Low Complexity User Selection and Power Allocation for Uplink NOMA Beamforming Systems

Abstract

In this paper, we develop user selection and power allocation methods for NOMA systems equipped with multi-antenna to enhance the sum capacity of the uplink. The system has the base station (BS) with $N$ antenna supporting 2N users in the same spectrum resource simultaneously, and successive interference cancellation (SIC) is applied at the BS. Because the superposition of multiple users in transmission within the same frequency resource block leads to the interference among users, we derive a user set selection algorithm and a suboptimal power control to mitigate the interference effect and to maximize the sum capacity. The user set selection algorithm is first applied by comparing the designed indicator which balances the factors affecting the sum capacity of the uplink. Second, the derived suboptimal power assignment algorithm is utilized with power control factors where four cases are evaluated. The simulation results show that the proposed schemes can significantly improve the sum capacity than the existing methods.

Appendix 1

1.1 The derivation of $\alpha_{2,1}^{*}$ and $\alpha_{2,2}^{*}$

In (15), for assuming that the antenna number is two, we represent the objective function as

$$ \begin{aligned}& {\mathop{\max}\limits_{\left[{\varvec{\alpha}_{1}^{*},\varvec{\alpha}_{2}^{*}} \right]}} \left({R_{1,1 -NOMA} + R_{1,2 - NOMA} + R_{2,1 - NOMA} + R_{2,2 - NOMA}}\right) \\ & \quad ={\mathop{\max}\limits_{\left[{\varvec{\alpha}_{1}^{*},\,\varvec{\alpha}_{2}^{*}}\right]}} BW\left[\log_{2} \left({1 + \frac{{A_{1,1}}}{{\alpha_{2,1}C_{2,1}^{\left(1 \right)} + \alpha_{2,2} C_{2,2}^{\left(1 \right)} + 1}}} \right) \right. \\ &\quad \left. \quad + \log_{2}\left({1 + \frac{{A_{1,2}}}{{\alpha_{2,1} C_{2,1}^{\left(2 \right)} + \alpha_{2,2} C_{2,2}^{\left(2 \right)} +1}}} \right) + \log_{2} \left({1 + \alpha_{2,1} A_{2,1}}\right) \right.\\ &\quad \left. \quad + \log_{2} \left({1 + \alpha_{2,2} A_{2,2}} \right) \right] \end{aligned} $$

(25)

where $A_{i,j} = \left| {{\mathbf{h}}_{i,j} } \right|^{2} \gamma_{i,j}$, $\gamma_{i,j} = P_{{i,{\text{j}}}} /\sigma^{2}$, and $C_{i,j}^{\left( n \right)} = \left| {{\mathbf{w}}_{1,n} {\mathbf{h}}_{i,j} } \right|^{2} \gamma_{i,j}$. And the constraints in (15) are $R_{1,j - NOMA} \ge R_{1,j - OMA}$ and $R_{2,j - NOMA} \ge R_{2,j - OMA}$ for $\forall j = 1,2$. They are also written as follows:

$$ \log_{2} \left( {1 + \frac{{A_{{1,{\text{j}}}} }}{{\alpha_{2,1} C_{2,1}^{\left( j \right)} + \alpha_{2,2} C_{2,2}^{\left( j \right)} + 1}}} \right) \ge \log_{2} \left( {1 + A_{{1,{\text{j}}}} } \right)^{{\frac{1}{2}}} , $$

(26)

$$ \log_{2} \left( {1 + \alpha_{{2,{\text{j}}}} A_{{2,{\text{j}}}} } \right) \ge \log_{2} \left( {1 + A_{{2,{\text{j}}}} } \right)^{{\frac{1}{2}}} . $$

(27)

If (26) and (27) are satisfied for $\forall j = 1,2$, (26) and (27) can be rearranged and then can be expressed below:

$$ A_{{1,{\text{j}}}} - \varphi_{{1,{\text{j}}}} \alpha_{2,1} C_{2,1}^{\left( j \right)} - \varphi_{{1,{\text{j}}}} \alpha_{2,2} C_{2,2}^{\left( j \right)} - \varphi_{{1,{\text{j}}}} \ge 0\quad \forall j = 1,2 $$

(28)

and

$$ \alpha_{{2,{\text{j}}}} \ge \frac{{\varphi_{{2,{\text{j}}}} }}{{A_{{2,{\text{j}}}} }}\quad \forall j = 1,2 $$

(29)

where $\varphi_{1,j} = \sqrt {1 + A_{1,j} } - 1$ and $\varphi_{2,j} = \sqrt {1 + A_{2,j} } - 1$ for $\forall j = 1,2$.

