Selfinterference suppression with imperfect channel estimation in a sharedantenna fullduplex massive MUMIMO system
 Pengbo Xing^{1},
 Ju Liu^{1}Email author,
 Chao Zhai^{1} and
 Zhiyuan Yu^{1}
https://doi.org/10.1186/s1363801708057
© The Author(s) 2017
Received: 3 June 2016
Accepted: 2 January 2017
Published: 19 January 2017
Abstract
In this paper, we consider a sharedantenna fullduplex massive MUMIMO system and prove that the selfinterference (SI) at the base station (BS) can be suppressed, though the directpath SI exists and the signal/SI channel is imperfectly estimated. The BS is assumed to employ zeroforcing (ZF) or maximalratio transmission/maximalratio combining (MRT/MRC) method to linearly process signals. We propose a precoded SI channel training scheme by employing orthogonal sequences and downlink precoding, so the SI channel can be estimated like the signal channel with a much lower dimension. Based on the channel estimation, we further analyze the SI suppression by combining the SI removal and the largescale antenna linear processing (LALP) method. Numerical results show that an additional 36 dB SI can be suppressed by combining the SI removal and the LALP suppression. We derive the tight closedform lower bounds of the achievable rates for the combined suppression. Finally, we maximize the system spectral efficiency and energy efficiency based on the simulation and closedform approximations.
Keywords
1 Introduction
Massive multipleinput multipleoutput (MIMO) and inband fullduplex (IBFD) techniques have attracted intensive research interest because they can be used for broadband green communications [1–8]. IBFD can realize simultaneous bidirectional (uplink and downlink) data transmissions using the same timefrequency resource. Massive MIMO system exploits a large number of antennas at the transmitter to focus the energy into a narrow beam to improve the signal strength and mitigate interference. Both techniques are promising to meet the evergrowing requirements of wireless data transmissions, but bring some design variations over the physical or higher layers [9–17].
The massive MIMO technique can significantly improve the spectral efficiency (SE) and energy efficiency (EE) while keeping a lower transmit power for both uplink and downlink [15, 18–20] by mitigating the detrimental effects of smallscale fading, noise, and interference [4]. With more antennas equipped, the signal processing becomes much easier than the traditional MIMO system [4, 19]. The massive MIMO system usually operates with timedivision duplexing (TDD) rather than frequencydivision duplexing (FDD). In the FDD system, channel state information (CSI) should be acquired by the feedback, and much overhead will be introduced with the increase of antennaarray scale [15], thus the antenna grouping and CSI compression techniques were studied to reduce the estimation overhead [16, 17]. While in the TDD system, channel reciprocity can be considered to reduce the overhead, as the downlink CSI can be acquired by the base station (BS) through the uplink training [18, 21].
The fullduplex (FD) technique can potentially double the spectral efficiency, but it faces the problem of selfinterference (SI), which refers to transmitted signals that can be directly or indirectly received by its own receivers. Recently, much effort has been made to suppress SI [22–25]. Riihonen et al. designed natural isolation, timedomain cancellation, and spatial SI suppression schemes [22]. Everett et al. proposed to exploit the antenna directional isolation and crosspolarization to suppress the SI in the wireless propagation domain [23]. Bharadia et al. proposed analog circuit domain cancellation methods that can keep a copy of the transmitted signal and subtract it from the received signal [24]. An analoganddigital hybrid SI cancellation method has been developed by Duarte et al. and can provide approximately 85 dB interference suppression [25]. Recently, Ngo et al. proposed the loop interference cancellation technique in a separateantenna relay system, where the SI can be suppressed with a largescale antenna linear processing (LALP) method in a random reflectedpath SI environment [26].
Sabharwal et al. discussed two methods of interfacing antennas to an IBFD wireless system: (i) separateantenna FD and (ii) sharedantenna FD [1]. In the sharedantenna model, a bidirection circulator is used to transmit and receive signals without modifying many architecture standards, thus the RF hardwares such as antenna, RF cable and filters can be saved [2]. At the same time, the nonlinearities of the RF hardware, for example the circulators and amplifiers, introduce weak nonlinear distortions in such systems.

