Joint Beamforming Design for RIS-aided AmBC-enabled MISO NOMA Networks

Abstract

In the context of “beyond 5 G" (B5G) mobile communication, a better quality of service (QoS) with a minimum power budget is one of the prime requirements. To improve QoS or sum rate, utilizing reconfigurable intelligent surfaces (RIS) in a non-orthogonal multiple access (NOMA) network is an effective approach. This paper focuses on a RIS-assisted backscatter-enabled NOMA network where the base station (BS) has to serve a backscatter device (BD) and dead-zone users simultaneously by achieving the maximum possible sum rate in a given power budget. With the help of joint optimization at BS, RIS, and BD, the beamforming vectors, RIS phase shift matrix (PSM), and power splitting (PS) ratio, respectively, are optimized such that the user’s QoS and BD’s energy harvesting constraints can be achieved for a given power constraint while increasing the sum rate for dead zone users. Particularly, the beamforming vectors and the phases of the RIS elements are alternately optimized using a lower bound, semidefinite relaxation (SDR), and Gaussian randomization techniques until they reach convergence. According to simulation results, the proposed scheme (RIS-BD-NOMA) always performs better than the baseline schemes, RIS-NOMA with random phases, and RIS-NOMA with optimized phases.

Appendix

1.1 Proof of (13b)

Using (9a), (9b), and assumptions made in Sect. 3.1.

$$\begin{aligned} R_1+R_2^1= & {} \log _2\left( 1+\frac{|\mathbf{h_{BU_1}^H w_1}|^2}{\sigma ^2}\right) +\log _2\left( 1+\frac{|\mathbf{h_{BU_1}^H w_2}|^2}{|\mathbf{h_{BU_1}^H w_1}|^2+\sigma ^2}\right) \nonumber \\= & {} \log _2\left( {\sigma ^2+|\mathbf{h_{BU_1}^H w_1}|^2}\right) -\log _2(\sigma ^2)+\log _2\left( {|\mathbf{h_{BU_1}^H w_1}|^2+\sigma ^2+|\mathbf{h_{BU_1}^H w_2}|^2}\right) \nonumber \\{} & {} -\log _2\left( |\mathbf{h_{BU_1}^H w_1}|^2+\sigma ^2\right) \nonumber \\= & {} \log _2 \left( 1+\frac{|\mathbf{h_{BU_1}^H w_1}|^2+|\mathbf{h_{BU_1}^H w_2}|^2}{\sigma ^2} \right) \end{aligned}$$

(20)

The constraint (12b) can be rewritten as,

$$\begin{aligned}{} & {} \log _2 \left( 1+\frac{|\mathbf{h_{BU_1}^H w_1}|^2+|\mathbf{h_{BU_1}^H w_2}|^2}{\sigma ^2} \right) \ge t_1 \end{aligned}$$

(21)

$$\begin{aligned}{} & {} \left( \frac{|\mathbf{h_{BU_1}^H w_1}|^2+|\mathbf{h_{BU_1}^H w_2}|^2}{\sigma ^2} \right) \ge 2^{t_1}-1 \end{aligned}$$

(22)

After introducing a slack variable $t_2$, the constraint (12b) is converted as

$$\begin{aligned} \left( \frac{|\mathbf{h_{BU_1}^H w_1}|^2+|\mathbf{h_{BU_1}^H w_2}|^2}{\sigma ^2} \right) \ge t_2 \end{aligned}$$

(23)

where $t_2=2^{t_1}-1$.

Now the constraint (12c) can be converted as

$$\begin{aligned} R_1+R_2^2= & {} \log _2\left( 1+\frac{|\mathbf{h_{BU_1}^H w_1}|^2}{\sigma ^2}\right) + \log _2\left( 1+\frac{|\mathbf{h_{BU_2}^H w_2}|^2}{|\mathbf{h_{BU_2}^H w_1}|^2+\sigma ^2}\right) \nonumber \\= & {} \log _2\left( \frac{\sigma ^2+|\mathbf{h_{BU_1}^H w_1}|^2}{\sigma ^2} \right) + \log _2\left( \frac{|\mathbf{h_{BU_2}^H w_1}|^2+\sigma ^2+|\mathbf{h_{BU_2}^H w_2}|^2}{|\mathbf{h_{BU_2}^H w_1}|^2+\sigma ^2} \right) \end{aligned}$$

(24)

It is assumed that near user $U_1$ has strong channel conditions than far user $U_2$ i.e.

$$\begin{aligned} |\mathbf{h_{BU_1}^H w_1}|^2 \ge |\mathbf{h_{BU_2}^H w_1}|^2 \end{aligned}$$

if we consider lower bound, i.e. $|\mathbf{h_{BU_1}^H w_1}|=|\mathbf{h_{BU_2}^H w_1}|$, Eq. (24) reduces as

$$\begin{aligned} R_1+R_2^2=\log _2 \left( 1+\frac{|\mathbf{h_{BU_2}^H w_1}|^2+|\mathbf{h_{BU_2}^H w_2}|^2}{\sigma ^2} \right) \end{aligned}$$

(25)

When we use (25) in (12c) and use the slack variable $t_2$, (12c) is converted as

$$\begin{aligned} \left( \frac{|\mathbf{h_{BU_2}^H w_1}|^2+|\mathbf{h_{BU_2}^H w_2}|^2}{\sigma ^2} \right) \ge t_2 \end{aligned}$$

(26)

Therefore, constraints in (12b) & (12c) can be converted into (13b) as shown. The proof is completed. $\blacksquare$

1.2 Proof of (17b)

Using (9a), (9b), and assumptions made in Sect. 3.2.

