Round robin, distributed and centralized relay selection for cognitive radio networks in the presence of Nakagami fading channels

Abstract

In this paper, we investigate the performance of different relay selection techniques for Cognitive Radio Networks (CRN). The network contains a primary transmitter $P_T$, a primary receiver ($P_R$), a secondary source and two secondary destinations. In the first transmission phase, the secondary source transmits a signal to the two secondary destinations only when it generate interference to $P_R$ less than $\beta$. In the secondary phase, one destination acts as relay and help the other destination. The generated interference by the chosen relay should be less than a prefixed threshold $\beta$. If the two destinations generate a lot of interference, there is no cooperation. If a single destination generate interference less than $\beta$, this node acts as relay. If the two destinations generate low interference, we use centralized (CRS), distributed (DRS) or round robin relay selection (RRRS). Centralized Relay Selection (CRS) chooses the relay having the highest end-to-end Signal to Noise Ratio (SNR). In DRS, a relay is chosen if its SNR is greater than threshold T. The value of T is optimized to have the best performance. RRRS activates both nodes with the same probability i.e. 0.5 without using the value of SNR. Most of previous studies dealt with best relay selection for CRN. Our main contribution is to suggest DRS for CRN and to optimize threshold T. We also derive the Bit Error Probability (BEP) of DRS, CRS and RRRS.

Appendices

Appendix A

When there is cooperation, the SNR at $D_i$ is the sum of SNRs direct and relayed link

$$\begin{aligned} \Gamma _{coop}= \Gamma _{S,D_i}+\Gamma _{S,D_j,D_i} \end{aligned}$$

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where $\Gamma _{S,D_{j},D_i}$ is the SNR between S, $D_{j}$ and $D_i$ [5]

$$\begin{aligned} \Gamma _{S,D_j,D_i}=\frac{\Gamma _{S,D_j}\Gamma _{D_j,D_i}}{1+\Gamma _{S,D_j}+\Gamma _{D_j,D_i}}, \end{aligned}$$

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The MGF of the SNR when there is cooperation is equal to

$$\begin{aligned}&M_{\Gamma _{coop}}(s)=E(e^{-s\Gamma _{coop}})=E(e^{-s(\Gamma _{S,D_i}+\Gamma _{S,D_j,D_i})})\nonumber \\&\quad =E(e^{-s\Gamma _{S,D_i}})E(e^{-s+\Gamma _{S,D_j,D_i}})=M_{\Gamma _{SD_{i}}}(s) M_{\Gamma _{SD_{j}D_{i}}}(s), \nonumber \\ \end{aligned}$$

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In order to obtain simple equations, the following lower and upper bounds will be used

$$\begin{aligned}&\Gamma _{S,D_{j},D_i}^{low}=\frac{1}{2}min\{\Gamma _{S,D_{j}},\Gamma _{D_{j},D_i}\}< \Gamma _{S,D_{j},D_i} < \Gamma _{S,D_{j},D_i}^{up}\nonumber \\&\quad =min\{\Gamma _{S,D_{j}},\Gamma _{D_{j},D_i}\} \end{aligned}$$

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We have

$$\begin{aligned}&\Gamma _{coop}^{low}=\Gamma _{S,D_i}+\Gamma _{S,D_{j},D_i}^{low}<\Gamma _{coop} < \Gamma _{coop}^{up}\nonumber \\&\quad =\Gamma _{S,D_i}+\Gamma _{S,D_{j},D_i}^{up} \end{aligned}$$

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Lower bound on BEP In the following, we derive a lower bound of the Symbol Error Probability (SEP) using the above upper bound on SNR.

