On the outage performance of decode-and-forward based relay ordering in cognitive wireless sensor networks

Abstract

In this paper, the issue of secondary network access in cognitive wireless sensor networks is investigated. An efficient scheme (in terms of both power and spectrum) named decode-and-forward based multi-relay scheme with relay ordering (DF-MRRO) is proposed for secondary network access. More important, this scheme is used to enhance the performance of secondary network. In an effort to assess the performance, the received signal-to interference-plus-noise ratio of both the primary and secondary links are derived, from which new exact closed-form expressions for the outage probability of primary and secondary networks are derived. Moreover, the diversity order is calculated for both the primary and secondary network. It is also shown that full diversity order can be achieved if the condition of power requirement is satisfied. Finally, simulation results obtained for the outage performance of the proposed scheme in primary and secondary networks prove the effectiveness of the proposed scheme. It also demonstrates that a maximum of 7 dB SNR gain can be achieved in the proposed DF-MRRO scheme.

Appendices

Appendix 1

At high SNR, Eq. (28) can be approximated as

$$P\left( {C_{j} } \right) = 1$$

(37)

$$P\left( {D_{j} } \right) = \left\{ {\begin{array}{*{20}l} {\varsigma_{1,C} \left( {\overline{\gamma }} \right)^{ - (N - n)} ,} \hfill & {m \le y + 1} \hfill \\ {\varsigma_{2,C} \left( {\overline{\gamma }} \right)^{ - (N - n)(m - x)} ,} \hfill & {m > y + 1} \hfill \\ \end{array} } \right.$$

(38)

$$P_{PR}^{out} = \varsigma_{3,C} \left( {\overline{\gamma }} \right)^{{ - \left( {n - y + 1} \right)}}$$

(39)

where

$$\begin{aligned} \varsigma_{1,C} & = \mathop \prod \limits_{b = 1}^{N - n} \lambda_{{PT_{O} ,ST_{qb} }} \gamma_{k} \\ \varsigma_{2,C} & = \mathop \prod \limits_{b = 1}^{N - n} \lambda_{{PT_{O} ,ST_{qb} }} \gamma_{k} \mathop \prod \limits_{g = y + 1}^{m - 1} \frac{{\lambda_{{PT_{O} ,ST_{qb} }} \gamma_{k} }}{{\alpha_{{P_{g} }} - \left( {1 - \alpha_{{P_{g} }} } \right)\gamma_{k} }} \\ \varsigma_{3,C} & = \lambda_{{PT_{O} ,PR}} \gamma_{k} \mathop \prod \limits_{g = y + 1}^{N - n} \frac{{\lambda_{{PT_{O} ,PR}} \gamma_{k} }}{{\alpha_{{P_{g} }} - \left( {1 - \alpha_{{P_{g} }} } \right)\gamma_{k} }} \\ \end{aligned}$$

The outage probability of primary link for each possible set of C can be approximated as

$$P\left( C \right)P\left( D \right)P_{out}^{{PR_{o} }} = \left\{ {\begin{array}{*{20}l} {\varsigma_{1,C} \varsigma_{3,C} \left( {\overline{\gamma }} \right)^{{ - \left( {N - y + 1} \right)}} ,} \hfill & {m \le y + 1} \hfill \\ {\varsigma_{1,C} \varsigma_{3,C} \left( {\overline{\gamma }} \right)^{{ - \left( {N - n} \right)\left( {m - y} \right)\left( {n - y + 1} \right)}} ,} \hfill & {m > y + 1} \hfill \\ \end{array} } \right.$$

If all secondary transmitters can decode successfully i.e.$C = \left\{ {ST_{1} ,ST_{2} , \ldots ,ST_{N} } \right\}$ and $D = \left\{ \emptyset \right\}$, the outage probability can be written as

$$P\left( C \right)P\left( D \right)P_{out}^{{PR_{o} }} = \left[ {1 - exp\left( { - \lambda_{{PT_{O} ,PR}} R} \right)\mathop \prod \limits_{j = 1}^{N} - exp\left( { - \lambda_{{ST_{j} ,PR}} R} \right)} \right]$$

At high SNR, the above equation can be approximated as

$$P\left( C \right)P\left( D \right)P_{out}^{{PR_{o} }} = \varsigma_{4,C} \left( {\overline{\gamma }} \right)^{{ - \left( {N - x + 1} \right)}}$$

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where

$$\varsigma_{4,C} = \lambda_{{PT_{O} ,PR}} \gamma_{k} \mathop \prod \limits_{j = x + 1}^{N} \lambda_{{ST_{j} ,PR}} \gamma_{k}$$

Moreover, we have

$$\left( {N - n} \right)\left( {m - y} \right) + n - y + 1 \ge N - x + 1$$

Hence, outage probability at high SNR can be given as

$$P_{primary}^{out} = \mathop \sum \limits_{D} P\left( {C_{j} } \right)P\left( {D_{j} } \right)P_{out}^{PR} = \varsigma \left( {\overline{\gamma }} \right)^{ - (N - x + 1)}$$

(41)

