for a two-user NOMA system in the absence of CSIT and CDIT. As shown in Section 5.5, this framework is a simple yet powerful tool for solving resource allocation problems in unknown environments, but is limited when dealing with only discrete sets of actions which causes optimality loss. Therefore, high resolution quantization, leading to more actions, was needed to approach the optimal action but at the cost of additional complexity and time to explore and exploit all actions.

The MAB framework could also be applied to the two-user NOMA system in the presence of an ambient backscatter device, assuming no knowledge on both the direct and the backscattered channels. In this case, the set of actions is expanded to include the reflection coefficient $\rho \in [0, 1]$ where each action would be defined as $\mathbf{a} = (i, \rho, \mathbf{p})$. Since the reflection coefficient is a continuous variable, an additional quantization is needed to make use of the MAB algorithms. This would require even more complexity and larger amount of time to reach the best performance. Therefore, we need to find ways to deal with continuous variable sets to overcome the quantization-caused loss and the computational complexity.

The use of deep learning techniques could be a great solution to deal with the aforementioned challenges where it has already been proved to be useful for various resource allocation problems [123, 124]. The investigation in Chapter 5 could also be extended to the multi-user case where user pairing is applied. In this case, the problem can be formulated as a Markov decision process (MDP) and deep reinforcement learning could be very promising [125]. Depending on the nature of the problem, one can compare the performance of classic MAB-based methods and deep learning or deep reinforcement learning methods with respect to the system performance and computational complexity similarly to [126].

**NOMA and multi-antenna systems:** while NOMA was originally intended for single-antenna deployments, it is inefficient in most multi-antenna deployments [127]. The first false assumption about NOMA is that it is an effective strategy in multi-antenna settings since it achieves capacity in single-antenna settings for the broadcast channels. In [128], massive multiple-input multiple-output (MIMO) was compared to NOMA where it was analytically proven that when the number of antennas at the transmitter is equal to the number of users, NOMA outperforms massive MIMO. However, when the number of antennas is much larger than the users, massive MIMO outperforms NOMA. Also in [129], the joint spectral efficiency and energy efficiency maximization was investigated where it was shown that the rate splitting multiple access (RSMA) technique outperforms NOMA in a multiple-input single-output setting.

Therefore, an interesting area of research is to extensively investigate in which situations NOMA performs better than other multi-antenna technologies. Additionally, a hybrid approach could be used to pair clusters by identifying the users that would benefit from NOMA while the rest are served via other multi-antenna technologies.
**Application of Reconfigurable Intelligent Surfaces within NOMA:** as discussed in Section 2.1, the performance gain of NOMA over OMA relies on the exploitation of the users' asymmetric channels condition. This gain becomes limited when the channel conditions are not suitable for its implementation (e.g., in the case of symmetric channels). The users' channel conditions have always been regarded as being fixed and non-tunable where they depend only on the propagation environment. Moreover, the users are assigned a decoding order that primarily depends on the order of their channel conditions. Therefore, users with lower decoding order, i.e., weak users, may only achieve poor data rates as a result of the uncontrolled inter-user interference, while users with higher decoding order, i.e., strong users, can obtain higher data rates by applying SIC to cancel the inter-user interference [125]. However in practice, and especially in heterogeneous networks, the QoS requirements of users may not line up with the order of their channels condition where, for example, a weak user may require a higher data rate than a strong user.

Reconfigurable Intelligent Surfaces (RIS) can help overcome these issues where, as defined in Section 2.2, they have the capability of smartly modifying the wireless communication environment. Therefore, implementing RIS with NOMA would be an attractive research direction, ensuring that the propagation environment of the multiplexed users is intelligently and efficiently customized to reap the benefits of NOMA [130]. The channel asymmetry between users can be increased rather than being dictated by the wireless environment by carefully choosing the location and the design of the reflection coefficients of the RIS, resulting in a higher gain over OMA. Moreover, the users decoding order can be arranged according to their QoS requirements while RIS can help achieve the corresponding desired arrangement of the channels condition [125].
# Appendix A

## Proof of Theorem 1

We denote $\theta_k(\mathbf{p}) = \sum_{i=1}^k p_k, \forall 1 \le k \le K$ with $\theta_0(\mathbf{p}) = 0$. The Lagrangian of the convex problem (3.10) is given by

$$ \mathcal{L} = \sum_{i=1}^{K} \frac{1}{2} \log_2 \left( \frac{1 + H_{i|0} \theta_i(\mathbf{p})}{1 + H_{i|0} \theta_{i-1}(\mathbf{p})} \right) - \alpha (\theta_K(\mathbf{p}) + P_c) + \lambda (P_{\max} - \theta_K(\mathbf{p})) \\ + \sum_{i=1}^{K} \beta_i \left( \theta_i(\mathbf{p}) - A_i \theta_{i-1}(\mathbf{p}) - \frac{A_i - 1}{H_{i|0}} \right), \qquad (A.1) $$

where $\lambda$ and $\beta = (\beta_1, \dots, \beta_K)$ are the positive Lagrange multipliers for the overall power constraint and the $K$ minimum rate constraints, respectively. The KKT optimality conditions are necessary and sufficient for optimality for convex optimization problems and they imply that $\frac{\partial \mathcal{L}}{\partial p_k} = 0, \forall k$, at the solution. Thus, we also have that the difference between two consecutive Lagrange derivatives equals zero: $\frac{\partial \mathcal{L}}{\partial p_{k+1}} - \frac{\partial \mathcal{L}}{\partial p_k} = 0$, which leads to the following equations

$$ A_{k+1} \beta_{k+1} = \frac{1}{2 \ln 2} \left( \frac{H_{k|0}}{1 + H_{k|0} \theta_k(\mathbf{p})} - \frac{H_{k+1|0}}{1 + H_{k+1|0} \theta_k(\mathbf{p})} \right) + \beta_k, \qquad (A.2) $$

for all $k \in \{1, \dots, K-1\}$. We discuss two cases below.

i) If $H_{1|0} > H_{2|0}$ then $\beta_1 \ge 0$ and $A_2 > 0$, the above equation for $k = 1$ implies that $\beta_2 > 0$. Since $H_{k|0} \ge H_{k+1|0}$ and $A_{k+1} > 0$, for all $k \ge 2$, the fact that $\beta_2 > 0$ has a cascading effect in the above equations and leads to $\beta_{k+1} > 0, \forall k \in \{2, \dots, K-1\}$. Hence, we have that $\beta_k > 0, \forall k \ge 2$. This means that all QoS constraints of all weaker users, $k \ge 2$, are active at the solution and they meet no more than their minimum rate requirement

$$ \theta_k(\mathbf{p}) = A_k \theta_{k-1}(\mathbf{p}) + \frac{A_k - 1}{H_{k|0}}, \quad \forall k > 1. \qquad (A.3) $$

By induction, we can show that

$$ \theta_k(\mathbf{p}) = \theta_1(\mathbf{p}) \prod_{i=2}^{k} A_i + \sum_{i=2}^{k-1} \frac{A_i - 1}{H_{i|0}} \prod_{j=i+1}^{k} A_j + \frac{A_k - 1}{H_{k|0}}, \quad \forall k > 1. \qquad (A.4) $$
Knowing that $p_k = \theta_k(\mathbf{p}) - \theta_{k-1}(\mathbf{p})$, for all $k > 1$, and that $\theta_1(\mathbf{p}) = p_1$, the above equations allow us to express all the powers allocated to the weaker users as functions of $p_1$, the power allocated to the strongest user as in (3.11). Hence, the multi-variable problem in (3.10) is reduced to the following single variable problem

$$ \max_{p_1} f_1(p_1) \quad \text{s.t.} \quad \ell \le p_1 \le u, \qquad (A.5) $$

where the new objective function is

$$ f_1(p_1) = \frac{1}{2}\log_2(1+H_{1|0}p_1) + \frac{1}{2}\log_2\left(\prod_{i=2}^{K} A_i - \alpha\left(p_1 \prod_{i=2}^{K} A_i + \sum_{i=2}^{K-1} \frac{A_i-1}{H_{i|0}} \prod_{j=i+1}^{K} A_j + \frac{A_K-1}{H_{K|0}} + P_c\right)\right), \qquad (A.6) $$

and the lower and upper bounds on $p_1$, coming from the QoS and maximum power constraints, are given in (3.12) and (3.13) respectively. The second-order partial derivative of (A.6) w.r.t $p_1$ equals

$$ \frac{\partial^2 f_1(p_1)}{\partial p_1^2} = \frac{-1}{2 \ln 2} \times \frac{(H_{1|0})^2}{(1 + H_{1|0}p_1)^2}. \qquad (A.7) $$

Hence, the objective function $f_1(p_1)$ is concave w.r.t $p_1$ and the optimal solution $p_1^*(\alpha)$ is either the critical point $\bar{p}_1(\alpha)$ in (3.14) canceling the first order derivative, or one of the borders of the new feasible set.

ii) If $H_{1|0} = H_{2|0}$, the main trick is to consider receiver 1 and 2 as a single entity, having the strongest channel gains $H'_{1|0} = H_{1|0} = H_{2|0}$, and whose achievable data rate and allocated power are considered as $R'_1 = (R_1 + R_2)$ and $p'_1 = p_1 + p_2$ respectively. Having done this variable change, the same discussion follows subsequently as i); two cases arise: a) $H'_{1|0} = H_{3|0}$, and b) $H'_{1|0} > H_{3|0}$. In the latter, we obtain the analytical closed-form expressions of $p_k^*, \forall k \ge 3$ as a function of $p'_1$ and $p'_1^*$ is obtained by solving the resulting problem in (A.5). Then, we can split the power $p'_1^*$ such that $p_2^*$ is the enough power required to satisfy the QoS requirement for receiver 2, and the remaining power $p_1^* = (p'_1)^* - p_2^*$. In the former case, a recurring reasoning can be applied etc.

