Fusion Rules Effects on Lifetime Maximization of Multi-channel Cooperative Spectrum Sensing

Abstract

Cooperative spectrum sensing (CSS) is an efficient method to detect the reliability of the free spectrum, however, with great overhead and energy consumption. In this paper, we study a much broader application of CSS, where the sensors are used to sense multiple channels. On the one hand, since the tiny and low-cost sensors do not have high-speed analog-to-digital-convertors, they cannot simultaneously sense more than one channel. Therefore, the simultaneous multi-channel CSS is an important issue in a cognitive sensor network (CSN). On the other hand, these tiny sensors do not have high-power batteries, which makes the network lifetime as an important metric. In this paper, node selection is proposed for the multi-channel CSS to maximize the lifetime of a CSN under some detection constraints with lower overhead than cooperative sensing by all the sensors simultaneously. We analyze the problem for the OR and the AND rules, which can be implemented at the fusion center. The problem is solved by using convex optimization methods where assignment indices for every sensor are assumed. We provide a performance analysis through simulations using MATLAB, which shows that the sensor selection scheme provides a significant long lifetime for a CSN compared to the case where sensors are selected randomly and where all sensors are just classified to sense the channels simultaneously.

Appendices

Appendix 1

Considering the Lagrange multipliers ${\uprho }_{\text{n}} ,{\uplambda }_{\text{m}} ,\,{\text{and}}\,{{\upxi }}_{\text{m}}$ for the constraints (20-1), (20-2), and (20-3), respectively, we form the Lagrange function as follows [33]:

$$\begin{aligned} & {\text{L}}\left( {{\varphi }_{{{\text{m}},{\text{n}}}} ,{\uprho }_{\text{n}} ,{\uplambda }_{\text{m}} ,{\upxi }_{\text{m}} } \right) = - \mathop \sum \limits_{\text{n}} {\uprho }_{\text{n}} \left( {{\varphi }_{{{\text{m}},{\text{n}}}} \left( {{\text{E}}_{\text{th}} - \left( {{\text{E}}_{{0,{\text{n}}}} - \mathop \sum \limits_{{{\text{m}} = 1}}^{\text{M}} {\varphi }_{{{\text{m}},{\text{n}}}} {\text{E}}_{{{\text{c}},{\text{n}},{\text{m}}}} } \right)} \right)} \right) - \mathop \sum \limits_{\text{m}} {\uplambda }_{\text{m}} \left( {{{\upbeta }}_{\text{m}} - 1 + \mathop \prod \limits_{{ {\text{n}} \in {\text{J}}_{\text{m}} }}^{{}} \left( {1 - {\varphi }_{{{\text{m}},{\text{n}}}} {\text{P}}_{{{\text{d}}_{{{\text{m}},{\text{n}}}} }} } \right)} \right) \\ & \quad - \mathop \sum \limits_{\text{m}} {{\upxi }}_{\text{m}} \left( {\left( {1 - \mathop \prod \limits_{{ {\text{n}} \in {\text{J}}_{\text{m}} }}^{{}} \left( {1 - {\varphi }_{{{\text{m}},{\text{n}}}} {\text{P}}_{{{\text{f}}_{{{\text{m}},{\text{n}}}} }} } \right)} \right) - {{\upalpha }}_{\text{m}} } \right) - \mathop \sum \limits_{\text{n}} {\upeta }_{\text{n}} \left( {\mathop \sum \limits_{{{\text{m}} = 1}}^{\text{M}} {\varphi }_{{{\text{m}},{\text{n}}}} - 1} \right) \\ \end{aligned}$$

(38)

Since we remove the selected nodes from the set of remained nodes, the constraints (20-4) and (20-5) are checked during the algorithm, therefore they are not considered in the Lagrange function. We can find the optimal values of ${\varphi }_{{{\text{m}},{\text{n}}}}^{*}$, by differentiating L with respect to ${\varphi }_{{{\text{m}},{\text{n}}}}$, as [33]:

