Optimal harvesting and sensing durations for cognitive radio networks

Abstract

In this paper, we optimize both harvesting and sensing durations for cognitive radio networks (CRNs). There are a primary source and destination $P_S$ and $P_D$. In the secondary network, there are a secondary source and destination $S_S$ and $S_D$. There are three time slots in secondary network. In the first one, secondary source $S_S$ harvests energy from radio frequency (RF) signal received from another node A. Node A can be a base station or any other node transmitting RF signal. In the second time slot, secondary source senses the channel using the energy detector to detect primary source activity. When $P_S$ is idle, the secondary source transmits data to secondary destination $S_D$ in the last time slot. We optimize harvesting and sensing durations to maximize the throughput. Our results are valid for quadrature amplitude modulation (QAM), phase shift keying (PSK) and amplitude shift keying (ASK) in the presence of Rayleigh or Nakagami fading channels.

Appendices

Appendix A

Let $X_{1}$ and $X_{2}$ be two exponential r.v. with respective mean 1/$\lambda _{1}$ and 1/$\lambda _{2}.$

The CDF of the product of two exponential rv $X=X_{1}X_{2}$ is expressed as

$$\begin{aligned} P_{X}(x)=P(X_{1}X_{2}\le x)=\int _{0}^{+\infty }P(X_{1}\le \frac{x}{y} )\lambda _{2}e^{-\lambda _{2}y}\mathrm{d}y.\nonumber \\ \end{aligned}$$

(30)

We deduce

$$\begin{aligned} P_{X}(x)= & {} \int _{0}^{+\infty }\left[ 1-e^{-\lambda _{1}\frac{x}{y}}\right] \lambda _{2}e^{-\lambda _{2}y}\mathrm{d}y \nonumber \\= & {} 1-\int _{0}^{+\infty }e^{-\lambda _{1}\frac{x}{y}}\lambda _{2}e^{-\lambda _{2}y}\mathrm{d}y. \end{aligned}$$

(31)

We have [26]

$$\begin{aligned} \int _{0}^{+\infty }e^{-\frac{c}{y}}e^{-\frac{y}{d}}\mathrm{d}y=2\sqrt{\frac{c}{d}} K_{1}\left( 2\sqrt{\frac{c}{d}}\right) . \end{aligned}$$

(32)

We use (31) and (32) with $c=\lambda _{1}x$ and $d=\frac{1}{ \lambda _{2}}$; we obtain the CDF of X

$$\begin{aligned} P_{X}(x)=1-2\sqrt{\lambda _{1}\lambda _{2}x}K_{1}\left( 2\sqrt{\lambda _{1}\lambda _{2}x}\right) . \end{aligned}$$

(33)

The PDF of X is given by

$$\begin{aligned} p_{X}(x)= & {} -\frac{\sqrt{\lambda _{1}\lambda _{2}}}{\sqrt{x}}K_{1}\left( 2\sqrt{ \lambda _{1}\lambda _{2}x}\right) \nonumber \\&-\,2\sqrt{\lambda _{1}\lambda _{2}x}K_{1}^{^{\prime }}\left( 2\sqrt{\lambda _{1}\lambda _{2}x}\right) \frac{\sqrt{\lambda _{1}\lambda _{2}}}{ \sqrt{x}}. \end{aligned}$$

(34)

Using [26],

$$\begin{aligned} K_{1}^{^{\prime }}(z)=-K_{0}(z)-\frac{K_{1}(z)}{z}. \end{aligned}$$

(35)

We obtain the PDF of X

$$\begin{aligned} p_{X}(x)=2\lambda _{1}\lambda _{2}K_{0}\left( 2\sqrt{\lambda _{1}\lambda _{2}x}\right) . \end{aligned}$$

(36)

Appendix B

The derivative of throughput is expressed as

$$\begin{aligned} \frac{\partial Thr}{\partial \alpha }= & {} -(1-\beta )\log _2(M)[1-\mathrm{PEP}]P_{idle}(1-P_f)\nonumber \\&-\,(1-\alpha )(1-\beta )\log _2(M)\frac{\partial Thr}{\partial \alpha }P_{idle}(1-P_f)\nonumber \\&-\,(1-\alpha )(1-\beta )\log _2(M)\frac{\partial P_f}{\partial \alpha }P_{idle}(1-\mathrm{PEP})\nonumber \\ \end{aligned}$$

