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A novel adaptive spectrum access protocol in cognitive radio with primary multicast network, secondary user selection and hardware impairments

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Abstract

In this paper, we propose an adaptive spectrum access protocol for secondary users (SUs) be used to access licensed bands in cognitive radio networks. Specifically, if the primary network, which uses multicast communication to transmit data from one primary source to multiple primary destinations, satisfies a required system quality of service (QoS), SUs can access the licensed bands, follows an underlay spectrum sharing. Otherwise, a secondary base (SB) station must assist the primary network in obtaining the QoS so that it can find opportunities to use the bands, i.e., cooperation-based spectrum access. To enhance the performance for the secondary network, in terms of outage probability (OP), various best-user selection methods are proposed. Moreover, we take into consideration the impact of hardware impairments on the OP of both primary and secondary networks. We derive exact and asymptotic closed-form expressions of the OP over Rayleigh fading channel. From the analytical results, an optimal value of maximal interference threshold and an optimal fraction of the SBs’ transmit power to the primary data are obtained when the secondary network operates on the underlay and the cooperation-based spectrum access modes, respectively. Finally, Monte Carlo simulations are performed to verify the theoretical derivations.

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Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.01-2014.33.

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Authors and Affiliations

  1. Posts and Telecommunications Institute of Technology, Ho Chi Minh City, Vietnam

    Tran Trung Duy

  2. Ho Chi Minh City University of Technology and Education, Ho Chi Minh City, Vietnam

    Pham Ngoc Son

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  1. Tran Trung Duy
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Correspondence to Tran Trung Duy.

Appendices

Appendix 1: Proof of Proposition 1

Firstly, the \(\mathrm{{SOP}}_\mathrm{{S}}^{\mathrm{{Ud}}}\) in (13) can be rewritten as

$$\begin{aligned} \mathrm{{SOP}}_\mathrm{{S}}^{\mathrm{{Ud}}}&= \underbrace{\Pr ( {{Y_{\max }} \le {\mathcal{I}_\mathrm{{P}}}/{\varPsi _\mathrm{{S}}},\,{Z_{1\max }}< {\rho _\mathrm{{S}}}} )}_{\mathrm{{OP}}1} \nonumber \\&\quad + \underbrace{\Pr ( {{Y_{\max }} > {\mathcal{I}_\mathrm{{P}}}/{\varPsi _\mathrm{{S}}},\,{Z_{2\max }} < {\rho _\mathrm{{S}}}} )}_{\mathrm{{OP2}}}, \end{aligned}$$
(60)

where,

$$\begin{aligned} {Z_{1\max }}&= \mathop {\max }\limits _{n = 1,\,2,\ldots ,N} \left( {\frac{{{\varPsi _\mathrm{{S}}}{\gamma _{4n}}}}{{{\kappa _\mathrm{{S}}}{\varPsi _\mathrm{{S}}}{\gamma _{4n}} + {\varPsi _\mathrm{{P}}}{\gamma _{3n}} + 1}}} \right) , \nonumber \\ {Z_{2\max }}&= \mathop {\max }\limits _{n = 1,\,2,\ldots ,N} \left( {\frac{{{\mathcal{I}_\mathrm{{P}}}{\gamma _{4n}}/{Y_{\max }}}}{{{\kappa _\mathrm{{S}}}{\mathcal{I}_\mathrm{{P}}}{\gamma _{4n}}/{Y_{\max }} + {\varPsi _\mathrm{{P}}}{\gamma _{3n}} + 1}}} \right) . \end{aligned}$$

Due to the independence between \({Y_{\max }}\) and \({Z_{1\max }},\) the probability OP1 in (60) can be rewritten as

$$\begin{aligned} \mathrm{{OP1}} = \Pr \left( {{Y_{\max }} \le {\mathcal{I}_\mathrm{{P}}}/{\varPsi _\mathrm{{S}}}} \right) \Pr \left( {{Z_{1\max }} < {\rho _\mathrm{{S}}}} \right) . \end{aligned}$$
(61)

By applying ([20], Eq. (22)) for \({Y_{\max }},\) we have

$$\begin{aligned} \Pr \left( {{Y_{\max }} \le \frac{{{\mathcal{I}_\mathrm{{P}}}}}{{{\varPsi _\mathrm{{S}}}}}} \right) = {\left( {1 - \exp \left( { - \frac{{{\lambda _2}{\mathcal{I}_\mathrm{{P}}}}}{{{\varPsi _\mathrm{{S}}}}}} \right) } \right) ^M}. \end{aligned}$$
(62)

