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Enabling Full-duplex in MEC Networks Using Uplink NOMA in Presence of Hardware Impairments

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Abstract

To satisfy the fast growth of Internet of Things (IoT) or development of applications in the fifth generation (5G) wireless networks, the expected systems need two main functions such as massive connectivity of IoT devices and the low latency. Fortunately, non-orthogonal multiple access (NOMA) has been recommended as a promising approach for 5G networks to satisfy concerned requirements and to significantly improve the network capacity. Together with advances of NOMA scheme, mobile edge computing (MEC) is recognized as one of the key emerging approaches to improve the quality of service and reduce the latency for 5G networks. In order to capture the potential gains of NOMA-aided MEC systems, this paper proposes design of multiple antennas and full-duplex for an edge computing aware NOMA architecture. As expected, such NOMA-aided MEC systems can enjoy the benefits of uplink NOMA in reducing MEC users’ uplink outage probability. To this end, we derive expression of outage probability for signals corresponding two-user model at uplink. Especially, degraded performance might happens in practice under impact of hardware impairment.

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Authors and Affiliations

  1. Faculty of Electronics Technology, Industrial University of Ho Chi Minh City (IUH), No.12, Nguyen Van Bao Street, Go Vap Dist., Ho Chi Minh City, Vietnam

    Minh-Sang Van Nguyen

  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, Thessaloniki, 54124, Greece

    Zaharias D. Zaharis

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

    Constandinos X. Mavromoustakis

  5. Department of Management Science and Technology, Hellenic Mediterranean University, Agios Nikolaos, 72100, Crete, Greece

    George Mastorakis

  6. Department of Electrical and Computer Engineering, Hellenic Mediterranean University, Heraklion, 71500, Crete, Greece

    Evangelos Pallis

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  1. Minh-Sang Van Nguyen
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Correspondence to Dinh-Thuan Do.

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Appendices

APPENDIX: A

Proof of Proposition 1

The closed-form of \({\Lambda _1}\) and \({\Lambda _2}\) in (10). The first component \({\Lambda _1}\) can be computed by

