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Energyefficient power allocation for massive MIMOenabled multiway AF relay networks with channel aging
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 206 (2018)
Abstract
In this paper, we consider a massive MIMOenabled multiway amplifyandforward relay network with channel aging, where multiple users mutually exchange information via an intermediate relay equipped with massive antennas. For this system, we propose an energyefficient power allocation scheme for the optimization of energy efficiency (EE). Specifically, we firstly derive accurate closedform expressions of the system sum rate with aged channel state information (CSI) and predicted CSI. Secondly, based on the derived analytical results, a unified power allocation optimization problem with aged/predicted CSI is formulated for maximizing the system EE. To solve this challenging problem, the successive convex approximation technique is invoked to transform the original optimization problem into a tractable concave fractional programming problem. Then, Dinkelbach’s algorithm and Lagrangian dual method are adopted to find the optimal solution. In addition, to strike a balance between the computational complexity and the optimality, the EE maximization problem using the equal power allocation scheme is solved by extreme value theorem, leading to a closedform optimal solution. Numerical results demonstrate the accuracy of our analytical results and the effectiveness of the proposed algorithms. Moreover, the impact of several important system parameters on the system performance achieved by the proposed algorithms is also illustrated.
1 Introduction
With the unprecedented growth of mobile data traffic volumes, the carbon emission of information and communication technologies is becoming an increasingly serious problem. Internationally, several academic and industrial research projects have been dedicated to maximize the overall network capacity and improve the energy efficiency (EE) of wireless communication systems (i.e., minimizing the amount of energy required to transmit data) [1–4]. More recently, as one of the major candidate technologies for fifthgeneration (5G) wireless systems, massive multipleinput multipleoutput (MIMO) has received tremendous attention from both academic and industry in wireless fields [5, 6].
More specifically, it was shown in [7] that massive MIMO systems (equipped with a very large number of antennas) are capable of achieving three orders of magnitude EE gains compared with singleantenna systems. The energyefficient design of massive MIMO systems has emerged as a new research trend for 5G wireless communications [8, 9]. For example, in [10], the EE was analyzed in massive MIMO systems, under the effect of a general transceiver hardware impairments. In [11], the EE was maximized in a multiuser massive MIMO system, and the optimal system parameters (includes the number of base station (BS)’s antennas and users). In [12], the BS density, transmit power levels and number of antennas are optimized for maximizing EE in massive MIMOenabled heterogeneous networks.
As another promising approach, multiway relay networks (MWRNs) have recently received plenty of research interest [13, 14]. In general, as compared to oneway relay networks (OWRNs) and twoway relay networks (TWRNs), MWRNs are capable of achieving higher capacity and spectral efficiency (SE) and thus can be employed to effectively deal with the ever increasing demand for higher data rate and SE in a multiuser scenario. Therefore, the integration of massive MIMO and MWRN is regarded as a promising network architecture to meet the significant demand for mobile data applications. Additionally, it was shown in [15] that using simple relay transceivers (e.g., linear zeroforcing (ZF) transceiver), a multiway MIMO relay system is capable of significantly alleviating the interference among different data streams/user equipments (UEs). Furthermore, similar to the observations in massive MIMOenabled OWRNs [16] and TWRNs [17, 18], it was shown in [19] that by invoking a largescale antenna arrayequipped relay and a lowcomplexity ZF transceiver, the SE of MWRNs is also proportional to the number of relay antennas. At present, the existing related research on massive MIMOenabled MWRNs mainly focused on analyzing the performance limits in various specific system configurations [19–24]. For instance, in [19], the asymptotic signaltointerferenceplusnoise ratio (SINR) of massive MIMOenabled MWRNs was studied. Later on, the authors of [20] further analyzed the asymptotic SINR and average error rate performance and obtained the optimal pilot sequence length for maximizing the SE of multicell massive MIMOenabled MWRNs. Moreover, the SE and asymptotic SINR of massive MIMOenabled MWRNs are first analyzed with maximumratio processing, and then the same authors derived a closed form expression of the SE of massive MIMOenabled MWRNs with ZF processing.
Moreover, prior works [19–22] considered the effect of channel imperfection due to channel estimation (CE), but ignored another important aspect of practical channel impairments known as channel aging, which refers to the phenomenon affected by the relative movement of users. This scenario is of high practical value in urban environments, where users move rapidly within a geographical area. Despite its significance, very few works have investigated its impact on the performance of massive MIMO systems. For pointtopoint massive MIMO system, the impact of channel aging on the SINR performance was firstly studied by assuming matched filtering [25]. The impact of channel aging was lately investigated with ZF precoders [26] and minimum mean square error (MMSE) receivers [27]. For massive MIMO relay system, the asymptotic impact of channel aging on the performance of massive MIMOenabled MWRNs was studied in [23]. Later on, the analysis was extended to the multicell massive MIMOenabled MWRNs scenario for simultaneous wireless information and power transfer [24]. To the best of the authors’s knowledge, there is a paucity of contributions on energyefficient transmission strategies of massive MIMOenabled MWRNS, considering the effect of channel aging. It is challenging to extend the existing energyefficient designs conceived for singlehop massive MIMO systems [10–12] to massive MIMOenabled relay systems. Due to this fact, compared to singlehop transmission schemes, both signal processing schemes and the performance analysis of massive MIMOenabled relay systems are fundamentally dependent on the more complex twohop channels. Therefore, it is important to design energyefficient transmission strategy for massive MIMOenabled relay systems. Furthermore, the consideration of channel aging is of paramount importance because it can provide the robustness against the practical setting of user mobility that results to delayed and degraded channel state information (CSI).
Motivated by the above discussions, in this paper, we investigate lowcomplexity energyefficient power allocation strategies for a massive MIMOenabled MWRN with channel aging. ^{Footnote 1} We assume that the CSI is estimated relying on the MMSE criterion, and the relay employs the lowcomplexity linear ZF transceivers. The main contributions of this paper are summarized as follows.

We respectively derive closedform expression of the achievable sum rate (SR) for aged and predicted CSI, which enables us to efficiently evaluate the system performance, thus facilitating the energyefficient power allocation strategies.

Based on the derived closedform expressions, we formulate a unified optimization problem that optimizes power allocation of all UEs for maximizing the system EE, subject to limited transmit power, and minimum qualityofservice (QoS) constraints. Because of the intractable nonconvexity of the formulated optimization problem, the successive convex approximation (SCA) technique is involved to transform the nonconvex problem into a concave fractional programming (CFP) problem, which is then efficiently solved by Dinkelbach’s algorithm and Lagrangian dual method.

Furthermore, to strike a balance between the computational complexity and the optimality, a closedform power control algorithm is provided under the assumption of equal power allocation (EPA) among multiple UEs, without requiring complicated iterative algorithms.

