Joint power allocation for MIMOOFDM fullduplex relaying communications
 Hoang D. Tuan^{1},
 Duy T. Ngo^{2}Email authorView ORCID ID profile and
 Ho H. M. Tam^{1}
https://doi.org/10.1186/s1363801608004
© The Author(s) 2017
Received: 29 June 2016
Accepted: 19 December 2016
Published: 19 January 2017
Abstract
In this paper, we address the problem of joint power allocation in a twohop MIMOOFDM network, where two fullduplex users communicate with each other via an amplifyandforward relay. We consider a general model in which the fullduplex relay can forward the received message in either oneway or twoway mode. Our aim is to maximize the instantaneous endtoend total throughput, subject to (i) the separate sumpower constraints at individual nodes or (ii) the joint sumpower constraint of the whole network. The formulated problems are largescale nonconvex optimization problems, for which efficient and optimal solutions are currently not available. Using the successive convex approximation approach, we develop novel iterative algorithms of extremely low complexity which are especially suitable for largescale computation. In each iteration, a simple closedform solution is derived for the approximated convex program. The proposed algorithms guarantee to converge to at least a local optimum of the nonconvex problems. Numerical results verify that the devised solutions converge quickly, and that our optimal power allocation schemes significantly improve the throughput of MIMOOFDM fullduplex oneway/twoway relaying over the conventional halfduplex relaying strategy.
Keywords
1 Introduction
The fifthgeneration (5G) wireless networks target a 1,000fold increase in the network capacity to meet the ever growing user demands for highspeed and ubiquitous network access. To support such an ambitious goal, multipleinput multipleoutput (MIMO) communications and cooperative orthogonal frequency division multiplexing (OFDM) relaying techniques play a key role in enhancing spectral efficiency and improving link reliability. MIMO transmission and reception increase the channel capacity through spatial multiplexing, modulation and coding. Cooperative OFDM relaying provides greater coverage areas without having to deploy costly additional base stations.
OFDM relays are traditionally designed for the halfduplexing (HD) mode, where signal transmission and reception take place in different time slots or frequency bands [1]. Only after fully receiving a data packet, the HD relay nodes forward it to the destination. Fullduplexing (FD) has recently been proposed as one of the key 5G transceiver techniques with the hope of doubling the spectral efficiency [2, 3]. With simultaneous signal transmission and reception in the same time slot and on the same frequency band, an FD relay node transmits a packet while receiving another packet, thereby significantly reducing the endtoend delay. Such bidirectional communication on the same radio resource block was assumed technically impossible, due to the huge selfinterference (SI) introduced by the transmit antenna to the receive antenna on the same device. Only recently, advances in hardware design have suppressed the SI to a level potentially suitable for practical FD applications [4–6].
Twoway relaying strategies [7] have attracted considerable research attention, due to their potential to provide substantially higher spectral efficiency than the conventional oneway relaying counterpart [8–15]. Under the FD twoway relaying, the overall network throughput could further be enhanced by exchanging data in only one time slot as opposed to four in the HD oneway relaying. However, the tradeoff is that an FD twoway relaying scheme suffers even more severe interferences than the FD oneway relaying. It is the direct result of allowing concurrent transmissions at both communication ends. Importantly, the rise in the interference level may null out any throughput gain achieved by using fewer time slots.
Efficient power allocation to realize the potential gains of the MIMOOFDM FD twoway relaying strategy remains an open research topic. Such an allocation is still underdeveloped even for the conventional MIMOOFDM HD oneway relaying networks. In [16], the problem of power allocation for amplifyandforward (AF) HD oneway relays is investigated with the aim of maximizing the instantaneous sum throughput. Since the objective function is not jointly but separately concave in the source and relay power variables, [16] proposes alternating optimization at the source and at the relay with individual pernode transmit power constraints. For the joint sumpower constraint at both the source and the relay, [16] resorts to a high signaltonoise ratio (SNR) approximation for the throughput to become a jointly concave function in the source and relay power variables ([17], Prop. 1 and Appendix B). Although a closedform optimal solution is available for the convex reformulation, such an approximation at high SNR regions does not always hold in practice. OFDM subchannels tend to be assigned with very different transmit power levels. Good subchannels are typically allocated more power to achieve high SNRs while the bad subchannels may even get zero SNRs. With the high SNR approximation of [16], the original nonconvex program is transformed to an inequivalent optimization problem. Indeed, [16, 17] provide upper bound maximization for the original nonconvex maximization, where lower bound maximization should always be preferred. Moreover, the solutions found by either alternating optimization or convex relaxation, in general, may not even satisfy the KarushKuhnTucker (KKT) necessary conditions for optimality.

