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Abstract

With the continuous development of fifth-generation technology, the number of mobile terminal Internet of Things devices has increased exponentially. How to effectively improve the throughput of fifth-generation systems has become a challenge. In the Internet of Things networks, ultra-dense networks and non-orthogonal multiple access technology have drawn extensive attention in recent years, because they can achieve multiplexing from the space domain and power domain. To improve the throughput of the system, this article combines non-orthogonal multiple access with ultra-dense networks technology and considers the orthogonal frequency division multiplexing non-orthogonal multiple access–based ultra-dense networks with multiple base stations and multiple Internet of Things devices. In particular, first, we build the network model and channel model. Second, we construct the downlink transmission rate maximizing problem subject to the total power. Then, to solve this problem, we divide it into three sub-problems: device grouping, inter-sub-band power allocation, and intra-sub-band power allocation problems. Solving these sub-problems, we obtain the optimal power allocation schemes by jointly employing channel-state sorting–pairing algorithm, water-filling algorithm, and convex optimization theory. Finally, numerical simulations are conducted to validate the performance of our proposed optimal downlink power allocation scheme. Experimental results show that the total throughput of the system has been significantly improved.

Keywords

5G IoT UDN NOMA total system throughput water-filling algorithm

Introduction

With the development of the fifth-generation (5G) mobile networks, the increasing demand for mobile communications services has led to a rapid growth in mobile data.^1–3 Recently, 5G has been deployed globally; it is predicted that by 2022, mobile data volumes are expected to increase 1000 times more than in 2012. Compared with the traditional second-generation (2G), third-generation (3G), and fourth-generation (4G) networks, 5G wireless communication networks have obvious characteristics such as high reliability, low latency, and large capacity.^4–6 With the goal of “Internet of Things (IoT),” we are creating a new era of networking by integrating it with people’s lives.⁷ Subsequently, non-orthogonal multiple access (NOMA) is proposed.

NOMA technology is considered as one of the most promising technologies for IoT to increase the total system capacity.^8,9 As a new space interface technology, comparing with orthogonal multiple access (OMA), NOMA has the potential to significantly improve the radio spectrum efficiency, which leads to multiplexing of power domain users to further increase the total system throughput. In addition, NOMA provides better fairness to users than traditional OMA systems.^10,11 The current research on NOMA technology is divided into two categories: power domain NOMA and code domain NOMA.¹² The power domain NOMA transmission technique allocates power for transmitters according to different channel states, that is, less power will be allocated to transmitters with better channel states and more power will be allocated to transmitters with worse channel states. At the receiver, we use successive interference cancellation (SIC) to decode user information. SIC technique is used at the receiver to decode the user information, thus, enabling multiple user transmission in the same sub-channel.^13,14 Although the power domain NOMA transmission technique suffers from error code accumulation phenomenon, this phenomenon can be reduced by a suitable power allocation scheme. Thus, power domain NOMA is more widely used in IoT scenario.

NOMA technology, as one of the key technologies in the field of IoT, has been studied by a large number of scholars.¹⁵ The literature¹⁶ describes the impact of user pairing on the performance of NOMA systems. The results show that NOMA systems with fixed power allocation can provide greater throughput than OMA systems. In the study by Liu et al.,¹⁷ the authors investigate the fractional order power allocation algorithm in NOMA systems, where the power received by each user is determined by its channel state. In this literature, the authors introduce fairness as a system optimization objective to maximize the data count rate. In the literature,¹⁸ to maximize the total system rate, the authors study the optimal channel assignment and power allocation, and solve the objective optimization problem by means of the proposed water-filling algorithm. However, this literature assumes that the number of subcarriers assigned to a single user is not limited. The literature¹⁹ used a greedy user selection algorithm to schedule the best set of multiplexed users satisfying the maximum proportional fairness metric and used the Karush–Kuhn–Tucker (KKT) conditions to solve the objective optimization problem.