Next, the probability \(\Pr ( {{Z_{1\max }} < {\rho _\mathrm{{S}}}} )\) can be formulated as

$$\begin{aligned}&\Pr \left( {{Z_{1\max }}< {\rho _\mathrm{{S}}}} \right) = {\left[ {\Pr \left( {\frac{{{\varPsi _\mathrm{{S}}}{\gamma _{4n}}}}{{{\kappa _\mathrm{{S}}}{\varPsi _\mathrm{{S}}}{\gamma _{4n}} + {\varPsi _\mathrm{{P}}}{\gamma _{3n}} + 1}} < {\rho _\mathrm{{S}}}} \right) } \right] ^N} \nonumber \\&\quad = {\left[ {\int _0^{ {+} \infty } {{f_{{\gamma _{3n}}}}( x ){F_{{\gamma _{4n}}}}\left( {\frac{{{\varPsi _\mathrm{{P}}}{\chi _\mathrm{{S}}}}}{{{\varPsi _\mathrm{{S}}}}}x + \frac{{{\chi _\mathrm{{S}}}}}{{{\varPsi _\mathrm{{S}}}}}} \right) dx} } \right] ^N}, \end{aligned}$$
(63)

where \({f_{{\gamma _{3n}}}}( x ) = {\lambda _3}\exp ( { - {\lambda _3}x} )\) is the probability density function (PDF) of the RV \({\gamma _{3n}}\) and \({F_{{\gamma _{4n}}}}( y ) = 1 - \exp ( { - {\lambda _4}y} )\) is the cumulative distribution function of the RV \({\gamma _{4n}}.\)

Substituting \({f_{{\gamma _{3n}}}}( x)\) and \({F_{{\gamma _{4n}}}}( y )\) into (63), after some manipulations, we obtain

$$\begin{aligned}&\Pr \left( {{Z_{1\max }} < {\rho _\mathrm{{S}}}} \right) \nonumber \\&\quad ={\left[ {1 - \frac{{{\lambda _3}}}{{{\lambda _3} + {\lambda _4}{\varPsi _\mathrm{{P}}}{\chi _\mathrm{{S}}}/{\varPsi _\mathrm{{S}}}}}\exp \left( { - {\lambda _4}\frac{{{\chi _\mathrm{{S}}}}}{{{\varPsi _\mathrm{{S}}}}}} \right) } \right] ^N}. \end{aligned}$$
(64)

Next, the probability OP2 in (60) can be given by

$$\begin{aligned} \mathrm{{OP2}} = \int _{{I_\mathrm{{P}}}/{\varPsi _\mathrm{{S}}}}^{ {+} \infty } {{f_{{Y_{\max }}}}( x){{[ {A( x )} ]}^N}dx}, \end{aligned}$$
(65)

where \(A( x) = \Pr ( {{\gamma _{4n}} < {\varPsi _\mathrm{{P}}}{\chi _\mathrm{{S}}}{\gamma _{3n}}x/{\mathcal{I}_\mathrm{{P}}} + {\chi _\mathrm{{S}}}x/{\mathcal{I}_\mathrm{{P}}}}),\) and \({f_{{Y_{\max }}}}( x )\) is the PDF of \({Y_{\max }},\) which is given as

$$\begin{aligned}&{f_{{Y_{\max }}}}( x ) \nonumber \\&\quad =\sum \limits _{b = 0}^{M - 1} {{{( { - 1} )}^b}C_{M - 1}^bM{\lambda _2}\exp \left( { - ( {b + 1} ){\lambda _2}x} \right) }, \end{aligned}$$
(66)

with \(C_{M - 1}^b = ( {M - 1} )!/b!/( {M - 1 - b} )!.\)

