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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 12, DECEMBER 2019 8273
Two-Way Massive MIMO Relaying Systems
With Non-Ideal Transceivers: Joint Power
and Hardware Scaling
Junjuan Feng , Shaodan Ma , Sonia Aïssa ,Fellow, IEEE, and Minghua Xia ,Member, IEEE
Abstract— Two-way massive MIMO amplify-and -forward
relaying systems with non-ideal transceivers are investigated in
this paper. To be general, multiple-antenna nodes and antenna
correlation at both the user equipments (UEs) and the relay are
considered, which differentiates the analysis from the prior ones.
The achievable rate is analyzed and derived deterministically
in closed-form. Joint scaling of the transmission powers and
hardware impairments is then particularly investigated. Feasible
scaling speeds for the transmission powers and hardware impair-
ments are discovered when the number of relay antennas grows
large. It is shown that down scaling of the transmission powers at
the UEs and the relay and up scaling of the hardware impairment
at the relay with the number of relay antennas are tolerable
without reducing the expected rate. However, UE hardware
impairment is a key limiting factor to the achievable rate and
is not allowed to scale up with the number of relay antennas in
order to achieve a non-vanishing rate. Moreover, ceiling effect on
the achievable rate is still observable and the ceiling rate varies
among different scaling cases. More interestingly, scalings of the
UEs transmission power and the relay hardware impairment are
found to be offsettable, which means that the relay hardware cost
and the UE transmission power are tradable. It is found that the
best tradeoff is achieved in the medium scalings of both the relay
hardware impairment and UE transmission power. Numerical
results are provided to verify the analysis and the tradability
between the relay hardware cost and the UE transmission power.
The analytical results thus provide solid foundation for flexible
system designs under various cost and energy constraints.
Manuscript received November 25, 2018; revised May 31, 2019 and July 26,
2019; accepted August 22, 2019. Date of publication September 10, 2019; date
of current version December 17, 2019. This work was supported in part by
National Natural Science Foundation of China under Grant 61601524, in part
by the Science and Technology Development Fund, Macau SAR (File no.
020/2015/AMJ and File no. SKL-IOTSC-2018-2020), in part by the Major
Science and Technology Special Project of Guangdong Province under Grant
2018B010114001, in part by the Research Committee of University of Macau
under Grant MYRG2018-00156-FST, and in part by a Discovery Grant from
the Natural Sciences and Engineering Research Council (NSERC) of Canada.
The associate editor coordinating the review of this article and approving it
for publication was B. Shim. (Corresponding author: Shaodan Ma.)
J. Feng and S. Ma are with the State Key Laboratory of Internet of Things
for Smart City and the Department of Electrical and Computer Engineering,
University of Macau, Macao, China (e-mail: junjuanfeng1989@gmail.com;
shaodanma@um.edu.mo).
S. Aïssa is with the Institut National de la Recherche Scientifique (INRS-
EMT), University of Quebec, Montreal, QC H5A 1K6, Canada (e-mail:
aissa@emt.inrs.ca).
M. Xia is with the School of Electronics and Information Tech-
nology, Sun Yat-sen University, Guangzhou 510006, China (e-mail:
xiamingh@mail.sysu.edu.cn).
Color versions of one or more of the figures in this article are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TCOMM.2019.2940634
Index Terms—Anten na correlation, massive MIMO, scaling
laws, transceiver hardware impairments, two-way relaying.
I. INTRODUCTION
WITH substantially high spectral efficiency, massive/
large-scale multiple-input multiple-out (MIMO) has
been recognized as a key enabling technique to achieve the
1000 times capacity improvement in future 5G communica-
tions [1], [2]. This technique also has the benefit of power
saving, that is, the transmission power can be cut down inverse
proportionally to the number of antennas while keeping a
desirable rate [3]. The benefits in both spectral efficiency and
energy efficiency have made massive MIMO gain considerable
research attention lately [4], [5].
With the deployment of a large number of antennas, hard-
ware cost which scales linearly with the number of antennas
becomes tremendously high. For practical implementation
of massive MIMO, low-cost hardware is desirable and the
transceiver is unavoidably non-ideal due to hardware impair-
ments coming from power amplifier non-linearities, oscilla-
tor phase noise, in-phase/quadrature-phase (I/Q) imbalance
and/or quantization errors [6]–[11]. Generally, lower cost
hardware may experience higher hardware impairments. These
impairments may degrade the system capacity and reduce the
power saving benefit brought by massive MIMO. How much
degradation/reduction may occur is of great importance to be
investigated for the success of massive MIMO. Some results
on the impact of hardware impairments have been reported
recently and are summarized next.
A. Related Works
For point-to-point communication systems, the impact of
hardware impairments has been investigated in [12]–[15].
Specifically, [12] has shown that radio frequency (RF) impair-
ments at the transceiver would introduce more than 20%
error in the outage probability performance for single-input
single-output (SISO) vehicle-to-vehicle communications over
cascaded fading channels. Considering hardware impairments
in single-input multiple-output (SIMO) systems, [13] revealed
that a nonzero outage floor and a ceiling of the achievable rate
occurred even under high signal-to-noise ratio (SNR), which
is in contrast to the systems with ideal transceivers. In regard
to MIMO systems, [14] investigated the impact of residual
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8274 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 12, DECEMBER 2019
transmit RF impairments (RTRI) on MIMO channel estimation
and found that the RTRI would cause an estimation error floor
and require long training sequence even under high SNR, but
the training sequence could be shortened by the deployment
of a large number of receive antennas. Then in [15], hardware
impairments at both the transmitter and the receiver were
considered and the achievable rates of regular and large-
scale MIMO systems subject to Rician fading were derived
in closed-form. It was discovered that a finite ceiling on the
achievable rate existed irrespective of the transmission power
and fading conditions, and that the influence of hardware
impairments could be removed in large-scale MIMO systems
under favorable propagation.
The investigation of hardware impairment effects has also
been extended to cellular networks with massive MIMO base
stations (BSs) and single-antenna users in [8], [16]–[18].
Particularly, single-cell and multi-cell massive MIMO cellular
systems operating over Rayleigh fading channels were consid-
ered in [8] and [16], respectively. It was found that the hard-
ware impairments would create non-zero channel estimation
error floor and finite capacity ceiling in both the uplink and the
downlink. Moreover, up scaling of the hardware impairment
at the BS proportionally to the square root of the number
of antennas at the BS was tolerable without sacrificing the
expected performance gain. Later, heterogeneous cellular net-
works with channel aging caused by users’ relative movement
were investigated in [17]. It was revealed that the hardware
impairment at the BS transmitter had more significant impact
than that at the user receiver side on the downlink, while
the transceiver hardware impairments had equal impact on
the uplink. When Rician fading channels were considered,
[18] showed that hardware impairments at the BS were
allowed to scale up linearly with the number of antennas at the
BS without rate reduction due to the presence of line-of-sight
propagation, which further justified the adoption of low cost
hardware at the massive MIMO BSs.
Considering signal relaying as another promising tech-
nique to enhance performance and extend the communication
range, various relaying systems have been proposed in the
literature. How hardware impairments affect the performance
in relaying systems was discussed in [19]–[26]. Among
them, [19] and [20] investigated one-way relaying systems
in which all nodes were equipped with single antenna. Both
of them analyzed the outage probability and ergodic capacity
under non-ideal transceiver, and showed that the end-to-end
signal-to-noise-plus-distortion ratio converged to a ceiling that
cannot be crossed by increasing the signal powers or chang-
ing the fading conditions, due to the hardware impairments.
Meanwhile, [21] considered a relaying system with one single-
antenna source, one full-duplex multiple-antenna relay and
one multiple-antenna destination. Its analytical results showed
that the full-duplex decode-and-forward (DF) relay was more
robust to hardware impairments than the full-duplex amplify-
and-forward (AF) relay. Introducing massive antennas to the
one-way relay, the analysis in [22] demonstrated similar ceil-
ing on the achievable rate, which was mainly due to hardware
impairments at the source and the destination, but not the
hardware impairments at the relay, when the number of relay
antennas tended to infinity. In [23], a transceiver scheme
aware of hardware impairments was proposed to mitigate the
distortion noises by exploiting the statistical knowledge of
channels and antenna arrays at the sources and destinations.
It was shown that the ceiling effect could be mitigated when
the numbers of antennas at sources and destinations scaled
with that at the relay. Moreover, [24]–[26] analyzed the perfor-
mance of two-way MIMO relaying systems with multiple pairs
of single-antenna users. Specifically, [24] considered multiple
MIMO relays with opportunistic relay selection and derived
outage probability and system throughput in closed-forms.
It showed that the system performance became better when the
number of relays grew larger. However, the outage floor and
the throughput bound appeared when hardware impairments
exist. On the other hand, [25], [26] considered the two-way
relaying systems with only one massive MIMO relay and mul-
tiple pairs of single-antenna users. They analyzed the hardware
scaling law and showed that the hardware impairment at the
relay can be scaled up proportionally to its number of antennas
when the antenna array became large.
B. Contributions
Clearly, most of the aforementioned works on mas-
sive MIMO relaying systems consider single-antenna user
equipments (UEs). Although [23] does consider multiple-
antenna UEs, the same data streams are assumed to be
transmitted through the different paths for spatial diversity.
In this paper, we take a step forward to analyze two-way mas-
sive MIMO relaying systems with two multiple-antenna UEs
and non-ideal transceivers. To boost the spectral efficiency
further, spatial multiplexing is adopted at each UE, that
is, multiple data streams are transmitted from each UE.
This kind of massive MIMO relaying systems has a wide
range of applications for device-to-device (D2D) communica-
tions [27], [28]. The main contributions are as follows:
•To be general, MIMO nodes and antenna correlation at
both the UEs and the relay node are considered. Notice
that they are rarely considered for massive MIMO relay-
ing systems in the literature due to serious challenges in
the analysis. Moreover, transceiver hardware impairments
at all nodes are taken into account in our analysis.
•The achievable rate is analyzed and derived determinis-
tically in closed-form. Joint scaling of the transmission
powers and the hardware impairments is then investi-
gated. Feasible scaling speeds for the powers and impair-
ments are discovered when the number of relay antennas
grows large. It is shown that the down scaling of the
transmission powers and up scaling of the hardware
impairments are tolerable without reducing the expected
rate. However, the down scaling of the transmission
powers at the UEs and the relay cannot be faster than
the inverse of the number of relay antennas, and the
up scaling of the relay hardware impairment cannot be
faster than a linear rate with its number of antennas.
Moreover, hardware impairment at UEs is a key limiting
factor to the achievable rate and is not allowed to scale
up with the number of relay antennas in order to achieve
a non-vanishing rate.
