Energy harvesting assisted cognitive radio: random location-based transceivers scheme and performance analysis

Abstract

We consider spectrum-sharing scenario where coexist two communication networks including primary network and secondary network using the same spectrum. While the primary network includes directional multi-transceivers, the secondary network consists of relaying-based transceiver forwarding signals by energy harvesting assisted relay node. In cognitive radio, signals transmitted from secondary network are sufficiently small so that all of primary network receivers have signal to noise ratio (SNR) greater than a given threshold. In contrast, the transmitted signals from primary network cause increasing noise which is difficult to demodulate at secondary network nodes and hence it leads to the peak power constraint. In this paper, we focus on the influence of random location of transceivers at primary network using decode-and-forward protocol. Specifically, we derive closed-form outage probability expression of the secondary network under random location of transceivers and peak power constraint of primary network. This investigation shows the relationship between the fraction of energy harvesting time $\alpha$ of time switching-based relaying protocol on outage probability of secondary network and throughput. In addition, we analyse the influence of the number of primary network transceivers as well as primary network’s SNR threshold on secondary network. Furthermore, the trade-off between increasing energy harvesting and rate was investigated under the effect of energy conversion efficiency. The accuracy of the expressions is validated via Monte-Carlo simulations. Numerical results highlight the trade-offs associated with the various energy harvesting time allocations as a function of outage performance.

Appendices

Appendix A: Proof of Proposition 1

Firstly, we can rewrite (9) as below expression

$$\begin{aligned} {\gamma _{SR}} = \frac{X}{{Y + 1}} \end{aligned}$$

(31)

where $X = \frac{{{P_{S\min }}}}{{{\sigma ^2}}}{\left| {{h_{sr}}} \right| ^2}$, $Y = \sum _{i = 1}^L {\frac{{{P_T}}}{{{\sigma ^2}}}} {\left| {{h_{pr[i]}}} \right| ^2}$. X is the exponential distributed random variable (ERV) with mean ${\lambda _{sr}} = \frac{{{P_{S\min }}}}{{{\sigma ^2}}}P{L_{sr}}$, Y is the sum of ERVs, in which each part has mean ${\lambda _{pr[i]}} = \frac{{{P_T}}}{{{\sigma ^2}}}P{L_{pr[i]}}$,respectively. By applying Lemma 1, the pdf(Y) and cdf(Y) are proven. Probability of (31) in (10) is solved by separating two cases:

$$\begin{aligned} \Pr \left( {\frac{X}{{Y + 1}} \le \gamma _s } \right)= & {} \left\{ {\begin{array}{*{20}l} {\Pr \left( {Y \ge \frac{{X - \gamma _s }}{{\gamma _s }}} \right) , \qquad X \ge \gamma _s } \\ {\Pr \left( {Y \ge \frac{{X - \gamma _s }}{{\gamma _s }}} \right) = 1 ,X< \gamma _s } \\ \end{array}} \right. \quad \nonumber \\= & {} 1 - \int \limits _{\gamma _s }^\infty {f_X \left( x \right) } \Pr \left( {Y < \frac{{X - \gamma _s }}{{\gamma _s }}} \right) dx\nonumber \\ \end{aligned}$$

(32)

and vice versa.

After reducing (32), we obtain the express as in (15).

Appendix B: Proof of Propostion 2

From (20) we can write:

$$\begin{aligned}&{F_{SD}}({\gamma _{SD}}) \nonumber \\&\quad = \underbrace{\Pr \left( {{P_{REH}} > {P_R}} \right) }_{1 - {\mathfrak {K}}}\underbrace{\Pr \left( {\frac{{\frac{{{P_R}}}{{{\sigma ^2}}}{{\left| {{h_{rd}}} \right| }^2}}}{{\sum \nolimits _{i = 1}^L {\frac{{{P_{PT[i]}}}}{{{\sigma ^2}}}{{\left| {{h_{pd[i]}}} \right| }^2}} + 1}} \leqslant {\gamma _s}} \right) }_{\mathfrak {M}}\nonumber \\&\qquad + \underbrace{\Pr \left( {{P_{REH}} \leqslant {P_R}} \right) }_{\mathfrak {K}}\underbrace{\Pr \left( {\frac{{\frac{{{P_{REH}}}}{{{\sigma ^2}}}{{\left| {{h_{rd}}} \right| }^2}}}{{\sum \nolimits _{i = 1}^L {\frac{{{P_{PT[i]}}}}{{{\sigma ^2}}}{{\left| {{h_{pd[i]}}} \right| }^2}} + 1}} \leqslant {\gamma _s}} \right) }_{\mathfrak {N}}\nonumber \\ \end{aligned}$$

(33)

We need to solve three subcases: (22), (23) and (24). Firstly, substituting the value of ${P_{REH}}$ in (18) to $\mathfrak {K}$ form in (33), we receive

$$\begin{aligned} {\mathfrak {K}}&= \Pr \left( {\frac{{2\alpha \delta {\sigma ^2}}}{{1 - \alpha }}\left( {{X_1} + {X_2} +\cdots + {X_L} + {X_{L + 1}} + 1} \right) \leqslant {P_R}} \right) \end{aligned}$$

(34)

$$\begin{aligned}&= \Pr \left( {{X_1} + {X_2} + \cdots + {X_L} + {X_{L + 1}} \leqslant \hat{w}} \right) \end{aligned}$$

(35)

where ${X_i}$: $i = 1 \div L$: $f\left( {{X_i}} \right) = \frac{1}{{{\lambda _{pr[i]}}}}{e^{ - \;\frac{x}{{{\lambda _{pr[i]}}}}}}$, ${X_{L + 1}}$: $f\left( {{X_{L + 1}}} \right) = \frac{1}{{{\lambda _{sr}}}}{e^{ - \;\frac{x}{{{\lambda _{sr}}}}}}$, $\hat{w} = \frac{{{P_R}\left( {1 - \alpha } \right) }}{{2\alpha \delta {\sigma ^2}}} - 1$. Applying Lemma 1, we obtain (22).

