Cross Layer Power Control and User Pairing for DL Multi-antenna NOMA

Abstract

In a previous study, Kim et al. proposed a suboptimal user pairing and optimal power control algorithm for maximizing sum capacity in downlink multi-antenna non-orthogonal multiple access (NOMA) systems. First, Kim et al. calculate the correlation between all users. If the channel correlation is greater than the threshold, the user’s channel gain difference is stored in the channel gain-difference set. Then, the first and second large channel gain-differences of the user pairs are selected from the set. Based on the user pairing of Kim et al. in the physical layer, we propose novel iterative user substitution algorithm where a user is substituted if the peak signal-to-noise ratio (PSNR), the video quality, in the application layer increases after substitution. And we propose an optimal power control that minimizes the sum video distortion to maximize PSNR in the application layer. The numerical results show that the proposed cross layer optimal power control and user substitution outperforms Kim et al. by 1.4 dB in PSNR and only 1 dB away from exhaustive upper bound, when SNR = 15 dB and the number of users is 16.

Appendix

From (8) and (9), the strong and weak user information rates are given by:

$$r_{n,1} \left( {\alpha_{n,1} } \right) = BW \times \log_{2} \left( {1 + \frac{{\eta \left| {\varvec{h}_{n,1} \varvec{w}_{n} } \right|^{2} \alpha_{n,1} P_{n,1} }}{{P_{N} }}} \right).$$

(A-1)

$$r_{n,2} \left( {1 - \alpha_{n,1} } \right) = BW \times \log_{2} \left( {1 + \frac{{\eta \left| {\varvec{h}_{n,2} \varvec{w}_{n} } \right|^{2} \left( {1 - \alpha_{n,1} } \right)P_{n,2} }}{{\left| {\varvec{h}_{n,2} \varvec{w}_{n} } \right|^{2} \alpha_{n,1} P_{n,1} + \left| {\varvec{h}_{n,2} \mathop \sum \nolimits_{k = 1,k \ne n}^{N} \varvec{w}_{k} x_{k} } \right|^{2} + P_{N} }}} \right).$$

(A-2)

where $\left| {\varvec{h}_{n,2} \varvec{w}_{n} } \right|^{2} \alpha_{n,1} P_{n,1}$ is from strong user interference.$\left| {\varvec{h}_{n,2} \mathop \sum \limits_{k = 1,k \ne n}^{N} \varvec{w}_{k} x_{k} } \right|^{2}$ is from interference for other groups.$P_{N}$ is power noise.

From (13), power control problem is formulated as follows (A-3):

$$\left[ {\alpha_{n,1}^{*} } \right] = arg\mathop {\hbox{min} }\limits_{{\left[ {\alpha_{n,1}^{*} } \right]}} \left\{ {f\left( {\alpha_{n,1} } \right)} \right\},\quad {\text{s}}.{\text{t }}g\left( {\alpha_{n,1} } \right) \le 0.$$

(A-3)

We use KKT conditions as follows (A-4):

• (i)
  $$\upmu \ge 0,$$
• (ii)
  $$D_{\alpha } f\left( {\alpha_{n} } \right) + \mu D_{\alpha } g\left( {\alpha_{n} } \right) = 0,$$
• (iii)
  $$\mu g\left( {\alpha_{n} } \right) = 0,$$
• (iv)
  $$g\left( {\alpha_{n} } \right) \le 0,$$

  (A-4)

Objective function (A-5) and restriction condition (A-6).

$$f\left( {\alpha_{n,1} } \right) = \alpha_{1} + \frac{{b_{1} }}{{r_{1} \left( {\alpha_{n,1} } \right) + c_{1} }} + \alpha_{2} + \frac{{b_{2} }}{{r_{2} \left( {1 - \alpha_{n,1} } \right) + c_{2} }}$$

(A-5)

$$g\left( {\alpha_{n,1} } \right) = \alpha_{1} + \frac{{b_{1} }}{{r_{1} \left( {\alpha_{n,1} } \right) + c_{1} }} + \alpha_{2} + \frac{{b_{2} }}{{r_{2} \left( {1 - \alpha_{n,1} } \right) + c_{2} }} - \left( {\alpha_{1} + \frac{{b_{1} }}{{\frac{{r_{1} \left( 1 \right)}}{2} + c_{1} }}} \right) \le 0.$$

(A-6)

(ii) Can be written as $\left( {1 + \mu } \right)\left( {D_{\alpha } f\left( {\alpha_{n} } \right) + D_{\alpha } g\left( {\alpha_{n} } \right)} \right) = 0$. According to (i), it can be divided by $\left( {1 + \mu } \right)$, (ii) becomes $D_{\alpha } f\left( {\alpha_{n} } \right) + D_{\alpha } g\left( {\alpha_{n} } \right) = 0$. The derivative of $f\left( {\alpha_{n,1} } \right)$ and $g\left( {\alpha_{n,1} } \right)$ with respect to $\alpha_{n,1}$, because the two differentials are the same result, so (ii) becomes $D_{\alpha } f\left( {\alpha_{n,1} } \right) = 0$.