By simulation, the impact of the maximal transmit power, of the QoS constraint, and of the transmit power of each pilot symbol on the optimum EE is quantified. Moreover, our numerical results show that the EPA schemebased power optimization strategies strike an attractive tradeoff between the achievable EE performance and the computational complexity imposed.
The remainder of this paper is organized as follows. The system model is described in Section 3. In Section 4, the closedform expressions for SR are derived under aged and predicted CSI scenarios. In Section 5, we present the energyefficient power allocation optimization problem under different performance criterions and constraints. The power allocation strategies are provided in Section 6, and the simulation results are given in Section 7. Finally, the conclusions are made in Section 8.
Notations: We use uppercase and lowercase boldface letters for denoting matrices and vectors, respectively. (·)^{∗}, (·)^{T}, and (·)^{H} denote the conjugate, transpose, and conjugate transpose, respectively. ·, tr{·}, \(\mathbb {E}[\cdot ]\), Cov(·), and Var[·] stand for the Euclidean norm, the trace of matrices, the expectation, covariation, and variance operators, respectively. The diag{x} denotes a diagonal matrix with the vector x being its diagonal entries, and the operators mod_{N}(x) denote the modulo N of x. [A]_{i,j} represents the entry at the ith row and the jth column of a matrix A. Finally, \(\mathcal {CN}(\mathbf {0},\mathbf {\Theta })\) denotes the circularly symmetric complex Gaussian distribution with zero mean and the covariance matrix Θ.
2 Method
This paper studies the energyefficient power allocation problem of massive MIMOenabled multiway relay systems, under channel aging. The performance of the proposed framework was in depth examined through a series of simulation experiments including different system parameters, whereas the superiority of the proposed approach was clearly demonstrated by comparing it with other research works in the literature. Specifically, it has been shown that the different implementations of the proposed algorithm succeed in providing considerably higher EE in all different system settings while at the same time maintaining QoS at high levels. Moreover, the impact of normalized Doppler shifts f_{D}T_{S} (i.e., channel aging) on the system achievable rate is also illustrated. The simulation code was written in MATLAB.
3 System model and transmission scheme
As shown in Fig. 1, we consider a massive MIMOenabled AF MWRN with nonpairwise ZF transmission,^{Footnote 2} where K spatially distributed singleantenna UEs (UE_{k}, k∈{1,⋯,K}) exchange their informationbearing signals in K time slots among one another via a shared relay (R) equipped with M antennas.^{Footnote 3} Without loss of significant generality, we assume that the number of relay antennas is greater than the number of UEs served at the same timefrequency resources (i.e., M>K). The system operates over a bandwidth of B Hz and the channels are static within the timefrequency coherence blocks composed of T=B_{C}T_{C} data symbols, where B_{C} and T_{C} are the coherence bandwidth and coherence time, respectively. It is assumed that the channel coefficients do not change within onesymbol duration, but vary slowly from symbol to symbol. We assume that the relay operates on the halfduplex TDD mode. Each coherence interval is divided into three time phases, i.e., the CE phase, the multipleaccess and broadcast phases. The multipleaccess phase consists of only one time slot, whereas the broadcast phase contains K−1 time slots.
3.1 Data transmission
In the multipleaccess phase, all K UEs simultaneously transmit their signals x_{U}[ n] to the relay. These signals can be expressed as \(\mathbf {x}_{\mathrm {U}}[\!n]=\mathbf {P}_{\mathrm u}^{1/2}\mathbf {s}[\!n]\in \mathbb {C}^{K\times 1}\), where s[ n]=[s_{1}[ n],…,s_{K}[ n]]^{T} is the informationbearing symbol vector with \(\mathbb {E}\left [\mathbf {s}[\!n]\mathbf {s}^{H}[\!n]\right ]=\mathbf {I}_{K}\) and P_{u}=diag{p_{1},⋯,p_{k},⋯,p_{K}}, p_{k} is the transmit power of the kth UE. The received signal \(\mathbf {y}_{\mathrm {R}}\in \mathbb {C}^{M\times 1}\) at the relay is given by
where \(\mathbf {G}[\!n]\in {\mathbb C}^{M\times K}\) represents the channel matrix from K UEs to the relay and n_{R}[ n] denotes the additive white Gaussian noise (AWGN) that obeys \(\mathcal {CN}\left (\mathbf {0},\sigma _{\mathrm r}^{2}\mathbf {I}_{M}\right)\) at the relay.
To be specific, the channel matrix G[ n] can be expressed as
where \(\mathbf {H}[\!n]\in {\mathbb C}^{M\times K}\) is the smallscale fading (SSF) channel matrix and their entries obey independent identically distributed (i.i.d.) Gaussian distribution as \(\mathcal {CN}\left (0,1\right)\). D is a K×K diagonal matrix with [D]_{k,k}=β_{k}, which models the largescale fading (LSF) capturing both pathloss and shadowing fading effects. Moreover, β_{k} is assumed to remain constant for all n and is assumed to be known a priori as it changes very slowly compared with SSF channel coefficients.
In the broadcast phase, the relay simply performs transceive processing, which firstly detects the signals received and transmitted to all UEs in K−1 subsequent time slots. Here, we consider an intermediate jth (j∈{1,⋯,K−1}) time slot of the broadcast phase for the sake of exposition. In the context, the relay transmitted signal in the jth time slot of the broadcast phase is given by
where F_{j}[ n]=W_{2}[ n]π_{j}W_{1}[ n] is the combined beamforming matrix at the relay, W_{1}[ n] is a ZF detection matrix, and W_{2}[ n] is a ZF precoding matrix. Moreover, π_{j} is the permutation matrix employed at the relay in the jth time slot of the broadcast phase, which is designed to ensure that the kth (k∈{1,⋯,K}) UE, receives the signal from the k^{′}th UE, with k^{′}=mod_{K}(k+j). Specifically, π_{j} is constructed as π_{j}=(π_{o})^{j}, and the K×K primary permutation matrix, π_{o}, can be written as
where e_{k} denotes a column vector of length K with 1 in the kth position and 0 in every other position. 𝜗_{j} is the amplification factor designed to constrain the longterm relay transmit power p_{r}, which is given as follows.
Then, the received signal vector at K UEs in the jth time slot of the broadcast phase can be written as follows.
where n_{u}[ n] is the AWGN vector satisfying \(\mathbf {n}_{\mathrm {u}}[\!n]\sim {\mathcal {CN}}\left (\mathbf {0}, \sigma _{\mathrm {u}}^{2}\mathbf {I}_{K}\right)\). The aforementioned broadcast phase continues until the completion of all K−1 relay transmissions in K−1 consecutive time slots.
Substituting (1) and (3) into (6), the received signal at the kth UE in the jth time slot of the broadcast phase is expressed as
where g_{k}[ n] is the kth column of G[ n] and n_{u,k}[ n] is the kth element of n_{u}[ n].
3.2 Channel estimation
The relay estimates the channel coefficients by transmitting orthogonal pilot sequences. All K UEs simultaneously transmit their pilot sequences of τ_{r}(τ_{r}≥K) symbols to the relay. The received pilot matrix at the relay is given by
where p_{p} is the transmit power of each pilot symbol and \(\mathbf {Z}[\!n]\in {\mathbb {C}}^{M\times \tau _{\mathrm {r}}}\) is a noise matrix whose elements are i.i.d \({\mathcal {CN}}\left (0,\sigma _{\mathrm {r}}^{2}\right)\). \(\boldsymbol {\Phi }\in \mathbb {C}^{K\times \tau _{\mathrm {s}}}\) is the pilot sequence matrix transmitted from K UEs, satisfying ΦΦ^{H}=I_{K}. Correlation of the received signal X[ n] with \(\frac {1}{\sqrt {\tau _{\mathrm {r}} p_{{\mathrm {p}}}}}\boldsymbol {\Phi }^{H}\) obtains
Therefore, the noisy observation of the channel vector from kth UE to the relay is expressed as
where \(\tilde {\mathbf {z}}_{k}[\!n]\) are the kth columns of the matrix \(\widetilde {\mathbf {Z}}[\!n]=\mathbf {Z}[\!n]\boldsymbol {\Phi }^{H}\). Since ΦΦ^{H}=I_{K}, \(\tilde {\mathbf {z}}_{k}[\!n]\sim {\mathcal {CN}}\left (\mathbf {0},\sigma _{\mathrm r}^{2}\mathbf {I}_{M}\right)\).
Exploiting the MMSE criterion [28], the estimate of g_{k}[ n], \(\widehat {\mathbf {g}}_{k}[\!n]\) is distributed as
where \(\widehat {\beta }_{k}=\frac {\tau _{{\mathrm {r}}}p_{{\mathrm {p}}}\beta _{k}^{2}}{\sigma _{\mathrm r}^{2}+{\mathrm \tau _{{\mathrm {r}}}}p_{{\mathrm {p}}}\beta _{k}}\).
Due to the orthogonality property of MMSE estimation, g_{k}[ n] can be decomposed into
where \(\tilde {\mathbf {g}}_{k}[\!n]\sim {\mathcal {CN}}\left (\mathbf {0},\left ({\beta }_{k}\widehat {\beta }_{k}\right)\mathbf {I}_{M}\right)\) is the CE error and is uncorrelated with \(\widehat {\mathbf {g}}_{k}[\!n]\).
3.3 Channel aging
To analyze the impact of channel aging, in our analysis, we adopt an autoregressive model of order 1 for approximating the temporally correlated fading channel coefficient. As such, the channel vectors for the kth UE at time n+1 can be expressed as [25]
where \(\boldsymbol {\epsilon }_{k}[n+1]\sim \mathcal {CN}\left (\mathbf {0},\left (1\alpha ^{2}\right)\beta _{k}\mathbf {I}\right)\) is a temporally uncorrelated complex Gaussian random process. We denote α=J_{0}(2πf_{D}T_{S}) as a temporal correlation parameter, where J_{0}(·) is the zeroorder firstkind Bessel function. T_{S} is the channel sampling duration and \(f_{\mathrm D}=\frac {vf_{c}}{c}\) is the maximum Doppler frequency shift, where v, f_{c}, and c are the UEs’ velocity, carrier frequency, and the speed of light, respectively. Without loss of generality, we assume that all users move with the same velocity. As a result, the time variation does not depend on the user index. While this seems not realistic, we stay very near to the practical case by considering the worstcase scenario where we set all users with the velocity corresponding to the most varying user.
To this end, a model accounting for the combined effects of the CE error and channel aging effect can be expressed as
