Optimal Power Allocation for Downlink NOMA Systems

Abstract

Non-orthogonal multiple access (NOMA) has been recognized as a promising multi-user access technique for the next generation cellular communication networks. In this chapter, we first review the basic concepts of downlink NOMA transmission and introduce the two-user, multi-user, and multi-channel NOMA schemes. Then, we investigate the optimal power allocation for these downlink NOMA schemes under different performance criteria, including the maximin fairness, sum rate, and energy efficiency. User weights and quality-of-service (QoS) constraints are taken into account. We show that in most cases the optimal NOMA power allocation can be analytically characterized, while in other cases the NOMA power allocation problems can be numerically solved via convex optimization methods.

Appendix

1.1 A. Proof of Theorem 1

Since the constraints in $TU_{c}^{\text {EE}1}$ are all linear, it suffices to investigate the concavity of $H\left( p_{1},q,\alpha \right)$. The second-order derivative of $H\left( p_{1},q,\alpha \right)$ with respect to $p_{1}$ is

$$\begin{aligned} \frac{\partial ^{2}H}{\partial p_{1}^{2}}=\frac{1}{\ln 2}\left( \frac{\varUpsilon \varTheta }{\left( p_{1}\varGamma _{1}+1\right) ^{2}\left( p_{1}\varGamma _{2}+1\right) ^{2}}\right) , \end{aligned}$$

(6.49)

where $\varUpsilon =\sqrt{W_{2}B}\varGamma _{2}\left( p_{1}\varGamma _{1}+1\right) +\sqrt{W_{1}B}\varGamma _{1}\left( p_{1}\varGamma _{2}+1\right)$ and $\varTheta =\sqrt{W_{2}B}\varGamma _{2}-\sqrt{W_{1}B}\varGamma _{1}+\sqrt{B}\varGamma _{1}\varGamma _{2}p_{1}\left( \sqrt{W_{2}}-\sqrt{W_{1}}\right)$. Given $\varGamma _{1}\ge \varGamma _{2}$, if $W_{1}\ge W_{2}$, then $\frac{\partial ^{2}H}{\partial p_{1}^{2}}\le 0$. On the other hand, with $q\le P$ and if C2 holds, we have

$$\begin{aligned} \sqrt{W_{2}B}\varGamma _{2}-\sqrt{W_{1}B}\varGamma _{1}+\sqrt{B}\varGamma _{1}\varGamma _{2}p_{1}\left( \sqrt{W_{2}}-\sqrt{W_{1}}\right) \sqrt{W_{2}B}\varGamma _{2}-\sqrt{W_{1}B}\varGamma _{1}+\left( \sqrt{W_{2}}-\sqrt{W_{1}}\right) \sqrt{B}\varGamma _{1}\varGamma _{2}P\le 0, \end{aligned}$$

(6.50)

also implying $\frac{\partial ^{2}H}{\partial p_{1}^{2}}\le 0$. Following the similar manner, it can be verified that $\frac{\partial ^{2}H}{\partial q^{2}}\le 0$, $\frac{\partial ^{2}H}{\partial q\partial p_{1}}=0$ and $\frac{\partial ^{2}H}{\partial p_{1}\partial q}=0$. Therefore, the Hessian matrix

$$\begin{aligned} \left( \begin{array}{cc} \frac{\partial ^{2}H}{\partial p_{1}^{2}} &{} \frac{\partial ^{2}H}{\partial q\partial p_{1}}\\ \frac{\partial ^{2}H}{\partial p_{1}\partial q} &{} \frac{\partial ^{2}H}{\partial q^{2}} \end{array}\right) \end{aligned}$$

is a negative semidefinite matrix, indicating that $H\left( p_{1},q,\alpha \right)$ is a concave in $(p_{1},q)$.

