 Research
 Open Access
On the energy/spectral efficiency of multiuser fullduplex massive MIMO systems with power control
 Ximing Wang^{1},
 Dongmei Zhang^{1},
 Kui Xu^{1}Email author and
 Wenfeng Ma^{1}
https://doi.org/10.1186/s1363801708649
© The Author(s) 2017
 Received: 18 November 2016
 Accepted: 11 April 2017
 Published: 3 May 2017
Abstract
In this paper, we consider a multiuser fullduplex (FD) massive multipleinput multipleoutput (MIMO) system. The FD base station (BS) is equipped with largescale antenna arrays, while each FD user is equipped with two antennas (one for transmission and the other one for reception). We assume that the channel state information (CSI) is imperfect, and no instantaneous CSI of loop interference (LI) channel is obtained. On the basis of lowcomplexity uplink beamforming and downlink precoding techniques, i.e. maximumratio combining/maximumratio transmission (MRC/MRT) and zeroforcing reception/zeroforcing transmission (ZFR/ZFT), the asymptotic expressions of signaltointerferenceplusnoise ratio (SINR) of uplink and downlink are derived respectively when the number of antennas of the BS tends to infinity. Based on the asymptotic SINR expressions, we first analyse the spectral efficiency (SE) performance assuming that the generalized power scaling scenario is adopted. Through the theoretical derivation, it is shown that the detrimental impact of the LI can be eliminated by the very large number of antennas at the BS if the power scaling scenario is appropriately applied, as well as the interference caused by the imperfect channel estimation, the multiuser interference (MUI) and interuser interference (IUI). Then, we propose power allocation (PA) scenarios to maximize the energy efficiency (EE) and the sum SE with the constraint of maximum powers at the BS and users. The simulation results verify the accuracy of the asymptotic method, we also validate the effectiveness of the EE and the sum SE maximization PA algorithms. We show that adopting the PA scheme to maximize the sum SE can make the multiuser FD massive MIMO system outperform the halfduplex (HD) counterpart regardless of the level of LI.
Keywords
 Multiuser
 Full duplex
 Imperfect channel state information (CSI)
 Maximumratio combining/maximumratio transmission (MRC/MRT)
 Zeroforcing reception/zeroforcing transmission (ZFR/ZFT)
 Power scaling
 Power allocation
1 Introduction
Now, the fourth generation (4G) wireless system has come to the mature stage, and with the booming development of all kinds of mobile smart terminals, the mobile Internet traffic in the recent years grows exponentially, which will increase by 1000 times beyond 2020. Meanwhile, the ratio of energy consumption in information technology system has been rapidly increasing, making it an urgent mission to decrease the energy consumption of the mobile communication systems. However, the current 4G is unable to satisfy the continuously increasing demands in the future mobile communication, which put forward a great challenge to the fifth generation (5G) wireless systems [1].
Adopting multipleinput multipleoutput (MIMO) techniques is an essential way to utilize the wireless spatial resources and improve the spectral efficiency (SE) and energy efficiency (EE). Massive MIMO has been deemed to be one key technology for 5G, which uses largescale antenna arrays (with orders of magnitude more antennas, e.g., 100 or more) to replace the currently adopted multiple antennas [2]. As the number of antennas grows larger, the simple linear precoder and detector tend to optimal, and the noise along with the uncorrelated interference can be ignored [3]. What is more, the spatial resolution of massive MIMO is remarkably enhanced compared with the conventional MIMO and can focus the signal beam within a narrow range; thus, the interference can be reduced by a wide margin [4], which as a result can significantly improve SE and EE [5].
On the parallel avenue, full duplex (FD) is another potential key technology for 5G. FD communication is the technology where signals are transmitted and received at the same timefrequency resource. In the conventional halfduplex (HD) systems, bidirection link is separated by orthogonal time or frequency resources (the corresponding systems are timedivision duplex (TDD) systems and frequencydivision duplex (FDD) systems), where theoretically wastes half of the timefrequency resource. Nevertheless, in FD systems, due to the large amplitude difference between received signals and transmitted signals, signals leaking from the output to the input lead to severe loop interference (LI), typically 100 dB [6]. As a result, the primary task to apply FD technique is LI cancellation. With the development of signal processing techniques and hardware processing techniques, various kinds of LI cancellation techniques have been developed, including analogdomain LI cancellation [7], the cancellation of known signals in digital domain, or the combination of them [8, 9]. Because of the considerable potential of theoretically doubling the SE and making the usage of spectrum more flexible, FD technique has become a hotspot of research at present. For example, [10] analyses the complete achievable rate regions of a smallarea radio system where a multipleantennas FD access point serves two multipleantennas HD devices, and the analytical results characterize the effect of selfinterference at the access point and interdevice interference on the achievable rate regions. In [11], the authors consider the relay selection schemes for FD relaying protocol, over Nakagamim fading channels, they show that the proposed schemes achieve better outage performance when compared with traditional schemes and are robust to the residual LI caused by the FD operation.