Novel algorithmic development. We address the nonconvex optimization problem formulations via solving a sequence of convex programs in the complete set of source and relay power variables. The proposed approach applies equally well to both separate and joint sumpower constraints. Our convex approximations are far from trivial even in the simplest scenario of MIMOOFDM HD oneway relaying considered in [16]. With a new bounding technique, the devised algorithms are novel even from an optimizationtheoretic perspective.

Lowcomplexity locally optimal algorithms for largescale computation. As each iteration of the algorithms yields an improved solution, they always guarantee to converge to at least a local optimum of the original nonconvex problems. Importantly, unlike [18, 19] we derive simple closedform suboptimal solutions for the convex program in each iteration, which require extremely low computational complexity. Numerical experiments demonstrate that the number of iterations required for our algorithms to converge is small. These two features make our algorithms particularly suitable for largescale computation.

Throughput performance improvement. The efficient power allocation schemes markedly enhance the throughput of the MIMOOFDM FD oneway/twoway relaying over the HD relaying strategies.
The rest of this paper is organized as follows. Section 2 presents the nonconvex problem formulations for power allocation in a MIMOOFDM relaying network. Sections 3 and 4 propose the optimal power allocation algorithms for twoway and oneway relaying cases, respectively. Section 5 verifies the performance of our proposed algorithms via numerical examples. Finally, Section 6 concludes the paper.
Notation. Boldfaced symbols are used for optimization variables whereas nonboldfaced symbols are for deterministic terms, regardless of whether they are matrix, vector or scalar. The dimensions of these symbols are interpreted from context, and they will be explicitly specified if there is any ambiguity.
2 System Model and Problem Formulations
2.1 MIMOOFDM FD TwoWay Relaying
where \(H^{(\ell)}_{S,k}\in \mathbb {C}^{N\times N}\) is the MIMO channel matrix between user ℓ and the relay on subcarrier k; \(e_{{\mathsf {LI}},k}\in \mathbb {C}^{N}\) is the relay FD loop interference on subcarrier k; and \(\tilde {w}_{R,k}\in \mathbb {C}^{N}\) is the additive zeromean Gaussian noise with covariance \({\mathcal {R}}_{R}\) encompassing all OFDM impairments such as intercarrier power leakage, narrow band interferences, channel estimation error and baseband noise [20–22]. The reader is referred to [23–25] and references therein for the suppression techniques of these impairments.
where \(H_{R,k}^{(\ell)}\in \mathbb {C}^{N\times N}\) is the MIMO channel matrix between the relay and user ℓ on subcarrier k; \(e^{(\ell)}_{\mathsf {LI},k}\in \mathbb {C}^{N}\) is the selfloop interference at user ℓ due to FD transmissions; and \(\tilde {w}_{D,k}^{(\ell)}\) is the zeromean Gaussian noise at user ℓ with covariance \(\mathcal {R}_{D}^{(\ell)}\) encompassing all impairments such as intercarrier power leakage, narrow band interferences, channel estimation error and baseband noise [20–22].
respectively. Denoting \(q^{(\ell)}_{k,n}\) as the diagonal entries of \(\left (Q^{(\ell)}_{S,k}\right)^{H}\left (Q^{(\ell)}_{S,k}\right)^{1}\), the transmit power from user ℓ∈{1,2} over subcarrier k is then \(\sum _{n=1}^{N}q^{(\ell)}_{k,n}\mathbf {p}^{(\ell)}_{S,k,n}\).