To obtain a better network performance, NOMA can be integrated with other key technologies.²⁰ For example, the combination of NOMA with ultra-dense networks (UDNs) technology can further increase the total transmission rate of the system from not only the power domain but also the space domain. The concentration of urban residential population has led to a gradual concentration of data traffic to hotspot areas, which puts higher demands on network capacity. UDNs with high-density spatial reuse of frequencies can effectively improve spatial reuse and meet the traffic demand of 5G, so it is considered as one of the key technologies to improve system capacity and spectrum efficiency in 5G mobile communications.^21,22 UDN is specifically the deployment of a large number of small base stations in a certain area, and the number of small base stations is much larger than the number of users. By increasing the number of small base stations per unit area to form a super-dense wireless communication network, UDN expands the spectrum resources of the system and increases the transmission rate of the system capacity greatly. However, due to the dense deployment of base stations in UDN and the small coverage area, there is a high possibility of overlap, resulting in serious interference between base stations.^23–26 The literature²⁷ uses inter-cell interference coefficients in UDNs to divide all base stations of the entire system under the constraint of maximum intra-cell interference and minimum inter-cell interference to divide the entire network into multiple cells, thus reducing the interference between cells. The literature²⁸ uses the coordinated multiple points (CoMP) technique to improve the data rate and total system throughput at edge terminals. CoMP technique can convert inter-cell interference signals into useful signals, thus mitigating inter-cell interference, and improving total system throughput. In the study by Zhang et al.,²⁹ the authors investigate the optimization of energy-efficient resources for downlink transmission in user-centric UDNs supported by wireless access, non-orthogonal multiple access, and beamforming wireless backhaul. The literature maximizes the energy efficiency of the system by jointly optimizing the scheduling, sub-channel assignment, and power allocation of users or access points. Simulation results show that the algorithm greatly improves the energy utilization of the system compared with the benchmark scheme. In the study by Khodmi et al.,³⁰ the authors propose joint access and backhaul resource allocation in 5G heterogeneous UDNs (H-UDNs) to maximize the overall throughput of the system. A decoupling approach is used to decompose the joint resource problem into two sub-problems. Most of these works only introduce techniques related to UDN or only improve the channel transmission rate in the spatial domain.

Motivated by the aforementioned problems, we consider the orthogonal frequency division multiplexing (OFDM)-NOMA–based UDN in IoT networks, and study the optimal downlink power allocation strategies optimal downlink. The specific works are as follows:

Considering the UDN scenario with multiple base stations and multiple IoT devices, we construct the problem to maximize system throughput;

To solve this problem, the original problem is equivalently converted into three sub-problems: device grouping algorithm, power allocation between sub-bands, and power allocation within sub-bands;

The simulations are conducted to validate the proposed power allocation scheme.

The remainder of this article is arranged as follows. The section “The system model” presents the downlink system model for OFDM-NOMA based UDN with multiple small base stations, and establishes the system throughput maximization problem. The section “Joint optimization strategies” divides the maximization problem into three sub-problems, and obtains the optimal downlink power allocation strategies by solving these sub-problems. The section “Performance evaluation” simulates our investigated optimal downlink power allocation scheme for OFDM-NOMA–based UDN. The section “Conclusion” concludes this article.

The system model

We consider a downlink NOMA-based UDN system with multiple small base stations and multiple IoT devices, as shown in Figure 1. We assume full knowledge of the channel-state information. There are $N$ small base stations, $U$ IoT devices, and each small base station has $M$ channels. We define the bandwidth of each channel as B. Small base stations and terminals are randomly distributed in the area, and the number of small base stations is far greater than the number of terminals. In this area, each small base station can serve multiple IoT groups, and each IoT group is allocated a sub-band. This article applies OFDM technology to make the sub-bands between the base stations orthogonal to each other. We divide the IoT devices of the entire cell into $K$ IoT groups. Taking into account the computational complexity of the receiver and the communication quality of the IoT device, we assume that each IoT group has two IoT devices denoted by $k_{1}$ and $k_{2}$ , respectively.

Figure 1.

OFDM-NOMA–based UDN system model.

