Outage behavior of the downlink reconfigurable intelligent surfaces-aided cooperative non-orthogonal multiple access network over Nakagami-m fading channels

Abstract

Although the non-orthogonal multiple access network (NOMA) network has already been used in 5G in wireless communication, it still needs to be supplied with large enough energy, especially for the amplify-and-forward (AF) process. Reconfigurable intelligence surface (RIS)-aided NOMA answers this weakness because it can relay without an AF process so that it could reduce the energy required for NOMA. This is conducted by vigorously relaying communication signals via transmitted signal reflection. In addition, the decode-and-forward process is dispatched to increase the signal coverage performance. It is thus expected that the RIS-aided NOMA will substantially ensure the obtainment of a lower outage probability towards the signal to noise ratio, relative channel estimation error, and distance between a base station and near users. Despite its potential, RIS might encounter a challenge to being effectively incorporated with communication networks regarding channel estimation error. This study evaluates the RIS-aided NOMA to address the issues mentioned above, and carefully derives the closest-form expressions of outage probability for a pair of users by applying perfect channel statistic information over Nakagami-m fading channel. The performance of cooperative relaying scenarios during outages is thoroughly analyzed. According to the simulation results, RIS-aided NOMA has a lower outage probability than conventional NOMA. Additionally, the optimal location for user relaying in RIS-aided NOMA and conventional NOMA networks should be closest to the base station.

Appendices

Appendix A

1.1 Proof of proposition: $y_{{SU_{1} }}$

It is defined, $s(t) = \sqrt {a_{1} P} x_{1} (t) + \sqrt {a_{2} P_{s} } x_{2} (t)$

$$y_{{SU_{1} }} = \left[ {h_{{D_{1} }} + h_{1l}^{H} \phi g_{1l} } \right]\left( {\sqrt {a_{1} P_{s} } x_{1} + \sqrt {a_{2} P_{s} } x_{2} } \right) + n_{{U_{1} }}$$

(56)

$$y_{{SU_{1} }} = \left[ {h_{{D_{1} }} + \alpha \sum\nolimits_{l = 1}^{L} {h_{1l}^{H} \phi g_{1l} } } \right]{\text{s}}({\text{t}}){\text{ + n}}_{{{\text{U}}_{{1}} }}$$

(57)

$$y_{{SU_{1} }} = \left[ {h_{{D_{1} }} + \alpha \sum\nolimits_{l = 1}^{L} {h_{1l} g_{1l} e^{{j\theta_{l} }} } } \right]{\text{s}}({\text{t}}){\text{ + n}}_{{{\text{U}}_{{1}} }}$$

(58)

$$y_{{SU_{1} }} = \, \left( {\left| {\hat{h}_{{D_{1} }} + e_{{D_{1} }} } \right|} \right){\text{s}}({\text{t}}){ + }\left( {\sum\nolimits_{{\text{l = 1}}}^{{\text{L}}} {\left| {{\hat{\text{h}}}_{{{\text{1l}}}} {\text{ + e}}_{{1}} } \right|\left| {{\hat{\text{g}}}_{{{\text{1l}}}} {\text{ + e}}_{{{\text{1l}}}} } \right|} } \right){\text{s}}({\text{t}}){\text{ + n}}_{{{\text{U1}}}}$$

(59)

Appendix B

2.1 Proof of proposition: $\tau_{1}$

If it is defined λ₁ a parameter which indicates path attenuation and noise due to the using of RIS-aided NOMA at U₁, where $\lambda_{1} = \eta_{{U_{1} }} d_{{U_{1} }}^{ - \chi } \rho_{s} + \eta_{SR} /L(d_{SR}^{ - \chi } )\eta_{{RU_{1} }} /L(d_{{RU_{1} }}^{ - \chi } )\rho_{s} + 1$.

