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Outage probability of MIMO cognitive radio networks with energy harvesting and adaptive transmit power

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Abstract

This paper derives the outage probability of cognitive radio networks (CRNs) with energy harvesting (EH). The primary nodes have a single antenna, whereas the secondary nodes have multiple antennas. A secondary source S harvests energy from radiofrequency (RF) signal received from primary transmitter PT using nr, S antennas. S also adapts its power so that the interference at primary receiver PR is less than threshold I. The main contribution of the paper is to derive the throughput of multiple input multiple output (MIMO) CRN with RF energy harvesting and adaptive transmit power. We also optimize the secondary throughput by choosing the optimal harvesting duration.

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Authors and Affiliations

  1. Higher College of Technology, Sharjah Women College, P.O. Box 7947, Sharjah, United Arab Emirates

    Nadhir Ben Halima

  2. COSIM Laboratory, SUPCOM, Ariana, Tunisia

    Hatem Boujemâa

Authors
  1. Nadhir Ben Halima
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  2. Hatem Boujemâa
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Correspondence to Nadhir Ben Halima.

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Appendix

Appendix

We have:

$$ \begin{array}{@{}rcl@{}} P_{\gamma_{SD,i,j}}(x)&=&P(\gamma_{SD,i,j} \leq x)=P(E_{Adaptive}|f_{SD,i,j}|^{2}\\ &\leq& (N_{0}+E_{P_{T}}|f_{P_{T},j}|^{2})x), \end{array} $$
(34)

Let \(Z=N_{0}+E_{P_{T}}|f_{P_{T},j}|^{2}\), then we deduce:

$$ P_{\gamma_{SD,i,j}}(x)={\int}_{N_{0}}^{+\infty}P(E_{Adaptive}|f_{SD,i,j}|^{2}\leq ux)p_{Z}(u)du. $$
(35)

For Rayleigh fading channels, \(E_{P_{T}}|f_{P_{T},j}|^{2}\) follows an exponential distribution with mean \(E_{P_{T}}\lambda _{P_{T},j}\) where \(\lambda _{P_{T},j}=E(|f_{P_{T},j}|^{2})\). We deduce:

$$ \begin{array}{@{}rcl@{}} P_{\gamma_{SD,i,j}}(x)&=&\frac{1}{\lambda_{P_{T},j}E_{P_{T}}}{\int}_{N_{0}}^{+\infty}P(E_{Adaptive}|f_{SD,i,j}|^{2}\\ &\leq& ux)e^{-\frac{(u-N_{0})}{\lambda_{P_{T},j}E_{P_{T}}}}du. \end{array} $$
(36)

The term P(EAdaptive|fSD, i, j|2 ≤ ux) was derived in Section 3.2 Eq. 22. We need to replace N0 by 1 and x by ux in Eq. 22:

$$ \begin{array}{@{}rcl@{}} &&P(E_{Adaptive}|f_{SD,i,j}|^{2}\leq ux)\\ &=& 1-\frac{2}{\lambda_{SD}^{2}}\sum\limits_{q=0}^{n_{r,S}-1}\frac{(ux)^{q}}{q!\lambda_{P_{T}S}^{2q}\mu^{q}} \left( \frac{ux\lambda_{SD}^{2}}{\mu\lambda^{2}_{P_{T}S}}\right)^{\frac{1-q}{2}}\\ &&K_{1-q}\left( 2\sqrt{\frac{xu}{\lambda_{SD}^{2}\mu \lambda^{2}_{P_{T}S}}}\right)\\ &&+\frac{2}{\lambda_{SD}^{2}}\sum\limits_{q=0}^{n_{r,S}-1}\frac{(ux)^{q}}{q!\lambda_{P_{T}S}^{2q}\mu^{q}} \left[\frac{u^{2}x^{2}}{\mu\lambda_{P_{T}S}^{2}\left( \frac{ux}{\lambda_{SD}^{2}}+\frac{I}{\lambda_{SP_{R}}^{2}}\right)}\right]^{\frac{1-q}{2}} \\ &&K_{1-q}\left( 2\sqrt{\frac{1}{\mu\lambda_{P_{T}S}^{2}}\left( \frac{ux}{\lambda_{SD}^{2}}+\frac{I}{\lambda_{SP_{R}}^{2}}\right)}\right) \end{array} $$
(37)