where \(\boldsymbol {\xi }_{{\mathrm {a}},k}[n+1]\sim {\mathcal {CN}}\left (\mathbf {0},\tilde {\beta }_{{\mathrm a},k}\mathbf {I}_{M}\right)\) is mutually independent of \(\bar {\mathbf {g}}_{{\mathrm {a}},k}[n+1]\sim {\mathcal {CN}}\left (\mathbf {0},\bar {\beta }_{{\mathrm {a}},k}\mathbf {I}_{M}\right)\) with \(\bar {\beta }_{{\mathrm {a}},k}=\alpha ^{2}\widehat {\beta }_{k}\), \(\tilde {\beta }_{{\mathrm a},k}\,=\,\beta _{k}\alpha ^{2}\widehat {\beta }_{k}\). Obviously, the combined error ξ_{a,k}[n+1] consists of both the CE error and aged CSI effects.
3.4 Channel prediction
Channel prediction is an important approach to alleviate the channel aging effect [29–31]. In this subsection, we focus on predicting g_{k}[n+1] based on the current and previous received training signals. The detailed procedure for predicting g_{k}[n+1] is given as follows.
We adopt a Wiener predictor. Then, g_{k}[n+1] is predicted according to \(\bar {\mathbf {x}}_{k}[\!n]\), where
with p being the predictor order. The predicted CSI is provided as follows.
where the optimal pth linear Wiener predictor is given as [31]
Specifically, we have δ(p,α)=[1,α,⋯,α^{p}] and
with
According to [30], the covariance matrix of \(\bar {\mathbf {g}}_{{\mathrm {p}},k}\left [n+1\right ]\) is given by α^{2}Θ_{k}(p,α), where
Thus, the real channel can be decomposed as [31]
where ξ_{p,k}[n+1] is the channel prediction error vector with covariance matrix β_{k}I_{M}−α^{2}Θ_{k}(p,α), which is independent of \(\bar {\mathbf {g}}_{{\mathrm p},k}\left [n+1\right ]\). According to [29], it can be obtained that
According to the result in ([31], Lemma 2), Θ_{k}(p,α) is a scaled identity matrix of size M×M, which can be straightforwardly shown as follows.
with \(\bar {\beta }_{{\mathrm p,}k}\,=\,\frac {1}{M}{\mathrm tr}\left (\alpha ^{2}\boldsymbol {\Theta }_{k}(p,\alpha)\right)\), \(\tilde {\beta }_{{\mathrm p,}k}\,=\,\frac {1}{M} {\text {tr}}\left (\beta _{k}\mathbf {I}_{M} \!\right. \!\left.\alpha ^{2}\boldsymbol {\Theta }_{k}(p,\alpha)\right)\).
4 Performance analysis for achievable sum rate
In this section, we consider two different scenarios, i.e., aged and predicted CSI. We first provide a unified achievable SR expression for two scenarios. Next, we derive closedform expressions for achievable SR under aged and predicted CSI scenarios, which are desirable for the subsequent energyefficient optimization problem formulation.
We assume that the temporal correlation parameter α and the LSF channel matrix D are known a priori at the relay. Hence, we can have the following CSI
Then, the ZFreceive matrix W_{1}[n+1] and ZFtransmit matrix W_{2}[n+1] are respectively expressed as
where \(\bar {\mathbf {G}}[n+1]\triangleq [\bar {\mathbf {g}}_{1}[n+1],\cdots,\bar {\mathbf {g}}_{K}[n+1]]\).
After the imperfect selfinterference cancelation (SIC), the received signal at the kth UE of the jth time slot of the broadcast phase can be rewritten as
where \(\lambda _{k}\,=\,\textbf {g}_{k}^{T}[n+1]\mathbf {F}_{j}[n+1]\textbf {g}_{k}[\!n]\bar {\textbf {g}}_{k}^{T}[n+1]\mathbf {F}_{j}[n+1] \bar {\textbf {g}}_{k}[n+1]\) is the SIC coefficient for the kth UE.
From (26), the ergodic achievable rate of the kth UE in the jth time slot of the broadcast phase can be expressed as
where
with \({\mathrm DS}_{k}^{(j)}\triangleq \vartheta _{j}^{2}p_{k^{\prime }} \left \mathbf {g}_{k}^{T}[n+1]\mathbf {F}_{j}[n+1]\mathbf {g}_{k^{\prime }}[n+1]\right ^{2}, {\mathrm RSI}_{k}^{(j)}\triangleq \vartheta _{j}^{2}p_{k}\lambda _{k}^{2}, {\mathrm NU}_{k}\triangleq \left n_{{\mathrm u},k}[n+1] \right ^{2}, {\mathrm IUI}_{k}^{(j)}\triangleq \vartheta _{j}^{2}p_{i} \left \mathbf {g}_{k}^{T}[n+1]\mathbf {F}_{j}[n+1]\mathbf {g}_{i}[n+1] \right \) and \({\mathrm NR}_{k}^{(j)}\triangleq \vartheta _{j}^{2} \left \mathbf {g}_{k}^{T}[n+1]\mathbf {F}_{j}[n+1]\mathbf {n}_{{\mathrm R}}[n+1] \right \).
However, further derivation of (27) is difficult because of the intractability to carry out the ensemble average analytically. Instead, we adopt another technique to derive a worstcase lower bound of achievable rate. According to [32], we can rewrite \(y_{{\mathrm u},k}^{(j)}[n+1]\) as
with
In (29), the first part \(\vartheta _{j}\sqrt {p_{k^{\prime }}}\mathbb {E} \left [\mathbf {g}_{k}^{T}[n+1]\mathbf {F}_{j}[n+1]\right. \left.\mathbf {g}_{k^{\prime }}[n+1]\right ] s_{k^{\prime }}[n+1]\) is considered as “desired signal,” and the second term \(\tilde {n}_{k}[n+1]\) is considered as “effective noise,” uncorrelated with the first term. Therefore, by approximating the effective noise as independent Gaussian noise of the same variance [32], we can obtain the statistical CSI based achievable rate of the kth UE in the jth time slot of the broadcast phase as
with
where \(\text {SI}_{k}^{(j)}\), \(\text {UI}_{k}^{(j)}\), \(\text {NR}_{k}^{(j)}\), and NU_{k} denote the residual selfinterference after SIC, the interuser interference, the amplified noise from the relay, and the noise at kth UE, respectively, i.e.,
Remark 1
The above worstcase lower bound of achievable rate in (31) is obtained by assuming that UE_{k} uses only statistical information of the channel gains (i.e., \(\mathbb {E}\left [\mathrm {g}_{k}^{T}[n+1]\mathbf {F}_{j}[n+1]\mathbf {g}_{k^{\prime }}[n+1]\right ]\)) to decode the signal transmitted by \(UE_{k^{\prime }}\). By contrast, the ergodic rate in (27) is obtained by a sophisticated receiver, i.e., UE_{k} knows perfectly \(\mathrm {g}_{k}^{T}[n+1]\mathbf {F}_{j}[n+1]\mathbf {g}_{k^{\prime }}[n+1]\). In Section 8, it is demonstrated via simulations that the performance gap between the achievable SRs given by (31) and (27) is rather small in massive MIMOenabled MWRNs. It is clear that (31) is a very useful metric for obtaining the achievable rate in practical applications where CSI is not available.
Accordingly, the statistical CSIbased achievable SR of the considered system with aged and predicted CSI is uniformly given as
In the following theorems, two accurate closedform expressions for the worstcase lower bound of achievable SR are derived under aged and predicted CSI scenarios.
Theorem 1
For ZF transceivers, with aged CSI, the worstcase lower bound of achievable SR in the considered massive MIMOenabled MWRNs is given by
where the closedform formula of \(\mathcal {R}_{\mathrm {a}k}^{(j)}\) is defined as
in which
where k^{′′}=mod_{K}(K+k−j), i^{′′}=mod_{K}(K+i−j), \(\mu _{1}=\frac {1}{MK1}\), \(\mu _{2}=\frac {(2+(MK)(MK3))}{(MK)(MK1)^{2}(MK3)}\) and \(\mathcal {A}_{\mathrm {a}j}={\sum \nolimits }_{k=1}^{K}\bar {\beta }_{\mathrm {a},k}^{1}\bar {\beta }_{\mathrm {a},k^{\prime \prime }}^{1}\).
Proof
Please see Appendix 1. □
For predicted CSI, a closedform expression for the statistical CSIbased achievable SR is derived as follows.
Theorem 2
For ZF transceivers, with predicted CSI, the worstcase lower bound of achievable SR in the considered massive MIMOenabled MWRNs is given by
where \(\widehat {\mathcal {R}}_{\mathrm {p}k}^{(j)}\) is derived as
in which
where \(\mathcal {A}_{\mathrm {p}j}={\sum \nolimits }_{k=1}^{K}\bar {\beta }_{\mathrm {p},k}^{1}\bar {\beta }_{\mathrm {p},k^{\prime \prime }}^{1}\).
Proof
Since the proof follows similar lines as the proof of Theorem 1, it is omitted. □
Remark 2
Through Theorems 1 and 2, two simple closedform expression of the SR of the considered system have been derived. The advantage of these expressions are that it only depends on the LSF channel coefficients and the configurable system parameters. Thus, complicated calculations involving largedimensional matrix variables that represent the SSF channel coefficients are avoided. In this way, the computational complexity which relates to the SSFbased signal processing is greatly reduced. It is underlined that these closedform expressions establish an explicit functional relationship between the SR, the transmit powers of UEs, thus facilitating the introduction of the following novel energyefficient resource allocation methodology.
5 EE optimization problem formulation
In the EE optimization, we employ a realistic power consumption model similar to those used in [11, 33]. The total power consumption of the considered system can be quantified as
where P_{PA} is the power consumed by power amplifiers (PAs) given as
in which η_{PA,U}∈(0,1) and η_{PA,R}∈(0,1) are the efficiency of PAs at the UEs and at the relay, respectively. P_{C} denotes the total circuit power consumption, to be more specific, we have
in which \(P_{\text {CE}}=M\frac {\log _{2}\left (\tau _{\mathrm {r}}\right)R_{\text {flops}}}{\eta _{\mathrm {C}}}\) is the power consumed for CE at the relay, R_{flops} is the floatingpoint operations per second (flops) per antenna for each user, and η_{C} is the power efficiency of computing measured in flops/W. \(P_{\text {LP}}=2M\frac {\left (K+K^{2}\right)R_{\text {flops}}}{\eta _{\mathrm {C}}}\) is the power consumed for the ZFreceive detector and ZFtransmit precoder. P_{RR} is the other baseband processing power (such as ADC/DAC, modulation/demodulation) at each antenna, P_{cR} and P_{cU} are the power consumed at circuit components of each antenna at the relay and each UE, respectively, and P_{f} is the fixed power consumption at the relay.
The power consumption in (41) can be rewritten as
where \(\upsilon _{1}\triangleq \frac {\left (1\frac {\tau _{\mathrm {r}}}{T}\right)}{\eta _{\mathrm {PA,U}}}\), \(P_{\text {fixed}}=\frac {\frac {\tau _{\mathrm {r}}}{T}Kp_{\mathrm {p}}}{\eta _{\mathrm {PA,U}}}+\frac {\left (1\frac {\tau _{\mathrm {r}}}{T}\right)\left (K1\right)p_{\mathrm {r}}}{\eta _{\text {PA},{\mathrm {R}}}}+KP_{\text {cU}}+P_{\text {CE}}+P_{\text {LP}}+M\left (P_{\text {RR}}+P_{\text {cR}}\right)+P_{\mathrm {f}}\).
Given the values of the other system parameters, the EE η_{EE} [bits/Joule] under aged and predicted CSI scenarios is unifiedly defined as
where
with \(\rho _{i,k}^{(j)}\), \(\mu _{k}^{(j)}\) are constant value (independent of transmit powers), which are different for aged CSI and predicted CSI. More precisely,