The channel gains from the $n$ th ( $n \in N$ ) small base station to the IoT device $k_{1}$ and IoT device $k_{2}$ are denoted by $h_{k 1}^{n}$ and $h_{k 2}^{n}$ , respectively. We set $h_{k 1}^{n} > h_{k 2}^{n}$ . Let us denote the channel matrices of strong IoT devices and weak IoT devices in the cell by $H^{K 1 \times N}$ and $H^{K 2 \times N}$ , which can be expressed as follows

\begin{matrix} H_{K 1} = {[\begin{matrix} h_{1}^{1} \dots h_{1}^{n} \dots h_{1}^{N} \\ ⋮ ⋱ ⋮ ⋱ ⋮ \\ h_{k 1}^{1} \dots h_{k 1}^{n} \dots h_{k 1}^{N} \\ ⋮ ⋱ ⋮ ⋱ ⋮ \\ h_{K 1}^{1} \dots h_{K 1}^{n} \dots h_{K 1}^{N} \end{matrix}]}_{K 1 \times N} \\ H_{K 2} = {[\begin{matrix} h_{1}^{1} \dots h_{1}^{n} \dots h_{1}^{N} \\ ⋮ ⋱ ⋮ ⋱ ⋮ \\ h_{k 2}^{1} \dots h_{k 2}^{n} \dots h_{k 2}^{N} \\ ⋮ ⋱ ⋮ ⋱ ⋮ \\ h_{K 2}^{1} \dots h_{K 2}^{n} \dots h_{K 2}^{N} \end{matrix}]}_{K 2 \times N} \end{matrix}

(1)

The signals received by the kth IoT group within the coverage of the nth small base station,³¹ denoted by $y_{k 1}^{n}$ and $y_{k 2}^{n}$ , corresponding to the strong IoT devices and the weak IoT devices, respectively, which are shown as follows

y_{k 1}^{n} = {| h_{k 1}^{n} |}^{2} p_{k 1}^{n} x_{k 1}^{n} + {| h_{k 1}^{n} |}^{2} p_{k 2}^{n} x_{k 2}^{n} + n_{k 1}

(2)

y_{k 2}^{n} = {| h_{k 2}^{n} |}^{2} p_{k 2}^{n} x_{k 2}^{n} + {| h_{k 2}^{n} |}^{2} p_{k 1}^{n} x_{k 1}^{n} + n_{k 2}

(3)

In equations (2) and (3), $x_{k 1}$ and $x_{k 2}$ are the expected reception signals of IoT devices $k_{1}$ and $k_{2}$ , respectively. $n_{k 1}$ and $n_{k 2}$ are Gaussian white noise with a mean value of zero and a variance of $z$ received by IoT devices $k_{1}$ and $k_{2}$ , respectively. $p_{k 1}$ and $p_{k 2}$ are the reception power of the two devices in the kth device group. In the system of the downlink, the small base station according to the power multiplexing, the IoT device with better channel state can be allocated less power. Meanwhile the IoT device with worse channel state can be allocated more power, that is, $p_{k 1} < p_{k 2}$ . The received power is subject to the total power constraint,^32,33 which are shown as follows

\sum_{k = 1}^{K} (p_{k 1}^{n} + p_{k 2}^{n}) \leq p_{\max}

(4)

When devices transmit information on the channel, SIC technology is implemented for IoT device $k_{1}$ , as shown in Figure 2. To eliminate the interference, we first demodulate the signal of IoT device $k_{2}$ . Eliminating the demodulated received signal of IoT device $k_{2}$ , we demodulate the received signal of IoT device $k_{1}$ . Assuming that there is no error propagation in the whole process, after IoT device $k_{1}$ eliminates the interference caused by IoT device $k_{2}$ ,^34,35 the SNR (signal-to-noise ratio) of IoT device $k_{1}$ , denoted by $SN R_{k 1}$ , is shown as follows

SN R_{k 1} = \frac{\sum_{n = 1}^{N} (p_{k 1}^{n} {| h_{k 1}^{n} |}^{2})}{σ^{2}}

(5)

Figure 2.

Illustration of two IoT devices downlink NOMA with SIC.

The SINR (signal to interference and noise ratio) of IoT device $k_{2}$ , denoted by $SIN R_{k 2}$ , is shown as follows

SIN R_{k 2} = \frac{\sum_{n = 1}^{N} (p_{k 2}^{n} {| h_{k 2}^{n} |}^{2})}{\sum_{n = 1}^{N} (p_{k 1}^{n} {| h_{k 2}^{n} |}^{2}) + σ^{2}}

(6)

where $\sum_{n = 1}^{N} (p_{k 1}^{n} {| h_{k 2}^{n} |}^{2})$ represents the interference of the IoT device $k_{1}$ in the kth IoT group to the IoT device $k_{2}$ .

