Wireless spectrum has become severely limited and the 5G system is expected to offer 10 times spectral efficiency (SE) and energy efficiency (EE) in comparison with the 4G system. The anticipated thousand-fold increase in wireless data traffic urgently calls for highly efficient spectrum utilization to achieve these challenging goals. Various new techniques have been proposed, among which multi-group multicast and non-orthogonal multiple access (NOMA) are two promising ones.
Wireless multi-group multicast is considered as an effective solution to improve SE and EE by exploiting the idiosyncracies of wireless medium. It utilizes the available bandwidth for delivering the same content to multiple users and further exploits the spatial diversity by simultaneously serving multiple user groups, which significantly promotes the efficient utilization of the spectrum. Besides, transmit beamforming makes it possible to focus energy into the directions of the target users within a specified geographical region.
NOMA is widely recognized as another emerging paradigm to enhance spectrum utilization for the 5G system. The major advantage of NOMA is to encourage spectrum sharing among users in a non-orthogonal way. Different from conventional orthogonal multiple access (OMA), NOMA realizes the simultaneous transmission of multiple data-streams via power domain division . At a transmitter, a superposition of different messages is broadcasted. At receivers, successive interference cancellation (SIC) is used to realize multi-user separation and detection.
Current designs from industry and academia consider multi-group multicast and NOMA independently one from the other and for different intentions. However, there is an important bridge between them: geographical locations of the target users plays a critical role in both the two techniques. For multi-group multicast, user grouping is quite important in satisfying concurrent users’ requests for the same content. Thus, it is natural to exploit multi-group multicast for transmitting different messages to user clusters in different geographical regions that could be intuitively characterized by the distances from the transmitter to the users. On the other hand, the nature of NOMA is to utilize the signal-to-interference-plus-noise ratio (SINR) imparity among users that results from either the natural near-far effect or the non-uniform power allocation at a transmitter [6, 7]. Accordingly, the difference in channel gains, caused by different signal attenuation due to different distances of users to the transmitter, is a natural advantage for NOMA. We can see that physical distance becomes a bridge that connects multi-group multicast and NOMA, because distance is not only an important geographical feature of regions, but also a crucial factor which makes the users in different regions have different channel gains.
Clearly, it is of paramount importance to use the geographical information for designing multi-group multicast and NOMA altogether. Out of this we porpose a novel multi-group multicast framework in this paper which is termed as NOMA assisted multi-region geocast111It is worth to point out that the term geocast derives from a technique for routing protocols which realizes the delivery of a common message to a group of users within a specified geographical region.. The novelty and particularity of this new framework consists in the exploitation of geographical features of the target users in different geographical regions and broadcast properties of wireless medium. To be specific, our proposed NOMA assisted multi-region geocast refers to the simultaneous transmission of different messages to user clusters located in different predetermined regions via NOMA principles, and the users in the same region are interested in a common message. By exploiting multiple antenna technology, the transmit beamforming, aided by the geographical information of users, can provide a new manner to fully utilize the limited spectrum and to serve users with the minimum possible energy consumption.
With NOMA assisted multi-region geocast, many new services and applications are feasible. These include geographical command systems, emergency communications for public alerting, and commercial geographical messaging applications, such as position-based advertising, weather forecasting, and traffic advisory services. These imply that our proposed NOMA assisted multi-region geocast is a promising location-based spectrum sharing framework, which holds tremendous potential to support the increasing demand for higher quality services with the minimum possible energy expenditure for the future 5G system.
I-a Related Works and Motivations
The transmit beamforming for multi-group multicast have been initially solved with the semi-definite relaxation (SDR) method in [3, 5]. Further, a weighted max-min fair multi-group multicast problem has been studied in  under per-antenna power constraints. In , the distributed beamforming design has been attempted for a multi-group multicasting relay network. However, all these works have neither exploited users’ geographical information nor took the advantage of NOMA to further improve the SE and EE of wireless transmissions.