where we define \(\gamma _{k,n}\left (\mathbf {p}_{S,k,n}^{(3\ell)},\mathbf {p}_{R,k,n}\right)\triangleq \mathbf {p}_{S,k,n}^{(3\ell)}/\left (\gamma _{\mathsf {LI},k,n}\mathbf {p}_{R,k,n}+1\right)\), \(\bar {h}_{R,k,n}^{(\ell)}\triangleq h_{R,k,n}/\mathcal {R}^{(\ell)}_{D,k}(n,n)\), and \(\xi ^{(\ell)}_{k,n}\left (\mathbf {p}_{R,k,n},\mathbf {p}_{S,k,n}^{(\ell)}\right)\triangleq \mathbf {p}_{R,k,n}/\left (\gamma _{\mathsf {LI},k,n}^{(\ell)}\mathbf {p}_{S,k,n}^{(\ell)}+1\right)\) with \(\gamma _{\mathsf {LI},k,n}^{(\ell)}\triangleq h^{S}_{\mathsf {LI},k,n}/\mathcal {R}^{(\ell)}_{D,k}(n,n)\).
Here, \(P \geq 0, P_{1}^{(\ell)}\geq 0, P_{2}\geq 0\) are the predefined power budgets. In practice, the users and the relay have separate power supplies constrained by (16) and the power allocation is performed at individual nodes. However, it is also important to consider the joint power allocation with the total power constraint (15) to gain meaningful insights into the power utilization of the whole system, and thereby realizing its full capacity.
subject to either (15) or (16). In (17), the pre log factor of 1/2 accounts for the two time slots needed to transmit one data packet. Note that joint power allocation in nonregenerative MIMO twoway relaying has previously been considered, e.g., in [27], under the reciprocity assumption \(H^{(\ell)}_{R,k}=\left (H^{(\ell)}_{S,k}\right)^{H}, \ \ell \in \{1, 2\}\), in which the corresponding channel (2) (with no amplified selfloop interference) is parallelized by precoding the data symbol \(s^{(\ell)}_{k}\) for subchannel alignment.
2.2 MIMOOFDM FD OneWay Relaying
for ℓ∈{1,2}.
for ℓ∈{1,2}.
This paper focuses on solving the general FD twoway/oneway relaying problems (14) s.t. (15)/(16) and (18) s.t. (19)/(20). The solutions to the HD twoway/oneway relaying problems (17) s.t. (15)/(16) and (22) s.t. (19)/(20) follow directly by replacing γ _{ LI,i }=0, and will be used as benchmarks for performance comparison between the FD and HD relaying strategies.
3 Proposed Solutions for TwoWay Relaying
To the best of our knowledge, the FD twoway relaying problems (14) s.t. (15)/(16) have never been considered before. From the definition of the fractional function \(\gamma _{i}(\mathbf {x}_{i}^{(\ell)},\mathbf {y}_{i})\) in (12), it is clear that the objective in (14) is a very complex nonconcave function with many fractional and cross terms in the source power variables \(\mathbf {x}\triangleq \left (\mathbf {x}^{(1)},\mathbf {x}^{(2)}\right)\) and the relay variable y. Moreover, this objective function is not individually concave in either x or y. Even performing the power allocation either at source node only or at the relay node only is already difficult. Although the objective function in (17) is simpler than (14), the same challenge remains in the HD twoway relaying problems (17) s.t. (15)/(16).

(P1): \(\ln \left (x_{1}+x_{2}\right)\leq \ln \left (x_{1}^{(0)}+x_{2}^{(0)}\right)+\frac {1}{x_{1}^{(0)}+x_{2}^{(0)}}\left [\left (x_{1}x_{1}^{(0)}\right)+\left (x_{2}x_{2}^{(0)}\right)\right ]\) for all \(x_{1}>0, x_{2}\geq 0, x_{1}^{(0)}>0, x_{2}^{(0)}\geq 0\).