Next, similar to (63) and (64), the probability A(x) can be computed as

$$\begin{aligned} A( x)&= \int _0^{ {+} \infty } {{f_{{\gamma _{3n}}}}( y ){F_{{\gamma _{4n}}}}\left( {\frac{{{\varPsi _\mathrm{{P}}}{\chi _\mathrm{{S}}}}}{{{\mathcal{I}_\mathrm{{P}}}}}xy + \frac{{{\chi _\mathrm{{S}}}}}{{{\mathcal{I}_\mathrm{{P}}}}}x} \right) } dy \nonumber \\&= 1 - \frac{{{\lambda _3}}}{{{\lambda _3} + {\lambda _4}{\varPsi _\mathrm{{P}}}{\chi _\mathrm{{S}}}x/{\mathcal{I}_\mathrm{{P}}}}}\exp \left( { - {\lambda _4}\frac{{{\chi _\mathrm{{S}}}}}{{{\mathcal{I}_\mathrm{{P}}}}}x} \right) \nonumber \\&= 1 - \frac{\omega }{{\omega + x}}\exp \left( { - {\lambda _4}\frac{{{\chi _\mathrm{{S}}}}}{{{\mathcal{I}_\mathrm{{P}}}}}x} \right) . \end{aligned}$$
(67)

Using binomial expansion for \({[ {A( x )} ]^N},\) we obtain

$$\begin{aligned}&{[ {A( x)} ]^N} \nonumber \\&\quad = 1 + \sum \limits _{c = 1}^N {{{( { - 1} )}^c}C_N^c\frac{{{\omega ^c}}}{{{{( {\omega + x} )}^c}}}\exp \left( { - c{\lambda _4}\frac{{{\chi _\mathrm{{S}}}}}{{{\mathcal{I}_\mathrm{{P}}}}}x} \right) }, \end{aligned}$$
(68)

where \(C_N^c = N!/c!/( {N - c})!.\)

Combining (65), (66) and (68), we arrive at

$$\begin{aligned}&\mathrm{{OP2}} = 1 - {\left( {1 - \exp \left( { - {\lambda _2}\frac{{{\mathcal{I}_\mathrm{{P}}}}}{{{\varPsi _\mathrm{{S}}}}}} \right) } \right) ^M} \nonumber \\&\quad + \sum \limits _{b = 0}^{M - 1} {\sum \limits _{c = 1}^N {{{( { - 1} )}^{b + c}}C_{M - 1}^bC_N^cM{\lambda _2}{\omega ^c}\int _{{I_\mathrm{{P}}}/{\varPsi _\mathrm{{S}}}}^{ {+} \infty } {\frac{{\exp ( { - \xi x} )}}{{{{( {\omega + x} )}^c}}}} dx} }. \end{aligned}$$
(69)

Then, by applying ([27], Eq. (3.351.4)) for the integral in (69), we can obtain (70). Finally, combining (60)–(62), (64) and (70) together, we obtain (14).

$$\begin{aligned}&\mathrm{{OP2}} = 1 - {\left( {1 - \exp \left( { - {\lambda _2}{\mathcal{I}_\mathrm{{P}}}/{\varPsi _\mathrm{{S}}}} \right) } \right) ^M} \nonumber \\&\quad + \sum \limits _{b = 0}^{M - 1} {\sum \limits _{c = 1}^N {{{( { - 1} )}^{b + c}}C_{M - 1}^bC_N^cM{\lambda _2}{\omega ^c}} } \exp ( {\xi \omega } ) \nonumber \\&\quad \times \left[ {\frac{{{{( { - 1} )}^{c + 1}}{\xi ^{c - 1}}}}{{( {c - 1})!}}{E_1}\left( {\xi \left( {\omega + \frac{{{\mathcal{I}_\mathrm{{P}}}}}{{{\varPsi _\mathrm{{S}}}}}} \right) } \right) }\right. \nonumber \\&\quad \left. {+ \frac{{\exp ( { - \xi ( {\omega + {\mathcal{I}_\mathrm{{P}}}/{\varPsi _\mathrm{{S}}}} )} )}}{{{{( {\omega + {\mathcal{I}_\mathrm{{P}}}/{\varPsi _\mathrm{{S}}}} )}^{c - 1}}}}\sum \limits _{k = 0}^{c - 2} {{{( { - 1} )}^k}\frac{{{\xi ^k}{{( {\omega + {\mathcal{I}_\mathrm{{P}}}/{\varPsi _\mathrm{{S}}}})}^k}}}{{\prod \nolimits _{t = 0}^k {( {c - 1 - t})} }}} } \right] . \end{aligned}$$
(70)