For aged CSI, \(\rho _{k,i}^{(j)}=p_{\mathrm {r}}\left (\mu _{1}\tilde {\beta }_{\mathrm {a},i}\bar {\beta }_{\mathrm {a},k^{\prime }}^{1}+\mu _{1}\tilde {\beta }_{\mathrm {a},k}\bar {\beta }_{\mathrm {a},i^{\prime \prime }}^{1}+\right. \left.\mu _{2}\tilde {\beta }_{\mathrm {a},i}\tilde {\beta }_{\mathrm {a},k}\mathcal {A}_{\mathrm {a}j}\right)+ \sigma _{\mathrm {u}}^{2}\left (\mu _{1}\bar {\beta }_{\mathrm {a},i^{\prime \prime }}^{1}+\mu _{2}\tilde {\beta }_{\mathrm {a},i}\mathcal {A}_{\mathrm {a}j}\right)\), \(\mu _{k}^{(j)}=p_{\mathrm {r}}\left (\mu _{1}\sigma _{\mathrm {r}}^{2}\bar {\beta }_{\mathrm {a},k^{\prime }}^{1}+\mu _{2}\sigma _{\mathrm {r}}^{2}\tilde {\beta }_{\mathrm {a},k}\mathcal {A}_{\mathrm {a}j}\right)+ \mu _{2}\sigma _{\mathrm {u}}^{2}\sigma _{\mathrm {r}}^{2}\mathcal {A}_{\mathrm {a}j}\).