(P2): \(\ln \left (x_{1}+x_{2}\right) \geq \ln \left (x_{1}^{(0)}+x_{2}^{(0)}\right)+\allowbreak \frac {1}{x_{1}^{(0)}+x_{2}^{(0)}}\left [x_{1}^{(0)}\left (\ln x_{1}\ln x_{1}^{(0)}\right)+\right.\allowbreak \left.x_{2}^{(0)}\left (\ln x_{2}\ln x_{2}^{(0)}\right)\right ]\) for all \(x_{1}>0, x_{2}>0, x_{1}^{(0)}>0, x_{2}^{(0)}> 0\).
property (P2) then follows after replacing \(x_{i}=e^{\tilde {x}_{i}}, x_{i}^{(0)}=e^{\tilde {x}_{i}^{(0)}}, \ i\in \{1,2\}\) in (24) and (25). This property is the key for the success of the SCALE algorithm in the multiuser OFDM spectrum balancing problem [29].

It matches the nonconcave objective function F(·) at (x ^{(κ)},y ^{(κ)}), i.e.,$$\begin{array}{*{20}l} F^{(\kappa)}\left(x^{(\kappa)}, y^{(\kappa)}\right) &= F\left(x^{(\kappa)}, y^{(\kappa)}\right). \end{array} $$(27)

It is a global lower bound of the nonconcave objective function F(·), i.e.,$$\begin{array}{*{20}l} F^{(\kappa)}(\mathbf{x},\mathbf{y}) &\leq F(\mathbf{x},\mathbf{y}), \ \forall (\mathbf{x},\mathbf{y}). \end{array} $$(28)
These properties guarantee that F ^{(κ)}(·) is both a local and a global concave approximation of F(·) at (x ^{(κ)},y ^{(κ)}). A proximity control is therefore not necessary.
as long as (x ^{(κ+1)},y ^{(κ+1)})≠(x ^{(κ)},y ^{(κ)}). In other words, (x ^{(κ+1)},y ^{(κ+1)}) is a better solution of the nonconvex program (26) than (x ^{(κ)},y ^{(κ)}). Moreover, the necessary optimality condition for (x ^{(κ)},y ^{(κ)}) is (x ^{(κ+1)},y ^{(κ+1)})=(x ^{(κ)},y ^{(κ)}). That is, for (x ^{(κ)},y ^{(κ)}) to be an optimal solution of the nonconvex program (26), it is necessary that (x ^{(κ)},y ^{(κ)}) is a globally optimal solution of the convex program (29). The efficiency is Algorithm 1 therefore hinges upon the computational tractability of (29).
Proposition 1
The inequality (35) follows from property (P2) whereas (36) from property (P1).
The iterative waterfilling algorithm that solves problems (14) s.t. (15)/(16) is described as follows.
ALGORITHM 2. Initialized by a feasible solution ( x ^{(1,0)},x ^{(2,0)},y ^{(0)} ) to problems ( 14 ) s.t. ( 15 )/( 16 ), generate a feasible solution ( x ^{(1,κ+1)},x ^{(2,κ+1)},y ^{(κ+1)}) at κ iteration for κ=0,1,…, according to formulae ( 41 )( 42 )/( 44 )( 45 ).
Proposition 2
Initialized from a feasible solution (x ^{(1,0)},x ^{(2,0)},y ^{(0)}) to problems (14) s.t. (15)/(16), the sequence {(x ^{(1,κ)},x ^{(2,κ)},y ^{(κ)})} generated by Algorithm 2 converges to at least a locally optimal solution of problems (14) s.t. (15)/(16).
The HD twoway relaying problems (17) s.t. (15)/(16) are particular cases of the FD twoway relaying problems (14) s.t. (15)/(16) with \(\gamma (\mathbf {x}_{i}^{(\ell)},\mathbf {y}_{i})=\mathbf {x}_{i}^{(\ell)}, \ \ell \in \{1, 2\}\). Therefore, Algorithm 2 can be employed to solve the former by setting γ _{ LI,i }=0 and using (17) to compute the achieved throughput.
