Throughput and Energy Efficiency of Wireless Powered Multi-tier MIMO HetNets

Abstract

Wireless powered communications have been recently proposed as a variable power solution for the heterogeneous networks (HetNets) with small and flexible deployments of low-power base stations (BSs). In this paper, we consider the wireless powered multi-tier multi-input multi-output (MIMO) HetNets, where the multi-antenna BSs perform downlink transmission. In particular, we consider two cases of interests, i.e., 1) energy harvesting (EH) without energy beamforming (EB) tier and 2) EH with EB tier. Using tools of stochastic geometry, we perform analysis on the throughput and energy efficiency (EE) of the considered network. Closed-form and tractable results are obtained to reveal interesting insights that the proposed wireless-powered MIMO HetNets can achieve higher EE as compared to the conventional HetNets without EH, and the EE performance can be further improved by introducing well-designed EB across tiers.

Appendices

Appendix A

Recalling the definition in (4), the Laplace transformation of E can be derived as

$$\begin{array}{@{}rcl@{}} \mathcal{L}_{E{{\left( \mathrm{s}\right)}}} & = & {\mathbb{E}_{{E}}\left[{e}^{{-s\beta{\sum}_{j\in\mathcal{K}}{\sum}_{x\in{\Phi}_{j}}P_{j}g_{j}\left( x\right)||x||^{-\alpha_{j}}}}\right]} \end{array} $$

(47)

$$\begin{array}{@{}rcl@{}} & \underset{=}{(a)} & {{\underset{{j\in\mathcal{K}}}{\prod}\mathbb{E}\left[{\underset{x\in{\Phi}_{j}}{\prod}\mathbb{E}_{{g_{j}\left( x\right)}}}\left( e^{{-s\beta P_{j}tg_{j}\left( x\right)||x||^{-\alpha_{j}}}}\right)\right]}} \end{array} $$

(48)

$$\begin{array}{@{}rcl@{}} & \underset{=}{(b)} & {\underset{{j\in\mathcal{K}}}{\prod}e^{-\lambda_{j}{\int}_{\mathbb{R}^{2}}\left( 1-{E_{{g_{j}}}\left( x\right)}\left( e^{{-s\beta P_{j}tg_{j}\left( x\right)||x||^{-\alpha_{j}}}}\right)\right)dx}} \end{array} $$

(49)

$$\begin{array}{@{}rcl@{}} & \underset{=}{(c)} & \underset{{j\in\mathcal{K}}}{\prod}e^{-\lambda_{j}{\int}_{\mathbb{R}^{2}}\left( 1-\frac{1}{\left( 1+{s\beta P_{j}||x||^{-\alpha_{j}}}\right)^{U_{j}}}\right)dx} \end{array} $$

(50)

$$\begin{array}{@{}rcl@{}} & \underset{=}{(d)} & \underset{{j\in\mathcal{K}}}{\prod}\exp\left( -{\lambda_{j}{{\int}_{\mathbb{R}^{2}}}{\sum\limits_{{m=\text{1}}}^{{U_{j}}}}\left( {\begin{array}{c} U_{j}\\ m \end{array}}\right)}\frac{{\left( s\beta P_{j}||x||^{-\alpha_{j}}\right)}^{{m}}}{{\left( \text{1}+{s\beta P_{j}||x||^{-\alpha_{j}}}\right)^{U_{j}}}}dx\right) \end{array} $$

(51)

$$\begin{array}{@{}rcl@{}} & \underset{=}{(e)} & \underset{{j\in\mathcal{K}}}{\prod}\exp\left( -2\pi{\lambda_{j}{\sum\limits_{m=\text{1}}^{{U_{j}}}}\left( {\begin{array}{c} U_{j}\\ m \end{array}}\right)}{{{\int}_{\text{0}}^{{{\infty}}}}}\frac{{\left( s\beta P_{j}R^{-\alpha_{j}}\right)}^{{m}}}{{\left( \text{1}+{s\beta P_{j}R^{-\alpha_{j}}}\right)^{U_{j}}}}{RdR}\right) \end{array} $$