Finally, the same reasoning applies in the fully symmetric case, in which $H_{1|0} = H_{2|0} = \dots = H_{K|0}$. Note that such an extreme case is quite unlikely to occur in practice. Moreover, NOMA in this case would likely not outperform OMA since it exploits the channel gain difference between different receivers as shown in Section 2.1. This concludes the proof.
# Appendix B

## Proof of Theorem 2

Since only the sum rate term in the energy-efficiency objective depends on the reflection coefficient $\rho$, we can only focus on optimizing the sum rate. Let $\theta_k(\mathbf{p}) = \sum_{i=1}^k p_k$ with $\theta_0(\mathbf{p}) = 0$ and $f(\rho)$ denote the sum rate as a function of the reflection coefficient $\rho$ for a fixed power allocation vector $\mathbf{p}$

$$f(\rho) = \sum_{k=1}^{K} \frac{1}{2} \log_2 \left( \frac{1 + H_{k|1}(\rho)\theta_k(\mathbf{p})}{1 + H_{k|1}(\rho)\theta_{k-1}(\mathbf{p})} \right). \quad (B.1)$$

Knowing that $H_{k|1} = \frac{(h_k + \sqrt{\rho}gg_k)^2}{\sigma_k^2}$, the first-order and second-order derivatives of $f(\rho)$ are respectively given by

$$\frac{\partial f(\rho)}{\partial \rho} = \frac{1}{2 \ln 2} \sum_{k=1}^{K} \frac{2h_k gg_k(\rho)^{-1/2} + (gg_k)^2}{\left(\frac{\sigma_k^2}{\theta_k(\mathbf{p})} + (h_k^2 + 2h_k gg_k \sqrt{\rho} + (gg_k)^2 \rho)\right)} - \frac{2h_k gg_k(\rho)^{-1/2} + (gg_k)^2}{\left(\frac{\sigma_k^2}{\theta_{k-1}(\mathbf{p})} + (h_k^2 + 2h_k gg_k \sqrt{\rho} + (gg_k)^2 \rho)\right)} \quad (B.2)$$
$$\geq 0,$$

$$\frac{\partial^2 f(\rho)}{\partial \rho^2} = \frac{1}{2 \ln 2} \sum_{k=1}^{K} \frac{-h_k gg_k(\rho)^{-3/2}}{\left(\frac{\sigma_k^2}{\theta_k(\mathbf{p})} + (h_k^2 + 2h_k gg_k \sqrt{\rho} + (gg_k)^2 \rho)\right) + \frac{h_k gg_k(\rho)^{-3/2}}{\left(\frac{\sigma_k^2}{\theta_{k-1}(\mathbf{p})} + (h_k^2 + 2h_k gg_k \sqrt{\rho} + (gg_k)^2 \rho)\right)}} - \frac{(h_k gg_k(\rho)^{-1/2} + (gg_k)^2)(2h_k gg_k(\rho)^{-1/2} + (gg_k)^2)}{\left(\frac{\sigma_k^2}{\theta_k(\mathbf{p})} + (h_k^2 + 2h_k gg_k \sqrt{\rho} + (gg_k)^2 \rho)\right)^2} \quad (B.3)$$
$$+ \frac{(h_k gg_k(\rho)^{-1/2} + (gg_k)^2)(2h_k gg_k(\rho)^{-1/2} + (gg_k)^2)}{\left(\frac{\sigma_k^2}{\theta_{k-1}(\mathbf{p})} + (h_k^2 + 2h_k gg_k \sqrt{\rho} + (gg_k)^2 \rho)\right)^2} \leq 0.$$

Since $\theta_k(\mathbf{p}) \geq \theta_{k-1}(\mathbf{p})$ by construction, the sum rate $f(\rho)$ is increasing and concave in $\rho$. Hence, in order to maximize the sum rate, the reflection coefficient $\rho$ must be chosen as large as possible while meeting the constraints of (EE1). In particular, the constraints (C2)-(C4) depend on $\rho$.
Since $H_{k|1}(\rho)$ is an increasing function of $\rho \in [0, 1]$, we can see that the higher the value of $\rho$ the less constrained (C2) becomes, implying a larger possible set for the other variables of the problem: $\theta_k(\mathbf{p})$, $\forall k$. Hence, choosing the largest value of $\rho$ is optimal in terms of (C2).

Now, to compute the largest possible value of $\rho$, we can focus only on the constraints (C3) and (C4) with no loss of optimality. These constraints require that $\gamma_{k \to i} \ge \gamma_{k \to k}$, $\forall k \ge 2$, $\forall i \le k-1$ and $0 \le \rho \le 1$.

First, note that for any user $k \ge 2$ and any $i \le k-1$, we have the following equivalencies

$$ \begin{aligned} \gamma_{k \to i} &\ge \gamma_{k \to k} \\ \Leftrightarrow \frac{H_{i|1}(\rho)p_k}{H_{i|1}(\rho) \sum_{j=1}^{k-1} p_j + 1} &\ge \frac{H_{k|1}(\rho)p_k}{H_{k|1}(\rho) \sum_{j=1}^{k-1} p_j + 1} \quad (B.4) \\ \Leftrightarrow H_{i|1}(\rho) &\ge H_{k|1}(\rho), \end{aligned} $$

where the first equivalence follows from our notations and the definitions of $\gamma_{k \to i}$ and $\gamma_{k \to k}$ in (4.6); and the second equivalence is obtained after some simple derivations. Hence, constraint (C3) is equivalent to the following set of $k-1$ constraints: $H_{1|1}(\rho) \ge H_{k|1}(\rho)$, ..., and $H_{k-1|1}(\rho) \ge H_{k|1}(\rho)$, for any user $k \ge 2$.

For example, for user $k=2$, the only constraint writes as $H_{1|1}(\rho) \ge H_{2|1}(\rho)$. For user $k=3$, the two constraints write as $H_{1|1}(\rho) \ge H_{3|1}(\rho)$, and $H_{2|1}(\rho) \ge H_{3|1}(\rho)$. All these constraints simplify to $H_{1|1}(\rho) \ge H_{2|1}(\rho) \ge H_{3|1}(\rho)$. The same reasoning can be extended up to user $K$, and all the constraints reduce to

$$ H_{1|1}(\rho) \ge H_{2|1}(\rho) \ge \dots \ge H_{K|1}(\rho). \quad (B.5) $$

Now we focus on a single inequality of the form $H_{k|1}(\rho) \le H_{k+1|1}(\rho)$ in the inequality chain above. The aim is to find the largest value of $\rho \in [0, 1]$ fulfilling this constraint, which can be equivalently expressed as

$$ \begin{aligned} H_{k|1}(\rho) &\le H_{k+1|1}(\rho) \\ \Leftrightarrow \frac{(h_k + \sqrt{\rho} g g_k)^2}{\sigma_k^2} &\le \frac{(h_{k+1} + \sqrt{\rho} g g_{k+1})^2}{\sigma_{k+1}^2} \quad (B.6) \\ \Leftrightarrow g\left(\frac{g_{k+1}}{\sigma_{k+1}} - \frac{g_k}{\sigma_k}\right)\sqrt{\rho} &\ge \left(\frac{h_k}{\sigma_k} - \frac{h_{k+1}}{\sigma_{k+1}}\right), \end{aligned} $$

where the first equivalence follows from the definition of $H_{k|1}(\rho)$ and $H_{(k+1)|1}(\rho)$ and the second one by simply rearranging the terms. Two cases can arise based on the order of $g_{k+1}$ and $g_k$:

a) If $g_{k+1} > g_k$, since $\frac{h_k}{\sigma_k} \ge \frac{h_{k+1}}{\sigma_{k+1}}$ by assumption, the constraints (B.6) and C(4) of the optimization problem lead to the following upper bounds

$$ \rho = \min \left\{ 1, \left( \frac{h_k}{\sigma_k} - \frac{h_{k+1}}{\sigma_{k+1}} \right)^2 / \left( g \left( \frac{g_{k+1}}{\sigma_{k+1}} - \frac{g_k}{\sigma_k} \right) \right)^2 \right\}, \quad \forall k > 1. \quad (B.7) $$
b) If $g_{k+1} \le g_k$, then (B.6) becomes trivial since $\frac{h_k}{\sigma_k} \ge \frac{h_{k+1}}{\sigma_{k+1}}$ by assumption. In this case, we only have the constraint $C(4)$ to be met: $0 \le \rho \le 1$.

Taking all this into account, the optimal reflection coefficient $\rho^*$ can be expressed as
in Theorem 2, which completes the proof.
# Appendix C

## Proof of Theorem 3

The objective function of (EE3) is increasing w.r.t $\rho_1$ for fixed $\mathbf{p}$ and $\rho_2$; and, is also increasing w.r.t $\rho_2$ for fixed $\mathbf{p}$ and $\rho_1$; the optimal reflection coefficients lie on the Pareto boundary, i.e., all feasible points such that none of their coordinates can be increased while remaining feasible, of the feasible set defined by the constraints (C3) – (C4). After some mathematical derivations, the constraint (C3) can be rewritten as $\sqrt{\rho_1}(G_{12}-G_{11}) + \sqrt{\rho_2}(G_{22}-G_{21}) \le G_1 - G_2$.

We now provide the optimal values of $(\rho_1, \rho_2)$ by analyzing geometrically the feasible set and its Pareto optimal boundary in the four possible cases given in Theorem 3.

**Case [H1]:** $(G_{12} - G_{11}) \le 0$ and $(G_{22} - G_{21}) \le 0$

In this case, since $(G_1 - G_2) \ge 0$ by assumption, the constraints are always fulfilled for all values of $\rho_1, \rho_2 \in [0, 1]$. Hence, the optimal solution is unique $\rho_1^* = \rho_2^* = 1$.

**Case [H2]:** $(G_{12} - G_{11}) \le 0$ and $(G_{22} - G_{21}) > 0$

Under [H2], one can show that $\overline{\rho}_1 < 0$; $\overline{\rho}_2 \ge 0$ and $\tilde{\rho}_2 > \overline{\rho}_2$. Several cases can arise depending on the value of $\overline{\rho}_2$ as depicted in Figure C.1.

[H21] If $\overline{\rho}_2 > 1$, which leads to $\tilde{\rho}_1 < 0$ and $\tilde{\rho}_2 > 1$, the optimal solution is unique: $\rho_1^* = \rho_2^* = 1$.

Otherwise $\overline{\rho}_2 \le 1$ leads to $\tilde{\rho}_1 \ge 0$ and $\tilde{\rho}_2 > \overline{\rho}_2 \ge 0$. Two sub-cases arise: either [H221] if $\tilde{\rho}_2 > 1$, then $\rho_1^* = \rho_2^* = 1$; or [H222] if $\tilde{\rho}_2 \le 1$, then the Pareto optimal boundary also reduces to a unique point $\rho_1^* = 1, \rho_2^* = \tilde{\rho}_2$.

Hence to summarize, the optimal values of the reflection coefficients are given as $\rho_1^* = 1; \rho_2^* = \min\{1, \tilde{\rho}_2\}$.
<figure><img src="image_30.png" /><figcaption>**Figure C.1:** The three cases that can arise under [H2]: the feasible set is depicted in red. The Pareto boundary reduces to a unique solution.</figcaption></figure>

**Case [H3]:** $(G_{12} - G_{11}) > 0$ and $(G_{22} - G_{21}) \le 0$

This case is similar to the previous case [H2], hence the detailed proof is omitted.

**Case [H4]:** $(G_{12} - G_{11}) > 0$ and $(G_{22} - G_{21}) > 0$

We can prove the following:

$$ \bar{\rho}_i \ge 0 \text{ and } \tilde{\rho}_i \le \bar{\rho}_i, \quad \forall i \in \{1, 2\}; $$

$$ \text{If } \bar{\rho}_i > 1, \text{ then } \tilde{\rho}_j > 0, \quad \forall i, j \in \{1, 2\} \text{ s.t. } j \neq i; $$

$$ \text{If } \bar{\rho}_i < 1, \text{ then } \tilde{\rho}_j < 0, \quad \forall i, j \in \{1, 2\} \text{ s.t. } j \neq i; $$

$$ \text{If } \tilde{\rho}_i > 1, \text{ then } \tilde{\rho}_j > 1, \quad \forall i, j \in \{1, 2\} \text{ s.t. } j \neq i; $$

$$ \text{If } \tilde{\rho}_i < 1, \text{ then } \tilde{\rho}_j < 1, \quad \forall i, j \in \{1, 2\} \text{ s.t. } j \neq i. $$

The optimal solution lies on the Pareto boundary of the feasible set depicted in red in Figure C.2. Note that except for the sub-case [H41], the Pareto boundary of the feasible set is not unique but a continuous segment (in blue) of the line between $\bar{\rho}_1$ and $\bar{\rho}_2$. This set can be characterized in a parametric manner as $\rho_1 = \beta\bar{\rho}_1$, $\rho_2 = (1 - \beta)\bar{\rho}_2$, where $\beta$ ranges over the interval specified in each subcases of Figure C.2. Note that except for subcase [H41], $H_{1|1,1}(\rho) = H_{2|1,1}(\rho)$.

