\min }\limits _{H_S ,R_S } \left\{ {f_{R,\max } \cos \left[ {\arctan \left( {H_S /R_S } \right) } \right] } \right\} \nonumber \\&= f_{R,\max } \cos \left[ {\arctan \left( {H_{S,\max } /R_{S,\min } } \right) } \right] . \end{aligned}$$

(65)

However, the SDPS should be sketched for the range

$$\begin{aligned} 0\le \left| {\nu -f_{T,\max } \cos \gamma _T } \right| \le \nu _{\max } , \end{aligned}$$

(66)

where

$$\begin{aligned} \nu _{\max }&= \mathop {\max }\limits _{H_S ,R_S } \left\{ {f_{R,\max } \cos \left[ {\arctan \left( {H_S /R_S } \right) } \right] } \right\} \nonumber \\&= f_{R,\max } \cos \left[ {\arctan \left( {H_{S,\min } /R_{S,\max } } \right) } \right] . \end{aligned}$$

(67)

Hence, the SDPS of the NLoS component can be written as in (32)–(36).

Appendix 2: The Relative PSDS of the NLoS Component

The relative PSDS of the NLoS component can be obtained by calculating the IFT of the SFCF of the NLoS component in (27)–(30)

$$\begin{aligned} P_{pl,qm}^{NLoS} \left( {\delta _T ,\delta _R ,\tau _r } \right) =\mathfrak I _{\Delta f}^{-1} \left\{ {R_{pl,qm}^{NLoS} \left( {\delta _T ,\delta _R ,\Delta t=0,\Delta f} \right) } \right\} . \end{aligned}$$

(68)

Considering that $\Delta t=0,$ the Bessel function $I_0 \left( {\sqrt{w_2^2 +w_3^2 }} \right)$ of (27) can be written as

$$\begin{aligned} I_0 \left( {\sqrt{w_2^2 +w_3^2 }} \right) \mathop =\limits _{\Delta t=0} J_0 \left[ {2\pi p_1 \sqrt{\left( {\Delta f-p_2 } \right) ^{2}+p_3^2 }} \right] , \end{aligned}$$

(69)

where

$$\begin{aligned} p_1&= \frac{R_S }{c_0 \cos \beta _T }, \nonumber \\ p_2&= \frac{\left( {m-l} \right) \delta _R \cos \theta _R \cos \psi _R c_0 \cos \beta _T \cos \left[ {\arctan \left( {H_S /R_S } \right) } \right] }{\lambda R_S }-\frac{jkc_0 \cos \mu \cos \beta _T }{2\pi R_S }, \nonumber \\ p_3&= \frac{\left( {q-p} \right) \delta _T \sin \theta _T c_0 }{\lambda D}+\frac{\left( {m-l} \right) \delta _R \sin \theta _R \cos \psi _R c_0 \cos \beta _T \cos \left[ {\arctan \left( {H_S /R_S } \right) } \right] }{\lambda R_S } \nonumber \\&-\frac{jkc_0 \sin \mu \cos \beta _T }{2\pi R_S }. \end{aligned}$$

(70)

Using (69) and (70), (68) becomes

$$\begin{aligned}&P_{pl,qm}^{NLoS} \left( {\delta _T ,\delta _R ,\tau _r } \right) \nonumber \\&\quad =\int \limits _{H_{S,\min } }^{H_{S,\max } } {\int \limits _{R_{S,\min } }^{R_{S,\max } } {A\int \limits _{-\infty }^\infty {e^{j2\pi \Delta f\left( {\tau _r -p_4 } \right) }J_0 \left( {2\pi p_1 \sqrt{\left( {\Delta f-p_2 } \right) ^{2}+p_3^2 }} \right) d\Delta fdR_S dH_S}, } } \end{aligned}$$

(71)

where

$$\begin{aligned} p_4 =\frac{R_S }{c_0 \cos \left[ {\arctan \left( {H_S /R_S } \right) } \right] }. \end{aligned}$$

(72)

Using the equality [27, eq. 6.677-3] and after extensive calculations, the relative PSDS of the NLoS component is derived as follows

$$\begin{aligned}&P_{pl,qm}^{NLoS} \left( {\delta _T ,\delta _R ,\tau _r } \right) \nonumber \\&\quad =\int \limits _{H_{S,\min } }^{H_{S,\max } } {\int \limits _{R_{S,\min } }^{R_{S,\max } } {\frac{Ae^{j2\pi p_2 \left( {\tau _r -p_4 } \right) }\cos \left( {2\pi p_3 \sqrt{p_1^2 -\left( {\tau _r -p_4 } \right) ^{2}}} \right) }{\pi \sqrt{p_1^2 -\left( {\tau _r -p_4 } \right) ^{2}}}dR _S dH_S } }. \end{aligned}$$

(73)

The double integral in (73) has to be evaluated numerically, since there is no closed-form solution. One observes that (73) is applicable for the range

$$\begin{aligned}&\left| {\tau _r -p_4 } \right| \le p_1 \nonumber \\&\quad \Rightarrow p_4 -p_1 \le \tau _r \le p_4 +p_1 \nonumber \\&\quad \Rightarrow \tau _1 \le \tau _r \le \tau _2, \end{aligned}$$

(74)

where

$$\begin{aligned} \tau _1&= \mathop {\max }\limits _{H_S ,R_S } \left\{ {p_4 -p_1 } \right\} \nonumber \\&= \frac{\sqrt{R_{S,\min }^2 +H_{S,\max }^2 }}{c_0 }-\frac{R_{S,\min } }{c_0 \cos \beta _T }, \end{aligned}$$

(75)

$$\begin{aligned} \tau _2&= \mathop {\min }\limits _{H_S ,R_S } \left\{ {p_4 +p_1 } \right\} \nonumber \\&= \frac{\sqrt{R_{S,\min }^2 +H_{S,\min }^2 }}{c_0 }+\frac{R_{S,\min } }{c_0 \cos \beta _T }. \end{aligned}$$

(76)

However, the relative PSDS should be sketched for the range

$$\begin{aligned} 0\le \tau _r \le \tau _{r,\max } , \end{aligned}$$

(77)

where $\tau _{r,\max } \approx \tau _{\max } -\tau _{LoS}$ and $\tau _{LoS}$ and $\tau _{\max }$ are defined in (8) and (11), respectively. Hence, the relative PSDS of the NLoS component can be written as in (38)–(42).