Approximate expressions of packet error probability, throughput and delay for cognitive radio networks using fixed and adaptive transmit power

Abstract

In this paper, we derive upper bound of Packet Error Probability (PEP), upper bound of delay and lower bound of throughput in cognitive radio networks. Our analysis is valid when the power of secondary source is fixed or adaptive. The secondary source can adapt its transmitting power so that interference to primary receiver is below a given threshold T. The analysis is carried in the absence and presence of interference from primary transmitter. PEP, throughput and delay are derived in closed form and validated using simulation results.

Appendices

Appendix A

The SINR (16) can be expressed as

$$\begin{aligned} \Gamma _{SD}=\frac{c_{1}Y_{1}}{Y_{2}(c_{2}+c_{3}Y_{3})} \end{aligned}$$

(34)

where $Y_{1}=|h_{SD}|^{2},Y_{2}=|h_{S \ddot{} P_{R}}|^{2},Y_{3}=|h_{P_{T}D}|^{2},c_{1}=T,c_{2}=N_{0},c_{3}=E_{P_{T}}.$

For Rayleigh fading channels, $Y_{1}$,$Y_{2}$ and $Y_{3}$ follow an exponential distribution distributed with mean $\mu _{i}=E(Y_{i})$.

We have

$$\begin{aligned}&P\left( \Gamma _{SD}<x|\frac{T}{|h_{SP_{R}}|^{2}}<E^{\max }\right) \nonumber \\&\quad =P\left( \Gamma _{SD}<x|Y_{2}>\frac{T}{E^{\max }}\right) \nonumber \\&\quad =P\left( c_{1}Y_{1}<xY_{2}(c_{2}+c_{3}Y_{3})|Y_{2}>\frac{T}{E^{\max }} \right) \end{aligned}$$

(35)

We define $Y_{4}=c_{2}+c_{3}Y_{3}$. The CDF of $Y_{4}$ is equal to

$$\begin{aligned} F_{Y_{4}}(w)=F_{Y_{3}}(\frac{w-c_{2}}{c_{3}}) \end{aligned}$$

(36)

We deduce the PDF of $Y_4$

$$\begin{aligned} f_{Y_{4}}(w)=\frac{1}{c_{3}}f_{Y_{3}}(\frac{w-c_{2}}{c_{3}}). \end{aligned}$$

(37)

Equation (35) is expressed as

$$\begin{aligned}&P\left( c_{1}Y_{1}<xY_{2}(c_{2}+c_{3}Y_{3})|Y_{2}>\frac{T}{E^{\max }}\right) \nonumber \\&\quad =\int _{\frac{T}{E^{\max }}}^{+\infty }\int _{c_{2}}^{+\infty }P(c_{1}Y_{1}<xvw)f_{Y_{2}|Y_{2}>\frac{T}{E^{\max }}}(v)f_{Y_{4}}(w)dvdw\nonumber \\&\quad =\int _{c_{2}}^{+\infty }\int _{\frac{T}{E^{\max }}}^{+\infty }e^{\frac{T}{ E_{\max }\mu _{2}}}\left[ 1-e^{-\frac{xvw}{c_{1}\mu _{1}}}\right] \frac{e^{-\frac{v}{\mu _{2}}}}{\mu _{2}}dv\frac{e^{-\frac{(w-c_{2})}{ c_{3}\mu _{3}}}}{c_{3}\mu _{3}}dw\nonumber \\ \end{aligned}$$

(38)

We have

$$\begin{aligned} \int _{\frac{T}{E^{\max }}}^{+\infty }e^{\frac{T}{E_{\max }\mu _{2}}} \left[ 1-e^{-\frac{xvw}{c_{1}\mu _{1}}}\right] \frac{e^{-\frac{v}{ \mu _{2}}}}{\mu _{2}}dv=1-\frac{e^{-\frac{Txw}{E^{\max }c_{1\mu _{1}}}}}{1+\frac{\mu _{2}xw}{\mu _{1}c_{1}}} \nonumber \\ \end{aligned}$$

(39)

Using (38) and (39), we have

$$\begin{aligned}&P\left( \Gamma _{SD}<x|\frac{T}{|h_{SP_{R}}|^{2}}<E^{\max }\right) \nonumber \\&\quad =\int _{c_{2}}^{+\infty }\left[ 1-\frac{e^{-\frac{Txw}{E^{\max }c_{1\mu _{1}}}}}{1+\frac{\mu _{2}xw}{\mu _{1}c_{1}}}\right] \frac{e^{-\frac{ (w-c_{2})}{c_{3}\mu _{3}}}}{c_{3}\mu _{3}}dw\nonumber \\&\quad =1-\frac{e^{\frac{c_{2}}{c_{3}\mu _{3}}}}{c_{3}\mu _{3}} \int _{c_{2}}^{+\infty }\frac{e^{-w\left( \frac{1}{c_{3}\mu _{3}}+\frac{xT }{E^{\max }c_{1}\mu _{1}}\right) }}{1+\frac{\mu _{2}xw}{\mu _{1}c_{1}}}dw \end{aligned}$$

