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Joint of full-duplex relay, non-linear energy harvesting and multiple access in performance improvement of cell-edge user in heterogeneous networks

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Abstract

To serve massive connections in heterogeneous networks with respect to higher energy efficiency, we focus on new paradigm in order to achieve multiple access and performance improvement at cell-edge area. The power domain based non-orthogonal multiple access is introduced to address such problem. In particular, this paper studies a self-energy relay together with full-duplex scheme to implement cooperative power domain based non-orthogonal multiple access in small-cell system of the heterogeneous networks. In such small-cell network, a nearby user can be employed as a decode-and-forward with self-energy recycling protocol to assist a far power domain based non-orthogonal multiple access user (cell-edge user). The relay harvests energy from dedicated energy signal sent by a base station, while it still reuses energy from loop self-interference signal. To characterize the performance of the proposed system with respect to where meets weak signal condition, numerous expressions of exact outage probability for far power domain based non-orthogonal multiple access user is derived. Several practical scenarios are performed in three different schemes related to how energy harvesting architecture can be achieved. Based on analytical results, the optimal throughput achieved by the cell-edge user in small-cell network can be observed. Numerical results are presented to validate the accuracy of the derived results.

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Appendices

Appendix 1

Proof of Proposition 1

The outage probability can be expressed as

We first recall expression of \( I_1 \) as

$$\begin{aligned} {I_1} = \Pr \left( {{{\left| {{h_r}} \right| }^2} < A,B \le {{\left| {{h_1}} \right| }^2} \le C,{{\left| {{h_1}} \right| }^2}{{\left| {{h_2}} \right| }^2} > D} \right) \end{aligned}$$
(38)

Then, after performing manipulations, it can be given by

$$\begin{aligned} {I_1} = \left\{ {\begin{array}{*{20}{c}} {0,{{\left| {{h_2}} \right| }^2}< \frac{D}{C}}\\ {\Pr \left( {{{\left| {{h_r}} \right| }^2}< A} \right) \Pr \left( {\frac{D}{{{{\left| {{h_2}} \right| }^2}}} \le {{\left| {{h_1}} \right| }^2} \le C} \right) ,\frac{D}{C} \le {{\left| {{h_2}} \right| }^2} \le \frac{D}{B}}\\ {\Pr \left( {{{\left| {{h_r}} \right| }^2} < A} \right) \Pr \left( {B \le {{\left| {{h_1}} \right| }^2} \le C} \right) ,{{\left| {{h_2}} \right| }^2} > \frac{D}{B}} \end{array}} \right. \end{aligned}$$
(39)

where \(A = \frac{{\left( {{a_2} - {\gamma _{th2}}{a_1}} \right) }}{{{\gamma _{th2}}a}}\), \(B = \frac{{d_1^m{P_{\min }}}}{{{P_S}}}\), \(C = \frac{{d_1^m{P_{{\mathrm{max}}}}}}{{{P_S}}}\) and \(D = \frac{{{\gamma _{th2}}d_1^m\xi _d}}{{a{P_S}}}\) . Considering x as integration variable, by employing exponential distribution for \({h_i}\), i.e. the CDF and PDF as considerations in (1),(2). As a result, \({I_1}\) can be formulated as

$$\begin{aligned} \begin{array}{l} {I_1} = \left[ {1 - \exp \left( { - \frac{A}{{{\lambda _r}}}} \right) } \right] \int \limits _{\frac{D}{C}}^{\frac{D}{B}} {\frac{1}{{{\lambda _2}}}\exp \left( { - \frac{x}{{{\lambda _2}}} - \frac{D}{{x{\lambda _1}}}} \right) } dx\\ \quad - \left[ {1 - \exp \left( { - \frac{A}{{{\lambda _r}}}} \right) } \right] \exp \left( { - \frac{C}{{{\lambda _1}}}} \right) \left[ \exp \left( { - \frac{D}{{{\lambda _2}C}}} \right) \right. \\ \left. \quad - \exp \left( { - \frac{D}{{{\lambda _2}B}}} \right) \right] \\ \quad + \left[ {1 - \exp \left( { - \frac{A}{{{\lambda _r}}}} \right) } \right] \exp \left( { - \frac{D}{{{\lambda _2}B}}} \right) \left( \exp \left( { - \frac{B}{{{\lambda _1}}}} \right) \right. \\ \left. - \exp \left( { - \frac{C}{{{\lambda _1}}}} \right) \right) \end{array} \end{aligned}$$
(40)

Similar to \({I_1}\), we can epress \({I_2}\) as:

$$\begin{aligned} {I_2} = \Pr \left( {{P_{input}}> {P_{{\mathrm{max}}}},{\gamma _{1,x2}}> {\gamma _{th2}},{\gamma _2} > {\gamma _{th2}}} \right) \end{aligned}$$
(41)