Those attractive benefits discussed above motivate researchers to combine massive MIMO and FD technique together so as to obtain better system performance. For example, FD massive MIMO relaying systems are considered in [12–14]. [12] considers a multipair FD decodeandforward (DF) relay with maximumratio combining/maximumratio transmission (MRC/MRT) to process the signals, and the authors propose the use of massive antenna arrays at the relay station to significantly reduce the LI effect. In [13], a twoway FD relay system with FD users under MRC/MRT and zeroforcing reception/zeroforcing transmission (ZFR/ZFT) processing is considered, and four typical power scaling schemes are applied, then the upper bounds of the sum rates for each power scaling schemes are derived. It is shown that the LI can be decreased through reducing the power of FD relay. However, both [12] and [13] assume that some novel LI mitigation scenarios in [15] are applied, which is impractical in massive MIMO systems, and the reason will be explained in Section 2. [14] considers a multipair FD amplifyandforward (AF) relaying system with ZFR/ZFT processing. It is shown that adopting lowcomplexity signal processing techniques can significantly decrease the detrimental impact of LI as well as multipair interference if the power scaling scheme is appropriately chosen, and FD systems combined with massive MIMO outperforms the HD counterpart in spectral efficiency even without active LI cancellation.

A new asymptotic method is used to derive the asymptotic expressions of uplink and downlink SINR for the kth user, which turns out to be effective.

We adopt general power scaling scenario at the BS and users, and based on the derived SINR expressions, we analyse how the number of BS antennas and the transmit power of BS and users impact the system performance. To be specific, the transmit powers of users and BS are scaled down by 1/N ^{ p } and 1/N ^{ q }, respectively, where 0≤p,q≤1, and N represents the number of BS transmit or receive antennas. We show that as N grows larger, the harmful effect of LI can be wiped out when q>0 and the multiuser interference (MUI), CSI estimation error and interuser interference (IUI) can be eliminated when p>0.

We consider a power allocation (PA) scenario to control the transmit power at the BS and users so as to maximize the SE and EE, which is under the constraint of the maximum of transmit power.
Notation: We use boldface uppercase letter A and boldface lowercase letter a to represent matrix and column vector, respectively. \({\mathbb {E}}(\cdot)\), ∥·∥, (·)^{∗}, (·)^{ H }, X(·), and Tr(·) stand for the expectation, the Euclidean norm, the conjugate, the conjugate transpose, the spectral radius of a matrix, and trace of a matrix, respectively. [A]_{ ij } denotes the ithrow and jthcolumn entry of matrix A. I _{ N } represents the N×N identity matrix. \({\mathcal {C}N} \left ({\mu,{\sigma ^{2}}}\right)\) stands for the complexGaussian distribution with mean of μ and variance of σ ^{2}. e _{ i } represents a vector whose ith entry is 1 and the other are 0.
2 System model
We consider a multiuser FD massive MIMO system, where K users are within a single cell. Both the users and the BS operate in FD mode. We assume that the BS is equipped with two separate largescale antennas arrays in which N for transmission and N for reception, while each user is equipped with two antennas in which one for transmitting signal and the other for receiving. All the users share the same timefrequency resource.
