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System Performance Analysis in Cognitive Radio-Aided NOMA Network: An Application to Vehicle-to-Everything Communications

Wireless Personal Communications volume 120, pages 1975–2000 (2021)Cite this article

Abstract

In this study, non-orthogonal multiple access (NOMA) together with cognitive radio (CR) benefit to the vehicle-to-everything (V2X) as promising application with high spectrum efficiency. We have higher priority to evaluate system performance of the secondary network in such CR-NOMA system operating in the context of V2X. We first arrange vehicles belonging to serving group in this CR-NOMA assisted V2X, and it is beneficial to serve massive connections for vehicles. There are two scenarios studied in this paper, with and without the support of CR scheme. In our proposed system, two system metrics need be investigated to evaluate performance of vehicles that need higher quality of service (QoS). Our results indicate that the outage performance gap among two vehicles exists since different transmit power allocation factors were assigned to them. In particular, the outage probability is first derived in exact forms and then the bit error rate (BER) can be further achieved. In specific situations, the optimal outage probability can be obtained by numerical simulations. Simulation results are also provided to verify the correctness of the derived expressions and it exhibits advantages of the proposed CR-NOMA assisted V2X system in terms of two main metrics such as outage probability and BER.

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Author information

Authors and Affiliations

  1. Faculty of Electronics Technology, Industrial University of Ho Chi Minh City (IUH), Ho Chi Minh City, 700000, Vietnam

    Chi-Bao Le

  2. Department of Computer Science and Information Engineering, Asia University, Taichung, 41354, Taiwan

    Dinh-Thuan Do

  3. Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece

    Zaharias D. Zaharis

  4. Department of Computer Science, University of Nicosia, Nicosia, Cyprus

    Constandinos X. Mavromoustakis

  5. Department of Electrical and Computer Engineering, Hellenic Mediterranean University, 71500, Estavromenos, Heraklion, Greece

    George Mastorakis & Evangelos K. Markakis

Authors
  1. Chi-Bao Le
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  2. Dinh-Thuan Do
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  3. Zaharias D. Zaharis
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  4. Constandinos X. Mavromoustakis
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Correspondence to Dinh-Thuan Do.

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Appendices

Appendix A

In (8) we have \(P_1\) is calculated

$$\begin{aligned} \begin{aligned} {P_1}&= 1 - \underbrace{\Pr \left( {{{\left| {{h_{S{D_1}}}} \right| }^2} \ge {{\bar{\chi }} _{max}}\left( {\kappa {\rho _{CUE}}{{\left| {{h_{C{D_1}}}} \right| }^2} + 1} \right) ,{{\left| {{h_{SP}}} \right| }^2} < \frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}}} \right) }_{{A_1}}\\&\quad - \underbrace{\Pr \left( {{{\left| {{h_{S{D_1}}}} \right| }^2} \ge {{\tilde{\chi }} _{max}}\left( {\kappa {\rho _{CUE}}{{\left| {{h_{C{D_1}}}} \right| }^2}{{\left| {{h_{SP}}} \right| }^2} + {{\left| {{h_{SP}}} \right| }^2}} \right) ,{{\left| {{h_{SP}}} \right| }^2} > \frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}}} \right) }_{{A_2}}, \end{aligned} \end{aligned}$$
(33)

where \({{\bar{\chi }} _{max}} = \max \left( {\frac{{{\varepsilon _2}}}{{{\rho _{RSU}}\left[ {\alpha - {\varepsilon _2}\left( {1 - \alpha } \right) } \right] }},\frac{{{\varepsilon _1}}}{{\left( {1 - \alpha } \right) {\rho _{RSU}}}}} \right)\) and \({{\tilde{\chi }} _{max}} = \max \left( {\frac{{{\varepsilon _2}}}{{{\rho _Q}\left[ {\alpha - {\varepsilon _2}\left( {1 - \alpha } \right) } \right] }},\frac{{{\varepsilon _1}}}{{{\rho _Q}\left( {1 - \alpha } \right) }}} \right)\)

The outage probability of fist term and second terms (33) can be obtained as \({A_1}\) and \({A_2}\), respectively in following proposition.