We consider both the nonrandom directpath and the random reflectedpath SI channels in this model. We prove that the Rician SI channel changes into Rayleigh channel when the downlink channel/SI channel is precoded. We also prove that the SI at BS with imperfect channel estimation can be asymptotically suppressed, though the uplinkdownlink channels are totally dependent and the line of sight (LOS) SI exists. It is shown that the capability of LALP suppression is proportional to the square root of the antenna array scale M, which is different from the separateantenna massive MIMO system. Because of the channel reciprocity, the signal channel estimation of the sharedantenna array is reduced by half compared with the separateantenna arrays.

We estimate the SI through the precoded SI channel with MMSE method, which is different from the traditional SI acquisition method. The estimation overhead and the dimension of pilots can be greatly suppressed compared with the direct estimation of the unprecoded SI channel. With the estimated SI, we can further analyze the SI suppression by combining the SI removal and the LALP method. It is shown that an additional 36 dB suppression capability can be achieved through combining the two methods. We also derive the closedform lower bounds of the uplink achievable rates with imperfect CSI. The optimal SE and EE are further studied based on the rate lower bounds.
The rest of this paper is organized as follows. The system model is described in Section 2. Section 3 studies the SI suppression and the uplink achievable rates. Section 4 analyzes the SE and EE. Numerical results and conclusions are presented in Section 5 and Section 6, respectively.
Notations: The boldface upper and lowercase letters represent matrices and vectors, respectively. The superscripts [·]^{∗},[·]^{ T }, and [·]^{ H } stand for conjugate, transpose, and Hermitian transpose of matrices or vectors, respectively. The mathematical expectation is denoted as \(\mathbb {E}\{\cdot \}\). ∥·∥ and represents the Euclidean norm. The (i,j)th entry of a matrix is denoted as [·]_{ ij }. The trace of a square matrix is denoted as tr(·). The \(\overset {a.s.}\longrightarrow \) represents almost sure convergence. u _{ n }=o(v _{ n }) and u _{ n }=O(v _{ n }) denote that there exists a constant C, such that u _{ n }=C v _{ n } and u _{ n }≤C v _{ n }, ∀n, respectively.
2 System model
where the first term represents the desired signal, and the second term represents the SI from the downlink transmission. G is the M×K channel matrix between the K users and the BS; x _{u} denotes the symbols transmitted by the K users with unit power \(\mathbb {E}\{{\boldsymbol {x}_{\mathrm {u}}\boldsymbol {x}{_{\mathrm {u}}^{H}}}\}=\boldsymbol {I}_{K}, p_{\mathrm {u}}\) is the transmit power of each user; G _{s} is the M×M SI channel matrix between the BS transceivers; s is the downlink precoded signal transmitted to the K users with unit power \(\mathbb {E}\{{\boldsymbol {s}^{H}\boldsymbol {s}}\}=1; p_{\mathrm {d}}\) is the downlink transmit power; and n represents the additive noise with zero mean and unit variance.
2.1 Channel model
where H is the M×K matrix with i.i.d entries [4], representing the smallscale fading between the K users and the BS, and the entry h _{ mk } is a circular symmetric complex Gaussian (CSCG) random variable with zero mean and unit variance, i.e., \(h_{mk}\sim \mathcal {CN}(0,1)\); D is a K×K diagonal matrix of the largescale pathloss with [D]_{ kk }=β _{ k }.
where \({\tilde {\boldsymbol {H}}}_{\mathrm {s}}\) is an M×M matrix with i.i.d. entries representing the smallscale fading. We suppose \([{\tilde {\boldsymbol {H}}}_{\mathrm {s}}]_{mj}={h}_{\mathrm {s},{mj}}\), where h _{s,m j } is a CSCG random variable with zero mean and unit variance, and \({\tilde {\boldsymbol {D}}}_{\mathrm {s}}\) is an M×M diagonal matrix of the largescale pathloss, where \([{\tilde {\boldsymbol {D}}}_{\mathrm {s}}]_{mm}={\beta }\), namely, all of the SI largescale pathloss coefficients are assumed to be identical. This assumption is reasonable, as the antenna array size is much smaller than the distance to the scatters. If these coefficients are not identical, we can set all the elements of SI channel matrix \({\tilde {\boldsymbol {G}}}_{\mathrm {s}}\) to be the maximum value β _{max}, which represents the worst case, then \(\tilde {\boldsymbol {G}_{s}}\) becomes a matrix with identical largescale fading.
