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Original citation:
Mi, De, Dianati, Mehrdad, Zhang, Lei, Muhaidat, Sami and Tafazolli, Rahim. (2017) Massive
MIMO performance with imperfect channel reciprocity and channel estimation error. IEEE
Transactions on Communications, 65 (9). pp. 3734-3749.
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1
Massive MIMO Performance with Imperfect
Channel Reciprocity and Channel Estimation Error
De Mi, Mehrdad Dianati, Lei Zhang, Sami Muhaidat and Rahim Tafazolli
Abstract—Channel reciprocity in time-division duplexing
(TDD) massive MIMO (multiple-input multiple-output) systems
can be exploited to reduce the overhead required for the
acquisition of channel state information (CSI). However, perfect
reciprocity is unrealistic in practical systems due to random
radio-frequency (RF) circuit mismatches in uplink and downlink
channels. This can result in a significant degradation in the
performance of linear precoding schemes which are sensitive
to the accuracy of the CSI. In this paper, we model and
analyse the impact of RF mismatches on the performance of
linear precoding in a TDD multi-user massive MIMO system,
by taking the channel estimation error into considerations.
We use the truncated Gaussian distribution to model the RF
mismatch, and derive closed-form expressions of the output
SINR (signal-to-interference-plus-noise ratio) for maximum ratio
transmission and zero forcing precoders. We further investigate
the asymptotic performance of the derived expressions, to provide
valuable insights into the practical system designs, including
useful guidelines for the selection of the effective precoding
schemes. Simulation results are presented to demonstrate the
validity and accuracy of the proposed analytical results.
Index Terms—Massive MU-MIMO, linear precoding, channel
reciprocity error, RF mismatch, imperfect channel estimation.
I. INTRODUCTION
M
ASSIVE (or large scale) MIMO (multiple-input
multiple-output) systems have been identified as en-
abling technologies for the 5th Generation (5G) of wireless
systems [1]–[5]. Such systems propose the use of a large
number of antennas at the base station (BS) side. A notable
advantage of this approach is that it allows the use of simple
processing at both uplink (UL) and downlink (DL) directions
[6], [7]. For example, for the DL transmission, two commonly
known linear precoding schemes, i.e., maximum ratio trans-
mission (MRT) and zero-forcing (ZF), have been extensively
investigated in the context of massive MIMO systems [8]–
[10]. It has been shown that both schemes perform well with
a relatively low computational complexity [8], and can achieve
a spectrum efficiency close to the optimal non-linear precoding
techniques, such as dirty paper coding [9], [11]. However, the
price to pay for the use of simple linear precoding schemes is
D. Mi, L. Zhang and R. Tafazolli are with the 5G Innovation Centre (5GIC),
Institute for Communication Systems (ICS), University of Surrey, Guildford,
GU2 7XH, U.K. (E-mail:{d.mi, lei.zhang, r.tafazolli}@surrey.ac.uk).
M. Dianati is with the Warwick Manufacturing Group, University of
Warwick, Coventry CV4 7AL, U.K, and also with the 5G Innovation Centre,
Institute for Communication Systems, University of Surrey, Guildford GU2
7XH, U.K. (E-mail:m.dianati@warwick.ac.uk).
S. Muhaidat is with the Department of Electrical and Computer Engineer-
ing, Khalifa University, Abu Dhabi, 127788, U.A.E., and also the 5G Innova-
tion Centre (5GIC), Institute for Communication Systems (ICS), University
of Surrey, Guildford, GU2 7XH, U.K. (E-mail: muhaidat@ieee.org).
the overhead required for acquiring the instantaneous channel
state information (CSI) in the massive MIMO systems [10],
[12].