We use the following result [25] (6.631.3):

$$ \begin{array}{@{}rcl@{}} &&{\int}_{0}^{+\infty}x^{a-0.5}e^{-bx}K_{2c}(2d\sqrt{x})dx\\ &=&\!\frac{\Gamma(a + c + 0.5){\Gamma}(a - c + 0.5)}{2d}e^{\frac{d^{2}}{2b}} b^{-a}W_{-a,c}\left( \frac{d^{2}}{b}\right) \end{array} $$
(38)

where We, d(x) is the Whittaker function [25].

Using Eqs. 36, 37, and 38, we obtain:

$$ \begin{array}{@{}rcl@{}} P_{\gamma_{SD,i,j}}(x)&=&1-\frac{1}{\lambda_{SD}^{2}\lambda_{P_{T},j}E_{P_{T}}}e^{\frac{N_{0}}{\lambda_{P_{T},j}E_{P_{T}}}} \sum\limits_{q=0}^{n_{r,S}-1}\frac{x^{q}}{\lambda_{P_{T}S}^{2q}\mu^{q}}\\ &&\left( \frac{x\lambda_{SD}^{2}}{\mu \lambda^{2}_{P_{T}S}}\right)^{\frac{1-q}{2}}\frac{1}{2\sqrt{\frac{x}{\lambda_{SD}^{2}\mu\lambda_{P_{T}S}^{2}}}} e^{\frac{x\lambda_{P_{T},j}E_{P_{T}}}{\lambda_{SD}^{2}\mu \lambda_{P_{T}S}}}\\ &&\times (\lambda_{P_{T},j}E_{P_{T}})^{1+q/2}W_{-1-q/2,0.5-q/2}\\ &&\left( \frac{x\lambda_{P_{T},j}E_{P_{T}}}{\lambda_{SD}^{2}\mu \lambda_{P_{T}S}}\right)\\ &&+{\int}_{0}^{N_{0}}\frac{2}{\lambda_{SD}^{2}\lambda_{P_{T},j}E_{P_{T}}}\sum\limits_{q=0}^{n_{r,S}-1}\frac{(ux)^{q}}{q! \lambda_{P_{T}S}^{2q}\mu^{q}}\\ &&\left( \frac{ux\lambda_{SD}^{2}}{\mu\lambda^{2}_{P_{T}S}}\right)^{\frac{1-q}{2}}K_{1-q}\left( 2 \sqrt{\frac{xu}{\lambda_{SD}^{2}\mu\lambda^{2}_{P_{T}S}}}\right)\\ &&e^{-\frac{(u-N_{0})}{\lambda_{P_{T},j}E_{P_{T}}}}du\\ &&+{\int}_{N_{0}}^{+\infty}\frac{2}{\lambda_{SD}^{2}\lambda_{P_{T},j}E_{P_{T}}}e^{\frac{-(u-N_{0})}{\lambda_{P_{T},j}E_{P_{T}}}} \sum\limits_{q=0}^{n_{r,S}-1} \end{array} $$
$$ \begin{array}{@{}rcl@{}} &&\frac{(ux)^{q}}{q!\lambda_{P_{T}S}^{2q}\mu^{q}}\left[\frac{u^{2}x^{2}}{\mu\lambda_{P_{T}S}^{2} (\frac{ux}{\lambda_{SD}^{2}}+\frac{I}{\lambda_{SP_{R}}^{2}})}\right]^{\frac{1-q}{2}}\\ &&K_{1-q}\left( 2\sqrt{\frac{1}{\mu\lambda_{P_{T}S}^{2}} \left( \frac{ux}{\lambda_{SD}^{2}}+\frac{I}{\lambda_{SP_{R}}^{2}}\right)}\right)du, \end{array} $$
(39)

where the last integrals are calculated numerically using MATLAB.

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Halima, N.B., Boujemâa, H. Outage probability of MIMO cognitive radio networks with energy harvesting and adaptive transmit power. Ann. Telecommun. 76, 355–362 (2021). https://doi.org/10.1007/s12243-020-00817-9

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