Spectrum-aware outage minimizing cooperative routing in cognitive radio sensor networks

Abstract

This paper investigates the optimal path selection problem for end-to-end (e2e) outage probability minimization in clustered cognitive radio sensor networks. In order to improve outage performance of the optimal path, under a high node density regime, we consider feasibility of virtual multiple-input single-output (v-MISO) links in addition to conventional single-input single-output (SISO) links in the path. Since sensor nodes in such networks are allowed to access the spectrum of the primary network only in an opportunistic manner, the path selection problem is studied under the constraints of probabilistic interference to PU receivers and only single use of any PU channel along the path. The above problem is formulated as a joint hop-constrained routing, spectrum assignment and transmit power control problem. A convex optimization framework is used to find a closed form expression for the optimal transmit power of each transmitting node along the optimal route. Extension of the analytical result facilitates design of a novel routing algorithm, called spectrum aware-minimum outage intelligent cooperative routing (SA-MOICR) algorithm, which not only selects the minimum outage path for a given routing session, but also determines the number of nodes and the unique PU channel to be used for transmission in each hop along the path. Simulation results are found to corroborate our analytical results and quantify the significant improvement of the SA-MOICR scheme over only SISO or only v-MISO based routing solutions in terms of the achievable e2e outage probability.

Appendices

Appendix 1: Proof of probability of interference in (8)

Assuming $K_{1} = P_{p}^{k}$, $K_{2} = P_{i}^{k}$, $K_{3} = N_{0}$, $X = \vert g_{k,k}^{k} \vert ^{2}$ with rate parameter $\lambda _{1}$, $Y = \sum \nolimits _{l=0}^{n_{i}-1} \vert g_{i,k}^{k}(l) \vert ^{2}$. The probability of interference following (7) can be expressed as,

$$\begin{aligned} Pr\left( \frac{K_{1}X}{K_{2}Y+K_{3}}<\gamma _{th}^{PU}\right) =1-Pr\left( Y\le \frac{K_{1}X}{K_{2}\gamma _{th}^{PU}}-\frac{K_{3}}{K_{2}}\right) \end{aligned}$$

(32)

Y is a chi-square distributed with $2n_{i}$ degrees of freedom, and the density function is given by,

$$\begin{aligned}&f\left( y \right) = \frac{y^{n_{i}-1} e^{-y}}{(n_{i}-1)!}, \quad y\ge 0 \end{aligned}$$

(33)

$$\begin{aligned}&Pr\left( Y\le \frac{K_{1}X}{K_{2}\gamma _{th}^{PU}}-\frac{K_{3}}{K_{2}}\right) \nonumber \\&\quad =\int \limits _{\frac{K_{3}\gamma _{th}^{PU}}{K_{1}}}^{\infty } \lambda _{1}e^{-\lambda _{1}x} \int \limits _{0}^{\frac{K_{1}x}{K_{2}\gamma _{th}^{PU}}-\frac{K_{3}}{K_{2}}} \frac{y^{n_{i}-1} e^{-y}}{(n_{i}-1)!} dy\,dx \end{aligned}$$

(34)

$$\begin{aligned}&=\int \limits _{\frac{K_{3}\gamma _{th}^{PU}}{K_{1}}}^{\infty }\lambda _{1}e^{-\lambda _{1}x}\,dx-\int \limits _{\frac{K_{3}\gamma _{th}^{PU}}{K_{1}}}^{\infty } \sum \limits _{l=0}^{n_{i}-1} \frac{\lambda _{1}}{l!} \left( \frac{K_{1}x}{K_{2}\gamma _{th}^{PU}}-\frac{K_{3}}{K_{2}}\right) ^{l}\nonumber \\&\quad \times e^{-\left\{ \left( \lambda _{1}+\frac{K_{1}}{K_{2}\gamma _{th}^{PU}}\right) x-\frac{K_{3}}{K_{2}} \right\} }\, dx \end{aligned}$$

(35)

Let us assume, $\left( \frac{K_{1}x}{K_{2}\gamma _{th}^{PU}}-\frac{K_{3}}{K_{2}}\right) =\phi$, then $dx = \frac{K_{2}\gamma _{th}^{PU}}{K_{1}} d\phi$. Now if $x = \frac{K_{3}\gamma _{th}^{PU}}{K_{1}}$, then $\phi = 0$.

