Resource Allocation in Heterogenous Fullduplex OFDMA Networks: Design and Analysis
Abstract
Recent studies indicate the feasibility of fullduplex (FD) bidirectional wireless communications. Due to its potential to increase the capacity, analyzing the performance of a cellular network that contains fullduplex devices is crucial. In this paper, we consider maximizing the weighted sumrate of downlink and uplink of an FD heterogeneous OFDMA network where each cell consists of an imperfect FD basestation (BS) and a mixture of halfduplex and imperfect fullduplex mobile users. To this end, first, the joint problem of subchannel assignment and power allocation for a single cell network is investigated. Then, the proposed algorithms are extended for solving the optimization problem for an FD heterogeneous network in which intracell and intercell interferences are taken into account. Simulation results demonstrate that in a single cell network, when all the users and the BSs are perfect FD nodes, the network throughput could be doubled. Otherwise, the performance improvement is limited by the intercell interference, internode interference, and selfinterference. We also investigate the effect of the percentage of FD users on the network performance in both indoor and outdoor scenarios, and analyze the effect of the selfinterference cancellation capability of the FD nodes on the network performance.
Index Terms: Fullduplex, selfinterference, resource allocation, OFDMA, femto cell, heterogeneous.
I Introduction
^{†}^{†}This paper has been presented in part at the IEEE International Conference on Communication (ICC), Kuala Lumpur, Malaysia, May 2016.In wireless communications, separation of transmission and reception in time or frequency has been the standard practice so far. However, through simultaneous transmission and reception in the same frequency band, wireless fullduplex has the potential to double the spectral efficiency. Due to this substantial gain, fullduplex technology has recently attracted noticeable interest in both academic and industrial worlds. The main challenge in fullduplex (FD) bidirectional communication is selfinterference (SI) cancellation. In recent years, many attempts have been made to cancel the selfinterference signal [1, 2, 3, 4]. In [5], it is shown that dB SI cancellation is achievable, and by jointly exploiting analog and digital techniques, SI may be reduced to the noise floor.
A fullduplex physical layer in cellular communications calls for a redesign of higher layers of the protocol stack, including scheduling and resource allocation algorithms. In [6], the performance of an FDbased cellular system is investigated and an analytic model to derive the average uplink and downlink channel rate is provided. A resource allocation problem for an FD heterogeneous orthogonal frequencydivision multiple access (OFDMA) network is considered in [7], in which the macro base station (BS) and small cell access points operate in either FD or halfduplex (HD) MIMO mode, and all mobile nodes operate in HD single antenna mode. In [8], using matching theory, a subchannel allocation algorithm for an FD OFDMA network is proposed. In both [7] and [8] only a single subchannel is assigned to each of the uplink users in which they transmit with constant power. Resource allocation solutions are proposed in [9] and [10] for FD OFDMA networks with perfect FD nodes (SI is canceled perfectly).
Recent research reports investigate resource allocation in multicell FD networks. In [11], a suboptimal resource management algorithm is presented for the sum rate maximization of a small multicell system, including FD base stations and HD mobile users. In [12], the problem of maximizing a networkwide ratebased utility function subject to uplink (UL) and downlink (DL) power constraints is studied in a flexible duplex system, in which UL/DL channels are allowed to have partial overlap via finetuned bandwidth allocation. For simplicity, it is assumed that the number of subchannels and the users are exactly the same. In [13], the problem of decoupled ULDL user association, which allows users to associate with different BSs for UL and DL transmissions, is investigated in a multitier FD Network. In [14], weighted sum rate maximization in a FD multiuser multicell MIMO network is studied. A user scheduling and power allocation method for ultradense FD smallcell networks is presented in [15]. In [13] [14] and [15], the subchannel allocation problem is not investigated since a single channel network is assumed. The most related work to the current research is [16], in which, a radio resource management solution for an OFDMA FD heterogeneous cellular network is presented. The algorithm jointly assigns the transmission mode, and the user(s) and their transmit power levels for each frequency resource block to optimize the sum of the downlink and uplink rates. The users are assumed to use a single class of service. A suboptimal resource allocation algorithm is then proposed which takes into account both intracell and intercell interferences. The suboptimal power adjustment algorithm is designed under the assumption of high SINR, where the rate of an FDFD or FDHD link is independent of power variations.
In this paper, we consider a general resource allocation problem in a heterogeneous OFDMAbased network consisting of imperfect FD macro BS and femto BSs and both HD and imperfect FD users. We aim to maximize the downlink and uplink weighted sumrate of femto users while protecting the macro users rates. The weights allow for users to utilize differentiated classes of service, accommodate both frequency or time division duplex for HD users, and prioritize uplink or downlink transmissions. To be more realistic, imperfect SI cancellation in FD devices is assumed and FD nodes suffer from their SI. A contribution of the current work is to consider the presence of a mixture of FD and HD users, which enables us to quantify the percentage of FD users needed to capture the full potential of FD technology in wireless OFDMA networks. We also analyze the effect of the SI cancellation level on the network performance, which to our knowledge has not been studied in prior works. We will show that when the SI cancellation capability is worse than a specified threshold, then the throughput of an all FD user network would not be larger than the throughput of an all HD user network. Moreover, we will analyze this threshold theoretically and compare its outcome with simulation results.
The remainder of this paper is organized as follows. In Section II, the basic system model of a single cell FD network is given and the optimization problem is formulated. In Section III, a subchannel allocation algorithm for selecting the best pair in each subchannel is presented. Power allocation is considered in Section IV. A theoretical approach for deriving the SI cancellation coefficient threshold is proposed in Section V. In Section VI, the optimization problem for an FD heterogeneous network is presented. Numerical results for the proposed methods are shown in Section VII. Finally, the paper is concluded in Section VIII.
Ii System Model And Problem Statement
We consider a single cell network that consists of a fullduplex basestation (BS) and a total of halfduplex and fullduplex users. For communications between the nodes and the BS, we assume that an OFDMA system with subchannels is used. All subcarriers are assumed to be perfectly synchronized, and so there is no interference between different subchannels. Since the basestation operates in fullduplex mode, it can transmit and receive simultaneously in each subchannel. In each timeslot the basestation is to properly allocate the subchannels to the downlink or uplink of appropriate users and also determine the associated transmission power in an optimized manner. We assume that the basestation and the FD users are imperfect fullduplex nodes that suffer from selfinterference. We define a selfinterference cancellation coefficient to take this into account in our model and denote it by , where indicates that SI is canceled perfectly and means no SI cancellation. For simplicity, we assume the same selfinterference cancellation coefficient for BS and FD users, but consideration of different coefficients would be possible. In this paper, the goal is to maximize the weighted sumrate of downlink and uplink users with a total power constraint at the basestation and a transmission power constraint for each user.
We define the downlink weighted sumrate as
(1) 
And the uplink weighted sumrate as
(2) 
The variables used in the above equations are introduced in Table I. We assume here that the channel is reciprocal, i.e., uplink and downlink channel gains are the same. We further assume that the receiver noise powers in different subchannels are the same. The term in (1) denotes the interference: When user is a FD device and both downlink and uplink of subchannel are allocated to it , , else is the channel gain between uplink user and downlink user . We assume that the basestation knows all the channel gains, the noise powers, and the SI cancellation coefficient and weights assigned to the downlink and uplink of all users.
Let and denote the maximum available transmit power for the basestation and for user , respectively. Then the proposed design optimization problem, denoted by P1, can be formulated as follows
weight assigned to the downlink of user  