\({A_1}\) can be calculate as

$$\begin{aligned} \begin{aligned} {A_1} =&\Pr \left( {{{\left| {{h_{S{D_1}}}} \right| }^2} \ge {{\bar{\chi }} _{max}}\left( {\kappa {\rho _{CUE}}{{\left| {{h_{C{D_1}}}} \right| }^2} + 1} \right) ,{{\left| {{h_{SP}}} \right| }^2} < \frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}}} \right) \\ =&\int \limits _0^{\frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}}} {{f_{{{\left| {{h_{SP}}} \right| }^2}}}\left( x \right) } \int \limits _0^\infty {{f_{{{\left| {{h_{C{D_1}}}} \right| }^2}}}\left( y \right) } \left[ {1 - {F_{{{\left| {{h_{S{D_1}}}} \right| }^2}}}\left( {{{\bar{\chi }} _{max}}\left( {\kappa {\rho _{CUE}}y + 1} \right) } \right) } \right] dxdy\\ =&\frac{{{e^{ - \frac{{{{\bar{\chi }} _{max}}}}{{{\Omega _{{h_{S{D_1}}}}}}}}}}}{{{\Omega _{{h_{SP}}}}{\Omega _{{h_{C{D_1}}}}}}}\int \limits _0^{\frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}}} {{e^{ - \frac{x}{{{\Omega _{{h_{SP}}}}}}}}} \int \limits _0^\infty {{e^{ - y\left( {\frac{1}{{{\Omega _{{h_{C{D_1}}}}}}} + \frac{{\kappa {{\bar{\chi }} _{max}}{\rho _{CUE}}}}{{{\Omega _{{h_{S{D_1}}}}}}}} \right) }}} dxdy\\ =&\left( {1 - {e^{ - \frac{{{\rho _Q}}}{{{\Omega _{{h_{SP}}}}{{\bar{\rho }}_{RSU}}}}}}} \right) \frac{{{\Omega _{{h_{S{D_1}}}}}{e^{ - \frac{{{{\bar{\chi }} _{max}}}}{{{\Omega _{{h_{S{D_1}}}}}}}}}}}{{\left( {{\Omega _{{h_{S{D_1}}}}} + {\Omega _{{h_{C{D_1}}}}}\kappa {\bar{\chi }_{max}}{\rho _{CUE}}} \right) }}. \end{aligned} \end{aligned}$$
(34)

Similarly, \({A_2}\) can be expressed as

$$\begin{aligned} \begin{aligned} {A_2} =&\Pr \left( {{{\left| {{h_{S{D_1}}}} \right| }^2} \ge {{\tilde{\chi }} _{max}}\left( {\kappa {\rho _{CUE}}{{\left| {{h_{C{D_1}}}} \right| }^2}{{\left| {{h_{SP}}} \right| }^2} + {{\left| {{h_{SP}}} \right| }^2}} \right) ,{{\left| {{h_{SP}}} \right| }^2} > \frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}}} \right) \\ =&\int \limits _{\frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}}}^\infty {{f_{{{\left| {{h_{SP}}} \right| }^2}}}\left( x \right) } \int \limits _0^\infty {{f_{{{\left| {{h_{C{D_1}}}} \right| }^2}}}\left( y \right) } \left[ {1 - {F_{{{\left| {{h_{S{D_1}}}} \right| }^2}}}\left( {{{\tilde{\chi }} _{max}}\left( {\kappa {\rho _{CUE}}yx + x} \right) } \right) } \right] dxdy\\ =&\frac{1}{{{\Omega _{{h_{SP}}}}{\Omega _{{h_{C{D_1}}}}}}}\int \limits _{\frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}}}^\infty {{e^{ - x\left( {\frac{1}{{{\Omega _{{h_{SP}}}}}} + \frac{{{{\tilde{\chi }} _{max}}}}{{{\Omega _{{h_{S{D_1}}}}}}}} \right) }}} \int \limits _0^\infty {{e^{ - y\left( {\frac{1}{{{\Omega _{{h_{C{D_1}}}}}}} + \frac{{\kappa {{\tilde{\chi }} _{max}}{\rho _{CUE}}x}}{{{\Omega _{{h_{S{D_1}}}}}}}} \right) }}} dxdy\\ =&\frac{{{\Omega _{{h_{S{D_1}}}}}}}{{{\Omega _{{h_{SP}}}}}}\int \limits _{\frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}}}^\infty {\frac{{{e^{ - x\left( {\frac{1}{{{\Omega _{{h_{SP}}}}}} + \frac{{{{\tilde{\chi }} _{max}}}}{{{\Omega _{{h_{S{D_1}}}}}}}} \right) }}}}{{\left( {{\Omega _{{h_{S{D_1}}}}} + {\Omega _{{h_{C{D_1}}}}}\kappa {{\tilde{\chi }} _{max}}{\rho _{CUE}}x} \right) }}} dx\\ =&\frac{\zeta }{{{\Omega _{{h_{SP}}}}}}\int \limits _{\frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}}}^\infty {\frac{{{e^{ - x\left( {\frac{1}{{{\Omega _{{h_{SP}}}}}} + \frac{{{{\tilde{\chi }} _{max}}}}{{{\Omega _{{h_{S{D_1}}}}}}}} \right) }}}}{{ {\zeta + x} }}} dx, \end{aligned} \end{aligned}$$
(35)

