# Novel Resource Allocation Techniques for Downlink Non-Orthogonal Multiple Access Systems

^{1}

^{2}

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^{*}

## Abstract

**:**

## 1. Introduction

- An optimal user pairing approach based HNG algorithm is proposed that considers pairing of only two users at each subchannel. This algorithm guarantees an optimal performance of both the sum rate and energy efficiency maximization process in a downlink NOMA system. For comparison, the performance of the proposed algorithm is compared with the center-based user pairing (CEB) method proposed in [16].
- The sum rate maximization problem is formulated by considering SIC for each user in the subchannel in which the constraints are the minimum acceptable data rate and the maximum available transmission power at the BS. To solve this problem and obtain the optimal power for each of the paired users in a subchannel, a power allocation solution is proposed based on Karush–Kuhn–Tucker (KKT) conditions (PKKT). Then, the closed-form optimal power allocation solution for the multiplexed users is obtained. The PKKT solution is applied after pairing users based on HNG to obtain the optimal sum rate in NOMA. Thus, we refer to this technique as NOMA-PKKT-HNG.
- The energy efficiency maximization problem is formulated with power constraint consideration, and SIC is applied to reduce complexity at the receiver. To solve this problem and obtain the optimal power for the multiplexed users, we propose a power allocation solution using PDKL. The formulated objective function is in fractional form. Therefore, the DKL algorithm is used to transform into linear form (subtractive function) and iteratively solve the problem with considerable error tolerance. The PDKL solution is applied after pairing users based on HNG to obtain the optimal energy efficiency in NOMA. Thus, we refer to this technique as NOMA-PDKL-HNG.

## 2. Related Works

## 3. System Model and Performance Metrics

#### 3.1. System Model

_{w,s}represents the wth user in the sth subchannel within the NOMA system. B

_{w}is the system bandwidth, and B

_{sch}= B

_{w}/S is the assigned bandwidth for each subchannel. P

_{max}and the P

_{s}denote the maximum transmit power for BS and the assigned power for each subchannel, respectively. With the assumption that the BS has an ideal knowledge of CSI, M

_{s}represents the number of the paired users on sth subchannel and S

_{i}represents their superposed signals. The superposition of the user‘s modulated symbols takes place in each of the S subchannels. Hence, the BS transmission symbols are expressed in Equations (1) and (3).

_{q}represents the modulated symbol of qth user, in which E [|S

_{q}|

^{2}= 1]. The transmit power allocated to the qth user on the sth subchannel is denoted by P

_{q,s}. The received signal for UE

_{w,s}is given as in [28].

_{w}is the distance between wth user and the BS, β is the path-loss exponent and ${n}_{w,s}\sim $$\mathcal{C}\mathcal{N}\left(0,{\sigma}_{s}^{2}\right)$ represents the additive white Gaussian noise (AWGN), with zero mean and variance ${\sigma}_{s}^{2}$. The power spectral density is symbolized by N

_{0}. The power constraints for each subchannel and the BS are limited to $\sum}_{i=1}^{{M}_{s}}{P}_{w,s}={P}_{s$ and $\sum}_{s=1}^{S}{P}_{s}={P}_{\mathrm{max}$, respectively, where P

_{w,s}is the power of UE

_{w,s}. Let ${H}_{w,s}={\left|{h}_{w,s}\right|}^{2}/{N}_{0}$ be the channel gain-to-noise ratio (CNR) of UE

_{w,s}. The arrangement of the users in a subchannel is based on their CNR as shown in Equation (4).

_{1,s}) decodes first and removes the weak user (UE

_{2,s}) signal before decoding its signal. On the contrary, the weak user (UE

_{2,s}) directly decodes its signal without adopting SIC. With the consideration of SIC at the receiver, the received signal to interference plus noise ratio (SINR) for UE

_{w,s}is represented in Equation (5).

#### 3.2. Performance Metrics

#### 3.2.1. Throughput

_{w,s}achievable rate in the NOMA system, the following mathematical representation is applied [27]:

#### 3.2.2. Energy Efficiency

_{w,s}energy efficiency (EE

_{w,s}) in a NOMA system can be determined with Equations (9) and (10). Accordingly, the sum of energy efficiency of the corresponding NOMA is estimated with the use of Equation (11).

#### 3.2.3. Fairness

## 4. Proposed Resource Allocation Techniques

#### 4.1. User Pairing Approach Based on HNG Algorithm

^{3}) is the computational complexity of the algorithm, which is especially high in a large-scale scenario; however, it ensures an optimal global performance [32,33]. Suppose there are two sets of users, with u = {1, 2, …, U} and v = {1, 2, …, V} denoting strong and weak users representing rows and columns, respectively. Therefore, a cost matrix (cost function) is formulated as m = u × v to attain the best optimal match for both sets of users. The element in the uth row and vth column characterizes the sum rate or energy efficiency of the cost matrix in the NOMA system. The proposed HNG solution is presented in Algorithm 1.