Current works on NOMA primarily focused on the SE improvement in either single-input single-output (SISO) systems[11, 12, 13, 14] or multiple-input multiple-output (MIMO) systems that can be decoupled into multiple SISO NOMA sub-systems by user clustering  or signal alignment . Only some works have directly tackled NOMA in multiple-antenna settings due to the difficulty of designing decoding order for SIC at receivers. In , multicast design has been investigated in a multiple-input single-output (MISO) NOMA system wherein a transmitter broadcasts only two different data streams. In , the beamforming design has been considered in MISO NOMA system where users are partitioned into multiple cluster with uniform power allocation. In , the optimal resource allocation for multicarrier downlink MISO NOMA system has been investigated. In , the worst-case achievable sum rate has been considered and an alternative optimization algorithm was proposed. In , a Hybrid NOMA precoding algorithm has been proposed by applying the quasi-degradation into downlink MISO NOMA system. Under bounded channel uncertainties, a robust beamforming techniques has been provided for MISO NOMA system in . Besides, in , the sum rate maximization problem has also been studied in a downlink MISO NOMA system, in which the power allocation policy is to allocate more power to users with worse channel conditions. However, this widely used policy is unnecessary and even degrades the overall performance of NOMA. Accordingly, in this work, we directly deal with NOMA in MISO systems and propose to guarantee users’ quality of service (QoS) with minimum SINR thresholds. Our previous work  has framed this NOMA assisted multi-region geocast by just focusing on its SE perspective.
In addition, NOMA can be viewed as a special form of cognitive radio (CR) networks, in which the user having stronger channel gains is viewed as a secondary user while the user having poorer channel gains is viewed as a primary one. In , the above described CR inspired NOMA framework was generalized by exploiting multiple antenna technologies and QoS guarantees for weaker users.
Currently, only some recent work have studied the EE issues in SISO and MIMO NOMA systems with statistical CSI [26, 27, 28], which greatly motivates us to study NOMA from the perspectives of both SE and EE. Furthermore, current designs of NOMA were generally compared to the traditional OMA scheme[16, 23, 26], which obviously demonstrates the superiority and advantage of NOMA. However, conventional spatial division multiple access (SDMA) should be considered as a strong competitor when studying NOMA in multiple-antenna settings.
Based on the above survey, this NOMA assisted multi-region geocast framework has a potential future for realizing higher efficient spectrum utilization. Further, comparisons between NOMA and SDMA are still not clear. Thereby, it is of great meaning to study this new framework from the perspectives of SE and EE for sake of further revealing the idiosyncrasy of NOMA.
I-B Our Work and Contributions
In this work, with advanced beamforming techniques, we apply NOMA to simultaneously transmit superposed messages to multiple geographical regions, which germinates a location-based spectrum sharing framework. The main contributions and motivation are summarized as below:
A novel multi-group multicast framework is proposed by exploiting both NOMA and the geographical information of users in transmit beamforming designs. Within the proposed framework, we initially minimize the total transmit power subject to an individual QoS constraint for each user. Secondly, we maximize the system sum rate under a predefined fair transmission policy that ensures constant ratios between the data rates of different messages. Finally, an EE maximization problem is investigated in the proposed framework under a dynamical access scenario. The majorization and minimization (MM) algorithms are employed to solve the non-convex and intractable problems therein.
Different from previous works on NOMA, this work abolishes the conventional power allocation policy that more transmit power should be given to users with worse channel conditions. Instead, a new decoding order for SIC is designed and guaranteed by users’ geographical information and their required QoS (SINR threshold) in the process of transmit beamforming designs.
Comprehensive numerical experiments are provided, with respect to different key parameters, to show that NOMA scheme holds tremendous promise for enhancing SE and EE but also certain limitations in wireless transmissions, compared with both conventional SDMA and OMA schemes. Our findings aim to provide not only insights on possibilities of NOMA but also guidance on applications of NOMA in future 5G systems.
I-C Organization and Notations
Organization: In Section II, we present the system and the signal model of multi-region geocast in a MISO NOMA system. In Section III, we minimize the total transmit power of the system subject to an individual QoS constraint for each user. In Section IV, we further maximize the sum rate of the system under a predefined fair transmission policy that ensures constant ratios between the data rates of different messages. In Section V, we lastly investigate an EE optimization problem in the previous system with an upgrade that an extra user group having better channel condition is simultaneously served. Section VI provides simulation results. Concluding remarks are made in Section VII.
: Boldface uppercase letters and boldface lowercase letters are used to denote matrices and column vectors, respectively., , , and are the transpose, conjugate, Hermitian transpose, trace, and expectation operators. x indicates that x is a circular symmetric complex Gaussian random vector whose mean vector is and covariance matrix is . represents the Euclidean norm of a vector. and is the real part and the imaginary part of a complex number, respectively. represents the number of linearly independent rows or columns of a matrix. denotes the complexity.