Remark 1
which is a convex but not necessarily computationally tractable problem. While the bound in (46) is not tight, applying the interior method to solve (46) as suggested in [27] is not computationally efficient.
4 Tailored Solutions for OneWay Relaying
Problems (18)/(22) s.t. (19)/(20) are also particular cases of the problem (14) s.t. (15)/(16). However, their simpler structures entail more computationally efficient algorithms as will be developed in this section.
4.1 FD OneWay Relaying
The iterative waterfilling algorithm that solves problems (18) s.t. (19)/(20) is described as follows.
ALGORITHM 3. Initialized by a feasible solution ( x ^{(ℓ,0)},y ^{(ℓ,0)}), ℓ∈{1,2} to problems ( 18 ) s.t. ( 19 )/( 20 ), generate a feasible solution ( x ^{(ℓ,κ+1)},y ^{(ℓ,κ+1)}) at κiteration for κ=0,1,…, according to formulae ( 60 )( 61 )/( 62 )( 63 ).
4.2 HD OneWay Relaying
The HD oneway relaying problems (22) s.t. (19)/(20) correspond to the FD oneway relaying problems (18) s.t. (19)/(20) with γ _{ LI,i }=0. Therefore, Algorithm 3 can be employed to solve the former by simply setting γ _{ LI,i }=0 and using (22) to compute the achieved throughput. In this special case, we can also derive even simpler waterfilling solutions for these problems as follows. First note that f ^{(κ)}(x ^{(ℓ)},y ^{(ℓ)})≡f(x ^{(ℓ)},y ^{(ℓ)}), so we do not need to approximate f(x ^{(ℓ)},y ^{(ℓ)}) and problem (29) then admits a closedform optimal solution.
where \(\lambda _{1}^{(\ell)}>0\) and \(\lambda _{2}^{(\ell)}>0\) are chosen such that the constraints (20) are met with equality for (x ^{(ℓ,κ+1)},y ^{(ℓ,κ+1)}). A bisection search similar to that in the Appendix can be used, where the initial values are set as \(\lambda ^{(\ell)}_{1,\text {lo}}=\lambda ^{(\ell)}_{2,\text {lo}}=0\) and \(\lambda ^{(\ell)}_{1,\text {hi}}= \max _{i=1,\dots,M}\left [1\big /\left (P_{1}^{(\ell)}/M+1/a_{i}^{(\ell)}\right)c_{i}^{(\ell,\kappa)}a_{i}^{(\ell)}\right ]\), \(\lambda ^{(\ell)}_{2,\text {hi}}= \max _{i=1,\dots,M}\left [1\big /\left (P_{2}/(2M)+b_{i}^{(\ell)}\right)c_{i}^{(\ell,\kappa)}b_{i}^{(\ell)}\right ]\).
The iterative waterfilling algorithm that solves problems (22) s.t. (19)/(20) is described as follows.
ALGORITHM 4. Initialized by a feasible solution (x ^{(ℓ,0)},y ^{(ℓ,0)}), ℓ∈{1,2} to problems ( 22 ) s.t. ( 19)/(20), generate a feasible solution (x ^{(ℓ,κ+1)},y ^{(ℓ,κ+1)}) at κ iteration for κ=0,1,…, according to formulae ( 64 )/( 62 )( 65 ).
Remark 2
problem (67) is in fact an upper bound maximization of problem (22) s.t. (19). However, it should be noted that a natural approximated optimization of the maximization problem (22) s.t. (19) should always be a lower bound maximization.