$$\begin{aligned} R_1+R_2^1= & {} \log _2\left( 1+\frac{|\mathbf{u^H b_{1,1}}+c_1\mathbf{u^H b_{b,1}}|^2)}{\sigma ^2}\right) \nonumber \\{} & {} +\log _2\left( 1+\frac{|\mathbf{u^H b_{1,2}}+c_1\mathbf{u^H b_{b,2}}|^2)}{|\mathbf{u^H b_{1,1}}+c_1\mathbf{u^H b_{b,1}}|^2+\sigma ^2}\right) \nonumber \\= & {} \log _2\left( \frac{\sigma ^2+|\mathbf{u^H b_{1,1}}+c_1\mathbf{u^H b_{b,1}}|^2)}{\sigma ^2}\right) \nonumber \\{} & {} +\log _2\left( \hspace{-1mm}\frac{|\mathbf{u^H b_{1,1}}\hspace{-1mm}+c_1\mathbf{u^H b_{b,1}}|^2+\sigma ^2\hspace{-1mm}+|\mathbf{u^H b_{1,2}}+\hspace{-1mm}c_1\mathbf{u^H b_{b,2}}|^2)}{|\mathbf{u^H b_{1,1}}+c_1\mathbf{u^H b_{b,1}}|^2+\sigma ^2}\hspace{-1mm}\right) \nonumber \\= & {} \log _2 \hspace{-1mm}\left( \hspace{-1mm}1+\frac{|\mathbf{u^H b_{1,1}}\hspace{-1mm}+c_1\mathbf{u^H b_{b,1}}|^2\hspace{-1mm}+|\mathbf{u^H b_{1,2}}\hspace{-1mm}+c_1\mathbf{u^H b_{b,2}}|^2)}{\sigma ^2} \hspace{-1mm}\right) \end{aligned}$$

(27)

The constraint (16b) is rewritten as

$$\begin{aligned} \log _2 \hspace{-1mm}\left( \hspace{-1mm}1+\frac{|\mathbf{u^H b_{1,1}}\hspace{-1mm}+c_1\mathbf{u^H b_{b,1}}|^2\hspace{-1mm}+|\mathbf{u^H b_{1,2}}\hspace{-1mm}+c_1\mathbf{u^H b_{b,2}}|^2)}{\sigma ^2}\hspace{-1mm} \right) \ge q_1 \end{aligned}$$

(28)

After introducing a slack variable $q_2$, (28) converted into (29), which is the required conversion corresponding to constraint (16b).

$$\begin{aligned} \left( \frac{|\mathbf{u^H b_{1,1}}+c_1\mathbf{u^H b_{b,1}}|^2+|\mathbf{u^H b_{1,2}}+c_1\mathbf{u^H b_{b,2}}|^2)}{\sigma ^2}\right) \ge q_2 \end{aligned}$$

(29)

where $q_2=2^{q_1}-1$.

The constraint (16c) can be rewritten as

$$\begin{aligned} R_1+R_2^2= & {} \log _2\left( 1+\frac{|\mathbf{u^H b_{1,1}}+c_1\mathbf{u^H b_{b,1}|^2)}}{\sigma ^2}\right) \nonumber \\{} & {} +\log _2\left( 1+\frac{|\mathbf{u^H b_{2,2}}+c_2\mathbf{u^H b_{b,2}}|^2)}{|\mathbf{u^H b_{2,1}}+c_2\mathbf{u^H b_{b,1}}|^2+\sigma ^2}\right) \nonumber \\= & {} \log _2 \left( \frac{\sigma ^2+|\mathbf{u^H b_{1,1}}+c_1\mathbf{u^H b_{b,1}}|^2)}{\sigma ^2}\right) \nonumber \\{} & {} +\log _2 \hspace{-1mm}\left( \hspace{-1mm}\frac{|\mathbf{u^H b_{2,1}} \hspace{-1mm}+c_2\mathbf{u^H b_{b,1}}|^2 \hspace{-1mm}+\sigma ^2\hspace{-1mm}+|\mathbf{u^H b_{2,2}}+c_2\mathbf{u^H b_{b,2}}|^2)}{|\mathbf{u^H b_{2,1}}+c_2\mathbf{u^H b_{b,1}}|^2+\sigma ^2}\hspace{-1mm}\right) \end{aligned}$$

(30)

It is assumed that near user $U_1$ has strong channel conditions than far user $U_2$, i.e.

$$\begin{aligned} |\mathbf{u^H b_{1,1}}+c_1\mathbf{u^H b_{b,1}}|^2\ge |\mathbf{u^H b_{2,1}}+c_2\mathbf{u^H b_{b,1}}|^2 \end{aligned}$$

by considering lower bound, Eq. (30) reduces as

$$\begin{aligned} R_1+R_2^2= \log _2 \left( \hspace{-1mm} 1+\frac{|\mathbf{u^H b_{2,1}}\hspace{-1mm}+c_2\mathbf{u^H b_{b,1}}|^2\hspace{-1mm}+|\mathbf{u^H b_{2,2}}\hspace{-1mm}+c_2\mathbf{u^H b_{b,2}}|^2)}{\sigma ^2} \hspace{-1mm}\right) \end{aligned}$$

(31)

After using the slack variable $q_2$, (31) converted into (32), which is the required conversion corresponding to constraint (16c).

$$\begin{aligned} \left( \frac{|\mathbf{u^H b_{2,1}}+c_2\mathbf{u^H b_{b,1}}|^2+|\mathbf{u^H b_{2,2}}+c_2\mathbf{u^H b_{b,2}}|^2)}{\sigma ^2}\right) \ge q_2 \end{aligned}$$

(32)

Therefore, constraints in (16b) & (16c) can be converted into (17b) as shown. The proof is completed. $\blacksquare$