# Resource Allocation for OFDM-NOMA Systems under Multiuser Scenarios with Different Speeds

Authors: Pengfei Wu; Wei Xiao; Zhicheng Dong; Jingjing Cui; Fuhua An; Weixi Zhou; Xiaoyu Li; Fangkai Cai
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

In this paper, we consider an OFDM-NOMA (orthogonal frequency division multiplexing-nonorthogonal multiple access) downlink system, which has multiple subcarriers and multiple users at different locations and speeds from the base station. The goal is to maximize energy efficiency (EE), which is the bit rate per unit of energy consumption, while meeting the user’s quality of service and power constraints. Since the optimization problem of joint user subcarrier allocation and power allocation is a mixed-integer nonlinear programming (MINLP) problem, the use of traversal search will produce an unacceptable amount of calculation. It should be noted that the optimization problem includes the variables of user and subcarrier allocation and power allocation coefficient. In order to effectively solve this problem, we divide the problem into two subproblems: the problem of user and subcarrier allocation and the problem of power allocation between users. We use a matching algorithm to solve the problem of user and subcarrier assignment. An iterative algorithm to ensure convergence is proposed to obtain the power budget between subcarriers. Finally, the effective binary search is used to obtain the power allocation among users. The simulation results show that the proposed algorithm has a faster convergence rate. At the same time, as the number of users and subcarriers increases, EE improves significantly. As the user speed increases, the EE gradually decreases; especially when the number of users is larger, the effect is more obvious. Compared with random matching, the proposed algorithm has higher EE under the same number of users, number of subcarriers, and speed.


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## Body

## 1. Introduction

The explosive growth of today’s data traffic has brought enormous challenges to the limited spectrum resources. Nonorthogonal multiple access (NOMA) technology is widely used in 5G mobile networks as a spectrum efficient multiple access technology [1]. NOMA technology allows multiple users to share spectral resources in the same spatial layer in both time and power domains. Compared with orthogonal multiple access (OMA), NOMA can support a large number of connections and achieve higher spectral efficiency [2]. NOMA allows multiple users to access the same orthogonal resource blocks, such as frequency bands, time slots, and spatial directions. Interuser interference can be eliminated at the receiving end by using a sequential interference cancellation (SIC) receiver for multiuser detection [3].Liang et al. [4] proposed user pairing based on one-to-one matching and studied the problem of power allocation in cognitive radio NOMA systems. In addition, [5] proposed a subchannel allocation method for a downlink NOMA system using the many-to-many matching theory. In heterogeneous networks, a suboptimal algorithm to alternately optimize the resource allocation of macrocells and small cells is proposed in [6]. Zhang et al. [7] designed a low-complexity subcarrier matching algorithm by decoupling subchannel allocation and power control. In incomplete CSI downlink NOMA heterogeneous networks, Song et al. [8] transformed the probabilistic problem into a nonprobabilistic problem through relaxation. Xu et al. [9] used joint optimization problem control involving user association, computing resource allocation, and transmit power. A Dinkelbach-like algorithm [10] is employed, by which each individual fractional EE function is transformed into a parametric function. In assisted cognitive radio nonorthogonal multiple access (CR-NOMA) networks, Zhao et al. [11] proposed a multiobjective iterative algorithm (MOIA) to achieve a joint optimal solution.Through the above research, we study the resource allocation of the OFDM-NOMA system in multiuser scenarios with different speeds. Since the optimization problem of joint user subcarrier allocation and power allocation is a mixed-integer nonlinear programming (MINLP) problem, the use of traversal search will produce an unacceptable amount of calculation; we divide the problem into two subproblems: the problem of user and subcarrier allocation and the problem of power allocation between users. The main contributions of this paper are as follows:(1)
This article considers the resource allocation of the OFDM-NOMA system in multiuser scenarios with different speeds. In order to solve the problem of grouped users and subcarriers, we adopted a matching algorithm. Firstly, users and suboperators establish their own preference lists and then perform matching exchanges through matching algorithms to obtain suboptimal solutions to solve the user grouping problem(2)
For the power allocation problem of different users on the same subcarrier, an iterative algorithm that guarantees convergence is proposed, which converts the nonconvex problem into a convex problem, obtains the power allocation for subcarriers, and then uses the effective binary search to solve the power allocation problem of different users on the same subcarrier(3)
The simulation results show that as the number of users and subcarriers increases, EE improves significantly. Compared with random matching, the algorithm proposed has higher energy efficiency (EE) under the same conditionsThe remaining sections of this article are arranged as follows: In Section2, we introduce system model and problem formulation. Resource allocations are reported in Section 3. Section 4 shows the simulation results and analysis. Finally, the conclusion is given in Section 5.