Next, it is defined $\tau_{1} = \left( {\left| {\left| {\hat{h}_{{D_{1} }} } \right| + \sum\nolimits_{l = 1}^{L} {\left| {\hat{h}_{l} } \right|\left| {\hat{g}_{1l} } \right|} } \right|^{2} } \right)$, then

$$\frac{{\left( {\left| {\left| {\hat{h}_{{D_{1} }} } \right| + \sum\nolimits_{l = 1}^{L} {\left| {\hat{h}_{l} } \right|\left| {\hat{g}_{1l} } \right|} } \right|^{2} } \right)a_{1} \rho_{s} }}{{\eta_{{SU_{1} }} \left( { \, d_{{SU_{1} }}^{ - \chi } } \right)\rho_{s} + \frac{{\eta_{{SR_{1} }} }}{L}\left( { \, d_{{SR_{1} }}^{ - \chi } } \right) \times \frac{{\eta_{{RU_{1} }} }}{L}\left( { \, d_{{RU_{1} }}^{ - \chi } } \right)\rho_{s} + 1}} \ge \rho_{{Th_{1} }}$$

(60)

$$\frac{{\tau_{1} a_{1} \rho_{s} }}{{\lambda_{1} }} \ge \rho_{{Th_{1} }}$$

(61)

$$\tau_{1} a_{1} \rho_{s} \ge \lambda_{1} \rho_{{Th_{1} }}$$

(62)

$$\tau_{1} \ge \frac{{\lambda_{1} \rho_{{Th_{1} }} }}{{a_{1} \rho_{s} }} \to \tau_{1} = \frac{{\lambda_{1} \rho_{{Th_{1} }} }}{{a_{1} \rho_{s} }}$$

(63)

Appendix C

3.1 Proof of proposition: $\tau_{2}$

By using the same assumption in Appendix B, it is defined a parameter $\lambda_{1} = \eta_{{U_{1} }} d_{{U_{1} }}^{ - \chi } \rho_{s} + \eta_{SR} /L(d_{SR}^{ - \chi } )\eta_{{RU_{1} }} /L(d_{{RU_{1} }}^{ - \chi } )\rho_{s} + 1$.

Next, it is defined $\tau_{2} = \left( {\left| {\left| {\hat{h}_{{D_{1} }} } \right| + \sum\nolimits_{l = 1}^{L} {\left| {\hat{h}_{l} } \right|\left| {\hat{g}_{1l} } \right|} } \right|^{2} } \right)$, then

$$\frac{{\left( {\left| {\left| {\hat{h}_{{D_{1} }} } \right| + \sum\nolimits_{l = 1}^{L} {\left| {\hat{h}_{l} } \right|\left| {\hat{g}_{1l} } \right|} } \right|^{2} } \right)a_{2} \rho_{s} }}{{\left( {\left| {\left| {\hat{h}_{{D_{1} }} } \right| + \sum\nolimits_{l = 1}^{L} {\left| {\hat{h}_{l} } \right|\left| {\hat{g}_{1l} } \right|} } \right|^{2} } \right)a_{1} \rho_{s} + \eta_{{SU_{1} }} \left( { \, d_{{SU_{1} }}^{ - \chi } } \right)\rho_{s} + \frac{{\eta_{{SR_{1} }} }}{L}\left( { \, d_{{SR_{1} }}^{ - \chi } } \right) \times \frac{{\eta_{{RU_{1} }} }}{L}\left( { \, d_{{RU_{1} }}^{ - \chi } } \right)\rho_{s} + 1}} \ge \rho_{{Th_{2} }}$$

(64)

$$\tau_{2} a_{2} \rho_{s} \ge \rho_{{Th_{2} }} \left( {\tau_{2} a_{1} \rho_{s} + \lambda_{1} } \right)$$

(65)

$$\tau_{2} a_{2} \rho_{s} \ge \rho_{{Th_{2} }} \tau_{2} a_{1} \rho_{s} + \rho_{{Th_{2} }} \lambda_{1}$$

(66)

$$\tau_{2} a_{2} \rho_{s} - \rho_{{Th_{2} }} \tau_{2} a_{1} \rho_{s} \ge \rho_{{Th_{2} }} \lambda_{1}$$

(67)

$$\tau_{2} \rho_{s} \left( {a_{2} - a_{1} \rho_{{Th_{2} }} } \right) \ge \rho_{{Th_{2} }} \lambda_{1}$$

(68)

$$\tau_{2} \rho_{s} \ge \frac{{\rho_{{Th_{2} }} \lambda_{1} }}{{\left( {a_{2} - a_{1} \rho_{{Th_{2} }} } \right)}}$$

(69)

$$\tau_{2} \ge \frac{{\rho_{{Th_{2} }} \lambda_{1} }}{{\left( {a_{2} - a_{1} \rho_{{Th_{2} }} } \right)\rho_{s} }} \to \tau_{2} = \frac{{\rho_{{Th_{2} }} \lambda_{1} }}{{\left( {a_{2} - a_{1} \rho_{{Th_{2} }} } \right)\rho_{s} }}$$