For predicted CSI, \(\rho _{k,i}^{(j)}\,=\,p_{\mathrm {r}}\left (\mu _{1}\tilde {\beta }_{\mathrm {p},i}\bar {\beta }_{\mathrm {p},k^{\prime }}^{1}+\mu _{1}\tilde {\beta }_{\mathrm {p},k}\bar {\beta }_{\mathrm {p},i^{\prime \prime }}^{1}+\right. \left.\mu _{2}\tilde {\beta }_{\mathrm {p},i}\tilde {\beta }_{\mathrm {p},k}\mathcal {A}_{\mathrm {p}j}\right)+ \sigma _{\mathrm {u}}^{2}\left (\mu _{1}\bar {\beta }_{\mathrm {p},i^{\prime \prime }}^{1}+\mu _{2}\tilde {\beta }_{\mathrm {p},i}\mathcal {A}_{\mathrm {p}j}\right)\), \(\mu _{k}^{(j)}=p_{\mathrm {r}}\left (\mu _{1}\sigma _{\mathrm {r}}^{2}\bar {\beta }_{\mathrm {p},k^{\prime }}^{1}+\mu _{2}\sigma _{\mathrm {r}}^{2}\tilde {\beta }_{\mathrm {p},k}\mathcal {A}_{\mathrm {p}j}\right)+\mu _{2}\sigma _{\mathrm {u}}^{2}\sigma _{\mathrm {r}}^{2}\mathcal {A}_{\mathrm {p}j}\).
Remark 3
In (45), the prelog factor \(\left (1\frac {\tau _{\mathrm {r}}}{T}\right)\) is due to the fact that during each coherence interval of T symbols, we spend τ_{r} symbols for pilotbased CE. Moreover, the numerator K−1 of the prelog factor \(\frac {K1}{K}\) is due to the fact that any user node receives signals from other K−1 user nodes, while the denominator K follows by the single time slot in the multipleaccess phase and K−1 time slots used for fulldata exchange in the broadcast phase.
It is seen from (45) that the EE η_{EE} is a function of the transmit powers of K UEs, \(\left \{p_{k}\right \}_{k=1}^{K}\). How to wisely allocate the transmit power among the K UEs is crucial for achieving the optimum EE in the context of green communications. Hence, the energyefficient power allocation is formulated as the following optimization problem:
where the objective function η_{EE} is defined by (45), and p_{max} is the maximum transmit power of each UE. The constraints C1 and C2 are the boundary values for the transmit powers of K UEs. The constraint C3 guarantees the transmission link quality by satisfying the minimum QoS requirement R_{0} for each UE at each time slot of the broadcast phase. Here, R_{0} denotes the required achievable rate for all UEs.
6 Energyefficient power allocation algorithm
6.1 Optimal power allocation (OPA) scheme
It is easy to observe that (47) is not a CFP optimization problem, because the numerator of the objection function η_{EE} and the QoS constraints C3 are nonconvex with respect to \(\left \{p_{k}\right \}_{k=1}^{K}\). Therefore, (47) cannot be directly solved by classic fractional programming tools. To overcome this difficulty, we employ the SCA technique proposed in [34–36] to sequently approximate \({\mathcal {R}}_{k}^{(j)}\) by using the following inequality:
The above inequation is tight at a particular value \(z_{k,j}=\bar {z}_{k,j}\) when the approximation constants a_{k,j} and b_{k,j} are chosen as
Motivated by the above convexity approximation, we employ the inequality (48) to approximate \(\widehat {\mathcal {R}}_{k}^{(j)}\), where z_{k,j} corresponds to \(\phantom {\dot {i}\!}\widehat {\gamma }_{k,j}=\frac {p_{\mathrm {r}}p_{k^{\prime }}}{{\sum \nolimits }_{i=1}^{K}p_{i}\rho _{k,i}^{(j)}+\mu _{k}^{j}}\). Then, the variable change \(p_{k}=2^{q_{k}}\phantom {\dot {i}\!}\), for ∀k was used. Finally, we arrive at the following approximated optimization problem
where Q=diag{q_{1}⋯,q_{k}⋯,q_{K}}, \(P_{\text {tot}}\left (\mathbf {Q}\right)=\upsilon _{1}\sum _{k=1}^{K}2^{q_{k}}+P_{\text {fixed}}\) and
with \(a_{k,j}=\frac {\bar {\gamma }_{k}^{(j)}}{1+\bar {\gamma }_{k}^{(j)}}\) and \(b_{k,j}=\log _{2}\left (1+\bar {\gamma }_{k}^{(j)}\right)\frac {\bar {\gamma }_{k}^{(j)}}{1+\bar {\gamma }_{k}^{(j)}} \log _{2}\bar {\gamma }_{k}^{(j)}\) being the approximation constants computed as (49), where \(\bar {\gamma }_{k}^{(j)}=\frac {p_{\mathrm {r}}2^{q_{k^{\prime }}}}{{\sum \nolimits }_{i=1}^{K}2^{q_{i}}\rho _{k,i}^{(j)}+\mu _{k}^{j}}\). For any fixed a_{k,j} and b_{k,j}, it can be easily verified that (51) is convex with respect to \(\left \{ q_{k}\right \}_{k=1}^{K}\).^{Footnote 4} Therefore, the optimization problem (50) is a CFP problem with a quasiconcave objective function \(\widetilde {\eta }_{\text {EE}}(\mathbf {Q})\)^{Footnote 5} and convex constraints, which can be transformed into a convex optimization in a subtractive by the Dinkelbach’s method as follows [37].
where
Here, λ is a nonnegative parameter, it can be noted that when λ→0, it implies that the energyefficient problem (52) is degenerated to an optimization problem for the SE maximization. The optimal factor λ^{∗} (i.e., the optimal objective function value of (50)) works as the optimal EE for the system. For fixed parameters a_{k,j}, b_{k,j}, and λ, the optimization problem (52) is a convex optimization problem, which can be efficiently solved using standard convex optimization tools, e.g., CVX [38]. Next, we derive an iterative algorithm for solving this optimization by applying the Lagrangian dual method.
Thus, the dual problem associated with the primal problem (52) can be written as
where μ={μ_{k}},∀k are the Lagrangian multipliers associated with the transmit power constraints C2^{′}. while ψ={ψ_{k,j}},∀k,j are the Lagrangian multipliers for QoS constraints C3^{′}.
In the following, we solve the dual problem (54) using Lagrangian dual approach, which alternates between a subproblem (inner problem), updating the power allocation variables Q by fixing the Lagrangian multipliers μ, ψ, and a master problem (outer problem), updating the Lagrangian multipliers μ, ψ for the obtained solution of the inner problem Q^{∗}. The Lagrangian dual approach is outlined as follows.
The optimization problem (54) is in a standard concave form, which can be efficiently solved by using standard optimization techniques and KKT conditions [38]. Thus, to obtain the optimal power allocation for users, we take the partial derivative of (54) with q_{k}, k=1,⋯,K, and equate the results to zero, thus the power allocation at the (m+1)th iteration is updated as follows.
where [x]^{+}= max{0,x}.
Since the dual problem in (54) is differentiable, the gradient method may be readily used for updating the Lagrangian dual variables μ_{k} and ψ_{k,j},∀k,j as follows [39].
where ε_{1}(m) and ε_{2}(m) are the step sizes used for moving in the direction of the negative gradient for the Lagrangian multipliers μ_{k} and ψ_{k,j}, respectively. The updated Lagrange multipliers are used for updating the power allocation policy. We repeat this process until convergence. The detailed iterative procedure is summarized in Algorithm 1.
To get a better insight into the computational complexity of our proposed algorithm, we perform an exhaustive complexity analysis. First, it is assumed that the network factor λ converges in W iterations. The optimization problem (52) consists of K×(K−1) subproblems due to K UEs operating on K−1 effective time slots. Besides, the computational complexity resulted by these constraints C1^{′}−C3^{′} is \({\mathcal {O}}\left (V^{3}+2\right)\), where V denotes each UE’s power level. Furthermore, the computational complexity of updating Lagrangian dual variables is given as \({\mathcal {O}}\left (K^{\varpi }\right)\) (for example ϖ=2 if the ellipsoid method is used [40]). Let us suppose if the dual objective function (54) converges in \(\mathcal {G}\) iterations, then the total complexity for the proposed OPA scheme becomes \({\mathcal {O}}\left (2W{\mathcal {G}}\left (K1\right)(K)^{\varpi +2}\left (V^{3}+2\right)\right)\).
6.2 Equal power allocation (EPA) scheme
To strike a balance between the computational complexity and the optimality, we propose another lowercomplexity power allocation scheme in this paper, i.e., an EPA scheme among all UEs. The samelevel transmit powers of K UEs is set as p_{k}=p_{u}, for ∀k; then, the optimization problem (47) under the EPA scheme is simplified as
where
To solve the above optimization problem (58), we firstly find the feasible region of p_{u} and then find the global extrema values. The detailed steps are shown as follows.
Firstly, we solve the Eq. \({\overline {\mathcal {R}}}_{k}^{(j)}\left (p_{\mathrm {u}}\right)={R}_{0}\) and get the solution \(p_{\mathrm {u},k,j}^{*}\) for ∀k,j. It can be easily determined that \({\overline {\mathcal {R}}}_{k}^{(j)}\left (p_{\mathrm {u}}\right)\) is a monotonically increasing function for p_{u}. Hence, the QoS constraints of (58) can be reset as \(p_{\mathrm {u}}\geq p_{\mathrm {u},k,j}^{*}\) for ∀k,j, i.e.,
where \(p_{\mathrm {u,\max }}^{*}=\max \left \{p_{\mathrm {u,1,1}}^{*}\cdots,p_{\mathrm {u,}k,j}^{*},\cdots p_{\mathrm {u,}K,K1}^{*}\right \} \). Considering both C5 and (61), the feasible region of p_{u} for (58) becomes \(\left [p_{\mathrm {u},\max }^{*},p_{\max }\right ]\). If pu,max∗>p_{max}, the optimization problem becomes infeasible. Namely, there is no solution of p_{u} satisfying the QoS constraints, so the algorithm should adjust p_{max}. If pu,max∗<p_{max}, (58) is feasible on \(\left [p_{\mathrm {u},\max }^{*},p_{\max }\right ]\).
Once feasible, we can find the global maximum of \(\overline {\eta }_{{\text {EE}}}\left (p_{{\mathrm {u}}}\right)\) in \(\left [p_{\mathrm {u},\max }^{*},p_{\max }\right ]\). To be more specific, it can be readily proved that \(\overline {\eta }_{{\text {EE}}}\left (p_{{\mathrm {u}}}\right)\) is quasiconcave in p_{u} and therefore has a unique stationary point \(\overline {p}_{\mathrm {u}}\), which coincides with its global maximizer and can be found from the firstorder derivative (i.e., \(\frac {\partial {\overline \eta }_{{\text {EE}}}\left (\overline {p}_{{\mathrm {u}}}\right)}{\partial \overline {p}_{\mathrm {u}}}=0\)). Then, since \(\overline {\eta }_{{\text {EE}}}\left (p_{{\mathrm {u}}}\right)\) is strictly increasing for \(p_{\mathrm {u}}\leq \overline {p}_{\mathrm {u}}\) and strictly decreasing for \(p_{\mathrm {u}}> \overline {p}_{\mathrm u}\). Therefore, the solution of (58), \(p_{\mathrm {u}}^{*}\) is obtained as follows.
Remark 4
Compared to the optimization problem (47), where K variables \(\left \{p_{k}\right \}_{k=1}^{K}\) are optimized, the new formulation (58) only uses p_{u} as the optimization variable. Hence, the computational complexity of the EPA scheme in (58) is significantly lower than that of the OPA scheme solving (47). Moreover, the OPA scheme based (47) is solved by the iterative Dinkelbach’s algorithm and Lagrangian dual method, which requires a complexity of \({\mathcal {O}}\left (2W{\mathcal {G}}\left (K1\right)(K)^{\varpi +2}\left (V^{3}+2\right)\right)\), the EPA scheme based (58) can obtain a closedform optimal solution by comparing extreme values and boundary values of the optimization problem (58), without iteration. In contrast to OPA scheme, the computational complexity of the EPA scheme can be negligible.