## 2. System Model and Problem Formulation

### 2.1. System Model

Consider an OFDM-NOMA (orthogonal frequency division multiplexing-nonorthogonal multiple access) downlink system withN subcarriers and K active users with different speeds and distances from base station (BS). We assume that the users’ velocities are also uniformly distributed. Different from the conventional OFDMA downlink system, a subcarrier can be shared by more than one user in the OFDM-NOMA system. The system model is shown in Figure 1. Let ηn,k∈0,1, where ηn,k=1 denotes that the nth subcarrier is assigned to the user k, ηn,k=0; if otherwise, then we can express the superposition symbol Xn on the subcarrier n as
(1)Xn=∑k=1Kηn,kPn,ksn,k,where Pn,k is the transmit power between the BS and the user k on the subcarrier n and sn,k denotes the unit energy Esn,k2=1, where E represents the expectation operation.The symbolsX1,⋯,XN are transformed into an OFDM symbol by inverse fast Fourier transformation (IFFT). A cyclic prefix (CP) is then added to eliminate the effect of the intersymbol interference (ISI). The time-domain transmitted signal can be written as xi=1/N∑n=0N−1Xnej2πin/N,−Lmax≤i≤N−1, where Lmax is the length of the guard interval for all users. The received signal yki in time-domain at the time i for user k can be expressed as
(2)yki=∑l=0Lhki,lxi−l+ωi,where hki,l is the channel impulse response of the lth tap at time i for the user k. We assume the Rayleigh fading channels, and a Jakes’ Doppler spectrum with a maximum Doppler frequency fkmax=fcυk/c for user k, where fc is the carrier frequency (Hz), vk is the speed of the mobile terminal (km/h) for user k, and c is the speed of light.Therefore, the autocorrelation function of the time-varying channels can be expressed asEhki1,l1hk∗i2,l2=c1e−l1/LJ02πfcvkTi2−i1/cNδl1−l2, where c1 is a constant, obtained to meet c1∑le−l/L=1. J0. denotes the zeroth-order Bessel function of the first kind, and T is the duration of an OFDM symbol. δ denotes the impulse function, and ∗ stands for conjugate transposition. Assume that the maximum channel delay spread L is less than or equal to the guard interval Lmax for all users, and ωi is a zero-mean, complex additive white Gaussian noise with variance σω2.
(3)Ykn=1N∑i=0N−1ykie−j2πnN=XnHnk+Ink+Wnk.Figure 1
NOMA systems under multiuser scenarios with different speeds.The second term in (3) denotes the intercarrier interference (ICI) caused by the time-varying nature of the channel. Since the data on each subcarrier are uncorrelated, the ICI power of the nth OFDM subcarrier for the user k with the velocity vk can be presented as follows [7, 12, 13].
(4)EInk2=∑m=0,m≠nN−1Pmρn,mk,where Pm=∑k=1Kηm,kPmk is the total power consumption on the subcarrier m. ρn,mk=1/N2N+2∑i=1N−1N−iJ02πfcvTi/cNcos2πin−m/N, and J0· denotes the zeroth-order Bessel function of the first kind [14]. T is the duration of an OFDM symbol. From (4), we can obtain that the ICI term monotone increases with increasing velocity. Then, gnk, regarded as an equivalent channel gain between the BS and the user k on the subcarrier n with the velocity vk, can be expressed as follows [15]:
(5)gnk=Hnk2ηn,k∑m=0,m≠nN−1Pmρn,mk+∑i=1,i≠kKHni2Pni+σω2.The NOMA protocol is used in each subcarrier in this article. Specifically, consider a pair of two usersk and j with different velocities vk and vj, in which the user k wants to decode and remove the user j′s signal by successive interference cancellation (SIC) at the receiver on the subcarrier n; then the following inequality holds: gnk≥gnj. In the NOMA system, SIC can be implemented for the users with stronger equivalent channel gains. Generally, it is assumed that all the equivalent channel gains on the subcarrier n follow the order as gnπ1≥⋯≥gnπk≥⋯≥gnπK, where πk presents the kth decoded the users' index. Therefore, gnπK first decodes the information of the users from K to k+1 and then successively subtracts these information to obtain its own information. In the conventional NOMA system, the order is sorted by the channel gains resulting from different distances between the BS and the users [3]. However, the different velocities can also be used to sort the equivalent channel gains’ order in this article. Therefore, based on the analysis of (4), users with slower velocities may guarantee that the equivalent channel gain is better than that of faster velocities’ users.Based on the above, the received signal to interference plus noise ratio (SINR) for the userπk at the subcarrier n with the velocity vk is given in (6).
(6)rnπk=Hnπk2ηn,πkPnπk∑i=1k−1Hnπk2ηn,πiPnπi+∑m=0,m≠nN−1Pmρn,mπk+σω2.The SINR performance on the subcarriern for the user k is impacted not only by noise and ICI but also by multiuser interference from SIC if the users have stronger equivalent channel gains than that of the kth decode user. Therefore, we can obtain the achievable rate of user πk at the subcarrier n with velocity vk, which is given by (7).
(7)Cnπk=BNlog1+rnπk.We define the total EE as the number of successfully transmitted bits per energy unit (bit/joule), or equivalently, the data rate per power unit (bps/watt). Therefore, we obtain the total EE of the small cell tier networks as(8)φ=RtotPtot,(9)Rtot=∑k=1K∑n=1NCnπk,(10)Ptot=∑k=1K∑n=1NPnk+Pc,where Pc is the circuit power consumption.


### 2.2. Problem Formulation EE

We formulate the problem to optimize the problem based on user, subcarrier assignment, and power allocation. The optimization problem can be expressed as follows:(11)maxPnπk,ηn,πkφPnπk,ηnπk=RtotPnπk,ηnπkPtotPnπk,ηnπk,such that
(12a)C1:gnπk≤gnπj,πk≥πj,∀n,k,C2:∑k=1K∑n=1NPnπkηn,πk≤Pmax,∀n,k,C3:Cnπk≥Rth,∀n,k,C4:∑k=1Kηn,πk≤Kf,C5:∑n=1Nηn,πk≤Nf,C6:ηn,πk=0,1,∀n,k,C7:Pnπk≥0,∀n,k,where C1 is used to guarantee that SIC can be implemented successfully for a specific order. Constraint C2 denotes the transmit power constraint for BS with the maximum power allowance Pmax. C3 represents the rate requirement of the kth decoded user on subcarrier n. C4 and C5 denote that the number of users on each subcarrier cannot be more than Kf and the number of subcarriers occupied by each user cannot be more than Nf.The optimization problem (8) by jointly designing user, subcarrier assignment, and power allocation for the OFDM-NOMA system is a mixed-integer nonlinear programming(MINLP) problem, which generally yields unacceptable computation burden with brute-force search. Note that the optimization problem (8) includes the binary optimization variables for the user, subcarrier assignment, and the continuous variables for the power allocation coefficients. To solve problem (8) effectively, we will divide the problem into two subproblems: (1) the problem of user and subcarrier assignment and (2) the problem of power allocation among users.