where \(\bar {\boldsymbol {D}}_{s}\) is an M×M deterministic matrix and \(\tilde {\boldsymbol {D}}_{s}\) is an M×M diagonal matrix with \([\tilde {\boldsymbol {D}}_{s}]_{mm} =\beta \). The proofs of (6) and (7) are given in Appendix.
2.2 Channel estimation
where P _{u} and P _{d} are the transmit powers of pilot sequences for the uplink and downlink, respectively. Suppose that the power of each pilot symbol equals the average power of data transmission, then we have P _{u}=τ p _{u} and P _{d}=τ p _{d};Φ _{u}=Φ _{d} are K×τ matrices whose rows are the pilot sequences for the estimation of the signal channel and SI channel. The pilot matrices satisfy \({\boldsymbol {\Phi }}_{\mathrm {u}}{\boldsymbol {\Phi }}{_{\mathrm {u}}^{H}}={\boldsymbol {\Phi }}_{\mathrm {d}}{\boldsymbol {\Phi }}{_{\mathrm {d}}^{H}} ={\boldsymbol {I}}_{K}\) and τ≥K;A is an M×K precoding matrix, which is used for downlink beamforming and updated from the uplink channel estimation; N and N _{s} are M×τ AWGN matrices with i.i.d \(\mathcal {CN} (0,1)\) entries.
2.2.1 Signal channel estimation
According to the property of statistician’s Pythagorean theorem of the MMSE estimation [30], the estimation error Δ is independent of \(\hat {\boldsymbol {G}}\). Therefore, the kth columns of \(\hat {\boldsymbol {G}}\) and Δ are mutually independent, and the elements of the kth vector are distributed as \(\mathcal {CN}(0,{\hat {\beta }_{k}})\) and \(\mathcal {CN}(0,{\tilde {\beta }_{k}})\), respectively, with \(\hat {\beta }_{k} \triangleq {\tau p_{\mathrm {u}}\beta {_{k}^{2}}\over {\tau p_{\mathrm {u}}\beta _{k} + 1}}\) and \(\tilde {\beta }_{k} \triangleq {\beta {_{k}}\over {\tau p_{\mathrm {u}}\beta _{k} + 1}}\).
Note that the sharedantenna array massive MIMO system has a single array, so the transmit and receive CSI are the same during the coherent time. We only need to estimate the uplink channel, and thus the estimation overhead is reduced by half. This is different from the separateantenna arrays system, where the CSI of the receive and transmit arrays are different, the uplink and downlink CSI have to be estimated separately.
2.2.2 SI channel estimation
The SI channel is estimated in the second training period τ. In fact, it is not necessary to estimate G _{s}, we can estimate the equivalent or precoded SI channel G _{sa} instead, which represents the product of G _{s} and A. The new precoding matrix A can be updated for the SI channel estimation after \(\hat {\boldsymbol {G}} \) is acquired. Because G _{sa} has the same dimension as the uplink channel G, the training pilot length and pilot amount depend on the user number K rather than the number of BS antennas M. Therefore, the dimension of the SI channel estimation can be decreased from M×M to M×K, and thus fewer orthogonal pilot sequences are needed. Note that the proposed training and estimation method is different from the traditional SI acquisition method where the SI is directly estimated with/without pilots [22].
where \(\alpha _{\text {ZF}} \triangleq \sqrt {MK\over \sum _{k=1}^{K}{\hat \beta }{_{k}^{1}}}\) and \(\alpha _{\text {MRT}} \triangleq \sqrt {1\over {M\sum _{k=1}^{K}{\hat \beta }{_{k}}}}\) are the normalization factors [26] satisfying \(\mathbb {E}\{{\boldsymbol {s}^{H}\boldsymbol {s}}\}=1\).