where \(\zeta = \frac{{{\Omega _{{h_{S{D_1}}}}}}}{{{\Omega _{{h_{C{D_1}}}}}\kappa {{\tilde{\chi }} _{max}}{\rho _{CUE}}}}\).

We using [29, Eq.(3.352.2)], \({A_2}\) is given as

$$\begin{aligned} \begin{aligned} {A_2} =&\frac{\zeta }{{{\Omega _{{h_{SP}}}}}}\int \limits _{\frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}}}^\infty {\frac{{{e^{ - x\left( {\frac{1}{{{\Omega _{{h_{SP}}}}}} + \frac{{{{\tilde{\chi }} _{max}}}}{{{\Omega _{{h_{S{D_1}}}}}}}} \right) }}}}{{ {\zeta + x} }}} dx\\ =&- \frac{\zeta }{{{\Omega _{{h_{SP}}}}}}{e^{\zeta \left( {\frac{1}{{{\Omega _{{h_{SP}}}}}} + \frac{{{{\tilde{\chi }} _{max}}}}{{{\Omega _{{h_{S{D_1}}}}}}}} \right) }}Ei\left( { - \left( {\frac{1}{{{\Omega _{{h_{SP}}}}}} + \frac{{{{\tilde{\chi }} _{max}}}}{{{\Omega _{{h_{S{D_1}}}}}}}} \right) \left( {\frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}} + \zeta } \right) } \right) , \end{aligned} \end{aligned}$$
(36)

where \(Ei\left( . \right)\) is exponential integral function.

Substituting (36) and (34) into (33), \({P_1}\) is given by

$$\begin{aligned} \begin{aligned} {P_1} =&1 - \left( {1 - {e^{ - \frac{{{\rho _Q}}}{{{\Omega _{{h_{SP}}}}{{{\bar{\rho }} }_{RSU}}}}}}} \right) \frac{{{\Omega _{{h_{S{D_1}}}}}{e^{ - \frac{{{{\bar{\chi }} _{max}}}}{{{\Omega _{{h_{S{D_1}}}}}}}}}}}{{\left( {{\Omega _{{h_{S{D_1}}}}} + {\Omega _{{h_{C{D_1}}}}}\kappa {{\bar{\chi }} _{max}}{\rho _{CUE}}} \right) }}\\ +&\frac{\zeta }{{{\Omega _{{h_{SP}}}}}}{e^{\zeta \left( {\frac{1}{{{\Omega _{{h_{SP}}}}}} + \frac{{{{\tilde{\chi }} _{max}}}}{{{\Omega _{{h_{S{D_1}}}}}}}} \right) }}Ei\left( { - \left( {\frac{1}{{{\Omega _{{h_{SP}}}}}} + \frac{{{{\tilde{\chi }} _{max}}}}{{{\Omega _{{h_{S{D_1}}}}}}}} \right) \left( {\frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}} + \zeta } \right) } \right) . \end{aligned} \end{aligned}$$
(37)

The proposition 1 is completed.