## 2.1. System Model

Consider an OFDM-NOMA (orthogonal frequency division multiplexing-nonorthogonal multiple access) downlink system withN subcarriers and K active users with different speeds and distances from base station (BS). We assume that the users’ velocities are also uniformly distributed. Different from the conventional OFDMA downlink system, a subcarrier can be shared by more than one user in the OFDM-NOMA system. The system model is shown in Figure 1. Let ηn,k∈0,1, where ηn,k=1 denotes that the nth subcarrier is assigned to the user k, ηn,k=0; if otherwise, then we can express the superposition symbol Xn on the subcarrier n as
(1)Xn=∑k=1Kηn,kPn,ksn,k,where Pn,k is the transmit power between the BS and the user k on the subcarrier n and sn,k denotes the unit energy Esn,k2=1, where E represents the expectation operation.The symbolsX1,⋯,XN are transformed into an OFDM symbol by inverse fast Fourier transformation (IFFT). A cyclic prefix (CP) is then added to eliminate the effect of the intersymbol interference (ISI). The time-domain transmitted signal can be written as xi=1/N∑n=0N−1Xnej2πin/N,−Lmax≤i≤N−1, where Lmax is the length of the guard interval for all users. The received signal yki in time-domain at the time i for user k can be expressed as
(2)yki=∑l=0Lhki,lxi−l+ωi,where hki,l is the channel impulse response of the lth tap at time i for the user k. We assume the Rayleigh fading channels, and a Jakes’ Doppler spectrum with a maximum Doppler frequency fkmax=fcυk/c for user k, where fc is the carrier frequency (Hz), vk is the speed of the mobile terminal (km/h) for user k, and c is the speed of light.Therefore, the autocorrelation function of the time-varying channels can be expressed asEhki1,l1hk∗i2,l2=c1e−l1/LJ02πfcvkTi2−i1/cNδl1−l2, where c1 is a constant, obtained to meet c1∑le−l/L=1. J0. denotes the zeroth-order Bessel function of the first kind, and T is the duration of an OFDM symbol. δ denotes the impulse function, and ∗ stands for conjugate transposition. Assume that the maximum channel delay spread L is less than or equal to the guard interval Lmax for all users, and ωi is a zero-mean, complex additive white Gaussian noise with variance σω2.
(3)Ykn=1N∑i=0N−1ykie−j2πnN=XnHnk+Ink+Wnk.Figure 1
NOMA systems under multiuser scenarios with different speeds.The second term in (3) denotes the intercarrier interference (ICI) caused by the time-varying nature of the channel. Since the data on each subcarrier are uncorrelated, the ICI power of the nth OFDM subcarrier for the user k with the velocity vk can be presented as follows [7, 12, 13].
(4)EInk2=∑m=0,m≠nN−1Pmρn,mk,where Pm=∑k=1Kηm,kPmk is the total power consumption on the subcarrier m. ρn,mk=1/N2N+2∑i=1N−1N−iJ02πfcvTi/cNcos2πin−m/N, and J0· denotes the zeroth-order Bessel function of the first kind [14]. T is the duration of an OFDM symbol. From (4), we can obtain that the ICI term monotone increases with increasing velocity. Then, gnk, regarded as an equivalent channel gain between the BS and the user k on the subcarrier n with the velocity vk, can be expressed as follows [15]:
(5)gnk=Hnk2ηn,k∑m=0,m≠nN−1Pmρn,mk+∑i=1,i≠kKHni2Pni+σω2.The NOMA protocol is used in each subcarrier in this article. Specifically, consider a pair of two usersk and j with different velocities vk and vj, in which the user k wants to decode and remove the user j′s signal by successive interference cancellation (SIC) at the receiver on the subcarrier n; then the following inequality holds: gnk≥gnj. In the NOMA system, SIC can be implemented for the users with stronger equivalent channel gains. Generally, it is assumed that all the equivalent channel gains on the subcarrier n follow the order as gnπ1≥⋯≥gnπk≥⋯≥gnπK, where πk presents the kth decoded the users' index. Therefore, gnπK first decodes the information of the users from K to k+1 and then successively subtracts these information to obtain its own information. In the conventional NOMA system, the order is sorted by the channel gains resulting from different distances between the BS and the users [3]. However, the different velocities can also be used to sort the equivalent channel gains’ order in this article. Therefore, based on the analysis of (4), users with slower velocities may guarantee that the equivalent channel gain is better than that of faster velocities’ users.Based on the above, the received signal to interference plus noise ratio (SINR) for the userπk at the subcarrier n with the velocity vk is given in (6).
(6)rnπk=Hnπk2ηn,πkPnπk∑i=1k−1Hnπk2ηn,πiPnπi+∑m=0,m≠nN−1Pmρn,mπk+σω2.The SINR performance on the subcarriern for the user k is impacted not only by noise and ICI but also by multiuser interference from SIC if the users have stronger equivalent channel gains than that of the kth decode user. Therefore, we can obtain the achievable rate of user πk at the subcarrier n with velocity vk, which is given by (7).
(7)Cnπk=BNlog1+rnπk.We define the total EE as the number of successfully transmitted bits per energy unit (bit/joule), or equivalently, the data rate per power unit (bps/watt). Therefore, we obtain the total EE of the small cell tier networks as(8)φ=RtotPtot,(9)Rtot=∑k=1K∑n=1NCnπk,(10)Ptot=∑k=1K∑n=1NPnk+Pc,where Pc is the circuit power consumption.


## 2.2. Problem Formulation EE

We formulate the problem to optimize the problem based on user, subcarrier assignment, and power allocation. The optimization problem can be expressed as follows:(11)maxPnπk,ηn,πkφPnπk,ηnπk=RtotPnπk,ηnπkPtotPnπk,ηnπk,such that
(12a)C1:gnπk≤gnπj,πk≥πj,∀n,k,C2:∑k=1K∑n=1NPnπkηn,πk≤Pmax,∀n,k,C3:Cnπk≥Rth,∀n,k,C4:∑k=1Kηn,πk≤Kf,C5:∑n=1Nηn,πk≤Nf,C6:ηn,πk=0,1,∀n,k,C7:Pnπk≥0,∀n,k,where C1 is used to guarantee that SIC can be implemented successfully for a specific order. Constraint C2 denotes the transmit power constraint for BS with the maximum power allowance Pmax. C3 represents the rate requirement of the kth decoded user on subcarrier n. C4 and C5 denote that the number of users on each subcarrier cannot be more than Kf and the number of subcarriers occupied by each user cannot be more than Nf.The optimization problem (8) by jointly designing user, subcarrier assignment, and power allocation for the OFDM-NOMA system is a mixed-integer nonlinear programming(MINLP) problem, which generally yields unacceptable computation burden with brute-force search. Note that the optimization problem (8) includes the binary optimization variables for the user, subcarrier assignment, and the continuous variables for the power allocation coefficients. To solve problem (8) effectively, we will divide the problem into two subproblems: (1) the problem of user and subcarrier assignment and (2) the problem of power allocation among users.