The MGF of $\Gamma _{coop}^{up}$ can be written as

$$\begin{aligned} \text {M}_{\Gamma _{coop}^{up}}(s)=\text {M}_{\Gamma _{S,D_i}}(s) \text {M}_{\Gamma _{S,D_{j},D_i}^{up}}(s) \end{aligned}$$

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where

$$\begin{aligned} \text {M}_{\Gamma _{S,D_i}}(s)=\left( 1+s\frac{\overline{\Gamma }_{S,D_i}}{m_{S,D_i} }\right) ^{-m_{S,D_i}} \end{aligned}$$

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The CDF is equal to

$$\begin{aligned} P_{\Gamma _{S,D_{j},D_i}^{up}}(\gamma )=P(min\{\Gamma _{S,D_{j}},\Gamma _{D_{j},D_i}\}\le \gamma ) \end{aligned}$$

We have

$$\begin{aligned}&P_{\Gamma _{S,D_{j},D_i}^{up}}(\gamma ) =1-P\left( \Gamma _{S,D_{j}}>\gamma \right) P\left( \Gamma _{D_{j},D_i}>\gamma \right) \nonumber \\&\quad =1-G^{up}\left( m_{S,D_{j}},\frac{m_{S,D_{j}}}{\overline{\Gamma } _{S,D_{j}}}\gamma \right) \nonumber \\&\quad G^{up}\left( m_{D_{j},D_i}, \frac{m_{D_{j},D_i}}{ \overline{\Gamma }_{D_{j},D_i}}\gamma \right) \end{aligned}$$

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where

$$\begin{aligned} G^{up}(x,y)= \frac{1}{\Gamma (x)} \int _{y}^{\infty }e^{-t}t^{x-1}dt \end{aligned}$$

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The PDF is obtained by a simple derivative of above equation

$$\begin{aligned}&p_{\Gamma _{S,D_{j},D_i}^{up}}(\gamma )=\left[ \frac{1}{\Gamma (m_{SD_j})}\left( \frac{m_{S,D_{j}}}{\overline{\Gamma }_{S,D_{j}}}\right) ^{m_{S,D_{j}}} \gamma ^{m_{S,D_{j}}-1} e^{\frac{-m_{S,D_{j}}}{\overline{\Gamma }_{S,D_{j}}} \gamma }\right. \nonumber \\&\times \left. G^{up}\left( {m_{D_{j},D_i},\frac{m_{D_{j},D_i}}{ \overline{\Gamma }_{D_{j},D_i}} \gamma }\right) +\frac{1}{\Gamma (m_{D_jD_i})}\left( \frac{m_{D_{j},D_i}}{ \overline{\Gamma }_{R_{k},D}}\right) ^{m_{D_{j},D_i}}\gamma ^{m_{D_{j},D_i}}\nonumber \right. \\&\left. \quad -1e^{\frac{-m_{D_{j},D_i}}{\overline{\Gamma }_{D_{j},D_i}}\gamma } G^{up}\left( m_{S,D_{j}},\frac{m_{S,D_{j}}}{ \overline{\Gamma }_{S,D_{j}}}\gamma \right) \right] \end{aligned}$$

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A Laplace Transform (LT) of the PDF gives the MGF (Moment Generating Function) [30]

