Investigation on outage capacity of spectrum sharing system using CSI and SSI under received power constraints

Abstract

In this paper, we have investigated the outage capacity of secondary user for opportunistic spectrum sharing under the joint peak and average received power constraints for Rayleigh fading environment. Under this communication scenario, on detecting the licensed primary user inactive, the secondary unlicensed users transmit data/information in the licensed frequency band such that no or minimum interference may be experienced by the primary user. The soft sensing information (SSI) and secondary user’s channel state information is used to obtain the closed form expressions for the ergodic and outage capacity using truncated channel inversion with fixed rate technique under the joint peak and average received power constraints. Numerically simulated results are provided to demonstrate the improvement in outage capacity of secondary user under the proposed spectrum sharing scheme. Moreover, the proposed scheme is also compared with other conventional spectrum sharing schemes to illustrate the benefits of SSI and received power constraints on the outage capacity of secondary user.

Appendix

For the optimum power allocation under constraints in Eqs. (4) and (5), we have used Lagrangian optimization technique. Thus, for the maximization problem in Eq. (7), the Lagrangian objective function $L_{{C_{er} }}$ is expressed as:

$$\begin{aligned} L_{{C_{er} }} = & \left[ {P\left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right), \lambda_{1} , \lambda_{2} \left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right),\lambda_{3} \left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right)} \right] = \bar{\alpha } E_{{{\raise0.7ex\hbox{${\gamma_{s} ,\gamma_{sp} ,\xi }$} \!\mathord{\left/ {\vphantom {{\gamma_{s} ,\gamma_{sp} ,\xi } {PU\_off}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${PU\_off}$}}}} \left[ {\log_{2} \left( {1 + \frac{{P\left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right) \gamma_{s} }}{{N_{0} B}}} \right)} \right] \\ & + \;\alpha E_{{{\raise0.7ex\hbox{${\gamma_{s} ,\gamma_{sp} ,\xi }$} \!\mathord{\left/ {\vphantom {{\gamma_{s} ,\gamma_{sp} ,\xi } {PU\_on}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${PU\_on}$}}}} \left[ {\log_{2} \left( {1 + \frac{{P\left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right) \gamma_{s} }}{{N_{0} B}}} \right)} \right] - \lambda_{1} E_{{{\raise0.7ex\hbox{${\gamma_{s} ,\gamma_{sp} ,\xi }$} \!\mathord{\left/ {\vphantom {{\gamma_{s} ,\gamma_{sp} ,\xi } {PU\_on}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${PU\_on}$}}}} \left[ {P\left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right)\gamma_{sp} - P_{Avg} } \right] \\ & + \;\mathop \int \limits_{0}^{\infty } \mathop \int \limits_{0}^{\infty } \mathop \int \limits_{0}^{\infty } \lambda_{2} \left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right)P\left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right)\gamma_{sp} d\gamma_{s} d\gamma_{sp} d\xi - \mathop \int \limits_{0}^{\infty } \mathop \int \limits_{0}^{\infty } \mathop \int \limits_{0}^{\infty } \lambda_{2} \left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right)\left[ {\left\{ {P\left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right)\gamma_{sp} } \right\}_{PU\_on} } \right. \\ & - \left. {P_{Peak} } \right]d\gamma_{s} d\gamma_{sp} d\xi . \\ \end{aligned}$$

(28)

By taking derivative of Eq. (28) with respect to $P\left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right)$ and setting it to zero yields:

$$\left( {\bar{\alpha } f_{off} \left( \xi \right) + \alpha f_{on} \left( \xi \right)} \right)\left( {\frac{{\gamma_{s} }}{{P\left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right)\gamma_{s} + N_{0} B}}} \right) - \lambda_{1} f_{on} \left( \xi \right)\gamma_{sp} + \lambda_{2} \left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right) {-}\lambda_{3} \left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right) \gamma_{sp} = 0.$$

(29)

Since the objective function $L_{{C_{er} }}$ is a concave function of $P\left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right)$, the Karush–Kuhn–Tucker (KKT) conditions are necessary and sufficient to find the optimum solution for Eq. (29). These conditions are:

$$\lambda_{1} E_{{{\raise0.7ex\hbox{${\gamma_{s} ,\gamma_{sp} ,\xi }$} \!\mathord{\left/ {\vphantom {{\gamma_{s} ,\gamma_{sp} ,\xi } {PU\_On}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${PU\_On}$}}}} \left[ {\left( {P\left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right)\gamma_{sp} } \right)_{PU\_On} - P_{Avg} } \right] = 0,$$

(30)

$$\lambda_{2} \left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right) P\left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right) = 0,$$

(31)

$$\lambda_{3} \left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right)\left( {\left( {\left( {P\left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right)\gamma_{sp} } \right)_{PU\_On} - P_{Peak} } \right)} \right) = 0.$$

(32)

Case I Suppose $P^{*} \left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right) = 0$ for some values of $\gamma_{s} ,\gamma_{sp}$ and $\xi$. In this case, Eq. (32) requires $\lambda_{3} \left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right) = 0$ and Eq. (31) implies that $\lambda_{2} \left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right) \ge 0$ which, when substituted into Eq. (29), yield

$$\left( {\bar{\alpha } f_{off} \left( \xi \right) + \alpha f_{on} \left( \xi \right)} \right)\left( {\frac{{\gamma_{s} }}{{N_{0} B}}} \right) - \lambda_{1} f_{on} \left( \xi \right)\gamma_{sp} \le 0,$$

$$\frac{{\gamma_{u} \left( \xi \right) }}{{\lambda_{1} N_{0} B}} \le \frac{{\gamma_{sp} }}{{\gamma_{s} }}.$$

(33)

Case II Suppose $P^{*} \left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right) = \frac{{P_{Peak} }}{{\gamma_{sp} }}$ for some values of $\gamma_{s} ,\gamma_{sp}$ and $\xi$. In this case, Eq. (31) requires $\lambda_{2} \left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right) = 0$ and Eq. (32) implies that $\lambda_{3} \left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right) \ge 0$ which, when substituted into Eq. (29), yield

$$\gamma_{u} \left( \xi \right) \left( {\frac{{\gamma_{s} }}{{P_{Peak} \frac{{\gamma_{s} }}{{\gamma_{sp} }} + N_{0} B}}} \right) - \lambda_{1} \gamma_{sp} \ge 0,$$

$$\frac{{\gamma_{v} \left( \xi \right)}}{{N_{0} B}} \ge \frac{{\gamma_{sp} }}{{\gamma_{s} }}.$$

(34)

Case III$0 \le P^{*} \left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right) = \frac{{P_{Peak} }}{{\gamma_{sp} }}$ for some values of $\gamma_{s} ,\gamma_{sp}$ and $\xi$. It requires $\lambda_{2} \left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right) = \lambda_{3} \left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right) = 0$ which, when substituted into Eq. (29), yield

$$\gamma_{u} \left( \xi \right) \left( {\frac{{\gamma_{s} }}{{P\left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right)\gamma_{s} + N_{0} B }}} \right){-} \lambda_{1} \gamma_{sp} = 0,$$

$$\frac{{\gamma_{u} \left( \xi \right) }}{{\lambda_{1} \gamma_{sp} }} - \frac{{N_{0} B}}{{\gamma_{s} }} = P\left( {\gamma_{s} ,\gamma_{sp} ,\xi } \right).$$

(35)

Finally, according to the results in Eqs. (33), (34) and (35), the power allocation policy that maximizes the ergodic capacity expression in Eq. (6), can be expressed according to Eq. (8), thus concluding the proof.