In principle, massive MIMO can be adopted in both
frequency-division duplexing (FDD) and time-division du-
plexing (TDD) systems. Nevertheless, the overhead of CSI
acquisition in FDD massive MIMO systems is considerably
higher than that of TDD systems, due to the need for a
dedicated feedback channel and the infeasible number of
pilots, which is proportional to the number of BS antennas
[13]. On the contrary, by exploiting the channel reciprocity in
TDD systems, the BS can estimate the DL channel by using
the UL pilots from the user terminals (UTs). Hence, there is
no feedback channel required, and the overhead of the pilot
transmission is proportional to the number of UTs antennas,
which is typically much less than the number of BS antennas
in massive MIMO systems [9]. Therefore, TDD operation has
been widely considered in the system with large-scale antenna
arrays [1], [7]–[9].
Most prior studies assume perfect channel reciprocity by
constraining that the time delay from the UL channel estima-
tion to the DL transmission is less than the coherence time
of the channel [1], [7], [8]. Such an assumption ignores two
key facts: 1) UL and DL radio-frequency (RF) chains are
separate circuits with random impacts on the transmitted and
received signals [2], [6]; 2) the interference profile at the BS
and UT sides may be significantly different [14]. The former
phenomenon is known as RF mismatch [15], which is the
main focus of this paper. RF mismatches can cause random
deviations of the estimated values of the UL channel from the
actual values of the DL channel within the coherent time of
the channel. Such deviations are known as reciprocity errors
that invalidate the assumption of perfect reciprocity.
The existing works on studying reciprocity errors can be
divided into two categories. In the first category, e.g. [16],
reciprocity errors are considered as an additive random un-
certainty to the channel coefficients. However, it is shown
in [15] that additive modelling of the reciprocity errors is
inadequate in capturing the full impact of RF mismatches.
Therefore, the recent works consider multiplicative reciprocity
errors where the channel coefficients are multiplied by ran-
dom complex numbers representing the reciprocity errors. For
example, the works in [17] and [18] model the reciprocity
errors as uniformly distributed random variables which are
multiplied by the channel coefficients. The authors model the
amplitude and phase of the multiplicative reciprocity error by
two independent and uniformly distributed random variables,
i.e., amplitude and phase errors. Rogalin et al. in [17] propose
2
a calibration scheme to deal with reciprocity errors. Zhang et
al. in [18] propose an analysis of the performance of MRT
and regularised ZF precoding schemes. Practical studies [19]–
[21] argue that the use of uniform distributions for modelling
phase and amplitude errors is not realistic. Alternatively, they
suggest the use of truncated Gaussian distributions instead.
However, these works do not provide an in-depth analysis
of the impact of reciprocity errors. In this paper, we aim to
fill this research gap and present an in depth analysis of the
impact of the multiplicative reciprocity errors for TDD massive
MIMO systems. In addition, we also take the additive channel
estimation error into considerations. The contributions of this
paper can be summarised as follows:
• Under the assumption of a large number of antennas at
BS and imperfect channel estimation, we derive closed-
form expressions of the output SINR for ZF and MRT
precoding schemes in the presence of reciprocity errors.
• We further investigate the impact of reciprocity errors on
the performance of MRT and ZF precoding schemes and
demonstrate that such errors can reduce the output SINR
by more than 10-fold. Note that all of the analysis is
considered in the presence of the channel estimation error,
to show the compound effects on the system performance
of the additive and multiplicative errors.
• We quantify and compare the performance loss of both
ZF and MRT analytically, and provide insights to guide
the choice of the precoding schemes for massive MIMO
systems in the presence of the reciprocity error and
estimation error.
The rest of the paper is organised as follows. In Section II,
we describe the TDD massive MIMO system model with
imperfect channel estimation and the reciprocity error model
due to the RF mismatches. The derivations of the output SINR
for MRT and ZF precoding schemes are given in Sections III.
In Section IV, we analyse the effect of reciprocity errors on
the output SINR when the number of BS antennas approaches
infinity. Simulation results and conclusions are provided in
Section V and Section VI respectively. Some of the detailed
derivations are given in the appendices.