The diversity order is calculated as follows: Diversity Order

$$= \mathop {\lim }\limits_{{\overline{\gamma } \to \infty }} \frac{{ - logP_{primary}^{out} }}{{log\left( {\overline{\gamma }} \right)}} = \mathop {{\text{lim}}}\limits_{{\overline{\gamma } \to \infty }} \frac{{ - log\left( {\varsigma \left( {\overline{\gamma }} \right)^{{ - \left( {N - x + 1} \right)}} } \right)}}{{log\left( {\overline{\gamma }} \right)}} = N - x + 1.$$

Appendix 2

At high SNR, Eqs. (28) and (30) can be approximated as

$$P\left( {C_{j} } \right) = 1$$

$$P\left( {D_{j} } \right) = \left\{ {\begin{array}{*{20}l} {\varsigma_{{1,C_{j} }} \left( {\overline{\gamma }} \right)^{{ - \left( {j - n - 1} \right)}} ,} \hfill & {m \le y + 1} \hfill \\ {\varsigma_{{2,C_{j} }} \left( {\overline{\gamma }} \right)^{{ - \left( {j - n - 1} \right)\left( {m - y} \right)}} ,} \hfill & {m > y + 1} \hfill \\ \end{array} } \right.$$

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where

$$\begin{aligned} \varsigma_{{1,C_{j} }} & = \mathop \prod \limits_{b = 1}^{j - n - 1} \lambda_{{PT_{O} ,ST_{qb} }} \gamma_{k} \\ \varsigma_{{2,C_{j} }} & = \mathop \prod \limits_{b = 1}^{j - n - 1} \lambda_{{PT_{O} ,ST_{qb} }} \gamma_{k} \mathop \prod \limits_{g = y + 1}^{n} \frac{{\lambda_{{ST_{pg} ,ST_{qb} }} \gamma_{k} }}{{\alpha_{{P_{g} }} - \left( {1 - \alpha_{{P_{g} }} } \right)\gamma_{k} }} \\ \end{aligned}$$

The outage probability for each possible set of $C_{j}$ is approximated as

$$P_{out}^{{SR_{j} }} = \mathop \sum \limits_{C} P\left( {C_{j} } \right)P\left( {D_{j} } \right)\left[ {1 - \left( {1 - P_{out}^{{ST_{j} }} } \right)\left( {1 - P_{{out,x_{s} }}^{{SR_{j} }} } \right)\left( {1 - P_{{out,P_{{ST_{1} }} }}^{{SR_{j} }} } \right)} \right]$$

$$P_{out}^{{SR_{j} }} = \left\{ {\begin{array}{*{20}l} {\varsigma_{{1,C_{j} }} \varsigma_{{3,C_{j} }} \left( {\overline{\gamma }} \right)^{{ - \left( {j - n} \right)}} ,} \hfill & {m \le y + 1} \hfill \\ {\varsigma_{{2,C_{j} }} \varsigma_{{4,C_{j} }} \left( {\overline{\gamma }} \right)^{{ - \left( {j - n - 1} \right)\left( {m - y} \right) - 1}} ,} \hfill & {m > y + 1} \hfill \\ \end{array} } \right.$$

where

$$\varsigma_{{3,C_{j} }} = \left( {\lambda_{{PT_{O} ,ST_{j } }} + \lambda_{{PT_{O} ,SR_{j } }} + \frac{{\lambda_{{ST_{j } ,SR_{j } }} }}{{1 - \alpha_{j} }}} \right)\gamma_{k}$$

$$\varsigma_{{4,C_{j} }} = \frac{{\lambda_{{ST_{j } ,SR_{j } }} }}{{1 - \alpha_{j} }}\gamma_{k}$$

If all the secondary transmitters $ST_{1} ,ST_{2} ,...,ST_{j - 1}$ can decode the primary signal $x_{s}$ successfully, the above equation can be approximated as

$$P_{out}^{{SR_{j} }} = \left\{ {\begin{array}{*{20}l} {\varsigma_{{1,C_{j} }} \varsigma_{{3,C_{j} }} \left( {\overline{\gamma }} \right)^{ - 1} ,} \hfill & {m \le y + 1} \hfill \\ {\varsigma_{{2,C_{j} }} \varsigma_{{4,C_{j} }} \left( {\overline{\gamma }} \right)^{ - 1} ,} \hfill & {m > y + 1} \hfill \\ \end{array} } \right.$$

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It is easy to see that $\left( {j - n - 1} \right)\left( {m - y} \right) + 1 \ge j - n \ge 1$. Hence the outage probability of the secondary network can be approximated at high SNR as

$$P_{out}^{{SR_{j} }} = \varsigma_{j} \left( {\overline{\gamma }} \right)^{ - 1}$$

(44)

where $\varsigma_{j}$ is a constant. The outage probability of secondary system can be calculated as

$$P_{Secondary}^{out} = \mathop \prod \limits_{j = 1}^{N} P_{out}^{{SR_{j} }} \approx \left( {\overline{\gamma }} \right)^{ - N} \mathop \prod \limits_{j = 1}^{N} \varsigma_{j}$$

(45)

From the definition of diversity order, the diversity gain of ST_j-SR_j link and secondary system are calculated as 1 and N respectively.