Based on the channel model, we can get the data rates of IoT device $k_{1}$ and IoT device $k_{2}$ as follows

{\begin{matrix} R_{k 1} = B \log_{2} (1 + SN R_{k 1}) \\ = B \log_{2} (1 + \frac{\sum_{n = 1}^{N} (p_{k 1}^{n} {| h_{k 1}^{n} |}^{2})}{σ^{2}}) \\ R_{k 2} = B \log_{2} (1 + SIN R_{k 2}) \\ = B \log_{2} (1 + \frac{\sum_{n = 1}^{N} (p_{k 2}^{n} {| h_{k 2}^{n} |}^{2})}{\sum_{n = 1}^{N} (p_{k 1}^{n} {| h_{k 2}^{n} |}^{2}) + σ^{2}}) \end{matrix}

(7)

The total throughput of the system is shown as follows

R_{total} = \sum_{k = 1}^{K} (R_{k 1} + R_{k 2})

(8)

In summary, we construct the OFDM-NOMA–based UDN throughput maximization problem P1, which can be expressed as follows

\begin{matrix} P 1 : \max_{p_{k 1}, p_{k 2}} \sum_{k = 1}^{K} (R_{k 1} + R_{k 2}), \\ s . t . C 1 : \sum_{k = 1}^{K} (p_{k 1}^{n} + p_{k 2}^{n}) \leq p_{\max}, \forall n, \\ C 2 : p_{k 1}^{n} \geq 0, p_{k 2}^{n} \geq 0, \forall n \end{matrix}

(9)

In problem P1, $C 1$ is the total power constraint; $C 2$ ensures that the power received by each IoT device is greater than or equal to zero.

To solve the above problems, we equivalently convert the total throughput maximization problem of the system into three sub-problems: device grouping algorithm, power allocation between sub-bands, and power allocation within sub-bands to design corresponding algorithms to solve them separately.

Joint optimization strategies

In this section, we employ the device grouping algorithm to group devices on each sub-band, allocate power to each sub-band, and then, obtain the power allocation coefficient of the IoT devices on the sub-band.

Device grouping algorithm

In the NOMA system, the greater the channel gain difference between IoT devices in the same group, the greater the sum rate of superimposed IoT devices and the better the grouping effect.³⁶ Thus, to group each sub-band, in channel-state sorting–pairing algorithm (CSS-PA), we sort IoT devices according to their channel states, and select the two IoT devices with larger channel gains difference as one IoT group. We denote the IoT device number by $U$ , the IoT device group number by $K$ . The specific process is as follows: The CSS-PA first sorts the IoT device’s channel conditions from low to high. If the number of IoT devices is even, we divide the entire sorted IoT devices into two groups, and the object matched by each IoT device is the IoT device whose distance differs by $K$ after sorting. If the number of IoT devices is odd, the intermediate IoT device becomes an IoT group, and the remaining IoT devices are still sorted according to the above method, as shown in Figure 3. According to CSS-PA, each group has no more than two IoT devices. The pseudo code of the algorithm is shown as follows.

Figure 3.

Grouping diagram for different number of IoT devices.

Algorithm 1.

Channel-state sorting–pairing algorithm.

1. Initialization: Initialize the number of IoT devices and generate two channel gain matrices

h 1

h 2

2. Ranking of channel gains

\hat{h} 1 = sort (h 1)

\hat{h} 2 = sort (h 2)

3. If the number of IoT devices is even
4. The part with strong channel gain forms the matrix

H 1

and the part with weak channel gain forms the matrix

H 2

5. Else
6. Generate matrix

H 1

for the intermediate IoT devices along with the part of the channel with strong gain and matrix

H 2

for the part of the channel with weak gain
7. End

Inter-sub-bands power allocation

On the grouped sub-band where the IoT devices have been grouped, we regard the two devices on the sub-band as an IoT group, and apply the water-filling algorithm to implement the power allocation for the inter-sub-bands. We first solve the sub-problem of problem P1, denoted by problem P2, as follows

\begin{matrix} P 2 : & \max_{p_{k}^{n}} \sum_{k = 1}^{K} B \log_{2} (1 + p_{k}^{n} H_{k}^{n}), \\ s . t . & \sum_{k = 1}^{K} p_{k}^{n} \leq p_{\max} \end{matrix}