(37)

$$\begin{aligned} \frac{\partial Thr}{\partial \beta }= & {} -(1-\alpha )\log _2(M)[1-\mathrm{PEP}]P_{idle}(1-P_f) \nonumber \\&-\,(1-\alpha )(1-\beta )\log _2(M)\frac{\partial Thr}{\partial \beta }P_{idle}(1-P_f) \nonumber \\&-\,(1-\alpha )(1-\beta )\log _2(M)\frac{\partial P_f}{\partial \beta }P_{idle}(1-\mathrm{PEP})\nonumber \\ \end{aligned}$$

(38)

where

$$\begin{aligned}&\frac{\partial P_f}{\partial \alpha }=\frac{-\beta k_0[\varGamma '(N,\zeta /2)\varGamma (N)-\varGamma (N,\zeta /2)\varGamma '(N)]}{\varGamma (N)^2} \end{aligned}$$

(39)

$$\begin{aligned}&\frac{\partial P_f}{\partial \beta }=\frac{(1-\alpha ) k_0[\varGamma '(N,\zeta /2)\varGamma (N)-\varGamma (N,\zeta /2)\varGamma '(N)]}{\varGamma (N)^2}\nonumber \\ \end{aligned}$$

(40)

$$\begin{aligned}&\frac{\partial \mathrm{PEP}}{\partial \alpha }=\frac{-1}{(m_{AS_S}-1)!(m_{S_SS_D}-1)!} \nonumber \\&\quad \times \, \left[ \frac{(1-\beta )N_0T_0m_{AS_S}m_{S_SS_D}\lambda _{AS_S}\lambda _{S_SS_D}}{\mu E_A}\right] ^{0.5m_{AS_S}+0.5m_{S_SS_D}} \nonumber \\&\quad \times \, \alpha ^{-0.5m_{AS_S}-0.5m_{S_SS_D}-1}(1-\alpha )^{0.5m_{AS_S}+0.5m_{S_SS_D}-1} \nonumber \\&\quad \times \, K_{m_{S_SS_D}-m_{AS_S}}\nonumber \\&\quad \left( 2\sqrt{\frac{(1-\alpha ) (1-\beta )N_0T_0m_{AS_S}m_{S_SS_D}\lambda _{AS_S}\lambda _{S_SS_D}}{\mu \alpha E_A}}\right) \end{aligned}$$

(41)

$$\begin{aligned}&\frac{\partial \mathrm{PEP}}{\partial \beta }=\frac{-1}{(m_{AS_S}-1)!(m_{S_SS_D}-1)!} \nonumber \\&\quad \times \, \left[ \frac{(1-\alpha )N_0T_0m_{AS_S}m_{S_SS_D}\lambda _{AS_S}\lambda _{S_SS_D}}{\alpha \mu E_A}\right] ^{0.5m_{AS_S}+0.5m_{S_SS_D}} \nonumber \\&\quad \times \, (1-\beta )^{0.5m_{AS_S}+0.5m_{S_SS_D}-1}\nonumber \\&\quad \times \, K_{m_{S_SS_D}-m_{AS_S}}\nonumber \\&\quad \left( 2\sqrt{\frac{(1-\alpha )(1-\beta )N_0T_0m_{AS_S}m_{S_SS_D} \lambda _{AS_S}\lambda _{S_SS_D}}{\mu \alpha E_A}}\right) \end{aligned}$$

(42)

$$\begin{aligned}&\varGamma '(N)=\int _0^{+\infty }ln(x)x^{N-1}e^{-x}\mathrm{d}x, \end{aligned}$$

(43)

and

$$\begin{aligned} \varGamma '(N,a)=\int _a^{+\infty }ln(x)x^{N-1}e^{-x}\mathrm{d}x, \end{aligned}$$

(44)