## 3. Resource Allocation

### 3.1. User and Subcarrier Assignment Based on Different Speeds

In this section, the problem of user and subcarrier assignment based on different speeds will be solved. Assume that the power allocated to the users in each subcarrier is a fixed value, i.e.,Pnk=Pn=Pmax/N,∀n∈1,⋯,N. Thus, we can obtain the subproblem of user and subcarrier assignment as the following:
(13)maxηn,πkφηnπk=RtotηnπkPtotηnπk,such that
(14)C1:gnπk≤gnπj,πk≥πj,∀n,k,C2:Cnπk≥Rth,∀n,k,C3:∑k=1Kηn,πk≤Kf,C4:∑n=1Nηn,πk≤Nf,C5:ηn,πk=0,1,∀n,k.Note that the complexity of the brute-force search increases exponentially with the number of users and subcarriers, which makes it impractical. To maximize the sum rate, we consider the user and subcarrier assignment as a many-to-many matching process between the sets of users and subcarriers. To solve this problem, we adopt the matching theory, which provides mathematically tractable and low-complexity solutions for the combinatorial problem of matching players in two distinct sets [16, 17]. We then propose an efficient algorithm to solve this problem.


### 3.2. Matching Theory for the User and Subcarrier Assignment

We design a low-complexity algorithm to solve the selection problem between users and subcarriers. First, we assume that the user set isK, N is used to represent the subchannel set, and the two sets are disjoint. We treat the two sets as selfish and rational players with the aim of maximizing their respective interests. Each player can exchange information with each other without additional signaling cost because the channel state information (CSI) and speed are known to the BS. This means that each player has complete information about each other. If a subchannel Nj is assigned to user Ki, then we say that Ki and Nj match each other, forming a matched pair. Matching is defined as pairing users in set K with subcarriers in set N, formally expressed as follows:Definition 1.
Given two disjoint sets,=1,2,⋯,K of the users, and N=1,2,⋯,N of the subchannels, a many-to many matching Ψ is a mapping from the set K∪N∪0 into the set of all subsets of K∪N such that
(1)
∣Ψk∣≤ln for every k∈K(2)
∣Ψn∣≤lk for every n∈N(3)
Ψk∈N if and only if Ψn∈K(4)
k=Ψn⇔n=Ψkwhere ln and lk are positive integers. Condition (1) states that each user could occupy at most ln subchannels, and condition (2) implies that each subchannel can be allocated to at most lk users.Then we introduce the conception of preference≻ for both users and subchannels, to better describe the competition behavior and decision process of each player. For any ki∈K, if ki prefers n1 to n2, we express it as n1≻kin2.It is not straightforward to define a stability concept in a many-to-many matching with externalities, as the gains from a matching pair depends on which players the other players have. However, a swap matching is defined asΨij=Ψ\i,k,j,n∪j,k,i,n, where Ψi=k and Ψj=n. Based on the swap operation, we introduce the two-sided exchange stability [18] as follows.Definition 2.
A matchingΨ is two-sided exchange stable if and only if there does not exist a pair of players i,j with k=Ψi and n=Ψj, such that
(1)
∀x∈i,j,k,n,UkΨij≥UkΨ(2)
∃x∈i,j,k,n, such that UkΨij≥UkΨ; then the swap matching Ψij is approved, and i,j is called a swap-blocking pair in Ψwhere UkΨ denotes the practicality for player k under matching Ψ. In this paper, it means the rate of user k in subcarrier n. In a nutshell, if any pair of players satisfied conditions (1) and (2), then it is called a swap-blocking pair.The characteristics of the swap-blocking pair ensure that if the exchange match is approved, the achieved rate of any participating user will not decrease, and the rate of at least one user will increase.As discussed above, the preference≻ can be formulated as the rate of each user over all subcarriers. Specifically, the sum rate of users associated to subcarriers can be expressed as
(15)Rnk=BNlog1+rnk.Then the preference of subcarriern on a set of users Ψk can be expressed as Rn=∑k∈ΨkRnk, and the preference of user k occupying a set of subcarriers Ψn can be expressed as Rk=∑n∈ΨnRnk.Therefore, for a given userk and any two subcarriers n,n′, with two different matchings Ψ and Ψ′, we have
(16)n,Ψ≻UEkn′,Ψ′⇔RkΨ>RkΨ′.This means that userk prefers subcarriers n to n′, because it can achieve a higher rate than subcarrier n′. Similarly, for a given subcarrier n and any two users k,k′ with two different matchings Ψ and Ψ′, we have
(17)k,Ψ≻SCnk′,Ψ′⇔RnΨ>RnΨ′.Then we introduce a matching Algorithm1 to achieve matching.Algorithm 1: Many-to-many matching based SC-US assignment.
1: Initialize.2: Construct the initial Subcarrier-User matching setM based on sum rate.3: Repeat.4: For any userk∈M, it searches for another subcarrier n with n∈M\MΨk. Let R=R0.5: For a givenk, calculate the candidate Rkn for the swapping pairs k,n.6:ifk,n, is a swapping-blocking pair then.7:R=R∪Rkn;8:end if.9: Keep the swapping-blocking pair withk,n∗=argmaxRknR, then Ψ=Ψkn∗. Set R0=Rkn∗.10: Until There are no swapping-blocking pairs.11: Output the stable matchingΨ.


