Energy Harvesting for MIMO Dual-Hop Cognitive Radio Networks

Abstract

In this paper, we optimize the throughput of cognitive radio networks (CRN) when all secondary nodes harvest energy from radio frequency signal received from primary transmitter ($P_T$). These secondary nodes are equipped with multiple antennas and adapt their power so that the interference at primary receiver $P_R$ is less than interference threshold I. Our results are valid for cooperative CRN with opportunistic amplify and forward, partial and reactive relay selection. We also choose an appropriate value of harvesting duration to enhance the throughput at secondary destination.

Appendices

Appendix A: CDF of SINR Between i-th Transmit Antenna of $R_k$ and j-th Receive Antenna of D

We have

$$\begin{aligned} P_{\Gamma _{R_kD}^{(i,j)}}(x)=P(E_{Adaptive}|h_{R_kD}^{(i,j)}|^{2}\le (N_{0}+E_{P_T}|h_{P_T,D}^{(j)}|^2)x), \end{aligned}$$

(57)

Let $Z=N_{0}+E_{P_{T}}|h_{P_{T},D}^{(j)}|^{2}$, we deduce

$$\begin{aligned} P_{\Gamma _{R_kD}^{(i,j)}}(x)=\int _{N_0}^{+\infty }P(E_{Adaptive}|h_{R_kD}^{(i,j)}|^{2}\le ux)p_Z(u)du, \end{aligned}$$

(58)

For Rayleigh fading channels, $E_{P_{T}}|h_{P_{T},D}^{(j)}|^{2}$ follows an exponential distribution with mean $E_{P_T}\beta _{P_T,D}^{(j)}$ where $\beta _{P_T,D}^{(j)}=E(|h_{P_{T},D}^{(j)}|^{2})$. Therefore, we have

$$\begin{aligned} P_{\Gamma _{R_kD}^{(i,j)}}(x)= & {} \frac{1}{\beta _{P_T,D}^{(j)}E_{P_T}}\int _{N_0}^{+\infty }P(E_{Adaptive}|h_{R_kD}^{(i,j)}|^{2} \nonumber \\\le & {} ux)e^{-\frac{(u-N_0)}{\beta _{P_T,D}^{(j)}E_{P_T}}}du. \end{aligned}$$

(59)

Using (22), we deduce

$$\begin{aligned}&P(E_{Adaptive}|h_{R_kD}^{(i,j)}|^{2}\nonumber \\&\quad \le ux)=1-\frac{2}{\rho _{R_kD}^{2(i,j)}}\sum _{q=0}^{n^r_{R_k}-1}\frac{(ux)^q}{q!\rho _{P_TR_k}^{2q}\mu ^q}\nonumber \\&\qquad \left( \frac{ux\rho _{R_kD}^{2(i,j)}}{\mu \rho ^2_{P_TR_k}}\right) ^{\frac{1-q}{2}}K_{1-q}\left( 2\sqrt{\frac{xu}{\rho _{R_kD}^{2(i,j)}\mu \rho ^2_{P_TR_k}}}\right) \nonumber \\&\qquad +\frac{2}{\rho _{R_kD}^{2(i,j)}}\sum _{q=0}^{n^r_{R_k}-1}\frac{(ux)^q}{q!\rho _{P_TR_k}^{2q}\mu ^q}\nonumber \\&\qquad \left[ \frac{u^2x^2}{\mu \rho _{P_TR_k}^2 \left( \frac{ux}{\rho _{R_kD}^{2(i,j)}}+\frac{I}{\rho _{R_kP_R}^2}\right) }\right] ^{\frac{1-q}{2}}K_{1-q} \nonumber \\&\qquad \left( 2\sqrt{\frac{1}{\mu \rho _{P_TR_k}^2}\left( \frac{ux}{\rho _{R_kD}^{2(i,j)}}+\frac{I}{\rho _{R_kP_R}^2}\right) }\right). \end{aligned}$$

(60)

We use Gradshteyn and Ryzhik (1994) (6.631.3)

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

(61)

where $W_{e,d}(x)$ is the Whittaker function Gradshteyn and Ryzhik (1994).

