I Introduction
Sparse Code Multiple Access (SCMA) [wei2016scma] illustrating a promising candidate for the 5G system, is a multidimensional codebook based nonorthogonal spreading technique that supplies a nearoptimal design of sparse codewords and achieves considerable progress in spectral efficiency and massive connectivity. Mobile communication networks contribute significantly towards the global carbon emission, thus, energy consumption is a critical concern when planning for 5G networks [wu2018noma, wu2014ofdma]. Therefore improving the energy efficiency (EE), which is defined as the achievable data rates over the total power consumption, has received much investigation recently.
In [zhang2014sparse], the EE of uplink SCMA network was analyzed and a comparison was made between the performance of average block error rate (BLER) and EE for the SCMA scheme and LTEA. The obtained simulation results demonstrate that the SCMA scheme is capable of aggregating more users in the uplink network to support the massive connectivity requirements for 5G systems with reasonable energy consumption. In addition to considering equal transmit power of nonzero elements for all users, a fixed factor graph matrix is also used in this reference for problem analysis. The codebook assignment and optimal power allocation for downlink SCMA is investigated in [li2016cost]. The obtained simulation results reveal that the SCMA network significantly outperforms the OFDMA network in term of EE. Moreover, in [jin2018resource], resource allocation for EE maximization of layered multicast in downlink SCMA network, by separating codebook assignment and power allocation was considered.
[zhai2015rate] investigated the rate and energy maximization problem in SCMA networks with simultaneous wireless information and power transfer. In [dong2016energy]
cooperative coevolutionary particle swarm optimization (CCPSO) was applied to solve the EE maximization problem for uplink SCMA network. In
[huang2018power] the Lagrange dual decomposition method and Dinkelbach theory to solve the nonconvex optimization problem to maximize the EE for SCMA downlink systems is employed, based on which a threelevel power allocation to improve the sum capacity for SCMA downlink system is proposed in [han2018optimal]. The authors in [d2018learning]proposed a computationally efficient algorithm to maximize the global energy efficiency with subject to both maximum power and minimum transmission rate constraints in the multicarrier wireless interference network. Finally, it is converted into a nonconvex fractional problem and was tackled through an interplay of fractional programming based on Dinkelbach algorithm, fixedpoint learning and the generalized game theory. In
[yu2019power] the EE problem was approximately decomposed into some independent convex subproblems by applying the fractional programming theory and using the coordinate descent method, and the closedform solutions of each subproblem are derived.[dong2016energy] evaluates EE of the network using the power allocation matrix without considering different pathloss. On the other hand, solving the resource allocation problem using separate subcarrier assignment and power allocation [li2015energy] is useful in analyzing downlink SCMA EE, especially in reducing the computational cost [li2016cost]. Moreover, [huang2018power] employs the waterfilling algorithm to solve the nonconvex power allocation optimization problem to evaluate the performance of downlink SCMA in term of EE with fixed factor graph. Briefly, we need to develop optimal resource allocation algorithms for uplink SCMA scheme by incorporating the factor graph matrix optimization to increase energy efficiency that meets 5G network requirements.
The aim of this paper is to maximize the EE for an uplink SCMA system where the the pathlosses between each user and the base station (BS) are not necessarily equal. The EE optimization problem involving subcarrier assignment and power allocation is a nonconvex mixedinteger nonlinear program (MINLP) problem [ghasemishabankareh2019genetic]. To reduce the computational cost, we separate subcarrier assignment and power allocation. A fast subcarrier assignment algorithm with power equally distributed is proposed to determine an energy efficient optimal and of lowcomplexity factor graph matrix. For specified factor graph matrix, a power allocation algorithm is proposed for EE maximization, which is decomposed into convex subproblems by applying the fractional programming theory based on Dinkelbach method.
Ii System model and problem formulation
Iia System model
We consider an SCMA based uplink communication network that is comprised of one based station (BS) and users, where an individual user represents a layer of SCMA. BS and all the users are equipped with a single antenna. There are totally subcarriers available where the users are multiplexed over these subcarriers. The number of codewords is equal to the number of users and the length of each codeword is . The transmitted codewords for the th user is represented by , which has nonzero elements. The zero and nonzero entries for all transmitted signals are represented by the factor graph matrix . According to the overloading feature of SCMA scheme, we have and is called overloading factor.
Let denote the
th column vector of matrix
, called indicator vector of the th user, where if , and otherwise. Let be the power of the th user in the th subcarrier, where if , and , if . Let be the channel vector of the th user, and is the channel gain of the th user in the th subcarrier. Let be the additive white Gaussian noise. Then the received signal in BS, which is a combination of all propagated signals through subcarriers, is written as follows.(1) 
where . Using the sparsity in the SCMA scheme, the message passing algorithm (MPA) detector can be applied to achieve a nearoptimal detection performance. With the knowledge of the channel state information (CSI), the data rate of the th user in the th subcarrier is [d2018learning],
(2) 
where is the noise power. The achievable rate in the th subcarrier is the sum of all users rates assigned to this subcarrier,
(3) 
Using (2) and (3), the sum rate of the network is
(4) 