7 Numerical results
In this section, we evaluate the EE performance of the considered massive MIMOenabled MWRN that uses the proposed energyefficient power allocation strategies, and demonstrate the accuracy of our analytical results as well as the impacts of several relevant parameters on the optimum EE via numerical simulations. Several key simulation parameters are set as Table 1 [11, 33]. Assume that the relay coverage area is modeled as a disc and the relay is located at the geometric center of the disc. Furthermore, all UEs are assumed to be randomly and uniformly distributed in the circular cell with a radius R, we assume that no UE is closer to the relay than R_{min}, and the lognormal shadowing \(\xi _{k}\sim \ln {\mathcal {N}}\left (0,\sigma _{k}^{2}\right)\).
7.1 Accuracy of analytical results
In this subsection, we evaluate the accuracy of analytical results given in (35) with aged CSI, as well as in (38) with predicted CSI for different f_{D}T_{S} and p. We use normalized Doppler shifts f_{D}T_{S} to characterize channel aging. Larger normalized Doppler shifts correspond to large CSI delays (i.e., the more serious channel aging effect). We choose \(\sigma _{\mathrm {r}}^{2}=\sigma _{\mathrm {u}}^{2}=1\) and τ_{r}=K. For the clarity of analysis, we assume that the EPA scheme used at UEs is considered, i.e., p_{k}=p_{u}. All the simulated values are obtained by averaging over 10^{6} independent Monte Carlo channel realizations.
Figure 2 shows the system’s achievable SR versus the number of antennas at the relay M for different normalized Doppler shifts f_{D}T_{S}. It can be clearly seen from Fig. 2 that the relative performance gaps between the analytical results (35) (marked as Analytical) and the simulated values (27) (marked as Simulated) are very small, which demonstrates analytical results’ accuracy. In addition, we can see a intuitive result that channel aging degrades the system’s achievable SR. Again, it is noted that increasing the number of relay antennas M improves the system’s achievable SR, as expected. This observation also implies that, when f_{D}T_{S} is relatively large, the contribution of the increasing of M diminishes quickly.
We now investigate the benefits of channel prediction on the achievable SR in Fig. 3. As can be readily, our analytical results (38) are in perfect agreement with the simulated curves (27), demonstrating the accuracy of analytical results. In addition, it is noted that, as the normalized Doppler shift f_{D}T_{S} becomes large, the achievable SR loss increases significantly. Apparently, when the channel prediction order grows large, the achievable SR gain improves considerably. We also observe that, when the channel aging effect is less severe (i.e., f_{D}T_{S} is small), channel prediction becomes more important. Finally, it can be observed that, the predicted CSI case achieves a higher SR than the current CSI (no channel aging) case when f_{D}T_{S} is small, while its performance degrades substantially when f_{D}T_{S} is large and becomes worse than that with the current CSI case.
7.2 Optimality of the proposed optimization strategy
In Fig. 4, we show the convergence behavior of the proposed power allocation strategies (including both the OPA and the EPA schemes) under different channel prediction orders p. It can be observed that the EEs of the OPA scheme (by solving (47)) are monotonically increased with the iteration number, then converge to the optimal EE value after only a few iterations. In addition, in order to further demonstrate the effectiveness of the proposed schemes, a performance comparison is given with other algorithm (i.e., CharnesCooper transformation (CCT)based method) for power allocation in [41]. From Fig. 4, we can observe that the OPA scheme with the CCTbased method is slightly superior to the proposed OPA scheme with lower iterations, but the CCTbased method involves perspective transformations, which increases the computational complexity.
At the same time, the lowercomplexity EPA schemes for solving (58) achieve nearoptimum performances, dispensing with any iteration. Moreover, it can be observed that the obtained EE performances of the EPA schemes are slightly worse those of the OPA schemes. Finally, in order to valid the accuracy of the derived lower bound and the optimality of the proposed power allocation strategies, in Fig. 4, we provide a performance benchmark that correspond to solving the problem (47) via the highcomplexity bruteforce searching relying on the ergodic achievable SR in (27). From Fig. 4, we can see that the EEs of the proposed methods are slightly inferior to the benchmarks, and this is mainly because the proposed schemes are suboptimal methods which involve iterations and convex approximation. When using OPA scheme, K variables \(\left \{p_{k}\right \}_{k=1}^{K}\) must be optimized. By contrast, with EPA scheme, we only need to optimize the single variable p_{u}. Furthermore, the OPA scheme obtains the optimal power allocation solution in virtue of the complicated iterative Dinkelbach’s algorithm and Lagrangian dual method. The EPA scheme can obtain a closedform optimal solution by only comparing extreme values and boundary values of the optimization problem (58), without iteration. Hence, compared with the the OPA scheme, the computational complexity of the EPA scheme is significantly reduced. Therefore, the EPA scheme is a good choice in terms of the tradeoff between the achievable EE performance and the computational complexity. Finally, numerical results also reveal that higher prediction order can obtain the improvement of EE performance.
Figure 5 illustrates the optimum EE achieved by the proposed power allocation strategies versus the transmit power constraint p_{max}. It is observed that the OPA scheme slightly outperforms the EPA scheme in terms of the optimum EE achieved. Furthermore, we can see that, when p_{max}≤26 dBm, the optimum EEs achieved by these proposed schemes can be substantially improved as p_{max} increases. This observation suggests that at this region [10,26]dBm, increasing the available power budget is an energyefficient choice. However, when p_{max}≥26 dBm, the optimum EEs of the proposed power allocation scheme converge to a certain stable level. This important observation suggests that, when p_{max} is large enough, the increasing of transmit power may not be a good choice from the perspective of EE. Finally, it is observed that the smaller f_{S}T_{D} achieves a higher EE for either power allocation strategy. This is rather expected, since the smaller f_{S}T_{D} means the less serious channel aging effect, and the EE loss becomes smaller accordingly.
Figure 6 illustrates the impact of the QoS threshold R_{0} on the optimum EEs achieved by the proposed power allocation schemes. It can be readily noted that when R_{0}≤ 4 bit/s/Hz, each optimum EE remains unchanged. This happens because when R_{0} takes small values, it is easy to satisfy the link’s QoS requirement. This observation suggests that, at the low QoS requirement region R_{0}≤ 4 bit/s/Hz, we can make the best use of all the available power to achieve the maximum EE, without having to waste more power on unfavorable links. Meanwhile, when R_{0}≥ 4 bit/s/Hz, the optimum EEs decrease as R_{0} increases. This is due to the fact that when R_{0} increases, an excess fraction of power has to be allocated to compensate for disadvantageous links, which results in a degradation of optimum EE. In other works, a higher minimum rate R_{0} is satisfied at the expense of a reduction of the optimum EE.
In Fig. 7, we show the impact of the transmit power of each pilot symbol p_{p} on the optimum EEs achieved by the proposed power allocation schemes. From these results and as it was expected, it can readily be observed that the the optimum EEs of all schemes increases with increasing p_{p}. Moreover, as p_{p} grows large, the growth of the achievable EE gradually slows down and saturates to the value that relies on perfect CSI estimation. This implies that although the system with high transmit power of each pilot symbol (i.e., p_{p}=50 dBm) is capable of improving the CE accuracy, then achieves a better EE performance, the extremely high CE accuracy is not a wise choice at the cost of consuming more power.
8 Conclusions
In this paper, we have provided the performance analysis of the system’s achievable SR and proposed lowcomplexity power allocation strategies for maximizing the EE of a massive MIMOenabled MWRN with channel aging. Specifically, we derived closedform expressions for the system’s achievable SR with/without channel prediction. Based on the derived analytical results, a unified power allocation optimization problem is established, under the transmit power and QoS constraints. Owing to the nonconvexity of the objective function and QoS constraints, the original nonconvex problem is sequently approximated as a solvable CFP problem with the aid of the SCA technique, which can be efficiently solved by the Dinkelbach’s algorithm and Lagrangian dual method. Moreover, we have proposed a closedform power control algorithm for the lowercomplexity EPA scheme. The impacts of normalized Doppler shifts f_{D}T_{S}, channel prediction order, and other relevant system parameters on the SR and EE performance are investigated via numerical simulations, which have verified the accuracy of our analytical results, and confirmed the effectiveness of the proposed power allocation schemes.
9 Appendix 1
10 Proof of Theorem 1
With aged CSI, \(\bar {\mathbf {g}}_{k}\left [n+1\right ]\sim {\mathcal {C}N}\left (\mathbf {0},\bar {\beta }_{{\mathrm a},k}\mathbf {I}_{K}\right)\) is independent of \({\boldsymbol \xi }_{{\mathrm a},k}\left [n+1\right ]\sim {\mathcal {C}N}\left (\mathbf {0},\tilde {\beta }_{{\mathrm a},k}\mathbf {I}_{K}\right)\).
where
with \({\widetilde {\mathbf {G}}}[n+1]=\left [{\boldsymbol {\xi }}_{{\mathrm {a}},1}[n+1],\cdots,{\boldsymbol {\xi }}_{{\mathrm {a}},K}[n+1]\right ]\).