Ii System and Signal Models
Ii-a System Model of NOMA Assisted Multi-Region Geocast
Consider a wireless transmission scenario which consists of a transmitter equipped with antennas and groups of users each equipped with a single antenna. Let denote the number of users in the -th group for . Note that the lowercase letter is used to represent the group index, and the lowercase letter is used to represent the user index in an arbitrary group, where and .
As shown in Fig. 1, each group is located in a predetermined region different from the other groups. Therefore, there is a one-to-one correspondence between groups and regions. It is worth to point out that these regions are distinct from each other by their average distances to the transmitter, which is explained as follows. We use to denote the minimum distance from the transmitter to the -th region and to denote the maximum distance. Without regard to the specific shape of the region, we suppose that the users are randomly deployed in the -th region. Then, we can define as the average distance between the transmitter and the -th region. Without loss of generality, assume that are sorted as , which indicates that the -th region is the nearest one to the transmitter and the 1st region is the farthest one.
The channel between the transmitter and each user is supposed to be a composition of large-scale path loss and quasi-static small-scale fading. Let represent the channel vector from the transmitter to the -th user in the -th group and be modeled as
where is the small-scale fading coefficient, is the distance from the transmitter to the -th user in the -th group, and is the path loss exponent. All users’ instantaneous CSI is supposed to be available at the transmitter.
Ii-B Signal Model of NOMA Assisted Multi-Region Geocast
We let denote the geocast information symbol intended for all the users in the -th group with , and represent the corresponding transmit beamformer, where . The transmitter broadcasts the summation of the weighted messages given by . The users in the same group (or region) are interested in a common message while different groups desire different messages. Thereby, the received signal at the -th user in the -th group can be expressed as
is the additive Gaussian noise with zero mean and variancefor all users. The second term at the right-hand side (R.H.S.) of (2) is the inter-group interference, namely the other groups’ target messages.
Unlike conventional multi-group multicast (i.e. SDMA) wherein the users decode their own desired messages by treating the messages for the other groups as noise , NOMA assisted multi-region geocast exploits the geographical information of users to aid the beamforming design at the transmitter. This is because, a remarkable difference in the average distances of the regions to the transmitter is likely to make the channel gains vary significantly among the users located in different regions, which is a natural advantage for the implementation of NOMA. Recall that in NOMA, users often apply SIC to eliminate partial co-channel interference. To realize SIC, the transmitter should establish a decoding order for all the users. In this work, the decoding order is predefined in accordance with the aforesaid order of the average distances of the regions to the transmitter. To be specific, the users in the -th group firstly decode the message for the -th group for , and then remove this message from their own received mixtures, in the order ; the messages for the -th group for will be treated as noise. It is worth to point out that the predefined decoding order herein may not be optimal, but it facilitates both the implementation of SIC and the beamforming design. This work focuses on the beamforming design under a predefined decoding order rather than optimizing the decoding order. Furthermore, no recent literature has reported the optimal decoding order in MISO settings due to its great difficulty.
A widely used power allocation policy in NOMA is to assign more power to users with worse channels[23, 29]. However, this policy is unnecessary and unfavorable for the overall performance of NOMA, which motivates us to annul the policy and propose to guarantee users’ QoS by minimum SINR thresholds. The specific processing is provided in the following.
With the signal model provided by (2) and the predefined decoding order of SIC, the decoding ability of any user can be characterized by its SINR. Let represent the SINR of the -th user in the -th group to decode its own desired message, i.e., , then can be given by
which holds if and only if all the users in the -th group are able to successfully decode the information for the -th group, . The guarantee of (3), namely the successful implementation of SIC given the predefined decoding order, is discussed in the sequel. We further denote as the SINR of the -th user in the -th group to decode the message for the -th group, where (the users in the -th group will treat the message for the -th group as noise for ), and then can be given by
According to (4), represent the ability of the -th user in the -th group to decode the messages intended for the groups located farther from the transmitter compared with itself. Let for represent the minimum required SINR for successfully decoding , then the successful implementation of SIC given the predefined decoding order is guaranteed if and only if the following constraints are satisfied:
We herein conclude that (4) is guaranteed by (5). In addition, it is worth to point out that the minimum required SINRs is actually the target SINR levels of messages with the corresponding rates for the groups of users. In other words, under our proposed NOMA framework, the transmission rates of all the data streams are predefined and fixed. Accordingly, based on the predefined data rates, the minimum SINRs of being able to successfully decode the data streams can be figured out and are eventually interpreted as these thresholds .