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FENG et al.: TWO-WAY MASSIVE MIMO RELAYING SYSTEMS WITH NON-IDEAL TRANSCEIVERS 8275
•Ceiling effect on the achievable rate is still observable
and the ceiling rate varies among different scaling cases.
More interestingly, scalings of the UE transmission power
and the relay hardware impairment are found to be offset-
table. This indicates that the relay hardware cost and the
UE transmission power are tradable. It is found that the
best tradeoff is achieved in the medium scalings of both
the relay hardware impairment and the UE transmission
power. Our analysis is the first to discover the joint
scaling laws of transmission powers and hardware impair-
ments and the tradability between the relay hardware cost
and the UE transmission power. It thus provides solid
foundation for flexible system designs under various cost
and energy constraints.
The remainder of the paper is organized as follows.
In Section II, hardware impairments, transmission process of
two-way massive MIMO relaying systems and the instan-
taneous rate are introduced. Preliminaries on matrix chain
products which are necessary for the rate analysis are given
in Section III. In Section IV, the deterministic rate under gen-
eral power and hardware scalings is derived. Then, the asymp-
totic rate under some special scaling cases is analyzed in
Section V. In Section VI, numerical results follow to verify the
analysis and find new insights. Finally, conclusions are drawn
in Section VII.
Notation:diag(A)denotes a diagonal matrix with diagonal
elements equal to the diagonal entries of matrix A,and
a.s.
−−−−→
M→∞ denotes almost sure convergence when Mapproaches
infinity. The notation AM→∞
−−−−→¯
Awhere Aand ¯
Aare
matrices denotes that lim
M→∞ A=¯
A. Here, lim
M→∞ is defined
as element-wise limit operation on a matrix. The notation
f(x)=O(g(x)) means that when x→∞,|f(x)
g(x)|≤c,
in which cis a constant, i.e., f(x)varies with the same speed
as g(x).
II. SYSTEM MODEL
Consider a MIMO relaying system with two multiple-
antenna UEs,1U1and U2, and one multiple-antenna relay, R.
Each UE is equipped with a small number of antennas Mu
while the relay has a large number of antennas Mr, with
2MuMr. The two UEs exchange information with each
other via the relay. The detailed signal model is given in the
following.
A. Hardware Impairments
Usually, low cost hardware is attractive for practical imple-
mentation of MIMO systems, especially large-scale MIMO
systems with a massive number of antennas. However, the low
cost implementation leads to non-negligible impairments in the
transceiver, including I/Q imbalance, amplifier nonlinearity,
phase noise, and/or quantization errors. As shown in [29], [30]
and references therein, the hardware impairments can be com-
positely modelled as independent additive distortion noises
1Please note that our analysis in this paper can be easily extended to massive
MIMO relaying systems with multiple pairs of MIMO users. The extension
is similar to that in [4].
Fig. 1. The transmission diagram.
at the transmitter and receiver sides. Specifically, denote the
signal to be transmitted as xd,whered={u1,u
2,r}and
E[xuix†
ui]=IMu,i=1,2.Whend={u1,u
2},xd∈C
Mu
while when d=r,xd∈C
Mr. With transmitter side hardware
impairments, the transmitted signal is distorted as
˜
xd=xd+td,(1)
where tddenotes the additive distortion noise vector
at the transmitter and follows Gaussian distribution as
td∼CN(0,κ
2
ddiag{E[xdx†
d]}).2Here, κ2
dcaptures the distor-
tion level at each antenna, i.e., error vector magnitude (EVM)
[8], [19]. Moreover, the signal vector xdand the distortion
noise vector tdare independent, that is, the autocorrelations
of the transmitted signals at the UEs and the relay satisfy
E[˜
xui(˜
xui)†]=(1+κ2
ui)IMufor i=1,2and E[˜
xr(˜
xr)†]=
E(xrx†
r)+κ2
rdiag{E(xrx†
r)}, respectively.
At the receiver side, the signal is also distorted by the
hardware impairments and the distorted signal can be written
as
˜
yd=yd+rd,(2)
where ydis the received signal and rddenotes the
independent Gaussian distortion noise vector with rd∼
CN(0,κ
2
ddiag{E[ydy†
d]}). Note that ydand rdhave the same
dimension as xdand td. It then follows that E[˜
yd˜
y†
d]=
E[ydy†
d]+κ2
ddiag{E[ydy†
d]}.
B. Transmission Process
As shown in Fig.1, the two UEs exchange information
via the relay through two phases, i.e., multiple access (MA)
and broadcasting (BC) phases. In the MA phase, the UEs
simultaneously transmit signals xui,i=1,2,totherelay
with transmission power Pu. With hardware impairments at
each transceiver, the received signal at the relay is distorted
as
˜
yr=Pu
Mu
i=1,2
Huir(xui+tui
=˜
xui
)+rr+nr,(3)
2Although the hardware impairment caused by the clipping of a power
amplifier (PA) is non-Gaussian [31], the typical nonlinear distortion within
the PA operation range with sufficient backoff can be modeled as Gaussian
by following the Bussgang’s theorem. Thus, the sum of all the hardware
impairments caused by multiple hardware components involved in the com-
munication systems, e.g., PA, antenna coupling, and I/Q imbalance, can be
accurately modeled as Gaussian distortion noise as shown in [7], [23], [32].
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8276 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 12, DECEMBER 2019
where Huir∈C
Mr×Muis the channel matrix from user Ui
to the relay and nr∈C
Mrdenotes the additive noise vector
at the relay with the distribution nr∼CN(0,σ
2
rIMr).
At the relay, the AF strategy is adopted such that the
distorted received signal is amplified with an amplifying
matrix A∈C
Mr×Mrbefore forwarding. Due to hardware
impairments at the relay, the relay’s transmitted signal is also
distorted as
˜
xr=xr+tr=βA˜
yr
=xr
+tr,(4)
where βis a power factor to satisfy the transmit
power constraint at the relay node, i.e., E{tr[xrx†
r]}=
tr{E(xrx†
r)}=Pr. In the BC phase, the distorted signal ˜
xr
is broadcasted to the two UEs. The received signal at Uican
be expressed as
˜
yui=Hrui˜
xr+rui+nui,(5)
where Hrui∈C
Mu×Mrdenotes the channel matrix from the
relaytotheuserUiand nui∈C
Muis the additive Gaussian
noise at user Uiwith nui∼CN(0,σ
2
uiIMu).
Putting (3) and (4) into (5), the received signal at
the ith UE, Ui, can be rewritten as
˜
yui=βPu
Mu
HruiAHu¯
irxu¯
i+βPu
Mu
HruiAHuirxui
+βPu
Mu
HruiAHu¯
irtu¯
i+βPu
Mu
HruiAHuirtui
+βHruiAnr+Hruitr+βHruiArr+rui+nui,(6)
where ¯
idenotes the other UE, with ¯
i=2
ifor i=1,2.
In (3)-(6), Hab signifies the channel matrix of the link from
node ato node b. With a massive number of antennas and lim-
ited spacing between antennas, the channel coefficients may
be spatially correlated. Considering the antenna correlations,
the channel matrix can be generally modeled based on a sto-
chastic Kronecker model as Hab =Ξ1/2
bWabΞ1/2
a[33], [34].
Here, the matrices Ξband Ξaare positive Hermitian deter-
ministic matrices with bounded spectrum and unity diagonal
elements. They denote the antenna correlation matrices at
nodes band arespectively, while Wab is a random matrix
with independent and identically distributed (i.i.d.) Gaussian
random elements with zero mean and variance γ2
ab.When
a=rand b=ui, the dimensions of these matrices are
Ξa∈C
Mr×Mr,Ξb∈C
Mu×Muand Wab ∈C
Mu×Mr,
while when a=uiand b=r, the dimensions become
Ξa∈C
Mu×Mu,Ξb∈C
Mr×Mrand Wab ∈C
Mr×Mu.This
stochastic Kronecker model is widely adopted for correlated
Rayleigh fading MIMO channels in centimeter communication
systems [33], [34]. It is also applicable to millimeter wave
communication systems under rich scattering environment,
as shown in [35] and [36].
Based on (3) and (4), the term E(xrx†
r)in the relay power
constraint can be written as
E(xrx†
r)=β2⎧
⎨
⎩
Pu
Mu
i=1,2
(1 + κ2
ui)AHuirH†
uirA†
+AE(rrr†
r)A†+σ2
rAA†⎫
⎬
⎭
,(7)
with E(rrr†
r)being
E(rrr†
r)
=κ2
rdiag{E[yry†
r]}
=κ2
r⎧
⎨
⎩
Pu
Mu
i=1,2
(1+ κ2
ui)diag[HuirH†
uir]+σ2
rIMr⎫
⎬
⎭
.(8)
Substituting (7) and (8) into the relay power constraint
tr{E(xrx†
r)}=Pr, the power factor can be determined as
per (9), shown at the top of the next page.
C. Amplifying Matrix
As shown in [37] and [38], the maximum-ratio-
combining/maximum-ratio-transmission (MRC/MRT) scheme
is one prevalent AF scheme with low-complexity. It simply
matches the relay transceiver with the user-to-relay and the
relay-to-user channels. It is nearly optimal and has been widely
adopted in massive MIMO AF relaying systems [37], [38].
We thus also adopt this scheme at the relay node. Then,
the amplifying matrix Ais given as
A=H†
ruiH†
ru¯
i
0Mu,MuP¯
i→i
Pi→¯
i0Mu,Mu
HuirHu¯
ir
†,(10)
where P¯
i→i,Pi→¯
i∈CMu×Muare permutation matrices with
only one element in each row and column being 1 while all the
other elements are 0. The permutation matrices are introduced
for the flexibility of information exchange on a per-antenna
basis between the two UEs.
D. Instantaneous Achievable Rate
In the received signal at node Ui(6), the UE’s own transmit-
ted signal xuiis also received through the two-way relaying.
With perfect channel state information at both the relay and
the users, this self-interference can be canceled out, but the
transceiver’s hardware impairments and noises remain. After
self-interference cancellation, the remaining signal becomes
(11), shown at the top of the next page. Here, (o)is the
desired signal; (a)and (b)come from the transmitter side
hardware impairments from nodes U¯
iand Ui, respectively;
(c)denotes the propagated noise from the relay; (d)and (e)
correspond to the relay hardware impairments at the transmit-
ter and the receiver sides, respectively; (f)is the receiver side
hardware impairment at Ui, while (g)is the additive Gaussian
noise at Ui. The received signal model in (11) is general to
accommodate various practical hardware impairments at all
the nodes. It can reduce to the signal models under special
cases with perfect hardware deployment by setting the corre-
sponding impairments to zero. This general model enables us
to thoroughly investigate the impacts of hardware impairments
from the UEs and the relay on the system performance and
provide solid theoretical foundation for system design in
practice.
