Development of coded-cooperation based multi-relay system for cognitive radio using mathematical modeling and its performance analysis

Abstract

Coded cooperation is a cooperative relaying technique which is bandwidth efficient and can provide better connectivity when used with cognitive radio. Earlier cognitive radio is implemented with cooperative schemes like decode and forward, amplify and forward, compress and forward, etc., which gives moderate spectrum efficiency and diversity with reduced fading problem of wireless communication. But these techniques are not bandwidth efficient so need huge bandwidth for transmission. In this paper, we have developed a mathematical model of new coded-cooperation based multi-relay system for cognitive radio and its outage probability is analytically derived for underlay mode. The performance analysis of proposed system is carried out in terms of interference temperature constraint, channel gain, number of relays through outage probability for single and multilink relays at the primary node for Rayleigh channel. The proposed system shows that the cognitive radio with coded cooperation outperforms the already existing techniques in terms of diversity, bandwidth, spectrum utilization efficiency and improve the communication quality. In addition, the theoretical analysis of the outage probability is validated by asymptotic analysis.

Appendix

For the computation of outage probability, $P_{out}^{a}$ is derived below as:

$$\begin{aligned} P_{out}^{a}=\int _{0}^{{2^{R/k}-1}}f_{\gamma _{sd}}(x) \left( \underbrace{\intop _{0}^{\alpha (x)}f_{\gamma _{r_{i}d}}(z)dz}_{I_{1}}\right) ^{L}dx \end{aligned}$$

(20)

where, $\alpha (x)=\frac{2^{R\varphi }}{(1+x)^{\beta }}-1$, $\varphi =\frac{1}{1-k}$ and $\beta =\frac{k}{1-k}$

To proceed further with the derivation, we need to solve the inner integral of Eq. (20). After some simple mathematical manipulation, the integral $(I_{1})$ can be expressed as

$$\begin{aligned} I_{1}=\left( \frac{A_{r_{i}d}\alpha (x)}{\left( \alpha (x)\frac{A_{r_{i}d}}{\mathcal {Q}/\sigma ^{2}} + 1 \right) }\right) ^{L} \end{aligned}$$

(21)

where, $A_{r_{i}d} = \frac{\lambda _{r_{i}d}}{\lambda _{up}}$.

By substituting the value of $(I_{1})$ together with the result of (5) in (20), from $(1+b)^{-L}=\sum _{l=0}^{\infty }\left( \begin{array}{c}L+l-1\\ l\end{array}\right) (-1)^{l}(b)^{l}$ and expanding it using the binomial expressions $(t-1)^{q}=\sum _{r=0}^{q}\left( \begin{array}{c}q\\ r\end{array}\right) (-1)^{q-r}(t)^{r}$, one can rewrite Eq. (20) as follows:

$$\begin{aligned} P_{out}^{a} =&\sum _{l=0}^{\infty } \left( \begin{array}{c}L+l-1\\ l\end{array}\right) (-1)^{l}\left( \frac{A_{r_{i}d}}{\mathcal {Q}/\sigma ^{2}}\right) ^{L+l}\sum _{r=0}^{L+l}\left( \begin{array}{c}L+l\\ r\end{array}\right) 2^{R \varphi r} (-1)^{L+l-r}\nonumber \\&\times \underbrace{\int _{0}^{{2^{R/k}-1}}\frac{dx}{\left( 1+x\right) ^{\beta r}\left( 1+\frac{x A_{sd}}{\mathcal {Q}/\sigma ^{2}}\right) ^{2}}}_{I_{2}} \end{aligned}$$

(22)

where, $A_{sd} = \frac{\lambda _{sd}}{\lambda _{up}}$, $\varphi =\frac{1}{1-k}$ and $\beta =\frac{k}{1-k}$

Finally using [11], eq. (3.211)] in integral $(I_{2})$, Eq. (22) can be further expressed as:

$$\begin{aligned} P_{out}^{a}= & {} \sum _{l=0}^{\infty } \left( \begin{array}{c}L+l-1\\ l\end{array}\right) \left( \frac{A_{r_{i}d}}{\mathcal {Q}/\sigma ^{2}}\right) ^{L+l}\sum _{r=0}^{L+l}\left( \begin{array}{c}L+l\\ r\end{array}\right) 2^{R \varphi r} (-1)^{L+2l-r}\nonumber \\&\times (2^{R/k}-1) F_{1}\left( 1;\beta r, 2; 2 ; -(2^{R/k}-1), -\frac{{(2^{R/k}-1)}A_{sd}}{\mathcal {Q}/\sigma ^{2}}\right) \end{aligned}$$

(23)