$$\frac{{\partial {\text{L}}}}{{\partial {{\upvarphi }}_{{{\text{m}},{\mathrm{n}}}} }} = {\uprho }_{\text{n}} \left( {E_{0,n} - {\text{E}}_{\mathrm{th}} - 2{{\upvarphi }}_{{{\text{m}},{\mathrm{n}}}} {\text{E}}_{{{\mathrm{c}},{\text{n}},{\mathrm{m}}}} } \right) + {\uplambda }_{\mathrm{m}} {\text{P}}_{{{\mathrm{d}}_{{{\text{m}},{\mathrm{n}}}} }} \mathop \prod \limits_{{\begin{array}{*{20}c} { l \in {\text{J}}_{\mathrm{m}} } \\ {l \ne n} \\ \end{array} }}^{{}} \left( {1 - {{\upvarphi }}_{{{\text{m}},\ell }} {\mathrm{P}}_{{{\text{d}}_{{{\mathrm{m}},\ell }} }} } \right) - {{\upxi }}_{\text{m}} {\mathrm{P}}_{{{\text{f}}_{{{\mathrm{m}},{\text{n}}}} }} \mathop \prod \limits_{{\begin{array}{*{20}c} { l \in {\text{J}}_{\mathrm{m}} } \\ {l \ne n} \\ \end{array} }}^{{}} \left( {1 - {{\upvarphi }}_{{{\text{m}},\ell }} {\mathrm{P}}_{{{\text{f}}_{{{\mathrm{m}},\ell }} }} } \right) = 0$$

(39)

Now, the priority of sensors to detect the m-th channel is calculated as:

$${{\upvarphi }}_{{{\text{m}},{\mathrm{n}}}} = \frac{{ - {{\upxi }}_{\text{m}} {\mathrm{P}}_{{{\text{f}}_{{{\mathrm{m}},{\text{n}}}} }} \mathop \prod \nolimits_{{\begin{array}{*{20}c} { l \in {\text{J}}_{\mathrm{m}} } \\ {l \ne n} \\ \end{array} }}^{{}} \left( {1 - {{\upvarphi }}_{{{\text{m}},\ell }} {\text{P}}_{{{\mathrm{f}}_{{{\text{m}},\ell }} }} } \right) + {\uplambda }_{\text{m}} {\mathrm{P}}_{{{\text{d}}_{{{\mathrm{m}},{\text{n}}}} }} \mathop \prod \nolimits_{{\begin{array}{*{20}c} { l \in {\text{J}}_{\mathrm{m}} } \\ {l \ne n} \\ \end{array} }}^{{}} \left( {1 - {{\upvarphi }}_{{{\text{m}},\ell }} {\mathrm{P}}_{{{\text{d}}_{{{\mathrm{m}},\ell }} }} } \right)}}{{2{\uprho }_{\text{n}} {\mathrm{E}}_{{{\text{c}},{\mathrm{n}},{\text{m}}}} }} + \frac{{{\uprho }_{\text{n}} \left( {E_{0,n} - {\text{E}}_{\mathrm{th}} } \right)}}{{2{\uprho }_{\text{n}} {\mathrm{E}}_{{{\text{c}},{\mathrm{n}},{\text{m}}}} }}$$

(40)

To obtain the optimum values of Lagrange multipliers in (40), the complementary slackness conditions are analyzed. Because it is easy to track, we don’t repeat them and just use the results.

Appendix 2

Considering the Lagrange multipliers ${\uprho }_{\text{n}} ,{\uplambda }_{\text{m}} \,{\text{and}}\,{{\upxi }}_{\text{m}}$ for the constraints (21-1), (21-2), and (21-3), respectively, we form the Lagrange function as follows [33]:

$$\begin{aligned} & {\text{L}}\left( {{{\upvarphi }}_{{{\text{m}},{\mathrm{n}}}} ,{\uprho }_{\mathrm{n}} ,{\uplambda }_{\text{m}} , {\upeta }_{\text{n}} } \right) \\ & \quad = - \mathop \sum \limits_{\text{n}} {\uprho }_{\text{n}} \left( {{{\upvarphi }}_{{{\text{m}},{\mathrm{n}}}} \left( {{\text{E}}_{\mathrm{th}} - \left( {{\text{E}}_{{0,{\mathrm{n}}}} - \mathop \sum \limits_{{{\text{m}} = 1}}^{\text{M}} {{\upvarphi }}_{{{\text{m}},{\mathrm{n}}}} {\text{E}}_{{{\mathrm{c}},{\text{n}},{\text{m}}}} } \right)} \right)} \right) \\ & \quad - \mathop \sum \limits_{\text{m}} {\uplambda }_{\text{m}} \left( {{{\upbeta }}_{\text{m}} - \mathop \prod \limits_{{ {\text{n}} \in {\text{J}}_{\mathrm{m}} }}^{{}} {{\upvarphi }}_{{{\text{m}},{\mathrm{n}}}} {\text{P}}_{{{\mathrm{d}}_{{{\text{m}},{\mathrm{n}}}} }} } \right) \\ & \quad - \mathop \sum \limits_{\text{m}} {{\upxi }}_{\text{m}} \left( {\mathop \prod \limits_{{ {\text{n}} \in {\text{J}}_{\mathrm{m}} }}^{{}} {{\upvarphi }}_{{{\text{m}},{\mathrm{n}}}} {\text{P}}_{{{\mathrm{f}}_{{{\text{m}},{\mathrm{n}}}} }} - {{\upalpha }}_{\text{m}} } \right) \\ \end{aligned}$$