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FENG et al.: TWO-WAY MASSIVE MIMO RELAYING SYSTEMS WITH NON-IDEAL TRANSCEIVERS 8277
β=Pr⎧
⎨
⎩
Pu
Mu
i=1,2
(1 + κ2
ui)tr(AHuirH†
uirA†)+κ2
rtr[Adiag(HuirH†
uir)A†]+(1+κ2
r)σ2
rtr(AA†)⎫
⎬
⎭
−1
2
(9)
¯
yui=˜
yui−βPu
Mu
HruiAHuirxui=βPu
Mu
HruiAHu¯
irxu¯
i
(o)
+βPu
Mu
HruiAHu¯
irtu¯
i
(a)
+βPu
Mu
HruiAHuirtui
(b)
+βHruiAnr
(c)
+Hruitr
(d)
+βHruiArr
(e)
+rui
(f)
+nui
(g)
(11)
Given (11), the instantaneous achievable rate of user Uican
be expressed as:3
Cui=1
2log2det ⎧
⎪
⎨
⎪
⎩
IMu+Ψui⎡
⎣
6
j=1
Ωui,j +σ2
uiIMu
⎤
⎦
−1
⎫
⎪
⎬
⎪
⎭
,(12)
where matrices Ψuiand Ωui,j ,j=1,...,6, correspond to
the terms (o)-(f)respectively, and are given as follows
Ψui=β2Pu
Mu
[HruiAHu¯
ir][H†
u¯
irA†H†
rui],(13)
Ωui,1=β2κ2
u¯
i
Pu
Mu
[HruiAHu¯
ir][H†
u¯
irA†H†
rui],(14)
Ωui,2=β2κ2
ui
Pu
Mu
[HruiAHuir][H†
uirA†H†
rui],(15)
Ωui,3=σ2
rβ2HruiAA†H†
rui,(16)
Ωui,4=Hrui
E[trt†
r]H†
rui=κ2
rHruidiag[E(xrx†
r)]H†
rui,(17)
Ωui,5=β2[HruiA]E[rrr†
r][A†H†
rui],(18)
Ωui,6=E[ruir†
ui]=κ2
uidiag{E[yuiy†
ui]}
=κ2
uidiag HruiE[˜
xr˜
x†
r]H†
rui+σ2
uiIMu
=κ2
uidiag[HruiE(xrx†
r)H†
rui]
+κ2
uiκ2
rdiag{Hruidiag[E(xrx†
r)]H†
rui}
+κ2
uiσ2
uiIMu.(19)
Please notice that the achievable rate in (12) is derived under
the assumption of perfect CSI at both the relay and the UEs.
In practice, CSI is usually estimated with the aid of
pilot sequences. Good channel estimation can be achieved
with low estimation errors by using advanced estimation
3The signal processing in our two-way massive MIMO relaying systems is
somehow similar to those in the conventional two-way AF MIMO relaying
systems [39] and [40], except that the antenna scale in the relay is much
larger than that in the conventional systems. However, our work is significantly
different from them since both transceiver hardware impairments and antenna
correlation are considered. Moreover, owing to the deployment of large-scale
antenna array at the relay, the random instantaneous achievable rate for the
massive MIMO relaying systems in (12) will approach to a constant when the
system becomes very large as analyzed later. The constant rate and the impact
of hardware impairment on the rate are our focus for analysis in this paper.
Their analysis is our key contribution and differentiates our work further from
that for the conventional two-way AF MIMO relaying systems.
algorithms [41], [42]. Similarly to [23], [25], here channel esti-
mation errors are not considered for simplicity. The rate in (12)
in fact can be regarded as an upper bound of the achievable
rate in practical systems with estimated channel information.
This upper bound is analyzed to extract meaningful insights
on the impact of hardware impairments.
Substituting the impairment statistics E(xrx†
r)and E[rrr†
r]
shown in (7) and (8) into (17)-(19), the terms Ωui,4,Ωui,5
and Ωui,6can be rewritten as (20)-(22), shown at the top of
the next page.
Apparently, the expression of the instantaneous achievable
rate is much different and more complicated than that for
massive MIMO relaying systems with single antenna users
in [19]–[23], [25], [26]. Since MIMO users are considered
here, the rate expression involves complicated matrix opera-
tions such as matrix inverse and determinant operations, matrix
chain products and nonlinear matrix operation diag{·},which
causes mathematical difficulties and makes the achievable
rate analysis challenging. Also, the consideration of antenna
correlation and transceiver hardware impairment here further
complicates the analysis. As far as we know, there are no
analytical results on the joint impact of antenna correlation
and hardware impairment available in the literature.
III. PRELIMINARIES ON MAT R I X CHAIN PRODUCTS
By substituting the amplifying matrix (10) into the instan-
taneous achievable rate (12), we find that various matrix chain
products and diagonal operation of the channel matrices are
involved. To proceed with the rate analysis, deterministic
equivalents of the random matrix chain products are necessary
and some preliminary results are first given in the following.
Theorem 1: Consider a random matrix with the form
Hvu =R1/2
uWvuT1/2
v,inwhichRu∈C
N×Nand Tv∈
CM×Mare Hermitian positive deterministic matrices with
bounded spectrum and diagonal elements being one, and
Wvu ∈C
N×Mis a random matrix with i.i.d. Gaussian
elements with zero mean and variance σ2
vu. Given a deter-
ministic matrix Gwith compatible dimension, the matrix chain
products almost surely converge as
1
N2(H†
vuHvu)G(H†
vuHvu)
a.s.
−−−−→
N→∞ σ4
vu
(TvGTv)+tr(R2
u)
N2tr(TvG)Tv,(23)
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8278 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 12, DECEMBER 2019
Ωui,4=κ2
rβ2Pu
Mu
j=1,2
(1 + κ2
uj)Hruidiag AHujrH†
ujrA†H†
rui+(1+κ2
r)κ2
rσ2
rβ2Hruidiag AA†H†
rui
+κ4
rβ2Pu
Mu
j=1,2
(1 + κ2
uj)Hruidiag Adiag HujrH†
ujrA†H†
rui,(20)
Ωui,5=κ2
rβ2Pu
Mu
j=1,2
(1 + κ2
uj)HruiAdiag HujrH†
ujrA†H†
rui+κ2
rσ2
rβ2HruiAA†H†
rui,(21)
Ωui,6=κ2
ui⎧
⎨
⎩
1+κ2
u¯
i
κ2
u¯
i
diag(Ωui,1)+ 1+κ2
ui
κ2
ui
diag(Ωui,2)+
j=3,4,5
diag(Ωui,j)+σ2
uiIMu⎫
⎬
⎭
(22)
1
NH†
v1uHv2uGH†
v2uHv1u
a.s.
−−−−→
N→∞ σ2
v1uσ2
v2u
tr(R2
u)
Ntr(Tv2G)Tv1,(24)
1
NH†
vuHvu
a.s.
−−−−→
N→∞ σ2
vuTv.(25)
Proof: See Appendix A.
Theorem 2: Given the same matrix definitions in
Theorem 1, the following results on the matrix chain
products with diagonal operations hold almost surely:
1
NH†
vudiag(HvuGH†
vu)Hvu
a.s.
−−−−→
N→∞ σ4
vu {TvGTv+tr(TvG)Tv},(26)
1
NH†
v1udiag(Hv2uGH†
v2u)Hv1u
a.s.
−−−−→
N→∞ σ2
v1uσ2
v2utr(Tv2G)Tv1.(27)
Proof: See Appendix B.
Notice that the diagonal operations of the channel matrices
here are introduced due to the consideration of transceiver
hardware impairments, which is one of the main challenges in
the analysis. Theorem 2 is the first to discover the deterministic
results of random matrix products with nonlinear diagonal
operations. This result will enable the asymptotic analysis of
the achievable rate as will be detailed later.
IV. DETERMINISTIC ACHIEVABLE RAT E
Due to the randomness in wireless channels, the instan-
taneous achievable rate in (12) is random. However, with a
massive number of antennas in the relay, the instantaneous
achievable rate will approach a deterministic rate when the
number of relay antennas becomes large (Mr→∞),asshown
later. To not only ease the analysis but also investigate the
scaling laws of powers and hardware impairments, we assume
the transmission powers and the hardware impairments to scale
with the number of relay antennas as follows: Pu=Eu
Mk
r
and Pr=Er
Mq
rwith Euand Erfixed; κ2
r=κ2
r,0Mm
rand
κ2
ui=κ2
ui,0Mn
rwith κ2
r,0and κ2
ui,0fixed. Here k, q, m, n are
non-negative rational numbers, i.e., k, q, n, m ≥0. Under this
scaling, the transmission powers at the UEs and the relay are
scaled down with the number of relay antennas Mr, while the
hardware impairments at the UEs and the relay are scaled up
with Mr, i.e., the hardware quality/cost is degraded/reduced
with the growth of Mr. Here, the factors k, q, m, n control
the speeds of the corresponding scalings; larger factors mean
faster scaling. Substituting the scaled powers and hardware
impairments into (9) and (12), and using Theorems 1 and 2,
we obtain the following deterministic result on the achievable
rate.
Theorem 3: When the number of antennas at the relay
approaches infinity (Mr→∞), the instantaneous achievable
rate in (12) for the massive MIMO relaying system almost
surely converges to a deterministic rate as
Cui
a.s.
−−−−−→
Mr→∞
¯
Cui,(28)
with ¯
Cuias (29), shown at the top of the next page, in
which ¯
Ωui,j for j=1,...,6are given as (30)-(35), shown at
the top of the next page, and ¯
βis given by
¯
β=Er Eu
Mu
¯
θ1+κ2
r,0Eu
Mu
1
M1−m
r
¯
θ2+(1
Mm
r+κ2
r,0)σ2
r
M1−k−m
r
¯
θ3
!−1
2
,
(36)
in which ¯
Ωui,4,j,j=1,2,3,¯
Ωui,5,1, and ¯
θjfor j=1,2,3,
are as defined in Appendix C. The terms (o)-(g)in (29)
correspond to the terms (o)-(g)in (12), respectively.
Proof: See Appendix C.
It is clear from (29) that the antenna correlations at the UEs
and the relay, Ξuiand Ξr, affect the achievable rate. The
impact of the relay antenna correlation Ξron the achievable
rate is clear since it is only involved through the scalar tr(Ξ2
r)
M2
r.