Secondly, $\mathfrak {M}$ form in (33) is given by

$$\begin{aligned} \mathfrak {M} = \Pr \left( {\frac{U}{{V + 1}} \leqslant {\gamma _s}} \right) \end{aligned}$$

(36)

where

$$\begin{aligned} V = \sum \limits _{i = 1}^L {{V_i}} = \sum \limits _{i = 1}^L {\frac{{{P_{PT[i]}}}}{{{\sigma ^2}}}{{\left| {{h_{pd[i]}}} \right| }^2}} \end{aligned}$$

(37)

is the sum of ERVs, $U = \frac{{{P_R}}}{{{\sigma ^2}}}{\left| {{h_{rd}}} \right| ^2}$ is ERV. (36) and (31) have similar in form. Thus, we use the same way to obtain (23).

Finally, $\mathfrak {N}$ form in (33) is expressed as

$$\begin{aligned} \mathfrak {N} = \Pr \left( {\frac{{\frac{{{P_{REH}}}}{{{\sigma ^2}}}{{\left| {{h_{rd}}} \right| }^2}}}{{\sum \nolimits _{i = 1}^L {\frac{{{P_T}}}{{{\sigma ^2}}}{{\left| {{h_{pd[i]}}} \right| }^2}} + 1}} \leqslant {\gamma _s}} \right) \end{aligned}$$

(38)

Let us denote $T = \frac{{2\alpha \delta }}{{\left( {1 - \alpha } \right) }}{\left| {{h_{rd}}} \right| ^2}$, $Z = \left( \frac{{{{\text {P}} _{S\min }}}}{{{\sigma ^2}}}{{\left| {{h_{sr}}} \right| }^2}\right. \left. + \sum \nolimits _{i = 1}^L {\frac{{{P_{PT[i]}}}}{{{\sigma ^2}}}{{\left| {{h_{pr[i]}}} \right| }^2} + \,} 1 \right)$,${P_{REH}}$ in (18), V in (37) can be substituted, we obtain

$$\begin{aligned} \mathfrak {N} = \Pr \left( {\frac{{TZ}}{{V + 1}} \leqslant {\gamma _s}} \right) \end{aligned}$$

(39)

Applying Rohatgi’s well-known results in [25] and formula Eq. 3.471.9 in [24], we have PDF of ($U=TZ$)

$$\begin{aligned} {f_U}(u)= & {} \left[ {\prod \limits _{i = 1}^{L + 1} {\frac{1}{{{\lambda _{pr[i]}}}}} } \right] \frac{1}{{{\lambda _{rdn}}}}\sum \limits _{j = 1}^{L + 1} {\frac{{2{e^{ + \;\frac{1}{{{\lambda _{pr[j]}}}}}}.{K_0}\left( {2\sqrt{\frac{u}{{{\lambda _{rdn}}.{\lambda _{pr[j]}}}}} } \right) }}{{\prod \nolimits _{k = 1,k \ne j}^{L + 1} {\left( {\frac{1}{{{\lambda _{pr[k]}}}} - \frac{1}{{{\lambda _{pr}}_{[j]}}}} \right) } }}}\nonumber \\ \end{aligned}$$

(40)

where ${\lambda _{^{rdn}}} = P{L_{rd}} \times \frac{{2\alpha \delta }}{{\left( {1 - \alpha } \right) }}$ and pdf(V) is similarly proved in Lemma 1. We can write $\mathfrak {N}$ as:

$$\begin{aligned} \mathfrak {N}= & {} \left[ {\prod \limits _{i = 1}^{L + 1} {\frac{1}{{{\lambda _{pr[i]}}}}} } \right] \left[ {\prod \limits _{i = 1}^L {\frac{1}{{{\lambda _{pd[i]}}}}} } \right] \frac{1}{{{\lambda _{rdn}}}}\nonumber \\&\times \sum \limits _{m = 1}^{L + 1} {\frac{{{e^{ + \;\frac{1}{{{\lambda _{pr[m]}}}}}}}}{{\prod \nolimits _{k = 1,k \ne m}^{L + 1} {\left( {\frac{1}{{{\lambda _{pr[k]}}}} - \frac{1}{{{\lambda _{pr}}_{[m]}}}} \right) } }}} \nonumber \\&\times \sum \limits _{n = 1}^L {\frac{{\overbrace{\int _0^\infty {\overbrace{\int _0^{{\gamma _s}v} {2{K_0}\left( {2\sqrt{\frac{u}{{{\lambda _{rdn}}{\lambda _{pr[m]}}}}} } \right) } }^{{I_1}}} \,{e^{ - \;\frac{{v - 1}}{{{\lambda _{pd[n]}}}}}}}^{{I_2}}.dudv}}{{\prod \nolimits _{k = 1,k \ne n}^L {\left( {\frac{1}{{{\lambda _{pd[k]}}}} - \frac{1}{{{\lambda _{pd[n]}}}}} \right) } }}}\nonumber \\ \end{aligned}$$

(41)

in which ${I_1}$ is calculated when using Eq. 5.522 in [24] where $p = -1$ and ${K_{ - v}} = {K_v}$.

Next, ${I_2}$ is solved when conducted from Eq. 6.643.6 in [24], where $m = 1$ and $\alpha = \frac{{{\lambda _{rdn}}{\lambda _{pr[m]}}}}{{{\gamma _s}{\lambda _{pd[n]}}}}$. After solving (22), (23), (24), Eq. (21) is clearly derived.