### 3.3. Power Allocation for the Subcarrier

In the previous section, we solved the user and subcarrier assignment problem. We decompose the problem into subchannel power allocation and user power allocation within the same subchannel to obtain the suboptimal solution. In this section, we solve the subcarrier power allocation problem. So the problem can be converted as(18)maxPnkφPnk=RtotPnkPtotPnk,(19)Cnk=BNlog21+rnk,such that
(20)C1:Cnk≥Rth,∀n,k,rnk=Hnk2Pnk∑i=1k−1Hni2Pni+∑m=0,m≠nN−1Pmρn,m+σω2.Problem (11) is still a nonconvex problem due to Pn. Thus, to solve the problem efficiently, we consider a lower bound and a variable transformation, which was also used in [19, 20]. Then we get a lower bound through the following inequality as follows.
(21)log21+ς≥Γlog2ς¯+ϖ,with
(22)Γ=ς¯1+ς¯,and,ϖ=log21+ς¯−ς¯log2ς¯1+ς¯,where, if ς=ς¯ at this time, the inequality becomes tight, and then log21+ς=Γlog2ς¯+ϖ. Inspired by (22) and as ς¯ corresponds to rn, then we can obtain the following approximation of the lower bound:
(23)∑n=1NBNlog2Γrnk+ϖ⏟φ¯≤φ,where φ¯ is the lower bound of (23), where Γ=rnk/1+rnk and ϖ=log21+rnk−rnklog2rnk/1+rnk. When ς=ς¯, we will have the following expression:
(24)∑n=1NB/Nlog2ΓHnk2PnkPtot+ϖ∑i=1k−1Hni2Pni+∑m=0,m≠nN−1Pmρn,m+σω2Ptot−∑n=1NB/Nlog2Γ∑i=1k−1∣Hni2Pni+∑m=0,m≠nN−1Pmρn,m+σω2Ptot≜φ¯.Finally, we use this variable substitutionPnk=2εnk, and then it becomes
(25)φ¯=∑n=1NB/Nlog2ΓHnk22εnkPtot+ϖ∑i=1k−1Hni22εni+∑m=0,m≠nN−1∑k=1K2εnkρn,m+σω2Ptot−∑n=1NB/Nlog2Γ∑i=1k−1Hni22εniPtot+∑m=0,m≠nN−1∑k=1K2εnkρn,m+σω2Ptot≜fεnkgεnk,where εnk=εn1,εn2,⋯,εnk,⋯,εNKT represent the power allocation vector for each user in each subchannel, and .T is denoted as the transpose operation. Then from(23), we can observe that fεnk is concave and gεnk is a convex due to the fact that the log-sum-exp function is convex [21, 22]. However, the user service quality (QOS) constraint Cnk≥Rth can be expressed as
(26)Hnk22εnk≥2NBRth−1∑i=1k−1∣Hni22εni+∑m=0,m≠nN−1∑k=1K2εnkρn,m+σω2.Then we apply the logarithmic operation at the both sides of (24), and we can obtain the expression of (25) as
(27)εnk+log2Hnk22N/BRth−1,−log2∑i=1k−1Hni22εni+∑m=0,m≠nN−1∑k=1K2εnkρn,m+σω2≥0.Equation (25) is convex with respect to the variable set of εnk. In order to obtain the power allocation vector of each user on each subcarrier, we first determine the power allocation budget εn of each subcarrier, where εn=∑k=1Kfεnk=Pns. To this end, we deal with the following objective function.
(28)maxφ¯εn=fεngεn,such that
(29a)C1:in18,C4:∑n=1N2εn≤Pmax,C6:2εn≥0,where εn=ε1,ε2,⋯,εn,⋯,εNT represents the power budget of each subchannel n. Recalling that the log-sum-exp function is convex, it is clear that (28) has a concave numerator and a convex denominator for the objective function u¯ over convex constraints [20]. In order to make the symbol more concise, we use D to represent the domain of formula (18) to obtain a feasible solution to the optimization problem. Thus, let εn∗ and θ∗ be the optimal solution and the maximum of (28), respectively. Then, we define the maximum φ¯, θ∗ of the considered system as
(30)θ∗=fεn∗gεn∗=maxεnfεngεn,∀εn∈D.To this end, the following proposition is introduced.Proposition 1.
The maximum sum rate is achieved if and only if(31)Fθ∗=maxεn∈Dfεn−θ∗gεn=maxεn∈Dfεn∗−θ∗gεn∗=0.Proof.
We assumeqn∗ is the optimal solution to (28); then
(32)θ∗=fεn∗gεn∗≥fεngεn,∀εn∈D.
This leads to(33)fεn−θ∗gεn≤0,∀εn∈D,fεn∗−θ∗gεn∗=0,∀εn∈D.
Therefore,εn∗ satisfies (31) and Fθ∗=0.
Conversely, letεn∗=argmaxfεn−θ∗gεn. This implies
(34)fεn−θ∗gεn≤fεn∗−θ∗gεn∗=Fθ∗=0,∀εn∈D,in which the two conditions can be rewritten as
(35)θ∗≥fεngεn,∀qn∈D,θ∗=fεn∗gεn∗.
Thus, the proof is completed.So, we can find the optimal solutions for fractional problem (28) by finding the unique zero of (30), i.e., the auxiliary function Fθ=fεn−δgεn.Therefore, we solve (11), and its equivalent objective function is (26), which can be achieved by designing an iterative algorithm. The algorithm is based on Dinkelbach’s algorithm, which is summarized as Algorithm 1. Given the t value of θt in each iteration, we calculate the power allocation εnt by solving the equivalent problem (28). Then we obtain εnt by updating θt+1=fεnt/gεnt. We pay attention to the Dinkelbach method based on Newton’s method in fractional programming since the update can be obtained in the following way:
(36)θt+1=θt−FθtF′θt=fεntgεnt.Therefore, as long as (28) is solved in each iteration, the iterative algorithm can guarantee to converge to the optimal value θ∗. The proposed algorithm is presented in Algorithm 2.Algorithm 2: Power allocation for subchannel.
Initialize2: The iterationi=0,θ0=0 and maximize tolerance α>0SetP0,ς¯0=ς0,t=0.ComputeΓ0,ϖ0 using (32).4:whileFθt>αdoqn∗=argmaxfεn−θgεn6:Fθt=fεn∗−θtgεn∗Setθt+1=fεn∗/gεn∗8:t=t+1,UpdatePnsi+1=2εni+1,10:  Updateς¯i+1=ςi+1,UpdateΓs,ki+1,ϖs,ki+1, using (25).12:i=i+1end whileThrough Algorithm2, we solve the problem of power allocation between subchannels, so we further solve the problem of power allocation among users of the same subchannel. At this point, we transform the problem into
(37)maxφ¯εnk=fεnkgεnk,such that
(38)C1:Cnk≥Rth,∀n,k,C4:∑n=1N∑k=1K2εnk≤Pmax,C6:2εnk≥0.Obviously, for the power allocation among users of the same subchannel, the power allocated by each subchannel cannot exceed the power allocation budgetPns. According to the theory in [23, 24], for a user with a strong channel gain, his assignable power is between 0 and Pns/2m−1, where m is the number of users allocated in the same subchannel and Pns is the power budget of each subchannel n[25]. We assume that at most Kn users are multiplexed on the same subchannel; in order to reduce the complexity of the receiver, we assume that each subchannel carries at most 2 users, that is, Kn=2. Then we use a binary search algorithm to solve the power allocation problem for users on the same subchannel, which was given in Algorithm 3.Algorithm 3: Power allocation between users on the same subchannel.
InitializeSetRth, and α as a small positive value.3. SetP0,x¯0=x0,t=0.Pn1,min=0,Pn1,max=Pns/2m−1whilePn1max−Pn1min>αdo6:   setPn1=Pn1max+Pn1min/2setPn2=Pns−Pn1if∑n∈NCnk≤Rththen9:Pn1max=Pn1,elsePn1,min=Pn1,12:end ifend whileoutputPn1∗=Pn1,Pn2∗=Pnmax−Pn1∗


