Coverage and network spectral efficiency analysis for UAV swarm under 3D beamforming in isolated regions

Abstract

Fifth-generation networks are facing coverage problems due to the vulnerability of millimeter-wave bands caused by blockage and penetration loss phenomena, resulting in shadow zones commonly called isolated regions (IRs). Unmanned aerial vehicles (UAVs) acting as aerial base stations, can be rapidly applied to provide communications coverage in IR. Existing research mainly adopts UAVs with conventional beamforming, but this scheme simply neglects the array gain in the vertical plane, and interference coordination becomes complicated. Considering a 3D beamforming approach that adjusts the tilt angle, this paper explores the coverage and network spectral efficiency (NSE) analysis using stochastic geometry, where UAVs are distributed as a binomial point process within a bounded circular area over the IR. Specifically, the association probability of an arbitrarily-located typical user in the finite region is computed based on the strongest averaged received power in line-of-sight/non-line-of-sight propagation. Furthermore, the signal-to-interference-plus-noise ratio coverage probability and NSE expressions are obtained under the hypothesis of Nakagami-m fading. Monte Carlo simulations are performed to illustrate the effectiveness of the proposed analysis method. Through the results we show the impact of antenna elevation on the coverage, the optimal tilt angle and the optimal number of simultaneously active UAVs to maximize performance.

Appendices

Appendix A

We recall that the K transmitting UAVs in a single swarm are i.i.d. with an arbitrary location $\{u_k\}_{k=1:K}$ according to the BPP, and $Y_{u_k}\le y_u$ is the distance between a UAV $u_k$ and typical user, thus that there is only one serving UAV, which is the one that gives the highest received power. Let $\mathcal {C}_L$ denote the event that the service UAV located at $u_i$ is in LoS connection with the user, the distance between the two being noted as r. $\mathcal {C}_L$ can be defined as

$$\begin{aligned}&\mathcal {C}_L\overset{(a)}{=} K P_L(Y_{u_i}) \underset{u_k\in \psi \backslash u_i}{\prod } \mathbb {P} (P_{e} \varphi _{L}^{-1} Y_{u_i}^{-\alpha _{L}}>P_{e} \varphi _{L}^{-1} Y_{u_k}^{-\alpha _{L}}\nonumber \\&\cup P_{e} \varphi _{L}^{-1} Y_{u_i}^{-\alpha _{L}}>P_{e} \varphi _{N}^{-1} Y_{u_k}^{-\alpha _{N}} \big )\nonumber \\ =&K P_L(r) \underset{u_k\in \psi \backslash u_i}{\prod }\ \mathbb {P} ( Y_{u_k}> r) + \mathbb {P} ( Y_{u_k}> r^{\frac{\alpha _{L}}{\alpha _{N}}}) \nonumber \\ =&K P_L(r) \left( \int _{r}^{y_u} P_{L}(y) f_{Y_{u_k}}(y) dy\right. \nonumber \\&\left. +\int _{R_{LN}(r)}^{y_u} P_{N}(y) f_{Y_{u_k}}(y) dy \right) ^{K-1}, \end{aligned}$$

(19)

where (a) follows from that there are K possibilities to choose the UAV $u_i$. and the event that the strongest UAV can be LoS or NLoS are mutually exclusive. (b) means that there is no UAV in NLoS located at a distance less than $R_{LN}(r)$. Integrating according to r results in the association probability $\mathcal {A}_L(y_0)$ given in (11).

Appendix B

The conditional coverage probability is the probability that the SINR at the user is greater than the SINR threshold conditioned on the serving UAV in $s \in \{L,N\}$ link, we denoted by $\hat{X}_s=r$ the serving distance conditioned on the event that UAV $u_i$ is s link, $P_s^C$ can be defined as follows

$$\begin{aligned} P_{s}^c =&\mathbb {E}_{{X}_s}\Bigg [\mathbb {P} \Bigg ( \frac{P_{e} \upsilon (\omega ,\theta _{u_i}) H_{u_i}^s \varphi _s^{-1} X_s^{-\alpha _s} }{I+\sigma _N^2}> \tau \nonumber \\ {}&|\text {the serving UAV is in} s \text {link}\Bigg )\Bigg ] \nonumber \\ =&\mathbb {E}_{\hat{X}_s}\left[ \mathbb {P} \left( H_{u_i}^s >\frac{ \tau \varphi _s {\hat{X}_s}^{\alpha _{s}}}{P_{e} \upsilon (\omega ,\theta _{u_i}) } (I+\sigma _N^2) \right) \right] \nonumber \\ =&\mathbb {E}_{\hat{X}_s} \left[ \bar{F}_{ H_{u_i}^s}\left( \frac{ \tau \varphi _s {\hat{X}_s}^{\alpha _{s}}}{P_{e} \upsilon (\omega ,\theta _{u_i}) } (I+\sigma _N^2)\right) \right] , \end{aligned}$$