(10)

where $H_{k}^{n} = \frac{{| h_{k}^{n} |}^{2}}{σ^{2}}$ and $h_{k}^{n}$ represents the equivalent water-filling channel gains for the kth IoT group within the coverage of the nth small base station. Assuming that the channel gain of a strong device in an IoT group is the channel gain of the entire IoT group, that is, $h_{k}^{n} = h_{k 1}^{n}$ .

To solve the problem P2, we construct the Lagrangian function for the optimization problem P2, denoted by $L_{1}$ , as follows

L_{1} = \sum_{k = 1}^{K} B \log_{2} (1 + H_{k}^{n} p_{k}^{n}) - λ (\sum_{k = 1}^{K} p_{k}^{n} - p_{\max})

(11)

where $λ$ is the introduced Lagrange multiplier. Based on the complementary relaxation principle of the Lagrange multiplier constraint, the KKT condition of the construction function is formulated as follows

{\begin{matrix} \frac{\partial L_{1}}{\partial p_{k}^{n}} = \frac{B}{\ln 2} - \frac{H_{k}^{n}}{1 + H_{k}^{n} p_{k}^{n}} - λ = 0, \\ \frac{\partial L_{1}}{\partial λ} = p_{\max} - \sum_{k = 1}^{K} p_{k}^{n} = 0 \end{matrix}

(12)

Solving equation (12), the optimal transmission power $p_{k}^{*}$ between inter-sub-bands is obtained as follows

p_{k}^{*} = {⌈ \frac{B}{λ \ln 2} - \frac{1}{H_{k}^{n}} ⌉}^{+}

(13)

$λ$ can be determined by substituting equation (13) into $p_{\max} = \sum_{k = 1}^{K} p_{k}^{*}$ . Observing equation (13), the inter-sub-band power allocation is the water-filling scheme, where $\frac{B}{λ \ln 2}$ is the water-filling level. To simplify the description, let us denote $δ = \frac{B}{λ \ln 2}$ . To derive the Lagrangian multiplier, the initial water-filling can be calculated according to the literature¹⁸ as follows

δ_{0} = \frac{1}{M} (p_{\max} + \sum_{m = 1}^{M} \frac{1}{H_{k}^{n}})

(14)

Substituting equation (14) into equation (13), we can obtain the initial optimal inter-sub-band power allocation scheme. Then, the iterative water-filling level is continuously updated with a certain step length until the water level converges. The iterative water-filling level is updated by the following equation

δ \leftarrow δ_{0} + ω \frac{1}{M} (p_{\max} - \sum_{m = 1}^{M} p_{k}^{n})

(15)

where $ω$ is the adjustment step length of the water-filling level.

The water-filling level converges when the condition $δ - δ_{0} \leq β$ is satisfied. Here, $β$ represents an infinitesimal positive number. The pseudo code of the algorithm is shown as follows.

Algorithm 2.

Iterative water-filling algorithm.

1. Initialization: Initial iterative water-filling level: equation (14)
2. From equation (13), we can obtain the power assigned to each sub-band
3. If the obtained power

<

0
4. Set this power value = 0 and delete it in the next iteration and update the iterative water-filling level equation (15)
5. Then
6. Return to step (1)
7. Else
8. End of iteration.

Let us denote $P_{S}$ the power matrix received by each IoT group as follows

P_{s} = {[\begin{matrix} p_{1}^{1} \dots p_{1}^{n} \dots p_{1}^{N / 2} \\ ⋮ ⋱ ⋮ ⋱ ⋮ \\ p_{k}^{1} \dots p_{k}^{n} \dots p_{k}^{N / 2} \\ ⋮ ⋱ ⋮ ⋱ ⋮ \\ p_{K}^{1} \dots p_{K}^{n} \dots p_{K}^{N / 2} \end{matrix}]}_{K \times N}

(16)

According to the above-mentioned iterative algorithm, the power allocation between the sub-bands is completed, and then the power allocation among the IoT devices in each sub-band is our main consideration.

