Uplink Performance of Time-Reversal MRC in
Massive MIMO Systems subject to Phase Noise
Antonios Pitarokoilis, Saif Khan Mohammed and Erik G. Larsson
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1
Uplink Performance of Time-Reversal MRC in
Massive MIMO Systems Subject to Phase Noise
Antonios Pitarokoilis, Saif Khan Mohammed and Erik G. Larsson
Abstract—Multi-user multiple-input multiple-output (MU-
MIMO) cellular systems with an excess of base station (BS) anten-
nas (Massive MIMO) offer unprecedented multiplexing gains and
radiated energy efficiency. Oscillator phase noise is introduced in
the transmitter and receiver radio frequency chains and severely
degrades the performance of communication systems. We study
the effect of oscillator phase noise in frequency-selective Massive
MIMO systems with imperfect channel state information (CSI).
In particular, we consider two distinct operation modes, namely
when the phase noise processes at the M BS antennas are identi-
cal (synchronous operation) and when they are independent (non-
synchronous operation). We analyze a linear and low-complexity
time-reversal maximum-ratio combining (TR-MRC) reception
strategy. For both operation modes we derive a lower bound on
the sum-capacity and we compare their performance. Based on
the derived achievable sum-rates, we show that with the proposed
receive processing an O(
√
M ) array gain is achievable. Due to
the phase noise drift the estimated effective channel becomes
progressively outdated. Therefore, phase noise effectively limits
the length of the interval used for data transmission and the
number of scheduled users. The derived achievable rates provide
insights into the optimum choice of the data interval length and
the number of scheduled users.
Index Terms—Receiver algorithns, MU-MIMO, phase noise.
I. INTRODUCTION
Multiple-input multiple-output (MIMO) technology offers
substantial performance gains in wireless links [1]. The spatial
degrees of freedom enable many users to share the same time-
frequency resources, paving the way for multi-user MIMO
(MU-MIMO) systems [2]. MU-MIMO systems with an excess
of BS antennas, termed as Massive MIMO or large-scale
MIMO, have recently attracted significant interest [3]–[5].
They promise a significant increase in the total cell throughput
by means of simple signal processing. At the same time, the
radiated power can be scaled down with the number of BS
antennas, M , while maintaining a desired sum-rate. More
specifically, in [6] the authors show that in a MU-MIMO
uplink with linear receivers and imperfect channel state infor-
mation (CSI), by increasing the number of BS antennas from
1 to M, one can reduce the total transmit power by a factor
O(
√
M) while maintaining a fixed per-user information rate.
A. Pitarokoilis and Erik G. Larsson are with the Department of Electri-
cal Engineering (ISY), Link¨oping University, 581 83 Link¨oping, Sweden,
{antonispit,erik.larsson}@isy.liu.se. Saif K. Mohammed
was with the Dept. of Electrical Engineering (ISY), Link¨oping University,
Sweden. He is now with the Dept. of Electrical Engineering, Indian Institute
of Technology (I.I.T.) Delhi, India, saifkm@ee.iitd.ac.in.
This work was supported by the Swedish Foundation for Strategic Research
(SSF) and ELLIIT. The work done by Saif K. Mohammed was supported by
the Science and Engineering Research Board (SERB), Department of Science
and Technology (DST), Government of India. This paper was presented in part
at the 50th Allerton Conference on Communication, Control and Computing,
Urbana-Champaign, IL, USA, Oct. 2012.
In [7] the authors report an improved result for channels with
arbitrary channel covariance matrices. The crucial assumption
in Massive MIMO is that the squared Euclidean norm of the
channel vector of each user grows as O(M), whereas the inner
products between channel vectors of different users grow at
a lesser rate. This assumption can be justified in the MU-
MIMO setting since the users are typically separated by many
wavelengths, which implies that their channel vectors become
asymptotically (in the number of BS antennas) orthogonal.
Extensive measurements have confirmed the validity of this
assumption [4], [5].