(70)

Appendix D

4.1 Proof of proposition: ${\rm E}\{ X_{v} \}$

It is defined, $X_{v} = |h_{{D_{v} }} + h_{vl}^{H} \phi g_{vl} |$, then its could be solved by the independence of channels and assuming arbitrary phase shifts as follows:

$${\rm E}\left\{ {X_{\upsilon } } \right\} = {\rm E}\left\{ {\left| {h_{{D_{\upsilon } }} + h_{\upsilon l}^{H} \Phi g_{\upsilon l} } \right|^{2} } \right\}$$

(71)

$${\rm E}\left\{ {X_{\upsilon } } \right\} = {\rm E}\left\{ {\left| {h_{{D_{\upsilon } }} } \right|^{2} } \right\} + {\rm E}\left\{ {\left| {h_{\upsilon l}^{H} \Phi g_{\upsilon l} } \right|^{2} } \right\}$$

(72)

$${\rm E}\left\{ {X_{\upsilon } } \right\} = \beta_{SU} + L\beta_{SR} \beta_{RU}$$

(73)

$${\rm E}\left\{ {\left| {X_{\upsilon } } \right|^{2} } \right\} = {\rm E}\left\{ {\left| {\left| {h_{{D_{\upsilon } }} } \right|^{2} + \left| {h_{\upsilon l}^{H} \Phi g_{\upsilon l} } \right|^{2} } \right|^{2} } \right\}$$

(74)

$${\rm E}\left\{ {\left| {X_{\upsilon } } \right|^{2} } \right\} = {\rm E}\left\{ {\left| {\left| {h_{{D_{\upsilon } }} } \right|^{2} + h_{{D_{\upsilon } }}^{*} h_{\upsilon l}^{H} \Phi g_{\upsilon l} + h_{{D_{\upsilon } }} g_{\upsilon l}^{H} \Phi^{H} h_{\upsilon l} + \left| {h_{\upsilon l}^{H} \Phi g_{\upsilon l} } \right|^{2} } \right|^{2} } \right\}$$

(75)

$${\rm E}\left\{ {\left| {X_{\upsilon } } \right|^{2} } \right\} = {\rm E}\left\{ {\left| {a + b + c + d} \right|^{2} } \right\} = {\rm E}\left\{ {\left| a \right|^{2} } \right\} + {\rm E}\left\{ {\left| b \right|^{2} } \right\} + {\rm E}\left\{ {\left| c \right|^{2} } \right\} + 2{\rm E}\left\{ {ad} \right\} + {\rm E}\left\{ {\left| d \right|^{2} } \right\}$$

(76)

$${\rm E}\left\{ {\left| a \right|^{2} } \right\} = 2\beta^{2}_{{SU_{1} }}$$

(77)

$${\rm E}\left\{ {\left| b \right|^{2} } \right\} = {\rm E}\left\{ {\left| c \right|^{2} } \right\} = L\beta_{SU} \beta_{SR} \beta_{RU}$$

(78)

$${\rm E}\left\{ {\left| {ad} \right|} \right\} = L\beta_{SU} \beta_{SR} \beta_{RU}$$

(79)

$${\rm E}\left\{ {\left| d \right|^{2} } \right\} = \left( {2L^{2} + 2L} \right)\beta_{RU}^{2} \beta_{SU}^{2}$$

(80)

$${\rm E}\left\{ {\left| {X_{\upsilon } } \right|^{2} } \right\} = 2\beta_{SU}^{2} + 4L\beta_{SU} \beta_{SR} \beta_{RU} + \left( {2L^{2} + 2L} \right)\beta_{RU}^{2} \beta_{SR}^{2}$$

(81)

$$Var\left\{ {X_{\upsilon } } \right\} = {\rm E}\left\{ {\left| {X_{\upsilon } } \right|^{2} } \right\} - \left| {{\rm E}\left\{ {X_{\upsilon } } \right\}} \right|^{2}$$

(82)

$$Var\left\{ {X_{\upsilon } } \right\} = \beta_{SU}^{2} + 2L\beta_{SU} \beta_{SR} \beta_{RU} + \left( {L^{2} + 2L} \right)\beta_{SR}^{2} \beta_{RU}^{2}$$

(83)

$$m = \frac{{{\rm E}\left\{ {\left| {X_{\upsilon } } \right|^{2} } \right\}}}{{Var\left\{ {X_{\upsilon } } \right\}}}$$