Notations: E{·} denotes the expectation operator, and var(·)
is the mathematical variance. Vectors and matrices are denoted
by boldface lower-case and upper-case characters, and the
operators (·)
∗
, (·)
T
and (·)
H
represent complex conjugate,
transpose and conjugate transpose, respectively. The M × M
identity matrix is denoted by I
M
, and diag(·) stands for the
diagonalisation operator to transform a vector to a diagonal
matrix. tr(·) denotes the matrix trace operation. |·| denotes the
magnitude of a complex number, while k·k is the Frobenius
norm of a matrix. The imaginary unit is denoted j, and <(·)
is the real part of a complex number. “,” is the equal by
definition sign. The exponential function and the Gauss error
function are defined as exp(·) and erf(·), respectively.
II. SYSTEM MODEL
We consider a Multi User (MU) MIMO system as shown
in Fig. 1 that operates in TDD mode. This system comprises
of K single-antenna UTs and one BS with M antennas,
Tx
Rx
BaseBand
Tx
Rx
Tx
Rx
Hbr Hbt
…
…
…
H
T
H
BS UTs
Propagation
Channel
Fig. 1. A massive MU-MIMO TDD System.
where M K. Each antenna element is connected to an
independent RF chain. We assume that the effect of antenna
coupling is negligible, and that the UL channel estimation and
the DL transmission are performed within the coherent time of
the channel. In the rest of this section, we model the reciprocity
errors caused by RF mismatches first, and then present the
considered system model in the presence of the reciprocity
error.
A. Channel Reciprocity Error Modelling
Due to the fact that the imperfection of the channel reci-
procity at the single-antenna UT side has a trivial impact
on the system performance [2], we focus on the reciprocity
errors at the BS side
1
. Hence, as shown in Fig. 1, the overall
transmission channel consists of the physical propagation
channel as well as transmit (Tx) and receive (Rx) RF frontends
at the BS side. In particular, considering the reciprocity of the
propagation channel in TDD systems, the UL and DL channel
matrices are denoted by H ∈ C
M×K
and H
T
, respectively.
H
br
and H
bt
represent the effective response matrices of
the Rx and Tx RF frontends at the BS, respectively. Unless
otherwise stated, subscript ‘b’ stands for BS, and ‘t’ and ‘r’
correspond to Tx and Rx frontends, respectively. H
br
and H
bt
can be modelled as M × M diagonal matrices, e.g., H
br
can
be given as
H
br
= diag(h
br,1
, ··· , h
br,i
, ··· , h
br,M
), (1)
with the i-th diagonal entry h
br,i
, i = 1, 2, ··· , M , represents
the per-antenna response of the Rx RF frontend. Considering
that the power amplitude attenuation and the phase shift for
each RF frontend are independent, h
br,i
can be expressed as
[15], [22]
h
br,i
= A
br,i
exp(jϕ
br,i
), (2)
where A and ϕ denote amplitude and phase RF responses,
respectively. Similarly, M × M diagonal matrix H
bt
can be
denoted as
H
bt
= diag(h
bt,1
, ··· , h
bt,i
, ··· , h
bt,M
), (3)
with i-th diagonal entry h
bt,i
given by
h
bt,i
= A
bt,i
exp(jϕ
bt,i
). (4)
1
The effective responses of Tx/Rx RF frontend at UTs are set to be ones.