Phase noise is inevitable in communication systems due to
imperfections in the circuitry of the local oscillators that are
used for the conversion of the baseband signal to passband and
vice versa. To be specific, phase noise is the instantaneous drift
of the phase of the carrier wave and results in a widening of the
power spectral density of the generated waveform. Phase noise
causes a partial loss of coherency between the channel estimate
and the true channel gain during data transmission. This can
result in severe degradation of the system performance.
In MIMO an array power gain is obtained by coherently
combining signals received by several antennas, using es-
timated channel responses. Since phase noise distorts the
received data, it is crucial to examine its effect on the perfor-
mance. Significant research work is available on phase noise.
However, most of it is concerned with single-user single-
antenna multi-carrier transmission, since multi-carrier trans-
mission is more sensitive to phase noise compared to single-
carrier transmission [8]. In [9] a method to calculate the bit-
error-rate (BER) of a single-user orthogonal frequency division
multiplexing (OFDM) system impaired with phase noise is
provided. Reference [10] studies the signal-to-interference-
and-noise-ratio (SINR) degradation in OFDM and proposes
a method to mitigate the effect of phase noise. In [11] a
method to characterize phase noise in OFDM systems is
developed and an algorithm to compensate for the degradation
is described. Finally, in [12] the authors propose a method to
jointly estimate the channel coefficients and the phase noise
in a single-user MIMO system and an associated phase noise
mitigation algorithm.
From an information-theoretic point of view, the calculation
of capacity of phase noise channels is challenging. To the
best of our knowledge, the exact capacity of typical phase
noise-impaired channels under realistic models is not known.
The behavior of the capacity of such channels is only known
asymptotically for some cases in the high signal-to-noise-ratio
(SNR) regime [13]. In [14] the authors derive a non-asymptotic
upper bound on the capacity of a single-user deterministic
MIMO channel impaired with Wiener phase noise, which is
2
tight in the high-SNR regime. In [15], the authors consider
the performance of Massive MIMO systems with hardware
impairments. Their model is suitable for the residual hardware
impairments after the application of appropriate compensation
algorithms.
To the authors’ knowledge, we present the first analysis
of the effect of Wiener phase noise in a multi-user multi-
antenna scenario with imperfect channel state information
where single-carrier transmission is used. Specifically, we
consider a single-cell frequency-selective MU-MIMO uplink,
where a number of non-cooperative users transmit independent
data streams to a base station having a large number of
antennas. Since the channel is assumed to be unknown, CSI
is acquired via uplink training. There are phase noise sources
both at the transmitters and at the receiver. In this paper
we extend the work presented in [16]. We consider and
compare two distinct cases. In the first case, which is termed
synchronous operation mode, the phase noise processes at the
BS antennas are identical. In the second case, which is termed
non-synchronous operation mode, the phase noise processes at
the BS antennas are independent. These two operation modes
correspond to the cases of a common phase reference versus
independent phase references, respectively. A time-reversal
maximum-ratio combining (TR-MRC) strategy is proposed
and achievable sum-rates are derived for both operation modes.
Based on the derived expressions of the achievable sum-
rates, we show that for a fixed desired per-user information
rate, by doubling the number of BS antennas, the total transmit
power can be reduced by a factor of
√
2. This is the same
scaling law as without phase noise [6]. We observe that
the use of independent phase noise sources can yield higher
sum-rate performance and we support this interesting result
by a simple toy example for which the exact capacity is
calculated. Furthermore, the achievable rate expressions reveal
a fundamental trade-off between the length of the time interval
spent on data transmission and the sum-rate performance. The
rate expressions also provide valuable insight into the optimum
number of scheduled users.
II. SYSTEM MODEL
We consider a frequency-selective MU-MIMO uplink chan-
nel with M BS antennas and K single-antenna users. The
channel between the k-th user and the m-th BS antenna is
modeled as a finite impulse response (FIR) filter with L
symbol-spaced channel taps. The l-th channel tap is given by
g
m,k,l
∆
=
p
d
k,l
h
m,k,l
, where h
m,k,l
and d
k,l
model the fast
and slow time-varying components, respectively. We assume
a block fading model where h
m,k,l
is fixed during the trans-
mission of a block of N
c
∆
= N
D
+ (K + 3)L −3 symbols and
varies independently from one block to another. N
D
denotes
the number of channel uses utilized for data transmission (see
Fig. 1). We further assume that the channel fading process
is ergodic. The parameters d
k,l
≥ 0, l = 0, . . . , L − 1
model the power delay profile (PDP) of the frequency-selective
channel for the k-th user. Since {d
k,l
} vary slowly with
time and spatial location, we assume them to be fixed for
the entire communication and independent of m. We further
assume h
m,k,l
to be independent and identically distributed
(i.i.d.) zero-mean and unit-variance proper complex random
variables. The i.i.d. assumption is justified in [4], [5], [17].