In [H41], both $\bar{\rho}_i$ and $\tilde{\rho}_i$, $i \in \{1, 2\}$, are larger than 1, hence the optimal solution is $\rho_1^* = \rho_2^* = 1$. In the other four cases, we need to find the reflection coefficients on the
<figure><img src="image_31.png" /><figcaption>**Figure C.2:** The five cases that can arise under [H4]: the feasible set is depicted in red. The Pareto boundary in cases [H42]-[H44] is the blue segment given by $\rho_1 = \beta\overline{\rho}_1$; $\rho_2 = (1 - \beta)\overline{\rho}_2$.</figcaption></figure>

Pareto boundary: $\rho_1 = \beta\overline{\rho}_1$, $\rho_2 = (1 - \beta)\overline{\rho}_2$, which maximize the objective function. Hence, the problem is reduced to a single variable optimization over $\beta$, whose objective is

$$f(\beta) = \log_2(1 + (p_1 + p_2)(G_2 + \beta\overline{\rho}_1 G_{12} + (1 - \beta)\overline{\rho}_2 G_{22})),$$

of derivative

$$\begin{align*}
\frac{\partial f(\beta)}{\partial \beta} &= \frac{(p_1 + p_2)(G_{12}\overline{\rho}_1 - G_{22}\overline{\rho}_2)}{\ln 2(1 + (p_1 + p_2)H_{2|1,1}(\beta))} \\
&\propto (G_{12}\overline{\rho}_1 - G_{22}\overline{\rho}_2) \\
&\propto \frac{(G_1 - G_2)(G_{22}G_{11} - G_{12}G_{21})}{(G_{12} - G_{11})(G_{22} - G_{21})}.
\end{align*}$$

Hence, the objective function is either decreasing or increasing w.r.t $\beta$ depending on the sign of $(G_{22}G_{11} - G_{12}G_{21})$.
a) If $(G_{22}G_{11} - G_{12}G_{21}) \ge 0$, the objective function is increasing in $\beta$, hence its optimal value is the upper-bound of its feasible interval, specified in each sub-case.

b) If $(G_{22}G_{11} - G_{12}G_{21}) < 0$, the objective function is decreasing in $\beta$, hence its optimal value is the lower-bound of its feasible interval, specified in each sub-case.

This completes the Proof.
# Appendix D

## Information-theoretic achievable rate region

### D.1 Proof of Theorem 4

First, we have to introduce the proper information-theoretic definitions needed for the coding and decoding procedures. We consider a two-sender K-receiver discrete memoryless channel defined by the tuple $(\mathcal{X} \times \mathcal{B}, p(y_1, \dots, y_K | x, b), \mathcal{Y}_1 \times \dots \times \mathcal{Y}_K)$ that consists of $K + 2$ finite sets: $\mathcal{X}$, $\mathcal{B}$, $\mathcal{Y}_1, \dots, \mathcal{Y}_K$ and a collection of conditional probability mass functions (pmfs) $p(y_1, \dots, y_2 | x, b)$ defined on $\mathcal{Y}_1 \times \dots \times \mathcal{Y}_K$. A $(2^{nR_0}, 2^{nR_1}, \dots, 2^{nR_K}, n)$ code consists of:

* $K + 1$ message sets $[1:2^{nR_0}], [1:2^{nR_1}], \dots, [1:2^{nR_K}]$,

* two encoders, where encoder 1, the source, assigns a codeword $x^n(m_1, \dots, m_K)$ to each message $(m_1, \dots, m_K) \in [1:2^{nR_1}] \times \dots \times [1:2^{nR_K}]$ and encoder 2, the backscatter device, assigns a codeword $b^n(m_0)$ to each message $m_0 \in [1:2^{nR_0}]$,

* $K$ decoders, where decoder $i \in \{1, \dots, K\}$ (or receiver, or user $i$) assigns an estimate $(\hat{m}_{0 \to i}, \hat{m}_{K \to i} \dots \hat{m}_{i+1 \to i}, \hat{m}_{i \to i}) \in [1:2^{nR_0}] \times [1:2^{nR_K}] \times \dots \times [1:2^{nR_{i+1}}] \times [1:2^{nR_i}]$, or an error message $e$ to each received sequence $y_i^n$, where $\hat{m}_{j \to i}, j \in \{K, \dots, i\}$ represents the estimation of $m_j$ when decoded by receiver $i$.

We assume that the message, which is a $K+1$ tuple: $(M_0, M_1, \dots, M_K)$, is uniformly distributed over $[1:2^{nR_0}] \times [1:2^{nR_1}] \times \dots \times [1:2^{nR_K}]$. The average probability of error is then defined as
$$P_e^n = \mathbb{P}\left\{\bigcup_{i=1}^n (\hat{M}_{0 \to i}, \hat{M}_{i \to i}) \neq (M_0, M_i)\right\}$$

#### Codebook generation

**Encoder 1:** Fix the pmfs $p(u_K)p(u_{K-1}|u_K) \dots p(x|u_2, \dots, u_K)$.

i) First, randomly and independently generate $2^{nR_K}$ sequences $u_K^n(m_K) \in [1:2^{nR_K}]$ each according to $\prod_{i=1}^n p_{U_K(u_i,K)}$.
ii) For each $m_{k+1} \in [1: 2^{nR_{k+1}}], k \in \{K-1, K-2, \dots, 2\}$ successively in this order, randomly and conditionally independently generate $2^{nR_k}$ sequences $u_k^n(m_k, m_{k+1}, \dots, m_K)$, for all $m_k \in [1: 2^{nR_k}]$, each according to $\prod_{i=1}^n p_{U_k|U_{k+1}, \dots, U_K}(u_{i,k}|u_{i,k+1}(m_{k+1}, \dots, m_K), \dots, u_{i,K}(m_K))$.

iii) For each $m_2 \in [1: 2^{nR_2}]$ randomly and conditionally independently generate $2^{nR_1}$ sequences $x^n(m_1, \dots, m_K), m_1 \in [1: 2^{nR_1}]$ each according to $\prod_{i=1}^n p_{X|U_{2+\cdot,K}}(x_i|u_{i,2}(m_2, \dots, m_K), \dots, u_{i,K}(m_K))$. To send message $(m_1, \dots, m_K)$, encoder 1 transmits codeword $x^n(m_1, \dots, m_K)$.

**Encoder 2:** Randomly and independently generate $2^{nR_0}$ sequences $b^n(m_0), m_0 \in [1: 2^{nR_0}]$, each according to $\prod_{i=1}^n p_B(b_i)$. To send message $m_0$, encoder 2 transmits codeword $b^n(m_0)$.

## Decoding procedure

Decoder $K$ decides that $(\hat{m}_{0 \to K}, \hat{m}_{K \to K})$ was sent, if it is the unique message pair such that $(u_K^n(\hat{m}_{K \to K}), b^n(\hat{m}_{0 \to K}), y_K^n) \in \mathcal{T}_\epsilon^{(n)}$; otherwise it declares an error.

Each decoder $i \in \{1, \dots, K-1\}$ successively decodes the messages $m_j, j \in \{K, \dots, i+1\}$ in this specific order by the following steps.

i) Each decoder $i$ first jointly decodes $U_K$ and $B$ and decides that $(\hat{m}_{0 \to i}, \hat{m}_{K \to i})$ was sent, if it is the unique pair such that $\{(u_K^n(\hat{m}_{K \to i}), b^n(\hat{m}_{0 \to i}), y_i^n) \in \mathcal{T}_\epsilon^n\}$; otherwise it declares an error.

ii) If such $(\hat{m}_{0 \to i}, \hat{m}_{K \to i})$ is found, decoder $i$ declares that $\hat{m}_{i \to i}$ is sent if it is the unique message such that

$$ \{(u_K^n(\hat{m}_{K \to i}), u_{K-1}^n(\hat{m}_{K-1 \to i}, \hat{m}_{K \to i}), \dots, u_i^n(\hat{m}_{i \to i}, \dots, \hat{m}_{K \to i}), b^n(\hat{m}_{0 \to i}), y_i^n) \in \mathcal{T}_\epsilon^n\}, $$ where the estimates $\hat{m}_{j \to i}, j \in \{K-1, \dots, i+1\}$ are obtained in a successive manner starting from $\hat{m}_{K-1 \to i}$ to $\hat{m}_{i+1 \to i}$ as decoder $i$ declares that $\hat{m}_{j \to i}, j \in \{K-1, \dots, i+1\}$ is sent if it is the unique message such that

$$ \{(u_K^n(\hat{m}_{K \to i}), u_{K-1}^n(\hat{m}_{K-1 \to i}, \hat{m}_{K \to i}), \dots, u_j^n(\hat{m}_{j \to i}, \dots, \hat{m}_{K \to i}), b^n(\hat{m}_{0 \to i}), y_i^n) \in \mathcal{T}_\epsilon^n\}; $$ otherwise, it declares an error. Note that whenever an error has been declared, the decoding process stops.

## Analysis of the probability of error

Assume without loss of generality that the message $(M_0 = M_1 = \dots = M_K = 1)$ was sent [40]. By the symmetry of code generation, the probability of error averaged over all possible codebooks and messages is: $P(\epsilon) = P(\epsilon|M_0 = M_1 = \dots = M_K = 1)$.