3
In practice, there might be differences between the Tx front
and the Rx front in terms of RF responses. We define the
RF mismatch between the Tx and Rx frontends at the BS by
calculating the ratio of H
bt
to H
br
, i.e.,
E , H
bt
H
−1
br
= diag(
h
bt,1
h
br,1
, ··· ,
h
bt,i
h
br,i
, ··· ,
h
bt,M
h
br,M
), (5)
where the M × M diagonal matrix E can be regarded as the
compound RF mismatch error, in the sense that E combines
H
bt
and H
br
. In (5), the minimum requirement to achieve
the perfect channel reciprocity is E = cI
M
with a scalar
2
c ∈ C
6=0
. The scalar c does not change the direction of
the precoding beamformer [15], hence no impact on MIMO
performance. Contrary to the case of the perfect reciprocity,
in realistic scenarios, the diagonal entries of E may be
different from each other, which introduces the RF mismatch
caused channel reciprocity errors into the system. Particularly,
considering the case with the hardware uncertainty of the RF
frontends caused by the various of environmental factors as
discussed in [2], [16], [23], the entries become independent
random variables. However, in practice, the response of RF
hardware components at the Tx front is likely to be inde-
pendent of that at the Rx front, which cannot be accurately
represented by the compound error model E in (5). Hence, the
separate modelling for H
bt
and H
br
is more accurate from a
practical point of view. Therefore, we focus our investigation
in this work on the RF mismatch caused reciprocity error by
considering this separate error model.
Next we model the independent random variables A
br,i
,
ϕ
br,i
, A
bt,i
and ϕ
bt,i
in (2) and (4) to reflect the randomness
of the hardware components of the Rx and Tx RF frontends.
Here, in order to capture the aggregated effect of the mismatch
on the system performance, the phase and amplitude errors
can be modelled by the truncated Gaussian distribution [20],
[21], which is more generalised and realistic comparing to
the uniformly distributed error model in [17] and [18]. The
preliminaries of the truncated Gaussian distribution are briefly
presented in Appendix A, and accordingly the amplitude and
phase reciprocity errors of the Tx front A
bt,i
, ϕ
bt,i
and the Rx
front A
br,i
, ϕ
br,i
can be modelled as
A
bt,i
∼ N
T
(α
bt,0
, σ
2
bt
), A
bt,i
∈ [a
t
, b
t
], (6)
ϕ
bt,i
∼ N
T
(θ
bt,0
, σ
2
ϕ
t
), ϕ
bt,i
∈ [θ
t,1
, θ
t,2
], (7)
A
br,i
∼ N
T
(α
br,0
, σ
2
br
), A
br,i
∈ [a
r
, b
r
], (8)
ϕ
br,i
∼ N
T
(θ
br,0
, σ
2
ϕ
r
), ϕ
br,i
∈ [θ
r,1
, θ
r,2
], (9)
where, without loss of generality, the statistical magnitudes
of these truncated Gaussian distributed variables are assumed
to be static, e.g., α
bt,0
, σ
2
bt
, a
t
and b
t
of A
bt,i
in (6) remain
constant within the considered coherence time of the channel.
Notice that the truncated Gaussian distributed phase error in
(7) and (9) becomes a part of exponential functions in (2) and
(4), whose expectations can not be obtained easily. Thus, we
provide a generic result for these expectations in the following
Proposition 1.
2
Particularly, the case with E = I
M
is equivalent to that with H
bt
= H
br
,
which means that the Tx/Rx RF frontends have the identical responses.
Proposition 1. Given x ∼ N
T
(µ, σ
2
), x ∈ [a, b], and the
probability density function f(x, µ, σ; a, b) as (59) in Ap-
pendix A. Then the mathematical expectation of exp(jx) can
be expressed as
E {exp(jx)} = exp
−
σ
2
2
+ jµ
×
erf
b−µ
√
2σ
2
− j
σ
√
2
− erf
a−µ
√
2σ
2
− j
σ
√
2
erf
b−µ
√
2σ
2
− erf
a−µ
√
2σ
2
. (10)
Proof. See Appendix B.
Then the phase-error-related parameters g
t
,
E {exp (jϕ
bt,i
)} and g
r
, E {exp (jϕ
br,i
)} can be given in
Appendix C by specialising Proposition 1. Also, based on (6),
(8) and Appendix A, the amplitude-error-related parameters
E {A
bt,i
}, E {A
br,i
}, var(A
bt,i
) and var(A
br,i
) can be given
by α
t
, α
r
, σ
2
t
and σ
2
r
respectively in Appendix C. Note that
these parameters can be measured from engineering points of
view, for example, by using the manufacturing datasheet of
each hardware component of RF frontends in the real system
[24].