Intra-sub-band power allocation

We assume that in the kth IoT group, the power allocation coefficient for the allocated IoT device $k 1$ is $α_{k}$ , that is, $p_{k 1}^{n} = α_{k} p_{k}^{n}$ . Then $p_{k 2}^{n} = (1 - α_{k}) p_{k}^{n}$ .

In the previous section, we have obtained the optimal inter-sub-band power allocation scheme $p_{k}^{n}$ . We need to derive the optimal intra-sub-band power allocation scheme $α$ . Thus, to obtain the optimal power allocation coefficient for the kth IoT group ( $k \in K$ ), we need to solve problem P3, which is shown as follows

\begin{matrix} P 3 : \max_{α_{1}} {R_{k 1} + R_{k 2}} \\ = \max_{α_{1}} {B \log_{2} (1 + \frac{α_{1} \sum_{n = 1}^{N} (p_{k}^{n} {| h_{k 1}^{n} |}^{2})}{σ^{2}}) \\ + B \log_{2} (1 + \frac{(1 - α_{1}) \sum_{n = 1}^{N} (p_{k}^{n} {| h_{k 2}^{n} |}^{2})}{α_{1} \sum_{n = 1}^{N} (p_{k}^{n} {| h_{k 2}^{n} |}^{2}) + σ^{2}})}, \\ s . t . \sum_{k = 1}^{K} (p_{k 1}^{n} + p_{k 2}^{n}) \leq p_{\max}, p_{k 1}^{n} \geq 0, p_{k 2}^{n} \geq 0 \end{matrix}

(17)

To guarantee the quality of service (QoS) for OFDM-NOMA–based UDNs, we set the IoT devices’ SINR not lower than the minimum SINR threshold. We reset the SINR thresholds of IoT devices $k 1$ and $k 2$ to be $β_{0}$ .^37–39 Then, the SINR requirements for IoT group $k$ are shown as follows

{\begin{matrix} \frac{\sum_{n = 1}^{N} ({| h_{k 1}^{n} |}^{2} α_{k} p_{k}^{n})}{σ^{2}} \geq β_{0}, \\ \frac{\sum_{n = 1}^{N} ({| h_{k 1}^{n} |}^{2} (1 - α_{k}) p_{k}^{n})}{\sum_{n = 1}^{N} ({| h_{k 2}^{n} |}^{2} α_{k} p_{k}^{n}) + σ^{2}} \geq β_{0} \end{matrix}

(18)

Simplifying equation (18), we can obtain the inequation as follows

\frac{β_{0}}{θ_{1}} \leq α_{k} \leq \frac{θ_{2} - β_{0}}{θ_{2} (1 + β_{0})}

(19)

where

θ_{1} = \frac{\sum_{n = 1}^{N} ({| h_{k 1}^{n} |}^{2} p_{k}^{n})}{σ^{2}}, θ_{2} = \frac{\sum_{n = 1}^{N} ({| h_{k 2}^{n} |}^{2} p_{k}^{n})}{σ^{2}}

(20)

We assume that $α_{\min} = \frac{β_{0}}{θ_{1}}, α_{\max} = \frac{θ_{2} - β_{0}}{θ_{2} (1 + β_{0})}$

To further derive the optimal intra-sub-band power distribution coefficient $α^{*}$ , we first give Lemma 1 corresponding to the convexity of problem P3.

Lemma 1

Problem P3 is a convex optimization problem.

Proof

Plugging equation (20) into the objective function of problem P3, it can be simplified to

\begin{matrix} f (α_{k}) = B \log_{2} (1 + α_{k} θ_{1}) \\ + B \log_{2} (1 + \frac{(1 - α_{k}) θ_{2}}{1 + α_{k} θ_{2}}) \end{matrix}

(21)

First, we construct the first-order derivative of $- f (α_{k})$ with respect to $α_{k}$ as follows

\frac{\partial (- f)}{\partial α_{k}} = - \frac{θ_{1} B}{(1 + α_{k} θ_{1}) \ln 2} + \frac{θ_{2} B}{(1 + α_{k} θ_{2}) \ln 2}

(22)