Let \({{\mathbf {G}}_{D}}=\,[{{\mathbf {\!g}}_{D,1}},\cdots,{{\mathbf {g}}_{D,K}}]\in {{\mathbb {C}}^{N\times K}}\) represent the downlink channel matrix between the transmit antenna array of the BS and the K users and let \({{\mathbf {G}}_{U}}=\,[{{\mathbf {\!g}}_{U,1}},\cdots,{{\mathbf {g}}_{U,K}}]\in {{\mathbb {C}}^{N\times K}}\) denote the uplink channel matrix between K users and the receive antenna array of the BS. G _{ a } (a∈{D,U}) is modelled as \({{\mathbf {G}}_{a}}={{\mathbf {H}}_{a}}\mathbf {D}_{a}^{1/2}\), where \({{\mathbf {H}}_{a}}\in {{\mathbb {C}}^{N\times K}}\) characterizes the smallscale fading of the channel which has independent and identically distributed (i.i.d.) \({\mathcal {C}N}(0,1)\) elements, while D _{ a } is the largescale fading diagonal matrix whose kth diagonal entry [ D _{ a }]_{ kk }=β _{ a,k } denotes the largescale fading coefficient. Moreover, \({{\mathbf {G}}_{\text {LI}}}=\,[\!{{\mathbf {g}}_{LI,1}},\cdots,{{\mathbf {g}}_{LI,N}}]\in {{\mathbb {C}}^{N\times N}}\) denotes the LI channel matrix between the transmit and receive antennas of the BS, whose entries are i.i.d. with distribution \({\mathcal {C}N}(0,{{\beta }_{\text {LI}}})\). \({{\mathbf {G}}_{IU}}=\,[{{\mathbf {\!g}}_{IU,1}},\cdots,{{\mathbf {g}}_{IU,K}}]\in {{\mathbb {C}}^{K\times K}}\) denotes the channel matrix between users whose entry [ G _{ IU }]_{ ij }=g _{ ij } represents the IUI channel coefficient from the ith user to the jth user, which are modelled as i.i.d. \({\mathcal {C}N}(0,{{\sigma }_{ij}})\). Note that [ G _{ IU }]_{ kk }=g _{ kk } denotes the coefficient of selfinterference channel for the kth user.
2.1 Imperfect channel estimation phase
In [12] and [13], it made an assumption that the residual LI can be regarded as additive noise after applying some novel LI mitigation scenarios, on the basis of knowing the CSI of G _{LI} to compute the filter matrices [15]. It may work in the conventional MIMO systems where the antennas deployed at the relay or BS are in a small quantity. Nevertheless, with the growth of the number of transmit and receive antennas at the relay or BS, the dimension of G _{LI} increases, and in order to obtain the CSI of G _{LI}, the length of pilot sequence τ should satisfy τ≥N which is unprocurable in the massive MIMO systems owing to the limited duration of channel coherent time. Hence, in this paper, we assume that the instantaneous knowledge of G _{LI} is unknown and the CSI of G _{ U } and G _{ D } are obtained by MMSE estimation.
respectively, where τ is the length of the pilots, P _{ P } denotes the transmit power of each pilot symbol; \({{\mathbf {G}}_{RR}}\in {{\mathbb {C}}^{N\times K}}\) and \({{\mathbf {G}}_{TT}}\in {{\mathbb {C}}^{N\times K}}\) denote the channels from the users’ transmit antennas to the transmit antennas of the BS and the users’ receive antennas to the receive antennas of the BS, respectively; \({{\mathbf {X }}_{R}}\in {{\mathbb {C}}^{K\times {\tau }}}\) and \({{\mathbf {X }}_{T}}\in {{\mathbb {C}}^{K\times {\tau }}}\) represent the pilot sequences which are transmitted from the receive antennas and transmit antennas of K users, respectively; N _{ R } and N _{ T } denote the AWGN matrices whose elements are i.i.d. \({\mathcal {C}N}(0,1)\). Note that all the pilot sequences are assumed to be orthogonal, i.e. \({{\mathbf {X }}_{a}}{{{\mathbf {X }}_{a}^{H}}} = {\mathbf{I}_{K}},a \in \left \{{R,T} \right \}\), and \({\mathbf {X }}_{R}{\mathbf {X }}_{T}^{H} = {{\mathbf {0}}_{K}}\).
where \({\boldsymbol {\Lambda }_{U}} = {\left ({{\mathbf {D}_{U}^{ 1}} / {\left ({\tau {P_{P}}} \right)} + {\mathbf {I}_{K}}} \right)^{ 1}}\) and \({\boldsymbol {\Lambda }_{D}} = ({\mathbf {D}_{D}^{ 1}}/ {({\tau {P_{P}}})} + {\mathbf {I}_{K}})^{ 1}\) and \({\tilde {\mathbf {N}}_{R}} = {\mathbf {N}_{R}}{\mathbf {X}}_{T}^{H}\) and \({\tilde {\mathbf {N}}_{T}} = {\mathbf {N}_{T}}{\mathbf {X }}_{R}^{H}\), both of which have i.i.d. \({\mathcal {C}N}(0,1)\) elements due to the orthogonal rows of X _{ T } and X _{ R }.
