Energy harvesting relay-antenna selection in cooperative MIMO/NOMA network over Rayleigh fading

Abstract

In this paper, a combination system of multi-antenna multiple input multiple output (MIMO) and non-orthogonal multiple access (NOMA) technologies is investigated, in which the source communicates with users using a multiple amplify-and-forward (AF) relaying network. These relay nodes are equipped with a single antenna and employ a power-splitting protocol to harvest energy from received signals, whereas the source and users are multiple-antenna nodes. In addition, two antenna-relay selection methods are considered to enhance the harvested energy at the relay including the maximum ratio transmission (MRT) and transmit antenna selection (TAS) at the source, with maximal-ratio combining at the users, these methods are compared to the performance of the random selection (RS) scheme. To evaluate the performance of the proposed system, we derive analytical expressions of the outage probability and throughput for the MRT and TAS schemes over Rayleigh fading channels, and use a Monte Carlo simulation to verify the accuracy of the analytical results. The results demonstrate the benefit of using MRT and TAS schemes, which provide a better performance than RS schemes, in a MIMO/NOMA system. Moreover, these results characterize the effects of various system parameters, such as power allocation factors, the numbers of antenna and relay nodes, power-splitting ratio, successive interference cancellation and energy-harvesting efficiency, on the system performance of two users of MIMO/NOMA. This is further compared with multiple-antenna conventional orthogonal multiple access (MIMO/OMA) schemes.

Appendices

Appendix 1: Proof of Theorem 1

In MRT schemes, by substituting (2b) and (3a) into (19), equation in (19) can be expressed as

$$\begin{aligned} OP_1^{MRT} = 1 - \frac{{K{e^{ - \frac{{{\theta _2}}}{{{\lambda _0}{\theta ^*}}}}}}}{{\varGamma \left( {{N_0}} \right) \lambda _0^{{N_0}}}} \int \limits _0^{ + \infty } {e^{ - \frac{1}{{{\lambda _1}{\theta ^*}t}} - \frac{t}{{{\lambda _0}}}}} \sum \limits _{j = 0}^{{N_1} - 1} {\left( \begin{array}{l} \frac{1}{{j!}}{\left( {\frac{1}{{{\lambda _1}{\theta ^*}t}}} \right) ^j}{\left( {t + \frac{{{\theta _2}}}{{\theta *}}} \right) ^{{N_0} - 1}}\\ {\left( {1 - {e^{ - \frac{t}{{{\lambda _0}}} - \frac{{{\theta _2}}}{{{\lambda _0}{\theta ^*}}}}}\sum \limits _{j = 0}^{{N_0} - 1} {\frac{1}{{j!}}{{\left( {\frac{t}{{{\lambda _0}}} + \frac{{{\theta _2}}}{{{\lambda _0}{\theta ^*}}}} \right) }^j}} } \right) ^{K - 1}} \end{array} \right) } dt \end{aligned}$$

(28)

Applying equation [29, Eq. 26.4.10] we have

$$\begin{aligned} {\left( {{x_1} + {x_2} +\cdots + {x_k}} \right) ^n} = \sum {\left( \begin{array}{c} n\\ {n_1},{n_2},\ldots ,{n_k} \end{array} \right) } x_1^{{n_1}}x_2^{{n_2}}\ldots x_k^{{n_k}} \end{aligned}$$

(29)

where the summation is over all nonnegative integers ${n_1},{n_2},\ldots ,{n_k}$ , such that ${n_1} + {n_2} +\cdots + {n_k} = n$ ; the $OP_1$ in MTR scheme in (28) can be formulated as