for which a closedform expression of the optimal solution can be obtained. One could see that the approximation (70) is poor for small values of γ _{ LI,i } (e.g., in FD relays with effective selfinterference cancelation). Likewise, because (71) ignores the essential term \(a_{i}^{(\ell)}(\mathbf {x}_{i}^{(\ell)}+\mathbf {y}_{i}^{(\ell)})\) in the denominator, this approximation is hardly valid for large values of \(a_{i}^{(\ell)}(\mathbf {x}_{i}^{(\ell)}+\mathbf {y}_{i}^{(\ell)})\). As more power is allocated to the good OFDM subcarriers to maximize the throughput, it is reasonable to expect a large value of \(a_{i}^{(\ell)}(\mathbf {x}_{i}^{(\ell)}+\mathbf {y}_{i}^{(\ell)})\) for some subcarrier i. Again (72) is upper bound maximization for the maximization problem (18) s.t. (19) while its lower bound maximization is always desirable. This kind of convex relaxation for the joint sumpower constraint is not applicable to the case of separate sumpower constraints in \(\mathbf {x}_{i}^{(\ell)}\) and \(\mathbf {y}_{i}^{(\ell)}\) because the power distribution (68) does not hold in the latter case.
5 Numerical Results
where d (in meters) is the transmitterreceiver distance and ψ (in dB) is a correction factor (e.g., to model the outdoor wall penetration loss). We model the effect of shadowing by a lognormal random variable with mean of zero and standard deviation of 6dB. To simulate the effect of frequency selectivity in each spatial channel, we assume an exponential power delay profile (PDP) with a rootmeansquare (RMS) delay spread of σ _{RMS}=3T _{ s } where T _{ s } is a constant. We model the magnitude of the timedomain channel corresponding to each tap of the PDP by either Rayleigh or Rician distribution. We further model the spatial correlation among the MIMO channels according to Case B of the 3GPP IMETRA MIMO channel model ([32], p.94).
The timedomain channels are converted to the frequency domain by the Fast Fourier transform (FFT) for the computation of the OFDM throughput. We use K=1,024 OFDM subcarriers, each of which occupies a bandwidth of Δ f=15kHz. Since we take T _{ s }=1/(K Δ f), Δ f is much smaller than the channel coherence bandwidth of 0.02/σ _{RMS} ([33], p.85). The OFDM subchannels are frequencyflat while there is correlation among the adjacent subchannels. In each subchannel, the power spectral density of additive white Gaussian background noise at each antenna is −174dBm/Hz, and the correlation between noise samples from different antennas is 0.2. The effect of all other impairments (including intercarrier power leakage) is modelled as additive Gaussian noise whose power is twice that of the background noise. We set the error tolerance for all algorithms as ε=10^{−4}. We repeat the simulation for 100 independent runs and average the results to get the final figures for spectral efficiency.
We evaluate the performance of FD/HD twoway relaying (by Algorithm 2) and FD oneway relaying (by Algorithm 3). We use the HD oneway relaying result (by Algorithms 3 and 4) as the baseline performance. For a fair comparison with twoway relaying, we average the throughput of the oneway relaying scheme in two different directions, i.e., one from user 1 to relay to user 2, and another from user 2 to relay to user 1. We set the maximum transmit power at the relay as P _{2} in the twoway relaying and as P _{2}/2 in each direction of the oneway relaying. We assume that \(P_{1}^{(1)} + P_{2}/2 = P_{2}^{(1)} + P_{2}/2 = P/2\). For simplicity, we set the instantaneous selfloop gain as \(h_{\mathsf {LI},k,n}=h^{S}_{\mathsf {LI},k,n} \equiv h_{\mathsf {LI}}\). Note that the presented value of h _{ LI } is not normalized with respect to noise power, and that h _{ LI }=0 in the HD twoway/oneway relaying. After normalization, the values of h _{ LI } would be consistent with those used in the literature, e.g., [17]. In all the results, we initialize Algorithm 2 by the feasible points \({x}^{(\ell)}_{i} = P_{1}^{(\ell)}\big /\left (M \max _{i} q_{i}^{(\ell)}\right), \ \ell \in \{1, 2\}\) and y _{ i }=P _{2}/M for all i=1,…,M (where M=NK=4,096). And we initialize Algorithms 3 and 4 by \({x}^{(\ell)}_{i} = P_{1}^{(\ell)}/M\) and y _{ i }=P _{2}/M for ℓ∈{1,2} and i=1,…,M.