Appendix 2: Proof of Corollary 1

Because \(\min ( {{\varPsi _\mathrm{{S}}},\,{\mathcal{I}_\mathrm{{P}}}/{Y_{\max }}})\mathop \approx \limits ^{{\varPsi _\mathrm{{S}}}> > {\mathcal{I}_\mathrm{{P}}}} {\mathcal{I}_\mathrm{{P}}}/{Y_{\max }},\) Eq. (11) can be approximated by

$$\begin{aligned} C_\mathrm{{S}}^{\mathrm{{Ud}},n}\mathop \approx \limits ^{{\varPsi _\mathrm{{S}}}> > {\mathcal{I}_\mathrm{{P}}}} {\log _2}\left( {1 + \frac{{{\mathcal{I}_\mathrm{{P}}}{\gamma _{4n}}/{Y_{\max }}}}{{{\kappa _\mathrm{{S}}}{\mathcal{I}_\mathrm{{P}}}{\gamma _{4n}}/{Y_{\max }} + {\varPsi _\mathrm{{P}}}{\gamma _{3n}} + 1}}} \right) . \end{aligned}$$
(71)

Then, similar to (65), the SOP of the secondary network can be formulated as

$$\begin{aligned}&\mathrm{{SOP}}_\mathrm{{S}}^{\mathrm{{Ud}}}\mathop \approx \limits ^{{\varPsi _\mathrm{{S}}}> > {\mathcal{I}_\mathrm{{P}}}} \nonumber \\&\Pr \left( {\mathop {\max }\limits _{n = 1,\,2,\ldots ,N} \left( {\frac{{{\mathcal{I}_\mathrm{{P}}}{\gamma _{4n}}/{Y_{\max }}}}{{{\kappa _\mathrm{{S}}}{\mathcal{I}_\mathrm{{P}}}{\gamma _{4n}}/{Y_{\max }} + {\varPsi _\mathrm{{P}}}{\gamma _{3n}} + 1}}} \right) < {\rho _\mathrm{{S}}}} \right) \nonumber \\&\mathop \approx \limits ^{{\varPsi _\mathrm{{S}}} \gg {\mathcal{I}_\mathrm{{P}}}} \int _0^{ {+} \infty } {{f_{{Y_{\max }}}}( x ){{[ {A( x)} ]}^N}dx}. \end{aligned}$$
(72)

Finally, with the same manner as (65)–(70), we can obtain (15).

Appendix 3: Proof of Corollary 2

At high \({\varPsi _\mathrm{{P}}}\) and \({\varPsi _\mathrm{{S}}},\) we can approximate (8) and (11), respectively as

$$\begin{aligned}&{\mathcal{I}_\mathrm{{P}}}\mathop \approx \limits ^{{\varPsi _\mathrm{{P}}},{\varPsi _\mathrm{{S}}} \rightarrow {+} \infty } \frac{{{\varPsi _\mathrm{{P}}}}}{{M{\lambda _0}{\chi _\mathrm{{P}}}}}\log \frac{1}{{1 - {Q_\mathrm{{P}}}}}, \end{aligned}$$
(73)
$$\begin{aligned}&C_\mathrm{{S}}^{\mathrm{{Ud}},n}\mathop \approx \limits ^{{\varPsi _\mathrm{{P}}},{\varPsi _\mathrm{{S}}} \rightarrow {+} \infty } \nonumber \\&\quad {\log _2}\left( {1 + \frac{{\min \left( {\mu ,\,\log \left( {\frac{1}{{1 - {Q_\mathrm{{P}}}}}} \right) /( {M{\lambda _0}{\chi _\mathrm{{P}}}{Y_{\max }}})} \right) {\gamma _{4n}}}}{{{\kappa _\mathrm{{S}}}\min \left( {\mu ,\,\log \left( {\frac{1}{{1 - {Q_\mathrm{{P}}}}}} \right) /( {M{\lambda _0}{\chi _\mathrm{{P}}}{Y_{\max }}} )} \right) {\gamma _{4n}} + {\gamma _{3n}}}}} \right) . \end{aligned}$$
(74)