B. Downlink Transmission with Imperfect Channel Estimation
In TDD massive MIMO systems, UTs first transmit the
orthogonal UL pilots to BS, which enables BS to estimate the
UL channel. In this paper, we model the channel estimation
error as the additive independent random error term [10], [12].
By taking the effect of H
br
into consideration, the estimate
ˆ
H
u
of the actual uplink channel response H
u
can be given by
ˆ
H
u
=
p
1 − τ
2
H
br
H + τV, (11)
where two M ×K matrices H and V represent the propagation
channel and the channel estimation error, respectively. We
assume the entries of both H and V are independent iden-
tically distributed (i.i.d.) complex Gaussian random variables
with zero mean and unit variance. In addition, the estimation
variance parameter τ ∈ [0, 1] is applied to reflect the accuracy
of the channel estimation, e.g., τ = 0 represents the perfect
estimation, whereas τ = 1 corresponds to the case that the
channel estimate is completely uncorrelated with the actual
channel response.
The UL channel estimate
ˆ
H
u
is then exploited in the DL
transmission for precoding. Specifically, by considering the
channel reciprocity within the channel coherence period, the
BS predicts the DL channel as
ˆ
H
d
=
ˆ
H
T
u
=
p
1 − τ
2
H
T
H
br
+ τV
T
. (12)
While the UL and DL propagation channels are reciprocal, the
Tx and Rx frontends are not, due to the reciprocity error. By
taking the effect of H
bt
into the consideration, the actual DL
channel H
d
can be denoted as
H
d
= H
T
H
bt
. (13)
Then, the BS performs the linear precoding for the DL
transmission based on the DL channel estimate
ˆ
H
d
instead
4
of the actual channel H
d
, and the received signal y for the K
UTs is given by
y =
√
ρ
d
λH
d
Ws + n =
√
ρ
d
λH
T
H
bt
Ws + n, (14)
where W represents the linear precoding matrix, which is
a function of the DL channel estimate
ˆ
H
d
instead of the
actual DL channel H
d
. The parameter ρ
d
denotes the av-
erage transmit power at the BS, and note that the power
is equally allocated to each UT in this work. The vector s
denotes the symbols to be transmitted to K UTs. We assume
that the symbols for different users are independent, and
constrained with the normalised symbol power per user. To
offset the impact of the precoding matrix on the transmit
power, it is multiplied by a normalisation parameter λ, such
that E
tr
λ
2
WW
H
= 1. This ensures that the transmit
power after precoding remains equal to the transmit power
budget that E
k
√
ρ
d
λWsk
2
= ρ
d
. In addition, n is the
additive white Gaussian noise (AWGN) vector, whose k-th
element is complex Gaussian distributed with zero mean and
covariance σ
2
k
, i.e., n
k
∼ CN(0, σ
2
k
). We assume that σ
2
k
= 1,
k = 1, 2, ··· , K. Therefore, ρ
d
can also be treated as the DL
transmit signal-to-noise ratio (SNR).
By comparing the channel estimate
ˆ
H
d
for the precoding
matrix in (12) with the actual DL channel H
d
in (13), we have
H
d
=
1
√
1 − τ
2
(
ˆ
H
d
− τV
T
) · H
−1
br
H
bt
| {z }
reciprocity errors
, (15)
where the term H
−1
br
H
bt
stands for the reciprocity errors, and
is equivalent to E defined in (5) (also corresponds to the
error model E
b
in [20]). The expression (15) reveals that the
channel reciprocity error is multiplicative, in the sense that
the corresponding error term H
−1
br
H
bt
is multiplied with the
channel estimate
ˆ
H
d
and the estimation error V. Based on the
discussion followed by (5), H
d
and
ˆ
H
d
can have one scale
difference in the case that H
−1
br
H
bt
= cI
M
, thus no reciprocity
error caused in this case. On the contrary, in the presence of
the mismatch between H
br
and H
bt
, the channel reciprocity
error can be introduced into the system. From (15), it is
also indicated that the integration between the multiplicative
reciprocity error and the additive estimation error brings a
compound effect on the precoding matrix calculation. We shall
analyse this effect in the following Section III.