weight assigned to the uplink of user  
transmission power from BS to user on subchannel  
transmission power from user to BS on subchannel  
Gaussian noise variance at the receiver of user  
Gaussian noise variance at the basestation receiver  
set of subchannels allocated to user for downlink  
set of subchannels allocated to user for uplink  
selfinterference cancellation coefficient  
channel gain between BS and user on subchannel  
channel gain between users and on subchannel  
equal to when , and to otherwise  
maximum available transmit power at BS  
maximum available transmit power at user 
(3) 
(4) 
(5) 
(6) 
(7) 
(8) 
(9) 
where (4) and (5) indicate the power constraint on the BS and the users, respectively. Constraint (6) shows the nonnegativity feature of powers; (7) come from the fact that a subchannel cannot be allocated to two distinct users simultaneously; (8) indicate that we have no more than subchannels, and the last constraint accounts for the halfduplex nature of the HD users.
The general resource allocation problem presented is combinatorial in nature because of the channel allocation issue and addressing it together with power allocation in an optimal manner is challenging, especially as the number of users and subchannels grow. Moreover, the nonconvexity of the rate function makes the power allocation problem itself challenging even for a fixed subchannel assignment. Here, we invoke a two step approximate solution. First, we determine the allocation of downlink and uplink subchannels to users and then determine the transmit power of the users and the basestation on their allocated subchannels. In other words, we first specify the sets and and then determine the variables , . In the next Section, we introduce our subchannel allocation algorithm.
Iii Subchannel Allocation
The subchannel allocation problem, denoted by P2, can be formulated as follows
subject to  (7)(11) 
To solve the problem P2, we should first solve the following power allocation problem, denoted by P3, to maximize the weighted sumrate in a single subchannel and for a fixed pair of uplink and downlink users. Since a single subchannel is being considered in P3, we have dropped the variable in the notation.
(10)  
(11) 
Here, and are the maximum allowable transmit powers.
Proposition 1.
For a fixed downlink user and uplink user , the optimal pair of powers that optimizes P3 belongs to the following set.
where
(12) 
and
(13)  
(14)  
(15)  
(16) 
Proof.
Computing the derivative with respect to and setting it to zero we have:
where , and are defined above. It is evident that , and if then . When the above quadratic equation either has no zeros in or has only one zero where the function changes sign from to indicating a local minimum for . Therefore, in both cases the maximum is attained at a boundary point or . But when , could be negative, and the smaller root of the quadratic equation could be positive. In this case, the maximum is attained at or . By similar analysis for one sees that if then the maximum is attained at a boundary point or and when the maximum is attained at or . As a result, when the optimal transmission powers belong to the following set,
Otherwise, if , they belong to the set below
The cases and cannot be the optimal solutions of P3 , because they are dominated by and which give a larger . Therefore, optimal powers could be found by checking the members of the set S and picking the one that corresponds to the largest . ∎
Based on the above Proposition one can find the best uplinkdownlink pair in each subchannel by choosing the one with the largest value of . This involves only operations. Now we can present our subchannel allocation algorithm to solve Problem P2, in which we employ a suboptimum power allocation scheme. First, for each subchannel , we find the best channel gain among all users and denote it by . Then, we sort the subchannels based on the value of . In other words. we find a subchannel permutation such that . Then, starting from subchannel , we seek and that maximize . At the first iteration, we set , and for iteration set and where and indicate the number of subchannels to be allocated to the BS’s downlink transmission and to user ’s uplink transmission, respectively, in the th iteration. The proposed subchannel allocation algorithm is summarized below.
Algorithm 1: Subchannel Allocation Algorithm 