where Δ G _{ i } represents the estimation error matrix. According to the property of MMSE channel estimation, we can conclude that \(\hat {\mathbf {G}}_{i}\) and Δ G _{ i } are independent and the rows of them are mutually independent which comply with distributions \({\mathcal {C}N}(0, \hat {\mathbf {D}}_{i})\) and \({\mathcal {C}N}(0,\Delta \mathbf {D}_{i})\). Both \(\hat {\mathbf {D}}_{i}\) and Δ D _{ i } are diagonal matrices whose kth diagonal entries are expressed as \({{\hat \beta }_{i,k}} = {\tau {P_{P}}\beta _{i,k}^{2}} / \left ({\tau {P_{P}}{\beta _{i,k}} + 1} \right)\) and Δ β _{ i,k }=β _{ i,k }/(τ P _{ P } β _{ i,k }+1), respectively [5].
2.2 Data phase
where \(\mathbf {F}=\,[{{\mathbf {\!f}}_{1}},\cdots,{{\mathbf {f}}_{K}}]\in {{\mathbb {C}}^{N\times K}}\) denotes the uplink beamforming matrix; \(\mathbf {W}=\,[{{\mathbf {w}}_{1}},\cdots,{{\mathbf {w}}_{K}}]\in {{\mathbb {C}}^{N\times K}}\) denotes the downlink precoding matrix; x _{ D }=[ x _{ D,1},⋯,x _{ D,K }]^{ T } and x _{ U }=[ x _{ U,1},⋯,x _{ U,K }]^{ T } represent the transmit signal vectors of the BS and users, respectively, with \({\mathbb {E}\{{\mathbf {x}_{a}}\mathbf {x}_{a}^{H}\}} = {\mathbf {I}_{K}}\), (a∈{D,U}); n _{ U }∼(0,σ _{ U } I _{ N }) denotes the additive white Gaussian noise (AWGN) vector at the BS; and P _{ D } and P _{ U } denote the transmit powers of the BS and users, respectively.
where n _{ D }∼(0,σ _{ D } I _{ K }) denotes AWGN vector at users.
The first term of the right side of (9) is the downlink signal for the kth user. The second term is AWGN at the kth user. The third term is the MUI which is the downlink signal transmitted for other users. And the last term denotes the IUI comes from the signals transmitted by other users.
Before the analysis, we introduce the following lemma which is useful in the field of massive MIMO system.
Lemma 1
almost surely as n→∞.
3 Analysis of asymptotic uplink and downlink SINR with MRC/MRT scenario
3.1 Analysis of asymptotic uplink SINR
3.1.1 Asymptotic expression of uplink SINR
where \(\xrightarrow [N\to \infty ]{a.s.}\) represents the almost sure convergence as N→∞ and \(\xrightarrow [N\to \infty ]{d}\) denotes the convergence in distribution when N tends to infinity.
3.1.2 Uplink power scaling analysis

Case 1, p=q=0: We assume there is no power scaling at the BS and users. It is obvious that under case 1, the uplink SINR tends to infinity as N→∞ when there is no power scaling scheme adopted. From (25), we can see that all components (including the MUI, channel estimation error (CEE), LI, and the noise at the BS) which have baneful impact on the kth user diminish when N→∞, while the desired signal is maintained. The reason is that in the regime of large N, the MUI is averaged. Similarly, due to the sufficient diversity gain and power constraint of the BS, the rest of the unexpected signals at the BS approach to zero when N tends to infinity.

Case 2, 0<p≤1,q=0: Case 2 is the power scaling scheme only adopted at the users. When 0<p<1, all the detrimental signals are averaged out as N→∞, which is similar to case 1. However, when the power of users are scaled down by N, i.e. p=1, although the harmful effect of MUI and CEE are eliminated as N grows larger, the LI and the noise at the BS cannot be wiped out, which makes the uplink SINR tends to a ceiling value as N→∞.

Case 3, p=0,0<q≤1: Case 3 denotes the power scaling scheme adopted at the BS. From Eq. (25), it can be seen that the asymptotic SINR of case 3 becomes infinite when N→∞, which is similar to case 1. What is more, the \({\frac {1}{N}}\) factor before LI indicates that, with power scaling at the BS, the harmful effect of LI which is amplified by the BS can be diminished when N approaches to infinity.