(41)

Similarly, the constraints (21-5) and (21-4) are not inserted in the Lagrange function. We differentiate ${\text{L}}$ with respect to ${{\upvarphi }}_{{{\text{m}},{\text{n}}}}$, and the priority of the sensors is calculated as:

$${{\upvarphi }}_{{{\text{m}},{\mathrm{n}}}} = \frac{{{\uplambda }_{\text{m}} {\mathrm{P}}_{{{\text{d}}_{{{\mathrm{m}},{\text{n}}}} }} \mathop \prod \nolimits_{{\begin{array}{*{20}c} { l \in {\text{J}}_{\mathrm{m}} } \\ {l \ne n} \\ \end{array} }}^{{}} {{\upvarphi }}_{{{\text{m}},\ell }} {\text{P}}_{{{\text{d}}_{{{\mathrm{m}},\ell }} }} - {{\upxi }}_{\text{m}} {\mathrm{P}}_{{{\text{f}}_{{{\mathrm{m}},{\text{n}}}} }} \mathop \prod \nolimits_{{\begin{array}{*{20}c} { l \in {\text{J}}_{\mathrm{m}} } \\ {l \ne n} \\ \end{array} }}^{{}} {{\upvarphi }}_{{{\text{m}},\ell }} {\mathrm{P}}_{{{\text{f}}_{{{\mathrm{m}},\ell }} }} }}{{2{\uprho }_{\text{n}} {\mathrm{E}}_{{{\text{c}},{\mathrm{n}},{\text{m}}}} }}{ + }\frac{{{\uprho }_{\text{n}} \left( {E_{0,n} - {\text{E}}_{\mathrm{th}} } \right) - {\upeta }_{\text{n}} }}{{2{\uprho }_{\text{n}} {\mathrm{E}}_{{{\text{c}},{\mathrm{n}},{\text{m}}}} }}$$

(42)

Also, the complementary slackness conditions are analyzed to find the optimum multipliers in (42).

Appendix 3

Lagrange function for the special case is formed as follows [33]:

$$\begin{aligned} {\text{L}}\left( {{{\upvarphi }}_{{{\text{m}},{\text{n}}}} ,{\uprho }_{\text{n}} ,{\uplambda }_{\text{m}} ,{\upeta }_{\text{n}} ,{{\upxi }}_{\text{m}} } \right) & = - \mathop \sum \limits_{\text{n}} {\uprho }_{\text{n}} \left( {{{\upvarphi }}_{{{\text{m}},{\text{n}}}} \left( {{\text{E}}_{\text{th}} - \left( {{\text{E}}_{{0,{\text{n}}}} - \mathop \sum \limits_{{{\text{m}} = 1}}^{\text{M}} {{\upvarphi }}_{{{\text{m}},{\text{n}}}} {\text{E}}_{{{\text{c}},{\text{n}},{\text{m}}}} } \right)} \right)} \right) \\ & \quad - \mathop \sum \limits_{\text{m}} {\uplambda }_{\text{m}} \left( {{{\upbeta }}_{\text{m}} - 1 + \mathop \prod \limits_{{ {\text{n}} \in {\text{J}}_{\text{m}} }}^{{}} \left( {1 - {{\upvarphi }}_{{{\text{m}},{\text{n}}}} {\text{P}}_{{{\text{d}}_{{{\text{m}},{\text{n}}}} }} } \right)} \right) - \mathop \sum \limits_{\text{n}} {\upeta }_{\text{n}} \left( {\mathop \sum \limits_{{{\text{m}} = 1}}^{\text{M}} {{\upvarphi }}_{{{\text{m}},{\text{n}}}} - 1} \right) \\ \end{aligned}$$

