Throughput and harvesting time trade-off in a energy harvesting cognitive radio network

Abstract

The performance analysis of a relay based cognitive radio (CR) network (non-cooperative and cooperative) with a combined (radio frequency (RF) and non-RF) energy harvesting under a collision constraint of a primary user (PU) is studied in this paper. Cooperative environment includes a network with multiple PUs, and CRs. The relay and CR at the destination employes multiple antenna for receive diversity. Both the CR source and relay node are considered as energy harvesting nodes. The CR source harvests the energy from RF signal of PU and non-RF signal over the detection cycle. In contrast, the relay node harvests the energy from non-RF signal and CR signal. Network performance is studied for instantaneous transmission and delay constraint transmission. Situations of relay node not transmitting due to insufficient harvested energy is captured, where harvested energy falls below a threshold causing energy outage. A probabilistic approach is adopted to redesign the expression of network throughput. Two protocols, namely, amplify-forward and decode-forward relaying are discussed. Novel analytical expressions are developed for outage probability and throughput. The impacts of several parameters of the network like collision probability, transmitting signal-to-noise ratio (SNR), fractional harvesting time parameter and fractional transmission time parameter on the system throughput are investigated and compared. An optimal value of fractional harvesting time parameter is estimated for maximization of throughput under a collision constraint of PU. Throughput increases as the sensing duration, non-RF energy mean, transmission SNR, and collision probability decrease. There is an optimal value of fractional harvesting time which maximizes the throughput. The combination of AND fusion rule and MRC data combining performs better compare to the other schemes.

Appendices

Appendix A

   (Proof of $Pr\left( E_{TH, r}\ge E_{th}\right)$):

$$\begin{aligned} Pr_{th}= & {} Pr\left( E_{TH, r}\ge E_{th}\right) \nonumber \\= & {} 1-Pr\left( E_{TH, r}< E_{th}\right) . \end{aligned}$$

(45)

Replacing the value of $E_{TH, r}$ in 45 from 17 and after some algebra 45 reduces to

$$\begin{aligned} Pr_{th}= & {} 1-\left[ |h_{sr}|^2<\frac{\left( E_{th}-\eta _r E_{non-RF, r}\right) d_1^p}{\eta _r E_s \alpha T_r}\right] \nonumber \\ {}= & {} 1-F_{|h_{sr}|^2}\left[ \frac{\left( E_{th}-\eta _r E_{non-RF, r}\right) d_1^p}{\eta _r E_s \alpha T_r} \right] . \end{aligned}$$

(46)

Since, $|h_{sr}|^2$ is exponentially distributed, $F_{|h_{sr}|^2}\left[ x\right] = 1- exp\left( -\frac{x}{\lambda _{sr}}\right)$. Hence, the expression obtained in , can be reduced to

$$\begin{aligned} Pr_{th}= & {} exp\left[ - \frac{\left( E_{th}-\eta _r E_{non-RF, r}\right) d_1^p}{\lambda _{sr} \eta _r E_s \alpha T_r}\right] . \end{aligned}$$

Appendix B

   (Proof of outage probability for AF relay):

Let $\gamma _1=\frac{P_{CR}|h_{sr}|^2}{d_1^p N_0}=a g_{sr}$ where $g_{sr}=|h_{sr}|^2$. Similarly, $\gamma _2=\frac{P_{relay}|h_{rd}|^2}{(P_{CR} |{\hat{h}}_{sr}|^2/d_1^p+N_0)d_2^p N_0}=b_{AF} g_{rd}$ where $g_{rd}=|h_{rd}|^2$. The PDF of $\gamma _1$ and $\gamma _2$ are $f_{\gamma _1}(x)=\frac{1}{a \lambda _{sr}}exp\left( -\frac{x}{a \lambda _{sr}}\right)$ and $f_{\gamma _2}(y)=\frac{1}{b_{AF} \lambda _{rd}}exp\left( -\frac{y}{b_{AF} \lambda _{rd}}\right)$ respectively.

Thus the outage probability for AF relaying can be expressed as,

$$\begin{aligned} Pr_{out}^{AF}= & {} Pr\left\{ \frac{\gamma _1 \gamma _2}{\gamma _1+\gamma _2+1}\le \gamma _{th}\right\} \nonumber \\= & {} 1-\int _0^\infty Pr\left\{ \gamma _1>\frac{\gamma _{th}\left( 1+ \gamma _{th}+t\right) }{t}\right\} \nonumber \\{} & {} \quad \quad \quad \quad \quad f_{\gamma _{2}}\left( \gamma _{th}+t \right) dt. \end{aligned}$$

(47)

Replacing $f_{\gamma _{2}}\left( \gamma _{th}+t \right) =\frac{1}{b_{AF} \lambda _{rd}}exp\left( -\frac{\gamma _{th}+t }{b_{AF} \lambda _{rd}}\right)$ in and after some algebra, $P_{out}$ can be obtained as,

$$\begin{aligned} Pr_{out}^{AF}= & {} 1-\frac{1}{b_{AF} \lambda _{rd}}exp\left[ -\gamma _{th}\left( \frac{1}{a\lambda _{sr}}+\frac{1}{b_{AF}\lambda _{rd}} \right) \right] \nonumber \\ {}{} & {} \quad \quad \quad \quad \quad \int _0^\infty exp\left( \frac{m_2}{4 t}-n_2 t \right) dt, \end{aligned}$$

(48)

where $m_2=\frac{4\gamma _{th}\left( \gamma _{th}+1 \right) }{a \lambda _{sr}}$ and $n_2=\frac{1}{b_{AF} \lambda _{sr}}$. Using [27, (3.324.1)] 48 reduces to

$$\begin{aligned} Pr_{out}^{AF}= & {} 1-\frac{1}{b_{AF} \lambda _{rd}}exp\left[ -\left( \frac{\gamma _{th}}{a\lambda _{sr}}+\frac{\gamma _{th}}{b_{AF}\lambda _{rd}} \right) \right] \nonumber \\ {}{} & {} \quad \quad \quad \quad \quad \sqrt{\frac{m_2}{n_2}}K_1\left( \sqrt{m_2n_2} \right) . \end{aligned}$$

Appendix C

   (Proof of outage probability for DF relay)

In the case of DF relaying, the PDF of $\gamma _2$ can be written as $f_{\gamma _2}(y)=\frac{1}{b_{DF} \lambda _{rd}}exp\left( -\frac{y}{b_{DF} \lambda _{rd}}\right)$ where $b_{DF}=\frac{P_{relay}}{P_{CR}d_2^p N_0}$. Hence, the outage probability can be described as,

$$\begin{aligned} Pr_{out}^{DF}= & {} P\left\{ min\left( \gamma _1, \gamma _2\right) \le \gamma _{th}\right\} \nonumber \\= & {} 1-P\left\{ min\left( \gamma _1, \gamma _2\right)> \gamma _{th}\right\} \nonumber \\= & {} 1-P\left\{ \gamma _1> \gamma _{th}\right\} P\left\{ \gamma _2 > \gamma _{th}\right\} \nonumber \\= & {} 1-\left[ 1-F_{\gamma _1}\left( \gamma _{th}\right) \right] \left[ 1-F_{\gamma _2}\left( \gamma _{th}\right) \right] . \end{aligned}$$

(49)

Using the CDF of $\gamma _1$ and $\gamma _2$ in 49, the outage probability for DF relaying can be reduced to

$$\begin{aligned} Pr_{out}^{DF}=1-exp\left[ -\gamma _{th}\left( \frac{1}{a}+\frac{1}{b_{DF}}\right) \right] . \end{aligned}$$