Cognitive IoT relaying NOMA networks with user clustering and imperfect SIC

Abstract

To implement machine type communications (MTCs) in next generation mobile networks, Narrowband-Internet of Things (NB-IoT) need be studied. Such NB-IoT has been released by the third generation partnership project (3GPP) as a promising method to exhibit extended coverage and low energy consumption, especially for cheap MTC devices. However, the existing NB-IoT using orthogonal multiple access (OMA) scheme which cannot provide connectivity for a massive MTC devices. In this paper, we consider NB-IoT using non-orthogonal multiple access (NOMA). We find that power allocation factors for devices in clustering users as main enabler to exhibit differences in term of performance of devices. The worse situation of imperfect successive interference cancellation (SIC) is further studied due to main reason regarding degraded system performance. We derive closed-form expressions of outage probability for devices in a cluster to highlight main parameters affecting quality of devices. We verify our mathematical analysis via Monte-Carlo simulations.

Appendices

Appendix A: Proof of proposition 1

With the help Eqs. 1, 4 and 5 we can write $P_{U_{2} }$ as below

$$ \begin{array}{@{}rcl@{}} P_{U_{2} }^{P} &=& 1 - {\text{OP}} \left( \frac{\left| {g_{B2}} \right|^{2} K_{1} \eta_{B}} {\left| {g_{B2} } \right|^{2} K_{2} \eta_{B} + \left| {g_{R2} } \right|^{2} \eta_{R} + 1}\right.\\ &&\left. > \gamma_{th1} ,\frac{\left| {g_{B2} } \right|^{2} K_{2} \eta_{BS}} {\left|g_{R2} \right|^{2} \eta_{R} + 1} > \gamma_{th2} \right) \\ &=& 1 - \underbrace{\text{OP} \left( \left|g_{BP} \right|^{2} < \frac{\eta_{D}}{\eta},\left| {g_{B2}} \right|^{2} > \frac{\phi \left( \left| g_{R2} \right|^{2} \eta_{R} + 1 \right)}{\eta} \right)_{A_{1}}} \\ &-& \underbrace{\text{OP} \left( \left|g_{BP} \right|^{2} > \frac{\eta_{D}} {\eta},\left| g_{B2} \right|^{2} > \frac{\phi \left( \left|g_{R2}\right|^{2} \eta_{R} + 1 \right) \left| g_{BP} \right|^{2}} {\eta_{D}} \right).}_{A_{2}} \end{array} $$

(34)

Moreover, A₁ can be rewritten as

$$ \begin{array}{@{}rcl@{}} A_{1} = {\text{OP}} \left( {\left| {g_{BP} } \right|^{2} < \frac{{\eta_{D}}} {\eta },\left| {g_{B2} } \right|^{2} > \frac{{\phi \left( {\left| {g_{R2} } \right|^{2} \eta_{R} + 1} \right)}} {\eta }} \right) \\ = \int\limits_{0}^{\frac{{\eta_{D}}} {\eta }} {f_{\left| {g_{BP} } \right|^{2} } \left( x \right)} \int\limits_{0}^{\infty} {f_{\left| {g_{R2} } \right|^{2} } \left( y \right)} \int\limits_{\frac{{\phi \left( {y\eta_{R} + 1} \right)}} {\eta }}^{\infty} {f_{\left| {g_{B2} } \right|^{2} } \left( z \right)} dzdydx. \end{array} $$

(35)

Putting Eq. 10 into Eq. 35, we get

$$ \begin{array}{@{}rcl@{}} A_{1} = \frac{{\left( {1 - e^{- \frac{{\eta_{D}}} {{G_{BP} \eta }}} } \right)e^{- \frac{\phi } {{G_{B2} \eta }}} }} {{G_{R2} }}\int\limits_{0}^{\infty} {e^{- \left( {\frac{{G_{B2} \eta + G_{R2} \phi \eta_{R} }} {{G_{R2} G_{B2} \eta }}} \right)x} } dx \\ = \frac{{\left( {1 - e^{- \frac{{\eta_{D}}} {{G_{BP} \eta }}} } \right)G_{B2} \eta e^{- \frac{\phi } {{G_{B2} \eta }}} }} {{\left( {G_{B2} \eta + G_{R2} \phi \eta_{R} } \right)}}. \end{array} $$

(36)

Then, the term A₂ is calculated as

$$ \begin{array}{@{}rcl@{}} A_{2} = {\text{OP}} \left( {\left| {g_{BP} } \right|^{2} > \frac{{\eta_{D}}} {\eta },\left| {g_{B2} } \right|^{2} > \frac{{\phi \left( {\left| {g_{R2} } \right|^{2} \eta_{R} + 1} \right)\left| {g_{BP} } \right|^{2} }} {{\eta_{D}}}} \right) \\ = \int\limits_{\frac{{\eta_{D}}} {\eta }}^{\infty} {f_{\left| {g_{BP} } \right|^{2} } \left( x \right)} \int\limits_{0}^{\infty} {f_{\left| {g_{R2} } \right|^{2} } \left( y \right)} \int\limits_{\frac{{\phi \left( {y\eta_{R} + 1} \right)x}} {{\eta_{D}}}}^{\infty} {f_{\left| {g_{R2} } \right|^{2} } \left( z \right)} dzdxdy. \end{array} $$

(37)