In order to investigate the effect of reciprocity errors on
the performance of the linearly precoded system in terms of
the output SINR for a given k-th UT, let M × 1 vectors h
k
and v
k
be the k-th column of the channel matrix H and the
estimation error matrix V respectively, as well as w
k
and s
k
represent the precoding vector and the transmit symbol for the
k-th UT, while w
i
and s
i
, i 6= k for other UTs, respectively.
Specifying the received signal y by substituting h
k
, w
k
and
w
i
into (14), we rewrite the received signal for the k-th UT
as
y
k
=
√
ρ
d
λh
T
k
H
bt
w
k
s
k
| {z }
Desired Signal
+
√
ρ
d
λ
K
X
i=1,i6=k
h
T
k
H
bt
w
i
s
i
| {z }
Inter-user Interference
+ n
k
|{z}
Noise
.
(16)
The first term of the received signal y
k
in (16) is related to the
desired signal for the k-th UT, and the second term represents
the inter-user interference among other K −1 UTs. Then, the
desired signal power P
s
and the interference power P
I
can be
expressed as
P
s
= |
√
ρ
d
λh
T
k
H
bt
w
k
s
k
|
2
, (17)
P
I
=
K
X
i=1,i6=k
√
ρ
d
λh
T
k
H
bt
w
i
s
i
2
, (18)
respectively. Considering (17), (18) and the third term in (16)
which is the AWGN, the output SINR for the k-th UT in the
presence of the channel reciprocity error can be given as in
[25]
SINR
k
= E
P
s
P
I
+ σ
2
k
≈ E {P
s
}E
1
P
I
+ σ
2
k
, (19)
thus we can approximate the output SINR by calculating
E {P
s
} and E
1/(P
I
+ σ
2
k
)
separately. In order to derive
the term E
1/(P
I
+ σ
2
k
)
and pursue the calculation of (19),
we provide one generalised conclusion as in the following
proposition.
Proposition 2. Let a random variable X
1
∈ C and X
1
6= 0,
∃
E{X
1
}, var(X
1
), E
n
1
X
1
o
∈ C, and E{X
1
} 6= 0, then
E
1
X
1
=
1
E{X
1
}
+ O
var(X
1
)
E{X
1
}
3
. (20)
Proof. Consider the Taylor series of E
n
1
X
1
o
, we have
E
1
X
1
= E
1
E{X
1
}
−
1
E{X
1
}
2
(X
1
− E{X
1
})
+
1
E{X
1
}
3
(X
1
− E{X
1
})
2
− ···
. (21)
Then one can easily arrive at (20).
From Proposition 2, it is expected that the approximation
in (19) can be more precise than the widely-used approximate
SINR expressions in the literatures, e.g., [18, Eq. (6)] and [26,
Eq. (6)], which are based on SINR
k
≈ E {P
s
}/E
P
I
+ σ
2
k
that is not accurate when the value of
var(X
1
)/E{X
1
}
3
is not negligible. We will verify the accuracy of (19) in the
analytical results in the following section.
III. SINR FOR MAXIMUM-RATIO TRANSMISSION AND
ZERO-FORCING PRECODING SCHEMES
In this section, we formulate and discuss the effect of the
reciprocity error on the performance of MRT and ZF precoding
schemes, in terms of the output SINR, by considering the reci-
procity error model with the truncated Gaussian distribution.
A. Maximum-Radio Transmission
Recall (12) and (14), for MRT, the precoding matrix W can
be given by
W
mrt
=
ˆ
H
H
d
=
p
1 − τ
2
H
∗
br
H
∗
+ τV
∗
. (22)