From (74), similar to Appendix 1, the SOP of the secondary network can be formulated by

$$\begin{aligned} \mathrm{{SOP}}_\mathrm{{S}}^{\mathrm{{Ud}}}\mathop \approx \limits ^{{\varPsi _\mathrm{{P}}},{\varPsi _\mathrm{{S}}} \rightarrow {+} \infty }&\underbrace{\Pr ( {{Y_{\max }} \le \varTheta ,\,{Z_{3\max }}< {\rho _\mathrm{{S}}}} )}_{\mathrm{{OP3}}} \nonumber \\&\quad + \underbrace{\Pr ( {{Y_{\max }} > \varTheta ,\,{Z_{4\max }} < {\rho _\mathrm{{S}}}} )}_{\mathrm{{OP4}}}, \end{aligned}$$
(75)

where,

$$\begin{aligned} {Z_{3\max }}&= \mathop {\max }\limits _{n = 1,\,2,\ldots ,N} \left( {\frac{{\mu {\gamma _{4n}}}}{{{\kappa _\mathrm{{S}}}\mu {\gamma _{4n}} + {\gamma _{3n}}}}} \right) , \nonumber \\ {Z_{4\max }}&= \mathop {\max }\limits _{n = 1,\,2,\ldots ,N} \left( {\frac{{\varTheta \mu {\gamma _{4n}}/{Y_{\max }}}}{{{\kappa _\mathrm{{S}}}\varTheta \mu {\gamma _{4n}}/{Y_{\max }} + {\gamma _{3n}}}}} \right) . \end{aligned}$$

Similarly, we can calculate the probabilities OP3 and OP4, respectively as

$$\begin{aligned} \mathrm{{OP3}}&= {\left( {1 - \exp \left( { - {\lambda _2}\varTheta } \right) } \right) ^M}{\left( {\frac{{{\lambda _4}{\chi _\mathrm{{S}}}}}{{{\lambda _3}\mu + {\lambda _4}{\chi _\mathrm{{S}}}}}} \right) ^N}, \end{aligned}$$
(76)
$$\begin{aligned} \mathrm{{OP4}}&= \int _\varTheta ^{ {+} \infty } {{f_{{Y_{\max }}}}( x ){{[ {B( x )} ]}^N}dx}, \end{aligned}$$
(77)

where,

$$\begin{aligned} B( x) = \Pr \left( {\frac{{\varTheta \mu {\gamma _{4n}}/x}}{{{\kappa _\mathrm{{S}}}\varTheta \mu {\gamma _{4n}}/x + {\gamma _{3n}}}} < {\rho _\mathrm{{S}}}} \right) = 1 - \frac{\delta }{{\delta + x}}, \end{aligned}$$

with \(\delta = {\lambda _3}\varTheta \mu /( {{\lambda _4}{\chi _\mathrm{{S}}}} ).\)

After some manipulations as in Appendices 1 and 2, we can obtain (78).

$$\begin{aligned}&\mathrm{{OP4}} = 1 - {\left( {1 - \exp \left( { - {\lambda _2}\varTheta } \right) } \right) ^M} \nonumber \\&\quad + \sum \limits _{b = 0}^{M - 1} {\sum \limits _{c = 1}^N {{{( { - 1} )}^{b + c}}C_{M - 1}^bC_N^cM{\lambda _2}{\delta ^c}\exp \left( {( {b + 1} ){\lambda _2}\delta } \right) } } \nonumber \\&\quad \times \left[ {\frac{{{{( { - 1})}^{c + 1}}{{( {( {b + 1} ){\lambda _2}} )}^{c - 1}}}}{{( {c - 1} )!}}{E_1}\left( {( {b + 1} ){\lambda _2}( {\delta + \varTheta } )} \right) }\right. \nonumber \\&\quad \left. {+ \frac{{\exp ( { - ( {b + 1} ){\lambda _2}( {\delta + \varTheta } )} )}}{{{{( {\delta + \varTheta })}^{c - 1}}}}\sum \limits _{k = 0}^{c - 2} {{{( { - 1} )}^k}\frac{{{{( {( {b + 1} ){\lambda _2}} )}^k}{{( {\delta + \varTheta } )}^k}}}{{\prod \nolimits _{t = 0}^k {( {c - 1 - t} )} }}} } \right] . \end{aligned}$$
(78)

Finally, plugging (75), (76) and (78) together, we obtain (16) and finish the proof here.

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Duy, T.T., Son, P.N. A novel adaptive spectrum access protocol in cognitive radio with primary multicast network, secondary user selection and hardware impairments. Telecommun Syst 65, 525–538 (2017). https://doi.org/10.1007/s11235-016-0251-6

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