It is noted that \({P_{input}} = \frac{{{P_S}{{\left| {{h_1}} \right| }^2}}}{{d_1^m}}\), (11) and (12), \({I_2}\) can be further expressed as

$$\begin{aligned} \begin{array}{l} {I_2} = \Pr \left( \frac{{{P_S}{{\left| {{h_1}} \right| }^2}}}{{d_1^m}}> {P_{{\mathrm{max}}}},\frac{{{a_2}{P_S}{{\left| {{h_1}} \right| }^2}}}{{{a_1}{P_S}{{\left| {{h_1}} \right| }^2} + d_1^ma{P_{{\mathrm{max}}}}{{\left| {{h_r}} \right| }^2}}} \right. \\ \quad \left.> {\gamma _{th2}},\frac{{a{P_{{\mathrm{max}}}}{{\left| {{h_2}} \right| }^2}}}{{\xi _d}}> {\gamma _{th2}} \right) \\ \quad = \Pr \left( {{{\left| {{h_1}} \right| }^2}> C,{{\left| {{h_1}} \right| }^2}> \frac{C}{A}{{\left| {{h_r}} \right| }^2},{{\left| {{h_2}} \right| }^2} > E} \right) \end{array} \end{aligned}$$
(42)

where \(E = \frac{{\xi _d{\gamma _{th2}}}}{{a{P_{{\mathrm{max}}}}}}\).

In this step, (41) can be transformed as

$$\begin{aligned} {I_2} = \left\{ {\begin{array}{*{20}{c}} {Pr\left( {{{\left| {{h_2}} \right| }^2}> E} \right) \Pr \left( {{{\left| {{h_1}} \right| }^2}> C} \right) ,{{\left| {{h_r}} \right| }^2} < A}\\ {Pr\left( {{{\left| {{h_2}} \right| }^2}> E} \right) \Pr \left( {{{\left| {{h_1}} \right| }^2}> \frac{C}{A}{{\left| {{h_r}} \right| }^2}} \right) ,{{\left| {{h_r}} \right| }^2} > A} \end{array}} \right. \end{aligned}$$
(43)

With help (1) and (2), \({I_2}\) can be shown in closed-form as:

$$\begin{aligned} {I_2}&= {} \exp \left( { - \frac{E}{{{\lambda _2}}}} \right) \left[ \exp \left( { - \frac{C}{{{\lambda _1}}}} \right) \left( {1 - \exp \left( { - \frac{A}{{{\lambda _r}}}} \right) } \right) \right. \\&\left. + \frac{1}{{1 + \frac{{{\lambda _r}C}}{{{\lambda _1}A}}}}\exp \left( { - \frac{A}{{{\lambda _r}}} - \frac{C}{{{\lambda _1}}}} \right) \right] \end{aligned}$$
(44)

This is end of the proof. \(\square \)

Appendix 2

In the HDNL mode, the components of outage probability can be clarified as below. It is noted that such outage event is given by

$$\begin{aligned} OP_{out}^{HD} = 1 - {L_1} - {L_2} \end{aligned}$$
(45)

To address such outage behavior, \(L_1\) and \(L_2\) can be defined as:

$$\begin{aligned} {L_1} = \Pr \left( {{P_{\min }} \le {P_{input}} \le {P_{\max }} \cap {\gamma _{1,x2}}> \gamma _{th2}^{HD} \cap {\gamma _2} > \gamma _{th2}^{HD}} \right) \end{aligned}$$
(46)

and

$$\begin{aligned} {L_2} = \Pr \left( {{P_{input}}> {P_{{\mathrm{max}}}},{\gamma _{1,x2}}> {\gamma _{th2}^{HD}},{\gamma _2} > {\gamma _{th2}^{HD}}} \right) \end{aligned}$$
(47)

With help (19), (20) and putting them into (46). Then, we can be obtained as:

$$\begin{aligned} {L_1} = \Pr \left( {B \le {{\left| {{h_1}} \right| }^2} \le C \cap {{\left| {{h_1}} \right| }^2}> F \cap {{\left| {{h_2}} \right| }^2} > \frac{D^{HD}}{{{{\left| {{h_1}} \right| }^2}}}} \right) \end{aligned}$$
(48)

where \({D^{HD}} = \frac{{\gamma _{th2}^{HD}d_1^m\xi _d}}{{a{P_S}}}\), \(F = \frac{{{\gamma _{th2}^{HD}}\xi _r}}{{{a_2}{P_S} - {\gamma _{th2}}{a_1}{P_S}}}\). Then, \(L_1\) can be written as