$$\begin{aligned} \begin{array}{l} {\Lambda _1} = 1 - \Pr \left( {\beta _{{U_{n*}}{B_1}}^{\left( {{v_1}} \right) } \ge \varphi _1^{\left( {FD} \right) }} \right) \\ = 1 - \Pr \left( {{{\left| {{f_{1n*}}} \right| }^2} \ge \frac{{\varphi _1^{\left( {FD} \right) }\left( {\left( {{\partial _2} + m_U^2} \right) \mu {{\left| {{f_{2n*}}} \right| }^2} + \sigma \mu {{\left| {{g_1}} \right| }^2} + 1} \right) }}{{\left( {{\partial _1} - \varphi _1^{\left( {FD} \right) }m_U^2} \right) \mu }}} \right) \\ = 1 - \int _0^\infty {\int _0^\infty {\left( {1 - {F_{{{\left| {{f_{1n*}}} \right| }^2}}}\left( {\frac{{\varphi _1^{\left( {FD} \right) }\left( {\left( {{\partial _2} + m_U^2} \right) \mu x + \sigma \mu y + 1} \right) }}{{\left( {{\partial _1} - \varphi _1^{\left( {FD} \right) }m_U^2} \right) \mu }}} \right) } \right) } {f_{{{\left| {{f_{2n*}}} \right| }^2}}}\left( x \right) } \\ \times {f_{{{\left| {{g_1}} \right| }^2}}}\left( y \right) dxdy\\ = 1 - \sum \limits _{n = 1}^N {\sum \limits _{t = 1}^N {\left( \begin{array}{l} N\\ n \end{array} \right) \left( \begin{array}{l} N\\ t \end{array} \right) {{\left( { - 1} \right) }^{n + t - 2}}} } \frac{t}{{{\alpha _{2n}}}}\frac{1}{{{\alpha _{g1}}}}\exp \left( { - \frac{{n\varphi _1^{\left( {FD} \right) }}}{{\left( {{\partial _1} - \varphi _1^{\left( {FD} \right) }m_U^2} \right) \mu {\alpha _{1n}}}}} \right) \\ \times \int _0^\infty {\exp \left( { - \left( {\frac{{\left( {{\partial _2} + m_U^2} \right) n\varphi _1^{\left( {FD} \right) }}}{{\left( {{\partial _1} - \varphi _1^{\left( {FD} \right) }m_U^2} \right) {\alpha _{1n}}}} + \frac{t}{{{\alpha _{2n}}}}} \right) x} \right) dx} \\ \times \int _0^\infty {\exp \left( { - \left( {\frac{{n\varphi _1^{\left( {FD} \right) }\sigma }}{{\left( {{\partial _1} - \varphi _1^{\left( {FD} \right) }m_U^2} \right) {\alpha _{1n}}}} + \frac{1}{{{\alpha _{g1}}}}} \right) y} \right) } dy\\ = 1 - \sum \limits _{n = 1}^N {\sum \limits _{t = 1}^N {\left( \begin{array}{l} N\\ n \end{array} \right) \left( \begin{array}{l} N\\ t \end{array} \right) {{\left( { - 1} \right) }^{n + t - 2}}} } \frac{{\left( {{\partial _1} - \varphi _1^{\left( {FD} \right) }m_U^2} \right) {\alpha _{1n}}t}}{{\left( {{\partial _2} + m_U^2} \right) n\varphi _1^{\left( {FD} \right) }{\alpha _{2n}} + \left( {{\partial _1} - \varphi _1^{\left( {FD} \right) }m_U^2} \right) {\alpha _{1n}}t}}\\ \times \frac{{\left( {{\partial _1} - \varphi _1^{\left( {FD} \right) }m_U^2} \right) {\alpha _{1n}}}}{{n\varphi _1^{\left( {FD} \right) }\sigma {\alpha _{g1}} + \left( {{\partial _1} - \varphi _1^{\left( {FD} \right) }m_U^2} \right) {\alpha _{1n}}}}\exp \left( { - \frac{{n\varphi _1^{\left( {FD} \right) }}}{{\left( {{\partial _1} - \varphi _1^{\left( {FD} \right) }m_U^2} \right) \mu {\alpha _{1n}}}}} \right) . \end{array} \end{aligned}$$
(A.1)

Then, it can be achieved \({\Lambda _2}\) as

$$\begin{aligned} \begin{array}{l} {\Lambda _2} = 1 - \Pr \left( {\beta _{{U_{n*}}{B_1}}^{\left( {{v_2}} \right) } \ge \varphi _2^{\left( {FD} \right) }} \right) \\ = 1 - \Pr \left( {{{\left| {{f_{2n*}}} \right| }^2} \ge \frac{{\varphi _2^{\left( {FD} \right) }\left( {m_U^2\mu {{\left| {{f_{1n*}}} \right| }^2} + \sigma \mu {{\left| {{g_1}} \right| }^2} + 1} \right) }}{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) \mu }}} \right) \\ = 1 - \int _0^\infty {\int _0^\infty {\left( {1 - {F_{{{\left| {{f_{2n*}}} \right| }^2}}}\left( {\frac{{\varphi _2^{\left( {FD} \right) }\left( {m_U^2\mu x + \sigma \mu y + 1} \right) }}{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) \mu }}} \right) } \right) {f_{{{\left| {{f_{1n*}}} \right| }^2}}}\left( x \right) } } \\ \times {f_{{{\left| {{g_1}} \right| }^2}}}\left( y \right) dxdy\\ = 1 - \sum \limits _{n = 1}^N {\sum \limits _{t = 1}^N {\left( \begin{array}{l} N\\ n \end{array} \right) \left( \begin{array}{l} N\\ t \end{array} \right) {{\left( { - 1} \right) }^{n + t - 2}}} } \frac{t}{{{\alpha _{1n}}}}\frac{1}{{{\alpha _{g1}}}}\exp \left( { - \frac{{n\varphi _2^{\left( {FD} \right) }}}{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) \mu {\alpha _{2n}}}}} \right) \\ \times \int _0^\infty {\exp \left( { - \left( {\frac{{n\varphi _2^{\left( {FD} \right) }m_U^2}}{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) {\alpha _{2n}}}} + \frac{t}{{{\alpha _{1n}}}}} \right) x} \right) dx} \\ \times \int _0^\infty {\exp \left( { - \left( {\frac{{n\varphi _2^{\left( {FD} \right) }\sigma }}{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) {\alpha _{2n}}}} + \frac{1}{{{\alpha _{g1}}}}} \right) y} \right) dy} \\ = 1 - \sum \limits _{n = 1}^N {\sum \limits _{t = 1}^N {\left( \begin{array}{l} N\\ n \end{array} \right) \left( \begin{array}{l} N\\ t \end{array} \right) {{\left( { - 1} \right) }^{n + t - 2}}} } \frac{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) {\alpha _{2n}}t}}{{n\varphi _2^{\left( {FD} \right) }m_U^2{\alpha _{1n}} + \left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) {\alpha _{2n}}t}}\\ \times \frac{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) {\alpha _{2n}}}}{{n\varphi _2^{\left( {FD} \right) }\sigma {\alpha _{g1}} + \left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) {\alpha _{2n}}}}\exp \left( { - \frac{{n\varphi _2^{\left( {FD} \right) }}}{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) \mu {\alpha _{2n}}}}} \right) . \end{array} \end{aligned}$$
(A.2)

