Optimal Power Allocation Strategies in
Fullduplex Relay Networks
Abstract
In this work, we consider a dualhop, decodeandforward network where the relay can operate in fullduplex (FD) or halfduplex (HD) mode. We model the residual selfinterference as an additive Gaussian noise with variance proportional to the relay transmit power, and we assume a Gaussian input distribution at the source. Unlike previous work, we assume that the source is only aware of the transmit power distribution adopted by the relay over a given time horizon, but not of the symbols that the relay is currently transmitting. This assumption better reflects the practical situation where the relay node also forwards signaling traffic, or data originated by other sources. Under these conditions, we identify the optimal power allocation strategy at the source and relay, which in some cases coincides with the half duplex transmission mode. In particular, we prove that such strategy implies either FD transmissions over an entire frame, or FD/HD transmissions over a variable fraction of the frame. We determine the optimal transmit power level at the source and relay for each frame, or fraction thereof. We compare the performance of the proposed scheme against reference FD and HD techniques, which assume that the source is aware of the symbols instantaneously transmitted by the relay. Our results highlight that our scheme closely approaches or outperforms such reference strategies.
I Introduction
Multihop relay communications are a key technology for next generation wireless networks, as they can extend radio access in case of coverage holes or users at the cell edge, as well as increase the potentialities of devicetodevice data transfers. The dualhop relay channel, in particular, has been widely investigated under different cooperative schemes, namely, decodeandforward (DF), compressandforward (CF) and amplifyandforward (AF) [1, 2, 3, 4, 5]. Most of this body of work has assumed the relay node to operate in halfduplex (HD) mode. Specifically, results on the capacity of the HD dualhop relay channel have appeared in [6, 7], where it was shown that the network capacity is achieved by a discrete input when no direct link between the source and the destination exists.
More recently, a number of studies [8, 9, 10, 11, 12] have addressed the case where the relay operates in fullduplex (FD) mode, i.e., it can transmit and receive simultaneously on the same frequency band. Indeed, advances in selfinterference suppression in FD systems have made such a technology very attractive for relay networks. The capacity of Gaussian twohop FD relay channels has been characterized in [13], under the assumption that the residual selfinterference can be neglected. The more realistic case where residual selfinterference ([14, 15]) is taken into account, has been instead addressed in [8, 9, 10, 11, 12]. In these works, the signal looping back from the relay output to its input is modeled as an additive Gaussian noise with variance proportional to the relay transmit power. In particular, [8] analyses the instantaneous and average spectral efficiency of a dualhop network with direct link between source and destination, and a relay node that can operate in either HD or FD mode. Interestingly, the authors propose hybrid FD/HD relaying policies that, depending on the channel conditions, optimally switch between the two operational modes when the FD relay transmit power is fixed to its maximum value, as well as when it can be reduced in order to mitigate selfinterference as needed. The FD mode only is considered in [10], which aims to maximize the signaltointerference plus noise ratio (SINR) as the relay transmit power varies, in the case where AF is used, the relay has multiple transmit antennas and a single receive antenna, and constraints on the average and maximum relay transmit power must hold. In [11], the maximum achievable rate and upper bounds on the capacity are obtained when the relay node operates in DF and CF and Gaussian inputs are considered at the source and the relay node.
The study in [12] is the first to derive the capacity of the Gaussian twohop FD relay channel with residual selfinterference, assuming the average transmit power at the source and the relay nodes to be limited to some maximum values. The study shows that the conditional probability distribution of the source input, given the relay input, is Gaussian while the optimal distribution of the relay input is either Gaussian or symmetric discrete with finite mass points. This result implies that a capacity achieving scheme requires the source to know at each time instant what the relay is transmitting. This can be realized with the aid of a buffer at the relay, which holds the data previously transmitted by the source and correctly decoded by the relay. The relay reencodes such data before forwarding it to the destination in the next available channel use. The source can use the same encoder as the relay, in order to predict what will be transmitted by the relay and hence guarantee a capacity achieving transmission.
In this work, we consider a scenario similar to [12], including a dualhop, DF network where the relay can operate in FD mode, and the residual selfinterference is modeled as an additive Gaussian noise, with variance proportional to the relay transmit power. Different from [12], in this paper we consider the case where the source does not know what symbols are transmitted by the relay and is aware only of the transmit power distribution adopted by the relay over a given time horizon. Therefore, our scenario can accommodate the case where the relay node has to handle multiple, simultaneous traffic flows, e.g., inband signaling as well as data traffic originated at the relay itself or previously received from other sources. The source knowledge about the relay power is exploited in order to optimally set the source transmit power and decide whether the relay should operate in HD or FD. Furthermore, we assume a Gaussian input distribution at both source and relay, with variance not exceeding a target maximum value.
Under this scenario, we formulate an optimization problem that aims at maximizing the achievable data rate, subject to the system constraints. We characterize different operational regions corresponding to optimal network performance, and provide conditions for their existence, as the values of the system parameters vary. Our analysis led to the following major results:

The distribution of the transmit power at the relay can be conveniently taken as the driving factor toward the network performance optimization. We prove that the optimal distribution of such a quantity is discrete and composed of either one or two delta functions, depending on the target value of average transmit power at the source and relay. We provide the expression of the above distribution for the whole range of the system parameter values, including the channel gains and the target values for the average transmit power at the source and the relay.

The above finding leads to the optimal communication strategy for the network under study, which implies either FD transmissions over an entire frame, or FD/HD transmissions over a fraction of the frame.

Given the optimal transmit power distribution at the relay and the system constraints, we derive the optimal power level to be used over time at the relay and the source. Such power allocation policy allows the system to achieve the maximum data rate. We remark that our policy establishes the time fractions during which the relay should work in FD and in HD, as well as the transmit power to be used at the source and the relay, given that only the average (not the instantaneous) relay transmit power needs to be known at the source.

We compare the results of our optimal power allocation to a reference FD and HD scheme, where the source knows the instantaneous relay transmit power. Interestingly, our scheme closely approaches the performance of such strategy in all the considered scenarios.
The remainder of the paper is organized as follows. Section II introduces our system model, while Section III presents the constrained optimization problem. The optimal communication strategy and our main analytical results are presented in Sections IV and V, for different values of the system parameters. Section VI shows the performance results, and Section VII discusses how to extend the analysis to the case where the average transmission power at the source is limited. Finally, Section VIII concludes the paper.
Ii System model
We consider a twohop, DF relay network including a source node , a relay and a destination . All network nodes are equipped with a single antenna, and the relay is assumed to be FD enabled. No direct link exists between source and destination, thus information delivery from the source to the destination necessarily takes place through the relay. As far as the channel is concerned, we consider independent, memoryless block fading channels with additive Gaussian noise, between source and relay as well as between relay and destination.
Source and relay operate on a frame basis, of constant duration , with being set so that channel conditions do not vary during a frame; without loss of generality, in the following we set . In general, the following modes of operations are possible for source and relay: (i) the source transmits while the relay receives only (HDRX mode); the source is inactive while the relay transmits (HDTX mode), (iii) the source transmits while the relay transmits and receives at the same time (FD mode).
We remark, however, that source and relay do not need to be synchronized on a persymbol basis, and that the relay can handle multiple (data or control) traffic streams originated at different network nodes, according to any scheduling scheme of its choice. This implies that, in order to select its operational mode, the source is not required to be aware of the information the relay is transmitting. We assume instead that the source has knowledge of the distribution of the transmit power adopted by the relay across a frame.
When the relay transmits to the destination, a residual selfinterference (after analog and digital suppression) adds up to what the relay receives from the source. Then the signal received at the relay and destination can be written as:
(1) 
where

and are the complex channel gains associated with, respectively, the sourcerelay and relaydestination links;

and are the input symbols transmitted by, respectively, the source and the relay. We assume the input at both source and relay to be zeromean complex Gaussian distributed with unit variance. From (1), we have that the levels of instantaneous power transmitted by source and relay, are and , respectively. In the most general case, and are timevarying continuous random variables ranging in and , respectively.