Then, the second-order derivative of $- f (α_{k})$ with respect to $α_{k}$ is shown as follows

\begin{matrix} \frac{\partial^{2} (- f)}{\partial {α_{k}}^{2}} = - \frac{{θ_{2}}^{2} B}{{(1 + α_{k} θ_{2})}^{2} \ln 2} + \frac{{θ_{1}}^{2} B}{{(1 + α_{k} θ_{1})}^{2} \ln 2} \\ = - \frac{({θ_{2}}^{2} {(1 + α_{k} θ_{1})}^{2} - {θ_{1}}^{2} {(1 + α_{k} θ_{2})}^{2}) B}{{(1 + α_{k} θ_{2})}^{2} {(1 + α_{k} θ_{1})}^{2} \ln 2} \end{matrix}

(23)

Since $θ_{1}$ is larger than $θ_{2}$ $(θ_{1} > θ_{2})$ , the second-order derivative of $- f (α_{1})$ is no smaller than zero $(\frac{\partial^{2} (- f)}{\partial {α_{1}}^{2}} \geq 0)$ .

Thus, the objective function of problem P3 is a convex. Also, the constraints of problem P3 are linear. Therefore, problem P3 is a convex optimization problem.▪

To solve problem P3, we construct the Lagrangian function of problem P3, denoted by $L_{2}$ as follows

\begin{matrix} L_{2} (α_{k}, γ_{1}, γ_{2}) = B \log_{2} (1 + α_{k} θ_{1}) \\ + B \log_{2} (1 + \frac{(1 - α_{k}) θ_{2}}{1 + α_{k} θ_{2}}) \\ + γ_{1} (α_{\min} - α_{k}) + γ_{2} (α_{k} - α_{\max}) \end{matrix}

(24)

where $γ_{1}$ and $γ_{2}$ are Lagrangian multipliers. Therefore, the established KKT conditions are

\frac{\partial L_{2}}{\partial α_{k}} = f' (α_{k}) - γ_{1} + γ_{2} = 0

(25)

γ_{1} (α_{\min} - α_{k}) = 0

(26)

γ_{2} (α_{k} - α_{\max}) = 0

(27)

Solving the above KKT conditions, we give some remarks corresponding to the optimal intra-sub-band power allocation coefficient under different Lagrangian multiplier.

1. When $γ_{1} > 0, γ_{2} > 0$ , the two KKT conditions of equations (26) and (27) need to be satisfied at the same time, which cannot hold

2. When $γ_{1} > 0, γ_{2} = 0$ , the optimal intra-sub-band power distribution coefficient can be calculated as follows

α^{*} = α_{\min} = \frac{β_{0}}{θ_{1}}

(28)

3. When $γ_{1} = 0, γ_{2} > 0$ , the optimal intra-sub-band power distribution coefficient can be calculated as follows

α^{*} = α_{\max} = \frac{θ_{2} - β_{0}}{θ_{2} (1 + β_{0})}

(29)

4. When $γ_{1} = 0, γ_{2} = 0$ , the optimal solution of the power distribution coefficient cannot be found.

In this section, we employ the channel -state sorting and pairing algorithm, iterative water-filling algorithm, and Lagrangian multiplier method to allocate the power for IoT groups, inter-sub-bands, and intra-sub-bands to maximize the transmission rate of the system.

Complexity analysis: the complexities for the algorithms I is $O (U)$ . Denote the number of iterations by $I$ . Then, we need to run $IU$ times the standard water-filling procedure, each requiring $O (M \log_{2} (M))$ operations. Therefore, the overall complexity is $O (IUM \log_{2} (M))$ .

Performance evaluation

In this section, the performance of our proposed scheme is evaluated through system-level simulation. In this simulation, we consider the downlink data transmission of the OFDM access (OFDMA)-NOMA system with multiple base stations. We assume that the maximum radius $R$ that the base station can cover is 50 m, the IoT devices in the cell are randomly and evenly distributed, the channel model conforms to the Rayleigh distribution, the system bandwidth $B$ is 1 MHz, the total transmission power $P$ of the base station is 5 W, and the noise power $N_{0}$ is 0.45 W.