Appendix B

In (20), we have \({F_X}\left( x \right)\) of \(P_{{D_1}}^{BER}\) calculated as

$$\begin{aligned} \begin{aligned} {F_X}\left( x \right) =&1 - \Pr \left( {{{\left| {{h_{S{D_1}}}} \right| }^2}> \frac{{{\mu _1}}x}{{\left( {1 - \alpha } \right) {{{\bar{\rho }} }_{RSU}}}},{{\left| {{h_{SP}}} \right| }^2} < \frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}}} \right) \\&- \Pr \left( {{{\left| {{h_{S{D_1}}}} \right| }^2}> \frac{{{\mu _1}{{\left| {{h_{SP}}} \right| }^2}x}}{{\left( {1 - \alpha } \right) {\rho _Q}}},{{\left| {{h_{SP}}} \right| }^2} > \frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}}} \right) \\ =&1 - \int \limits _0^{\frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}}} {{f_{{{\left| {{h_{SP}}} \right| }^2}}}\left( y \right) } \left[ {1 - {F_{{{\left| {{h_{S{D_1}}}} \right| }^2}}}\left( {\frac{{{\mu _1}x}}{{\left( {1 - \alpha } \right) {{{\bar{\rho }} }_{RSU}}}}} \right) } \right] dy\\&- \int \limits _{\frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}}}^\infty {{f_{{{\left| {{h_{SP}}} \right| }^2}}}\left( y \right) } \left[ {1 - {F_{{{\left| {{h_{S{D_1}}}} \right| }^2}}}\left( {\frac{{{\mu _1}yx}}{{\left( {1 - \alpha } \right) {{{\bar{\rho }} }_{RSU}}}}} \right) } \right] dy\\ =&1 - \left( {1 - {e^{ - \frac{{{\rho _Q}}}{{{\Omega _{{h_{SP}}}}{{{\bar{\rho }} }_{RSU}}}}}}} \right) {e^{ - \frac{{{\mu _1}x}}{{{\Omega _{{h_{S{D_1}}}}}\left( {1 - \alpha } \right) {{{\bar{\rho }} }_{RSU}}}}}}\\&- \frac{{{\Omega _{{h_{S{D_1}}}}}\left( {1 - \alpha } \right) {{\bar{\rho }}_{RSU}}{e^{ - \left( {\frac{1}{{{\Omega _{{h_{SP}}}}}} + \frac{{{\mu _1}x}}{{{\Omega _{{h_{S{D_1}}}}}\left( {1 - \alpha } \right) {{{\bar{\rho }} }_{RSU}}}}} \right) \frac{{{\rho _Q}}}{{{{\bar{\rho }}_{RSU}}}}}}}}{{\left( {{\Omega _{{h_{S{D_1}}}}}\left( {1 - \alpha } \right) {{{\bar{\rho }} }_{RSU}} + {\mu _1}{\Omega _{{h_{SP}}}}x} \right) }}, \end{aligned} \end{aligned}$$
(38)

Substituting (38) into (20), the BER of \(D_1\) is written as

$$\begin{aligned} \begin{aligned} P_{{D_1}}^{BER} =&\frac{{\sqrt{{\bar{b}}} }}{{2\sqrt{2\pi } }}\left[ {\int \limits _0^\infty {\frac{{{e^{ - \frac{{{\bar{b}}}}{2}x}}}}{{\sqrt{x} }}} dx - \left( {1 - {e^{ - \frac{{{\rho _Q}}}{{{\Omega _{{h_{SP}}}}{{{\bar{\rho }} }_{RSU}}}}}}} \right) \int \limits _0^\infty {\frac{{{e^{ - \psi x}}}}{{\sqrt{x} }}dx} } \right. \\&\left. { - \Phi {e^{ - \frac{{{\rho _Q}}}{{{\Omega _{{h_{SP}}}}{{{\bar{\rho }} }_{RSU}}}}}}\int \limits _0^\infty {\frac{{{e^{ - \xi x}}}}{{\left( {\Phi + x} \right) \sqrt{x} }}} dx} \right] , \end{aligned} \end{aligned}$$
(39)

where \(\Phi = \frac{{{\Omega _{{h_{S{D_1}}}}}\left( {1 - \alpha } \right) {{{\bar{\rho }} }_{RSU}}}}{{{\mu _1}{\Omega _{{h_{SP}}}}}}\), \(\psi = \left( {\frac{{{\bar{b}}}}{2} + \frac{{{\mu _1}}}{{{\Omega _{{h_{S{D_1}}}}}\left( {1 - \alpha } \right) {{\bar{\rho }}_{RSU}}}}} \right)\) and \(\xi = \left( {\frac{{{\bar{b}}}}{2} + \frac{{{\mu _1}{\rho _Q}}}{{{\Omega _{{h_{S{D_1}}}}}\left( {1 - \alpha } \right) {\bar{\rho }} _{RSU}^2}}} \right)\)

