Performance Analysis of Opportunistic, Reactive and Partial Relay Selection with Adaptive Transmit Power for Cognitive Radio Networks

Abstract

In this paper, we derive the packet error probability of cognitive radio networks. Our analysis is valid when the powers of secondary source and relays are adaptive. The secondary source and relays can adapt their transmitting power so that interference to primary receiver is below a given threshold T. The analysis takes into account interference from primary transmitter. Different relay selection techniques are investigated such as opportunistic amplify and forward (AF) relaying, partial and reactive relay selection. In opportunistic AF relaying, the selected relay offers the highest end-to-end signal to interference plus noise ratio (SINR). Partial relay selection activates the relay with the largest SINR of first hop. Reactive relay selection activates the relay with the largest SINR of second hop.

Appendices

Appendix A

We can write

$$\begin{aligned}&P\left( \frac{T|g_{SD}|^{2}}{|g_{SP_{{R}}}|^{2}N_{0}}<x||g_{SP_{{R}}}|^{2}>\frac{T}{E^{\max }}\right) \nonumber \\&\quad =P\left( |g_{SD}|^{2}<\frac{x|g_{SP_{{R}}}|^{2}N_{0} }{T}||g_{SP_{{R}}}|^{2}>\frac{T}{E^{\max }}\right) \nonumber \\&\quad =e^{\frac{T}{\lambda _{SP_{{R}}}^{2}E^{\max }}}\int _{\frac{T}{E^{\max }} }^{+\infty }P\left( |g_{SD}|^{2}<\frac{xuN_{0}}{T}\right) \frac{e^{-\frac{u}{ \lambda _{SP_{{R}}}^{2}}}}{\lambda _{SP_{{R}}}^{2}}du \nonumber \\&\quad =e^{\frac{T}{\lambda _{SP_{{R}}}^{2}E^{\max }}}\int _{\frac{T}{E^{\max }} }^{+\infty }\left[ 1-e^{-\frac{xuN_{0}}{T\lambda _{SD}^{2}}}\right] \frac{e^{- \frac{u}{\lambda _{SP_{{R}}}^{2}}}}{\lambda _{SP_{{R}}}^{2}}du \nonumber \\&\quad =1-\frac{e^{-\frac{N_{0}x}{\lambda _{SD}^{2}E^{\max }}}}{1+\frac{\lambda _{SP_{{R}}}^{2}xN_{0}}{T\lambda _{SD}^{2}}} \end{aligned}.$$

(45)

Appendix B

When the adaptive power exceeds $P_{max}$, the secondary source transmit power is equal to $P_{max}$. The SINR between S and D is expressed as

$$\Gamma _{S,D}=\frac{E^{max}|g_{SD}|^{2}}{E_{P_{T}}|g_{P_{T}D}|^{2}+N_{0}}$$

(46)

where $E^{max}=T_sP^{max}$, $T_s$ is the symbol period, $E_{P_{T}}$ is the transmitted energy per symbol of primary transmitter $P_{T}$, $g_{P_{T}D}$ is the channel coefficient between $P_T$ and node D. $E_{P_{T}}|g_{P_{T}D}|^{2}$ is the interference at node D from $P_{T}$.

The CDF of the SINR is expressed as

$$F_{\Gamma _{SD}}(\gamma )=P(E^{max}|g_{SD}|^{2}<\gamma (E_{P_{T}}|g_{P_{T}D}|^{2}+N_{0}))$$

(47)

For Rayleigh channels, $Z_1=E^{max}|g_{SD}|^{2}$ follows an exponential distribution with mean $E^{max}\lambda _{SD}^{2}$ where $\lambda _{SD}^{2}=E(|g_{SD}|^{2})$. E(X) is the expectation of X. Also, $Z_2=E_{P_{T}}|g_{P_{T}D}|^{2}$ follows an exponential distribution with mean $E_{P_T}\lambda _{P_{T},D}^{2}$ where $\lambda _{P_{T},D}^{2}=E(|g_{P_{T}D}|^{2})$