Case 4, 0<p≤1,0<q≤1: Case 4 represents the power scaling scenario applied both at the BS and users. When q=1, it can be observed from (25) that the detrimental effect of MUI, CEE, and LI disappear as N→∞ while the noise at the BS cannot be eliminated, under which situation the noise at the BS poses greater influence to the uplink SE performance when N is large. When 0<q<1, all the unexpected signals vanish when N approaches to infinity.
3.2 Analysis of asymptotic downlink SINR
3.2.1 Asymptotic expression of downlink SINR
3.2.2 Downlink power scaling analysis

Case 1, p=q=0: (33) implies that the downlink SINR of case 1 tends to infinity when N→∞. All the MUI, CEE, IUI, and the noise at the kth user are wiped out as N goes to infinity, and only the desired signal is maintained.

Case 2, 0<p≤1,q=0: From (33), we can see that the downlink SINR of case 2 goes to infinity when N is large enough. Different from case 1, we can conclude that scaling down the transmit power of users can decrease the detrimental effect of IUI which is amplified by it.

Case 3, p=0,0<q≤1: In case 3, MUI and CEE are eliminated when N is large enough. Nevertheless, when q=1, due to the remained baneful effect of IUI and AWGN at the kth user, the asymptotic downlink SINR for case 3 tends to be deterministic when N approaches infinity. If 0<q<1, the SINR increases without upper bound as N grows.

Case 4, 0<p≤1,0<q≤1: When N→∞, MUI, CEE, and IUI tend to be eliminated. Similar to uplink case 4, when q=1, the detrimental effect of noise at kth user still remains, the AWGN at the users has more influence to the downlink SE for large N. When 0<q<1, all the harmful signals disappear and the downlink SINR tends to infinity when N grows infinite.
Remark: From the above analysis, we can conclude that scaling down the BS power can decrease the adverse effect of LI and cutting down the user power can minish the MUI, CEE, and IUI. However, decrease P _{ U } and P _{ D } can also decrease the SE, which as a result exits a tradeoff between EE and SE, and we will discuss it in the simulation section.
4 Analysis of asymptotic uplink and downlink SINR with ZFR/ZFT processing
4.1 Analysis of asymptotic uplink SINR
where \({\tilde n_{U,k}}\) obeys \({\mathcal {C}N}(0,{{{\hat \beta }_{U,k}}{\sigma _{U}}})\).
Since the method of derivation and analysis for the powerscaled ZFR/ZFT scheme is similar to the MRC/MRT cases, therefore, we omit them.
4.2 Analysis of asymptotic downlink SINR
respectively. In order to validate the accuracy of the asymptotic method in this paper, we compare our asymptotic SE with the ergodic SE. In Section 6, MonteCarlo simulation method is adopted to generate the ergodic SE curve in the figures of the simulation results, and we will show that the approximate results are quiet tight even when N is small (e.g., N=25).
5 Power control
Obviously, the primary intention of applying FD techniques in wireless communication systems is to enhance the SE performance. It is worth noting that since the uplink and downlink channels transmit simultaneously and LI cancellation techniques are adopted in FD systems; therefore, more energy is consumed than the HD counterpart. Due to the increasing attention to green communications, EE is another metric widely used in many systems to evaluate the performance, especially in massive MIMO systems where enhancing the EE performance is significant [5].
Observing (24), (32), (43), and (49), we can see that the SINR of each user is different due to the different fading channels each user experiences, and in the previous sections, the transmit powers of K users, i.e. P _{ U }, are all assumed to be the same, and the powers allocated to the K streams of downlink signals are fixed, which as a result may not achieve the optimal SE/EE performance. Hence, in this section, we assume the transmit power of kth user and the power allocated to the kth downlink stream are different and propose the PA schemes to maximize the sum SE and EE subject to power constraints, which are called SE PA and EE PA for short.
5.1 Maximization of the SE
where ={P _{ U,1},⋯,P _{ U,K },P _{ D,1},⋯,P _{ D,K }} and C _{sum} is defined in (50).
where p _{1}, p _{2}, and p _{3} are power constraints given by (51).
In this way, the SE optimization problem in (57) becomes a standard GP which can be solved by the Algorithm 1.
5.2 Maximization of the EE
where P _{cir} is the power consumed by the circuits.
where \({{\bar C}_{\text {sum}}}\) denotes the required sum SE.