Compute \(\varPsi _{1}^{(j)}\): According to the definition of F_{j}[n+1], we have
where \(\mu _{1}=\frac {1}{MK1}\) and k^{′′}=mod_{K}(K+k−j). (a) results from the property Tr{AB}=Tr{BA}. As to the detailed derivation of (b), we use the identity as follows [44]: \(\boldsymbol {\Omega }\triangleq \left (\bar {{\mathbf {G}}}^{H}[n+1]\bar {\mathbf {G}}[n+1]\right)^{1}\) is an inverted Wishart matrix, i.e., \({\boldsymbol {\Omega }}\sim {\mathcal {W}}_{K}^{1}\left (M+K+1,\widehat {\mathbf {D}}_{{\mathrm {a}}}^{1}\right)\)with \(\widehat {\mathbf {D}}_{{\mathrm {a}}}^{1}={\text {diag}}\left \{ \bar {\beta }_{{\mathrm {a}},1}^{1},\cdots,\bar {\beta }_{{\mathrm {a}},K}^{1}\right \} \). Hence, we have [45]

Compute \(\varPsi _{2}^{(j)}\): Since \(\bar {\mathbf {G}}[n+1]\) and \(\widetilde {\mathbf {G}}[n+1]\) are independent, we
obtain
in which
where m^{′′}=mod_{K}(K+m−j), \(\mu _{2}=\frac {(2+(MK)(MK3))}{(MK)(MK1)^{2}(MK3)}\) and \({\mathcal {A}}_{{\mathrm {a}} j}={\sum \nolimits }_{k=1}^{K}\bar {\beta }_{{\mathrm {a}},k}^{1}\bar {\beta }_{{\mathrm {a}},k^{\prime \prime }}^{1}\). The detailed derivation of (c) is given as follows [45]:
and for m≠k
According to (71) and (68), we get