(43)

The optimal value of ${{\upvarphi }}_{{{\text{m}},{\text{n}}}}^{ *}$ are calculated by differentiating ${\text{L}}$ with respect to ${{\upvarphi }}_{{{\text{m}},{\text{n}}}}$. Now, the sensors priority to detect the m-th channel, are calculated as:

$${{\upvarphi }}_{{{\text{m}},{\text{n}}}} = \frac{{ - {\upeta }_{\text{n}} + {\uprho }_{\text{n}} \left( {E_{0,n} - {\text{E}}_{\text{th}} } \right) + {\uplambda }_{\text{m}} {\mathrm{P}}_{{{\text{d}}_{{{\mathrm{m}},{\text{n}}}} }} \mathop \prod \nolimits_{{\begin{array}{*{20}c} { l \in {\text{J}}_{\mathrm{m}} } \\ {l \ne n} \\ \end{array} }}^{{}} \left( {1 - {{\upvarphi }}_{{{\text{m}},\ell }} {\mathrm{P}}_{{{\text{d}}_{{{\mathrm{m}},\ell }} }} } \right)}}{{2{\uprho }_{\text{n}} {\mathrm{E}}_{{{\text{c}},{\mathrm{n}},{\text{m}}}} }}$$

(44)

The optimum values of Lagrange multipliers in (44) are calculated from analyzing the complementary slackness conditions [24]. Also, we check the number of the selected sensors, hence ${{\upxi }}_{\text{m}}$ is removed.

Appendix 4

Lagrange function for AND fusion rule in the CFP scenario is formed as [33]:

$${\text{L}}\left( {{{\upvarphi }}_{{{\text{m}},{\text{n}}}} ,{\uprho }_{\text{n}} ,{\uplambda }_{\text{m}} ,{\upeta }_{\text{n}} } \right) = - \mathop \sum \limits_{\text{n}} {\uprho }_{\text{n}} \left( {{{\upvarphi }}_{{{\text{m}},{\text{n}}}} \left( {{\text{E}}_{\text{th}} - \left( {{\text{E}}_{{0,{\text{n}}}} - \mathop \sum \limits_{{{\text{m}} = 1}}^{\text{M}} {{\upvarphi }}_{{{\text{m}},{\text{n}}}} {\text{E}}_{{{\text{c}},{\text{n}},{\text{m}}}} } \right)} \right)} \right) - \mathop \sum \limits_{\text{m}} {\uplambda }_{\text{m}} \left( {{{\upbeta }}_{\text{m}} - \mathop \prod \limits_{{ {\text{n}} \in {\text{J}}_{\text{m}} }}^{{}} {{\upvarphi }}_{{{\text{m}},{\text{n}}}} {\mathrm{P}}_{{{\text{d}}_{{{\mathrm{m}},{\text{n}}}} }} } \right)$$

(45)

Now the priority of the sensors is calculated by differentiating ${\text{L}}$ with respect to ${{\upvarphi }}_{{{\text{m}},{\text{n}}}}$, as:

$${{\upvarphi }}_{{{\text{m}},{\mathrm{n}}}} = \frac{{{\uprho }_{\text{n}} \left( {E_{0,n} - {\text{E}}_{\mathrm{th}} } \right) + {\uplambda }_{\text{m}} {\mathrm{P}}_{{{\text{d}}_{{{\mathrm{m}},{\text{n}}}} }} \mathop \prod \nolimits_{{\begin{array}{*{20}c} { l \in {\text{J}}_{\mathrm{m}} } \\ {l \ne n} \\ \end{array} }}^{{}} {{\upvarphi }}_{{{\text{m}},\ell }} {\mathrm{P}}_{{{\text{d}}_{{{\mathrm{m}},\ell }} }} }}{{2{\uprho }_{\text{n}} {\mathrm{E}}_{{{\text{c}},{\mathrm{n}},{\text{m}}}} }}$$

(46)

To obtain the optimum Lagrange multipliers, the complementary slackness conditions are analyzed [33].