Substituting (A-1) and (A-2) into (A-5), and we get:

$$\begin{aligned} f\left( {\upalpha_{n,1} } \right) & = a_{1} + \frac{{b_{1} }}{{BW \times \log_{2} \left( {1 + \frac{{\eta \left| {\varvec{h}_{n,1} \varvec{w}_{n} } \right|^{2} \alpha_{n,1} P_{n,1} }}{{P_{N} }}} \right) + c_{1} }} + a_{2} \\ & \quad + \frac{{b_{2} }}{{BW \times \log_{2} \left( {1 + \frac{{\eta \left| {\varvec{h}_{n,2} \varvec{w}_{n} } \right|^{2} \left( {1 - \alpha_{n,1} } \right)P_{n,2} }}{{\left| {\varvec{h}_{n,2} \varvec{w}_{n} } \right|^{2} \alpha_{n,1} P_{n,1} + \left| {\varvec{h}_{n,2} \mathop \sum \nolimits_{k = 1,k \ne n}^{N} \varvec{w}_{k} x_{k} } \right|^{2} + P_{N} }}} \right) + c_{2} }} \\ \end{aligned}$$

(A-7)

Differential with respect to $\upalpha_{n,1}$ and get

$$\begin{aligned} D_{\alpha} f\left( {{\upalpha }_{n,1} } \right) & = - b_{1} \frac{{\partial \left( {BW \times \log_{2} \left( {1 + \frac{{\eta \left| {\varvec{h}_{n,1} \varvec{w}_{n} } \right|^{2} \alpha_{n,1} P_{n,1} }}{{P_{N} }}} \right)} \right)}}{{\left( {BW \times \log_{2} \left( {1 + \frac{{\eta \left| {\varvec{h}_{n,1} \varvec{w}_{n} } \right|^{2} \alpha_{n,1} P_{n,1} }}{{P_{N} }}} \right) + c_{1} } \right)^{2} }} \\ & \quad - \;b_{2} \frac{{\partial \left( {BW \times \log_{2} \left( {1 + \frac{{\eta \left| {\varvec{h}_{n,2} \varvec{w}_{n} } \right|^{2} \left( {1 - \alpha_{n,1} } \right)P_{n,2} }}{{\left| {\varvec{h}_{n,2} \varvec{w}_{n} } \right|^{2} \alpha_{n,1} P_{n,1} + \left| {\varvec{h}_{n,2} \mathop \sum \nolimits_{k = 1,k \ne n}^{N} \varvec{w}_{k} x_{k} } \right|^{2} + P_{N} }}} \right)} \right)}}{{\left( {BW \times \log_{2} \left( {1 + \frac{{\eta \left| {\varvec{h}_{n,2} \varvec{w}_{n} } \right|^{2} \left( {1 - \alpha_{n,1} } \right)P_{n,2} }}{{\left| {\varvec{h}_{n,2} \varvec{w}_{n} } \right|^{2} \alpha_{n,1} P_{n,1} + \left| {\varvec{h}_{n,2} \mathop \sum \nolimits_{k = 1,k \ne n}^{N} \varvec{w}_{k} x_{k} } \right|^{2} + P_{N} }}} \right) + c_{2} } \right)^{2} }} = 0 \\ \end{aligned}$$

(A-8)