(52)

$$\begin{array}{@{}rcl@{}} & \underset{=}{(f)} & \underset{{j\in\mathcal{K}}}{\prod}\exp\left( -2\pi{\lambda_{j}}(s\beta P_{j})^{\frac{2}{\alpha_{j}}}{\sum\limits_{m=\text{1}}^{{U_{j}}}}\left( {\begin{array}{c} U_{j}\\ m \end{array}}\right){{{\int}_{\text{0}}^{{{\infty}}}}}\frac{{r}^{{-\alpha_{j}m}}}{{\left( \text{1}+r^{-\alpha_{j}}\right)^{U_{j}}}}{rdr}\right) \end{array} $$

(53)

$$\begin{array}{@{}rcl@{}} & \underset{=}{(g)} & \exp\left( -2\pi{\sum}_{j\in\mathcal{K}}\frac{\lambda_{j}}{\alpha_{j}}(s\beta P_{j})^{\frac{2}{\alpha_{j}}}{\sum}_{m = 1}^{U_{j}}\left( {\begin{array}{c} U_{j}\\ m \end{array}}\right){{\int}_{0}^{1}}(1-t)^{m-\frac{2}{\alpha_{j}}-1}t^{\frac{2}{\alpha_{j}}-m+U_{j}-1}dt\right) \end{array} $$

(54)

$$\begin{array}{@{}rcl@{}} & \underset{=}{(h)} & \exp\left( -2\pi{\sum}_{j\in\mathcal{K}}\frac{\lambda_{j}}{\alpha_{j}}(s\beta P_{j})^{\frac{2}{\alpha_{j}}}{\sum}_{m = 1}^{U_{j}}\left( {\begin{array}{c} U_{j}\\ m \end{array}}\right)B\left( U_{j}-m+\frac{2}{\alpha_{j}},m-\frac{2}{\alpha_{j}}\right)\right) \end{array} $$

(55)

where (a) follows from the independence of tiers, (b) uses the probability generating function (PGFL) of PPP [22], and $\mathbb {R}^{2}$ is presented as a two-dimensional plane. (c) follows from the Laplace transform of g _j (x) ∼Γ(U _j ,1), (d) follows from Binomial theorem where $\left ({\begin {array}{c} U_{j}\\ m \end {array}}\right )$ represents ${C_{U_{j}}^{m}}={\frac {U_{j}!}{m!(U_{j}-m)!}}$, (e) uses the transformation to the polar coordinates where x = (R,𝜃), (f) follows by $(s\beta P_{j})^{\alpha _{j}}R=r$, then $R=r(s\beta P_{j})^{-\alpha _{j}},dR=(s\beta P_{j})^{-\alpha _{j}}dr$, (g) follows from $\frac {1}{\text {1}+r^{-\alpha _{j}}}=t$ and then (h) employs the Beta function $B(x,y)={{\int }_{0}^{1}}u^{x-1}(1-u)^{y-1}du$.

Appendix B

From the viewpoint of a typical EB BS, the PMF of the number of EH BSs $N_{E}^{\prime }$ in tier K associated with one EB BS in tier i can refer to [34]

$$ P\left( N_{E}^{\prime}=n\right)=\frac{3.5^{3.5}{\Gamma}\left( n + 3.5\right)\left( \lambda_{K}/p\lambda_{i}\right)^{n}\ }{{\Gamma}\left( n + 3.5\right)n!\left( \lambda_{K}/p\lambda_{i}+ 3.5\right)^{n + 3.5}} $$

(56)

From the viewpoint of a typical EH BS, there is a tagged EB BS that the typical EH BS is associated with. Besides, there are other EH BSs associated with the tagged EB BS. With reference to [27], the PMF of the number of the other EH BSs (apart from the typical EH BS that are associated with the tagged EB BS) is the same with Eq. 56. Then, the PMF of the number of EH BSs N _E including the typical EH BS that are associated with one tagged EB BS can be written as

$$ P\left( N_{E}=n\right)=P\left( N_{E}^{\prime}=n + 1\right) $$

(57)

Therefore, N _E and $N_{E}^{\prime }$ can be approximated by the mean value $1 + 1.28\frac {\lambda _{K}}{p\lambda _{i}}$ and $\frac {\lambda _{K}}{p\lambda _{i}}$ [33]. respectively.