## 3.1. User and Subcarrier Assignment Based on Different Speeds

In this section, the problem of user and subcarrier assignment based on different speeds will be solved. Assume that the power allocated to the users in each subcarrier is a fixed value, i.e.,Pnk=Pn=Pmax/N,∀n∈1,⋯,N. Thus, we can obtain the subproblem of user and subcarrier assignment as the following:
(13)maxηn,πkφηnπk=RtotηnπkPtotηnπk,such that
(14)C1:gnπk≤gnπj,πk≥πj,∀n,k,C2:Cnπk≥Rth,∀n,k,C3:∑k=1Kηn,πk≤Kf,C4:∑n=1Nηn,πk≤Nf,C5:ηn,πk=0,1,∀n,k.Note that the complexity of the brute-force search increases exponentially with the number of users and subcarriers, which makes it impractical. To maximize the sum rate, we consider the user and subcarrier assignment as a many-to-many matching process between the sets of users and subcarriers. To solve this problem, we adopt the matching theory, which provides mathematically tractable and low-complexity solutions for the combinatorial problem of matching players in two distinct sets [16, 17]. We then propose an efficient algorithm to solve this problem.


## 3.2. Matching Theory for the User and Subcarrier Assignment

We design a low-complexity algorithm to solve the selection problem between users and subcarriers. First, we assume that the user set isK, N is used to represent the subchannel set, and the two sets are disjoint. We treat the two sets as selfish and rational players with the aim of maximizing their respective interests. Each player can exchange information with each other without additional signaling cost because the channel state information (CSI) and speed are known to the BS. This means that each player has complete information about each other. If a subchannel Nj is assigned to user Ki, then we say that Ki and Nj match each other, forming a matched pair. Matching is defined as pairing users in set K with subcarriers in set N, formally expressed as follows:Definition 1.
Given two disjoint sets,=1,2,⋯,K of the users, and N=1,2,⋯,N of the subchannels, a many-to many matching Ψ is a mapping from the set K∪N∪0 into the set of all subsets of K∪N such that
(1)
∣Ψk∣≤ln for every k∈K(2)
∣Ψn∣≤lk for every n∈N(3)
Ψk∈N if and only if Ψn∈K(4)
k=Ψn⇔n=Ψkwhere ln and lk are positive integers. Condition (1) states that each user could occupy at most ln subchannels, and condition (2) implies that each subchannel can be allocated to at most lk users.Then we introduce the conception of preference≻ for both users and subchannels, to better describe the competition behavior and decision process of each player. For any ki∈K, if ki prefers n1 to n2, we express it as n1≻kin2.It is not straightforward to define a stability concept in a many-to-many matching with externalities, as the gains from a matching pair depends on which players the other players have. However, a swap matching is defined asΨij=Ψ\i,k,j,n∪j,k,i,n, where Ψi=k and Ψj=n. Based on the swap operation, we introduce the two-sided exchange stability [18] as follows.Definition 2.
A matchingΨ is two-sided exchange stable if and only if there does not exist a pair of players i,j with k=Ψi and n=Ψj, such that
(1)
∀x∈i,j,k,n,UkΨij≥UkΨ(2)
∃x∈i,j,k,n, such that UkΨij≥UkΨ; then the swap matching Ψij is approved, and i,j is called a swap-blocking pair in Ψwhere UkΨ denotes the practicality for player k under matching Ψ. In this paper, it means the rate of user k in subcarrier n. In a nutshell, if any pair of players satisfied conditions (1) and (2), then it is called a swap-blocking pair.The characteristics of the swap-blocking pair ensure that if the exchange match is approved, the achieved rate of any participating user will not decrease, and the rate of at least one user will increase.As discussed above, the preference≻ can be formulated as the rate of each user over all subcarriers. Specifically, the sum rate of users associated to subcarriers can be expressed as
(15)Rnk=BNlog1+rnk.Then the preference of subcarriern on a set of users Ψk can be expressed as Rn=∑k∈ΨkRnk, and the preference of user k occupying a set of subcarriers Ψn can be expressed as Rk=∑n∈ΨnRnk.Therefore, for a given userk and any two subcarriers n,n′, with two different matchings Ψ and Ψ′, we have
(16)n,Ψ≻UEkn′,Ψ′⇔RkΨ>RkΨ′.This means that userk prefers subcarriers n to n′, because it can achieve a higher rate than subcarrier n′. Similarly, for a given subcarrier n and any two users k,k′ with two different matchings Ψ and Ψ′, we have
(17)k,Ψ≻SCnk′,Ψ′⇔RnΨ>RnΨ′.Then we introduce a matching Algorithm1 to achieve matching.Algorithm 1: Many-to-many matching based SC-US assignment.
1: Initialize.2: Construct the initial Subcarrier-User matching setM based on sum rate.3: Repeat.4: For any userk∈M, it searches for another subcarrier n with n∈M\MΨk. Let R=R0.5: For a givenk, calculate the candidate Rkn for the swapping pairs k,n.6:ifk,n, is a swapping-blocking pair then.7:R=R∪Rkn;8:end if.9: Keep the swapping-blocking pair withk,n∗=argmaxRknR, then Ψ=Ψkn∗. Set R0=Rkn∗.10: Until There are no swapping-blocking pairs.11: Output the stable matchingΨ.