(20)

Since $H_{u_i}^s$ is a normalized gamma random variable, $\bar{F}_{ H_{u_i}^s}(x)$ can be approximated using the gamma approximation [33] [34] to simplify the evaluation process, which yields

$$\begin{aligned} P_{s}^c=&\mathbb {E}_{\hat{X}_s,I} \left[ 1-\left( 1- e^{\frac{ \tau \varphi _s {\hat{X}_s}^{\alpha _{s}}}{P_{e} \upsilon (\omega ,\theta _{u_i}) } (I+\sigma _N^2) } \right) ^{m_s}\right] \nonumber \\ \overset{(a)}{=}&\sum _{n=1}^{m_s} (-1)^{n+1} \begin{pmatrix} m_s\\ n \end{pmatrix} \mathbb {E}_{\hat{X}_s,I}\left[ e^{\frac{- n \varepsilon _s \tau \varphi _s {\hat{X}_s}^{\alpha _{s}}}{P_{e} \upsilon (\omega ,\theta _{u_i})} (I+\sigma _N^2)} \right] \nonumber \\ =&\int _{h}^{y_u} \sum _{n=1}^{m_s} (-1)^{n+1} \begin{pmatrix} m_s\\ n \end{pmatrix} e^{-\delta \sigma _N^2} \nonumber \\&\times \mathcal {L}_I\left( \delta ,R_{sL}(r),R_{sN}(r)\right) f_{\hat{X}_s}(r)dr \end{aligned}$$

(21)

where (a) comes from the binomial theorem, $\delta =\frac{n \varepsilon _s \tau \varphi _s {r}^{\alpha _{s}}}{P_{e} \upsilon (\omega ,\theta _{u_i})}$ and $\mathcal {L}_I(\delta ,R_{sL}(r),R_{sN}(r))$ is the Laplace transform of the interference. This completes the proof of (14).

Appendix C

Denote the event that the interfering SA UAV located at $u_k \in \psi _a \backslash u_i$ in $\ell \in \{L,N\}$ link by $\mathcal {I}_{\ell }$. Since the number of UAVs is fixed we can write $\mathcal {I}_{L} \cup \mathcal {I}_{N}$. Hereafter we denote by $\mathcal {Q}_{sL}$, $\mathcal {Q}_{sN}$ the probability that the interfering UAV is in LoS,