Therefore, the CDF can be expressed as

$$\begin{aligned} F_{\Gamma _{SD}}(\gamma )& = P(Z_2<\gamma (N_{0}+Z_2)) \nonumber \\ & = \int _{N_{0}}^{+\infty }F_{Z_1}(\gamma u)f_{Z_2}(u-N_{0})du \end{aligned}$$

(48)

where $F_{Z_1}(u)=P(Z_1<u)$ is the CDF of $Z_1$ and $f_{Z_2}(u)$ is the PDF of $Z_2$.

Since $Z_1$ and $Z_2$ follow an exponential distribution, we have

$$\begin{aligned} F_{\Gamma _{SD}}(\gamma )& = \int _{N_{0}}^{+\infty }\left[ 1-e^{-\frac{ \gamma u}{E^{max}\lambda _{SD}^{2}}}\right] e^{-\frac{(u-N_{0})}{E_{P_T}\lambda _{P_{T},D}^{2}}}\frac{1}{E_{P_T}\lambda _{P_{T}D}^{2}}du \nonumber \\ & = 1-\frac{E^{max}\lambda _{SD}^{2}}{E^{max}\lambda _{SD}^{2}+\gamma E_{P_T}\lambda _{P_{T}D}^{2}} e^{-\frac{N_{0}\gamma }{E^{max}\lambda _{SD}^{2}}} \end{aligned}$$

(49)

Appendix C

The SINR (13) can be written as

$$\Gamma _{SD}=\frac{b_{1}U_{1}}{U_{2}(b_{2}+b_{3}U_{3})}$$

(50)

where $U_{1}=|g_{SD}|^{2},U_{2}=|g_{S \ddot{} P_{R}}|^{2},U_{3}=|g_{P_{T}D}|^{2},e_{1}=T,e_{2}=N_{0},e_{3}=E_{P_{T}}.$

For Rayleigh channels, $U_{1}$,$U_{2}$ and $U_{3}$ are exponentially distributed with mean $\lambda _{i}=E(U_{i})$.

We have to compute

$$\begin{aligned} P\left( \Gamma _{SD}<x|\frac{T}{|g_{SP_{R}}|^{2}}<E^{\max }\right)& = P\left( \Gamma _{SD}<x|U_{2}>\frac{T}{E^{\max }}\right) \nonumber \\ & = P\left( e_{1}U_{1}<xU_{2}(e_{2}+e_{3}U_{3})|U_{2}>\frac{T}{E^{\max }} \right) \end{aligned}$$

(51)

Let $U_{4}=e_{2}+e_{3}U_{3}$. The CDF of $U_{4}$ is equal to

$$F_{U_{4}}(w)=F_{U_{3}}\left( \frac{w-e_{2}}{e_{3}}\right)$$

(52)

We deduce the PDF

$$f_{U_{4}}(w)=\frac{1}{e_{3}}f_{U_{3}}\left( \frac{w-e_{2}}{e_{3}}\right) .$$

(53)

Equation (51) can be expressed as

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

(54)

We have

$$\int _{\frac{T}{E^{\max }}}^{+\infty }e^{\frac{T}{E_{\max }\lambda _{2}}} \left[ 1-e^{-\frac{xvw}{e_{1}\lambda _{1}}}\right] \frac{e^{-\frac{v}{ \lambda _{2}}}}{\lambda _{2}}dv=1-\frac{e^{-\frac{Txw}{E^{\max }e_{1\lambda _{1}}}}}{1+\frac{\lambda _{2}xw}{\lambda _{1}e_{1}}}$$

(55)

Using (54) and (55), we deduce

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

(56)

Let

$$z=1+\frac{\lambda _{2}xw}{\lambda _{1}e_{1}}$$

(57)

We deduce

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

(58)

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

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

(59)