Performance of a Relay Antenna Selection Based Dynamic Spectrum Sharing Protocol with Opportunistic Selection of Users

Abstract

We consider a cellular scenario in which a base-station primary transmitter communicates to its primary receivers (PRs) using a set of multiple antenna relay stations (RS). One of several RS that is selected is assumed to employ antenna selection. The primary network allows one of its users to access the secondary network simultaneously in the same two phases (by using a selected RS) through dynamic spectrum sharing principles. Such a framework allows the cellular system to increase its frequency utilization. We derive expressions for the outage and ergodic rate performance of such spectrum sharing communication links. Systems performing spectrum sharing are assumed to have a peak power constraint and use peak interference control (to limit interference to the PR). For a fixed primary outage, it is demonstrated that increasing the number of users improves the performance at medium values of SNR, whereas increasing the number of RSs improves the performance at high SNRs.

Appendices

Appendix 1

In reference to $Y={\dfrac{I_{p}}{|g^{II}_{j^{\dagger }l^{\dagger }}|^{2}}}$ in equation (21). CDF of Y can be written as:

$$\begin{aligned} \text{Pr}\left\{ \dfrac{I_{p}}{|g^{II}_{j^{\dagger }l^{\dagger }}|^{2}}<y \right\}&= {} \displaystyle \int \limits _{0}^{\infty }\int \limits _{-\sigma ^{2}}^{y\,x} p_{|g^{II}_{j^{\dagger }l^{\dagger }}|^{2}}(x)\,p_{I_{p}}(z)dx\,dz\nonumber \\&= {} \displaystyle \int \limits _{0}^{\propto }\dfrac{N_{r}\,j_{s}}{\lambda _{g}} \text{exp}\left( -\dfrac{N_{r}\,j_{s}\,x}{\lambda _{g}}\right) \left( 1-\exp \left( \dfrac{-\gamma _{thp}(y\,x+\sigma ^2)}{P_{rp}\lambda _{h}}\right) \right) ^{j_{p}\,N_{r}} dx \end{aligned}$$

(50)

Applying the binomial expansion of the term in the integral, and integrating yields the CDF. Finally, derivative of the CDF yields the PDF expression in (22). The CDF $F_{X}(x)$ of X in Eq. (23) can be derived similarly.

Appendix 2

On using the PDF of X and Y in (24), it can be seen that ${\mathbb{I}}$ is given by:

$$\begin{aligned} {\mathbb{I}}&= {} -B\int \limits _{0}^{P_{s}^{max}}\dfrac{e^{-\frac{\gamma _{ths}\,m\,\sigma ^{2}}{\lambda _f\,y}}}{\left( \alpha + B\,y\right) ^{2}\left( \dfrac{d}{y}+\dfrac{1}{\lambda _e}\right) }\,dy\nonumber \\&= {} -\dfrac{B}{\alpha ^{2}d}\displaystyle \int \limits _{\frac{1}{P_{s}^{max}}}^{\infty }e^{-\frac{\gamma _{ths}m\sigma ^{2}}{\lambda _f}t} \left( \dfrac{R_{1}}{\left( t+\frac{B}{\alpha }\right) } + \dfrac{R_{2}}{\left( t+\frac{B}{\alpha }\right) ^{2}} + \dfrac{R_{3}}{\left( t+\frac{1}{d\lambda _e}\right) }\right) dt\nonumber \\&\text{where} \nonumber \\ R_{1}&= {} -\dfrac{1}{\left( \dfrac{1}{d\,\lambda _e} - \dfrac{B}{\alpha } \right) ^{2}} ,\;R_{2}=\dfrac{1}{\left( \dfrac{1}{d\,\lambda _e} - \dfrac{B}{\alpha } \right) }, R_{3}=\dfrac{1}{\left( \dfrac{1}{d\,\lambda _e} - \dfrac{B}{\alpha } \right) ^{2}} \end{aligned}$$

(51)

Using Eq. 3.351-4 of [2]), we obtain (27).