By plugging (A.1), (A.2) into (10) it can be calculated outage probability \({\mathrm{P}}_{BS1}^{\left( {FD} \right) }\). The proof is completed. \(\square \)

APPENDIX: B

Proof of Proposition 2

The closed-form expression of \({\Xi _1}\) in (15) is given by

$$\begin{aligned} \begin{array}{l} {\Xi _1} = 1 - \Pr \left( {{\beta _{{B_1}{B_2}}} \ge \varphi _2^{\left( {FD} \right) }} \right) \\ = 1 - \Pr \left( {{{\left| q \right| }^2} \ge \frac{{\varphi _2^{\left( {FD} \right) }\left( {\sigma \mu {{\left| {{g_2}} \right| }^2} + 1} \right) }}{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_B^2} \right) \mu }}} \right) \\ = 1 - \int _0^\infty {\left( {1 - {F_{{{\left| q \right| }^2}}}\left( {\frac{{\varphi _2^{\left( {FD} \right) }\left( {\sigma \mu x + 1} \right) }}{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_B^2} \right) \mu }}} \right) } \right) {f_{{{\left| {{g_2}} \right| }^2}}}\left( x \right) dx} \\ = 1 - \frac{t}{{{\alpha _{g2}}}}\exp \left( { - \frac{{\varphi _2^{\left( {FD} \right) }}}{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_B^2} \right) \mu {\alpha _q}}}} \right) \\ \times \int _0^\infty {\exp \left( { - \left( {\frac{{\varphi _2^{\left( {FD} \right) }\sigma }}{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_B^2} \right) {\alpha _q}}} + \frac{1}{{{\alpha _{g2}}}}} \right) x} \right) dx} \\ = 1 - \frac{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_B^2} \right) {\alpha _q}}}{{\varphi _2^{\left( {FD} \right) }\sigma {\alpha _{g2}} + \left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_B^2} \right) {\alpha _q}}}\exp \left( { - \frac{{\varphi _2^{\left( {FD} \right) }}}{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_B^2} \right) \mu {\alpha _q}}}} \right) . \end{array} \end{aligned}$$
(B.1)

Replacing (A.1), (A.2), (B.1) into (15). This completes the proof. \(\square \)