and represent zeromean complex Gaussian noise over, respectively, the sourcerelay and the relaydestination link, with variance ;

represents the instantaneous residual selfinterference at the relay. As typically done in previous studies [16, 17, 11, 12], we model as a Gaussian noise with variance proportional to the instantaneous transmission power at the relay, i.e., and variance . In these expressions, denotes the selfinterference attenuation factor at the relay and is the expectation operator with respect to . Also, we remark that, as shown in [12], assuming as a zeromean i.i.d. Gaussian random variable represents the worstcase linear residual selfinterference model.
We define as the probability density function of , with support in .
Finally, we consider that the average power over a frame at the source and at the relay is constrained to given target values, denoted by and , respectively. The average power at the source and relay is therefore given by:
(2)  
(3) 
where the expression in (3) is due to the fact that the source selects its transmission power based on its knowledge of , hence depends on . In order to highlight this dependency, in the above expression and in the following, we use the notation.
Iii Problem formulation
In our study, we aim at determining the power allocation at the source and relay that maximizes the achievable rate of the dualhop network described above. To this end, we start by recalling some fundamental concepts:

the network rate will be determined by the minimum between the rate achieved over the sourcerelay link and over the relaydestination link, hereinafter referred to as and , respectively;

depends on the source transmit power, the Gaussian noise, as well as on the residual selfinterference at the relay, which, in turn, depends on the relay transmit power;

depends on the relay transmit power and the noise at the destination;

the transmit power at source and relay may vary over time. Whenever and , the relay works in FD mode, while, when and the relay is receiving in HD mode. When and , the relay is transmitting in HD mode while the source is silent.
Based on (b) and (c), the residual selfinterference introduces a dependency between the performance of the first and second hop. Thus, in order to maximize the network rate, source and relay should coordinate their power allocation strategies. In our study, we optimize the power allocation, hence the network rate, by controlling the distribution of the transmit power, , at the relay. As a first step, we fix and derive the expressions of the rates and as detailed below.
Iiia Optimal power distribution at the source
Given the system model introduced above and fixed the value of , the rate on the sourcerelay and relaydestination links are given by and , respectively. Then the average rates over a frame can be written as:
(4) 
where .
For a given distribution , the rate can be maximized with respect to . It can be shown (see Appendix A) that, given , the power distribution at the source maximizing is given by where is a parameter defined as (see Appendix A):
with being the Lagrange multiplier used in the constrained maximization of . In the following, we assume that is large enough so that
(5) 
We will remove this assumption and discuss the impact on the obtained results in Section VII.
IiiB Optimal power distribution at the relay
Having expressed the source power as a function of , and the rates and as functions of , we need to find the optimal distribution that maximizes the network data rate . We therefore formulate the following optimization problem, subject to the system constraints:
P1:  
In the above formulation,

constraints (a) and (b) represent the average rates achieved on the sourcerelay and relaydestination links, respectively;

(c) is the average power constraint at the source;

(d) is the average power constraint at the relay;

(e) imposes that , being a distribution, integrates to 1;

(f) constraints to not exceed .
Iv Optimal power allocation for
In order to solve problem P1 we first consider the case . By using such assumption in the constraints (c), (d) and (e) of P1, we obtain . Then the constraint implies that a solution to problem P1 exists if . Moreover, by using in (a) and in (5), we obtain
(8) 
and . Since , , is a concave function of and has average , we can apply Lemma B.1 reported in Appendix B and write:
(9)  
(10) 
with the equality holding when where is the Dirac delta function. Similarly, by applying again Lemma B.1, we get:
(11) 
with the equality holding when . Now, after having bounded the rates and , we consider the following three cases.

If , then and the optimal relay power distribution is . Solving for the inequality , we obtain
and

If , then and the optimal relay power distribution is . Solving for the inequality , we get
(12) and

Otherwise, we find solutions for such that . Indeed, for , problem P1 becomes:
P2: In this case the minimizer of the functional can be found by applying the following theorem.
Theorem IV.1
Consider the following constrained minimization problem:
where , , , and is a probability distribution with ranging in , . Moreover, , , and are constant parameters. Then the minimizer has the following expression
(14) where the constants and are obtained by replacing (14) in the constraint (a) in (IV.1).
Proof:
The proof is given in Appendix C. \qed
Through the above theorem and considering , the maximizer of the rate in P2 is given by
(15) where is obtained by replacing with in constraint (a) in P2, i.e., by solving
(16) with . When instead , the maximizer of the rate in P2 is given by
(17) where is obtained again using in constraint (a), i.e., by solving
(18)
Given the optimal distribution , which is related to the power transmitted at the relay, the optimal power allocation at the source node can be obtained by using (5).
From the above results, some important observations can be made:

even more importantly, the optimal power allocation at the relay is discrete, with either one or two probability masses depending on the number of functions appearing in the expression of ;

the above finding implies that source and relay should operate according to a time division strategy consisting of transmissions over either the entire frame (when includes one probability mass only), or two fractions of the frame (when two probability masses appear in ). Hereinafter, we will refer to such fractions as, respectively, phase A and phase B; clearly, they reduce to one phase when includes only one probability mass. An example where two phases exist is depicted in Figure IV(top).