Based on [29, Eq. (3.361.2)], [29, Eq. (3.383.10)] and after few steps, (39) can then be further derived as

$$\begin{aligned} \begin{aligned} P_{{D_1}}^{BER} =&\frac{{\sqrt{{\bar{b}}} }}{{2\sqrt{2\pi } }}\left[ {\sqrt{\frac{{2\pi }}{{{\bar{b}}}}} - \left( {1 - {e^{ - \frac{{{\rho _Q}}}{{{\Omega _{{h_{SP}}}}{{{\bar{\rho }} }_{RSU}}}}}}} \right) \sqrt{\frac{\pi }{\psi }} } \right. \\&\left. { - \sqrt{\Phi }{e^{ - \frac{{{\rho _Q}}}{{{\Omega _{{h_{SP}}}}{{{\bar{\rho }} }_{RSU}}}} + \xi \Phi }}\Gamma \left( {\frac{1}{2}} \right) \Gamma \left( {\frac{1}{2},\xi \Phi } \right) } \right] , \end{aligned} \end{aligned}$$
(40)

where \(\Gamma \left( . \right)\) is the Gamma function and \(\Gamma \left( {.,.} \right)\) is the upper incomplete Gamma function

The proposition 2 is completed.

Appendix C

Similar to (22), we have

$$\begin{aligned} \begin{aligned} {F_Y}\left( x \right)&= 1 - \int \limits _0^{\frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}}} {{f_{{{\left| {{h_{SP}}} \right| }^2}}}\left( y \right) } \left[ {1 - {F_{{{\left| {{h_{S{D_2}}}} \right| }^2}}}\left( {\frac{{x{\mu _2}}}{{{{{\bar{\rho }} }_{RSU}}\left( {\alpha - x\left( {1 - \alpha } \right) } \right) }}} \right) } \right] dx\\&\quad - \int \limits _{\frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}}}^\infty {{f_{{{\left| {{h_{SP}}} \right| }^2}}}\left( y \right) } \left[ {1 - {F_{{{\left| {{h_{S{D_2}}}} \right| }^2}}}\left( {\frac{{xy{\mu _2}}}{{{\rho _Q}\left( {\alpha - x\left( {1 - \alpha } \right) } \right) }}} \right) } \right] dx\\&= 1 - \left( {1 - {e^{ - \frac{{{\rho _Q}}}{{{\Omega _{{h_{SP}}}}{{{\bar{\rho }} }_{RSU}}}}}}} \right) {e^{ - \frac{{x{\mu _2}}}{{{\Omega _{{h_{S{D_2}}}}}{{{\bar{\rho }} }_{RSU}}\left( {\alpha - x\left( {1 - \alpha } \right) } \right) }}}}\\&\quad - \frac{{{{\mathcal {X}}}\left( {\alpha - x\left( {1 - \alpha } \right) } \right) }}{{\left( {{{\mathcal {X}}}\left( {\alpha - x\left( {1 - \alpha } \right) } \right) + x} \right) }}{e^{ - \left( {\frac{1}{{{\Omega _{{h_{SP}}}}}} + \frac{{x{\mu _2}}}{{{\Omega _{{h_{S{D_2}}}}}{\rho _Q}\left( {\alpha - x\left( {1 - \alpha } \right) } \right) }}} \right) \frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}}}}dx, \end{aligned} \end{aligned}$$
(41)

where \({{\mathcal {X}}} = \frac{{{\Omega _{{h_{S{D_2}}}}}{\rho _Q}}}{{{\Omega _{{h_{SP}}}}{\mu _2}}}\)