According to [li2015energy, li2016cost, dong2016energy], the total power consumption of the whole system is
(5) 
where is the fixed circuit power consumption of each user. In (5), the total power consumption for the uplink is linearly increasing with , and obviously, which has a critical role on the total power consumption of the system. As per [li2015energy, li2016cost, dong2016energy], EE is the total uplink sum rate of all users over the total power consumption. Thus, from (4) and (5), the energy efficiency is
(6) 
where is the power allocation matrix of all users of the network.
IiB Problem formulation
As mentioned above, it is crucial to obtain the factor graph matrix and transmit power allocation matrix to improve the EE of the network. This is more important when SNRs are different for network users and users are distributed in different locations within the network. The EE maximization problem can be formulated as
(7) 
where is the maximum power of user . The constraint shows the number of subcarriers that each user occupies, and according to the conditions of the SCMA network, is the number of nonzero symbols of users. The constraint shows the feature of the factor graph matrix. The constraint denotes that have no two same columns. There are possible states for the first column, and there is one state decrement for the subsequent columns. Hence, the number of all different possible is
(8) 
The constraint shows the limitation of each user’s transmit power, and shows that each power allocation coefficient must be nonnegative. Optimization problem (7) is a nonconvex mixedinteger non linear programming (nonconvex MINLP) problems [ghasemishabankareh2019genetic].
Iii Fast subcarriers assignment and power allocation
Considering the received signal in each subcarrier, the sum capacity of uplink network can be rewritten as
(9) 
Eq. (9) implies that both and have important roles on the SCMA sum rate. The problem (7) can be written as
(10) 
Iiia Fast Subcarrier Assignment Algorithm
This algorithm is to allocate the subcarrier to users one by one. At each stage, calculate the network EE increment. When the th user access to the network, the th column of is updated to form the new factor graph . The EE increment is calculated as
(11) 
Determine the optimal under the condition that the transmit power is equally distributed for user symbols, meaning that power is equally distributed among codebooks. Generate a factor graph matrix randomly. When , can be obtained by allocating orthogonal column of . When , the is calculated as follows.
(12) 
Then can be obtained by solving the following optimization problem over the rest columns of , i.e.,
(13) 
This algorithm is summarized in the following Algorithm .
IiiB Energy efficient power allocation
There is a tradeoff between the sum rate of the network and the total power consumption for the EE maximization problem while users have limitation on the transmit power. The optimization problem (10) is a nonconvex problem due to the fractionalform of the objective function. The fractional programming based on Dinklebach’s algorithm [dinkelbach1967nonlinear] can be exploited for solving this problem [d2018learning]. It is observed that the numerator and denominator of the objective function in (10) are both concave. After is determined according to subcarrier algorithm in the previous section. The optimization problem (10) can be written as
(14) 
Define an auxiliary function
(15) 
where is a nonnegative factor. Based on Dinklebach method, we can transform problem (14) into a series of parametric subtractiveform subproblems as given as follows [yu2019power]
(16) 
According to Dinklebach algorithm, the value of starts from and it is updated to its optimal value by using an iterative procedure. The procedure is terminated when the value of auxiliary function is smaller than threshold , i.e., . As increases, the value of decreases. The term in denotes the updated power after solving (16). Once the iterative procedure is terminated, the maximum value of is .
Since is a concave function with respect to , we can use Lagrangian multiplier method to solve problem (16). The Lagrangian function is defined as
(17) 
where denotes the Lagrange multiplier. In addition, in terms of the determined , we have . Then the derived transmit power satisfying KarushKuhnTucker conditions [boyd2004convex] is
(18) 
Once is derived, the Lagrangian multiplier can be updated using the following subgradient algorithm [huang2018power],
(19) 
where is the iteration step size. Subsequently, after a limited number of iterations, (19) will converge. Additionally, to prevent the reduction in the sum rate of the SCMA network (9), it can be assumed that each user utilizes the maximum transmit power. In other words, the sum of transmit power of nonzero elements of each user is . To prevent the reduction in the sum rate of the network in addition to optimizing EE, we can address the calculation of (18) via initialization of the uplink network power into a feasible values, while the Lagrange multiplier satisfies the equation . Having the derived , subcarrier assignment to nonzero elements of users is determined. Therefore, it is merely required to design in order to satisfy for different users. The energy efficient power allocation algorithm is summarized in the following Algorithm .
Iv Simulation results
We consider a singlecell uplink network where users are randomly distributed in a circular area of m radius. In the simulation, both smallscale fading and largescale path loss have been considered. The channel gain where represents the Rayleigh fading channel gain of the th user on the th subcarrier, and is the path loss exponent [yang2017uplink]. The numbers of subcarriers and nonzeros elements in each SCMAcodeword are and , respectively. The bandwidth of each subcarrier, the power density of noise and the fixed circuit power consumption at each user are , and , respectively.
The simulation results are averaged over experiments. The computational complexity of Algorithm is . In the case , , and , the number of iterations is , while, according to (8), the number of iterations is for the exhaustive search.
Fig. 1 and Fig. 2, present the energy efficient SCMA uplink system for four different cases of resource allocation defined as follows.