1
Further, the PDP for every user is normalized such that the
average received power is independent of the length of the
channel impulse response, L. Therefore, it holds that
L−1
X
l=0
E
h
|
p
d
k,l
h
m,k,l
|
2
i
=
L−1
X
l=0
d
k,l
= α
k
, (1)
for 1 ≤ k ≤ K. The positive constants, α
k
, account for
different propagation losses between users and are assumed to
be fixed throughout the communication. The BS is assumed to
have perfect knowledge of all the PDPs. Finally, we assume
exact knowledge of the channel statistics at the BS, but not of
the particular channel realizations.
A. Phase Noise Model
Phase noise is introduced at the transmitter during up-
conversion, when the baseband signal is multiplied with the
carrier generated by the local oscillator. The phase of the gen-
erated carrier drifts randomly, resulting in a phase distortion
of the transmitted signal. A similar phenomenon also happens
at the receiver side during down-conversion of the bandpass
signal to baseband. In the following, θ
k
, k = 1, . . . , K denotes
the phase noise process at the k-th single-antenna user. Since
the users have different local oscillators, the transmitter phase
noise processes are assumed to be mutually independent. On
the other hand, at the receiver side two distinct operation
modes are considered. We term these operation modes as
synchronous and non-synchronous operation depending on
whether the phase noise processes at the BS antennas are
identical or independent. For the synchronous case, all BS
antennas are subject to the same phase noise process and
φ denotes this common phase noise process at each BS
antenna. This models the scenario of a centralized BS with
a single oscillator feeding the down-conversion module in
each receiver. For the case of non-synchronous operation,
φ
m
, m = 1, . . . , M denotes the phase noise process at the m-
th BS antenna. This models a completely distributed scenario
where each BS antenna uses a distinct oscillator for down-
conversion. We further assume that the phase noise processes
θ
k
, k = 1, . . . , K and φ (or φ
m
, m = 1, . . . , M ) for the
case of synchronous (or non-synchronous) operation mode are
mutually independent.
In this study each phase noise process is modeled as
an independent Wiener process, which is a well-established
model [11], [18]. Therefore, the discrete-time phase noise
process at the k-th user at time i is given by
2
θ
k
[i] = θ
k
[i − 1] + w
t
k
[i], (2)
1
We note that with the i.i.d. assumption on the channel gains, the captured
energy increases linearly with the number of BS antennas, M . This is not
reasonable if M grows unbounded. However, this deficiency of the model
takes effect only for exorbitantly large values of M which do not lie in the
regime of our interest [5], [4], [7].
2
The discrete-time phase noise model is used since we will be working
with the discrete-time complex baseband representation of the transmit and
receive signals.
3
where w
t
k
[i] ∼ N(0, σ
2
θ
) are independent identically dis-
tributed zero-mean Gaussian increments with variance σ
2
θ
∆
=
4π
2
f
2
c
c
θ
T
s
, f
c
is the carrier frequency, T
s
is the symbol
interval and c
θ
is a constant that depends on the oscillator.
Depending on the operation mode, the phase noise processes
φ[i] and φ
m
[i] at the M BS antennas are defined in a
manner similar to (2), where the increments have variance
σ
2
φ
∆
= 4π
2
f
2
c
c
φ
T
s
.
B. Received Signal
Let x
k
[i] be the symbol transmitted from the k-th user at
time i. The received sample at the m-th BS antenna element
at time i is then given by, for the non-synchronous operation
y
m
[i] =
√
P
K
X
k=1
L−1
X
l=0
e
−jφ
m
[i]
g
m,k,l
e
jθ
k
[i−l]
x
k
[i − l] + n
m
[i],
(3)
where n
m
[i] ∼ CN(0, σ
2
) represents noise at the m-th
receiver at time i, which is distributed as circularly symmetric
complex Gaussian.