Iii Transmit Power Minimization Under Quality of Service Constraints
In this section, we focus on the problem of minimizing the total transmit power subject to an individual QoS constraint for each user. There are two categories of constraints in this optimization problem: one is the QoS constraints which guarantee the predefined target SINR levels for all users and the other is the SIC constraints which ensure the successful implementation of SIC given the predefined decoding order. Based on the signal model established in the previous section, this optimization problem can be formulated as
where denotes the minimum required transmit power that satisfies the QoS requirements222From the perspective of practical implementation, the QoS requirements should be bounded. Otherwise, the total transmit power may be too large to be feasible. Nevertheless, in this work, the optimization is supposed to be performed under a reasonable QoS requirements setting since its design is an interesting but challenging topic which is out of the scope of this work. for all users, (6b) represents the QoS constraints and (6c) represents the SIC constraints. For example, consider that there are 3 regions () and there are 3 users in each region (). Then there should be 9 QoS constraints and 9 SIC constraints. Particularly, there are several special cases of problem (6): when , problem (6) degrades into a single-group multicast beamforming design problem, in which the application of NOMA is not needed; when for , problem (6) degrades into a common broadcast beamforming design problem in a MISO NOMA system.
Unlike those transmit beamforming optimization problems considered in conventional multi-group multicast , the extra constraints given in (6c), i.e., the SIC constraints, are due to the application of NOMA. To further explore problem (6), we rewrite it by substituting and with the R.H.S. of (3) and (4), respectively. Thus, problem (6) is rewritten as
It can be easily verified that problem (7) is non-convex due to the quadratic terms at the left-hand sides (L.H.S.) of the inequality constraints in (7b) and (7c). In fact, problem (7) is NP-hard according to . Therefore, it is difficult to solve this problem directly, which leads us to use convex approximation techniques to find an approximate solution for the original problem. In the following, an efficient algorithm is proposed to solve problem (7) based on the majorization and minimization (MM)333The MM is termed as minorization-maximization when the original problem is a maximization problem. algorithms[30, 31].
Iii-a MM-based Method
In this subsection, we develop a MM-based method to solve problem (7). The basic idea of the MM method is to substitute a simple optimization problem for a difficult one and iteration is the price for simplifying the original problem . Specifically, in each iteration, every non-convex constraint is substituted with its inner convex approximation which serves as its tangent plane at a certain point. A local optimal solution is ensured by the MM method[31, 32]. Thus, we are to analyze the non-convexity of problem (7) and find the approximation functions.
According to the formulation of problem (7), its main difficulty lies in the quadratic terms on the L.H.S. of the inequalities constraints in (7b) and (7c). Noting that those quadratic terms are the squared norms of complex numbers, we decouple the complex numbers into their real parts and imaginary parts in order to unveil the hidden concavity or linearity. Consequently, an equivalent transformation of the quadratic terms on the L.H.S. of (7b) is given as below:
We further create a two-dimensional real-valued column vector given as
Therefore, is equivalently transformed to the squared norm of the created real-valued vector, which is shown as
where is the first-order Taylor approximation of around , the setting of which will be given in the following. minorizes at point , which means that lies under the surface of and is tangent to it at the point . Thus, the linear expression can be used to approximate the original quadratic term . In a similar way, the quadratic terms in the L.H.S. of (7c) can be also approximated as below:
With the above approximations, the constraints in (7b) and (7c) can be approximated by more stringent but convex constraints, which makes the original problem (7) become an iterative convex program. Specifically, the -th iteration is to solve the following convex optimization problem:
Here, the settings of and are given by
where denotes the optimal solution to problem (13). Accordingly, the MM-based method for solving problem (7) is outlined in Algorithm 1. Moreover, the approach used to generate the initial feasible solution and the convergence analysis of our proposed iterative algorithm will be discussed as follows, respectively.
Generation of Initial Feasible Solution: We now propose an efficient method to seek out an initial feasible solution for our proposed MM-based method. In fact, problem (7) can be approximated by a second-order cone program (SOCP), which is presented as follows444Another approach of generating an initial feasible solution for the general conic quadratic problems are also provided in .. We firstly rewrite the non-convex constraints in (7b) as
Further, the conservative approximation proposed in  is applied as below:
which can be further rewritten under the second-order convex (SOC) representations given as
In a similar way, the inner convex approximations of the constraints in (7c) can be given by
As a result, problem (7) is approximated by the following standard SOCP:
Let us denote the optimal solution to problem (20) as and set it as the initial feasible solution for Algorithm 1, then our proposed MM-based method can solve problem (7) in the vicinity of this initial solution by finding a better solution. In particular, problem (20) can be easily solved with general interior-point methods or the advanced SOCP solvers, e.g., MOSEK , ECOS .