## 3.3. Power Allocation for the Subcarrier

In the previous section, we solved the user and subcarrier assignment problem. We decompose the problem into subchannel power allocation and user power allocation within the same subchannel to obtain the suboptimal solution. In this section, we solve the subcarrier power allocation problem. So the problem can be converted as(18)maxPnkφPnk=RtotPnkPtotPnk,(19)Cnk=BNlog21+rnk,such that
(20)C1:Cnk≥Rth,∀n,k,rnk=Hnk2Pnk∑i=1k−1Hni2Pni+∑m=0,m≠nN−1Pmρn,m+σω2.Problem (11) is still a nonconvex problem due to Pn. Thus, to solve the problem efficiently, we consider a lower bound and a variable transformation, which was also used in [19, 20]. Then we get a lower bound through the following inequality as follows.
(21)log21+ς≥Γlog2ς¯+ϖ,with
(22)Γ=ς¯1+ς¯,and,ϖ=log21+ς¯−ς¯log2ς¯1+ς¯,where, if ς=ς¯ at this time, the inequality becomes tight, and then log21+ς=Γlog2ς¯+ϖ. Inspired by (22) and as ς¯ corresponds to rn, then we can obtain the following approximation of the lower bound:
(23)∑n=1NBNlog2Γrnk+ϖ⏟φ¯≤φ,where φ¯ is the lower bound of (23), where Γ=rnk/1+rnk and ϖ=log21+rnk−rnklog2rnk/1+rnk. When ς=ς¯, we will have the following expression:
(24)∑n=1NB/Nlog2ΓHnk2PnkPtot+ϖ∑i=1k−1Hni2Pni+∑m=0,m≠nN−1Pmρn,m+σω2Ptot−∑n=1NB/Nlog2Γ∑i=1k−1∣Hni2Pni+∑m=0,m≠nN−1Pmρn,m+σω2Ptot≜φ¯.Finally, we use this variable substitutionPnk=2εnk, and then it becomes
(25)φ¯=∑n=1NB/Nlog2ΓHnk22εnkPtot+ϖ∑i=1k−1Hni22εni+∑m=0,m≠nN−1∑k=1K2εnkρn,m+σω2Ptot−∑n=1NB/Nlog2Γ∑i=1k−1Hni22εniPtot+∑m=0,m≠nN−1∑k=1K2εnkρn,m+σω2Ptot≜fεnkgεnk,where εnk=εn1,εn2,⋯,εnk,⋯,εNKT represent the power allocation vector for each user in each subchannel, and .T is denoted as the transpose operation. Then from(23), we can observe that fεnk is concave and gεnk is a convex due to the fact that the log-sum-exp function is convex [21, 22]. However, the user service quality (QOS) constraint Cnk≥Rth can be expressed as
(26)Hnk22εnk≥2NBRth−1∑i=1k−1∣Hni22εni+∑m=0,m≠nN−1∑k=1K2εnkρn,m+σω2.Then we apply the logarithmic operation at the both sides of (24), and we can obtain the expression of (25) as
(27)εnk+log2Hnk22N/BRth−1,−log2∑i=1k−1Hni22εni+∑m=0,m≠nN−1∑k=1K2εnkρn,m+σω2≥0.Equation (25) is convex with respect to the variable set of εnk. In order to obtain the power allocation vector of each user on each subcarrier, we first determine the power allocation budget εn of each subcarrier, where εn=∑k=1Kfεnk=Pns. To this end, we deal with the following objective function.
(28)maxφ¯εn=fεngεn,such that
(29a)C1:in18,C4:∑n=1N2εn≤Pmax,C6:2εn≥0,where εn=ε1,ε2,⋯,εn,⋯,εNT represents the power budget of each subchannel n. Recalling that the log-sum-exp function is convex, it is clear that (28) has a concave numerator and a convex denominator for the objective function u¯ over convex constraints [20]. In order to make the symbol more concise, we use D to represent the domain of formula (18) to obtain a feasible solution to the optimization problem. Thus, let εn∗ and θ∗ be the optimal solution and the maximum of (28), respectively. Then, we define the maximum φ¯, θ∗ of the considered system as
(30)θ∗=fεn∗gεn∗=maxεnfεngεn,∀εn∈D.To this end, the following proposition is introduced.Proposition 1.
The maximum sum rate is achieved if and only if(31)Fθ∗=maxεn∈Dfεn−θ∗gεn=maxεn∈Dfεn∗−θ∗gεn∗=0.Proof.
We assumeqn∗ is the optimal solution to (28); then
(32)θ∗=fεn∗gεn∗≥fεngεn,∀εn∈D.
This leads to(33)fεn−θ∗gεn≤0,∀εn∈D,fεn∗−θ∗gεn∗=0,∀εn∈D.
Therefore,εn∗ satisfies (31) and Fθ∗=0.
Conversely, letεn∗=argmaxfεn−θ∗gεn. This implies
(34)fεn−θ∗gεn≤fεn∗−θ∗gεn∗=Fθ∗=0,∀εn∈D,in which the two conditions can be rewritten as
(35)θ∗≥fεngεn,∀qn∈D,θ∗=fεn∗gεn∗.
Thus, the proof is completed.So, we can find the optimal solutions for fractional problem (28) by finding the unique zero of (30), i.e., the auxiliary function Fθ=fεn−δgεn.Therefore, we solve (11), and its equivalent objective function is (26), which can be achieved by designing an iterative algorithm. The algorithm is based on Dinkelbach’s algorithm, which is summarized as Algorithm 1. Given the t value of θt in each iteration, we calculate the power allocation εnt by solving the equivalent problem (28). Then we obtain εnt by updating θt+1=fεnt/gεnt. We pay attention to the Dinkelbach method based on Newton’s method in fractional programming since the update can be obtained in the following way:
(36)θt+1=θt−FθtF′θt=fεntgεnt.Therefore, as long as (28) is solved in each iteration, the iterative algorithm can guarantee to converge to the optimal value θ∗. The proposed algorithm is presented in Algorithm 2.Algorithm 2: Power allocation for subchannel.
Initialize2: The iterationi=0,θ0=0 and maximize tolerance α>0SetP0,ς¯0=ς0,t=0.ComputeΓ0,ϖ0 using (32).4:whileFθt>αdoqn∗=argmaxfεn−θgεn6:Fθt=fεn∗−θtgεn∗Setθt+1=fεn∗/gεn∗8:t=t+1,UpdatePnsi+1=2εni+1,10:  Updateς¯i+1=ςi+1,UpdateΓs,ki+1,ϖs,ki+1, using (25).12:i=i+1end whileThrough Algorithm2, we solve the problem of power allocation between subchannels, so we further solve the problem of power allocation among users of the same subchannel. At this point, we transform the problem into
(37)maxφ¯εnk=fεnkgεnk,such that
(38)C1:Cnk≥Rth,∀n,k,C4:∑n=1N∑k=1K2εnk≤Pmax,C6:2εnk≥0.Obviously, for the power allocation among users of the same subchannel, the power allocated by each subchannel cannot exceed the power allocation budgetPns. According to the theory in [23, 24], for a user with a strong channel gain, his assignable power is between 0 and Pns/2m−1, where m is the number of users allocated in the same subchannel and Pns is the power budget of each subchannel n[25]. We assume that at most Kn users are multiplexed on the same subchannel; in order to reduce the complexity of the receiver, we assume that each subchannel carries at most 2 users, that is, Kn=2. Then we use a binary search algorithm to solve the power allocation problem for users on the same subchannel, which was given in Algorithm 3.Algorithm 3: Power allocation between users on the same subchannel.
InitializeSetRth, and α as a small positive value.3. SetP0,x¯0=x0,t=0.Pn1,min=0,Pn1,max=Pns/2m−1whilePn1max−Pn1min>αdo6:   setPn1=Pn1max+Pn1min/2setPn2=Pns−Pn1if∑n∈NCnk≤Rththen9:Pn1max=Pn1,elsePn1,min=Pn1,12:end ifend whileoutputPn1∗=Pn1,Pn2∗=Pnmax−Pn1∗