Algorithm 1. User pairing based on Hungarian algorithm |

Step 1: Formulation of cost matrix. |

a) Formulate the cost matrix m. |

b) Find the largest possible value from the entire cost matrix and subtract it from each cost matrix element. |

Step 2: Reduction of row and column. |

a) Find the lowest value from each row and subtract it from every value of the same row. |

b) Find the lowest value from each column and subtract it from every value of the same column. |

Step 3: Optimization procedure. |

a) Checking of row. |

i If exactly one zero in any row is found, mark the zero with a circle and draw a vertical line across it, but if not, skip that row. |

ii If all zeros are traversed with lines, then proceed to Step 4. But if not, do column checking. |

b) Checking of column. |

i If exactly one zero in any column is found, mark the zero with a circle and draw a horizontal line across it, but if not, skip that column. |

ii Make confirmation whether all the zeros are traversed with lines. |

Step 4: If the number of circles is equal to the number of cost matrix rows, then proceed to |

Step 5, but if not, proceed to Step 3. |

Step 5: The marked circles are the objective function solutions for all marked users pairs. |

_{s,w}represents the total data rate, and E

_{s,w}represents the total energy efficiency for strong and weak user pairings). The main aim is to maximize the user’s sum rate and energy efficiency. Therefore, the largest possible value is first determined and then deducted through the whole cost matrix (W

_{s,w}or E

_{s,w}). Table 1 and Table 2 illustrate the W

_{s,w}and E

_{s,w}for 10 Users (UEs) available in the BS. The size of rows and columns is decreased by determining the minimum value in each of them in the second step. The determined value is deducted from corresponding row or column.

_{1}, UE

_{6}), (UE

_{2}, UE

_{9}) and (UE

_{3}, UE

_{8}).

#### 4.2. Power Allocation Solution Using KKT Conditions (PKKT)

_{w}indicates the minimum achieved data rate as the quality of service (QoS) requirement for the corresponding user. The objective function in Equation (13) is a convex problem [28]. To find the solution of this problem, the KKT conditions are proposed to obtain the optimal power for the multiplexed user in a closed-form solution [34]. The corresponding Lagrange function of the problem expressed in Equation (13) is determined as illustrated in Equation (16).

_{w}is the target data rate. Let that,

**Lemma**

**1.**

#### 4.3. Power Allocation Solution Using DKL Algorithm

_{w}is a real number, which can be written as in Equation (35):

**Proposition**

**1.**

Algorithm 2. Dinkelbach method for energy efficiency maximization |

Step 1: Initialize ${q}_{w}=0$, iteration number $w=0$ and $\epsilon =0.01$ |

Step 2: while$Z\left({q}_{w}\right)=T\left({P}_{w,s}\right)-{q}_{w}B\left({P}_{w,s}\right)>\epsilon $do |

${P}_{w,s}^{*}\leftarrow \underset{{P}_{w,s>0}}{{\displaystyle argmax}}\left(Z\left({q}_{w}\right)\right)$ |

${q}_{w+1}\leftarrow \frac{T\left({P}_{w,s}^{*}\right)}{B\left({P}_{w,s}^{*}\right)}$ |

$w\leftarrow w+1$ |

Step 3: If$\left|Z\left({q}_{w}\right)\right|\le \epsilon $ |

break |

end if |

end while |

Step 4: Output $\left({P}_{w,s}^{*},{q}_{w}\right)$ |

## 5. Results and Discussion

_{max}) of 41 dBm and the circuit power consumption (P

_{c}) of 1 W is considered. The BS is located at the center of a circle with radius R = 500 m, and there is a random distribution of M users in the cell coverage. To reduce computational complexity, NOMA assigns only two users to each subchannel, and OMA assigns a single user to each subchannel. The CSI is known at the BS. The minimum distance between BS and users is 50 m, and the minimum distance between users is 40 m. The bandwidth of the NOMA system (B

_{w}) is 5 MHz and the path-loss exponent β = 2. The noise power ${\sigma}_{s}^{2}={B}_{w}{N}_{0}/S$, and the power spectral density N

_{0}= –174dBm/Hz. The obtained results are compared with OFDMA, FTPA [11] and the DC algorithm [28].

#### 5.1. Complexity Analysis

^{3}), where N refers to the number of users available in the NOMA system that need to be paired to achieve their maximum sum rate or energy efficiency. However, this algorithm shows a higher complexity in large-scale scenarios with a great number of users. In the Dinkelbach algorithm, the computational complexity is mainly based on both the convergence rate of the sequence of the subproblem and the individual complexity of each individual subproblem. It is noted that the Dinkelbach algorithm shows superlinear convergence. Based on this optimization problem expression of Equation (31), it is assumed that the convergence is achieved with M

_{1}and M

_{2}iterations for outer and inner loops, respectively, with the computational complexity of variables (Q

_{1}) and constraints (Q

_{2}) due to polynomial characteristics. Therefore, the inner loop complexity corresponds to O(M

_{1}M

_{2}(Q

_{1}+Q

_{2})). The updates of q

_{w}require O(K) operations. Hence, the total computational complexity for the proposed Dinkelbach algorithm corresponds to O(M

_{1}(M

_{2}(Q

_{1}+Q

_{2}) + O(K))).