3
Each user transmits a stream of i.i.d.
CN(0, 1) information symbols (i.e., x
k
[i] ∼ CN(0, 1)), that
are independent of the information symbols of the other users.
P denotes the average uplink transmitted power from each
user.
III. TRANSMISSION SCHEME AND RECEIVE PROCESSING
We consider a block-based uplink transmission scheme.
A transmission block of N
c
channel uses consists of KL
channel uses dedicated to uplink channel training followed by
a preamble of L −1 channel uses, where i.i.d. CN(0, 1) non-
information symbols are sent. The data interval of N
D
channel
uses comes after that and a postamble of L − 1 channel uses
is appended at the end of the coherence interval, where i.i.d.
CN(0, 1) non-information symbols are sent. The inclusion of
the preamble and postamble accounts for the edge effects
introduced due to the intersymbol interference. This way the
subsequent analysis is valid for all the N
D
channel uses during
data transmission and no separate analysis for the edges of the
data interval is required. At the beginning of each coherence
interval an all-zero block of L −1 channel uses is prepended
to eliminate inter-block interference (IBI) (see Fig. 1).
A. Channel Estimation
For coherent demodulation, the BS needs to estimate the
uplink channel. This is facilitated through the transmission
of uplink pilot symbols during the training phase of each
transmission block.
4
The users transmit uplink training signals
3
In the following we will present only the expressions of the non-
synchronous mode. The expressions for the synchronous operation are ob-
tained easily by substituting φ
1
[i] ≡ . . . ≡ φ
M
[i] ≡ φ[i]. In Sections IV–VI,
when the expressions of the two distinct modes differ in a non-obvious way,
both expressions will be given explicitly.
4
In this paper we deal only with uplink transmission. In Massive MIMO
Time Division Duplex (TDD) operation pilots are transmitted on the uplink.
The number of required pilots scales with the number of terminals, K, but
not the number of BS antennas, M , making Massive MIMO scalable with
respect to M [3], [4].
KL
L − 1L − 1
L − 1
N
D
Training
Data phase
Preamble Postamble
IBI
Fig. 1: The transmission block is assumed to span a coherence
interval, N
c
∆
= N
D
+ (K + 3)L − 3. In each block, the first
KL channel uses (cu) are utilized for pilot based channel
estimation and N
D
cu are utilized for data transmission. An
all-zero block, a preamble and a postamble of L − 1 cu each
are added due to the edge effects of the channel.
sequentially in time, i.e., at any given time only one user
is transmitting uplink training signals and all other users
are silent. To be precise, the k-th user sends an impulse of
amplitude
p
P
p
KL at the (k − 1)L-th channel use and is
idle for the remaining portion of the training phase. Here,
P
p
is the average power transmitted by a user during the
training phase. We choose the proposed training sequence
since it allows for a very simple channel estimation scheme
at the BS and since it facilitates our derivation of achievable
rates. However, many of our results, such as partial loss of
coherency due to Wiener phase noise and monotonic decrease
in performance with increased variance of the phase noise
increments, are expected to be qualitatively valid also for other
(but not necessarily all possible) training schemes. Therefore,
using (3), the signal received at the m-th BS receiver at time
(k −1)L + l, l = 0, . . . , L −1, k = 1, . . . , K is given by, for
non-synchronous operation
y
m
[(k − 1)L + l]=
p
P
p
KLg
m,k,l
e
j(θ
k
[(k−1)L]− φ
m
[(k−1)L+l])
+ n
m
[(k −1)L + l]. (4)
Based on (4), we derive the maximum like-
lihood (ML) estimate of the effective channel
g
m,k,l
e
j(θ
k
[(k−1)L]− φ
m
[(k−1)L+l])
. The corresponding channel
estimates are then given by, for non-synchronous operation
ˆg
m,k,l
=
1
p
P
p
KL
y
m
[(k − 1)L + l]
= g
m,k,l
e
−jφ
m
[(k−1)L+l]
e
jθ
k
[(k−1)L]
+
1
p
P
p
KL
n
m
[(k − 1)L + l]. (5)
We observe that the channel estimate is distorted by the
AWGN and by the phase noise of the local oscillators at the
user and at the BS.