Proposition 1
In the sharedantenna FD massive MUMIMO system, the SI equivalent channel G _{sa}=G _{s} A is an M×K complex Gaussian matrix if K is fixed and M→∞. The (m,i)th entry follows the distribution \(\mathcal {CN}\left (0, \rho _{m,i}\right)\), where \(\rho _{m,i}={1\over M^{2}}\alpha _{\text {ZF}}^{2}\hat \beta _{i}^{\!1}\!\sum _{j=1}^{M}\left (\left c_{mj}\right ^{2}\,+\,\beta \right)\) for ZF precoding and \(\rho _{m,i}=\alpha _{\text {MRT}}^{2}\hat \beta _{i}\!\sum _{j=1}^{M}\left (\left c_{mj}\right ^{2}\,+\,\beta \right)\) for MRT precoding, \(\hat \beta _{i} \triangleq {\tau p_{\mathrm {u}}\beta {_{i}^{2}}\over {\tau p_{\mathrm {u}}\beta _{i} + 1}}\), i=1,…,K.
Proof.
Please see Appendix. □
the (m,i)th entry of \({\hat {\boldsymbol {G}}}_{\text {sa}}\) and \(\boldsymbol {\mathcal {E}}\) are distributed as \(\mathcal {CN}(0,{\hat {\rho }_{m,i}})\) and \(\mathcal {CN}(0,{\tilde {\rho }_{m,i}})\), respectively, with \(\hat \rho _{m,i} \triangleq {\tau p_{\mathrm {d}}\rho {_{m,i}^{2}}\over {\tau p_{\mathrm {d}}\rho _{m,i} + 1}}\), \(\tilde \rho _{m,i} \triangleq {\rho {_{m,i}}\over {\tau p_{\mathrm {d}}\rho _{m,i} + 1}}\), m=1,...,M, i=1,...,K.
2.3 Uplink decoding under precoded SI
Precoding can be used in the training phase for the SI channel estimation, and it can also be used in the data transmission phase for beamforming, which can help enhance the strength of the desired signal and reduce the interference to others. During the data transmission period, the ZF or MRT scheme is adopted by the BS for the downlink beamforming. Correspondingly, the BS receives the precoded SI from the transmitters, and it uses the ZF or MRC linear processing method for the uplink decoding.
where \(\sqrt {p_{\mathrm {u}}} \boldsymbol {W}^{T}{\boldsymbol {G}}{\boldsymbol {x}_{\mathrm {u}}}\) is the desired signal, \(\sqrt {p_{\mathrm {d}}}\boldsymbol {W}^{T} {\boldsymbol {G}_{\mathrm {s}}}\boldsymbol {A} \boldsymbol {x}_{\mathrm {d}}\) is the SI between the BS transceivers, and W ^{ T } n is the additive noise.
where \(\sqrt {p_{\mathrm {u}}} \boldsymbol {W}^{T}\hat {\boldsymbol {G}}{\boldsymbol {x}_{\mathrm {u}}}\) is the estimated signal, \(\sqrt {p_{\mathrm {d}}}\boldsymbol {W}^{T} {\hat {\boldsymbol {G}}_{\text {sa}}}\boldsymbol {x}_{\mathrm {d}}\) is the SI from the estimation, \(\sqrt {p_{\mathrm {u}}} \boldsymbol {W}^{T}{\boldsymbol {\Delta }}{\boldsymbol {x}_{\mathrm {u}}}\) and \(\sqrt {p_{\mathrm {d}}}\boldsymbol {W}^{T} \boldsymbol {\mathcal {E}}\boldsymbol {x}_{\mathrm {d}}\) are the estimation errors of the desired signal and SI, respectively.
3 SI suppression and achievable rate
In the massive MIMO system, the white noise and the interuser interference can be eliminated by an LALP method [15, 19, 31]. In this section, we consider the sharedantenna FD massive MUMIMO system and discuss the combined SI suppression based on the estimated channels of signal and SI. We also derive the closedform lower bounds of the uplink achievable rates, which can be used as a system performance metric.