SCMAPAPPC: The subcarrier assignment uses the proposed assignment (PA) Algorithm , and the power allocation uses the proposed power allocation Algorithm under the power constraints (PPC): .

SCMAPAPMP: The subcarrier assignment uses the proposed assignment (PA) Algorithm , but the power allocation uses the proposed power allocation Algorithm under the maximum power constraint (PMP): .

SCMARAPMP: Random assignment (RA) method refers to the method that randomly assigns subcarriers to users [li2016cost, xiong2018optimal]. The power allocation uses Algorithm under the maximum power constraint (PMP).

SCMAFAPMP: In fixed subcarrier assignment (FA) method we use a factor graph matrix that it’s given by [zhang2014sparse] and [yang2017uplink], and the power allocation uses Algorithm under the maximum power constraint (PMP).
Fig. 1 shows the EE for the four different subcarrier assignments and power allocations, where all the user distances are equal to m. It is shown that SCMAPAPPC and SCMAPAPMP outperforms SCMARAPMP and SCMAFAPMP. Therefore the proposed subcarrier assignment performs better. Moreover both SCMAPAPPC and SCMAPAPMP converge to a similar maximum level as increases. It is also shown that SCMAPAPPC has a better performance compared to SCMAPAPMP. This is because that each user using the maximum power has more fairness but loses EE performance.
Fig. 2 considers the influence of user distances to the uplink SCMA EE. Two different conditions are considered for users with the same average location, which are listed in the following table.
User distance  Condition 1  Condition 2 

User distance  m  m 
User distance  m  m 
User distance  m  m 
User distance  m  m 
User distance  m  m 
User distance  m  m 
Average user distance  m  m 
From Fig. 2, it is observed that SCMAPAPPC has almost the same performance for the user distance condition 1 and condition 2. So does SCMAPAPMP. This is due to the proposed method which offers the optimal factor graph that leads to the maximum EE via applying an equal maximum transmit power to each nonzeros element of all users. It is clear that there is a significant increase in EE for SCMAPAPPC and SCMAPAPMP over SCMARAPMP and SCMAFAPMP. The EE for both SCMARAPMP and SCMAFAPMP has better performance for the user distance condition 1 than that for condition 2.
From Fig. 2, it can be seen, the EE performance for condition 1 is better than that for condition 2, which implies that the EE performance is mainly influenced by the users close to the base station. These users close to base station are more likely to have the most important contribution to optimize the energy use.
V Conclusion
In this letter, we proposed a fast subcarrier assignment algorithm for energy efficient uplink SCMA. Then for a determined SCMA factor graph matrix, we propose a power allocation algorithm to solve the nonconvex optimization problem using Dinklebach method. By the proposed algorithms, the SCMA EE is well optimized based on subcarrier assignment and power allocation, while the complexity of Algorithm is much decreased compared to the exhaustive search. Simulations show that the proposed algorithm outperforms the random and fixed factor graph matrix, and users closed to base station contribute more to the SCMA EE.
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