Convergence Analysis: Following the general convergence principle of the MM method in [30, 32], we can prove the convergence of algorithm 1. According to [32, Theorem 1] and , it can be verified that the first-order Taylor approximations in (11) and (12) satisfy the constraints for assuring the convergence of the MM method. As a result, it is assured that the solution obtained by Algorithm 1 converges to a Karush-Kuhn-Tucker solution of the original problem (7). Besides, our simulation results in Section VI further show that Algorithm 1 converges fast in a finite number of iterations.
Complexity Analysis: In each iteration of Algorithm 1, the SOCP problem (13) can be solved by interior-point methods, of which the complexity depends on the number of constraints . As mentioned before, there are QoS constraints and SIC constraints in problem (13). By setting , the worst-case complexity for the SOCP problem (13) is . This complexity scales linearly with the number of iterations, which is shown to be a small number by the simulation results provided in Section VI.
Iii-B Lower Bound of : Classical SDR Method
In this subsection, we use the classical SDR method to obtain the lower bound of the minimum required transmit power . Since problem (7) is a standard quadratically constrained quadratic programming (QCQP) problem, its suitable reformulation can realize the application of SDR. By introducing for , problem (7) can be equivalently transformed as
By dropping the rank-one constraints in (21e), which are the only existing non-convex constraints, problem (21) is thereby relaxed to a standard semi-definite programming (SDP) problem that can be solved by modern SDP solvers, such as SeDuMi. However, due to the relaxation, the obtained solution to the SDP problem may not be rank one. 555The drawback of this relaxation method is that the solution of the SDP problem is not guaranteed to be feasible for its original problem. In some cases, the Gaussian randomization method (see, e.g.,  and references therein) can generate a feasible approximate solution from the solution of the SDP problem. Thus, the optimal solution to the relaxed problem can serve as a lower bound of . In Section VI where the numerical results are provided, we use this SDR method as a benchmark for our proposed MM-based method.
Iv System Sum Rate Maximization Under Fair Transmission Policy
For many communication scenarios, the system sum rate is a more important criterion to evaluate the performance of spectrum sharing, compared with the transmit power. Therefore, in this section, based on the previous transmit power minimization problem, we further consider a related problem which is to maximize the achievable sum rate of the system subject to a total transmit power constraint. This investigation aims to explore the NOMA assisted multi-region geocast framework from the perspective of system throughput. In particular, a fair transmission policy is proposed and adopted in this scenario, which is elaborated in the following:
We denote the target transmit data rate of the message for the -th group as and meanwhile define as the prescribed grades of service for the groups (greater is, higher transmit data rate will be). Then, our proposed fair transmission policy requires that all ratios between the data rates of groups should be the ratios between their grades of service, which are mathematically characterized by the following equations:
Herein, (, , , ) could be interpreted as a set of baseline data rates. Under the proposed fair transmission policy characterized by (22), we can further define as a scale coefficient of the grades of service, which means that (bits/s/Hz) is set to be the new target transmit data rate of the message for the -th group so that (22) is always satisfied. Accordingly, each target data rate will be multiplied by a factor and the newly desired set of rates becomes (, , , ) where could be either greater or less equal than 1. As a result, the minimum required SINR for successful decoding the message for -th group, denoted by , can be expressed as
Let represent the total transmit power available at the transmitter, then the problem of maximizing the achievable sum rate of the system, subject to the total transmit power constraint and under the predefined fair transmission policy, can be formulated as
The analysis and the solution of the above problem are discussed in the following.
Iv-a Bisection Method Combined with MM-based Method
Comparing problem (24) with problem (6) studied in Section III, we can see that the QoS and SIC constraints of both the problems preserve the same structure except that the R.H.S. of the constraints in (24b) and (24c) are not constants but the functions of argument . Accordingly, a straightforward strategy to solve (24) is proposed as below: by fixing to a constant , we firstly solve problem (6) with by Algorithm 1. Then by comparing to , we judge whether the fixed scale coefficient is feasible under the total transmit power constraint in (24d). If it is, we further increase the value of to make the most use of the available power at the transmitter. If not, we should decrease the value of since cannot satisfy the current target transmit data rates of all user groups.