1.for to do 
2. 
3.end for 
4.Find a subchannel permutation , , such that 
5. set for and 
6.for to do 
7. Set and 
8. for to do 
9. for to (if is an HD user ) 
10. In subchannel solve the problem P3 
11. end for 
12. end for 
13. Using the obtained optimal powers, find the best pair in the 
subchannel that has the largest value of 
14. , 
15. if then ; 
16. if then ; 
17.end for 
The complexity of finding the best user in each subchannel is and for subchannels is . Similarly, the complexity of finding the best pair in each subchannel is and doing so for subchannels requires operations. Since the complexity of sorting values is , then the overall computational complexity of the proposed subchannel allocation algorithm is .
Iv Power Allocation
The power allocation problem, denoted by P4, can be formulated as follows
subject to  (4)(6) 
Due to the interference terms, the power allocation problem is nonconvex. Here, we use the “difference of two concave functions/sets” (DC) programming technique [17] to convexify this problem. In this procedure, the nonconcave objective function is expressed as the difference of two concave functions, and the discounted term is approximated by its first order Taylor series. Hence, the objective becomes concave and can be maximized by known convex optimization methods. This procedure runs iteratively, and after each iteration the optimal solution serves as an initial point for the next iteration until the improvement diminishes in iterations. In [18], the DC approach is used to formulate optimized power allocation in a multiuser interference channel, and in [19], the DC optimization method is used to optimize the energy efficiency of an OFDMA device to device network. Here, we rewrite the objective function of P4 in DC form as follows
where
is the downlink and uplink transmitted power vector, and and denote the uplink and downlink users that are selected for the th subchannel after the subchannel allocation phase. Now, the objective is a DC function. To write the Taylor series of the discounted function ), we need its gradient, that can be easily derived as follows.
To make the problem convex, is approximated with its first order approximation at point . We start from a feasible at the first iteration, and at the th iteration is generated as the optimal solution of the following convex program
Since is a concave function, its gradient is also its super gradient so we have
and we can deduce
Then it can be proved that in each iteration the solution of problem P4 is improved as follows
According to the above equations, the objective value after each iteration is either unchanged or improved and since the constraint set is compact it can be concluded that the above DC approach converges to a local maximum.
V Analyzing SelfInterference Cancellation Coefficient Threshold
In [20], through simulations it has been observed that in a network that contains an imperfect FD BS and some imperfect FD and HD users, when the selfinterference cancellation coefficient is larger than a specified threshold, there is no difference between the throughput of an all HD user network and an all FD user network. Here we wish to analyze this threshold.
Recall that in FD networks there are four possible types of connections in a given subchannel:

HD downlink

HD uplink

Joint downlink and uplink for two distinct users over an FD BS

A fullduplex bidirectional connection between an FD user and an FD BS
Employing an FD user in a cell with an FD capable BS can increase the throughput when the 4th case is more appealing than the other cases in at least one subchannel. Considering subchannel , we assume that user is the best downlink user, user is the best uplink user, users and are the best downlink and uplink pair, and user is the best FD node for FD communication with the BS. Here the best user, is the user who gives the highest weighted rate with the same power than the rest. The rate of the four previous cases are presented below (we drop the subchannel index for simplicity)
(17) 
(18) 
(19) 
(20) 
where, and are the transmission powers form BS to user and from user to the BS, respectively. The other variables were introduced in Table I. In short, using an FD user in the network could be beneficial when these conditions hold in at least one subchannel:
(21) 
(22) 
(23) 
Since we wish to focus on parameter , and to avoid dealing with other parameters, we introduce some simplifications. First, we assume the sum rate case where, . Second, we assume that the noise powers at the BS and at the users are the same. Third, we assume that the transmission power from the BS to all users is the same and is equal to the average BS power, i.e., . Fourth, we assume that the transmission powers from different users to the BS are the same and are equal to the average user power, i.e., . Fifth, the channel gains , , , , , are random variables and in the sum rate case the best downlink user, the best uplink user and the best FD user are all the same , because the channel is reciprocal and the user with maximum channel gain is selected for all of these three cases. If we assume that the number of users in the network is , then random variable can be defined as: , where is the random channel gain between the BS and the user , that itself is a multiplication of an exponential random variable, , with unit power and a path loss random variable that depends on the path loss model and the distance between the BS and the th user whose pdf is shown by (for ). We assume is a random channel gain between two users and which are distributed uniformly in a circle with radius .We also assume that and are the maximum and the second maximum channel gain between users.
Due to the randomness of the channel gains, itself is a random variable and here we wish to derive its distribution. According to conditions (21)  (23), we have:

The FD rate should be bigger than the HD downlink rate, so we have:
After some manipulations, this is reduced to the inequality , where:
(24) (25) (26) which holds for . Therefore, due to condition (21) .

By writing the condition (23) and doing the same procedure as the previous parts we arrive at the inequality , where:
(30) (31) (32) (33) The above cubic function has the following three roots:
(34) (35) (36) where
(37) (38) (39) (40) (41) It is evident that . Also, it can be shown that the value of is always negative, therefore, it is deduced that the cubic function has at least one positive real root. Therefore, due to the condition (23), is the smallest positive real root of this cubic function, where is given by:
Finally, we arrive at the following proposition:
Proposition 2.
For a wireless cell with an FD BS and users with imperfect SI cancellation factor , FD operation is advantageous from the perspective of the network throughput performance if .
In Section VII, we will compare the outcome of this analysis with simulation results.
Vi TwoTier Heterogeneous Full Duplex Network
In this section, we consider a twotier heterogeneous fullduplex OFDMA network. This system includes a macrocell FD BS and multiple femto cell FD BSs along with their associated HD and FD users. Our goal is to maximize the uplink and downlik weighted sum rate of femto cell users while provisioning for the macrocell user’s uplink and downlink data rate. Assume that the numbers of femto cells and available subchannels are and , respectively, and the number of users related to the th BS is . We denote the set of all BSs as , where the macro BS is indexed by 0. The variables used in the following equations are summarized in Table II.
The downlink rate in cell is given by:
(42) 
where and are the downlink and uplink intercell interference on subchannel in the th cell for user , i.e.:
(43) 
(44) 
Similarly the uplink rate in cell is given by:
(45) 
where and are the downlink and uplink intercell interference on subchannel at the th BS, i.e.:
(46) 
(47) 
weight assigned to the downlink of user in the th cell  

weight assigned to the uplink of user in the th cell  
downlink transmission power from the BS to user on subchannel in the th cell  
uplink transmission power from user to the BS on subchannel in the th cell  
downlink transmission power from the th BS on subchannel  
channel gain between user in the th cell and the th BS on subchannel  
channel gain between user in the th cell and user in the th cell on subchannel  
channel gain between the BS and user on subchannel in the th cell  
channel gain between the th BS and the 