B. Time-Reversal Maximum Ratio Combining (TR-MRC)
Using (3), the received signal during the data phase is given
by, for non-synchronous operation
y
m
[i] =
p
P
D
K
X
k=1
L−1
X
l=0
e
−jφ
m
[i]
g
m,k,l
e
jθ
k
[i−l]
x
k
[i − l] + n
m
[i],
(6)
where i ∈ I
d
, I
d
∆
= {(K + 1)L −1, . . . , (K + 1)L + N
D
−2}
and P
D
is the per-user average transmit power constraint dur-
ing the data phase. Motivated by the need for low-complexity
4
detection, we consider the TR-MRC receiver at the BS. The
TR-MRC receiver convolves the received symbols, y
m
[i], with
the complex conjugate of the time-reversed estimated channel
impulse response. The detected symbol, ˆx
k
[i], is given by
ˆx
k
[i] =
L−1
X
l=0
M
X
m=1
ˆg
∗
m,k,l
y
m
[i + l], (7)
where (·)
∗
denotes the complex conjugation operation.
IV. ACHIEVABLE SUM-RATE
We use the information sum-rate as the performance metric
for quantifying the effects of phase noise. To this end, using
(5) and (6) for the non-synchronous operation, (7) is written
as
ˆx
k
[i] = A
k
[i]x
k
[i] + ISI
k
[i] + MUI
k
[i] + AN
k
[i], (8)
where it holds for the non-synchronous operation that
A
k
[i]
∆
=
p
P
D
M
X
m=1
L−1
X
l=0
|g
m,k,l
|
2
ϑ
m,k,k
i,l,l
(9)
ISI
k
[i]
∆
=
p
P
D
M
X
m=1
L−1
X
l=0
L−1
X
p=0
p6=l
g
∗
m,k,l
g
m,k,p
ϑ
m,k,k
i,l,p
x
k
[i+l−p]
(10)
MUI
k
[i]
∆
=
p
P
D
M
X
m=1
K
X
q=1
q6=k
L−1
X
l=0
L−1
X
p=0
g
∗
m,k,l
g
m,q,p
×
ϑ
m,k,q
i,l,p
x
q
[i + l − p] (11)
AN
k
[i]
∆
=
s
P
D
P
p
KL
M
X
m=1
K
X
q=1
L−1
X
l=0
L−1
X
p=0
g
m,q,p
×
e
−j(φ
m
[i+l]−θ
q
[i+l−p])
n
m
[(k − 1)L + l]x
q
[i+l−p]
+
M
X
m=1
L−1
X
l=0
ˆg
∗
m,k,l
n
m
[i + l], (12)
where ϑ
m,k,q
i,l,p
∆
=e
j(θ
q
[i+l−p]−θ
k
[(k−1)L]− φ
m
[i+l]+φ
m
[(k−1)L+l])
.
In (8), A
k
[i]x
k
[i] is the desired signal term for the k-th user,
ISI
k
[i] stands for the intersymbol interference for user k
at time i, caused by the information symbols of the k-th
user transmitted at other time instances, MUI
k
[i] denotes the
multi-user interference due to the information symbols of the
other users and finally AN
k
[i] is an aggregate noise term that
incorporates the effects of the channel estimation error and
the receiver AWGN noise, n
m
[i]. The expressions for the
terms in (8) for the synchronous operation are obtained from
(9)-(12) by substituting φ
1
[i] ≡ . . . ≡ φ
M
[i] ≡ φ[i].