3.1 SI suppression with LALP
In the TDD halfduplex (HD) system, the interferences are asymptotically orthogonal to the subspace spanned by the desired signals when the antennaarray scale M approaches infinity. The desired signal can be recovered by projecting the received signal in a desired subspace with linear processing [19, 26]. In the sharedantenna FD massive MUMIMO system, the asymptotic orthogonality property still holds true. Unlike the separateantenna array system [26], the SI in our model includes the nonrandom directpath interference, and the uplink channel is totally dependent on the downlink channel, as the same antenna array is employed to transmit and receive signals simultaneously.
Theorem 1
In the sharedantenna FD massive MUMIMO system, when the signal and SI channels are not perfectly estimated and ZF or MRT/MRC linear processing is adopted, the SI at the BS will vanish if K is fixed and M→∞.
Proof.
for ZF and MRT/MRC linear processing schemes. □
Remark: In the sharedantenna FD massive MUMIMO system, the decoding matrix W ^{ T } and the precoding matrix A in the SI term are totally dependent because of the channel reciprocity between uplink and downlink. This is different from the FD relay massive MIMO system where the sourcerelay and relaydestination channels are independent. According to Lemma 13 in [32] and Theorem 3.7 in [33], the LALP suppression capability is O(1/M ^{1/2}), and it is different from [26] where the LALP suppression capability is O(1/M) for the separateantenna arrays.
Proposition 2
for the MRT/MRC precoding and decoding, where \([\hat {\boldsymbol {D}}]_{kk}=\hat \beta _{k}\) and \(\hat \beta _{k} \triangleq {\tau p_{\mathrm {u}}\beta {_{k}^{2}}\over {\tau p_{\mathrm {u}}\beta _{k} + 1}}\), k=1,…,K.
3.2 Combined SI suppression
Practically, the number of antennas cannot be infinite, the LALP method cannot ideally suppress the SI, especially for the dependent sharedantenna array. As mentioned above, the SI in the FD massive MUMIMO system can be estimated by the MMSE through a precoded SI channel training process, thus the known SI can be removed before the LALP suppression. After the SI removal, the remaining interference becomes much weaker, and the LALP can be further adopted to suppress the remaining SI. Therefore, the suppression capability is greatly improved by combining the two suppression methods.
where w _{ k }, \(\hat {\boldsymbol {g}}_{k}\), δ _{ i }, and ε _{ i } are the kth or ith column of W, \(\hat {\boldsymbol {G}}\), Δ, and \(\boldsymbol {\mathcal {E}}\), respectively; x _{ k } and \(x{_{i}^{\mathrm {d}}}\) are the kth and ith element of x _{u} and x _{d}, respectively; \(\sqrt {p_{\mathrm {u}}}\boldsymbol {w}{_{k}^{T}}{\hat {\boldsymbol {g}}_{k}}{x_{k}}\) is the desired signal from the kth user; \(\sqrt {p_{\mathrm {u}}}\sum _{i=1,i\neq k}^{K}{\boldsymbol {w}{_{k}^{T}}}{\hat {\boldsymbol {g}}_{i}}{x_{i}}\) is the interuser interference; \(\sqrt {p_{\mathrm {u}}}\sum _{i= 1}^{K}{\boldsymbol {w}{_{k}^{T}}}{\boldsymbol {\delta }_{i}}{x_{i}}\) is the estimation error from the signal channel G; \(\sqrt {p_{\mathrm {d}}}\sum _{i= 1}^{K}{\boldsymbol {w}{_{k}^{T}}}\boldsymbol {{\epsilon }}_{i} x{_{i}^{\mathrm {d}}}\) is the estimation error from the SI equivalent channel G _{sa} and \(\boldsymbol {w}{_{k}^{T}}\boldsymbol {n}\) is the additive noise.
Theoretically, the SI and other interferences and noise can be suppressed as much as possible, and the achievable rate of users can approach infinity with the increase of the antenna array scale. Practically, tens or hundreds of antennas can be a good approximation of infinity [4, 34].