$$\begin{aligned} {L_1}&= {} \left\{ {\begin{array}{*{20}{c}} {0,{{\left| {{h_1}} \right| }^2} \le \max \left( {{B^{HD}},F} \right) \buildrel \varDelta \over = W}\\ {{{\left| {{h_2}} \right| }^2}> \frac{{{D^{HD}}}}{{{{\left| {{h_1}} \right| }^2}}},W \le {{\left| {{h_1}} \right| }^2} \le C}\\ {{{\left| {{h_2}} \right| }^2}> \frac{{{D^{HD}}}}{{{{\left| {{h_1}} \right| }^2}}},{{\left| {{h_1}} \right| }^2} > C} \end{array}} \right. \end{aligned}$$
(49)
$$\begin{aligned} {L_1}&= {} \int \limits _W^C {\frac{1}{{{\lambda _{1}}}}} {e^{ - \frac{x}{{{\lambda _{1}}}}}}{e^{ - \frac{D}{{{\lambda _{2}}x}}}}dx + \underbrace{\int \limits _C^\infty {{e^{ - \frac{y}{{{\lambda _{1}}}}}}{e^{ - \frac{D}{{{\lambda _{2}}y}}}}dy} }_\psi \end{aligned}$$
(50)

By using the expanded Taylor’s series [43, Eq. 5.90], it follows that \({e^{\left( { - \frac{a}{x}} \right) }} = \sum \nolimits _{k = 0}^{{N_t}} {\frac{{{{\left( { - 1} \right) }^k}}}{{k!}}} {\left( {\frac{a}{x}} \right) ^k}\), where \({N_t} \in \left\{ {1, \ldots ,\infty } \right\} \). Then, \(\psi \) can be calculated as [44]

$$\begin{aligned} \psi&= {} \sum \limits _{k = 0}^{{N_t}} {\frac{{{{\left( { - 1} \right) }^k}}}{{k!}}} \int \limits _C^\infty {{{\left( {\frac{D^{HD}}{{{\lambda _{2}}}}} \right) }^k}{e^{ - \frac{y}{{{\lambda _{1}}}}}}dy} \\&= {} \sum \limits _{k = 0}^{{N_t}} {\frac{{{{\left( { - 1} \right) }^k}}}{{k!}}} {\left( {\frac{D^{HD}}{{{\lambda _{2}}}}} \right) ^k}{\left( {\frac{1}{C}} \right) ^{k - 1}}{E_k}\left( {\frac{C}{{{\lambda _{1}}}}} \right) \end{aligned}$$
(51)

Then, we have new manipulation as:

$$\begin{aligned} \begin{array}{l} {L_1} = \int \limits _0^C {\frac{1}{{{\lambda _1}}}\exp \left( { - \frac{x}{{{\lambda _1}}} - \frac{D^{HD}}{{x{\lambda _2}}}} \right) } dx\\ + \sum \limits _{k = 0}^{{N_t}} {\frac{{{{\left( { - 1} \right) }^k}}}{{k!}}} {\left( {\frac{D^{HD}}{{{\lambda _{2}}}}} \right) ^k}{\left( {\frac{1}{C}} \right) ^{k - 1}}{E_k}\left( {\frac{C}{{{\lambda _{1}}}}} \right) . \end{array} \end{aligned}$$
(52)

Similarly, with help \({P_{input}} = \frac{{{P_S}{{\left| {{h_1}} \right| }^2}}}{{d_1^m}}\), (19) and (20), \(L_2\) can be expressed as

$$\begin{aligned} {L_2}&= {} \Pr \left( \frac{{{P_S}{{\left| {{h_1}} \right| }^2}}}{{d_1^m}}> {P_{{\mathrm{max}}}},\frac{{{a_2}{P_S}{{\left| {{h_1}} \right| }^2}}}{{{a_1}{P_S}{{\left| {{h_1}} \right| }^2} + d_1^m\sigma _r^2}}\right. \\&\left.> {\gamma _{th2}},\frac{{a{P_{{\mathrm{max}}}}{{\left| {{h_2}} \right| }^2}}}{{\xi _d}} > {\gamma _{th2}} \right) . \end{aligned}$$
(53)

It is noted that equation (53) is rewritten by

$$\begin{aligned} {L_2} = \Pr \left( {{{\left| {{h_1}} \right| }^2}> \max \left( {C,F} \right) \buildrel \varDelta \over = \zeta } \right) \Pr \left( {{{\left| {{h_2}} \right| }^2} > {E^{HD}}} \right) , \end{aligned}$$
(54)

where \({E^{HD}} = \frac{{\xi _d\gamma _{th2}^{HD}}}{{a{P_{{\mathrm{max}}}}}}\). Furthermore, equation (54) can be formulated as

$$\begin{aligned} {L_2} = \left( {\frac{1}{{{\lambda _1}{\lambda _2}}}\exp \left( { - \frac{\zeta }{{{\lambda _1}}} - \frac{{{E^{HD}}}}{{{\lambda _2}}}} \right) } \right) \end{aligned}$$
(55)

Finally, we have outage probability for the cell-edge user.

It completes the proof.

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Do, DT., Le, CB. & Le, AT. Joint of full-duplex relay, non-linear energy harvesting and multiple access in performance improvement of cell-edge user in heterogeneous networks. Wireless Netw 26, 6253–6266 (2020). https://doi.org/10.1007/s11276-020-02436-7

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