Massive MIMO with Non-Ideal Arbitrary Arrays: Hardware Scaling
Laws and Circuit-Aware Design
Björnson, E., Matthaiou, M., & Debbah, M. (2015). Massive MIMO with Non-Ideal Arbitrary Arrays: Hardware
Scaling Laws and Circuit-Aware Design.
IEEE Transactions on Wireless Communications
,
14
(8), 4353-4368.
https://doi.org/10.1109/TWC.2015.2420095
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IEEE Transactions on Wireless Communications
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 1
Massive MIMO with Non-Ideal Arbitrary Arrays:
Hardware Scaling Laws and Circuit-Aware Design
Emil Bj
¨
ornson, Member, IEEE, Michail Matthaiou, Senior Member, IEEE, and M
´
erouane Debbah, Fellow, IEEE
Abstract—Massive multiple-input multiple-output (MIMO)
systems are cellular networks where the base stations (BSs) are
equipped with unconventionally many antennas, deployed on co-
located or distributed arrays. Huge spatial degrees-of-freedom
are achieved by coherent processing over these massive arrays,
which provide strong signal gains, resilience to imperfect channel
knowledge, and low interference. This comes at the price of more
infrastructure; the hardware cost and circuit power consumption
scale linearly/affinely with the number of BS antennas N . Hence,
the key to cost-efficient deployment of large arrays is low-cost
antenna branches with low circuit power, in contrast to today’s
conventional expensive and power-hungry BS antenna branches.
Such low-cost transceivers are prone to hardware imperfections,
but it has been conjectured that the huge degrees-of-freedom
would bring robustness to such imperfections. We prove this
claim for a generalized uplink system with multiplicative phase-
drifts, additive distortion noise, and noise amplification. Specifi-
cally, we derive closed-form expressions for the user rates and a
scaling law that shows how fast the hardware imperfections can
increase with N while maintaining high rates. The connection
between this scaling law and the power consumption of different
transceiver circuits is rigorously exemplified. This reveals that one
can make the circuit power increase as
√
N , instead of linearly,
by careful circuit-aware system design.
Index Terms—Achievable user rates, channel estimation, mas-
sive MIMO, scaling laws, transceiver hardware imperfections.
I. INTRODUCTION
Interference coordination is the major limiting factor in
cellular networks, but modern multi-antenna base stations
(BSs) can control the interference in the spatial domain
by coordinated multipoint (CoMP) techniques [1]–[3]. The
cellular networks are continuously evolving to keep up with
the rapidly increasing demand for wireless connectivity [4].
E. Bj
¨
ornson was with the Alcatel-Lucent Chair on Flexible Radio, Sup
´
elec,
Gif-sur-Yvette, France, and with the Department of Signal Processing, KTH
Royal Institute of Technology, Stockholm, Sweden. He is currently with the
Department of Electrical Engineering (ISY), Link
¨
oping University, Link
¨
oping,
Sweden (email: emil.bjornson@liu.se).
M. Matthaiou is with the School of Electronics, Electrical Engineering
and Computer Science, Queen’s University Belfast, Belfast, U.K. and with
the Department of Signals and Systems, Chalmers University of Technology,
Gothenburg, Sweden (email: m.matthaiou@qub.ac.uk).
M. Debbah is with the Alcatel-Lucent Chair on Flexible Radio, Sup
´
elec,
Gif-sur-Yvette, France (email: merouane.debbah@supelec.fr).
Parts of this work were published at the IEEE Conference on Acoustics,
Speech, and Signal Processing (ICASSP), Florence, Italy, May 2014 and at
the IEEE International Symposium on Communications, Control, and Signal
Processing (ISCCSP), Athens, Greece, May 2014.
This research has received funding from the EU 7th Framework Programme
under GA n
o
ICT-619086 (MAMMOET). This research has been supported by
ELLIIT, the International Postdoc Grant 2012-228 from the Swedish Research
Council and the ERC Starting Grant 305123 MORE (Advanced Mathematical
Tools for Complex Network Engineering).
Massive densification, in terms of more service antennas per
unit area, has been identified as a key to higher area throughput
in future wireless networks [5]–[7]. The downside of densifi-
cation is that even stricter requirements on the interference co-
ordination need to be imposed. Densification can be achieved
by adding more antennas to the macro BSs and/or distributing
the antennas by ultra-dense operator-deployment of small
BSs. These two approaches are non-conflicting and represent
the two extremes of the massive MIMO paradigm [7]: a
large co-located antenna array or a geographically distributed
array (e.g., using a cloud RAN approach [8]). The massive
MIMO topology originates from [9] and has been given many
alternative names; for example, large-scale antenna systems
(LSAS), very large MIMO, and large-scale multi-user MIMO.
The main characteristics of massive MIMO are that each cell
performs coherent processing on an array of hundreds (or even
thousands) of active antennas, while simultaneously serving
tens (or even hundreds) of users in the uplink and downlink.
In other words, the number of antennas, N , and number of
users per BS, K, are unconventionally large, but differ by
a factor two, four, or even an order of magnitude. For this
reason, massive MIMO brings unprecedented spatial degrees-
of-freedom, which enable strong signal gains from coherent
reception/transmit beamforming, give nearly orthogonal user
channels, and resilience to imperfect channel knowledge [10].
Apart from achieving high area throughput, recent works
have investigated additional ways to capitalize on the huge
degrees-of-freedom offered by massive MIMO. Towards this
end, [5] showed that massive MIMO enables fully distributed
coordination between systems that operate in the same band.