APPENDIX: C

Proof of Proposition 3

The closed-form formula of \({\Theta _1}\) in (20) is computed by

$$\begin{aligned} \begin{array}{l} {\Theta _1} = 1 - \Pr \left( {\beta _{{U_{n*}}{B_1}}^{\left( {{v_2} - ip} \right) } \ge \varphi _2^{\left( {FD} \right) }} \right) \\ = 1 - \Pr \left( {{{\left| {{f_{2n*}}} \right| }^2} \ge \frac{{\varphi _2^{\left( {FD} \right) }\left( {{\partial _1}\mu {{\left| {\widetilde{{f_{1n*}}}} \right| }^2} + m_U^2\mu {{\left| {{f_{1n*}}} \right| }^2} + \sigma \mu {{\left| {{g_1}} \right| }^2} + 1} \right) }}{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) \mu }}} \right) \\ = 1 - \int _0^\infty {\int _0^\infty {\int _0^\infty {\left( {1 - {F_{{{\left| {{f_{2n*}}} \right| }^2}}}\left( {\frac{{\varphi _2^{\left( {FD} \right) }\left( {{\partial _1}\mu x + m_U^2\mu y + \sigma \mu z + 1} \right) }}{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) \mu }}} \right) } \right) } {f_{{{\left| {\widetilde{{f_{1n*}}}} \right| }^2}}}\left( x \right) } } \\ \times {f_{{{\left| {{f_{1n*}}} \right| }^2}}}\left( y \right) {f_{{{\left| {{g_1}} \right| }^2}}}\left( z \right) dxdydz\\ = 1 - \sum \limits _{n = 1}^N {\sum \limits _{t = 1}^N {\sum \limits _{r = 1}^N {\left( \begin{array}{l} N\\ n \end{array} \right) \left( \begin{array}{l} N\\ t \end{array} \right) \left( \begin{array}{l} N\\ r \end{array} \right) {{\left( { - 1} \right) }^{n + t + r - 3}}} } } \\ \times \frac{t}{{\kappa {\alpha _{ip}}}}\frac{r}{{{\alpha _{1n}}}}\frac{1}{{{\alpha _{g1}}}}\exp \left( { - \frac{{n\varphi _2^{\left( {FD} \right) }}}{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) \mu {\alpha _{2n}}}}} \right) \\ \times \int _0^\infty {\exp \left( { - \left( {\frac{{n\varphi _2^{\left( {FD} \right) }{\partial _1}}}{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) {\alpha _{2n}}}} + \frac{t}{{\kappa {\alpha _{ip}}}}} \right) x} \right) dx} \\ \times \int _0^\infty {\exp \left( { - \left( {\frac{{n\varphi _2^{\left( {FD} \right) }m_U^2}}{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) {\alpha _{2n}}}} + \frac{r}{{{\alpha _{1n}}}}} \right) y} \right) dy} \\ \times \int _0^\infty {\exp \left( { - \left( {\frac{{n\varphi _2^{\left( {FD} \right) }\sigma }}{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) {\alpha _{2n}}}} + \frac{1}{{{\alpha _{g1}}}}} \right) z} \right) } dz\\ = 1 - \sum \limits _{n = 1}^N {\sum \limits _{t = 1}^N {\sum \limits _{r = 1}^N {\left( \begin{array}{l} N\\ n \end{array} \right) \left( \begin{array}{l} N\\ t \end{array} \right) \left( \begin{array}{l} N\\ r \end{array} \right) {{\left( { - 1} \right) }^{n + t + r - 3}}} } } \\ \times \frac{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) {\alpha _{2n}}t}}{{n\varphi _2^{\left( {FD} \right) }{\partial _1}\kappa {\alpha _{ip}} + \left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) {\alpha _{2n}}t}}\\ \times \frac{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) {\alpha _{2n}}r}}{{n\varphi _2^{\left( {FD} \right) }m_U^2{\alpha _{1n}} + \left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) {\alpha _{2n}}r}}\\ \frac{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) {\alpha _{2n}}}}{{n\varphi _2^{\left( {FD} \right) }\sigma {\alpha _{g1}} + \left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) {\alpha _{2n}}}}\exp \left( { - \frac{{n\varphi _2^{\left( {FD} \right) }}}{{\left( {{\partial _2} - \varphi _2^{\left( {FD} \right) }m_U^2} \right) \mu {\alpha _{2n}}}}} \right) . \end{array} \end{aligned}$$
(C.1)

By plugging (A.1), (C.1) into (20) it can be calculated the OP at BS1 in term of imperfect SIC. The proof is completed. \(\square \)

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Van Nguyen, MS., Do, DT., Zaharis, Z.D. et al. Enabling Full-duplex in MEC Networks Using Uplink NOMA in Presence of Hardware Impairments. Wireless Pers Commun 120, 1945–1973 (2021). https://doi.org/10.1007/s11277-021-08102-1

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