Similarly, it is rewritten as

$$ \begin{array}{@{}rcl@{}} A_{2} = \frac{1} {{G_{BP} G_{R2} }}\int\limits_{\frac{{\eta_{D}}} {\eta }}^{\infty} {e^{- \frac{\phi } {{\eta_{D}G_{B2} }}x - \frac{x} {{G_{BP} }}} } \int\limits_{0}^{\infty} {e^{- \frac{{\phi \eta_{R} x}} {{\eta_{D}G_{B2} }}y - \frac{y} {{G_{R2} }}} } dxdy \\ = \frac{{G_{B2} \eta_{D}}} {{G_{BP} G_{R2} \phi \eta_{R} }}\int\limits_{\frac{{\eta_{D}}} {\eta }}^{\infty} {\frac{{e^{- \left( {\frac{{G_{B2} \eta_{D} + G_{BP} \phi }} {{G_{BP} G_{B2} \eta_{D}}}} \right)x} }} {{x + \frac{{G_{B2} \eta_{D}}} {{G_{R2} \phi \eta_{R} }}}}} dy. \end{array} $$

(38)

Based on [36, Eq. 3.352.2], A₂ can be obtained as

$$ \begin{array}{@{}rcl@{}} &&A_{2} = - \frac{{G_{B2} \eta_{D}e^{\frac{{G_{B2} \eta_{D} + G_{BP} \phi }} {{G_{BP} G_{R2} \phi \eta_{R} }}} }} {{G_{BP} G_{R2} \phi \eta_{R} }}\\ &&Ei\left( { - \frac{{G_{B2} \eta_{D} + G_{BP} \phi }} {{G_{BP} G_{B2} \eta }} - \frac{{G_{B2} \eta_{D} + G_{BP} \phi }} {{G_{BP} G_{R2} \phi \eta_{R} }}} \right). \end{array} $$

(39)

Substituting Eqs. 36 and 38 into Eq. 34, the proof is completed.

Appendix B: Proof of proposition 2

To compute the outage probability for user U₁, B₁ can be first written as

$$ \begin{array}{@{}rcl@{}} B_{1} = {\text{OP}} \left( {\left| {g_{BP} } \right|^{2} < \frac{{\eta_{D}}} {\eta },\left| {g_{BR} } \right|^{2} > \frac{{\varepsilon_{1} \left( {\eta_{R} \left| {g_{R} } \right|^{2} + 1} \right)}} {\eta }} \right) \\ + {\text{OP}} \left( {\left| {g_{BP} } \right|^{2} > \frac{{\eta_{D}}} {\eta },\left| {g_{BR} } \right|^{2} > \frac{{\varepsilon_{1} \left( {\eta_{R} \left| {g_{R} } \right|^{2} + 1} \right)\left| {g_{BP} } \right|^{2} }} {{\eta_{D}}}} \right) \\ = \int\limits_{0}^{\frac{{\eta_{D}}} {\eta }} {f_{\left| {g_{BP} } \right|^{2} } \left( x \right)} \int\limits_{0}^{\infty} {f_{\left| {g_{R} } \right|^{2} } \left( y \right)} \int\limits_{\frac{{\varepsilon_{1} \left( {\eta_{R} y + 1} \right)}} {\eta }}^{\infty} {f_{\left| {g_{BR} } \right|^{2} } \left( z \right)} dzdydx \\ + \int\limits_{\frac{{\eta_{D}}} {\eta }}^{\infty} {f_{\left| {g_{BP} } \right|^{2} } \left( x \right)} \int\limits_{0}^{\infty} {f_{\left| {g_{R} } \right|^{2} } \left( y \right)} \int\limits_{\frac{{\varepsilon_{1} \left( {\eta_{R} y + 1} \right)x}} {{\eta_{D}}}}^{\infty} {f_{\left| {g_{BR} } \right|^{2} } \left( z \right)} dzdydx. \end{array} $$

(40)

Similarly, we can obtain B₁ as

$$ \begin{array}{@{}rcl@{}} B_{1} = \frac{{\left( {1 - e^{- \frac{{\eta_{D}}} {{G_{BP} \eta }}} } \right)G_{BR} \eta e^{- \frac{{\varepsilon_{1} }} {{G_{BR} \eta }}} }} {{\left( {G_{BR} \eta + G_{R} \varepsilon_{1} \eta_{R} } \right)}} - \frac{{G_{BR} \eta_{D}e^{\frac{{G_{BR} \eta_{D} + G_{BP} \varepsilon_{1} }} {{G_{BP} G_{R} \varepsilon_{1} \eta_{R} }}} }} {{G_{BP} G_{BR} \varepsilon_{1} \eta_{R} }}\\ \times Ei\left( { - \frac{{G_{BR} \eta_{D} + G_{BP} \varepsilon_{1} }} {{G_{BP} G_{BR} \eta }} - \frac{{G_{BR} \eta_{D} + G_{BP} \varepsilon_{1} }} {{G_{BP} G_{R} \varepsilon_{1} \eta_{R} }}} \right). \end{array} $$

(41)

Furthermore, B₂ can be calculated by

$$ B_{2} = {\text{OP}} \left( {\left| {g_{R1} } \right|^{2} > \frac{{\gamma_{th1} }} {{\eta_{R} }}} \right) = e^{- \frac{{\gamma_{th1} }} {{\eta_{R} G_{R1} }}} . $$

(42)

Finally, combining Eqs. 16 and 17 the closed-form for outage probability of U₁ is given as Eq. 27.

It completes the proof.