Solving the EE maximization problem (59) can be separated into three steps: (a) Check whether the optimal solution of problem Q_{1}, i.e., Q_{1}∗ satisfies the SE constraint. If it does, go to (b). If not, then the problem is infeasible. (b) Check whether the optimal solution of problem Q_{2} satisfies the SE constraint. If it satisfies, then Q_{2}∗ is the solution of (59). Otherwise, go to (c), i.e. obtain Q_{3}∗ of problem Q_{3} which is the solution of (59). In this paper, the required sum SE \({{\bar C}_{\text {sum}}}\) and the power constraints are set to make the EE maximization problem feasible. Next, the detailed solutions to the subproblems Q_{2} and Q_{3} are introduced in the following subsections.
5.2.1 Solution to problem Q_{2}
where \({z_{U,k}} = \sum \limits _{i = 1}^{K} {{P_{U,i}}{B_{k,i}}} + \sum \limits _{i = 1}^{K} {{P_{D,i}}C_{i}} \), \(z_{U,k}^{\left ({n  1} \right)} = \sum \limits _{i = 1}^{K} {P_{U,i}^{\left ({n  1} \right)}{B_{k,i}}} + \sum \limits _{i = 1}^{K} {P_{D,i}^{\left ({n  1} \right)}C_{i}} \), \({z_{D,k}} = \sum \limits _{i = 1}^{K} {{P_{D,i}}{E_{k,i}}} + \sum \limits _{i = 1}^{K} {{P_{U,i}}} {F_{k,i}}\), and \(z_{D,k}^{\left ({n  1} \right)} = \sum \limits _{i = 1}^{K} {P_{D,i}^{\left ({n  1} \right)}{E_{k,i}}} + \sum \limits _{i = 1}^{K} {P_{U,i}^{\left ({n  1} \right)}} {F_{k,i}}\). Here, \({\{ P_{U,i}^{\left ({n  1} \right)}\} }\) and \({\{ P_{D,i}^{\left ({n  1} \right)}\} }\) denote the value of {P _{ U,i }} and {P _{ D,i }} at the (n−1)th iteration.
In conclusion, the method to solving the subproblem Q_{2} is given in detail by Algorithm 2.
5.2.2 Solution to problem Q_{3}
6 Simulation results
In this section, we examine the simulation results of the derivation and analysis conducted in the above sections. In order to validate the accuracy of the asymptotic method in this paper, we compare our asymptotic SE with the ergodic SE \({{\tilde {C}}_{U}}\) and \({{\tilde {C}}_{D}}\) generated by MonteCarlo simulation method. The SE of multiuser HD massive MIMO system proposed in [5] is also simulated for comparison, where the BS and users operate in the HD mode, and the sum SE of the system is defined as \({{C}_{\text {sum}}^{\text {HD}}}=C_{D}^{\text {HD}}+C_{U}^{\text {HD}}\) where \(C_{a}^{\text {HD}}=\mathrm {}\frac {1}{2}\left [\frac {{T  \tau }}{T}\sum \limits _{k=1}^{K}{{{\log }_{2}}(1+\gamma _{a,k}^{\text {HD}})} \right ]\), a∈{U,D}, and the factor 1/2 is for the HD BS. Since the FD systems take only one time slot to transmit while two time slots are needed in the HD systems, the transmit powers of the BS and users in HD systems are set to 2P _{ D } and 2P _{ U } for a fair comparison. The length of channel coherence time is set to T=300 (symbols), and the length of pilot sequence is set to τ=2K. The number of users is assumed to be K=10 in this system, and P _{cir}=1W, σ _{ U }=σ _{ D }=σ _{ ik }=1.
Note that we compare the asymptotic method adopted in this paper with the one used in [12,13,16,25]. From Figs. 2 and 3, we can see that our asymptotic results matches the exact results perfectly even when N is small, which means we can predict the exact results of SE through the asymptotic analysis method adopted in this paper even when N is small (N=25 for example). Nevertheless, the upper bound of each power scaling scheme provided in [13,16,25] matches the exact curve only in the large N scenario. This is due to the fact that we use two kinds of law of large number, i.e. Lemma 1 in this paper and Lemma 1 in [16], to make the asymptotic results quiet tight, while [13,16,25] only use Lemma 1 in [16] to get the limit results. From the simulation results, it can also be observed that our asymptotic result is closer to the exact result than the one provided in [12].
