I Introduction
Fullduplexing (FD) [1, 2, 3, 4, 5] is a technique for simultaneous transmission and reception in the same time slot and over the same frequency band while twoway relaying (TWR) [6, 7, 8, 9] allows pairs of users to exchange their information in one step. FD deployed at both users and relays thus enables the users to exchange information via relays within a single timeslot [10]. This is in contrast to the conventional oneway relaying which needs four time slots, and the halfduplexing (HD) TWR [11, 12, 8, 13], which needs two time slots for the same task. Thus, FD TWR seems to be a very attractive tool for devicetodevice (D2D) and machinetomachine (M2M) communications [14, 15] and low latency communication [16, 17, 18] for Internet of Things (IoT) applications.
The major issue in FD is the loop selfinterference (SI) due to the colocation of transmit antennas and receive antennas. Despite considerable progress [3, 4, 5], it is still challenging to attenuate the FD SI to a level such that FD can use techniques of signal processing to outperform the conventional halfduplexing in terms of spectral and energy efficiencies [19, 20]. Similarly, it is not easy to manage TWR multichannel interference, which becomes double as compared to oneway relaying [21, 22]. The FDbased TWR suffers even more severe interference than the FD oneway relaying, which may reduce any throughput gain achieved by using fewer time slots [10].
There is another approach to implement halfduplexing (HD) TWR within a single time slot, which avoids FD at both users and relays. In a fraction of a time slot, the HD users send the information intended for their partners to the relays and then the relays send the beamformed signals to the users within the remaining fraction of the time slot. In contrast to FD relays, which use half of their available antennas for simultaneous transmission and reception, the HD relays now can use all their antennas for separate transmissions and receptions. Thus, compared with FD users, which need two antennas for simultaneous transmission and reception, the HD users now need only one antenna for separate transmission and reception.
In this paper, we consider the problem of joint design of users’ power allocation and relays’ beamformers to either maximize the user exchange information throughput or the network energy efficiency [23] subject to user qualityofservice (QoS) constraints in terms of minimal rate thresholds. As they constitute optimization of nonconvex objective functions subject to nonconvex constraints under both these approaches, finding a feasible point is already challenging computationally. Nevertheless, we develop efficient pathfollowing algorithms for their computation, which not only converge rapidly but also invoke a lowcomplexity convex quadratic optimization problem at each iteration for generating a new and better feasible point. The numerical examples demonstrate the full advantage of the second approach over the first approach.
The rest of this paper is organized as follows. Section II considers the two aforementioned nonconvex problems under a FDbased TWR setting. Section III considers them under the timefraction (TF)wise HD TWR setting. Section IV verifies the full advantage of the TFwise HD TWR over FDbased TWR via numerical examples. Section V concludes the paper. The appendix provides some fundamental inequalities, which play a crucial role in the development of the pathfollowing algorithms in the previous sections.
Notation.
Boldfaced characters denote matrices and column vectors, with upper case used for the former and lower case for the latter.
represents the th row of the matrix while is its th entry. is the trace of the matrix . The inner product between vectors and is defined as . is referred either to the Euclidean vector squared norm or the Frobenius matrix squared norm. Accordingly, for any complex . Lastly, meansis a vector of Gaussian random variables with mean
and covariance .Ii Fullduplexing based twoway relaying
Fig. 1 illustrates a FD TWR network consisting of pairs of FD users (UEs) and FD relays indexed by . Each FD user (UE) uses one transmit antenna and one receive antenna, while each FD relay uses receive antennas and transmit antennas. Without loss of generality, the th UE (UE ) and th UE (UE ) are assumed to exchange information with each other via the relays. The pairing operator is thus defined as for and if . For each , define the set of UEs, which are in the same side with th UE as
Under simultaneous transmission and reception, FD UEs in interfere each other. Such kind of interference is called interFDuser interference.
Let be the vectors of information symbols transmitted from UEs, which are independent and have unit energy, i.e. . For as the vector of channels from UE to relay , the received signal at relay is
(1) 
where is the background noise, and is a vector of UE power allocation, while models the effect of analog circuit nonideality and the limited dynamic range of the analogtodigital converter (ADC) at FD relay .
The transmit power at UEs is physically limited by as
(2) 
The total transmit power of UEs is bounded by to prevent their excessive interference to other networks as
(3) 
Relay processes the received signal by applying the beamforming matrix for transmission:
(4) 
For simplicity it is assumed that with the relay channel’s instantaneous residual SI attenuation level .^{1}^{1}1It is more practical to assume so resulting in . Usually can be assumed so for This gives
in calculating the transmit power at relay by a closedform as
(5)  
This transmit power at relay must be physically limited by a physical parameter as
(6) 
and their sum is also bounded by to control the network emission to other networks:
(7) 
The relays transmit the processed signals to all UEs. For the vector channel from relay to UE and channel from UE to UE , the received signal at UE is given by
(8)  
where is the background noise, and as represents the loop interference at UE . We can rewrite (8) as
(9)  
Note that the first term in (9) is the desired signal component, the third term is the interpair interference and the last two terms are noise. UE can cancel the selfinterference by the second term using the channel state information of the forward channels from itself to the relays and backward channels from the relays to itself as well as the beamforming matrix . The challenges here is that the loop SI term , which may be strong due to the proximity of UEs in , cannot be nulled out. This means more power should be given to the relays but it leads to more FD SI at the relays.
Furthermore, for , the signaltointerferenceplusnoise ratio () at UE ’ receiver can be calculated as
(10)  
Under the definitions
(11) 
it follows that
(12)  
In FD TWR, the performance of interest is the exchange information throughput of UE pairs:
(13) 
The problem of maximin exchange information throughput optimization subject to transmit power constraints is then formulated as
(14a)  
s.t.  (14b) 
Another problem, which attracted recent attention in 5G [23, 24] is the following problem of maximizing the network energyefficiency (EE) subject to UE QoS in terms of the exchange information throughput thresholds:
(15a)  
s.t.  (15c)  
where , and are the reciprocal of drain efficiency of power amplifier, the circuit powers of the relay and UE, respectively, and sets the exchange throughput threshold for UE pairs.
The next two subsections are devoted to computational solution for problems (14) and (15), respectively.
Iia FD TWR maximin exchange information throughput optimization
By introducing new nonnegative variables
(16) 
and functions
(17) 
which are convex [25], (12) can be reexpressed by
(18)  
Similarly to [26] and [13, Th. 1] we can prove the following result.
Theorem 1
The optimization problem (14), which is maximization of nonconcave objective function over a nonconvex set, can be equivalently rewritten as the following problem of maximizing a nonconcave objective function over a set of convex constraints:
(19a)  
(19b)  
(19c)  
(19d)  
(19e)  
(19f) 
The main issue now is to handle the nonconcave objective function in (19a) of (19), which is resolved by the following theorem.
Theorem 2
Proof: (2) follows by applying inequality (59) in the Appendix for
and
and then the inequality
(23) 
over the trust region (21).
By Algorithm 1 we propose a pathfollowing procedure for computing (19), which solves the following convex optimization problem of inner approximation at the th iteration to generate the next feasible point :
(24) 
Similarly to [13, Alg. 1], it can be shown that the sequence generated by Algorithm 1 at least converges to a local optimal solution of (19).^{2}^{2}2As mentioned in [27, Remark] this desired property of a limit point indeed does not require the differentiability of the objective function
IiB FD TWR energyefficiency maximization
We return to consider the optimization problem (15), which can be shown similarly to Theorem 1 to be equivalent to the following optimization problem under the variable change (16):
(25a)  
(25b) 
for
and
(26)  
The objective function in (25a) is nonconcave and constraint (25b) is nonconvex.
Suppose that is a feasible point for (25) found from the th iteration. Applying inequality (Appendix) in the Appendix for
and
and using inequality (23) yield the following bound for the terms of the objective function in (25a):
(27) 
over the trust region (21), where
and
(28) 
Furthermore, we use defined from (20) to provide the following inner convex approximation for the nonconvex constraint (25b):
(29) 
By Algorithm 2 we propose a pathfollowing procedure for computing (25), which solves the following convex optimization problem at the th iteration to generate the next feasible point :
(30a)  
(30b) 
Analogously to Algorithm 1, the sequence generated by Algorithm 2 at least converges to a local optimal solution of (25).
Iii Timefractionwise HD twoway relaying
Through the FDbased TWR detailed in the previous section one can see the following obvious issues for its practical implementations:

It is difficult to attenuate FD SI at the UEs and relays to a level in realizing the benefits by FD. The FD SI is even more severe at the relays, which are equipped with multiple antennas;

InterFDuser interference cannot be controlled;

It is technically difficult to implement FD at UEs, which particularly requires two antennas per UE.
We now propose a new way for UE information exchange via HD TWR within the time slot as illustrated by Fig. 2, where at timefraction all UEs send information to the relays and at the remaining time fraction the relays send the beamformed signals to UEs. This alternative has the following advantages:

Each relay uses all available antennas for separated receiving and transmitting signals;

UEs need only a single antenna to implement the conventional HD, which transmits signal and receive signals in separated time fractions.
Suppose that UE uses the power to send information to the delay. The following physical limitation is imposed:
(32) 
where is a physical parameter to signify the hardware limit in transmission during timefractions. Typically, for defined from (2).
As in (3), the power budget of all UEs is :
(33) 
The received signal at relay can be simply written as
(34) 
where and is the vector of channels from UE to relay .
Relay processes the received signal by applying the beamforming matrix for transmission:
(35) 