The phases durations are given by the coefficients of the functions composing (see Figure IV(bottom)). Note that now takes on a new meaning, as it represents the average level of transmission power to be used at the relay during a phase of the frame. The values of , hence of the average transmission power at the relay over each phase, are given by the arguments of the functions in . Likewise, through (5), the average level of transmitted power at the source is determined by the arguments of the functions in .
To summarize, Table I reports the solution of problem P1 for , along with the corresponding power allocation at the source and relay.
Phase A  Phase B  
0  
–  –  –  1  
Phase A  Phase B  
0  
1  –  –  – 
Rate  

;  
;  
Looking at the top tables, we remark that:

for , both source and relay transmit during phase A and thus the relay operates in FD. In phase B, the relay is silent and only receives (HDRX mode);

for , two cases are possible. For the relay always operates in FD but source and relay use different power levels in the two phases. Otherwise, the relay uses the same scheme as for , i.e., FD in phase A and HDRX in phase B, but its transmit power in phase A should be set to ;

for , the relay continuously operates in FD, and source and relay always transmit at their average power.
V Optimal power allocation for
Here, we consider the solution of the problem P1 when . We first fix and then rewrite as the weighted sum of two distributions, i.e.,
(19) 
where the distributions and have support in and , respectively. is the cumulative distribution function of , given by . By imposing constraint (2), we get
(20) 
If we define
(21) 
where , from (20) it immediately follows that
(22) 
For simplicity, from now on we drop the dependence on from and . Also, using the above definition, constraint (6) can be rewritten as
(23) 
We also need to impose that the averages in (21) and (22) lie in the support of the distributions and , respectively. In other words,
All the above conditions can be rewritten in terms of and as follows:
(24) 
where we recall that and .
Note that the condition implies and . Furthermore, in order to ensure that takes positive values, we must have , i.e., . Since by assumption , the above condition is less restrictive than . In the light of these considerations, our conditions on reduce to
Clearly, a solution of the above inequalities exists if , i.e., if . Summarizing, these inequalities represent a region defined as
with vertices
(25) 
The region is depicted in Figure V, where the edge – has equation while the edge – has equation .
It turns out that the maximization problem P1 is equivalent to the maximization of the rate over the region . To this end we substitute (19) in the expressions of the rates and and obtain
(26)  
where the inequality follows from Lemma B.1. The upper bound, , is achieved for
(27) 
Similarly the rate in (7) can be rewritten as
Thus, the maximization problem can be recast as
P3  
where the last two constraints come from (21), (23) and the fact that is a probability distribution. In order to solve , we first apply Lemma B.1 to and . By doing so, we obtain:
(31)  
(32) 
The above bounds hold with equality when
(33) 
Similarly, we can write
(34)  
(35) 
In the above expressions, equality holds for
(36) 
Va Breaking the solution space into subregions
In order to maximize the rate over , we exploit the above bounds and define the following subregions.

Let . Then in the problem P3 reduces to maximizing . The maximum rate will be denoted by . We observe that can be viewed as the set of points where (i.e., ). Then the implicit curve is one of the edges of (see Figure V). Also, the intersection point between and the edge –, whose equation is , is . The value of can be computed numerically by solving .
The intersection between and the edge –, whose equation is , is . The value of can be computed numerically by solving . Moreover, we observe that the curve intersects the line at most in a single point. The proof is given in Appendix D. Finally, as shown in Appendix E, decreases with while increases with . Thus, we conclude that is located on the left of the curve (see Figure V).

Let . Then in the problem P3 reduces to maximizing . The maximum rate achieved in this subregion will be denoted by . We observe that is given by the set of points where (i.e., ). Then the implicit curve is one of the edges of . From the results obtained in Appendix E, we conclude that increases with while decreases with . By consequence, the curve defined by the implicit equation , has positive derivative:
Moreover, the curve intersects the edge – in , as it can be easily proven by observing that