1.2 B. Proof of Proposition 4

The Lagrange of $TU_{c}^{\text {EE}1}$ is given by

$$\begin{aligned} L=W_{1}R_{1}(p_{1})+W_{2}R_{2}(p_{1},q)-\alpha \left( P_{T}+q\right) +\mu \left( q-2p_{1}\right) -\lambda \left( q-P\right) \end{aligned}$$

(6.51)

with Lagrange multipliers $\mu$ and $\lambda \ge 0$. According to Theorem 1, $TU_{c}^{\text {EE}1}$ is a convex problem under condition C1 or C2. Therefore, its optimal solution is characterized by the following Karush–Kuhn–Tucker (KKT) conditions:

$$\begin{aligned} \frac{\partial L}{\partial p_{1}}=\frac{W_{1}B\varGamma _{1}}{\ln 2\left( 1+p_{1}\varGamma _{1}\right) }-\frac{W_{2}B\varGamma _{2}}{\ln 2\left( 1+p_{1}\varGamma _{2}\right) }-2\mu =0, \end{aligned}$$

(6.52)

$$\begin{aligned} \frac{\partial L}{\partial q}=\frac{W_{2}B\varGamma _{2}}{\ln 2\left( 1+q\varGamma _{2}\right) }-\alpha +\mu -\lambda =0, \end{aligned}$$

(6.53)

$$\begin{aligned} \mu \left( q-2p_{1}\right) =0, \end{aligned}$$

(6.54)

$$\begin{aligned} \lambda \left( q-P\right) =0. \end{aligned}$$

(6.55)

According to Definition 1, if $p_{1}=q/2$, then the NOMA system is SIC-unstable. Therefore, from (6.54), considering the SIC stability, we have $\mu =0$. Hence, from (6.52) we obtain the optimal $p_{1}^{\star }=\varOmega$. It follows from (6.55) that if $q<P$, then $\lambda =0$. Then, from (6.53) we obtain

$$\begin{aligned} 2\varOmega \le q=\frac{W_{2}B}{\alpha \ln 2}-\frac{1}{\varGamma _{2}}<P. \end{aligned}$$

(6.56)

On the other hand, if $q=P$, then from (6.53) we have

$$\begin{aligned} \lambda =\frac{W_{2}B\varGamma _{2}}{\ln 2\left( 1+P\varGamma _{2}\right) }-\alpha \ge 0, \end{aligned}$$

(6.57)

which leads to

$$\begin{aligned} \frac{W_{2}B}{\alpha \ln 2}-\frac{1}{\varGamma _{2}}\ge P. \end{aligned}$$

(6.58)

Therefore, the optimal q is given by $q^{\star }=\left[ \frac{W_{2}B}{\alpha \ln 2}-\frac{1}{\varGamma _{2}}\right] _{2\varOmega }^{P}$.

1.3 C. Proof of Proposition 5

The Lagrange of $TU_{c}^{\text {EE}2}$ is given by

$$\begin{aligned} L&=R_{1}(p_{1})+R_{2}(p_{1},q)-\alpha \left( P_{T}+q\right) +\mu \left( q-2p_{1}\right) -\lambda \left( q-P\right) \\&+\sigma _{1}\left( p_{1}-\frac{A_{1}-1}{\varGamma _{1}}\right) +\sigma _{2}\left( 1+q\varGamma _{2}-A_{2}-A_{2}p_{1}\varGamma _{2}\right) \nonumber , \end{aligned}$$

(6.59)

where $\mu$, $\lambda$, $\sigma _{1}$, and $\sigma _{2}$ are the Lagrange multipliers. The optimal solution is characterized by the following KKT conditions:

$$\begin{aligned} \frac{\partial L}{\partial p_{1}}=\frac{B\varGamma _{1}}{\ln 2\left( 1+p_{1}\varGamma _{1}\right) }-\frac{B\varGamma _{2}}{\ln 2\left( 1+p_{1}\varGamma _{2}\right) }-2\mu +\sigma _{1}-\sigma _{2}A_{2}\varGamma _{2}=0, \end{aligned}$$

(6.60)

$$\begin{aligned} \frac{\partial L}{\partial q}=\frac{B\varGamma _{2}}{\ln 2\left( 1+q\varGamma _{2}\right) }-\alpha +\mu -\lambda +\sigma _{2}\varGamma _{2}=0, \end{aligned}$$

(6.61)

$$\begin{aligned} \mu \left( q-2p_{1}\right) =0, \end{aligned}$$

(6.62)

$$\begin{aligned} \lambda \left( q-P\right) =0, \end{aligned}$$

(6.63)

$$\begin{aligned} \sigma _{1}\left( p_{1}-\frac{A_{1}-1}{\varGamma _{1}}\right) =0, \end{aligned}$$

(6.64)

$$\begin{aligned} \sigma _{2}\left( 1+q\varGamma _{2}-A_{2}-A_{2}p_{1}\varGamma _{2}\right) =0. \end{aligned}$$