Let us focus on the decoder $i \in \{1, \dots, K\}$. In the SIC decoding procedure, decoder $i$ first decodes messages destined to receivers $j \in \{K, K-1, \dots, i+1\}$ before decoding its own message. Hence, decoder $i$ declares an error if at least one of the following error events occurs
$$
\epsilon_{i,1} = \{(u_K^n(1), b^n(1), y_i^n) \notin \mathcal{T}_{\epsilon}^{(n)}\}, \text{ non-joint typicality;}
$$

$$
\epsilon_{i,2} = \{(u_K^n(1), b^n(m_{0 \to i}), y_i^n) \in \mathcal{T}_{\epsilon}^{(n)}, m_{0 \to i} \neq 1\}, m_{0 \to i} \text{ not successfully decoded;}
$$

$$
\epsilon_{i,3} = \{(u_K^n(m_{K \to i}), b^n(1), y_i^n) \in \mathcal{T}_{\epsilon}^{(n)}, m_K \neq 1\}, m_{K \to i} \text{ not successfully decoded;}
$$

$$
\epsilon_{i,4} = \{(u_K^n(m_{K \to i}), b^n(m_{0 \to i}), y_i^n) \in \mathcal{T}_{\epsilon}^{(n)}, m_{0 \to i} \neq 1, m_{K \to i} \neq 1\}, \\
m_{0 \to i} \text{ and } m_{K \to i} \text{ not successfully decoded;}
$$

$$
\epsilon_{i,j,5} = \{(u_K^n(1), \dots, u_j^n(1, \dots, 1), b^n(1), y_i^n) \notin \mathcal{T}_{\epsilon}^{(n)}, \quad i+1 \le j \le K-1\}, \text{ non-joint typicality;}
$$

$$
\epsilon_{i,j,6} = \{(u_K^n(1), \dots, u_j^n(m_{j \to i}, 1, \dots, 1), b^n(1), y_i^n) \in \mathcal{T}_{\epsilon}^{(n)}, m_{j \to i} \neq 1\}, m_{j \to i} \text{ not successfully decoded;}
$$

$$
\epsilon_{i,i,7} = \{(u_K^n(1), \dots, u_i^n(m_{i \to i}, 1, \dots, 1), b^n(1), y_i^n) \notin \mathcal{T}_{\epsilon}^{(n)}\}, \text{ non-joint typicality;}
$$

$$
\epsilon_{i,i,8} = \{(u_K^n(1), \dots, u_i^n(m_{i \to i}, 1, \dots, 1), b^n(1), y_i^n) \in \mathcal{T}_{\epsilon}^{(n)}, m_{i \to i} \neq 1\}, m_{i \to i} \text{ not successfully decoded.}
$$

By the union bound of events, we obtain $\mathbb{P}(\epsilon_i) \le \mathbb{P}(\epsilon_{i,1}) + \dots + \mathbb{P}(\epsilon_{i,i,8})$. Let us now bound each term individually. By the law of large numbers (LLN), the probabilities $\mathbb{P}(\epsilon_{i,1})$, $\mathbb{P}(\epsilon_{i,j,5})$ and $\mathbb{P}(\epsilon_{i,i,7})$ can be shown to tend to zero as $n \to \infty$. By the packing lemma [40, Lemma 3.1], $\mathbb{P}(\epsilon_{i,2})$ tends to zero as $n \to \infty$ if $R_{0 \to i} \le \mathbb{I}(B; U_K, Y_i) - \delta(\epsilon)$. Furthermore, since $B$ is independent of $U_K$, then $\mathbb{I}(B; U_K, Y_i) = \mathbb{I}(B; Y_i|U_K)$. Hence, $\mathbb{P}(\epsilon_{i,2})$ tends to zero as $n \to \infty$ if $R_{0 \to i} \le \mathbb{I}(B; Y_i|U_K) - \delta(\epsilon)$, $\forall 1 \le i \le K$. Similarly, the error probabilities $\mathbb{P}(\epsilon_{i,3})$, $\mathbb{P}(\epsilon_{i,4})$, $\mathbb{P}(\epsilon_{i,j,6})$, and $\mathbb{P}(\epsilon_{i,i,8})$ tend to zero as $n \to \infty$, if the following conditions are met

$$
R_{K \to i} \le \mathbb{I}(U_K; Y_i|B) - \delta(\epsilon), \quad \forall 1 \le i \le K, \qquad (D.1)
$$

$$
R_{0 \to i} + R_{K \to i} \le \mathbb{I}(U_K, B; Y_i) - \delta(\epsilon), \quad \forall 1 \le i \le K, \qquad (D.2)
$$

$$
R_{j \to i} \le \mathbb{I}(U_j; Y_i|B, U_K, \dots, U_{j+1}) - \delta(\epsilon), \quad \forall 1 \le j \le K-1, \forall i < j, \qquad (D.3)
$$

$$
R_{i \to i} \le \mathbb{I}(U_i; Y_i|B, U_K, \dots, U_{i+1}) - \delta(\epsilon), \quad \forall 1 \le i \le K-1. \qquad (D.4)
$$

The message intended for receiver $j \in \{1, \dots, K-1\}$ is decoded by receivers $i \in \{1, \dots, j-1\}$ and receiver $j$ with a data rate $R_{j \to i}$ and $R_{j \to j}$, respectively. Therefore, the achievable data rate for decoding $M_j$ is $R_j \le \min_{i<j} (R_{j \to i}, R_{j \to j})$, $\stackrel{(a)}{=} \min_{i \le j} \mathbb{I}(U_j; Y_i|B, U_K, \dots, U_{i+1})$, $\forall 1 \le j \le K-1$, where (a) follows from (D.3) and (D.4). Further, the achievable rate for decoding $M_0$ and $M_K$ are defined as $R_0 \le \min_{1 \le i \le K} R_{0 \to i}$ and $R_K \le \min_{1 \le i \le K} R_{K \to i}$ and are constrained as

$$
R_0 \le \min_{1 \le i \le K} \mathbb{I}(B; Y_i|U_K), \quad R_K \le \min_{1 \le i \le K} \mathbb{I}(U_K; Y_i|B), \quad R_0 + R_K \le \min_{1 \le i \le K} \mathbb{I}(U_K, B; Y_i), \quad (D.5)
$$

which completes the proof of Theorem 4.
## D.2 Proof of Theorem 5

From Lemma (1), the achievable rate for decoding the message destined to receiver $j$ satisfies

$$R_j \le \min_{i \le j} \mathbb{I}(U_j; Y_i | B, U_K, \dots, U_{j+1}), \quad \forall 1 \le j \le K-1, \qquad (D.6)$$

$$\stackrel{(b)}{=} qC \left( \min_{i \le j} (\gamma_{j \to i|1}) \right) + (1-q)C \left( \min_{i \le j} (\gamma_{j \to i|0}) \right), \quad \forall 1 \le j \le K-1,$$

where (b) follows from $B \sim \text{Bern}(q)$, $U_k \sim \mathcal{N}(0, p_k)$, $\forall k \in \{2, \dots, K\}$ and $X = \sum_{k=2}^{K} U_k + V$, with $V \sim \mathcal{N}(0, p_1)$, which maximizes the mutual information expressions in (D.6) and yields the capacity of the point-to-point additive white Gaussian noise (AWGN) channel when decoding the message intended to receiver $j$ at decoder $i$ in the reflecting state ($B=1$) and transparent state ($B=0$) respectively.

Following similar steps as above, we obtain the achievable rates $R_K$ and $R_0$ as

$$R_K \le qC \left( \min_{1 \le i \le K} (\gamma_{K \to i|1}) \right) + (1-q)C \left( \min_{1 \le i \le K} (\gamma_{K \to i|0}) \right), \qquad (D.7)$$

$$R_0 \le \min_{1 \le i \le K} \mathbb{H}(Y_i | U_K) - \frac{q}{2} \log_2 \left( 2\pi e \sigma_i^2 \left( H_{i|1}(\rho) \sum_{k=1}^{K-1} p_k + 1 \right) \right) - \frac{1-q}{2} \log_2 \left( 2\pi e \sigma_i^2 \left( H_{i|0} \sum_{k=1}^{K-1} p_k + 1 \right) \right), \qquad (D.8)$$

which concludes the proof of Theorem 5.
# Appendix E

## Proof of Theorem 6

### E.1 Convexity of (EE7)

For the sake of simplicity of presentation, we denote $H_{k|1}(\rho^*)$ simply by $= H_{k|1}$. We also recall that $\theta_k(\mathbf{p}) = \sum_{i=1}^k p_i, \forall 1 \le k \le K$ with $\theta_0(\mathbf{p}) = 0$.

We start by showing that (EE7) is a convex optimization problem. For this, we analyze its objective function $\eta_{EE}(\mathbf{p})$ whose second-order partial derivative w.r.t $p_i, p_j, \forall i, j$, equals

$$ \frac{\partial^2 \eta_{EE}(\mathbf{p})}{\partial p_j \partial p_i} = \begin{cases} d_j, & \text{if } j \ge i \\ d_i, & \text{otherwise,} \end{cases} $$

where

$$ d_j = \sum_{k=j}^{K-1} \frac{q}{2 \ln 2} \left( \frac{(H_{k+1|1})^2}{(1+H_{k+1|1}\theta_k(\mathbf{p}))^2} - \frac{(H_{k|1})^2}{(1+H_{k|1}\theta_k(\mathbf{p}))^2} \right) \quad (E.1) $$

$$ \begin{aligned} &+ \sum_{k=j}^{K-1} \frac{(1-q)}{2 \ln 2} \left( \frac{(H_{k+1|0})^2}{(1+H_{k+1|0}\theta_k(\mathbf{p}))^2} - \frac{(H_{k|0})^2}{(1+H_{k|0}\theta_k(\mathbf{p}))^2} \right) \\ &- \frac{q}{2 \ln 2} \times \frac{(H_{K|1})^2}{(1+H_{K|1}\theta_K(\mathbf{p}))^2} - \frac{(1-q)}{2 \ln 2} \times \frac{(H_{K|0})^2}{(1+H_{K|0}\theta_K(\mathbf{p}))^2}. \end{aligned} \quad (E.2) $$

Let $\mathbf{D}$ be the Hessian matrix which has the following special structure

$$ \mathbf{D} = \begin{pmatrix} d_1 & d_2 & d_3 & \dots & d_K \\ d_2 & d_2 & d_3 & \dots & d_K \\ d_3 & d_3 & d_3 & \dots & d_K \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ d_K & d_K & d_K & \dots & d_K \end{pmatrix}. $$

We define $\mathbf{T} = -\mathbf{D}$ and show that it is positive semi-definite by proving that all of its leading principal minors are positive, i.e., $\det \mathbf{T}[1:j, 1:j] \ge 0, \forall j \in \{1, \dots, K\}$.
The first leading principal minor is $det T[1, 1] = -d_1 \ge 0$. For $1 < j \le K$, the j-th leading principal minor equals

$$ \det T[1:j, 1:j] = \begin{vmatrix} d_2 - d_1 & d_3 - d_2 & d_4 - d_3 & \dots & d_j - d_{j-1} & -d_j \\ 0 & d_3 - d_2 & d_4 - d_3 & \dots & & -d_j \\ 0 & 0 & d_4 - d_3 & \dots & & -d_j \\ \vdots & \vdots & \vdots & \dots & & \vdots \\ 0 & 0 & 0 & \dots & & -d_j \end{vmatrix} \quad (E.3) $$
$$ = -d_j \prod_{k=1}^{j-1} (d_{k+1} - d_k). \quad (E.4) $$

We write the terms $d_{k+1} - d_k$, for all $1 \le k < K$

$$ \begin{aligned} d_{k+1} - d_k &= \frac{q}{2 \ln 2} \left( \frac{(H_{k|1})^2}{(1 + H_{k|1} \theta_k(\mathbf{p}))^2} - \frac{(H_{k+1|1})^2}{(1 + H_{k+1|1} \theta_k(\mathbf{p}))^2} \right) \\ &\quad + \frac{(1-q)}{2 \ln 2} \left( \frac{(H_{k|0})^2}{(1 + H_{k|0} \theta_k(\mathbf{p}))^2} - \frac{(H_{k+1|0})^2}{(1 + H_{k+1|0} \theta_k(\mathbf{p}))^2} \right) \\ &\ge 0, \end{aligned} \quad (E.5) $$

Note that since $H_{k|1} \ge H_{k+1|1}$ and $H_{k|0} \ge H_{k+1|0}$, $\forall k \in \{1, \dots, K-1\}$, we can show that $d_j \le 0$, for all $j$ and that $d_{k+1} - d_k \ge 0$, $\forall k$. The first leading principal minor is $det T[1, 1] = -d_1 \ge 0$. For $1 < j \le K$, the j-th leading principal minor equals $det T[1:j, 1:j] = -d_j \prod_{k=1}^{j-1} (d_{k+1} - d_k) \ge 0$. Therefore, all leading principal minors of $T$ are positive, which implies that $T \ge 0$ equivalent to the Hessian matrix $D \le 0$. The latter means that the objective function is jointly concave w.r.t $\mathbf{p} = (p_1, \dots, p_K)$. Now, given the above and that all the inequality constraints are affine w.r.t $\mathbf{p}$, the optimization problem (EE7) is convex.