Figure 4 shows a schematic diagram of the relationship between the total system throughput and the number of requested IoT devices. As shown in Figure 4, as the number of IoT devices increases, the total throughput of the NOMA system gradually increases, and is always higher than the total throughput of the OMA system. This is because compared with the traditional OMA system, NOMA systems can transmit information over the same time slots and frequency points with less resources, thus, greatly improving the utilization of spectrum resources. In addition, the gap between the curves of the NOMA system and the curves of the OMA system gradually increases as the number of IoT devices increases. When the number of IoT devices is 26, the total throughput of the NOMA system is 31.68% higher than that of the OMA system. This also further shows that the NOMA system has better spectrum utilization than the OMA system when the number of IoT devices is large.

Figure 4.

Impact of the number of IoT devices on total transmission rate. IoT: Internet of things.

Figure 5 displays the relationship between the total throughput of the NOMA system and the OMA system and the number of small base stations. At this time, we set the number of IoT devices to 10. From Figure 5, we can see that the total throughputs of NOMA and OMA system increase as the number of base stations increases, and the throughput of NOMA in both cases is far greater than the OMA system throughput. At the same time, it can be seen from the figure that the throughput of Plan 2 is higher than the throughput of Plan 1, so the power distribution coefficient obtained in the second case is greater than the power distribution in the first case coefficient.

Figure 5.

Impact of the number of SBS on total transmission rate.

Figure 6 plots the comparison between the NOMA system and the OMA system at different noise powers when the number of IoT devices is 2, 4, 6 and the total transmit power of the base station is $P = 0.5$ . We can see that the transmission rate of both NOMA and OMA systems tends to decrease as the noise increases, and that the smaller the number of devices, the slower the decrease. When the number of devices $U = 2$ , the difference between the values of Plan 2 and Plan 1 is small, but as the number of devices increases, the gap between them increases gradually. We can also see that the transmission rate of OMA system at device number U = 6 will gradually be smaller than the transmission rate of NOMA system at Plan 2 $U = 4$ as the noise gradually increases. The results are a good illustration of the effectiveness of the strategy we have proposed.

Figure 6.

Impact of the noise on total transmission rate.

Figure 7 demonstrates the comparison between NOMA and OMA systems with different transmit powers when the number of IoT devices is 2, 4, 6 and the noise power is 0.35. From the figure we can see that as the power increases, the gap between Plan1, Plan2 and OMA systems becomes greater, and the total system throughput in Plan2 of this article is greater than Plan1 and both cases are greater than the total throughput of the OMA system for a certain number of IoT devices. At power $P = 0.5$ , the transmission rate of NOMA at Plan2 $U = 4$ is greater than that of the OMA system at $U = 6$ , but as the power increases, the transmission rate of the OMA system gradually becomes greater than that of NOMA. In the case of varying only the number of IoT devices, as the power increases, the greater the number of IoT devices, the greater the total system throughput.

Figure 7.

Impact of the power on total transmission rate.

Conclusion

This article mainly studied the problem of maximizing the total throughput for OFDM-NOMA system with multiple base stations and multiple IoT devices. This maximization problem was equivalently converted into three sub-problems, that is, device grouping, inter-sub-band power allocation, and intra-sub-band power allocation. To solve these problems, we employ the channel state sorting, water-filling algorithm, and Lagrangian function. Then, the optimal joint power allocation strategies are obtained. The simulation results showed that the proposed optimal power allocation strategies can improve the performance compared with the OMA system, which verified the effectiveness of the proposed algorithms.

Footnotes

Handling Editor: Miguel Acevedo

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the Natural Science Foundation of Henan (No. 202300410292), the Key Scientific Projects of Henan Higher Education Institutions (No. 19A510018, No. 20A510008, and No. 21A510008), the Key Scientific and Technological Projects (No. 202102210120 and No. 212102210553), the Foundation for Young Backbone Teachers in Higher Education Institutions (No. 2018GGJS126), and the Henan key Laboratory for Big Data Processing and Analytics of Electronic Commerce (2020-KF-6).

ORCID iDs

Ya Gao

Yongpeng Shi

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Optimal downlink power allocation schemes for OFDM-NOMA-based Internet of things

Abstract

Keywords

Introduction

The system model

Joint optimization strategies

Device grouping algorithm

Inter-sub-bands power allocation

Intra-sub-band power allocation

Lemma 1

Proof

Performance evaluation

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References