(6.65)

In (6.62), considering the SIC stability, we have $q>2p_{1}$ and hence $\mu =0$. Note that $\sigma _{2}\ne 0$. To see this, if $\sigma _{2}=0$, according to (6.60), we have

$$\begin{aligned} \frac{B\varGamma _{1}}{\ln 2\left( 1+p_{1}\varGamma _{1}\right) }-\frac{B\varGamma _{2}}{\ln 2\left( 1+p_{1}\varGamma _{2}\right) }+\sigma _{1}=0 \end{aligned}$$

(6.66)

which, however, does not hold since $\frac{B\varGamma _{1}}{\ln 2\left( 1+p_{1}\varGamma _{1}\right) }-\frac{B\varGamma _{2}}{\ln 2\left( 1+p_{1}\varGamma _{2}\right) }+\sigma _{1}>0$ with $\varGamma _{1}\ge \varGamma _{2}$.

We consider two cases: (1) $\sigma _{1}\ne 0,\sigma _{2}\ne 0$; and (2) $\sigma _{1}=0,\sigma _{2}\ne 0$. First, if $\sigma _{1}\ne 0,\sigma _{2}\ne 0$, the optimal solution can be easily obtained as

$$\begin{aligned} p_{1}^{\star }=\frac{1+q^{\star }\varGamma _{2}-A_{2}}{A_{2}\varGamma _{2}},~q^{\star }=\varUpsilon \end{aligned}$$

(6.67)

from (6.64) and (6.65). Then, if $\sigma _{1}=0,\sigma _{2}\ne 0$, according to (6.60) and (6.61), we have

$$\begin{aligned} \frac{A_{2}\varGamma _{2}}{\left( 1+q\varGamma _{2}\right) }+\left( \frac{1}{1/\varGamma _{1}+p_{1}}-\frac{1}{1/\varGamma _{2}+p_{1}}\right) =\left( \alpha +\lambda \right) A_{2}. \end{aligned}$$

(6.68)

From (6.65), we obtain $p_{1}=\frac{1+q\varGamma _{2}-A_{2}}{A_{2}\varGamma _{2}}$, which along with (6.68) leads to

$$\begin{aligned} q^{\star }=\frac{1}{\alpha +\lambda }-\frac{A_{2}}{\varGamma _{1}}+\frac{A_{2}-1}{\varGamma _{2}}. \end{aligned}$$

(6.69)

It follows from (6.63) that if $q<P$, then $\lambda =0$. From (6.69), we obtain

$$\begin{aligned} \Upsilon \le q=\frac{1}{\alpha }-\frac{A_{2}}{\varGamma _{1}}+\frac{A_{2}-1}{\varGamma _{2}}<P. \end{aligned}$$

(6.70)

On the other hand, if $q=P$, then from (6.53) we have

$$\begin{aligned} \lambda =\frac{\varGamma _{1}\varGamma _{2}}{A_{2}\varGamma _{2}-\left( A_{2}-1\right) \varGamma _{1}+P\varGamma _{2}\varGamma _{1}}-\alpha \ge 0, \end{aligned}$$

(6.71)

which leads to

$$\begin{aligned} \frac{1}{\alpha }-\frac{A_{2}}{\varGamma _{1}}+\frac{A_{2}-1}{\varGamma _{2}}\ge P. \end{aligned}$$

(6.72)

Therefore, optimal q is given by $q^{\star }=\left[ \frac{W_{2}B}{\alpha \ln 2}-\frac{1}{\varGamma _{2}}\right] _{\Upsilon }^{P}$.

1.4 D. Proof of Theorem 2

Let $q_{i}=\sum _{j=1}^{i}p_{j}$, then $q_{N}=P$ and $p_{i}=\frac{2^{t}-1}{\varGamma _{i}}\left( \sum _{j=1}^{i-1}p_{j}\varGamma _{i}+1\right)$ can be transformed into $q_{i}=q_{i-1}2^{t}+\frac{2^{t}-1}{\varGamma _{i}}$. Thus, we obtain $P=q_{N}=\sum _{i=1}^{N}\frac{\left( 2^{t}-1\right) 2^{(N-i)t}}{\varGamma _{i}}\ge \chi$, implying $t\ge 1$ and $p_{i}\ge p_{i-1}$ for $i=2,\ldots ,N$. Therefore, this solution satisfies the power order constraint.