$$\begin{aligned}&M_{\Gamma _{S,D_{j},D_i}^{up}}\!\!(s)=\left( \!\!\frac{m_{S,D_{j}}}{ \overline{\Gamma }_{S,D_{j}}}\!\!\right) ^{m_{S,D_{j}}} \!\!\!\!\left( \!\! \frac{m_{D_{j},D_i}}{\overline{\Gamma }_{D_{j},D_i}}\!\!\right) ^{m_{D_{j},D_i}} \!\!\!\frac{\left( m_{S,D_{j}}+m_{D_{j},D_i}-1\right) !}{ \left( m_{S,D_{j}}-1\right) !\left( m_{D_{j},D_i}-1\right) !}\nonumber \\&\quad \left[ \frac{1}{m_{S,D_{j}}}\text {} _{2}F_{1}\!\!\left( \!\!1,m_{S,D_{j}}+m_{D_{j},D_i};m_{S,D_{j}}+1;\frac{\frac{ m_{S,D_{j}}}{\overline{\Gamma }_{S,D_{j}}}+s}{\frac{m_{S,D_{j}}}{\overline{ \Gamma }_{S,D_{j}}}+\frac{m_{D_{j},D_i}}{\overline{\Gamma }_{D_{j},D_i}}+s} \!\!\right) +\right. \nonumber \\&\quad \left. \frac{1}{m_{D_{j},D_i}}\text {} _{2}F_{1}\left( 1,m_{S,D_{j}}+m_{D_{j},D_i};m_{D_{j},D_i}+1;\frac{\frac{ m_{R_{j},D_i}}{\overline{\Gamma }_{D_{j},D_i}}+s}{\frac{m_{S,D_{j}}}{\overline{ \Gamma }_{S,D_{j}}}+\frac{m_{D_{j},D_i}}{\overline{\Gamma }_{D_{j},D_i}}+s}\right) \right] \nonumber \\&\quad \times \frac{1}{\left( \frac{m_{S,D_{j}}}{\overline{\Gamma }_{S,D_{j}}}+ \frac{m_{D_{j},D_i}}{\overline{\Gamma }_{D_{j},D_i}} +s\right) ^{m_{S,D_{j}}+m_{D_{j},D_i}}} \end{aligned}$$

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where $_{2}F_{1}(.,.,.)$ is Gauss’ hypergeomeric function defined in [25].

Upper bound on BEP

We explain how we can easily plot an upper bound of the SEP using the above lower bound on SNR.

Let $Y=\Gamma _{S,D_{j},D_i}^{low}$ and $X=\Gamma _{S,D_{j},D_i}^{up}$. We have Y=X/2. Therefore, the CDF, PDF and MGF of Y can be deduced from the following simple equations $P_Y(u)=P_X(2u)$, CDF $p_Y(u)=2p_X(2u)$, PDF $M_Y(s)=E(e^{-sY})=M_X(s/2)$, MGF

Therefore, the MGF of the lower bound of SNR is given by

$$\begin{aligned} \text {M}_{\Gamma _{coop}^{low}}(s)= & {} \text {M}_{\Gamma _{S,D_i}}(s) \text {M}_{\Gamma _{S,D_{j},D_i}^{low}}(s)\nonumber \\ {}= & {} \text {M}_{\Gamma _{S,D_i}}(s) \text {M}_{\Gamma _{S,D_{j},D_i}^{up}}(s/2) \end{aligned}$$

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Appendix B

The derivative of function A(T) is given by

$$\begin{aligned} \frac{\partial A(T)}{\partial T}= & {} -0.5p_{\Gamma _{SD_{j}}}(T)\int _{0}^{T}p_{\Gamma _{coop}}(x)erfc(\sqrt{x })dx\nonumber \\&+\,0.5[1-P_{\Gamma _{SD_{j}}}(T)]p_{\Gamma _{coop}}(T)erfc(\sqrt{T }) \nonumber \\&+\,0.5p_{\Gamma _{SD_{j}}}(T)\int _{0}^{T}p_{\Gamma _{SD_{i}}}(x)erfc(\sqrt{x})dx\nonumber \\&+0.5P_{\Gamma _{SD_{j}}}(T)p_{\Gamma _{SD_{i}}}(T)erfc(\sqrt{T})\nonumber \\&-\,0.5p_{\Gamma _{SD_{j}}}(T)\int _{T}^{+\infty }p_{\Gamma _{SD_i}}(x)erfc(\sqrt{x })dx\nonumber \\&-\,0.5[1-P_{\Gamma _{SD_{j}}}(T)]p_{\Gamma _{SD_{i}}}(T)erfc(\sqrt{T}) \nonumber \\&+\,0.5p_{\Gamma _{SD_{j}}}(T)\int _{T}^{+\infty }p_{\Gamma _{max}}(x)erfc(\sqrt{x})dx\nonumber \\&-\,0.5P_{\Gamma _{SD_{j}}}(T)p_{\Gamma _{max}}(T)erfc(\sqrt{T}) \end{aligned}$$

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