$$\begin{aligned} OP_1^{MRT} = 1 - \frac{{K{e^{ - \frac{{{\theta _2}}}{{{\lambda _0}{\theta ^*}}}}}}}{{\varGamma \left( {{N_0}} \right) \lambda _0^{{N_0}}}} \int \limits _0^{ + \infty } {\left[ \begin{array}{l} {e^{ - \frac{1}{{{\lambda _1}{\theta ^*}t}} - \frac{t}{{{\lambda _0}}}}}\sum \limits _{j = 0}^{{N_1} - 1} {\frac{1}{{j!}}{{\left( {\frac{1}{{{\lambda _1}{\theta ^*}t}}} \right) }^j}} {\left( {t + \frac{{{\theta _2}}}{{{\theta ^*}}}} \right) ^{{N_0} - 1}}\sum {\left( \begin{array}{c} K - 1\\ {n_0},{n_1},\ldots ,{n_{{N_0}}} \end{array} \right) {{\left( { - 1} \right) }^{K - {n_0} - 1}}} \\ {e^{ - \left( {^{K - {n_0} - 1}} \right) \left( {\frac{t}{{{\lambda _0}}} + \frac{{{\theta _2}}}{{{\lambda _0}{\theta ^*}}}} \right) }}\prod \limits _{i = 0}^{{N_0} - 1} {{{\left( {\frac{1}{{i!}}} \right) }^{{n_{i + 1}}}}} {\left( {\frac{t}{{{\lambda _0}}} + \frac{{{\theta _2}}}{{{\lambda _0}{\theta ^*}}}} \right) ^{i{n_{i + 1}}}} \end{array} \right] dt} = 1 - \sum \limits _{j = 0}^{{N_1} - 1} \sum \limits _{\left( {{n_0} + {n_1} +\cdots + {n_{{N_0}}} = K - 1} \right) } {\left( \begin{array}{c} K - 1\\ {n_0},{n_1},\ldots ,{n_{{N_0}}} \end{array} \right) } \frac{{K{e^{ - \frac{{{\theta _2}\left( {K - {n_0}} \right) }}{{{\lambda _0}{\theta ^*}}}}}}}{{\varGamma \left( {{N_0}} \right) \lambda _0^{\sum \limits _{i = 0}^{{N_0} - 1} {i{n_{i + 1}}} + {N_0}}}} \frac{1}{{j!}}{\left( {\frac{1}{{{\lambda _1}{\theta ^*}}}} \right) ^j}{\left( { - 1} \right) ^{K - {n_0} - 1}} \prod \limits _{i = 0}^{{N_0} - 1} {{{\left( {\frac{1}{{i!}}} \right) }^{{n_{i + 1}}}}} \int \limits _0^{ + \infty } {e^{ - \frac{1}{{{\lambda _1}{\theta ^*}t}} - \frac{{t\left( {K - {n_0}} \right) }}{{{\lambda _0}}}}} {{\left( {\frac{1}{t}} \right) }^j}{{\left( {t + \frac{{{\theta _2}}}{{{\theta ^*}}}} \right) }^{\sum \limits _{i = 0}^{{N_0} - 1} {i{n_{i + 1}}} + {N_0} - 1}}dt \end{aligned}$$

(30)

Here, ${n_0} + {n_1} +\cdots + {n_{{N_0}}} = K - 1$.

Applying the Newton binomial ${\left( {a + b} \right) ^k} = \sum \nolimits _{i = 0}^k {\left( \begin{array}{l} k\\ i \end{array} \right) } {a^{k - i}}{b^i}$ , (30) can be expressed as