## 4. Simulation

In this part, we analyze the simulation performance of the system model; in Figure2, we, respectively, set K=128, N=64, and V=100km/h; K=64, N=32, and V=100km/h; and K=32, N=16, and V=100km/h. It can be seen from the simulation results of Figure 2 that as the power increases, EE first increases and then decreases. It shows that within a certain power range, the system model has an optimal power distribution situation.Figure 2
EE of different subcarriers and users at different powers.Figure3 reflects the EE at different speeds for different subcarriers and users. From the figure, we can see that with the increase of user speed, EE gradually decreases; especially when the number of users and the number of subcarriers are large, the effect is more obvious. This is because as the number of subcarriers increases, the spacing between carriers becomes smaller, and the greater the total ICI generated by the system. Figure 4 presents the comparison of random matching plus power distribution and matching plus power distribution under EE conditions at zero speed and different powers. It can be seen that the proposed algorithm has a significant improvement on EE.Figure 3
EE of different subcarriers and users at different speeds.Figure 4
EE and power under different matching algorithms.Figure5 shows the relationship between the convergence of the proposed algorithm and EE. It can be seen from Figure 5 that when the number of iterations reaches 3, the algorithm has completely converged.Figure 5
Iterative convergence of EE and algorithm for different subcarriers and users.Table1 shows the performance comparison of various algorithms, where V and V′ represent the number of iterations of Algorithms 2 and 3. As can be seen from the table, the algorithm proposed in this paper has higher computational complexity than random matching, but it has better EE. Compared with SCA, the proposed algorithm has relatively low computational complexity and can obtain approximate performance.Table 1
Comparison of various algorithms.
AlgorithmsComplexityOptimalityMatch+POK+N2+VK+V′KfNfN+OV¯NSuboptimalRandmatch+POK+N2+OV¯NSuboptimalMatch+SCA [26]OK+N2+VK+V′KfNfN+OV¯NK3SuboptimalReference [24] does not consider the impact of ICI on EE generated by users at different speeds. Compared with reference [26], although the branch and bound (BB) approach can be used to obtain the optimal solution, it will produce unacceptable computational complexity. The algorithm proposed in this paper can only obtain suboptimal solutions, but it can greatly reduce the calculation time.


## 5. Conclusion

In this paper, we mainly study the resource allocation of the OFDM-NOMA system in multiuser scenarios with different speeds. The resource allocation problem we describe is NP-hard, so we decompose the resource allocation problem into two subproblems for solving, namely the BS-SC matching problem and the power allocation problem. First, BS-SC is matched according to the matching theory, and then power allocation is carried out for the BS-SC pairs successfully matched. Two suboptimal algorithms are proposed to solve the nonconvex optimization problem.



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*Source: 1388097-2022-03-03.xml*