Compute \(\varPsi _{3}^{(j)}\): Similarly, according to (68), we can obtain
Substituting (65), (71), and (72) into (8), we have

Derive \(\widehat {\mathcal {R}}_{k}^{(j)}\): From (31), we need to compute \({\mathbb E}\left [{\mathbf {g}}_{k}^{T}[n+1]\mathbf {F}_{j}[n+1]{\mathbf {g}}_{k^{\prime }}[n+1]\right ]\), \({\text {SI}}_{k}^{(j)}\), \({\text {UI}}_{k}^{(j)}\), \({\text {NR}}_{k}^{(j)}\), NU_{k}, and \({\text {Var}}\left [{\mathbf {g}}_{k}^{T}[n+1]\mathbf {F}_{j}[n+1]\textbf {g}_{k^{\prime }}[n+1]\right ]\).

Compute \({\mathbb E}\left [{\mathbf {g}}_{k}^{T}[n+1]\mathbf {F}_{j}[n+1]{\mathbf {g}}_{k^{\prime }}[n+1]\right ]\): We have
with
where (d) results from the independence between \(\bar {\mathbf {g}}_{i}[n+1]\) and ξ_{a,j}[n+1] for ∀i,j. Hence, we have

Compute \({\mathrm Var}\left [\boldsymbol {{\mathrm {g}}}_{k}^{T}[n+1]\mathbf {F}_{j}[n+1]\mathbf {g}_{k^{\prime }}[n+1]\right ]\): According to the definition of variance, we have
where \(\varsigma _{k,j}=\mathbf {g}_{k}^{T}[n+1]\mathbf {F}_{j}[n+1]\mathbf {g}_{k^{\prime }}[n+1]\), \(\mathbb {E}\left [\left \varsigma _{k,j}\right ^{2}\right ]\) can be decomposed into the following parts:
with
According to (77)–(79), we obtain

Compute \({\text {SI}}_{k}^{(j)}\): We have
where (e) results from the property \(\bar {\mathbf {g}}_{k}^{T}[n+1] \mathbf {F}_{j}[n+1] \bar {\mathbf {g}}_{k}[n+1]={\boldsymbol {e}}_{k}^{T}{\boldsymbol {\pi }}_{j}{\boldsymbol {e}}_{k}=0\), and
Substituting (82) into (81), we obtain

Compute \({\text {UI}}_{k}^{(j)}\): Following the same methodology used for computing \(\mathbb {E}\left [\lambda _{k}^{2}\right ]\), we can easily derive
with
where i^{′′}=mod_{K}(K+i−j).
According to (84) and (85), we can obtain
Therefore, we can conclude

Compute \({\mathrm NR}_{k}^{(j)}\): We have
with
Substituting (90) into (89), we can obtain

Compute NU_{k}: We have
Substituting (73), (76), (80), (83), (88), (91), and (92) into (32), we can obtain (35) after some simple algebraic manipulations. Thus, the proof of Theorem 1 is complete.
Notes
Although we respectively studied the resource allocation for EE maximization in the massive MIMOenabled OWRNs and MWRNS in [42, 43], the works in [42, 43] only considered the channel estimation (CE) error and ignored the effect of channel aging. Contrast to the transmission schemes proposed in [42, 43], the performance analysis and power allocation algorithms in this paper have stronger robustness over the practical communication scenario.
This setup is general enough to model a variety of communication scenarios. Certain practical applications such as multimedia teleconferencing via a satellite or mutual data exchange between sensor nodes and the data fusion center in wireless sensor networks require mutual data exchange among more than just two terminals.
In this paper, for simplicity and tractability, we assumed that the direct link between any two UEs is ignored due to severe path loss. This assumption has been widely made in multiway relay systems [13, 14], and it is easily extended to complex system models with a direct communication link between two UEs.
The logsumexp is convex [38].
The numerator is concave, and the denominator is convex [38].
Abbreviations
 5G:

Fifthgeneration
 AWGN:

Additive white Gaussian noise
 BS:

Base station
 CCT:

CharnesCooper transformation
 CE:

Channel estimation
 CFP:

Concave fractional programming
 CSI:

Channel state information
 EE:

Energy efficiency
 EPA:

Equal power allocation
 LSF:

Largescale fading
 MIMO:

Multipleinput multipleoutput
 MMSE:

Minimum mean square error
 MWRN:

Multiway relay network
 OPA:

Optimal power allocation
 OWRN:

Oneway relay network
 QoS:

Qualityofservice
 SCA:

Successive convex approximation
 SE:

Spectral efficiency
 SINR:

Signaltointerferenceplusnoise ratio
 SR:

Sum rate
 SSF:

Smallscale fading
 TWRN:

Twoway relay network
 UE:

User equipment
 ZF:

Zeroforcing
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Funding
This work was supported by the National Natural Science Foundation of China (61471135, 61671165); the Guangxi Natural Science Foundation (2016GXNSFGA380009); the PhD Research Startup Fund of Guilin University of Electronic Technology (UF17048Y); the Fund of Key Laboratory of Cognitive Radio and Information Processing (Guilin University of Electronic Technology), China; and the Guangxi Key Laboratory of Wireless Wideband Communication and Signal Processing (CRKL160105, CRKL170101).
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FT was responsible for mathematical derivation, numerical simulation, and paper writing. HC was responsible for problem formulation and paper revision. FZ was responsible for problem and result discussion. XL was responsible for model validation and result check. All authors read and approved the final manuscript.
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Fangqing Tan received the M.Eng. degree in communication and information systems from Chongqing University of Post and Telecommunications, China, in 2012, and the Ph.D. degree in communication and information systems from Beijing University of Posts and Telecommunications, China, in 2017. He is currently a lecturer with the School of Information and Communication, Guilin University of Electronic Technology, Guilin, China. His research interests include massive MIMO systems, cooperative communications, and energyefficient wireless communications. Hongbin Chen received the B.Eng. degree in electronic and information engineering from Nanjing University of Posts and Telecommunications, Nanjing, China, in 2004 and the Ph.D. degree in circuits and systems from South China University of Technology, Guangzhou, China, in 2009. From October 2006 to May 2008, he was a Research Assistant with the Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hong Kong. From March to April 2014, he was a Research Associate with the same department. From May 2015 to May 2016, he was a Visiting Scholar with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. He is currently a Professor with the School of Information and Communication, Guilin University of Electronic Technology, Guilin, China. His research interests include energyefficient wireless communications. Feng Zhao received the Ph.D. degree in communication and information systems from Shandong University, China in 2007. Now, he is a Professor with the School of Information and Communication, Guilin University of Electronic Technology, China. His research interests include wireless communications, signal processing, and information security. Xiaohuan Li received the B.Eng. and M.Sc. degrees from Guilin University of Electronic Technology, China, in 2006 and 2009, respectively, and the Ph.D. degree from South China University of Technology, China, in 2015. He is currently an Associate Professor with the School of Information and Communication, Guilin University of Electronic Technology, China. His research interests include vehicular ad hoc networks.
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Tan, F., Chen, H., Zhao, F. et al. Energyefficient power allocation for massive MIMOenabled multiway AF relay networks with channel aging. J Wireless Com Network 2018, 206 (2018). https://doi.org/10.1186/s1363801812222
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DOI: https://doi.org/10.1186/s1363801812222