Expanding the numerator part:

$$\begin{aligned} D_{\alpha } f\left( {{\upalpha }_{n,1} } \right) & = - b_{1} \frac{{BW \times \frac{1}{\ln 2}\frac{{\frac{{\eta \left| {\varvec{h}_{n,1} \varvec{w}_{n} } \right|^{2} P_{n,1} }}{{P_{N} }}}}{{1 + \frac{{\eta \left| {\varvec{h}_{n,1} \varvec{w}_{n} } \right|^{2} \alpha_{n,1} P_{n,1} }}{{P_{N} }}}}}}{{\left( {BW \times \log_{2} \left( {1 + \frac{{\eta \left| {\varvec{h}_{n,1} \varvec{w}_{n} } \right|^{2} \alpha_{n,1} P_{n,1} }}{{P_{N} }}} \right) + c_{1} } \right)^{2} }} \\ & \quad - \; b_{2} \frac{{BW \times \frac{1}{\ln 2}\frac{{\frac{{\eta \left| {\varvec{h}_{n,2} \varvec{w}_{n} } \right|^{2} \left( { - 1} \right)P_{n,2} }}{{\left| {\varvec{h}_{n,2} \varvec{w}_{n} } \right|^{2} \alpha_{n,1} P_{n,1} + \left| {\varvec{h}_{n,2} \mathop \sum \nolimits_{k = 1,k \ne n}^{N} \varvec{w}_{k} x_{k} } \right|^{2} + P_{N} }}}}{{1 + \frac{{\eta \left| {\varvec{h}_{n,2} \varvec{w}_{n} } \right|^{2} \left( {1 - \alpha_{n,1} } \right)P_{n,2} }}{{\left| {\varvec{h}_{n,2} \varvec{w}_{n} } \right|^{2} \alpha_{n,1} P_{n,1} + \left| {\varvec{h}_{n,2} \mathop \sum \nolimits_{k = 1,k \ne n}^{N} \varvec{w}_{k} x_{k} } \right|^{2} + P_{N} }}}}}}{{\left( {BW \times \log_{2} \left( {1 + \frac{{\eta \left| {\varvec{h}_{n,2} \varvec{w}_{n} } \right|^{2} \left( {1 - \alpha_{n,1} } \right)P_{n,2} }}{{\left| {\varvec{h}_{n,2} \varvec{w}_{n} } \right|^{2} \alpha_{n,1} P_{n,1} + \left| {\varvec{h}_{n,2} \mathop \sum \nolimits_{k = 1,k \ne n}^{N} \varvec{w}_{k} x_{k} } \right|^{2} + P_{N} }}} \right) + c_{2} } \right)^{2} }} = 0 \\ \end{aligned}$$

(A-9)

Simplify the numerator part:

$$\begin{aligned} D_{\alpha } f\left( {\alpha_{n,1} } \right) & = - b_{1} \frac{{\frac{1}{\ln 2} \frac{{\eta \left| {\varvec{h}_{n,1} \varvec{w}_{n} } \right|^{2} P_{n,1} }}{{ P_{N} + \eta \left| {\varvec{h}_{n,1} \varvec{w}_{n} } \right|^{2} \alpha_{n,1} P_{n,1} }}}}{{\left( {BW \times \log_{2} \left( {1 + \frac{{\eta \left| {\varvec{h}_{n,1} } \right|^{2} \alpha_{n,1} P_{n,1} }}{{P_{N} }}} \right) + c_{1} } \right)^{2} }} \\ & \quad + b_{2} \frac{{\frac{1}{{\ln 2}}~~\frac{{\eta \left| {\user2{h}_{{n,2}} \user2{w}_{n} } \right|^{2} P_{{n,2}} }}{{\left| {\user2{h}_{{n,2}} \user2{w}_{n} } \right|^{2} \alpha _{{n,1}} P_{{n,1}} + \left| {\user2{h}_{{n,2}} \mathop \sum \nolimits_{{k = 1,k \ne n}}^{N} \user2{w}_{k} x_{k} } \right|^{2} + \eta \left| {\user2{h}_{{n,2}} \user2{w}_{n} } \right|^{2} \left( {1 - \alpha _{{n,1}} } \right)P_{{n,2}} + P_{N} }}}}{{\left( {BW \times \log _{2} \left( {1 + \frac{{\eta \left| {\user2{h}_{{n,2}} \user2{w}_{n} } \right|^{2} \left( {1 - \alpha _{{n,1}} } \right)P_{{n,2}} }}{{\left| {\user2{h}_{{n,2}} \user2{w}_{n} } \right|^{2} \alpha _{{n,1}} P_{{n,1}} + \left| {\user2{h}_{{n,2}} \mathop \sum \nolimits_{{k = 1,k \ne n}}^{N} \user2{w}_{k} x_{k} } \right|^{2} + P_{N} }}} \right) + c_{2} } \right)^{2} }} = 0\\ \end{aligned}$$

(A-10)