Secondary Spectrum Access in Cognitive Radio Networks Using Rateless Codes over Rayleigh Fading Channels

Abstract

In this paper, we propose secondary relaying schemes in cognitive spectrum leasing. In the proposed protocols, a primary transmitter uses rateless code to transmit its data to a primary receiver. In the secondary network, $M$ secondary transmitters are ready to help the primary transmitter forward the data to a primary receiver so that they can find opportunities to transmit their data. For performance evaluation, we derive the average outage probability, the average number of encoded packets transmitted by the primary transmitter, the average number of remaining time slots for secondary network and the average capacity of the secondary network over Rayleigh fading channels. Various Monte-Carlo simulations are presented to verify the derivations.

Appendix

Derivation of (12)

First, we set the random variable $X$ as $X=\sum _{j=1,j\ne i}^M {\gamma _{4ij}}$. Next, we use the moment generating function (MGF) to find the CDF of $X$. In addition, the MGF of $\gamma _{4ij}$ is $\lambda _{4ij}/\left( {\lambda _{4ij}+s}\right)$. Hence, due to the independence of $\gamma _{4ij}$, the MGF of $X$ is given by

$$\begin{aligned} \hbox {MGF}_X \left( s \right) = \prod _{j=1,\,j\ne i}^M {\frac{\lambda _{ST_j,SR_i}}{\lambda _{ST_j,SR_i}+s}} \end{aligned}$$

(46)

Using partial fraction expansion and under the condition that the $\gamma _{4ij}$ are different from each other, (46) can be decomposed into

$$\begin{aligned} \hbox {MGF}_X (s)=\sum _{j=1,\,j\ne i}^M {\frac{\alpha _j }{\lambda _{ST_j,SR_i}+s}} \end{aligned}$$

(47)

where $\alpha _j =\lambda _{ST_j ,SR_i } \prod _{k=1,k\ne j}^M {\frac{\lambda _{ST_k ,SR_i } }{\lambda _{ST_k ,SR_i } -\lambda _{ST_j ,SR_i } }}$.

Applying the inverse Laplace transform for (47), we can get the PDF of $X$ as follows:

$$\begin{aligned} f_X \left( x \right) =\sum _{j=1,j\ne i}^M {\alpha _j \exp \left( {-\lambda _{ST_j ,SR_i } x} \right) } \end{aligned}$$

(48)

Now, considering the random variable $Y=1+X$, it is easy to obtain the PDF of $Y$ as

$$\begin{aligned} f_Y \left( y \right) =\sum _{j=1,\,j\ne i}^M {\alpha _j \exp \left( {-\lambda _{ST_j ,SR_i } \left( {y-1} \right) } \right) } \end{aligned}$$

(49)

Since $\gamma _{SR_i } =\gamma _{4i} /Y$, the CDF of $\gamma _{SR_i}$ is given as

$$\begin{aligned} F_{\gamma _{SR_i}}(z)=\Pr \left[ {\gamma _{SR_i } <z} \right] =\Pr \left[ {\gamma _{4i} <Yz} \right] =\int _1^{+\infty } {f_Y (y)dy\int _0^{yz} {f_{\gamma _{4i} } \left( t \right) dt}} \end{aligned}$$

(50)

Substituting (49) into (50) and after some manipulation, we get

$$\begin{aligned} F_{\gamma _{SR_i } } \left( z \right) =1-\exp \left( {-\lambda _{ST_i ,SR_i } z} \right) \sum _{j=1,j\ne i}^M {\frac{\alpha _j }{\lambda _{ST_j ,SR_i } +\lambda _{ST_i ,SR_i } z}} \end{aligned}$$

(51)

Differentiating (51) with respect to $z$, we obtain (12).