$$\begin{aligned} \mathcal {L}_{I}(\delta ,&R_{sL}(r),R_{sN}(r))= \mathbb {E}_{I|\mathcal {I}_{L}\cup \mathcal {I}_{N}}\left[ \exp \left( {-\delta I}\right) \right] \nonumber \\ =&\mathbb {E}_{{W_{u_k},H_{u_k}^{\ell }|\mathcal {I}_{L}\cup \mathcal {I}_{N}}}\left[ \exp \left( {-\delta \underset{{{u_k} \in \psi _a \backslash u_i}}{\sum }P_{e} \upsilon (\omega ,\theta _{u_k}) H_{u_k}^L \varphi _L^{-1} W_{{u_k}}^{-\alpha _L} + P_{e} \upsilon (\omega ,\theta _{u_k}) H_{u_k}^N \varphi _N^{-1} W_{{u_k}}^{-\alpha _N} }\right) \right] \nonumber \\ \overset{(a)}{=}&\underset{{{u_k} \in \psi _a \backslash u_i}}{\prod }\mathbb {E}_{W_{u_k}|\mathcal {I}_{sL}}\left[ \left( \frac{m_{L}}{m_{L}+\delta P_{e} \upsilon (\omega ,\theta _{u_k}) \varphi _{L}^{-1} {W_{u_k}}^{-\alpha _{L}}}\right) ^{m_{L}}\dfrac{\int _{R_{sL}(r)}^{y_u}f_{Y_{u_k}}(w)P_{L}(w)}{\underset{q}{\sum }{ \int _{R_{sq}(r)}^{y_u} f_{Y_{u_k}}(w) P_{q}(w) dw}}\right] \nonumber \\&+ \mathbb {E}_{W_{u_k}|\mathcal {I}_{sN}}\left[ \left( \frac{m_{N}}{m_{N}+\delta P_{e} \upsilon (\omega ,\theta _{u_k}) \varphi _{N}^{-1} {W_{u_k}}^{-\alpha _{N}}}\right) ^{m_{N}}\dfrac{\int _{R_{sN}(r)}^{y_u}f_{Y_{u_k}}(w)P_{N}(w)}{\underset{q}{\sum }{ \int _{R_{sq}(r)}^{y_u} f_{Y_{u_k}}(w) P_{q}(w) dw}}\right] \nonumber \\ \overset{(b)}{=}&\Bigg [ \int _{R_{sL}(r)}^{y_u} \left( \frac{m_{L}}{m_{L}+\delta P_{e} \upsilon (\omega ,\theta _{u_k}) \varphi _{L}^{-1} {W_{u_k}}^{-\alpha _{L}}}\right) ^{m_{L}}\dfrac{\int _{R_{sL}(r)}^{y_u}f_{Y_{u_k}}(w)P_{L}(w)}{\underset{q}{\sum }{ \int _{R_{sq}(r)}^{y_u} f_{Y_{u_k}}(w) P_{q}(w) dw}} f_{W_{u_k}}(w|\mathcal {I}_{L}) dw \nonumber \\&+ \int _{R_{sN}(r)}^{y_u} \left( \frac{m_{N}}{m_{N}+\delta P_{e} \upsilon (\omega ,\theta _{u_k}) \varphi _{N}^{-1} {W_{u_k}}^{-\alpha _{N}}}\right) ^{m_{N}}\dfrac{\int _{R_{sN}(r)}^{y_u}f_{Y_{u_k}}(w)P_{N}(w)}{\underset{q}{\sum }{ \int _{R_{sq}(r)}^{y_u} f_{Y_{u_k}}(w) P_{q}(w) dw}} f_{\hat{W}_{u_k}}(w|\mathcal {I}_{N}) dw \Bigg ]^{K_a -1}, \end{aligned}$$

(22)

NLoS link respectively conditioned on the two events $\mathcal {I}_{L}$ and $\mathcal {I}_{N}$ are mutually exclusive and given that the service UAV is in s link. $\mathcal {Q}_{s\ell }$ is given by

$$\begin{aligned} \mathcal {Q}_{s\ell }=&\mathbb {P}(\mathcal {I}_{\ell }|\mathcal {I}_{\ell }\cup \mathcal {I}_{b}) =\dfrac{\mathbb {P}(\mathcal {I}_{\ell }|\mathcal {C}_s)}{\mathbb {P}(\mathcal {I}_{\ell }|\mathcal {C}_s+\mathcal {I}_{b}|\mathcal {C}_s)}\nonumber \\ =&\dfrac{\int _{R_{s\ell }(r)}^{y_u}f_{Y_{u_k}}(w)P_{\ell }(w)}{\underset{q}{\sum }{ \int _{R_{sq}(r)}^{y_u} f_{Y_{u_k}}(w) P_{q}(w) dw}}, \end{aligned}$$

(23)

where $s,\ell ,b\in \{L,N\}$ and $b\ne \ell$. Using the aggregate interference equation in (8) and applying the Laplace transform definition, $\mathcal {L}(\delta ,R_{sL}(r),R_{sN}(r))$ is given in (22).

Where (a) follows from the gamma distribution of the channel gain $H_{u_k}^{\ell }$ using the moment generating function, and by substituting the equation (23), (b) is due to the fact that the $K_a -1$ interfering UAVs are i.i.d. over $D_a$. $f_{W_{u_k}}(w|\mathcal {I}_{\ell })$ is the PDF of the interfering distance $W_{u_k}$ conditioned on the event $\mathcal {I}_{\ell }$, it can be given as [31]

$$\begin{aligned} f_{W_{u_k}}(w|\mathcal {I}_{\ell })= \dfrac{f_{Y_{u_k}}(w) P_{\ell }(w)}{\int _{R_{s\ell }(r)}^{y_u} f_{Y_{u_k}}(w) P_{\ell }(w) dw}, \end{aligned}$$

(24)

Substitution (24) in (22) results in (16).