As proved in [43], the term tr(Ξ2
r)
M2
ris Schur-convex with
respect to the eigenvalue vector of the antenna correlation
matrix Ξr.4This means that with the increase of the relay
antenna correlation, the achievable rate will decrease. How-
ever, as shown in Lemma 2 in Appendix A, the term tr(Ξ2
r)
M2
r
approaches zero when Mr→∞, i.e., tr(Ξ2
r)
M2
r−→ 0. As such,
the negative impact of the relay antenna correlation becomes
insignificant when the number of relay antennas is large. With
respect of the impact of the UE’s antenna correlation Ξui
4Assuming the eigenvalues of matrix Ξrin descending order are λΞr
1≥
λΞr
2≥,...,≥λΞr
Mr, the eigenvalue vector is defined as ΛΞr
(λΞr
1,λ
Ξr
2,...,λ
Ξr
Mr).
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FENG et al.: TWO-WAY MASSIVE MIMO RELAYING SYSTEMS WITH NON-IDEAL TRANSCEIVERS 8279
¯
Cui=1
2log2det ⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
IMu+(¯
β)2Eu
Mu
γ4
ruiγ4
u¯
irΞuiP¯
i→iΞ2
u¯
iP†
¯
i→iΞui
(o)
⎡
⎢
⎢
⎢
⎣
κ2
u¯
i,0Mn
r(¯
β)2Eu
Mu
¯
Ωui,1
(a)
+κ2
ui,0Mn
r(¯
β)2Eu
Mu
¯
Ωui,2
(b)
+σ2
r(¯
β)21
M1−k
r
¯
Ωui,3
(c)
+
j=4,5,6
¯
Ωui,j
(d),(e),(f)
+1
M1−q
r
σ2
uiIMu
(g)
⎤
⎥
⎥
⎥
⎥
⎥
⎦
−1⎫
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎭
,(29)
¯
Ωui,1=γ4
ruiγ4
u¯
irΞuiP¯
i→iΞ2
u¯
iP†
¯
i→iΞui+tr(Ξ2
r)
M2
rtr(Ξu¯
i)ΞuiP¯
i→iΞu¯
iP†
¯
i→iΞui
+tr(ΞuiP¯
i→iΞ2
u¯
iP†
¯
i→i)Ξui+$tr(Ξ2
r)
M2
r%2
tr(Ξu¯
i)γ2
ruiγ2
u¯
ir⎡
⎣
i=1,2
γ2
ruiγ2
u¯
irtr(ΞuiP¯
i→iΞu¯
iP†
¯
i→i)⎤
⎦Ξui,(30)
¯
Ωui,2=γ2
uirγ2
rui
tr(Ξ2
r)
M2
rγ2
ruiγ2
u¯
irtr(Ξui)ΞuiP¯
i→iΞu¯
iP†
¯
i→iΞui
+⎡
⎣γ2
uirγ2
ru¯
itr(Ξu¯
iPi→¯
iΞ2
uiP†
i→¯
i)+[tr(Ξ2
r)
M2
r
tr(Ξui)]
i=1,2
γ2
ruiγ2
u¯
irtr(ΞuiP¯
i→iΞu¯
iP†
¯
i→i)⎤
⎦Ξui⎫
⎬
⎭
,(31)
¯
Ωui,3=γ2
rui⎧
⎨
⎩
γ2
ruiγ2
u¯
irΞuiP¯
i→iΞu¯
iP†
¯
i→iΞui+tr(Ξ2
r)
M2
r⎡
⎣
i=1,2
γ2
ruiγ2
u¯
irtr(ΞuiP¯
i→iΞu¯
iP†
¯
i→i)⎤
⎦Ξui⎫
⎬
⎭
,(32)
¯
Ωui,4=κ2
r,0(¯
β)2Eu
Mu
1
M1−m
r
¯
Ωui,4,1+κ4
r,0(¯
β)2Eu
Mu
1
M2−2m
r
¯
Ωui,4,2
+( 1
Mm
r
+κ2
r,0)κ2
r,0σ2
r(¯
β)21
M2−2m−k
r
¯
Ωui,4,3,(33)
¯
Ωui,5=κ2
r,0(¯
β)2Eu
Mu
1
M1−m
r
¯
Ωui,5,1+κ2
r,0σ2
r(¯
β)21
M1−k−m
r
¯
Ωui,3,(34)
¯
Ωui,6=(κ2
ui,0Mn
r)Eu
Mu
(¯
β)2&(1 + κ2
u¯
i,0Mn
r)diag( ¯
Ωui,1)+(1+κ2
ui,0Mn
r)diag( ¯
Ωui,2)'
+(κ2
ui,0Mn
r)σ2
r(¯
β)21
M1−k
r
diag( ¯
Ωui,3)+diag( ¯
Ωui,4)+diag( ¯
Ωui,5)+σ2
ui
1
M1−q
r
IMu(35)
on the achievable rate, it is quite complicated since the antenna
correlation matrices Ξuiat the two UEs are coupled together
and intertwined with other matrices through matrix operations.
It is thus difficult to investigate the impact of UE antenna cor-
relation analytically. But with the expression of the achievable
rate in (29), the impact can be evaluated numerically and will
be discussed later in the simulation section.
By further analyzing the exponents k, q, m, n with base the
number of relay antennas Mrin (29), we can also obtain
the following result on the feasible power and hardware
impairment scaling factors k,q,mand n.
Corollary 1: To achieve a non-vanishing rate in (29) when
Mr→∞, the power and hardware impairment scaling factors
should satisfy the following conditions: 0≤k, q ≤1,n=0
and 0≤m≤1−k≤1.
Proof: As shown in (29), the term (o)is a con-
stant irrelevant to the number of relay antennas Mr.When
Mr→∞, to achieve a non-vanishing rate, i.e., ¯
Cui0,
the scalar coefficients in the terms (a)-(g)corresponding to
various impairments and noises should not approach infinity.
In other words, the exponents k, q, m, n of Mrin (a)-(g)
should satisfy the following conditions:
n≤0,m≥0,1−k≥0,1−q≥0,
1−m≥0,2−2m−k≥0,1−k−m≥0.(37)
Together with the assumptions of k, q, m, n ≥0, solving the
inequalities in (37) yields the feasible regions for the scaling
factors, which completes the proof.
From Corollary 1, the following observations can be made.
1) Although massive MIMO technique has the benefit of
power saving without degrading capacity [44], [45],
the transmission powers (Prand Pu) at the relay and
the UEs in the system under consideration cannot be
scaled down faster than 1
Mrwith the increase of the
number of relay antennas Mr,i.e.,k, q ≤1.Otherwise,
the achievable rate will become zero. This is consistent
with the results for massive MIMO systems [4], [5].
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8280 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 12, DECEMBER 2019
2) The feasible scaling factor for the hardware impairments
at the UEs (n=0) means that the corresponding
hardware impairments κ2
uiare not allowed to increase
with the number of relay antennas so as to achieve
the non-vanishing rate. In other words, even with a
large number of antennas at the relay, the quality of
the hardware in the UEs cannot be degraded with the
increase of Mr, i.e., the UE hardware cost cannot be
cut proportionally.
3) The scaling factor mcorresponding to the hardware
impairment at the relay should not be larger than one,
specifically, 0≤m≤1. It means that with the
deployment of massive relay antennas, low cost/quality
hardware can be applied at the relay and the hardware
impairment increasing with the number of relay antennas
is tolerable. However, the increasing speed cannot be
faster than linear rate.
4) The power scaling factor kfor the UEs and the relays’
hardware impairment scaling factor mshould satisfy
0≤m+k≤1. This somehow implies that the
scalings of the transmission power at the UEs and
the hardware impairment at the relay may be offset-
table. This will be further investigated in the next
section.
V. J OINT SCALING OF TRANSMISSION POWERS
AND HARDWARE IMPAIRMENTS
Based on the results in Theorem 3 and Corollary 1,
the scaling laws of the transmission powers and the hardware
impairments can be investigated. To reveal the relationship
between the scalings of the transmission power at the UEs
and the hardware impairment at the relay, we take special
cases with q=1,n=0and m+k=1. Under these cases,
the relay transmission power is scaled down as Pr=Er
Mr,and
the UE hardware impairment level is constant as κ2
ui=κ2
ui,0.
Moreover, the channel variances corresponding to the MAC
and BC phases are assumed equal, γrui=γuir=γur ,for
i=1,2, and the hardware impairments at the UEs are the
same, κ2
ui,0=κ2
u¯
i,0=κ2
u,0. Sufficient antenna spacing is
also assumed so that no antenna correlation occurs at the
UEs, i.e., Ξui=Ξu¯
i=IMu. Three scenarios corresponding
to various combinations of the scalings of the transmission
power at the UEs and the hardware impairment at the relay
are discussed in the following. Notice that the results can be
easily extended to general cases with arbitrary scaling factors
within the feasible region in Corollary 1.
A. Constant User Transmit Power and Up-Scaled Relay
Hardware I mp airment: k=0, m=1
In this case, the UE transmission power is constant as
Pu=Eu, and the relay’s hardware impairment is linearly
scaled up with its number of antennas, i.e., κ2
r=κ2
r,0Mr.
In other words, the quality/cost of relay hardware is scaled
down when the number of relay antennas increases.