(40)

For

$$\begin{aligned} z=1+\frac{\mu _{2}xw}{\mu _{1}c_{1}} \end{aligned}$$

(41)

We can write

$$\begin{aligned}&P\left( \Gamma _{SD}<x|\frac{T}{|h_{SP_{R}}|^{2}}<E^{\max }\right) \nonumber \\&\quad =1-\frac{ e^{\frac{c_{2}}{c_{3}\mu _{3}}}}{c_{3}\mu _{3}}\frac{\mu _{1}c_{1}}{\mu _{2}x}\nonumber \\&\qquad \times \int _{1+\frac{\mu _{2}xc_{2}}{\mu _{1}c_{1}} }^{+\infty }\frac{e^{-\left( z-1\right) \frac{\mu _{1}c_{1}}{\mu _{2}x}\left( \frac{1}{c_{3}\mu _{3}}+\frac{xT}{E^{\max }c_{1}\mu _{1} }\right) }}{z}dz \nonumber \\&\quad =1-\frac{e^{\frac{c_{2}}{c_{3}\mu _{3}}}}{c_{3}\mu _{3}}\frac{ \mu _{1}c_{1}}{\mu _{2}x}e^{\frac{\mu _{1}c_{1}}{c_{3}\mu _{3}\mu _{2}x}+\frac{T}{E^{\max }\mu _{2}}}\nonumber \\&\qquad \times E_{i}\left( \left( \frac{ \mu _{1}c_{1}}{\mu _{2}x}+c_{2}\right) \left( \frac{1}{c_{3}\mu _{3}}+\frac{xT}{E^{\max }c_{1}\mu _{1}}\right) \right) \end{aligned}$$

(42)

where $E_{i}(x)$ is the exponential integral function

$$\begin{aligned} E_{i}(x)=\int _{x}^{+\infty }\frac{e^{-t}}{t}dt. \end{aligned}$$

(43)

Appendix B

We have

$$\begin{aligned}&P\left( \Gamma _{P_{T}P_{R}}<x,|h_{SP_{R}}|^{2}<\frac{T}{E_{S}}\right) \nonumber \\&\quad =P(E_{P_{T}}|h_{P_{T}P_{R}}|^{2}<x(N_{0}+E_{S}|h_{SP_{R}}|^{2}),|h_{SP_{R}}|^{2}< \frac{T}{E_{S}})\nonumber \\ \end{aligned}$$

(44)

Let $Z=E_{P_{T}}|h_{P_{T}P_{R}}|^{2}$ and $W=N_{0}+E_{S}|h_{SP_{R}}|^{2}|$$|h_{SP_{R}}|^{2}<\frac{T}{E_{S}},$ we deduce

$$\begin{aligned}&P\left( E_{P_{T}}|h_{P_{T}P_{R}}|^{2}<x\left( N_{0}+E_{S}|h_{SP_{R}}|^{2}\right) | |h_{SP_{R}}|^{2}<\frac{T}{E_{S}}\right) \nonumber \\&\quad =\int _{N_{0}}^{N_{0}+T}F_{Z}(xu)f_{W}(u-N_{0})du \end{aligned}$$

(45)

where

$$\begin{aligned} f_{W}(u-N_{0})= & {} \frac{\exp (-\frac{(u-N_{0})}{E_{S}\sigma _{SP_{R}}^{2}})}{ E_S\sigma _{SP_{R}}^{2}\left[ 1-\exp (-\frac{T}{E_{S }\sigma _{SP_{R}}^{2}})\right] } \end{aligned}$$

(46)

$$\begin{aligned} F_{Z}(xu)= & {} 1-\exp (-\frac{xu}{E_{P_{T}}\sigma _{P_{T}P_{R}}^{2}}) \end{aligned}$$

(47)

We finally obtain

$$\begin{aligned}&P(\Gamma _{P_{T}P_{R}}<x,|h_{SP_{R}}|^{2}<\frac{T}{E_{S}})=\left[ 1-\exp (-\frac{T}{E_{S }\sigma _{SP_{R}}^{2}})\right] \nonumber \\&\quad \times \left[ 1-\exp (-\frac{N_{0}x}{E_{P_{T}}\sigma _{P_{T}P_{R}}^{2}})\frac{ E_{P_{T}}\sigma _{P_{T}P_{R}}^{2}}{E_{P_{T}}\sigma _{P_{T}P_{R}}^{2}+xE_{S }\sigma _{SP_{R}}^{2}}\right] \end{aligned}$$

(48)