Based on the proposed strategy, the bisection method can be exploited to solve problem (24). To be more specific, the combination of the one-dimensional search on and the MM-based method can solve problem (24) approximately.
Due to the application of the bisection method, it is necessary to determine an appropriate searching range for . Let and represent the lower bound and the upper bound of the searching range for , respectively. Apparently, should be initialized as 0. As for , assuming that the total transmit power is directed towards an arbitrary single user without regarding to the QoS guarantees of the other users, we can obviously find inaccessible upper bounds for , which are given by
In order to minimize the number of iterations of the bisection method, we choose the smallest one among (25) to be , which is given by
As a result, the algorithm which combines the bisection method and the MM-based method proposed in Section III, for solving problem (24) is outlined in Algorithm 2 where is the tolerance of the optimal scale coefficient .
V Energy Efficiency Optimization of the Upgraded System With an Extra User Group
As the 5G system must be more compatible and flexible, our proposed NOMA framework is supposed to provide multiple options for the wireless access under different criteria, allowing the users with better channel condition to access the channel dynamically is one of the most important features. Accordingly, in this subsection, we consider a scenario wherein an additional user group with delay-sensitive data locating in a region most closely to the transmitter intends to access the system, when there are already multiple user groups being served. This scenario may widely appear in 5G applications with delay sensitive requirements, such as vehicle-to-vehicle networking and Internet of Things applications. In this applications, there is no rigorous requirement on data rate but the energy efficiency of the system is a great concern due to the severe constraint on power supplement.
Based on the above discussions, in this section, an EE maximization problem is investigated in our proposed framework with an upgrade that an extra user group having better channel condition is simultaneously served. To be specific, the transmitter serves the original regions (groups) and the newly joined region simultaneously, subject to the constraint that the QoS is always guaranteed for the users in the original regions. The objective is to maximize the EE of the system. In particular, the EE is evaluated by the commonly adopted metric “bits per Joule per Hertz (bits/Joule/Hz)” [26, 27, 28], which is defined as the ratio of the system sum rate to the total power expenditure.
V-a Problem Formulation
Based on the above discussion, the sum of the achievable data rate of the original user groups is a constant given by
With regard to the newly joined user group, its achievable data rate is restricted by the minimum SINR among its users, since a common message is transmitted to the users in the same group. We use as the index of the newly joined user group. Furthermore, since the region locates most closely to the transmitter, it is supposed to perform the SIC to eliminate all the inter-group interference, the achievable data rate of the can be expressed as
As a result, the EE of this upgraded multi-region geocast system can be given by
where is the total transmit power and represents the fixed power expenditure of the system. Consequently, the EE optimization problem is formulated as
The constraints in (30b) and (30c) guarantee the minimum required QoS of the users in the original user groups and its implementation of SIC, respectively, the constraints in (30c) ensure that the users in the newly joined group can successful perform SIC to eliminate all the inter-group interference, and the constraint in (30e) is the total transmit power constraint. Different from the previous optimization problems (6) and (24), the difficulty of problem (30) lies in its complicated objective function that is fractional. Most of the current solutions to maximize a fractional expression are turning to the Dinkelbach’s method, the computational complexity of which could be very high. Hereinafter, an efficient beamforming design algorithm is proposed by using the MM method.
V-B MM-based Method for EE Maximization
Note that the approximation methods of the constraints in (30b), (30c), and (30d) have already been studied in Subsection III-A by linearizing quadratic terms, thus the difficulty of resolving the problem only rests with its fractional objective in (30a). We decompose (30a) by introducing some auxiliary variables to further expose its hidden concavity. Then, problem (30) is rewritten as
where and are the auxiliary variables which could be explained as the squared power consumption and the squared EE, respectively. The equivalence between (30) and (32) is ensured by that the constraints in (32b) and (32c) should hold with equality at optimum, and that maximizing is equivalent to maximizing . Note that (32c) is convex while (32b) is non-convex due to the bilinear product term . We equivalently transform (32b) as below:
where is the auxiliary variable which can be interpreted as an attainable SINR level for the users in the group to decode their target message, and is the auxiliary variable which is interpreted as the data rate of , namely the achievable rate . Then, problem (32) has been equivalently transformed into