3.3 Achievable rate
In this subsection, we will discuss the uplink achievable rate of the sharedantenna FD massive MIMO system for the LALP and the combined suppression.
3.3.1 Combined method
3.3.2 LALP only
3.3.3 Perfect channel estimation
3.4 Achievable rate lower bounds of the combined method
From (28), we can derive the lower bounds of the achievable rates for the combined method with ZF and MRT/MRC linear processing.
Lemma 1
In the sharedantenna FD massive MUMIMO system, the random variable \(\tilde g_{i}={{\boldsymbol {w}}{_{k}^{T}}{\hat {\boldsymbol {g}}_{i}}\over \left \{\boldsymbol {w}}{_{k}^{T}}\right \}\) is a zeromean complex Gaussian variable with variance \(\hat \beta _{i}\), which is independent on \({\boldsymbol {w}}{_{k}^{T}}\) if K is fixed and M→∞, where \(\hat \beta _{i}={\tau p_{\mathrm {u}}\beta {_{i}^{2}}\over \tau p_{\mathrm {u}}\beta {_{i}}+1}\), i=1,...,K, i≠k.
Proof.
Please see Appendix A of [19]. □
Lemma 2
In the sharedantenna FD massive MUMIMO system, the random variable \(\tilde \delta _{i}={{\boldsymbol {w}}{_{k}^{T}}{\boldsymbol {\delta }_{i}}\over \left \{\boldsymbol {w}}{_{k}^{T}}\right \}\) is a zeromean complex Gaussian variable with variance \(\tilde \beta _{i}\), which is independent on \({\boldsymbol {w}}{_{k}^{T}}\) if K is fixed and M→∞, where \(\tilde \beta _{i}={\beta {_{i}}\over \tau p_{\mathrm {u}}\beta {_{i}}+1}\), i=1,...,K.
Proof.
According to the optimum MMSE estimation property, δ _{ i } is independent of \(\hat {\boldsymbol {g}}_{i}\) for i=1,...,K. Similarly, as the proof of Lemma 1, we can obtain the result. □
Lemma 3
In the sharedantenna FD massive MUMIMO system, the random variable \(\tilde \epsilon _{i}={{\boldsymbol {w}}{_{k}^{T}}{\boldsymbol {\epsilon }_{i}}\over \left \{\boldsymbol {w}}{_{k}^{T}}\right \}\) is a zeromean complex Gaussian variable with variance \(\tilde \rho _{i}\), which is independent on \({\boldsymbol {w}}{_{k}^{T}}\) if K is fixed and M→∞, where \(\tilde \rho _{i}={1\over M}\!\sum _{m=1}^{M}{\rho _{m,i}\over {\tau p_{\mathrm {d}}\rho _{m,i}+1}}\), i=1,...,K.
Proof.
Please see Appendix. □
Proposition 3
where \(\hat \beta _{i}={\tau p_{\mathrm {u}}\beta {_{i}^{2}}\over \tau p_{\mathrm {u}}\beta {_{i}}+1}\), \(\tilde \beta _{i}={\beta {_{i}}\over \tau p_{\mathrm {u}}\beta {_{i}}+1}\), and \(\tilde \rho _{i}={1\over M}\!\sum _{m=1}^{M}{\rho _{m,i}\over {\tau p_{\mathrm {d}}\rho _{m,i}+1}}\).
Proof.
Proposition 4
where \(\hat \beta _{i}={\tau p_{\mathrm {u}}\beta {_{i}^{2}}\over \tau p_{\mathrm {u}}\beta {_{i}}+1}\) and \(\tilde \rho _{i}={1\over M}\!\sum _{m=1}^{M}{\rho _{m,i}\over {\tau p_{\mathrm {d}}\rho _{m,i}+1}}\).
Proof.
From Lemma 1, Lemma 2, and Lemma 3, by substituting (42) into (41), we obtain the result. □
4 Spectral efficiency and energy efficiency
The SE and EE are important performance metrics for nextgeneration mobile communication systems. In this section, we discuss the SE and EE of a sharedantenna FD massive MUMIMO system with the combined SI suppression method.