$$\begin{aligned} OP_1^{MRT} = 1 - \sum \limits _{j = 0}^{{N_1} - 1} \sum \limits _{\left( {{n_0} + {n_1} +\cdots + {n_{{N_0}}} = K - 1} \right) } \frac{{K{e^{ - \frac{{{\theta _2}\left( {K - {n_0}} \right) }}{{{\lambda _0}{\theta ^*}}}}}}}{{\varGamma \left( {{N_0}} \right) \lambda _0^{\sum \limits _{i = 0}^{{N_0} - 1} {i{n_{i + 1}}} + {N_0}}}}\frac{1}{{j!}}{{\left( {\frac{1}{{{\lambda _1}{\theta ^*}}}} \right) }^j} \left( \begin{array}{c} K - 1\\ {n_0},{n_1},\ldots ,{n_{{N_0}}} \end{array} \right) {{\left( { - 1} \right) }^{K - {n_0} - 1}} \prod \limits _0^{{N_0} - 1} {{{\left( {\frac{1}{{i!}}} \right) }^{{n_{i + 1}}}}} \int \limits _0^{ + \infty } {e^{ - \frac{1}{{{\lambda _1}{\theta ^*}t}} - \frac{{y\left( {K - {n_0}} \right) }}{{{\lambda _0}}}}}{{\left( {\frac{1}{t}} \right) }^j} \sum \limits _{m = 0}^{\sum \limits _0^{{N_0} - 1} {i{n_{i + 1}}} + {N_0} - 1} {\left( \begin{array}{c} \sum \limits _0^{{N_0} - 1} {i{n_{i + 1}}} + {N_0} - 1\\ m \end{array} \right) } {t^{\sum \limits _0^{{N_0} - 1} {i{n_{i + 1}}} + {N_0} - 1 - m}}{{\left( {\frac{{{\theta _2}}}{{{\theta ^*}}}} \right) }^m}dt = 1 - \sum \limits _{j = 0}^{{N_1} - 1} \sum \limits _{\left( {{n_0} + {n_1} +\cdots + {n_{{N_0}}} = K - 1} \right) } \sum \limits _{m = 0}^{\sum \limits _{i = 0}^{{N_0} - 1} {i{n_{i + 1}}} + {N_0} - 1} \frac{{K{e^{ - \frac{{{\theta _2}\left( {K - {n_0}} \right) }}{{{\lambda _0}{\theta ^*}}}}}}}{{\varGamma \left( {{N_0}} \right) \lambda _0^{\sum \limits _{i = 0}^{{N_0} - 1} {i{n_{i + 1}}} + {N_0}}}} \frac{1}{{j!}}{{\left( {\frac{1}{{{\lambda _1}{\theta ^*}}}} \right) }^j}\left( \begin{array}{c} K - 1\\ {n_0},{n_1},\ldots ,{n_{{N_0}}} \end{array} \right) \left( \begin{array}{c} \sum \limits _0^{{N_0} - 1} {i{n_{i + 1}}} + {N_0} - 1\\ m \end{array} \right) {\left( {\frac{{{\theta _2}}}{{{\theta ^*}}}} \right) ^m}{\left( { - 1} \right) ^{K - {n_0} - 1}}\prod \limits _0^{{N_0} - 1} {{{\left( {\frac{1}{{i!}}} \right) }^{{n_{i + 1}}}}} \int \limits _0^{ + \infty } {e^{ - \frac{1}{{{\lambda _1}{\theta ^*}t}} - \frac{{t\left( {K - {n_0}} \right) }}{{{\lambda _0}}}}} {t^{\sum \limits _{i = 0}^{{N_0} - 1} {i{n_{i + 1}}} + {N_0} - 1 - m - j}}dt \end{aligned}$$

(31)

By considering [28, Eq. (3.471.9)] as ${\int \nolimits _0^\infty {{x^{v - 1}}{e^{ - \frac{\beta }{x} - \gamma x}}dx = 2\left( {\frac{\beta }{\gamma }} \right) } ^{\frac{v}{2}}}{K_v}\left( {2\sqrt{\beta \gamma } } \right)$, we obtain $OP_1^{MRT}$ as in (20).

In TAS schemes, by substituting (2b) and (4b) into (19), equation in (19) can be expressed as