Substituting (41) into (22), \(P_{{D_2}}^{BER}\) is given by

$$\begin{aligned} \begin{aligned} P_{{D_2}}^{BER}&=\frac{{\sqrt{{\bar{b}}} }}{{2\sqrt{2\pi } }}\left[ {\int \limits _0^{\frac{\alpha }{{1 - \alpha }}} {\frac{{{e^{ - \frac{{{\bar{b}}}}{2}x}}}}{{\sqrt{x} }}} dx - \left( {1 - {e^{ - \frac{{{\rho _Q}}}{{{\Omega _{{h_{SP}}}}{{{\bar{\rho }} }_{RSU}}}}}}} \right) } \right. \\&\quad \times \int \limits _0^{\frac{\alpha }{{1 - \alpha }}} {\frac{{{e^{ - \frac{{{\bar{b}}}}{2}x}}}}{{\sqrt{x} }}{e^{ - \frac{{x{\mu _2}}}{{{\Omega _{{h_{S{D_2}}}}}{{{\bar{\rho }} }_{RSU}}\left( {\alpha - x\left( {1 - \alpha } \right) } \right) }}}}dx} - \int \limits _0^{\frac{\alpha }{{1 - \alpha }}} {\frac{{{e^{ - \frac{{{\bar{b}}}}{2}x}}}}{{\sqrt{x} }}} dx\\&\quad \left. { \times {e^{ - \left( {\frac{1}{{{\Omega _{{h_{SP}}}}}} + \frac{{x{\mu _2}}}{{{\Omega _{{h_{S{D_2}}}}}{\rho _Q}\left( {\alpha - x\left( {1 - \alpha } \right) } \right) }}} \right) \frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}}}}\frac{{{{\mathcal {X}}}\left( {\alpha - x\left( {1 - \alpha } \right) } \right) }}{{\left( {{{\mathcal {X}}}\left( {\alpha - x\left( {1 - \alpha } \right) } \right) + x} \right) }}} \right] , \end{aligned} \end{aligned}$$
(42)

The first integral in (42) can be easily obtained as in [29, Eq. (3.361.1)]. For the second integral, setting \(t = \frac{{2\left( {1 - \alpha } \right) x}}{\alpha }\) results in \(x = \frac{{\alpha \left( {t + 1} \right) }}{{2\left( {1 - \alpha } \right) }}\). Hence, (42) is given as

$$\begin{aligned} \begin{array}{l} P_{{D_2}}^{BER} = \frac{{\sqrt{{\bar{b}}} }}{{2\sqrt{2\pi } }}\left[ {\sqrt{\frac{{2\pi }}{{{\bar{b}}}}} \Phi \left( {\sqrt{\frac{{\alpha {\bar{b}}}}{{2\left( {1 - \alpha } \right) }}} } \right) - \frac{{\phi \alpha }}{{2\left( {1 - \alpha } \right) }}\int \limits _{ - 1}^1 {\frac{{{e^{ - \frac{{{\bar{b}}{{\mathcal {S}}}\left( t \right) }}{2} - \frac{{{{\mathcal {S}}}\left( t \right) {\mu _2}}}{{{\Omega _{{h_{S{D_2}}}}}{{{\bar{\rho }} }_{RSU}}\left( {\alpha - {{\mathcal {S}}}\left( t \right) \left( {1 - \alpha } \right) } \right) }}}}}}{{\sqrt{{{\mathcal {S}}}\left( t \right) } }}dt} } \right. \\ \left. { - \frac{\alpha }{{2\left( {1 - \alpha } \right) }}\int \limits _{ - 1}^1 {\frac{{{{\mathcal {X}}}\left( {\alpha - {{\mathcal {S}}}\left( t \right) \left( {1 - \alpha } \right) } \right) {e^{ - \frac{{{\bar{b}}{{\mathcal {S}}}\left( t \right) }}{2} - \left( {\frac{1}{{{\Omega _{{h_{SP}}}}}} + \frac{{{{\mathcal {S}}}\left( t \right) {\mu _2}}}{{{\Omega _{{h_{S{D_2}}}}}{\rho _Q}\left( {\alpha - {{\mathcal {S}}}\left( t \right) \left( {1 - \alpha } \right) } \right) }}} \right) \frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}}}}}}{{\left[ {\mathcal{X}\left( {\alpha - {{\mathcal {S}}}\left( t \right) \left( {1 - \alpha } \right) } \right) + {{\mathcal {S}}}\left( t \right) } \right] \sqrt{\mathcal{S}\left( t \right) } }}} dt} \right] , \end{array} \end{aligned}$$
(43)