## E.2 Solving the KKT conditions for (EE7)

The associated Lagrangian function of the optimization problem (EE7) is given by

$$ \mathcal{L}(\rho^*, \mathbf{p}) = \sum_{k=1}^{K} R_k(\rho^*, \mathbf{p}) - \alpha(\theta_K(\mathbf{p}) + P_c) + \lambda(P_{\max} - \theta_K(\mathbf{p})) + \sum_{k=1}^{K} \beta_k \left( \theta_k(\mathbf{p}) - A_k \theta_{k-1}(\mathbf{p}) + \frac{(A_k - 1)}{H_{k|0}} \right), \quad (E.6) $$

where $\lambda$ and $\beta = (\beta_1, \dots, \beta_K)$ are the non-negative Lagrange multipliers associated with the constraints (C1) and (C2a) respectively. The necessary and sufficient KKT optimality
conditions write as

$$ \left\{ \begin{array}{l} \frac{\partial \mathcal{L}}{\partial p_i} \triangleq \sum_{k=i}^{K-1} \frac{q}{2 \ln 2} \left( \frac{H_{k|1}}{1+H_{k|1} \theta_k(\mathbf{p})} - \frac{H_{k+1|1}}{1+H_{k+1|1} \theta_k(\mathbf{p})} \right) + \sum_{k=i}^{K-1} \frac{(1-q)}{2 \ln 2} \left( \frac{H_{k|0}}{1+H_{k|0} \theta_k(\mathbf{p})} - \frac{H_{k+1|1}}{1+H_{k+1|1} \theta_k(\mathbf{p})} \right) \\ + \frac{q}{2 \ln 2} \times \frac{H_{K|1}}{1+H_{K|1} \theta_K(\mathbf{p})} + \frac{(1-q)}{2 \ln 2} \times \frac{H_{K|0}}{1+H_{K|0} \theta_K(\mathbf{p})} - (\alpha + \lambda) + \sum_{k=i}^{K} \beta_k - \sum_{k=i}^{K-1} (A_{k+1} \beta_{k+1}) = 0, \quad (K1) \end{array} \right. $$

$$ \lambda(P_{max} - \theta_K(\mathbf{p})) = 0, \quad (K2) $$

$$ \beta_k \left( \theta_k(\mathbf{p}) - A_k \theta_{k-1}(\mathbf{p}) + \frac{(A_k - 1)}{H_{k|0}} \right) = 0, \quad (K3) $$

$$ (C1'), (C2'), \lambda \geq, \beta \geq 0 \quad (K4) $$

From (K1), and by using the difference between two consecutive derivatives, $(\frac{\partial \mathcal{L}}{\partial p_i} - \frac{\partial \mathcal{L}}{\partial p_{i+1}}) = 0$, we obtain

$$ \frac{q}{2 \ln 2} \left( \frac{H_{i|1}}{1+H_{i|1} \theta_i(\mathbf{p})} - \frac{H_{i+1|1}}{1+H_{i+1|1} \theta_i(\mathbf{p})} \right) + \frac{(1-q)}{2 \ln 2} \left( \frac{H_{i|0}}{1+H_{i|0} \theta_i(\mathbf{p})} - \frac{H_{i+1|0}}{1+H_{i+1|0} \theta_i(\mathbf{p})} \right) + \beta_i = A_{i+1} \beta_{i+1}. \quad (E.7) $$

We know that $H_{i|1} \geq H_{i+1|1}$, $H_{i|0} \geq H_{i+1|0}$, $A_{i+1} > 0$ and $\beta_{i+1} \geq 0$, $\forall i \in \{1, \dots, K-1\}$. We discuss two cases:

a) If $H_{1|0} > H_{2|0}$, by replacing $i=1$ in (E.7), since $A_2 > 0$, we have $\beta_2 > 0$. For $i=2$, since $\beta_2 > 0$ and $A_3 > 0$ we obtain $\beta_3 > 0$. Recursively, for all $i \geq 2$ we find $\beta_i > 0$. Using (K3), we obtain $\theta_k(\mathbf{p}) = A_k \theta_{k-1}(\mathbf{p}) - \frac{(A_k-1)}{H_{k|0}}$, $\forall k \geq 2$. This means that all receivers $k \geq 2$ will be allocated a power to meet exactly their minimum rate requirement. By induction, we find that

$$ \theta_k(\mathbf{p}) = \theta_1(\mathbf{p}) \prod_{i=2}^{k} A_i + \sum_{i=2}^{k-1} \frac{(A_i - 1)}{H_{i|0}} \prod_{j=i+1}^{k} A_j + \frac{(A_k - 1)}{H_{k|0}}, \forall k \geq 2. \quad (E.8) $$

Notice that all variables $\theta_k$ are expressed in terms of $\theta_1(\mathbf{p}) = p_1$. Hence, we obtain the expressions of the optimal powers $p_k^*$, $\forall k \geq 2$ in closed-form as functions of $p_1$ as in Theorem 6.

b) In the case where $H_{1|0} = H_{2|0}$, the first order partial derivatives of $\eta_{EE}(\mathbf{p})$ write as

$$ \frac{\partial \eta_{EE}}{\partial p_i} = \left\{ \begin{array}{ll} \Upsilon(\mathbf{p}) \triangleq & \sum_{k=i}^{K-1} \frac{q}{2 \ln 2} \left( \frac{H_{k|1}}{1+H_{k|1} \theta_k(\mathbf{p})} - \frac{H_{k+1|1}}{1+H_{k+1|1} \theta_k(\mathbf{p})} \right) \\ & + \sum_{k=i}^{K-1} \frac{(1-q)}{2 \ln 2} \left( \frac{H_{k|0}}{1+H_{k|0} \theta_k(\mathbf{p})} - \frac{H_{k+1|0}}{1+H_{k+1|0} \theta_k(\mathbf{p})} \right) \\ & + \frac{q}{2 \ln 2} \times \frac{H_{K|1}}{1+H_{K|1} \theta_K(\mathbf{p})} + \frac{(1-q)}{2 \ln 2} \times \frac{H_{K|0}}{1+H_{K|0} \theta_K(\mathbf{p})}, & \text{if } i \geq 2 \\ \Upsilon(\mathbf{p}) + \frac{q}{2 \ln 2} \left( \frac{H_{1|1}}{1+H_{1|1} \theta_1(\mathbf{p})} - \frac{H_{2|1}}{1+H_{2|1} \theta_1(\mathbf{p})} \right), & \text{if } i = 1. \end{array} \right. $$

In this case, the difference between two consecutive derivatives of the Lagrange
function gives

$$ \frac{q}{2 \ln 2} \left( \frac{H_{1|1}}{1 + H_{1|1} \theta_1(\mathbf{p})} - \frac{H_{2|1}}{1 + H_{2|1} \theta_1(\mathbf{p})} \right) + \beta_1 = A_2 \beta_2, \qquad (E.9) $$

$$ \frac{q}{2 \ln 2} \left( \frac{H_{i|1}}{1 + H_{i|1} \theta_i(\mathbf{p})} - \frac{H_{i+1|1}}{1 + H_{i+1|1} \theta_i(\mathbf{p})} \right) + \frac{(1-q)}{2 \ln 2} \left( \frac{H_{i|0}}{1 + H_{i|0} \theta_i(\mathbf{p})} - \frac{H_{i+1|0}}{1 + H_{i+1|0} \theta_i(\mathbf{p})} \right) + \beta_i = A_{i+1} \beta_{i+1}. \qquad (E.10) $$

We discuss two cases:

i) If $H_{1|1} > H_{2|1}$ then, since $\beta_1 \ge 0$ and $A_2 > 0$, we have $\beta_2 > 0$ and recursively we find $\beta_i > 0, \forall i \ge 2$, which leads to the analytical closed-form expressions of $p_k^*, \forall k \ge 2$ as functions of $p_1$ and given in Theorem 6, as in the case $H_{1|0} > H_{2|0}$.

ii) If $H_{1|1} = H_{2|1}$, the analysis of the solution $\mathbf{p}^*$ is delegated to the end of the appendix.

### E.2.1 Finding the optimal power allocated to user 1 when $H_{1|0} > H_{2|0}$ or $H_{1|1} > H_{2|1}$

Using the above optimal expressions $p_k^*, \forall k \ge 2$ as functions of $p_1$, we can reformulate the multi-variable optimization problem (EE7) into a simple single-variable optimization problem

$$ \max_{\ell \le p_1 \le u} f_1(p_1) \triangleq \sum_{k=1}^{K} \frac{q}{2} \log_2 \left( \frac{1 + H_{k|1} \theta_k(\mathbf{p})}{1 + H_{k|1} \theta_{k-1}(\mathbf{p})} \right) + \frac{(1-q)}{2} \log_2 (1 + H_{1|0} p_1) + \sum_{k=2}^{K} (1-q) R_k^{\min} \\ - \alpha \left( p_1 \prod_{i=2}^{K} A_i + \sum_{i=2}^{K-1} \frac{(A_i - 1)}{H_{i|0}} \prod_{j=i+1}^{K} A_j + \frac{(A_K - 1)}{H_{K|0}} + P_c \right) \qquad (E.11) $$

$$ \text{where } \ell = \frac{(A_1 - 1)}{H_{1|0}} \text{ and } u = \left( P_{\max} - P_{\min} + \ell \prod_{j=2}^{K} A_j \right) / \prod_{i=2}^{K} A_i. $$

The optimal solution of (E.11) is given by $p_1^*(\alpha) = \max(\min(\bar{p}_1(\alpha); u); \ell)$ where $\bar{p}_1$ is the critical point of the objective satisfying $\frac{\partial f_1(p_1)}{\partial p_1} = 0$ and can be found numerically.