Putting these scaling factors and Ξui=Ξu¯
i=IMu
into (36), when Mr→∞, the power factor ¯
βapproaches
to
¯
βMr→∞
−−−−−→¯
β∞
H
=Er2(1+κ2
u,0)(1+κ2
r,0)γ6
urEu
+2 γ4
urκ2
r,0Mu[2(1+κ2
u,0)γ2
ur Eu+σ2
r]−1
2
.(38)
Together with the asymptotic result of the relay antenna cor-
relation matrix, i.e., tr(Ξ2
r)
M2
r−→ 0, we can derive the asymptotic
results of the terms (a)-(g)in the deterministic rate (29) as:
(a)=κ2
u¯
i,0Mn
r(¯
β)2Eu
Mu
¯
Ωui,1
Mr→∞
−−−−−→κ2
u,0γ8
ur(¯
β∞
H)2Eu
Mu
IMu,
(39)
(b)=κ2
ui,0Mn
r(¯
β)2Eu
Mu
¯
Ωui,2
Mr→∞
−−−−−→0Mu,(40)
(c)=σ2
r(¯
β)21
Mr
¯
Ωui,3
Mr→∞
−−−−−→0Mu,(41)
(d)Mr→∞
−−−−−→¯
¯
ΩH,∞
ui,4=(2Mu+1)γ6
urκ2
r,0(¯
β∞
H)2
×Eu
Mu
γ2
ur(1+ κ2
u,0)&1+κ2
r,0(2Mu+1)
'+κ2
r,0σ2
rIMu,
(42)
(e)Mr→∞
−−−−−→¯
¯
ΩH,∞
ui,5=γ6
urκ2
r,0(¯
β∞
H)2
×Eu
Mu
γ2
ur(1 + κ2
u,0)(2Mu+1)+σ2
rIMu,(43)
(f)Mr→∞
−−−−−→κ2
u,0⎧
⎪
⎨
⎪
⎩
(1 + κ2
u,0)γ8
ur(¯
β∞
H)2Eu
Mu
IMu
+
j=4,5
diag( ¯
¯
ΩH,∞
ui,j )+σ2
uiIMu⎫
⎪
⎬
⎪
⎭
,(44)
(g)=σ2
uiIMu.(45)
Here, Mr→∞
−−−−−→in (39)-(44) corresponds to element-wise limit
operation. Recalling the definitions of the terms (a)-(g)
in (11), it is clear from the results in (39)-(45) that when the
number of antennas at the relay goes to infinity, the transmitter
side hardware impairment from the user itself (i.e., term (b))
and the propagated noise from the relay (i.e., term (c)) are
eliminated while the transmitter side hardware impairment
from the other user, the transceiver hardware impairments from
the relay, the receiver side hardware impairments from the user
itself and the additive Gaussian noise at the user are retained
in this case. Substituting (38)-(45) into (29), the achievable
rate asymptotically approaches a constant:
¯
Cui
Mr→∞
−−−−−→¯
C∞
ui,H
=Mu
2log21+&IH,1κ4
r,0+IH,2κ2
r,0+IH,3'−1,(46)
with IH,j ,j=1,2,3,givenby
IH,1=(2Mu+1)
2(1 + κ2
u,0)2
+(2Mu+ 1)(1 + κ2
u,0)Mu
σ2
r
γ2
urEu
,(47)
IH,2=2(2Mu+ 1)(1 + κ2
u,0)2σ2
ui
γ2
urEr
Mu+1
+(2 σ2
ui
γ2
urEr
Mu+ 1)(1 + κ2
u,0)Mu
σ2
r
γ2
urEu
,(48)
IH,3=2(1+κ2
u,0)2Mu
σ2
ui
γ2
urEr
+κ2
u,0(2 + κ2
u,0).(49)
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FENG et al.: TWO-WAY MASSIVE MIMO RELAYING SYSTEMS WITH NON-IDEAL TRANSCEIVERS 8281
This result shows that a constant rate can be achieved even
when the relay transmission power scales down at a rate
of 1
Mrand the relay hardware impairment scales up linearly
with the number of relay antennas simultaneously. This means
that, when massive antennas are deployed at the relay, its
transmission power and hardware cost can be reduced simul-
taneously while keeping constant achievable rate.
In high SNR regime, i.e., Er
σ2
ui→∞and Eu
σ2
r→∞,theterms
caused by the noises vanish and the asymptotic achievable rate
shown in (46) becomes (50), shown at the bottom of the next
page. It is seen from (50) that even with high SNRs at the
UEs and the relay, the instantaneous rate can not approach
infinity, and a ceiling exists due to the hardware impairments
at all nodes.
B. Down-Scaled User Transmit Power and Constant Relay
Hardware I mp airment: k=1,m=0
In this scenario, the user transmission power scales down at
arateof 1
Mras Pu=Eu
Mr, and the relay hardware impairment
is constant as κ2
r=κ2
r,0. Similar to the previous case, putting
the scaling factors m=n=0and k=q=1along with
Ξui=Ξu¯
i=IMuinto the deterministic result in Theorem 3
leads to the asymptotic result of the power factor ¯
βas
¯
βMr→∞
−−−−−→¯
β∞
L=Er2γ6
urEu(1 + κ2
u,0)
+2σ2
rγ4
urMu(1 + κ2
r,0)−1
2.(51)
Moreover, the asymptotic results in (39), (40) and (45), for the
terms (a),(b)and (g), still hold by replacing ¯
β∞
Hin (39) with
¯
β∞
Lin (51), while the other terms (c)-(f)vary as follows:
(c)Mr→∞
−−−−−→σ2
r(¯
β∞
L)2γ6
urIMu,(52)
(d)=κ2
r,0(¯
β)2⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
1
Mr
Eu
Mu
¯
Ωui,4,1
+1
M2
r
κ2
r,0
Eu
Mu
¯
Ωui,4,2
+1
Mr
(1+ κ2
r,0)σ2
r¯
Ωui,4,3
⎫
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎭
Mr→∞
−−−−−→0Mu,(53)
(e)Mr→∞
−−−−−→κ2
r,0σ2
r(¯
β∞
L)2γ6
urIMu,(54)
(f)Mr→∞
−−−−−→κ2
u,0⎧
⎨
⎩
(1+ κ2
u,0)( ¯
β∞
L)2Eu
Mu
γ8
ur
+( 1 + κ2
r,0)σ2
r(¯
β∞
L)2γ6
ur +σ2
ui
⎫
⎬
⎭
IMu.(55)
Clearly in this case, the transmitter side hardware impairments
from the user itself and the relay can be eliminated while
the other terms are retained when the number of relay anten-
nas tends to infinity. Then putting these asymptotic results
into (29) of Theorem 3 yields the asymptotic rate as (56),
shown at the bottom of the next page. This indicates that
with constant hardware impairments at the UEs and the
relay, the transmission powers at these nodes can be scaled
down simultaneously at the rate of 1
Mrwithout degrading the
achievable rate when the number of relay antennas increases.
When the SNRs are high, i.e., Er
σ2
ui→∞and Eu
σ2
r→∞,
the rate will approach a ceiling as
¯
C∞
ui,L
SNR→∞
−−−−−−→¯
C∞
ui,L,HSN R=Mu
2log2 1+ 1
κ2
u,0(2+κ2
u,0)!.
(57)
Different from (50), this ceiling depends on the UEs’ hardware
impairment κ2
u,0only and grows linearly with their number of
antennas Mu.
C. Down-Scaled User Transmit Power and Up-Scaled Relay
Hardware I mp airment: k=1/2, m=1/2
In this case, the UE’s transmission power scales down as
Pu=Eu
√Mrand the relay hardware impairment scales up
as κ2
r=κ2
r,0√Mr. Following a similar approach, the power
factor ¯
βin (36) asymptotically becomes
¯
βMr→∞
−−−−−→¯
β∞
M=Er2(1+κ2
u,0)γ6
urEu+2γ4
urκ2
r,0σ2
rMu
−1
2.
(58)
When Mr→∞,theterms(a),(b)and (g)in (29) still
approach asymptotically to (39), (40) and (45), respectively,
by replacing ¯
β∞
Hin (39) with ¯
β∞
Min (58), while the asymp-
totical results for the terms (c)-(f)change as
(c)=σ2
r(¯
β)21
√Mr
¯
Ωui,3
Mr→∞
−−−−−→0Mu,(59)
(d)Mr→∞
−−−−−→0Mu,(60)
(e)=κ2
r,0(¯
β)21
√Mr
Eu
Mu
¯
Ωui,5,1+σ2
r¯
Ωui,3
Mr→∞
−−−−−→κ2
r,0σ2
r(¯
β∞
M)2γ6
urIMu,(61)
(f)Mr→∞
−−−−−→κ2
u,0(1 + κ2
u,0)( ¯
β∞
M)2Eu
Muγ8
ur
+κ2
r,0σ2
r(¯
β∞
M)2γ6
ur +σ2
uiIMu.(62)
Different from the previous cases, the transmitter side hard-
ware impairments from the user itself and the relay, the propa-
gated noise from the relay vanish while the other terms remain
even with infinity number of relay antennas.
With the asymptotic results in (39), (40), (45) and (59)-(62),
the asymptotic achievable rate can be derived as (63), shown
at the bottom of the next page.
Under high SNR, the achievable rate further approaches a
ceiling as
¯
C∞
ui,M
SNR→∞
−−−−−−→¯
C∞
ui,M,HS N R
=Mu
2log2 1+ 1
κ2
u,0(2 + κ2
u,0)!,(64)
which is the same as (57) and depends on the UE hardware
impairment κ2
u,0only. Clearly, these results show that a
constant rate can still be achieved even when the transmission
power of the relay and the UEs scale down as Pr=Er
Mrand
Pu=Eu
√Mr, respectively, and the relay hardware impairment
scales up as κ2
r=κ2
r,0√Mrsimultaneously.
D. Summary and Discussion
With the above analytical results, the scaling laws of the
transmission powers and the hardware impairments can be
summarized as follows. The transmission powers at the UEs
and the relay can be scaled down at a rate of 1
Mrwhile
keeping a constant achievable rate, when the number of relay
antennas grows large. In regard to the hardware impairment
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8282 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 12, DECEMBER 2019
scaling, the relay’s hardware impairment can be scaled up
linearly with its number of antennas while achieving a non-
zero constant rate, however the UE’s hardware impairment
cannot be scaled up with the number of relay antennas. This
means that the UE hardware impairment is a key limiting
factor to the achievable rate and, thus, high quality hardware is
suggested to be deployed at the UEs. This is in fact affordable
since only a small number of antennas and RF chains are
deployed at the UEs, and the hardware cost will not increase
with the number of relay antennas. On the other hand, since a
massive number of antennas are deployed at the relay, the relay
hardware cost is high and will increase significantly with the
increase of the number of antennas. In fact, our analytical
result on the scaling law of the relay hardware impairment
provides a solid theoretical support to adopt massive antennas
at the relay and reduce the relay hardware cost by using low
quality hardware without degrading the achievable rate when
the number of relay antennas grows large.
More interestingly, we found that the scalings of UE trans-
mission power and the relay hardware impairment should
satisfy k+m≤1and they are offsettable. Three cases with
different scaling combinations have been particularly investi-
gated. The first is with k=0and m=1, and corresponds
to the case with high UE transmission power and high relay
hardware impairment. The second case is with k=1and
m=0, and corresponds to low user transmission power and
low relay hardware impairment. The third one is with k=1/2
and m=1/2and denotes a case with medium user trans-
mission power and medium relay hardware impairment. From
the asymptotic results in (46), (56) and (63), we can see that
all three cases can achieve constant asymptotic rates, which
indicates that relay hardware impairment is offsettable with UE
transmission power and high relay hardware impairment can
be compensated by high UE transmission power. Moreover,
it is easy to show that ¯
C∞
ui,M >¯
C∞
ui,H and ¯
C∞
ui,M >¯
C∞
ui,L.
This means that although the UE transmission power increase
is tradable with the relay hardware cost reduction (i.e., relay
impairment increase), the tradeoff between them should be
properly designed to maximize the achievable rate. The best
TAB L E I
SIMULATION SETTING
tradeoff may be in the region with medium UE transmission
power and medium relay impairment.