Besides, in order to satisfy the SE requirement for both uplink and downlink instead of the sum SE, we replace the required sum SE in (69) with the required uplink and downlink SE which are set to be equal (the practical case in the Fig. 5). From Fig. 5, we can see that when C _{sum}<8, the EEs with PA and without PA are nearly the same and increase as the sum SE grows, which is because P _{cir} dominates the consumed power when the sum SE is small. With the increasing of the sum SE, the users and the BS become the main energy consumer; thus, there is a tradeoff between the sum SE and the EE when the sum SE is high.
The performance improvement after adopting the EE maximization algorithms is encouraging. The optimal EE achieved by (62) can be viewed as the achievable EE, which is improved remarkably, especially for ZFR/ZFT scheme. Besides, the improvement of the practical PA case is also significant. For example, when C _{sum}=23.3, 28.3 b/s/Hz under ZFR/ZFT processing and C _{sum}=30 b/s/Hz under MRC/MRT processing, the performance improvement obtained by the PA algorithm is about 104.1, 649.3, and 122.2%, respectively. What is more, from the performance improvement of ZFT/ZFR processing we can see that ZFT/ZFR suffers more from the largescale fading than MRC/MRT, especially the practical largescale fading like (71). Due to the inversion operation in the ZFT/ZFT processing (34), the largescale fading factors which are close to zero (for example, 0.018 in (71)) have a strong impact on the SE performance, in which case the PA scheme is particularly meaningful.
7 Conclusions
In this paper, we consider a multiuser FD massive MIMO system, where the BS is equipped with largescale antenna arrays, while each FD user is equipped with two antennas (one for transmission and the other one for reception). The linear processing techniques of MRC/MRT and ZFR/ZFT are adopted at the BS to process the signals. Based on the signals processing, the asymptotic expressions of SINR are derived. With a general power scaling scenario at each downlink stream and user, it is shown that the detrimental effect of MUI, CEE, LI, and AWGN can be wiped out as N goes large if the scenario is appropriately chosen. On the basis of the asymptotic expressions of SINR, the PA schemes at the BS and users are proposed to maximize the EE and the sum SE of the multiuser FD massive MIMO system. In the simulation part, we verify the accuracy of the analysis and show the ZFR/ZFT processing is more susceptible to the largescale fading. Then, we show the effectiveness of our PA schemes. Finally, we show that the SE PA can make the multiuser FD massive MIMO system outperform the HD counterpart no matter how big β _{LI} is.
8 Endnotes
^{1} Due to the separate antenna configuration at the BS, therefore, the reciprocity of uplink and downlink channels does not hold.
^{2} In order to accomplish the channel estimation, the receive antenna of each user and the transmit antenna array of the BS are assumed to be shared antennas, i.e. the transmit and receive radiofrequency chains share the same antenna [8].
Declarations
Funding
This research was supported by National Natural Science Foundation of China (No. 61671472), Jiangsu Province Natural Science Foundation (BK20160079), and National Natural Science Foundation of China (No. 61371123, No. 91438115).
Authors’ contributions
XW is the main writer of this paper. XW proposed the main idea, conducted the simulations, and analysed it. DZ, KX, and WM assisted in the review of this manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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
References
 CX Wang, et al, Cellular architecture and key technologies for 5G wireless communication networks. IEEE Commun. Mag. 52(2), 122–130 (2014).View ArticleGoogle Scholar
 EG Larsson, O Edfors, F Tufvesson, TL Marzetta, Massive MIMO for next generation wireless systems. IEEE Commun. Mag. 52(2), 186–195 (2014).View ArticleGoogle Scholar
 T Marzetta, Noncooperative cellular wireless with unlimited numbers of base station antennas. IEEE Trans. Wirel. Commun. 9(11), 3590–3600 (2010).View ArticleGoogle Scholar