### E.2.2 The case in which $H_{1|0} = H_{2|0}$ and $H_{1|1} = H_{2|1}$

The expression of $\eta_{EE}(\mathbf{p})$ simplifies in this case. Similar to our discussion in Appendix A, we can consider receiver 1 and 2 as a single entity, having the strongest channel gains $H'_{1|0} = H_{1|0} = H_{2|0}$ and $H'_{1|1} = H_{1|1} = H_{2|1}$, and whose achievable data rate and allocated power are considered as $R'_1 = (R_1 + R_2)$ and $p'_1 = p_1 + p_2$ respectively. The same discussion follows subsequently as in the previous subsections of this Appendix E; two cases arise: a) $H'_{1|0} = H_{3|0}$ and $H'_{1|1} = H_{3|1}$, and b) $H'_{1|0} > H_{3|0}$ or $H'_{1|1} > H_{3|1}$. In the latter, we obtain the analytical closed-form expressions of $p_k^*, \forall k \ge 3$ as a function of $p'_1$ and $(p'_1)^*$ is obtained by solving the resulting (EE7) problem. Then, we can split the power $(p'_1)^*$ such that $p_2^*$ is the enough power required to satisfy the minimum rate requirement for receiver 2, and the remaining power $p_1^* = (p'_1)^* - p_2^*$. In the
former case, a recurring reasoning can be applied etc. Finally, the same reasoning applies in the
extreme symmetric case in which $H_{1|0} = H_{2|0} = \dots = H_{K|0}$ and $H_{1|1} = H_{2|1} = \dots = H_{K|1}$,
which unlikely to occur.

For the sake of clarity and simplicity of presentation, and given that the main ideas and
the procedure behind the proof is explained in the previous subsections of this Appendix E, we
have decided to include only the case $H_{1|0} > H_{2|0}$ or $H_{1|1} > H_{2|1}$ in the main text of Theorem 6.
Bibliography

[1] A. A. Huurdeman, *The worldwide history of telecommunications*. John Wiley & Sons, 2003.

[2] A. Goldsmith, *Wireless communications*. Cambridge university press, 2005.

[3] M. Wendler and S. Guillemaut, “ETCS in PS-mode GPRS/EGPRS guideline,” *Kapsch CarrierCom, Nokia: UIC ERTMS/GSM-R Operators Group*, 2014.

[4] I. R. Sector, “Requirements related to technical performance for IMT-advanced radio interface (s),” *Report ITU*, pp. 2134–2008, 2008.

[5] L. Horwitz, “The future of IoT miniguide: The burgeoning IoT market continues,” *CISCO, San Jose, CA, USA, Tech. Rep*, 2019.

[6] M. Series, “Imt vision–framework and overall objectives of the future development of IMT for 2020 and beyond,” *Recommendation ITU*, vol. 2083, p. 0, 2015.

[7] M. Vaezi, Z. Ding, and H. V. Poor, *Multiple access techniques for 5G wireless networks and beyond*. Springer, 2019, vol. 159.

[8] X. Chen, D. W. K. Ng, W. Yu, E. G. Larsson, N. Al-Dhahir, and R. Schober, “Massive access for 5G and beyond,” *IEEE Journal on Selected Areas in Communications*, vol. 39, no. 3, pp. 615–637, 2020.

[9] V. W. Wong, R. Schober, D. W. K. Ng, and L.-C. Wang, *Key technologies for 5G wireless systems*. Cambridge university press, 2017.

[10] Y. Saito, Y. Kishiyama, A. Benjebbour, T. Nakamura, A. Li, and K. Higuchi, “Non-orthogonal multiple access (NOMA) for cellular future radio access,” in *2013 IEEE 77th vehicular technology conference (VTC Spring)*. IEEE, 2013, pp. 1–5.

[11] K. Higuchi and A. Benjebbour, “Non-orthogonal multiple access (NOMA) with successive interference cancellation for future radio access,” *IEICE Transactions on Communications*, vol. 98, no. 3, pp. 403–414, 2015.

[12] S. R. Islam, N. Avazov, O. A. Dobre, and K.-S. Kwak, “Power-domain non-orthogonal multiple access (NOMA) in 5G systems: Potentials and challenges,” *IEEE Communications Surveys & Tutorials*, vol. 19, no. 2, pp. 721–742, 2016.
[13] Q. Wu, W. Chen, D. W. K. Ng, and R. Schober, "Spectral and energy-efficient wireless powered IoT networks: NOMA or TDMA?" *IEEE Transactions on Vehicular Technology*, vol. 67, no. 7, pp. 6663-6667, 2018.

[14] Z. Ding, X. Lei, G. K. Karagiannidis, R. Schober, J. Yuan, and V. K. Bhargava, "A survey on non-orthogonal multiple access for 5G networks: Research challenges and future trends," *IEEE Journal on Selected Areas in Communications*, vol. 35, no. 10, pp. 2181-2195, 2017.

[15] J. Malmodin and D. Lundén, "The energy and carbon footprint of the global ICT and E&M sectors 2010–2015," *Sustainability*, vol. 10, no. 9, p. 3027, 2018.

[16] F. M. Al-Turjman, M. Imran, and S. T. Bakhsh, "Energy efficiency perspectives of femtocells in Internet of Things: Recent advances and challenges," *IEEE Access*, vol. 5, pp. 26808–26818, 2017.

[17] V. Liu, A. Parks, V. Talla, S. Gollakota, D. Wetherall, and J. R. Smith, "Ambient backscatter: Wireless communication out of thin air," *ACM SIGCOMM Computer Communication Review*, vol. 43, no. 4, pp. 39–50, 2013.

[18] G. Wang, F. Gao, R. Fan, and C. Tellambura, "Ambient backscatter communication systems: Detection and performance analysis," *IEEE Transactions on Communications*, vol. 64, no. 11, pp. 4836–4846, 2016.

[19] N. Van Huynh, D. T. Hoang, X. Lu, D. Niyato, P. Wang, and D. I. Kim, "Ambient backscatter communications: A contemporary survey," *IEEE Communications surveys & tutorials*, vol. 20, no. 4, pp. 2889–2922, 2018.

[20] F. Jameel and S. A. Hassan, *Wireless-Powered Backscatter Communications for Internet of Things*. Springer, 2020.

[21] R. Fara, D.-T. Phan-Huy, and M. Di Renzo, "Ambient backscatters-friendly 5G networks: creating hot spots for tags and good spots for readers," in *IEEE WCNC*, 2020, pp. 1–7.

[22] Y. Kokar, D.-T. Phan-Huy, R. Fara, K. Rachedi, A. Ourir, J. de Rosny, M. Di Renzo, J.-C. Prévotet, and M. Helard, "First experimental ambient backscatter communication using a compact reconfigurable tag antenna," in *IEEE Globecom Workshops*, 2019, pp. 1–6.

[23] C. Song, Y. Ding, A. Eid, J. G. Hester, X. He, R. Bahr, A. Georgiadis, G. Goussetis, and M. M. Tentzeris, "Advances in wirelessly powered backscatter communications: From antenna/RF circuitry design to printed flexible electronics," *Proceedings of the IEEE*, vol. 110, no. 1, pp. 171–192, 2021.
[24] J. Hoydis, F. A. Aoudia, A. Valcarce, and H. Viswanathan, "Toward a 6G AI-native air interface," *IEEE Communications Magazine*, vol. 59, no. 5, pp. 76-81, 2021.

[25] B. Rong, "6G: The next horizon: From connected people and things to connected intelligence," *IEEE Wireless Communications*, vol. 28, no. 5, pp. 8-8, 2021.

[26] S. Ali, W. Saad, N. Rajatheva, K. Chang, D. Steinbach, B. Sliwa, C. Wietfeld, K. Mei, H. Shiri, H.-J. Zepernick et al., "6G white paper on machine learning in wireless communication networks," *arXiv preprint arXiv:2004.13875*, 2020.

[27] Y. Saito, A. Benjebbour, Y. Kishiyama, and T. Nakamura, "System-level performance evaluation of downlink non-orthogonal multiple access (NOMA)," in *2013 IEEE 24th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC)*. IEEE, 2013, pp. 611-615.

[28] Z. Ding, Z. Yang, P. Fan, and H. V. Poor, "On the performance of non-orthogonal multiple access in 5G systems with randomly deployed users," *IEEE signal processing letters*, vol. 21, no. 12, pp. 1501-1505, 2014.

[29] J. Choi, "Power allocation for max-sum rate and max-min rate proportional fairness in NOMA," *IEEE Communications Letters*, vol. 20, no. 10, pp. 2055-2058, 2016.

[30] S. Timotheou and I. Krikidis, "Fairness for non-orthogonal multiple access in 5G systems," *IEEE signal processing letters*, vol. 22, no. 10, pp. 1647-1651, 2015.

[31] J. Choi, "Non-orthogonal multiple access in downlink coordinated two-point systems," *IEEE Communications Letters*, vol. 18, no. 2, pp. 313-316, 2014.

[32] Q. Wu, W. Chen, D. W. K. Ng, and R. Schober, "Spectral and energy-efficient wireless powered IoT networks: NOMA or TDMA?" *IEEE Transactions on Vehicular Technology*, vol. 67, no. 7, pp. 6663-6667, 2018.

[33] D. Tse and P. Viswanath, *Fundamentals of wireless communication*. Cambridge university press, 2005.

[34] L. Song, R. C. de Lamare, and A. G. Burr, "Successive interference cancellation schemes for time-reversal space-time block codes," *IEEE transactions on vehicular technology*, vol. 57, no. 1, pp. 642-648, 2008.

[35] P. Li, R. C. De Lamare, and R. Fa, "Multiple feedback successive interference cancellation detection for multiuser MIMO systems," *IEEE Transactions on Wireless Communications*, vol. 10, no. 8, pp. 2434-2439, 2011.

[36] L. Dai, B. Wang, Y. Yuan, S. Han, I. Chih-Lin, and Z. Wang, "Non-orthogonal multiple access for 5G: solutions, challenges, opportunities, and future research trends," *IEEE Communications Magazine*, vol. 53, no. 9, pp. 74-81, 2015.
[37] S. Vanka, S. Srinivasa, Z. Gong, P. Vizi, K. Stamatiou, and M. Haenggi, "Superposition coding strategies: Design and experimental evaluation," *IEEE Transactions on Wireless Communications*, vol. 11, no. 7, pp. 2628-2639, 2012.

[38] M. Vaezi, R. Schober, Z. Ding, and H. V. Poor, "Non-orthogonal multiple access: Common myths and critical questions," *IEEE Wireless Communications*, vol. 26, no. 5, pp. 174-180, 2019.

[39] T. M. Cover, "Thomas. elements of information theory," *Wiley Series in Telecommunications*, 1991.

[40] A. El Gamal and Y.-H. Kim, *Network information theory*. Cambridge university press, 2011.

[41] J. Axnäs, Y.-P. E. Wang, M. Kamuf, and N. Andgart, "Successive interference cancellation techniques for LTE downlink," in *2011 IEEE 22nd International Symposium on Personal, Indoor and Mobile Radio Communications*. IEEE, 2011, pp. 1793-1797.