In the following, we derive an achievable information rate
for the k-th user. Similar capacity bounding techniques have
been used earlier in e.g. [19], [20]. In (8), we add and subtract
the term E [A
k
[i]] x
k
[i], where the expectation is taken over
the channel gains, g
m,k,l
, and the phase noise processes,
θ
k
, φ for the synchronous operation and θ
k
, φ
m
for the non-
synchronous operation. We relegate the variation around this
term, i.e., IF
k
[i]
∆
= (A
k
[i] − E [A
k
[i]])x
k
[i], to an effective
noise term. This results in the following equivalent expression
ˆx
k
[i] = E [A
k
[i]] x
k
[i] + EN
k
[i], (13)
where
EN
k
[i]
∆
= IF
k
[i] + ISI
k
[i] + MUI
k
[i] + AN
k
[i], (14)
is the effective additive noise term. In (13) the detected
symbol, ˆx
k
[i], is a sum of two uncorrelated terms (i.e.,
E
(E[A
k
[i]]x
k
[i]) (EN
k
[i])
∗
= 0). The importance of the
equivalent representation in (13) is that the scaling factor
E[A
k
[i]]x
k
[i] of the desired information symbol is a constant,
which is known at the BS since the BS has knowledge of the
channel statistics. The exact probability distribution of EN
k
[i]
is difficult to compute. However, its variance can be easily
calculated given that the channel statistics is known at the BS.
Therefore, (13) describes an effective single-user single-input
single-output (SISO) additive noise channel, where the noise
is zero mean, has known variance and is uncorrelated with the
desired signal term. From the expressions for A
k
[i] and EN
k
[i]
in (9) and (14), the mean value of A
k
[i] and the variance of
EN
k
[i] is given by two propositions that follow.
Proposition 1. The mean value of A
k
[i] in both operation
modes is given by
E[A
k
[i]] =
p
P
D
Mα
k
e
−
σ
2
φ
+σ
2
θ
2
(i−(k−1)L)
. (15)
Proof: We prove the statement for the non-synchronous
operation. The proof for the synchronous operation is nearly
identical. From (9), we have
E[A
k
[i]] = E
"
p
P
D
M
X
m=1
L−1
X
l=0
|g
m,k,l
|
2
ϑ
m,k,k
i,l,l
#
(a)
=
p
P
D
E
h
e
−j(θ
k
[(k−1)L]− θ
k
[i])
i
M
X
m=1
L−1
X
l=0
E
h
|g
m,k,l
|
2
i
· E
h
e
−j(φ
m
[i+l]−φ
m
[(k−1)L+l])
i
(b)
=
p
P
D
e
−
σ
2
θ
2
(i−(k−1)L)
M
X
m=1
L−1
X
l=0
d
k,l
e
−
σ
2
φ
2
(i−(k−1)L)
(c)
=
p
P
D
Mα
k
e
−
σ
2
φ
+σ
2
θ
2
(i−(k−1)L)
.
In (a) we have used the fact that the channel realizations,
g
m,k,l
, the phase noise at the BS, φ
m
, and the phase noise at
the k-th user, θ
k
, are mutually independent random processes.
The equality (b) is a consequence of the Wiener phase noise
model. That is, after a time interval, ∆t = i − (k − 1)L, the
phase drift of an oscillator is a zero mean Gaussian random
variable with variance that is proportional to ∆t,
U
φ
m
∆
= φ
m
[i+l]−φ
m
[(k−1)L+l]∼N(0, σ
2
φ
(i − (k − 1)L)),
U
θ
k
∆
= θ
k
[i] − θ
k
[(k −1)L] ∼ N(0, σ
2
θ
(i −(k − 1)L)).
Henceforth E
e
−jU
φ
m
= ϕ
φ
m
(−1) = e
−
σ
2
φ
2
(i−(k−1)L)
and
![TABLE II: Gap in requiredPD σ2 due to phase noise forND = 1000, σφ = σθ = 0.49o, K = 10 users andM = 500 BS antennas for various values of the desired per-user information rate in bits per channel use [bpcu].](/figures/table-ii-gap-in-requiredpd-s2-due-to-phase-noise-fornd-1000-35lle5pg.webp)
![Fig. 3: Sum-rate as a function ofPD σ2 [dB] for M = 200, K = 10, L = 20 andND = 1000. The dotted vertical lines denote the high SNR asymptotic values of the achievable sumrates.](/figures/fig-3-sum-rate-as-a-function-ofpd-s2-db-for-m-200-k-10-l-20-rc4b0h1a.webp)