where \(\phi = \left( {1 - {e^{ - \frac{{{\rho _Q}}}{{{\Omega _{{h_{SP}}}}{{{\bar{\rho }} }_{RSU}}}}}}} \right)\), \({{\mathcal {S}}}\left( t \right) = \frac{{\alpha \left( {t + 1} \right) }}{{2\left( {1 - \alpha } \right) }}\) and \(\Phi \left( . \right)\) is the Error function [29, Eq. (3.321.1)]

Using the Gaussian-Chebyshev quadrature method in [32], we can obtain \(P_{{D_2}}^{BER}\) as

$$\begin{aligned} \begin{aligned} P_{{D_2}}^{BER} \approx&\frac{{\sqrt{{\bar{b}}} }}{{2\sqrt{2\pi } }}\left[ {\sqrt{\frac{{2\pi }}{{{\bar{b}}}}} \Phi \left( {\sqrt{\frac{{\alpha {\bar{b}}}}{{2\left( {1 - \alpha } \right) }}} } \right) - \phi \sum \limits _{t = 1}^T {\frac{{\pi \alpha \sqrt{1 - \varphi _t^2} }}{{2T\left( {1 - \alpha } \right) \sqrt{{{\mathcal {S}}}\left( {{\varphi _t}} \right) } }}} } \right. \\&\times {e^{ - {{\mathcal {S}}}\left( {{\varphi _t}} \right) \left( {\frac{{{\bar{b}}}}{2} + \frac{{{\mu _2}}}{{{\Omega _{{h_{S{D_2}}}}}{{{\bar{\rho }} }_{RSU}}\left( {\alpha - {{\mathcal {S}}}\left( {{\varphi _t}} \right) \left( {1 - \alpha } \right) } \right) }}} \right) }} - \sum \limits _{t = 1}^T {\frac{{\pi \alpha \sqrt{1 - \varphi _t^2} {{\mathcal {X}}}}}{{2T\left( {1 - \alpha } \right) \sqrt{{{\mathcal {S}}}\left( {{\varphi _t}} \right) } }}} \\&\left. { - \frac{{\left[ {\alpha - {{\mathcal {S}}}\left( {{\varphi _t}} \right) \left( {1 - \alpha } \right) } \right] {e^{ - \frac{{\bar{b}{{\mathcal {S}}}\left( {{\varphi _t}} \right) }}{2} - \left( {\frac{1}{{{\Omega _{{h_{SP}}}}}} + \frac{{{{\mathcal {S}}}\left( {{\varphi _t}} \right) {\mu _2}}}{{{\Omega _{{h_{S{D_2}}}}}{\rho _Q}\left( {\alpha - {{\mathcal {S}}}\left( {{\varphi _t}} \right) \left( {1 - \alpha } \right) } \right) }}} \right) \frac{{{\rho _Q}}}{{{{{\bar{\rho }} }_{RSU}}}}}}}}{{\left[ {{{\mathcal {X}}}\left( {\alpha - {{\mathcal {S}}}\left( {{\varphi _t}} \right) \left( {1 - \alpha } \right) } \right) + \mathcal{S}\left( {{\varphi _t}} \right) } \right] }}} \right] , \end{aligned} \end{aligned}$$
(44)

where \({\varphi _t} = \cos \left( {\frac{{2t - 1}}{{2T}}\pi } \right)\)

The proposition 3 is completed.

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Le, CB., Do, DT., Zaharis, Z.D. et al. System Performance Analysis in Cognitive Radio-Aided NOMA Network: An Application to Vehicle-to-Everything Communications. Wireless Pers Commun 120, 1975–2000 (2021). https://doi.org/10.1007/s11277-021-08273-x

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Keywords

  • Bit error rate (BER)
  • Non-orthogonal multiple access (NOMA)
  • Vehicle-to-everything
  • Outage probability