Coverage Performance in Cognitive Radio Networks with Self-sustained Secondary Transmitters

Abstract

In this paper, we investigate the opportunistic spectrum access (OSA) of self-sustained secondary transmitters (STs) in cognitive radio (CR) network to improve both the spectral efficiency and energy efficiency. Particularly, by utilizing energy harvesting, the STs are assumed to be able to collect and store ambient powers for data transmission. An energy harvesting based OSA protocol, namely the EH-PRA protocol, is considered, under which a ST is eventually allowed to launch the transmission only if its battery level is larger than the transmit power and the estimated interference perceived at the active primary receivers (PRs) is lower than a threshold $N_{ra}$. Given that the battery capacity of STs is infinite, we derive the transmission probability of STs. We then characterize the coverage performance of the CR network. Finally, simulation results are provided for the validation of our analysis.

A Proof of Theorem 2

Proof

With the energy harvesting based PRA protocol, given a typical PR located at the origin, the received SIR is given by

$$\begin{aligned} \mathrm {SIR}_p = \frac{P_p h_0 d_p^{- \alpha }}{\sum \limits _{i \in \Pi _p^t}P_p h_i |\mathbf{{X}}_i|^{-\alpha } + \sum \limits _{j \in \Pi _{s}^{ra}}P_s g_j|\mathbf{{Y}}_j|^{-\alpha }}. \end{aligned}$$

(21)

It is worth noting that under the EH-PRA protocol, at the typical PR, the received interference from the j-th active ST is constrained as $P_s g_j |\mathbf{{Y}}_j|^{- \alpha }\le \frac{P_s{N_{ra}}}{P_p}$. Therefore, under Assumption 1, the coverage probability of the primary network with the EH-PRA protocol is given by

$$\begin{aligned} \begin{aligned} \tau _p^{ra} =&\Pr \left\{ {\mathrm {SIR}}_p \ge \theta _p \Bigg | g_j |\mathbf{{Y}}_j|^{-\alpha } \le \frac{N_{ra} }{P_p}\right\} \\ =&\Pr \left\{ \frac{P_p h_0 d_p^{- \alpha }}{\sum \limits _{i \in \Pi _p^t}P_p h_i |\mathbf{{X}}_i|^{-\alpha } + \sum \limits _{j \in \Pi _{s}^{ra}}P_s g_j|\mathbf{{Y}}_j|^{-\alpha }} \ge \theta _p \Bigg | g_j |\mathbf{{Y}}_j|^{-\alpha } \le \frac{N_{ra} }{P_p}\right\} \\ =&{\mathbb {E}}_\mathbf{{X}}\left[ \prod \limits _{i \in \Pi _p^t} {\mathbb {E}}_h\left[ e^{-\frac{\theta _p h_i |\mathbf{{X}}_i|^{-\alpha }}{d_p^{- \alpha }}}\right] \right] \\&\times \, {\mathbb {E}}_\mathbf{{Y}} \left[ \prod \limits _{j \in \Pi _{s}^{ra}} {\mathbb {E}}_g \left[ e^{-\frac{\theta _p P_s g_j |\mathbf{{Y}}_j|^{-\alpha }}{P_p d_p^{- \alpha }}}\Bigg | g_j \le \frac{N_{ra} |\mathbf{{Y}}_j|^{\alpha }}{P_p}\right] \right] \\ \buildrel {(a)}\over =&\exp \left\{ - \frac{2 \pi ^2}{\alpha \sin \left( \frac{2 \pi }{\alpha }\right) } \mu _p\theta _p^{\frac{2}{\alpha }} d_p^2 \right\} \times \exp \left\{ - 2 \pi \int _0^{\infty } \mathcal {G} \cdot \lambda _{ra}^{\mathbf {R}}(u) u \text {d}u\right\} \\ \buildrel {(b)}\over =&\exp \left\{ - \frac{2 \pi ^2}{\alpha \sin \left( \frac{2 \pi }{\alpha }\right) } \theta _p^{\frac{2}{\alpha }} d_p^2 \mu _p\right\} \times \exp \left\{ - 2 \pi \lambda _0 \frac{\nu _e^s}{P} \int _\zeta ^{\infty }\left( 1 - \varrho (u)\right) u d u\right\} \\&\times \, \exp \left\{ - 2 \pi \lambda _0 Q_{ra}\int _0^\zeta \left( 1 - \varrho (u)\right) \mathcal {P}(u) u d u\right\} . \end{aligned} \end{aligned}$$

(22)

where (a) follows from the fact that the probability density function of g conditioned on $g \le t$ is given by

$$\begin{aligned} f(g|g \le t) = \frac{e^{-g}}{1 - e^{-t}}, \end{aligned}$$

$$ \mathcal {G} = 1 - \int _0^{\frac{N_{ra} u^{\alpha }}{P_p}} e^{- \frac{\theta _p P_s g u^{- \alpha }}{P_p d_p^{- \alpha }}} \times \frac{e^{-g}}{1 - e^{- {\frac{N_{ra} u^{\alpha }}{P_p}}}}dg, $$

and (b) follows from (8). This thus completes the proof of Theorem 2.