VI. SIMULATION RESULTS
Numerical results are provided to verify the correctness
of the theoretical analysis and, at the same time, find new
insights which are otherwise hard to obtain analytically. The
simulation results are obtained based on the system setting
shown in Table I unless otherwise stated.
A. Verification of the Deterministic and Asymptotic Results
Fig. 2 shows the deterministic rate in (29) and the asymp-
totic rates in (46), (56) and (63) for different scaling cases.
The deterministic rates match well with the simulated instan-
taneous rates, and both approach the corresponding asymptotic
rates (dashed line plots) when the number of relay antennas
Mrgrows large. Also, the convergence speed for the case
k=m=1
2is slower than the other cases. This is consistent
with the previous analysis in (29). By putting the values
of k, m, q, n into the terms (a)−(g), we can find that the
convergence speed of the terms is O(1
√Mr)for the case
k=m=1
2but O(1
Mr)for the other two cases. Furthermore,
Fig. 2 confirms that higher rate can be achieved when the
transmit powers of the UEs scale down and the hardware
impairments at the relay scale up proportionally to the square
root of the number of relay antennas, i.e., k=m=1
2.
Meanwhile, a ceiling rate can be observed even for the case
k=q=0with an infinite number of the relay antennas.
¯
C∞
ui,H
SNR→∞
−−−−−−→¯
C∞
ui,H,HSNR =Mu
2log21+(1 + κ2
u,0)2(2Mu+1)&(2Mu+1)κ4
r,0+2κ2
r,0'+κ2
u,0(2 + κ2
u,0)−1(50)
¯
Cui
Mr→∞
−−−−−→¯
C∞
ui,L =Mu
2log2⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
1+⎡
⎢
⎢
⎣
(1 + κ2
u,0)(1 + κ2
r,0)σ2
r
γ2
urEu
Mu2σ2
ui
γ2
urEr
Mu+1
+2(1 + κ2
u,0)2σ2
ui
γ2
urEr
Mu+κ2
u,0(2 + κ2
u,0)
⎤
⎥
⎥
⎦
−1⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
(56)
¯
Cui
Mr→∞
−−−−−→¯
C∞
ui,M
=Mu
2log2⎧
⎨
⎩
1+ (1 + κ2
u,0)(κ2
r,0σ2
r
γ2
urEu
2σ2
ui
γ2
urEr
M2
u+(κ2
r,0σ2
r
γ2
urEu
+2σ2
ui(1 + κ2
u,0)
γ2
urEr)Mu)+κ2
u,0(2 + κ2
u,0)!−1⎫
⎬
⎭
=Mu
2log2⎧
⎨
⎩
1+((1 + κ2
u,0)( 2σ2
ui
γ2
urEr
Mu+1)Mu
σ2
r
γ2
urEu
κ2
r,0+2σ2
ui(1+ κ2
u,0)2
γ2
urEr
Mu+κ2
u,0(2 + κ2
u,0))−1⎫
⎬
⎭
(63)
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FENG et al.: TWO-WAY MASSIVE MIMO RELAYING SYSTEMS WITH NON-IDEAL TRANSCEIVERS 8283
Fig. 2. The deterministic and asymptotic rates versus the number of relay
antennas Mr.
Fig. 3. The asymptotic rate versus the hardware impairment κr, 0at the relay
with κu,0=0.175.
This is different from the result in [4], [5], due to the presence
of hardware impairments at the relay and the UEs.
B. Impact of Hardware Impairments
1) Effect of the Hardware Impairments at the Relay and
the UEs: By setting lu=0, Fig. 3 and Fig. 4 illustrate the
effects of impairments at the relay and the UEs respectively
on the asymptotic rate under various scalings and SNRs.
It is shown in Fig. 3 that the relay impairment significantly
degrades the achievable rate in the scaling case with k=0and
m=1. However, it barely affects the achievable rate under
the other two cases with k=1,m =0and k=m=1/2.
These results are consistent with the previous analysis and
indicate that low-to-medium up scaling of relay’s hardware
impairment with its number of antennas is tolerable even when
the initial relay impairment κr,0is high. On the other hand,
as shown in Fig. 4, the hardware impairments at UEs κu,0
always have non-ignorable negative impact on the achievable
rate no matter the scaling schemes applied. This suggests high
quality hardware applied at the UEs, which is affordable since
UEs only have a small number of antennas.
2) Trade-Off Between Relay Hardware Cost and UE
Transmit Power: As shown in Corollary 1, the scalings of
the UE transmission power and the relay hardware impair-
ment should satisfy m+k≤1. Meanwhile, the analysis
in Section V shows that the scalings of UE transmission
power and relay hardware impairment are offsettable, and
different combinations of these two scalings lead to different
Fig. 4. The asymptotic rate versus the hardware impairment κu,0at users
with κr,0=0.175.
Fig. 5. Achievable rate versus relay hardware impairment scaling factor m
with m+k=1.
achievable rates. To verify these analytical results and further
investigate the tradeoff between the relay hardware cost and
the UE transmission power, we show the achievable rates
under varying scaling of relay hardware impairment with the
constraint m+k=1in Fig. 5. It is clear that the achievable
rate is concave with respect to the relay scaling factor m,and
is roughly symmetric with respect to the line with medium
scaling m. This symmetric rate further verifies the tradability
between the relay hardware cost and the UE transmission
power, and demonstrates that the best tradeoff is achieved in
the medium scaling region.
C. Effect of Antenna Correlations
The achievable rate versus the relay antenna correlation
index is first shown in Fig. 6(a). It is clear that the relay
antenna correlation has consistently negative impact on the
achievable rate no matter how many antennas are deployed at
the UEs. However, the negative impact is negligible for low
and median correlation level. This is in accordance with our
analytical result in Section IV. On the other hand, the effect
of UE antenna correlation on the achievable rate is revealed
in Fig. 6(b). As expected, the UE antenna correlation has a
complicated impact on the rate and the impact trend varies
with the number of antennas. Interestingly, the user antenna
correlation can boost the achievable rate under certain situ-
ations, especially with a large number of antennas at UEs.
Moreover, increasing the number of UE antennas may not be
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8284 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 12, DECEMBER 2019
Fig. 6. The achievable rate versus antenna correlations.
always beneficial to the achievable rate especially in the low
correlation region.
VII. CONCLUSION
This paper investigated the impact of transceiver hardware
impairments on the achievable rate in two-way massive MIMO
AF relaying systems with MIMO UEs. After deriving the
achievable rate in closed-form, joint scaling of the trans-
mission powers and hardware impairments was investigated.
It was found that in order to achieve non-vanishing rate,
the feasible down scaling factors for the transmission powers
at the UEs and the relay (Pu=Eu
Mk
rand Pr=Er
Mq
r)are
within the range 0≤k, q ≤1, and the feasible up scaling
factor for the relay hardware impairment (κ2
r=κ2
r,0Mm
r)is
within 0≤m≤1. However, up scaling of the UE hardware
impairment with the number of relay antennas is not tolerable.
Moreover, it was found that scalings of the UE transmission
power and the relay hardware impairment are tradable and the
best tradeoff to achieve the highest achievable rate corresponds
to medium scalings of the relay hardware impairment and the
UEs’ transmission power. The impact of antenna correlations
on the achievable rate was also discovered. It was interestingly
shown that UE antenna correlation had a complicate impact
on the achievable rate. In certain situations, the antenna
correlation may be beneficial to the achievable rate. However,
increasing the number of UE antennas may not be always
appreciated.
APPENDIX A
PROOF OF THEOREM 1
Before proceeding to the proof, two necessary lemmas are
presented.
Lemma 1: Given a complex random matrix W∼
CNN,M(0N,M,IM⊗IN), the following results hold
E[(W∗)ab(W)cd]=(IN)ac(IM)bd ,(65)
E[(W∗)ab(W)cd(W∗)ef (W)gh]
=(IN)ac(IM)bd (IN)ge(IM)hf
+(IN)ag(IM)bh(IN)ce(IM)df ,(66)
where (W)ab denotes the (a, b)th element of the matrix W.
Proof: The results can be easily extended from
Theorem 2.3.3 in [49] for real random matrices.
Lemma 2 (Property 1, [5]): For a Hermitian positive def-
inite deterministic antenna correlation matrix Ru∈C
N×N
with bounded spectrum and all diagonal entries being 1,
we have
lim
N→∞
tr(R2
u)
N=c, (67)
where constant cis independent of the matrix dimension N.
Now, the expectation of the (i, j )th element of the Hermitian
random matrix 1
N2(H†
vuHvu)G(H†
vuHvu)in (23) can be
written as (68), shown at the top of the next page. Substituting
the result (66) of Lemma 1 into (68) yields
E1
N2[(H†
vuHvu)G(H†
vuHvu)]ij
=1
N2σ4
vu[tr(Ru)]2(TvGTv)ij +σ4
vutr(R2
u)tr(TvG)(Tv)ij ,
(69)
where the fact that (Ru)ii =(Tv)jj =1for i=1,...,N
and j=1···,M is exploited. Meanwhile, we can also get
(70), shown at the top of the next page, in which Kis a
constant and the inequality is obtained after tedious derivations
by using H¨
older’s Inequality and Lemma 2. Detailed derivation
is omitted here due to the limit of space. Clearly the term K
N2
is bounded, i.e., K
N2<∞, and also summable. Then applying
Markov Inequality and First Borel-Cantelli Lemma in Theo-
rem 3.5 and Theorem 3.6 respectively in [34], the following
almost sure (a.s.) result can be directly obtained
1
N2[(H†
vuHvu)G(H†
vuHvu)]ij
a.s.
−−−−→
N→∞
1
N2σ4
vu[tr(Ru)]2(TvGTv)ij
+σ4
vutr(R2
u)tr(TvG)(Tv)ij .(71)
Writing (71) into matrix form, the almost sure (a.s.)results
in (23) can then be obtained based on Proposition 306 in [50].
For the other two results (24) and (25), similarly, we have
E[1
N(H†
v1uHv2uGH†
v2uHv1u)ij ]= 1
Nσ2
v1uσ2
v2utr(R2
u)
×tr(Tv2G)(Tv1)ij ,
(72)
E[1
N(H†
vuHvu)ij ]=1
Nσ2
vu
tr(Ru)( Tv)ij .(73)
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FENG et al.: TWO-WAY MASSIVE MIMO RELAYING SYSTEMS WITH NON-IDEAL TRANSCEIVERS 8285
E1
N2[(H†
vuHvu)G(H†
vuHvu)]ij =1
N2
M
a,d,e,h
N
b,c,f,g (T1/2
v)†
ia(Ru)bc (T1/2
vG(T1/2
v)†)de(Ru)fg(T1/2
v)hj
×E[(W∗
vu)ba(Wvu)cd(W∗
vu)fe(Wuv )gh](68)
E ****
1
N2[(H†
vuHvu)G(H†
vuHvu)]ij −1
N2σ4
vu[tr(Ru)]2(TvGTv)ij +σ4
vutr(R2
u)tr(TvG)(Tv)ij ****
4!