Appendix 3

The assumption taken in asymptotic analysis (34) and ergodic capacity analysis (38) follows $P_{rp}\gg\sigma ^2$. Based on this assumption the CDFs of $Y=\frac{I_p}{|g^{II}_{{j^\dagger }{l^\dagger }}|^2} = \frac{P_{rp}|h^{II}_{j^{*}l^{*}}|^{2}}{\gamma _{thp}|g^{II}_{{j^\dagger }{l^\dagger }}|^2}$ and $X=\frac{ |f_{{k^\dagger }{l^\dagger }}^{II}|^2}{P_{rp} |e_{j^*l^*}^{II}|^2}$ are as follows:

$$\begin{aligned} F_X(x) &= {} \sum \limits _{m=0}^{K}(-1)^m{K \atopwithdelims ()m}\dfrac{1}{(1 + C_1x)}\nonumber \\ F_Y(y)&= {} \sum \limits _{l=0}^{j_p\,N_r}(-1)^l{j_p\,N_r \atopwithdelims ()l}\dfrac{1}{(1 + C_2y)} \end{aligned}$$

(52)

where $C_1=\frac{m\,P_{rp}\,\lambda _e}{\lambda _f}$, and $C_2=\frac{\lambda _g\,l\,\gamma _{thp}}{\lambda _h\,N_r\,P_{rp}\,j_s}$.

Using the above expressions to evaluate $p_{os}^{pno}$ in (17), we get:

$$\begin{aligned}&\lim \underset{P_s^{max}\longrightarrow \infty }{p_{os}^{pno}(\gamma _{ths})} =\displaystyle \int \limits _{0}^{\infty }\int \limits _{0}^{\frac{\gamma _{ths}}{y}}p_Y(y)p_X(x)\,dx\,dy\nonumber \\&\quad = \sum \limits _{l=1}^{j_{p}\,N_r}(-1)^l{j_{p}\,N_r \atopwithdelims ()l}\sum \limits _{m=0}^{K}(-1)^m{K \atopwithdelims ()m}{\mathbb{K}} \end{aligned}$$

(53)

where

$$\begin{aligned} {\mathbb{K}}&= {} -C_2\int \limits _{0}^{\infty }\dfrac{dx}{(1+C_2x)^2(1+C_1\frac{\gamma _{ths}}{x})}\nonumber \\&= {} -\dfrac{C_1C_2\gamma _{ths}}{(1-C_1C_2\gamma _{ths})^2}\ln (C_1C_2\gamma _{ths}) - \dfrac{1}{(1-C_1C_2\gamma _{ths})}\nonumber \\&= {} -\dfrac{u}{(1-u)^2}\ln u - \dfrac{1}{(1-u)} \end{aligned}$$

(54)

where $u=C\gamma _{ths}; \, C=C_1C_2$. Since $\lim \underset{P_s^{max}\longrightarrow \infty }{p_{os}^{po}}=0$, we have:

$$\begin{aligned}&\lim \limits _{P_{s}^{max}\rightarrow \infty }{p_{os}(\gamma _{ths})}\nonumber \\&\quad =\sum \limits _{j_{p}=1}^{J_p}\text{Pr}\left\{ |\mathcal{J}_{dp}|=j_p\right\} \sum \limits _{j_{s}=1}^{J_s}\text{Pr}\left\{ |\mathcal{J}_{ds}|=j_s\right\} \lim \limits _{P_s^{max}\longrightarrow \infty }{{p_{os}^{pno}}(\gamma _{ths})} \end{aligned}$$

(55)

Appendix 4

At high SNRs when $\frac{I_p}{|g^{II}_{j^\dagger l^\dagger }|^2}<P_s^{max}$ the expression in (48) is derived as:

$$\begin{aligned} \displaystyle \int \limits _{1}^{\infty }\text{p}_{12}(\gamma )\log _2(\gamma )d\gamma = \sum \limits _{l=1}^{j_pNr}{j_pNr\atopwithdelims ()l}(-1)^{l} \sum \limits _{m=1}^{K}{K\atopwithdelims ()m}(-1)^{m}{\mathbb{N}} \end{aligned}$$

(56)

where

$$\begin{aligned} {\mathbb{N}}&= {} \displaystyle \int \limits _{1}^{\infty }\log _2(1+\gamma )\dfrac{d\left\{ -C_2\displaystyle \int \limits _{0}^{P_s^{max}} \dfrac{dx}{(1 + C_2 x)^2 (1 + C_1 \frac{\gamma }{x})} \right\} }{d\gamma }d\gamma \nonumber \\&= {} \dfrac{\log _2\left\{ \frac{C_1}{C_1 + P_s^{max}}\right\} }{(1 + C_2 P_s^{max})} + \dfrac{\log _2\left\{ \frac{C_1 + C_1C_2P_s^{max}}{C_1+P_s^{max}}\right\} }{(-1 + C_1C_2)} \end{aligned}$$

(57)

Note that $C_1$ and $C_2$ are as defined in Sect. 6.