(84)

$$m = \frac{{2\beta_{SU}^{2} + 4L\beta_{SU} \beta_{SR} \beta_{RU} + \left( {2L^{2} + 2L} \right)\beta_{SR}^{2} \beta_{RU}^{2} }}{{\beta_{SU}^{2} + 2L\beta_{SU} \beta_{SR} \beta_{RU} + \left( {L^{2} + 2L} \right)\beta_{SR}^{2} \beta_{RU}^{2} }}$$

(85)

If $X_{\upsilon } = \hat{X}_{\upsilon } + e_{\upsilon }$ then the shape factor of the channel using RIS-aided NOMA in the system model in Fig. (1) by modifying is obtained as follows.

It is defined, $\kappa_{v1} = \beta_{{SU_{\upsilon } }} + \eta_{{SU_{\upsilon } }} \left( {d_{{SU_{\upsilon } }} } \right)^{ - x} ;\kappa_{v2} = \beta_{{SR_{\upsilon } }} + \eta_{{SR_{\upsilon } }} \left( {d_{{SR_{\upsilon } }} } \right)^{ - x} ;\kappa_{v3} = \beta_{{RU_{\upsilon } }} + \eta_{{RU_{\upsilon } }} \left( {d_{{RU_{\upsilon } }} } \right)^{ - x}$, so thatby modifying Eq. (84) could be rewritten as Eq. (86).

$$m_{\upsilon } = \frac{{2\left( {\kappa_{v1} } \right)^{2} + 4L\left( {\kappa_{v1} } \right)\left( {\kappa_{v2} } \right)\left( {\kappa_{v3} } \right) + \left( {2L^{2} + 2L} \right)\left( {\kappa_{v2} } \right)^{2} \left( {\kappa_{v3} } \right)^{2} }}{{\left( {\kappa_{v1} } \right)^{2} + 2L\left( {\kappa_{v1} } \right)L\left( {\kappa_{v2} } \right)\left( {\kappa_{v3} } \right) + \left( {L^{2} + 2L} \right)\left( {\kappa_{v2} } \right)^{2} \left( {\kappa_{v3} } \right)^{2} }}$$

(86)

Appendix E

5.1 Proof of proposition: ${\text{P}}_{{U_{{2}} }}$

According to Eq. (35), it could be derived as follows.

$$P_{{U_{2} }} = \Pr \left( {\rho_{{1,U_{2} }} < \rho_{{Th_{2} }} } \right) + \Pr \left( {\rho_{{2,U_{2} }} < \rho_{{Th_{2} }} ,\rho_{{1,U_{2} }} > \rho_{{Th_{2} }} } \right)$$

(87)

$$P_{{U_{2} }} = \Pr \left( {\left| {X_{1} } \right|^{2} < \tau_{2} } \right) + \Pr \left( {\left| {X_{1} } \right|^{2} > \tau_{2} ,\left| {X_{1} } \right|^{2} < \tau_{3} } \right)$$

(88)

$$P_{{U_{2} }} = \Pr \left( {\left| {X_{1} } \right|^{2} < \tau_{2} } \right) + \Pr \left( {\left| {X_{1} } \right|^{2} > \tau_{2} } \right).\Pr \left( {\left| {X_{1} } \right|^{2} < \tau_{3} } \right)$$

(89)

$$P_{{U_{2} }} = 1 - \Pr \left( {\left| {X_{1} } \right|^{2} > \tau_{2} } \right)\left( {1 - \Pr \left( {\left| {X_{1} } \right|^{2} < \tau_{3} } \right)} \right)$$

(90)

$$P_{{U_{2} }} = 1 - \Pr \left( {\left| {X_{1} } \right|^{2} > \tau_{2} } \right).\Pr \left( {\left| {X_{1} } \right|^{2} > \tau_{3} } \right)$$

(91)

$$P_{{U_{2} }} = 1 - e^{{ - \delta_{1} \tau_{2} }} \sum\limits_{j - 0}^{{m_{1} - 1}} {\frac{{\left( {\delta_{1} \tau_{2} } \right)^{j} }}{j!}} .e^{{ - \delta_{2} \tau_{3} }} \sum\limits_{k - 0}^{{m_{1} - 1}} {\frac{{\left( {\delta_{2} \tau_{3} } \right)^{k} }}{k!}}$$

(92)