Both HD twoway relaying and HD oneway relaying are not subjected to any SI. However, Fig. 3 shows that HD twoway relaying does not double the throughput of HD oneway relaying. This is because in the twoway relaying, (i) our precoding design may have amplified noise, (ii) the available power is further constrained by (15) and (16), and (iii) the denominator of the SINR expression is \(a_{i}\left (\mathbf {x}_{i}^{(1)}+\mathbf {x}_{i}^{(2)}\right)\) which is greater than or equal to \(a_{i}^{(\ell)} \mathbf {x}_{i}^{(\ell)}, \ \ell \in \{1, 2\}\) of the oneway relaying. The gain from using half the number of time slots by twoway relaying may not be sufficient to offset the loss in the achieved throughput in each way. Similar reasons apply to FD twoway relaying where this strategy is further subjected to the SI at the users. Therefore, although FD tworelaying uses only one time slot for transmissions in both link directions, it does not double the throughput of FD oneway relaying. When SI is large (i.e., h _{ LI }≥−100dB), Fig. 3 reveals that FD twoway relaying is even slightly outperformed by FD oneway relaying.
Average number of iterations for h _{ LI }=−140dB
UE relaying (P _{2}=20dBm)  BS relaying (P _{2}=40dBm)  BS relaying (P _{2}=46dBm)  

Joint power  Sep. power  Joint power  Sep. power  Joint power  Sep. power  
Alg. 2  14.26  14.78  17.56  16.45  11.81  20.76 
Alg. 3  17.75  17.99  23.90  14.90  19.80  29.38 
Alg. 4  14.49  15.86  20.57  21.14  17.56  23.75 
In Figs. 3, 4 and 5, it is observed that the performance of the concerned algorithms in the joint sumpower constraint case is better than that in the separate sumpower constraint case. The reason is that the feasible set of the former problem contains that of the latter problem.
6 Conclusions
This paper addresses the problem of joint power allocation for a MIMOOFDM network consisting of two FD users and one FD amplifyandforward relay. The aim is to maximize the instantaneous total network throughput, subject to (i) the separate sumpower constraints at individual nodes or (ii) the joint sumpower constraint. To solve the highly nonconvex problem formulations, we have employed the successive convex approximation approach to develop novel iterative algorithms of extremely low complexity. A simple closedform solution is available for the approximated convex program at each iteration. The proposed algorithms are shown to always converge to at least a local optimum. Our approach applies to the general case where any combination of oneway/twoway and HD/FD relaying is allowed. The advantages of our novel solutions have been confirmed by numerical examples.
7 Endnotes
^{1} Noises are possibly amplified here. Moreover, \({\mathcal {R}}_{D,k}^{(\ell)}\) is no longer diagonal even when \(\mathcal {R}_{D}^{(\ell)}\) is diagonal.
^{2} As the noises in (7) are correlated, the subchannelwise decoding is not optimal. Therefore, the objective function in (14) provides a lower bound instantaneous capacity of the network.
^{3} One should use the obtained solution to compute the objective (14) in order to have the actual throughput performance.
8 Appendix: Bisection Search
The initial values λ _{lo} and λ _{hi} are given. For λ:=(λ _{lo}+λ _{hi})/2, use (41)(42) to find (x ^{(1,κ+1)},x ^{(2,κ+1)},y ^{(κ+1)}). If \(\sum _{i=1}^{M} \left (q_{i}^{(1)}\mathbf {x}_{i}^{(1)}+q_{i}^{(2)}\mathbf {x}_{i}^{(2)}+\mathbf {y}_{i}\right)>P\), set λ _{lo}:=λ. If \(\sum _{i=1}^{M}\left (q_{i}^{(1)}\mathbf {x}_{i}^{(1)}+q_{i}^{(2)}\mathbf {x}_{i}^{(2)}+\mathbf {y}_{i}\right)<P\), set λ _{hi}:=λ. Proceed to the next bisection until \(\sum _{i=1}^{M} \left (q_{i}^{(1)}\mathbf {x}_{i}^{(1)}+q_{i}^{(2)}\mathbf {x}_{i}^{(2)}+\mathbf {y}_{i}\right)=P\).
Declarations
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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