Using (58), (59) and (60), we have

$$\begin{aligned} P_{\Gamma _{R_kD}^{(i,j)}}(x)= & {} 1-\frac{1}{\rho _{R_kD}^{2(i,j)}\beta _{P_T,D}^{(j)}E_{P_T}}e^{\frac{N_0}{\beta _{P_T,D}^{(j)}E_{P_T}}}\nonumber \\&\sum _{q=0}^{n^r_{R_k}-1}\frac{x^q}{\rho _{P_TR_k}^{2q}\mu ^q}\left( \frac{x\rho _{R_kD}^{2(i,j)}}{\mu \rho ^2_{P_TR_k}}\right) ^{\frac{1-q}{2}}\nonumber \\&\frac{1}{2\sqrt{\frac{x}{\rho _{R_kD}^{2(i,j)}\mu \rho _{P_TR_k}^2}}}e^{\frac{x\beta _{P_T,D}^{(j)}E_{P_T}}{\rho _{R_kD}^{2(i,j)}\mu \rho _{P_TR_k}}}\nonumber \\&\times (\beta _{P_T,D}^{(j)}E_{P_T})^{1+q/2}W_{-1-q/2,0.5-q/2}\nonumber \\&\left( \frac{x\beta _{P_T,D}^{(j)}E_{P_T}}{\rho _{R_kD}^{2(i,j)}\mu \rho _{P_TR_k}}\right) \nonumber \\&+\int _{0}^{N_0}\frac{2}{\rho _{R_kD}^{2(i,j)}\beta _{P_T,D}^{(j)}E_{P_T}}\sum _{q=0}^{n^r_{R_k}-1}\frac{(ux)^q}{q!\rho _{P_TR_k}^{2q}\mu ^q}\nonumber \\&\left( \frac{ux\rho _{R_kD}^{2(i,j)}}{\mu \rho ^2_{P_TR_k}}\right) ^{\frac{1-q}{2}}K_{1-q}\nonumber \\&\left( 2\sqrt{\frac{xu}{\rho _{R_kD,}^{2(i,j)}\mu \rho ^2_{P_TR_k}}}\right) e^{-\frac{(u-N_0)}{\beta _{P_T,D}^{(j)}E_{P_T}}}du\nonumber \\&+\int _{N_0}^{+\infty }\frac{2}{\rho _{R_kD}^{2(i,j)}\beta _{P_T,D}^{(j)}E_{P_T}}e^{\frac{-(u-N_0)}{\beta _{P_T}^{(j)}E_{P_T}}}\nonumber \\&\sum _{q=0}^{n^r_{R_k}-1}\frac{(ux)^q}{q!\rho _{P_TR_k}^{2q}\mu ^q}\left[ \frac{u^2x^2}{\mu \rho _{P_TR_k}^2\left( \frac{ux}{\rho _{R_kD}^{2(i,j)}}+\frac{I}{\rho _{R_kP_R}^2}\right) }\right] ^{\frac{1-q}{2}}\nonumber \\&\times K_{1-q}\left( 2\sqrt{\frac{1}{\mu \rho _{P_TR_k}^2}\left( \frac{ux}{\rho _{R_kD}^{2(i,j)}}+\frac{I}{\rho _{R_kP_R}^2}\right) }\right) du, \end{aligned}$$

(62)

where two integrals are evaluated numerically using MATLAB.

Appendix B: Probability to Select Relay $R_k$ using PRS

For PRS, the probability to select relay $R_{k}$ is expressed as follows

$$\begin{aligned} a_{k}=P(\Gamma _{SR_{k}}>\max _{j\ne k}\Gamma _{SR_{j}}). \end{aligned}$$

(63)

Let $X=\max _{j\ne k}\Gamma _{SR_{j}},$ we can write

$$\begin{aligned}&a_{k}=\int _{0}^{+\infty }P(\Gamma _{SR_{k}}>x)p_{X}(x)dx \nonumber \\&\quad =\int _{0}^{+\infty }p_{X}(x)[1-P_{\Gamma _{SR_{k}}}(x)]dx. \end{aligned}$$

(64)

If $\Gamma _{SR_{p}}$ are independent, we have

$$\begin{aligned} P_{X}(x)=\prod _{p=1,p\ne k}^{K}P_{\Gamma _{SR_{p}}}(x). \end{aligned}$$

(65)

The PDF of X is written as

$$\begin{aligned} p_{X}(x)=\sum _{n=1,n\ne k}^{K}p_{\Gamma _{SR_{n}}}(x)\prod _{p=1,p\ne k,p\ne n}^{K}P_{\Gamma _{SR_{p}}}(x). \end{aligned}$$

(66)

Appendix C: Probability to Select Relay $R_k$ using RRS

For RRS, the probability to select relay $R_{k}$ is expressed as follows

$$\begin{aligned} b_{k}=P(\Gamma _{R_{k}D}>\max _{j\ne k}\Gamma _{R_{j}D}). \end{aligned}$$

(67)

Let $Y=\max _{j\ne k}\Gamma _{R_{j}D},$ we can write

$$\begin{aligned}&b_{k}=\int _{0}^{+\infty }P(\Gamma _{R_{k}D}>x)p_{Y}(x)dx \nonumber \\&\quad =\int _{0}^{+\infty }p_{Y}(x)[1-P_{\Gamma _{R_{k}D}}(x)]dx. \end{aligned}$$

(68)

If $\Gamma _{R_{p}D}$ are independent, we have

$$\begin{aligned} P_{Y}(x)=\prod _{p=1,p\ne k}^{K}P_{\Gamma _{R_{p}D}}(x). \end{aligned}$$

(69)

We deduce

$$\begin{aligned} p_{Y}(x)=\sum _{n=1,n\ne k}^{K}p_{\Gamma _{R_{n}D}}(x)\prod _{p=1,p\ne k,p\ne n}^{K}P_{\Gamma _{R_{p}D}}(x). \end{aligned}$$

(70)