 C Sun, X Gao, S Jin, M Matthaiou, Z Ding, C Xiao, Beam division multiple access transmission for massive MIMO communications. IEEE Trans. Commun. 63(6), 2170–2184 (2015).View ArticleGoogle Scholar
 HQ Ngo, E Larsson, T Marzetta, Energy and spectral efficiency of very large multiuser MIMO systems. IEEE Trans. Commun. 61(4), 1436–1449 (2013).View ArticleGoogle Scholar
 S Hong, et al, Applications of selfinterference cancellation in 5G and beyond. IEEE Commun. Mag. 52(2), 114–121 (2014).View ArticleGoogle Scholar
 Z Zhang, X Chai, K Long, AV Vasilakos, L Hanzo, Full duplex techniques for 5G networks: selfinterference cancellation, protocol design, and relay selection. IEEE Commun. Mag. 53(5), 128–137 (2015).View ArticleGoogle Scholar
 D Kim, H Lee, D Hong, A survey of inband fullduplex transmission: from the perspective of PHY and MAC layers. IEEE Commun. Surv. Tutorials. 17(4), 2017–2046 (2015).View ArticleGoogle Scholar
 A Sabharwal, P Schniter, D Guo, DW Bliss, S Rangarajan, R Wichman, Inband fullduplex wireless: challenges and opportunities. IEEE J. Sel. Areas Commun. 32(9), 1637–1652 (2014).View ArticleGoogle Scholar
 M Vehkaperä, MA Girnyk, T Riihonen, R Wichman, LK Rasmussen, On achievable rate regions at largesystem limit in fullduplex wireless local access. 2013 First International Black Sea Conference on Communications and Networking (BlackSeaCom), Batumi, 7–11 (2013).Google Scholar
 Y Wang, Y Xu, N Li, W Xie, K Xu, X Xia, Relay selection of fullduplex decodeandforward relaying over Nakagamim fading channels. IET Commun. 10(2), 170–179 (2016).View ArticleGoogle Scholar
 HQ Ngo, HA Suraweerat, M Matthaiou, EG Larsson, in 2014 IEEE International Conference on Communications (ICC). Multipair massive MIMO fullduplex relaying with MRC/MRT processing (Sydney, 2014), pp. 4807–4813.Google Scholar
 Z Zhang, Z Chen, M Shen, B Xia, Spectral and energy efficiency of multipair twoway fullduplex relay systems with massive MIMO. IEEE J. Sel. Areas Commun. 34(4), 848–863 (2016).View ArticleGoogle Scholar
 X Xia, Y Xu, K Xu, D Zhang, W Ma, Fullduplex massive MIMO AF relaying with semiblind gain control. IEEE Trans. Veh. Technol. 65(7), 5797–5804 (2016).View ArticleGoogle Scholar
 T Riihonen, S Werner, R Wichman, Mitigation of loopback selfinterference in fullduplex MIMO relays. IEEE Trans. Signal Process. 59(12), 5983–5993 (2011).MathSciNetView ArticleGoogle Scholar
 H Cui, L Song, B Jiao, MultiPair twoway amplifyandforward relaying with very large number of relay antennas. IEEE Trans. Wireless Commun. 13(5), 2636–2645 (2014).View ArticleGoogle Scholar
 S Jin, X Liang, KK Wong, X Gao, Q Zhu, Ergodic rate analysis for multipair massive MIMO twoway relay networks. IEEE Trans. Wirel. Commun. 14(3), 1480–1491 (2015).View ArticleGoogle Scholar
 J Evans, DNC Tse, Large system performance of linear multiuser receivers in multipath fading channels. IEEE Trans. Inf. Theory. 46(6), 2059–2078 (2000).MathSciNetView ArticleMATHGoogle Scholar
 H Yang, T Marzetta, Performance of conjugate and zeroforcing beamforming in largescale antenna systems. IEEE J. Sel. Areas Commun. 31(2), 172–179 (2013).View ArticleGoogle Scholar
 S Boyd, L Vandenberghe, Convex Optimization (Cambridge University Press, 2004).Google Scholar
 PC Weeraddana, M Codreanu, M Latvaaho, A Ephremides, Resource allocation for crosslayer utility maximization in wireless networks. IEEE Trans. Veh. Technol. 60(6), 2790–2809 (2011).View ArticleMATHGoogle Scholar
 H Ren, N Liu, C Pan, C He, Energy Efficiency optimization for MIMO distributed antenna systems. IEEE Trans. Veh. Technol. PP(99), 1–1. doi:10.1109/TVT.2016.2574899.
 JB Frenk, S Schaible, Handbook of Generalized Convexity and Eneralized Monotonicity (SpringerVerlag, New York, 2006).Google Scholar
 Inc CVX Research, CVX: Matlab Software for Disciplined Convex Programming, version 2.0 (2011). http://cvxr.com/cvx. 5 Mar 2017.
 HA Suraweera, HQ Ngo, TQ Duong, C Yuen, EG Larsson, in Proc. 2013 IEEE Int. Conf. on Commun. (ICC). Multipair amplifyandforward relaying with very large antenna arrays, (2013), pp. 4635–4640.Google Scholar