[42] G. Zhou, W. Xu, and G. Bauch, "Network assisted inter-cell codeword cancellation for interference-limited LTE-A and beyond," in *2014 IEEE Wireless Communications and Networking Conference Workshops (WCNCW)*. IEEE, 2014, pp. 52-57.

[43] S. Moon, H. Choe, M. Chu, C. You, H. Liu, J.-H. Kim, J. Kim, D. J. Kim, and I. Hwang, "Design and performance analysis of soft decision-based network-assisted interference cancellation and suppression scheme," *Wireless Personal Communications*, vol. 96, no. 4, pp. 5849-5863, 2017.

[44] A. Davydov, G. Morozov, and S. Han, "Network assisted interference cancellation and suppression with respect to interfering control channel transmissions," Jun. 28 2016, uS Patent 9,380,090.

[45] J.-Y. Hwang and S.-W. Lee, "Advanced receiver for interference cancellation in 3GPP LTE-A," *The Journal of Korean Institute of Communications and Information Sciences*, vol. 39, no. 7, pp. 389-397, 2014.

[46] L. Xiu, "Time moore: Exploiting moore's law from the perspective of time," *IEEE Solid-State Circuits Magazine*, vol. 11, no. 1, pp. 39-55, 2019.

[47] M. L. Memon, N. Saxena, A. Roy, S. Singh, and D. R. Shin, "Ambient backscatter communications to energize IoT devices," *IETE Technical Review*, vol. 37, no. 2, pp. 196-210, 2020.
[48] S. S. Afzal, R. Ghaffarivardavagh, W. Akbar, O. Rodriguez, and F. Adib, "Enabling higher-order modulation for underwater backscatter communication," in *Global Oceans 2020: Singapore-US Gulf Coast*. IEEE, 2020, pp. 1-6.

[49] R. Long, H. Guo, L. Zhang, and Y.-C. Liang, "Full-duplex backscatter communications in symbiotic radio systems," *IEEE Access*, vol. 7, pp. 21597-21608, 2019.

[50] R. Long, Y.-C. Liang, H. Guo, G. Yang, and R. Zhang, "Symbiotic radio: A new communication paradigm for passive Internet of Things," *IEEE Internet of Things Journal*, vol. 7, no. 2, pp. 1350-1363, 2019.

[51] H. Guo, Y.-C. Liang, R. Long, S. Xiao, and Q. Zhang, "Resource allocation for symbiotic radio system with fading channels," *IEEE Access*, vol. 7, pp. 34333-34347, 2019.

[52] Q. Zhang, L. Zhang, Y.-C. Liang, and P.-Y. Kam, "Backscatter-NOMA: A symbiotic system of cellular and internet-of-things networks," *IEEE Access*, vol. 7, pp. 20000-20013, 2019.

[53] G. Yang, Q. Zhang, and Y.-C. Liang, "Cooperative ambient backscatter communications for green internet-of-things," *IEEE Internet of Things Journal*, vol. 5, no. 2, pp. 1116-1130, 2018.

[54] M. Dai, P. Wang, S. Zhang, B. Chen, H. Wang, X. Lin, and C. Sun, "Survey on cooperative strategies for wireless relay channels," *Transactions on Emerging Telecommunications Technologies*, vol. 25, no. 9, pp. 926-942, 2014.

[55] A. Savard and C. Weidmann, "On the multiway relay channel with direct links," in *2014 IEEE Information Theory Workshop (ITW 2014)*. IEEE, 2014, pp. 651-655.

[56] S. I. Bross and M. A. Wigger, "On the relay channel with receiver-transmitter feedback," *IEEE Transactions on Information Theory*, vol. 55, no. 1, pp. 275-291, 2008.

[57] M. Di Renzo, K. Ntontin, J. Song, F. H. Danufane, X. Qian, F. Lazarakis, J. De Rosny, D.-T. Phan-Huy, O. Simeone, R. Zhang et al., "Reconfigurable intelligent surfaces vs. relaying: Differences, similarities, and performance comparison," *IEEE Open Journal of the Communications Society*, vol. 1, pp. 798-807, 2020.

[58] Y.-C. Liang, R. Long, Q. Zhang, J. Chen, H. V. Cheng, and H. Guo, "Large intelligent surface/antennas (LISA): Making reflective radios smart," *Journal of Communications and Information Networks*, vol. 4, no. 2, pp. 40-50, 2019.
[59] E. Basar, M. Di Renzo, J. De Rosny, M. Debbah, M.-S. Alouini, and R. Zhang, "Wireless communications through reconfigurable intelligent surfaces," *IEEE access*, vol. 7, pp. 116 753–116 773, 2019.

[60] M. Di Renzo, A. Zappone, M. Debbah, M.-S. Alouini, C. Yuen, J. De Rosny, and S. Tretyakov, "Smart radio environments empowered by reconfigurable intelligent surfaces: How it works, state of research, and the road ahead," *IEEE journal on selected areas in communications*, vol. 38, no. 11, pp. 2450–2525, 2020.

[61] Q. Wu and R. Zhang, "Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming," *IEEE Transactions on Wireless Communications*, vol. 18, no. 11, pp. 5394–5409, 2019.

[62] C. Huang, A. Zappone, M. Debbah, and C. Yuen, "Achievable rate maximization by passive intelligent mirrors," in *2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)*. IEEE, 2018, pp. 3714–3718.

[63] C. Huang, A. Zappone, G. C. Alexandropoulos, M. Debbah, and C. Yuen, "Reconfigurable intelligent surfaces for energy efficiency in wireless communication," *IEEE Transactions on Wireless Communications*, vol. 18, no. 8, pp. 4157–4170, 2019.

[64] Q.-U.-A. Nadeem, A. Kammoun, A. Chaaban, M. Debbah, and M.-S. Alouini, "Asymptotic analysis of large intelligent surface assisted MIMO communication," *Arxiv*, vol. 2019, 2019.

[65] R. Masmoudi, E. V. Belmega, I. Fijalkow, and N. Sellami, "A unifying view on energy-efficiency metrics in cognitive radio channels," in *EUSIPCO, Lisbon, Portugal*, Sep. 2014.

[66] E. Belmega, S. Lasaulce, M. Debbah, and A. Hrungnes, "A new energy efficiency function for quasi-static MIMO channels," in *International Wireless Communications and Mobile Computing Conference (IWCMC), Leipzig, Germany*, invited paper, 2009.

[67] G. Bacci, L. Sanguinetti, M. Luise, and H. V. Poor, "Energy-efficient power control for contention-based synchronization in OFDMA systems with discrete powers and limited feedback," *EURASIP Journal on Wireless Communications and Networking*, vol. 2013, no. 1, pp. 1–14, 2013.

[68] D. Goodman and N. Mandayam, "Power control for wireless data," *IEEE Personal Communications*, vol. 7, no. 2, pp. 48–54, 2000.

[69] S. Verdú, "On channel capacity per unit cost," *IEEE Transactions on Information Theory*, vol. 36, no. 5, pp. 1019–1030, 1990.
[70] A. Zappone, E. Jorswieck et al., "Energy efficiency in wireless networks via fractional programming theory," *Foundations and Trends in Communications and Information Theory*, vol. 11, no. 3-4, pp. 185-396, 2015.

[71] C. Isheden, Z. Chong, E. Jorswieck, and G. Fettweis, "Framework for link-level energy efficiency optimization with informed transmitter," *IEEE Transactions on Wireless Communications*, vol. 11, no. 8, pp. 2946-2957, 2012.

[72] S. Boyd and L. Vandenberghe, *Convex Optimization*. Cambridge University Press, 2004.

[73] J. Tang, J. Luo, D. So, E. Alsusa, K.-K. Wong, and N. Zhao, "Energy-efficient resource allocation in SWIPT enabled NOMA systems," in *2018 IEEE Global Communications Conference (GLOBECOM)*. IEEE, 2018, pp. 1-7.

[74] Y. Zhang, H.-M. Wang, T.-X. Zheng, and Q. Yang, "Energy-efficient transmission design in non-orthogonal multiple access," *IEEE Transactions on Vehicular Technology*, vol. 66, no. 3, pp. 2852-2857, 2016.

[75] R. S. Prabhu and B. Daneshrad, "An energy-efficient water-filling algorithm for OFDM systems," in *2010 IEEE International Conference on Communications*. IEEE, 2010, pp. 1-5.

[76] W. Dinkelbach, "On nonlinear fractional programming," *Management science*, vol. 13, no. 7, pp. 492-498, 1967.

[77] R. Jagannathan, "On some properties of programming problems in parametric form pertaining to fractional programming," *Management Science*, vol. 12, no. 7, pp. 609-615, 1966.

[78] R. S. Sutton and A. G. Barto, *Reinforcement learning: An introduction*. MIT press, 2018.

[79] E. V. Belmega, P. Mertikopoulos, and R. Negrel, "Online convex optimization in wireless networks and beyond: The feedback-performance trade-off," in *RAWNET 2022-International Workshop on Resource Allocation and Cooperation in Wireless Networks*, 2022, pp. 1-8.

[80] S. Bubeck and N. Cesa-Bianchi, "Regret analysis of stochastic and nonstochastic multi-armed bandit problems," *arXiv preprint arXiv:1204.5721*, 2012.

[81] P. Auer, N. Cesa-Bianchi, Y. Freund, and R. E. Schapire, "Gambling in a rigged casino: The adversarial multi-armed bandit problem," in *IEEE 36th Annual Foundations of Computer Science*, 1995, pp. 322-331.
[82] W. U. Khan, F. Jameel, T. Ristaniemi, S. Khan, G. A. S. Sidhu, and J. Liu, "Joint spectral and energy efficiency optimization for downlink NOMA networks," *IEEE Transactions on Cognitive Communications and Networking*, 2019.

[83] J. Zhu, J. Wang, Y. Huang, S. He, X. You, and L. Yang, "On optimal power allocation for downlink non-orthogonal multiple access systems," *IEEE Journal on Selected Areas in Communications*, vol. 35, no. 12, pp. 2744-2757, 2017.

[84] Z. Yang, W. Xu, C. Pan, Y. Pan, and M. Chen, "On the optimality of power allocation for NOMA downlinks with individual QoS constraints," *IEEE Communications Letters*, vol. 21, no. 7, pp. 1649-1652, 2017.

[85] F. Fang, H. Zhang, J. Cheng, S. Roy, and V. C. Leung, "Joint user scheduling and power allocation optimization for energy-efficient NOMA systems with imperfect CSI," *IEEE Journal on Selected Areas in Communications*, vol. 35, no. 12, pp. 2874-2885, 2017.

[86] S. D'Oro, A. Zappone, S. Palazzo, and M. Lops, "A learning approach for low-complexity optimization of energy efficiency in multicarrier wireless networks," *IEEE Transactions on Wireless Communications*, vol. 17, no. 5, pp. 3226-3241, 2018.

[87] Z. Chen, Z. Ding, X. Dai, and R. Zhang, "An optimization perspective of the superiority of NOMA compared to conventional OMA," *IEEE Transactions on Signal Processing*, vol. 65, no. 19, pp. 5191-5202, 2017.