=1
N8E|[(H†
vuHvu)G(H†
vuHvu)]ij −σ4
vu[tr(Ru)]2(TvGTv)ij +σ4
vutr(R2
u)tr(TvG)(Tv)ij |4
≤1
N8N6K=K
N2(70)
Based on (72), (73) and Lemma 2, and following a similar
approach as proving the bounded and summable property in
(70) and the elementary almost sure result in (71), the almost
sure (a.s.) results in (24) and (25) can also be obtained, which
completes the proof.
APPENDIX B
PROOF OF THEOREM 2
The expectation of the (i, j)th element of matrix
1
NH†
vudiag(HvuGH†
vu)Hvu in (26) can be expressed as (74),
shown at the top of the next page. Substituting the result (66)
of Lemma 1 into (74), we further have the result in (75), shown
at the top of next page, in which (Ru)ii =1,fori=1,...,N,
is used. Similarly, we have
E[1
N(H†
v1udiag(Hv2uGH†
v2u)Hv1u)ij ]
=1
Nσ2
v1uσ2
v2utr(Ru)tr(Tv2G)(Tv1)ij .(76)
Now following a similar approach as proving the bounded and
summable property in (70) and the elementary almost sure
result in (71), the almost sure results in (26) and (27) can be
obtained based on (75) and (76) respectively.
APPENDIX C
PROOF OF THEOREM 3
Recalling the instantaneous rate in (12) and taking mathe-
matical manipulations on it yield
Cui=1
2log2det ⎧
⎨
⎩
IMu
+( 1
M1−q
r
Ψui)[
6
j=1
(1
M1−q
r
Ωui,j)+ σ2
ui
M1−q
r
IMu]−1⎫
⎬
⎭
.
(77)
Clearly from the definition of Ωui,1in (14), there involves
a matrix term HruiAHu¯
irH†
u¯
irA†H†
rui. Substituting the
amplifying matrix Ain (10) into it and with (69), (72)
and (73), the equation (78), shown at the top of the next page
can be obtained.
Then substituting the scaled powers and hardware impair-
ments into 1
M1−q
r
Ωui,1yields
1
M1−q
r
Ωui,1=⎧
⎨
⎩
[(βM
q−k+3
2
r)2](κ2
u¯
i,0Mn
r)
×Eu
Mu
1
M4
r
[HruiAHu¯
ir][H†
u¯
irA†H†
rui]⎫
⎬
⎭
.
(79)
From (78) and Theorem 1, the following a.s. result can be
obtained for the matrix term in (79)
1
M4
r
[HruiAHu¯
ir][H†
u¯
irA†H†
rui]a.s.
−−−−−→
Mr→∞
¯
Ωui,1,(80)
where ¯
Ωui,1is given as (30).
Regarding to the term βM
q−k+3
2
rin (79), recalling the
coefficient definition of βin (9) and substituting the scaled
powers and hardware impairments into (9) yield (81), shown
at the top of the next page. With the results in Theorems 1
and 2, the terms involved in (81) almost surely converge as
1
M3
r
i=1,2
(1+ κ2
ui,0Mn
r)tr &AHuirH†
uirA†'a.s.
−−−−−→
Mr→∞
¯
θ1,(82)
1
M3−m
r
i=1,2
(1 + κ2
ui,0Mn
r)tr Adiag[HuirH†
uir]A†
a.s.
−−−−−→
Mr→∞
1
M1−m
r
¯
θ2,(83)
1
M3−k−m
r
tr(AA†)a.s.
−−−−−→
Mr→∞
1
M1−k−m
r
¯
θ3,(84)
where
¯
θ1=
i=1,2
γ2
uir(1+ κ2
ui,0Mn
r)⎧
⎨
⎩
γ2
ru¯
iγ2
uirtr(Ξu¯
i
Pi→¯
iΞ2
uiP†
i→¯
i)
+[ tr(Ξ2
r)tr(Ξui)
M2
r
]
⎡
⎣
i=1,2
γ2
ru¯
iγ2
uirtr(Ξu¯
iPi→¯
iΞuiP†
i→¯
i)
⎤
⎦⎫
⎬
⎭
,
(85)
¯
θ2=
i=1,2
γ2
uir(1+ κ2
ui,0Mn
r)⎧
⎨
⎩
γ2
ru¯
iγ2
uirtr(Ξu¯
iPi→¯
iΞ2
uiP†
i→¯
i)
+tr(Ξui)⎡
⎣
i=1,2
γ2
ru¯
iγ2
uirtr(Ξu¯
iPi→¯
iΞuiP†
i→¯
i)
⎤
⎦⎫
⎬
⎭
,
(86)
¯
θ3=
i=1,2
γ2
uirγ2
ru¯
itr(Ξu¯
iPi→¯
iΞuiP†
i→¯
i).(87)
Putting (82)-(87) into (81) yields the following a.s. result
βM
q+3−k
2
ra.s.
−−−−−→
Mr→∞
¯
β, (88)
with ¯
βgiven in (36). Then putting (80) and (88) into (79),
we have
1
M1−q
r
Ωui,1
a.s.
−−−−−→
Mr→∞ κ2
u¯
i,0Mn
r(¯
β)2Eu
Mu
¯
Ωui,1.(89)
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8286 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 12, DECEMBER 2019
E1
N[H†
vudiag(HvuGH†
vu)Hvu]ij
=E⎧
⎨
⎩
1
N
a,b,c,d,e
(T1/2
v)†
ia(W∗
vu)ba(R1/2
u)†
bc (
c1,c2,c3,c4
(R1/2
u)cc1(Wvu)c1c2
×+T1/2
vG(T1/2
v)†,c2c3
(W∗
vu)c4c3(R1/2
u)†
c4c%(R1/2
u)cd(Wvu)de (T1/2
v)ej
=1
N
a,b,c,d,e
c1,c2,c3,c4
(T1/2
v)†
ia(R1/2
u)†
bc(R1/2
u)cc1+T1/2
vG(T1/2
v)†,c2c3
(R1/2
u)†
c4c(R1/2
u)cd(T1/2
v)ej
×E[(W∗
vu)ba (Wvu)c1c2(W∗
vu)c4c3(Wvu)de ](74)
E1
N[H†
vudiag(HvuGH†
vu)Hvu]ij
=1
Nσ4
vu
a,b,c,d,e
(T1/2
v)†
ia(R1/2
u)†
bc(R1/2
u)cb +T1/2
vG(T1/2
v)†,ae (R1/2
u)†
dc(R1/2
u)cd(T1/2
v)ej
+1
Nσ4
vu
a,b,c
c1,c2
(T1/2
v)†
ia(R1/2
u)†
bc(R1/2
u)cc1+T1/2
vG(T1/2
v)†,c2c2
(R1/2
u)†
c1c(R1/2
u)cb(T1/2
v)aj
=1
Nσ4
vutr(Ru){(TvGTv)ij +tr(TvG)(Tv)ij }(75)
E&(HruiAHu¯
irH†
u¯
irA†H†
rui)ij '
=γ4
ruiγ4
u¯
ir[tr(Ξr)]4(ΞuiP¯
i→iΞ2
u¯
iP†
¯
i→iΞui)ij
+γ4
ruiγ4
u¯
irtr(Ξ2
r)[tr(Ξr)]2tr(Ξu¯
i)(ΞuiP¯
i→iΞu¯
iP†
¯
i→iΞui)ij +tr(ΞuiP¯
i→iΞ2
u¯
iP†
¯
i→i)(Ξui)ij
+&tr(Ξ2
r)'2tr(Ξu¯
i)γ2
ruiγ2
u¯
ir⎡
⎣
i=1,2
γ2
ruiγ2
u¯
irtr(ΞuiP¯
i→iΞu¯
iP†
¯
i→i)⎤
⎦(Ξui)ij (78)
βM
q+3−k
2
r=Er⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
Eu
Mu
i=1,2
(1 + κ2
ui,0Mn
r)$1
M3
r
tr(AHuirH†
uirA†)
+κ2
r,0
M3−m
r
tr[Adiag(HuirH†
uir)A†])+(1
Mm
r+κ2
r,0)σ2
r
M3−k−m
r
tr(AA†)
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
−1
2
(81)
With respect to the desired signal term 1
M1−q
r
Ψui, compar-
ing (13) with (14), we can find that 1
M1−q
r
Ψui=1
M1−q
r
Ωui,1
κ2
u¯
i
.
By ignoring the higher-order infinity small term o(1
M2
r)+
o(1
M4
r)in the matrix ¯
Ωui,1in (30) when Mr→∞, i.e.,
¯
Ωui,1γ4
ruiγ4
u¯
irΞuiP¯
i→iΞ2
u¯
iP†
¯
i→iΞuiwhere denotes the
matrix approximation, the following result can be obtained
based on (89)
1
M1−q
r
Ψui
a.s.
−−−−−→
Mr→∞ (¯
β)2Eu
Mu
γ4
ruiγ4
u¯
irΞuiP¯
i→iΞ2
u¯
iP†
¯
i→iΞui.
(90)
Similarly, the a.s. results of 1
M1−q
r
Ωui,2and 1
M1−q
r
Ωui,3
in (77) can be obtained as follows
1
M1−q
r
Ωui,2
a.s.
−−−−−→
Mr→∞ κ2
ui,0Mn
r(¯
β)2Eu
Mu
¯
Ωui,2,(91)
1
M1−q
r
Ωui,3
a.s.
−−−−−→
Mr→∞ σ2
r(¯
β)21
M1−k
r
¯
Ωui,3,(92)
where ¯
Ωui,2and ¯
Ωui,3are given as (31) and (32), respectively.
Moreover, we have
1
M1−q
r
Ωui,4
a.s.
−−−−−→
Mr→∞ κ2
r,0(¯
β)2
×⎧
⎪
⎨
⎪
⎩
Eu
Mu
1
M1−m
r
¯
Ωui,4,1+κ2
r,0Eu
Mu
1
M2−2m
r
¯
Ωui,4,2
+( 1
Mm
r+κ2
r,0)σ2
r
1
M2−2m−k
r
¯
Ωui,4,3
⎫
⎪
⎬
⎪
⎭
,(93)
1
M1−q
r
Ωui,5
a.s.
−−−−−→
Mr→∞ κ2
r,0(¯
β)2
×Eu
Mu
1
M1−m
r
¯
Ωui,5,1+σ2
r
1
M1−k−m
r
¯
Ωui,3,(94)
1
M1−q
r
Ωui,6
a.s.