Afterward, (25) is also reconstructed into

$$ \begin{aligned} \mathop { \hbox{max} }\limits_{{\left[ {\varvec{\alpha}_{1}^{ *} ,\varvec{\alpha}_{2}^{ *} } \right]}} \left[ {\begin{array}{*{20}c} {\log_{2} \left( {\begin{array}{*{20}c} {1 + \frac{{A_{1,1} }}{{\alpha_{2,1} C_{2,1}^{\left( 1 \right)} + \alpha_{2,2} C_{2,2}^{\left( 1 \right)} + 1}} + \alpha_{2,1} A_{2,1} } \\ { + \frac{{A_{1,1} \alpha_{2,1} A_{2,1} }}{{\alpha_{2,1} C_{2,1}^{\left( 1 \right)} + \alpha_{2,2} C_{2,2}^{\left( 1 \right)} + 1}}} \\ \end{array} } \right)} \\ { + \log_{2} \left( {\begin{array}{*{20}c} {1 + \frac{{A_{1,2} }}{{\alpha_{2,1} C_{2,1}^{\left( 2 \right)} + \alpha_{2,2} C_{2,2}^{\left( 2 \right)} + 1}} + \alpha_{2,2} A_{2,2} } \\ { + \frac{{A_{1,2} \alpha_{2,2} A_{2,2} }}{{\alpha_{2,1} C_{2,1}^{\left( 2 \right)} + \alpha_{2,2} C_{2,2}^{\left( 2 \right)} + 1}}} \\ \end{array} } \right)} \\ \end{array} } \right] \hfill \\ = \mathop { \hbox{max} }\limits_{{\left[ {\varvec{\alpha}_{1}^{ *} ,\varvec{\alpha}_{2}^{ *} } \right]}} \left[ {\begin{array}{*{20}c} {\log_{2} \left( {1 + f_{1} \left( {\alpha_{2,1} ,\alpha_{2,2} } \right)} \right)} \\ { + \log_{2} \left( {1 + f_{2} \left( {\alpha_{2,1} ,\alpha_{2,2} } \right)} \right)} \\ \end{array} } \right]. \hfill \\ \end{aligned} $$

(30)

Moreover, (31) can be maximized if $f_{1} \left( {\alpha_{2,1} ,\alpha_{2,2} } \right)$ and $f_{2} \left( {\alpha_{2,1} ,\alpha_{2,2} } \right)$ are both maximized. However, $f_{1} \left( {\alpha_{2,1} ,\alpha_{2,2} } \right)$ and $f_{2} \left( {\alpha_{2,1} ,\alpha_{2,2} } \right)$ all involve parameters $\alpha_{2,1}$ and $\alpha_{2,2}$ and can not be maximized separately. For $f_{1}$, the partial derivatives are taken with respect to $\alpha_{2,1}$ and $\alpha_{2,2}$. From the partial derivative with respect to $\alpha_{2,1}$, we have the expression as follows:

$$ \frac{{\partial f_{1} \left( {\alpha_{2,1} ,\alpha_{2,2} } \right)}}{{\partial \alpha_{2,1} }} > 0\quad {\text{if}}\quad \alpha_{2,2} > \frac{{C_{2,1}^{\left( 1 \right)} - A_{2,1} }}{{C_{2,2}^{\left( 1 \right)} A_{2,1} }}. $$

(31)

As $\alpha_{2,2} \ge \frac{{\varphi_{2,2} }}{{A_{2,2} }}$ is the constraint in (29), (32) can be filled. That is, $f_{1}$ is a monotonic increasing function of $\alpha_{2,1}$. Hence, we set $\alpha_{2,1} = 1$ ($0 \le \alpha_{2,1} \le 1$) in $f_{1}$ to maximize $f_{1}$. Then, for the partial derivative with respect to $\alpha_{2,1}$, we have

$$ \frac{{\partial f_{1} \left( {1,\alpha_{2,2} } \right)}}{{\partial \alpha_{2,2} }} < 0, $$

(32)

$f_{1}$ is a monotonic decreasing function of $\alpha_{2,2}$; nevertheless, the minimum $\alpha_{2,2}$ is restricted by (29). For these reasons,

$$ \alpha_{2,2} = \frac{{\varphi_{2,2} }}{{A_{2,2} }}. $$

(33)

The other case of maximizing $f_{2} \left( {\alpha_{2,1} ,\alpha_{2,2} } \right)$ can be obtained by using the similar approach.