^{S}) is needed to achieve all possible user pairing processes. S refers to subchannels available in the network. It is concluded that the proposed algorithms have lower complexity than the exhaustive search algorithm. The complexity comparison between the proposed and exhaustive search algorithms is summarized in Table 3.

#### 5.2. Performance Evaluation

_{1}, UE

_{6}), (UE

_{2}, UE

_{7}), (UE

_{3}, UE

_{8}), (UE

_{4}, UE

_{9}) and (UE

_{5}, UE

_{10})). In FTPA, the allocated power is obtained based on the condition of the user’s channel. Hence, the transmission power of the wth user on the sth subchannel (UE

_{w,s}) is expressed as in Equation (37).

_{w,s}denotes the channel gain normalized by the noise of user UE

_{w,s}, P

_{s}represents the subchannel’s power and α is the decay factor with a value between 0 and 1. Increasing α indicates that high power is assigned to user with weak channel gain. The user with poorer channel gain is allocated with a greater α, which means a high power increase. This approach exhibits a low computational complexity, with degraded performance as a result of a user’s allocation power instability in proportion to the path loss [28]. The default simulation parameter values are given in Table 4.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Lemma 1.**

## Appendix B

**Proof**

**of**

**Proposition 1.**

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# | UE_{6} | UE_{7} | UE_{8} | UE_{9} | UE_{10} |
---|---|---|---|---|---|

UE_{1} | 4.9913 | 5.1440 | 4.9890 | 5.1128 | 4.9877 |

UE_{2} | 0.0469 | 0.1997 | 0.0446 | 0.1684 | 0.0433 |

UE_{3} | 0.3331 | 0.4859 | 0.3307 | 0.4546 | 0.3294 |

UE_{4} | 0.0218 | 0.1746 | 0.0195 | 0.1433 | 0.0182 |

UE_{5} | 0.2218 | 0.3746 | 0.2194 | 0.3433 | 0.2181 |

# | UE_{6} | UE_{7} | UE_{8} | UE_{9} | UE_{10} |
---|---|---|---|---|---|

UE_{1} | 25.759 | 31.111 | 25.769 | 30.725 | 25.769 |

UE_{2} | 0.0057 | 5.3476 | 0.0054 | 4.9617 | 0.0053 |

UE_{3} | 6.8344 | 1.2176 | 6.8341 | 11.79 | 6.8339 |

UE_{4} | 0.0025 | 5.3444 | 0.0022 | 4.9585 | 0.0021 |

UE_{5} | 5.8915 | 11.233 | 5.8912 | 10.847 | 5.891 |

Algorithm | Complexity |
---|---|

Hungarian algorithm | O(N^{3}) |

Dinkelbach algorithm | O(M_{1} (M_{2}(Q_{1}+Q_{2}) + O(K))) |

Exhaustive search | O(2S!/2^{S}) |

**Table 4.**Parameter specifications [28].

Parameter | Default Value |
---|---|

Radius of the cell (R) | 500 m |

Maximum BS transmit power (P_{max}) | 41 dBm |

Circuit power consumption (P_{c}) | 1 W |

System bandwidth (B_{w}) | 5 MHz |

Noise power spectral density (N_{0}) | −174 dBm/Hz |

Decay factor (α) for FTPA | 0.4 |

Path-loss exponent (β) | 2 [8] |

Minimum data rate for strong user | 200 Kbps |

Minimum data rate for weak user | 20 Kbps |

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**MDPI and ACS Style**

Ali, Z.J.; Noordin, N.K.; Sali, A.; Hashim, F.; Balfaqih, M.
Novel Resource Allocation Techniques for Downlink Non-Orthogonal Multiple Access Systems. *Appl. Sci.* **2020**, *10*, 5892.
https://doi.org/10.3390/app10175892

**AMA Style**

Ali ZJ, Noordin NK, Sali A, Hashim F, Balfaqih M.
Novel Resource Allocation Techniques for Downlink Non-Orthogonal Multiple Access Systems. *Applied Sciences*. 2020; 10(17):5892.
https://doi.org/10.3390/app10175892

**Chicago/Turabian Style**

Ali, Zuhura J., Nor K. Noordin, Aduwati Sali, Fazirulhisyam Hashim, and Mohammed Balfaqih.
2020. "Novel Resource Allocation Techniques for Downlink Non-Orthogonal Multiple Access Systems" *Applied Sciences* 10, no. 17: 5892.
https://doi.org/10.3390/app10175892