Therefore, from (35)

$$\begin{aligned}&=e^{-\lambda _{1}\frac{K_{3}\gamma _{th}^{PU}}{K_{1}}}-\int \limits _{0}^{\infty } \sum \limits _{l=0}^{n_{i}-1} \frac{\lambda _{1}\phi ^{l}K_{2}\gamma _{th}^{PU}}{l!K_{1}} \nonumber \\&\quad \times e^{-\left\{ \phi \left( 1+\frac{\lambda _{1}K_{2}\gamma _{th}^{PU}}{K_{1}}\right) + \frac{\lambda _{1}K_{3}\gamma _{th}^{PU}}{K_{1}}\right\} }d\phi \end{aligned}$$

(36)

$$\begin{aligned}&=e^{-\frac{\lambda _{1}K_{3}\gamma _{th}^{PU}}{K_{1}}}- \frac{\lambda _{1}K_{2}\gamma _{th}^{PU}}{K_{1}} \nonumber \\&\quad \times e^{-\frac{\lambda _{1}K_{3}\gamma _{th}^{PU}}{K_{1}}} \sum \limits _{l=0}^{n_{i}-1} \left\{ 1+\frac{\lambda _{1}K_{2}\gamma _{th}^{PU}}{K_{1}} \right\} ^{-\left( l+1\right) } \end{aligned}$$

(37)

$$\begin{aligned}&=e^{-\frac{\lambda _{1}K_{3}\gamma _{th}^{PU}}{K_{1}}}\left[ 1-\frac{\frac{\lambda _{1}K_{2}\gamma _{th}^{PU}}{K_{1}}}{1+\frac{\lambda _{1}K_{2}\gamma _{th}^{PU}}{K_{1}}} \left\{ \frac{1-\left( 1+\frac{\lambda _{1}K_{2}\gamma _{th}^{PU}}{K_{1}}\right) ^{-n_{i}}}{1-\left( 1+\frac{\lambda _{1}K_{2}\gamma _{th}^{PU}}{K_{1}}\right) ^{-1}} \right\} \right] \end{aligned}$$

(38)

Appendix 2: Proof of e2e SU outage probability (15)

Using (13), (14) is represented as,

$$\begin{aligned}&\prod _{i=1}^{L}a_{i}^{k} {\mathcal {P}}\left( {\mathcal {H}}_{0}\right) \left( 1-{\mathcal {P}}_{f}^{i,k}\right) \times \left\{ 1-\frac{1}{n_{i}!}\left[ \frac{\gamma _{th}^{SU} N_{0}d_{i,i+1}^{\alpha }}{P_{i}^{k}}\right] ^{n_{i}}\right\} \nonumber \\&\quad =1- {\mathcal {P}}_{out}^{e2e}\left( CH_{1},n_{i},\omega _{L},P_{i}^{k}\right) \end{aligned}$$

(39)

Taking $\log$ in both sides,

$$\begin{aligned} \sum \limits _{i=1}^{L} \log \left[ a_{i}^{k} {\mathcal {P}}\left( {\mathcal {H}}_{0}\right) \left( 1-{\mathcal {P}}_{f}^{i,k}\right) \times \left\{ 1-\frac{1}{n_{i}!}\left[ \frac{\gamma _{th}^{SU} N_{0}d_{i,i+1}^{\alpha }}{P_{i}^{k}}\right] ^{n_{i}}\right\} \right] \nonumber \\ =\log \left[ 1- {\mathcal {P}}_{out}^{e2e}\left( CH_{1},n_{i},\omega _{L},P_{i}^{k}\right) \right] \end{aligned}$$

(40)

As we assumed in the paper $\frac{P_{i}^{k}d_{i,j}^{-\alpha }}{N_{0}}>> \gamma _{th}^{SU}$. Therefore, $\log \left( 1-\frac{\gamma _{th}^{SU} N_{0}d_{i,i+1}^{\alpha }}{P_{i}^{k}}\right) \approx -\frac{\gamma _{th}^{SU} N_{0}d_{i,i+1}^{\alpha }}{P_{i}^{k}}$ and $\frac{1}{n_{i}!} < 1$ for any real number.

$$\begin{aligned}&\sum \limits _{i=1}^{L} \log \left\{ a_{i}^{k} {\mathcal {P}} \left( {\mathcal {H}}_{0}\right) \left( 1-{\mathcal {P}}_{f}^{i,k}\right) \right\} - \sum \limits _{i=1}^{L} \frac{1}{n_{i}!}\left\{ \frac{\gamma _{th}^{SU} N_{0}d_{i,i+1}^{\alpha }}{P_{i}^{k}}\right\} ^{n_{i}} \nonumber \\&\quad =\log \left[ 1- {\mathcal {P}}_{out}^{e2e}\left( CH_{1},n_{i},\omega _{L},P_{i}^{k}\right) \right] \end{aligned}$$

(41)

$$\begin{aligned}&\mathcal {P}_{out}^{e2e}\left( CH_{1},n_{i},\omega _{L},P_{i}^{k}\right) =1-exp \left[ \sum \limits _{i=1}^{L}\log \left\{ a_{i}^{k} {\mathcal {P}}\left( {\mathcal {H}}_{0}\right) \left( 1-{\mathcal {P}}_{f}^{i,k}\right) \right\} \right. \nonumber \\&\quad \left. -\sum \limits _{i=1}^{L}\frac{1}{n_{i}!}\left\{ \frac{\gamma _{th}^{SU} N_{0}d_{i,i+1}^{\alpha }}{P_{i}^{k}}\right\} ^{n_{i}}\right] \end{aligned}$$

(42)