−−−−−→
Mr→∞
¯
Ωui,6,(95)
where ¯
Ωui,4,1,¯
Ωui,4,2,¯
Ωui,4,3and ¯
Ωui,5,1are given
in (96)–(99), shown at the top of the next page while ¯
Ωui,6
is given in (35). Finally, substituting (89)-(92), (93), (94)
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FENG et al.: TWO-WAY MASSIVE MIMO RELAYING SYSTEMS WITH NON-IDEAL TRANSCEIVERS 8287
¯
Ωui,4,1=γ2
rui
i=1,2
(1 + κ2
u¯
i,0Mn
r)γ2
ruiγ4
u¯
irtr(ΞuiP¯
i→iΞ2
u¯
iP†
¯
i→i)Ξui
+γ4
ruiγ4
u¯
ir(1 + κ2
u¯
i,0Mn
r)ΞuiP¯
i→iΞ2
u¯
iP†
¯
i→iΞui
+⎧
⎨
⎩
γ2
rui
tr(Ξ2
r)
M2
r⎡
⎣
i=1,2
γ2
u¯
ir(1 + κ2
u¯
i,0Mn
r)tr(Ξu¯
i)⎤
⎦⎡
⎣
i=1,2
γ2
ruiγ2
u¯
irtr(ΞuiP¯
i→iΞu¯
iP†
¯
i→i)⎤
⎦⎫
⎬
⎭
Ξui
+γ4
ruiγ2
u¯
ir⎧
⎨
⎩
tr(Ξ2
r)
M2
r⎡
⎣
i=1,2
γ2
u¯
ir(1 + κ2
u¯
i,0Mn
r)tr(Ξu¯
i)⎤
⎦ΞuiP¯
i→iΞu¯
iP†
¯
i→iΞui⎫
⎬
⎭
(96)
¯
Ωui,4,2=γ2
rui
i=1,2
(1 + κ2
u¯
i,0Mn
r)γ2
ruiγ4
u¯
irtr(ΞuiP¯
i→iΞ2
u¯
iP†
¯
i→i)Ξui
+γ4
ruiγ4
u¯
ir(1 + κ2
u¯
i,0Mn
r)ΞuiP¯
i→iΞ2
u¯
iP†
¯
i→iΞui
+⎧
⎨
⎩
γ2
rui⎡
⎣
i=1,2
γ2
u¯
ir(1 + κ2
u¯
i,0Mn
r)tr(Ξu¯
i)⎤
⎦⎡
⎣
i=1,2
γ2
ruiγ2
u¯
irtr(ΞuiP¯
i→iΞu¯
iP†
¯
i→i)⎤
⎦⎫
⎬
⎭
Ξui
+γ4
ruiγ2
u¯
ir⎧
⎨
⎩⎡
⎣
i=1,2
γ2
u¯
ir(1 + κ2
u¯
i,0Mn
r)tr(Ξu¯
i)⎤
⎦ΞuiP¯
i→iΞu¯
iP†
¯
i→iΞui⎫
⎬
⎭
(97)
¯
Ωui,4,3=γ2
rui⎡
⎣
i=1,2
γ2
ruiγ2
u¯
irtr +ΞuiP¯
i→iΞu¯
iP†
¯
i→i,⎤
⎦Ξui+γ4
ruiγ2
u¯
irΞuiP¯
i→iΞu¯
iP†
¯
i→iΞui(98)
¯
Ωui,5,1=γ2
rui
tr(Ξ2
r)
M2
r
i=1,2
(1 + κ2
u¯
i,0Mn
r)γ2
ruiγ4
u¯
irtr(ΞuiP¯
i→iΞ2
u¯
iP†
¯
i→i)Ξui
+γ4
ruiγ4
u¯
ir(1 + κ2
u¯
i,0Mn
r)ΞuiP¯
i→iΞ2
u¯
iP†
¯
i→iΞui
+⎧
⎨
⎩
γ2
rui
tr(Ξ2
r)
M2
r⎡
⎣
i=1,2
γ2
u¯
ir(1 + κ2
u¯
i,0Mn
r)tr(Ξu¯
i)⎤
⎦⎡
⎣
i=1,2
γ2
ruiγ2
u¯
irtr(ΞuiP¯
i→iΞu¯
iP†
¯
i→i)⎤
⎦⎫
⎬
⎭
Ξui
+γ4
ruiγ2
u¯
ir⎧
⎨
⎩⎡
⎣
i=1,2
γ2
u¯
ir(1 + κ2
u¯
i,0Mn
r)tr(Ξu¯
i)⎤
⎦ΞuiP¯
i→iΞu¯
iP†
¯
i→iΞui⎫
⎬
⎭
(99)
and (95) into (77), and using the continuity of the function
log2det(·), the deterministic result in Theorem 3 can be
obtained based on Continuous Mapping Theorem in [50],
which completes the proof.
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Junjuan Feng received the bachelor’s and mas-
ter’s degrees from the School of Mathematics
and Statistics, Lanzhou University, Lanzhou, China,
in 2011 and 2014, respectively, and the Ph.D. degree
from the Department of Electrical and Computer
Engineering, University of Macau, Macao, in 2019.
From September 2017 to February 2018, she was
a Visiting Ph.D. Student with the Institut National
de la Recherche Scientifique-Energy, Materials and
Telecommunications Center (INRS-EMT), Univer-
sity of Quebec, Montreal, QC, Canada. Her research
interests include massive MIMO, and full-duplex and two-way relaying, with
the tools of random matrix theory and majorization theory.
Shaodan Ma received the double bachelor’s degree
in science and economics and the master’s degree in
engineering from Nankai University, Tianjin, China,
in 1999 and 2002, respectively, and the Ph.D. degree
in electrical and electronic engineering from The
University of Hong Kong, Hong Kong, in 2006.
From 2006 to 2011, she was a Post-Doctoral Fel-
low with The University of Hong Kong. Since
August 2011, she has been with the University of
Macau, where she is currently an Associate Pro-
fessor. She was a Visiting Scholar with Princeton
University in 2010. Her research interests are in the general areas of signal
processing and communications, particularly, transceiver design, resource
allocation, and performance analysis.
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FENG et al.: TWO-WAY MASSIVE MIMO RELAYING SYSTEMS WITH NON-IDEAL TRANSCEIVERS 8289
Sonia Aïssa (S’93–M’00–SM’03–F’19) received the
Ph.D. degree in Electrical and Computer Engineer-
ing from McGill University, Montreal, QC, Canada,
in 1998. Since then, she has been with the Institut
National de la Recherche Scientifique-Energy, Mate-
rials and Telecommunications Center (INRS-EMT),
University of Quebec, Montreal, QC, Canada, where
she is a Full Professor.
From 1996 to 1997, she was a Researcher with
the Department of Electronics and Communications
of Kyoto University, and with the Wireless Systems
Laboratories of NTT, Japan. From 1998 to 2000, she was a Research Associate
at INRS-EMT. In 2000–2002, while she was an Assistant Professor, she
was a Principal Investigator in the major program of personal and mobile
communications of the Canadian Institute for Telecommunications Research,
leading research in radio resource management for wireless networks. From
2004 to 2007, she was an Adjunct Professor with Concordia University,
Canada. She was Visiting Invited Professor at Kyoto University, Japan, in
2006, and at Universiti Sains Malaysia, in 2015. Her research interests include
the modeling, design and performance analysis of wireless communication
systems and networks.
Dr. Aïssa is the Founding Chair of the IEEE Women in Engineering
Affinity Group in Montreal, 2004–2007; acted as TPC Symposium Chair or
Cochair at IEEE ICC ’06 ’09 ’11 ’12; Program Cochair at IEEE WCNC
2007; TPC Cochair of IEEE VTC-spring 2013; TPC Symposia Chair of
IEEE Globecom 2014; TPC Vice-Chair of IEEE Globecom 2018; and
serves as the TPC Chair of IEEE ICC 2021. Her main editorial activities
include: Editor, IE E E TRANSACTIONS ON WIRELES S COMMUNICATIONS,
2004–2012; Associate Editor and Technical Editor, I EEE C OMMUNICATIONS
MAGAZINE, 2004–2015; Technical Editor, I EEE W IRELESS COMMUNICA-
TIONS MAGAZINE, 2006–2010; and Associate Editor, Wiley Security and
Communication Networks Journal, 2007–2012. She currently serves as Area
Editor for the IEE E TRANSACTIONS ON WIR ELESS COMMUNICATIONS.
Awards to her credit include the NSERC University Faculty Award in
1999; the Quebec Government FRQNT Strategic Faculty Fellowship in
2001–2006; the INRS-EMT Performance Award multiple times since 2004,
for outstanding achievements in research, teaching and service; and the
Technical Community Service Award from the FRQNT Centre for Advanced
Systems and Technologies in Communications, 2007. She is co-recipient of
five IEEE Best Paper Awards and of the 2012 IEICE Best Paper Award; and
recipient of NSERC Discovery Accelerator Supplement Award. She served as
Distinguished Lecturer of the IEEE Communications Society and Member of
its Board of Governors in 2013–2016 and 2014–2016, respectively. Professor
Aïssa is a Fellow of the Canadian Academy of Engineering.
Minghua Xia (M’12) received the Ph.D. degree in
telecommunications and information systems from
Sun Yat-sen University, Guangzhou, China, in 2007.
From 2007 to 2009, he was with the Elec-
tronics and Telecommunications Research Institute
(ETRI), South Korea, and the Beijing Research and
Development Center, Beijing, China, where he was a
member and a Senior Member of Engineering Staff.
From 2010 to 2014, he was a Post-Doctoral Fellow
with The University of Hong Kong, Hong Kong,
the King Abdullah University of Science and Tech-
nology, Jeddah, Saudi Arabia, and the Institut National de la Recherche
Scientifique (INRS), University of Quebec, Montreal, Canada. Since 2015,
he has been a Professor with Sun Yat-sen University. His research interests
are in the general areas of wireless communications and signal processing.
Dr. Xia was a recipient of the Professional Award at the IEEE TENCON,
Macau, in 2015. He has served as the TPC Symposium Chair for IEEE
ICC’2019. He was recognized as an Exemplary Reviewer by IEEE TRANSAC-
TIONS ON COMMUNICATIONS in 2014, IEE E COMMUNICATIONS LETTERS
in 2014, and IE E E WIRELESS COMMUNICATIONS LETTERS in 2014 and
2015. He also serves as an Associate Editor for I EEE T RANSACTIONS ON
COGNITIVE COMMUNICATIONS AND NETWORKING and IET Smart Cities.
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