4.1 SE and EE
4.1.1 SE and EE for the FD
4.1.2 SE and EE for the HD
We can see that the SE of the HD system is smaller than that of the FD system when the antenna scale is large enough, because it transmits data only half of the time. In contrast, the EE is slightly higher because there is no SI and fewer training overhead in the HD system.
4.1.3 Lower bounds of SE and EE
where the superscript A is “Z” or “M”, which represents ZF or MRT/MRC. The lower bounds are in the closed form, so they can be easily used to show the system performance.
4.2 Maximizing SE and EE
4.2.1 Maximizing SE
This problem can be numerically solved by the linear search, the simulation in the next section shows that this solution is global optimal when the antenna scale M is large enough.
4.2.2 Maximizing the EE
It can be numerically solved by the linear search, the simulation in the next section shows that this solution is also global optimal.
4.2.3 Tradeoff of EE versus SE
The maximal EE can be obtained at the point \((\tau ^{*},\; p{{\!~\!}_{\mathrm {u}}^{*}})\). However, the SE decreases monotonically with the decreasing p _{u}. To keep a reasonable uplink throughput, the SE is lower constrained. Therefore, the optimal \(p{{\!~\!}_{\mathrm {u}}^{*}}\) should be constrained by the SE requirement. The optimal value \(p{{\!~\!}_{\mathrm {u}}^{*}}\) is adopted if the minimum SE is satisfied; otherwise, the tradeoff value \(p^{\prime }{{\!~\!}_{\mathrm {u}}^{*}}\),which meets the SE requirement, is used.
5 Numerical and simulation results
In this section, we present the performance results of the sharedantenna FD massive MUMIMO system. All the users work in FD mode with a single antenna, and the insertion loss of the circulators is neglected. The number of users is given as K=4. Unless stated otherwise, the system parameters are set as follows: The coherent time T of the uplink data transmission is 400 (symbol length), and the training length τ of each channel is 8. The normalized uplink power p _{u} and the downlink power p _{d} are set as 10 dB and 13 dB, respectively. Similarly, as [26], the largescale pathloss β _{ k } (k=1,2,3,4) of the signal channel is set as 0.749, 0.246, 0.125, and 0.635, respectively; The largescale pathloss β of the SI channel is set as 0.6, and the directpath SI coefficient c _{ mj } is set as arbitrary but deterministic values in a circle on the complex plane with radius 0.33.
5.1 Achievable rate and lower bounds
5.2 SE and EE with SI suppression
The EE of the FD system is slightly lower than that of the HD system because of the residual SI in the FD system, but the gaps are very small, so it is worth acquiring twice the SE by losing some EE. Furthermore, the EE grows with the increase of the antenna array scale M, so the performance is much better than that of the traditional FD system when M is large enough.
5.3 Maximizing SE and EE
6 Conclusion
We introduced and analyzed a fullduplex massive MUMIMO system with a single sharedantenna array at the BS. Either the ZF or MRT/MRC method can be adopted to linearly process signals, and a novel training scheme via the precoded SI channel is proposed to estimate the SI with a much lower dimension. We proved that the SI can vanish by using the largescale antenna linear processing method even if the SI/signal channels are not perfectly estimated and the uplink/downlink channels completely depend on each other. By combining the SI removal and the LALP method, we show that the SI can be sufficiently suppressed. We also derived the closedform lower bounds of the achievable rates and showed that they are very close to the simulation results. With the combination of the SI suppression, the “large enough” number of antennas can be scaled down, and the FD massive MUMIMO system significantly outperforms the halfduplex system in terms of SE.
7 \thelikesection Appendix
7.1 \thelikesubsection Derivations of (6) and (7)
7.2 \thelikesubsection Proof of Proposition 1
where g _{s,m } is the mth row of G _{s}.
for the MRT precoding.
7.3 \thelikesubsection Proof of Lemma 3
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (61371188), Research Fund for the Doctoral Program of Higher Education (20130131110029), China Postdoctoral Science Foundation (2014M560553), Special Funds for Postdoctoral Innovative Projects of Shandong Province (201401013).
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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