[88] X. Kang, Y.-C. Liang, and J. Yang, "Riding on the primary: A new spectrum sharing paradigm for wireless-powered IoT devices," *IEEE Transactions on Wireless Communications*, vol. 17, no. 9, pp. 6335-6347, 2018.

[89] Y. Ye, L. Shi, R. Qingyang Hu, and G. Lu, "Energy-efficient resource allocation for wirelessly powered backscatter communications," *IEEE Communications Letters*, vol. 23, no. 8, pp. 1418-1422, 2019.

[90] B. Lyu, C. You, Z. Yang, and G. Gui, "The optimal control policy for RF-powered backscatter communication networks," *IEEE Transactions on Vehicular Technology*, vol. 67, no. 3, pp. 2804-2808, 2017.

[91] W. Chen, H. Ding, S. Wang, D. B. da Costa, F. Gong, and P. H. J. Nardelli, "Backscatter cooperation in NOMA communications systems," *IEEE Transactions on Wireless Communications*, vol. 20, no. 6, pp. 3458-3474, 2021.

[92] Y. Xu, Z. Qin, G. Gui, H. Gacanin, H. Sari, and F. Adachi, "Energy efficiency maximization in NOMA enabled backscatter communications with QoS guarantee," *IEEE Wireless Communications Letters*, vol. 10, no. 2, pp. 353-357, 2020.
[93] W. U. Khan, X. Li, M. Zeng, and O. A. Dobre, “Backscatter-enabled NOMA for future 6G systems: A new optimization framework under imperfect SIC,” *IEEE Communications Letters*, vol. 25, no. 5, pp. 1669–1672, 2021.

[94] W. U. Khan, M. A. Javed, T. N. Nguyen, S. Khan, and B. M. Elhalawany, “Energy-efficient resource allocation for 6G backscatter-enabled NOMA IoT networks,” *IEEE Transactions on Intelligent Transportation Systems*, 2021.

[95] S. Zhou, W. Xu, K. Wang, C. Pan, M.-S. Alouini, and A. Nallanathan, “Ergodic rate analysis of cooperative ambient backscatter communication,” *IEEE Wireless Communications Letters*, vol. 8, no. 6, pp. 1679–1682, 2019.

[96] X. Jia and X. Zhou, “Performance characterization of relaying using backscatter devices,” *IEEE Open Journal of the Communications Society*, vol. 1, pp. 819–834, 2020.

[97] S. Zeb, Q. Abbas, S. A. Hassan, A. Mahmood, R. Mumtaz, S. H. Zaidi, S. A. R. Zaidi, and M. Gidlund, “NOMA enhanced backscatter communication for green IoT networks,” in *2019 16th International Symposium on Wireless Communication Systems (ISWCS)*. IEEE, 2019, pp. 640–644.

[98] A. W. Nazar, S. A. Hassan, H. Jung, A. Mahmood, and M. Gidlund, “BER analysis of a backscatter communication system with non-orthogonal multiple access,” *IEEE Transactions on Green Communications and Networking*, vol. 5, no. 2, pp. 574–586, 2021.

[99] S. Gong, X. Huang, J. Xu, W. Liu, P. Wang, and D. Niyato, “Backscatter relay communications powered by wireless energy beamforming,” *IEEE Transactions on Communications*, vol. 66, no. 7, pp. 3187–3200, 2018.

[100] J. Qian, F. Gao, G. Wang, S. Jin, and H. Zhu, “Noncoherent detections for ambient backscatter system,” *IEEE Transactions on Wireless Communications*, vol. 16, no. 3, pp. 1412–1422, 2016.

[101] W. Liu, Y.-C. Liang, Y. Li, and B. Vucetic, “Backscatter multiplicative multiple-access systems: Fundamental limits and practical design,” *IEEE Transactions on Wireless Communications*, vol. 17, no. 9, pp. 5713–5728, 2018.

[102] J. K. Devineni and H. S. Dhillon, “Ambient backscatter systems: Exact average bit error rate under fading channels,” *IEEE Transactions on Green Communications and Networking*, vol. 3, no. 1, pp. 11–25, 2018.

[103] Z. Ding, R. Schober, and H. V. Poor, “Unveiling the importance of SIC in NOMA systems—part ii: New results and future directions,” *IEEE Communications Letters*, vol. 24, no. 11, pp. 2378–2382, 2020.
[104] W. Shin, M. Vaezi, B. Lee, D. J. Love, J. Lee, and H. V. Poor, "Non-orthogonal multiple access in multi-cell networks: Theory, performance, and practical challenges," *IEEE Communications Magazine*, vol. 55, no. 10, pp. 176-183, 2017.

[105] L. Zhu, J. Zhang, Z. Xiao, X. Cao, and D. O. Wu, "Optimal user pairing for downlink non-orthogonal multiple access (NOMA)," *IEEE Wireless Communications Letters*, vol. 8, no. 2, pp. 328-331, 2018.

[106] F. Fang, H. Zhang, J. Cheng, and V. C. Leung, "Energy efficiency of resource scheduling for non-orthogonal multiple access NOMA wireless network," in *2016 IEEE International Conference on Communications (ICC)*. IEEE, 2016, pp. 1-5.

[107] J. Cui, Z. Ding, and P. Fan, "A novel power allocation scheme under outage constraints in NOMA systems," *IEEE Signal Processing Letters*, vol. 23, no. 9, pp. 1226-1230, 2016.

[108] X. Wang, J. Wang, L. He, and J. Song, "Outage analysis for downlink NOMA with statistical channel state information," *IEEE Wireless Communications Letters*, vol. 7, no. 2, pp. 142-145, 2017.

[109] J. Choi, "Joint rate and power allocation for NOMA with statistical CSI," *IEEE Transactions on Communications*, vol. 65, no. 10, pp. 4519-4528, 2017.

[110] P. Xu, Y. Yuan, Z. Ding, X. Dai, and R. Schober, "On the outage performance of non-orthogonal multiple access with 1-bit feedback," *IEEE Transactions on Wireless Communications*, vol. 15, no. 10, pp. 6716-6730, 2016.

[111] S. Li, M. Derakhshani, S. Lambotharan, and L. Hanzo, "Outage probability analysis for the multi-carrier NOMA downlink relying on statistical CSI," *IEEE Transactions on Communications*, vol. 68, no. 6, pp. 3572-3587, 2020.

[112] M. A. Adjif, O. Habachi, and J.-P. Cances, "Joint channel selection and power control for NOMA: A multi-armed bandit approach," in *IEEE WCNCW*, 2019, pp. 1-6.

[113] Z. Tian, J. Wang, J. Wang, and J. Song, "Distributed NOMA-based multi-armed bandit approach for channel access in cognitive radio networks," *IEEE Wireless Communications Letters*, vol. 8, no. 4, pp. 1112-1115, 2019.

[114] H. Sun, Q. Wang, R. Q. Hu, and Y. Qian, "Outage probability study in a NOMA relay system," in *2017 IEEE Wireless Communications and Networking Conference (WCNC)*. IEEE, 2017, pp. 1-6.

[115] H. Zhang, J. Zhang, and K. Long, "Energy efficiency optimization for NOMA UAV network with imperfect CSI," *IEEE Journal on Selected Areas in Communications*, vol. 38, no. 12, pp. 2798-2809, 2020.
[116] D. Ghosh, M. K. Hanawal, and N. Zlatanov, "Learning to optimize energy efficiency in energy harvesting wireless sensor networks," *IEEE Wireless Communications Letters*, vol. 10, no. 6, pp. 1153-1157, 2021.

[117] P. Auer, N. Cesa-Bianchi, and P. Fischer, "Finite-time analysis of the multiarmed bandit problem," *Machine learning*, vol. 47, no. 2-3, pp. 235-256, 2002.

[118] P. Perick, D. L. St-Pierre, F. Maes, and D. Ernst, "Comparison of different selection strategies in monte-carlo tree search for the game of tron," in *IEEE CIG*, 2012, pp. 242-249.

[119] A. Garivier and E. Moulines, "On upper-confidence bound policies for non-stationary bandit problems," *arXiv preprint arXiv:0805.3415*, 2008.

[120] Z. Ding, "Harvesting devices' heterogeneous energy profiles and QoS requirements in IoT: WPT-NOMA vs BAC-NOMA," *IEEE Transactions on Communications*, vol. 69, no. 5, pp. 2837-2850, 2021.

[121] Z. Ding, L. Lv, F. Fang, O. A. Dobre, G. K. Karagiannidis, N. Al-Dhahir, R. Schober, and H. V. Poor, "A state-of-the-art survey on reconfigurable intelligent surface-assisted non-orthogonal multiple access networks," *Proceedings of the IEEE*, 2022.

[122] E. V. Belmega, P. Mertikopoulos, R. Negrel, and L. Sanguinetti, "Online convex optimization and no-regret learning: Algorithms, guarantees and applications," *arXiv preprint arXiv:1804.04529*, 2018.

[123] H. Zhang, H. Zhang, K. Long, and G. K. Karagiannidis, "Deep learning based radio resource management in NOMA networks: User association, subchannel and power allocation," *IEEE Transactions on Network Science and Engineering*, vol. 7, no. 4, pp. 2406-2415, 2020.

[124] N. Yang, H. Zhang, K. Long, H.-Y. Hsieh, and J. Liu, "Deep neural network for resource management in NOMA networks," *IEEE Transactions on Vehicular Technology*, vol. 69, no. 1, pp. 876-886, 2019.

[125] Y. Liu, S. Zhang, X. Mu, Z. Ding, R. Schober, N. Al-Dhahir, E. Hossain, and X. Shen, "Evolution of NOMA toward next generation multiple access (NGMA) for 6G," *IEEE Journal on Selected Areas in Communications*, vol. 40, no. 4, pp. 1037-1071, 2022.

[126] I. Chafaa, R. Negrel, E. V. Belmega, and M. Debbah, "Self-supervised deep learning for mmwave beam steering exploiting sub-6 GHz channels," *IEEE Transactions on Wireless Communications*, 2022.
[127] B. Clerckx, Y. Mao, R. Schober, E. A. Jorswieck, D. J. Love, J. Yuan, L. Hanzo, G. Y. Li, E. G. Larsson, and G. Caire, “Is NOMA efficient in multi-antenna networks? a critical look at next generation multiple access techniques,” *IEEE Open Journal of the Communications Society*, vol. 2, pp. 1310–1343, 2021.

[128] K. Senel, H. V. Cheng, E. Björnson, and E. G. Larsson, “What role can NOMA play in massive MIMO?” *IEEE Journal of Selected Topics in Signal Processing*, vol. 13, no. 3, pp. 597–611, 2019.

[129] G. Zhou, Y. Mao, and B. Clerckx, “Rate-splitting multiple access for multi-antenna downlink communication systems: Spectral and energy efficiency tradeoff,” *IEEE Transactions on Wireless Communications*, 2021.

[130] Z. Ding, L. Lv, F. Fang, O. A. Dobre, G. K. Karagiannidis, N. Al-Dhahir, R. Schober, and H. V. Poor, “A state-of-the-art survey on reconfigurable intelligent surface-assisted non-orthogonal multiple access networks,” *Proceedings of the IEEE*, 2022.