$$\begin{aligned} OP_1^{T{{AS}}} = 1 - \int \limits _0^{ + \infty } \left( {{e^{ - \frac{1}{{{\lambda _1}}}\frac{1}{{{\theta ^*}t}}}}\sum \limits _{j = 0}^{{N_1} - 1} {\frac{1}{{j!}}{{\left( {\frac{1}{{{\lambda _1}}}\frac{1}{{{\theta ^*}t}}} \right) }^j}} } \right) \frac{{K{N_0}}}{{{\lambda _0}}}{e^{ - \frac{t}{{{\lambda _0}}} - \frac{{{\theta _2}}}{{{\lambda _0}{\theta ^*}}}}} {{\left( {1 - {e^{ - \frac{t}{{{\lambda _0}}} - \frac{{{\theta _2}}}{{{\lambda _0}{\theta ^*}}}}}} \right) }^{K{N_0} - 1}}dt = 1 - \frac{{K{N_0}}}{{{\lambda _0}}}{e^{ - \frac{{{\theta _2}}}{{{\lambda _0}{\theta ^*}}}}} \int \limits _0^{ + \infty } {e^{ - \frac{1}{{{\lambda _1}{\theta ^*}t}} - \frac{t}{{{\lambda _0}}}}} \sum \limits _{j = 0}^{{N_1} - 1} {\frac{1}{{j!}}{{\left( {\frac{1}{{{\lambda _1}{\theta ^*}t}}} \right) }^j}} {{\left( {1 - {e^{ - \frac{1}{{{\lambda _0}}}\left( {t + \frac{{{\theta _2}}}{{{\theta ^*}}}} \right) }}} \right) }^{K{N_0} - 1}}dt= 1 - \frac{{K{N_0}}}{{{\lambda _0}}}\sum \limits _{j = 0}^{{N_1} - 1} \sum \limits _{i = 0}^{K{N_0} - 1} \frac{1}{{j!}}{{\left( { - 1} \right) }^i}\left( \begin{array}{c} K{N_0} - 1\\ i \end{array} \right) {e^{ - \frac{{\left( {i + 1} \right) {\theta _2}}}{{{\lambda _0}{\theta ^*}}}}} {\left( {\frac{1}{{{\lambda _1}{\theta ^*}}}} \right) ^j}\int \limits _0^{ + \infty } {{e^{ - \frac{1}{{{\lambda _1}{\theta ^*}t}} - \frac{{\left( {i + 1} \right) }}{{{\lambda _0}}}t}}{{\left( {\frac{1}{t}} \right) }^j}dt} \end{aligned}$$

(32)

Applying the Newton binomial and using [28, Eq. (3.471.9)], we obtain $OP_1^{T{{AS}}}$ as in (21).

In RS schemes, by substituting (2b) and the PDF of X as ${f_{{g_{m,k,i}}}}\left( x \right) = {e^{ - \frac{x}{{{\lambda _m}}}}}/{\lambda _m}$ in (19), we can formulate (19) as

$$\begin{aligned} OP_1^{RS} = 1 - \int \limits _0^{ + \infty } {e^{ - \frac{1}{{{\lambda _1}{\theta ^*}t}}}} \sum \limits _{j = 0}^{{N_1} - 1} {\frac{1}{{j!}}{{\left( {\frac{1}{{{\lambda _1}{\theta ^*}t}}} \right) }^j}} \frac{1}{{{\lambda _0}}}{e^{ - \frac{t}{{{\lambda _0}}} - \frac{{{\theta _2}}}{{{\lambda _0}{\theta ^*}}}}}dt = 1 - \frac{1}{{{\lambda _0}}}{e^{ - \frac{{{\theta _2}}}{{{\lambda _0}{\theta ^*}}}}}\sum \limits _{j = 0}^{{N_1} - 1} {\frac{1}{{j!}}} \int \limits _0^{ + \infty } {{e^{ - \frac{1}{{{\lambda _1}{\theta ^*}t}} - \frac{t}{{{\lambda _0}}}}}{{\left( {\frac{1}{{{\lambda _1}{\theta ^*}t}}} \right) }^j}dt} = 1 - \frac{1}{{{\lambda _0}}}{e^{ - \frac{{{\theta _2}}}{{{\lambda _0}{\theta ^*}}}}}\sum \limits _{j = 0}^{{N_1} - 1} {\frac{1}{{j!}}} {\left( {\frac{1}{{{\lambda _1}{\theta ^*}}}} \right) ^j}\int \limits _0^{ + \infty } {{e^{ - \frac{1}{{{\lambda _1}{\theta ^*}t}} - \frac{t}{{{\lambda _0}}}}}{t^{ - j}}dt} \end{aligned}$$

(33)

By using [28, Eq. (3.471.9)] as ${\int \nolimits _0^\infty {{x^{v - 1}}{e^{ - \frac{\beta }{x} - \gamma x}}dx = 2\left( {\frac{\beta }{\gamma }} \right) } ^{\frac{v}{2}}}{K_v}\left( {2\sqrt{\beta \gamma } } \right)$ , we obtain $OP_1^{RS}$ as in (22).

This ends the Proof of Theorem 1.

Appendix 2: Proof of Theorem 2

Similar to formulating the outage probability at $U_1$, the OP at $U_2$ in three MRT, TAS and RS schemes can be obtained, as in (24), (25) and (26), respectively. This ends the Proof of Theorem 2.