Moreover, it was shown in [11] and [12] that the transmit
uplink/downlink powers can be reduced as
1
√
N
with only a
minor loss in throughput. This allows for major reductions
in the emitted power, but is actually bad from an overall
energy efficiency (EE) perspective—the EE is maximized by
increasing the emitted power with N to compensate for the
increasing circuit power consumption [13].
This paper explores whether the huge degrees-of-freedom
offered by massive MIMO provide robustness to transceiver
hardware imperfections/impairments; for example, phase
noise, non-linearities, quantization errors, noise amplification,
and inter-carrier interference. Robustness to hardware imper-
fections has been conjectured in overview articles, such as [7].
Such a characteristic is notably important since the deployment
cost and circuit power consumption of massive MIMO scales
linearly with N , unless the hardware accuracy constraints can
be relaxed such that low-power, low-cost hardware is deployed
which is more prone to imperfections. Constant envelope
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 2
precoding was analyzed in [14] to facilitate the use of power-
efficient amplifiers in the downlink, while the impact of phase-
drifts was analyzed and simulated for single-carrier systems
in [15] and for orthogonal frequency-division multiplexing
(OFDM) in [16]. A preliminary proof of the conjecture was
given in [17], but the authors therein considered only additive
distortions and, thus, ignored other important characteristics
of hardware imperfections. That paper showed that one can
tolerate distortion variances that increase as
√
N with only
minor throughput losses, but did not investigate what this
implies for the design of different transceiver circuits.
In this paper, we consider a generalized uplink massive
MIMO system with arbitrary array configurations (e.g., co-
located or distributed antennas). Based on the extensive liter-
ature on modeling of transceiver hardware imperfections (see
[3], [4], [15], [18]–[24] and references therein), we propose
a tractable system model that jointly describes the impact
of multiplicative phase-drifts, additive distortion noise, noise
amplification, and inter-carrier interference. This stands in
contrast to the previous works [15]–[17], which each inves-
tigating only one of these effects. The following are the main
contributions of this paper:
• We derive a new linear minimum mean square error
(LMMSE) channel estimator that accounts for hardware
imperfections and allows the prediction of the detrimental
impact of phase-drifts.
• We present a simple and general expression for the
achievable uplink user rates and compute it in closed-
form, when the receiver applies maximum ratio combin-
ing (MRC) filters. We prove that the additive distortion
noise and noise amplification vanish asymptotically as
N → ∞, while the phase-drifts remain but are not
exacerbated.
• We obtain an intuitive scaling law that shows how fast
we can tolerate the levels of hardware imperfections
to increase with N, while maintaining high user rates.
This is an analytic proof of the conjecture that massive
MIMO systems can be deployed with inexpensive low-
power hardware without sacrificing the expected major
performance gains. The scaling law provides sufficient
conditions that hold for any judicious receive filters.
• The practical implications of the scaling law are exem-
plified for the main circuits at the receiver, namely, the
analog-to-digital converter (ADC), low noise amplifier
(LNA), and local oscillator (LO). The main components
of a typical receiver are illustrated in Fig. 1. The scaling
law reveals the tradeoff between hardware cost, level of
imperfections, and circuit power consumption. In partic-
ular, it shows how a circuit-aware design can make the
circuit power consumption increase as
√
N instead of N.
• The analytic results are validated numerically in a realis-
tic simulation setup, where we consider different antenna
deployment scenarios, common and separate LOs, dif-
ferent pilot sequence designs, and two types of receive
filters. A key observation is that separate LOs can provide
better performance than a common LO, since the phase-
drifts average out and the interference is reduced. This is
DSP
Filter
Receive
antenna 1
ADC
LNA
Mixer
LO
Filter
Receive
antenna N
ADC
LNA
Mixer
LO
Fig. 1. Block diagram of a typical N-antenna receiver. The main circuits are
shown, but these can be complemented with additional intermediate filters
and amplifiers depending on the implementation. Most of the circuits affect
only one antenna, whilst the LO can be either common for all antennas or
different.
also rigorously supported by the analytic scaling law.
This paper extends substantially our conference papers [25]
and [26], by generalizing the propagation model, generalizing
the analysis according to the new model, and providing more
comprehensive simulations. The paper is organized as follows:
In Section II, the massive MIMO system model under consid-
eration is presented. In Section III, a detailed performance
analysis of the achievable uplink user rates is pursued and the
impact of hardware imperfections is characterized, while in
Section IV we provide guidelines for circuit-aware design in
order to minimize the power dissipation of receiver circuits.
Our theoretical analysis is corroborated with simulations in
Section V, while Section VI concludes the paper.
Notation: The following notation is used throughout the
paper: Boldface (lower case) is used for column vectors, x,
and (upper case) for matrices, X. Let X
T
, X
∗
, and X
H
denote the transpose, conjugate, and conjugate transpose of X,
respectively. A diagonal matrix with a
1
, . . . , a
N
on the main
diagonal is denoted as diag(a
1
, . . . , a
N
), while I
N
is an N ×N
identity matrix. The set of complex-valued N × K matrices
is denoted by C
N×K
. The expectation operator is denoted
E{·} and , denotes definitions. The matrix trace function is
tr(·) and ⊗ is the Kronecker product. A Gaussian random
variable x is denoted x ∼ N(¯x, q), where ¯x is the mean and
q is the variance. A circularly symmetric complex Gaussian
random vector x is denoted x ∼ CN(
¯
x, Q), where
¯
x is the
mean and Q is the covariance matrix. The big O notation
f(x) = O(g(x)) means that
f(x)
g(x)
is bounded as x → ∞.
II. SYSTEM MODEL WITH HARDWARE IMPERFECTIONS
We consider the uplink of a cellular network with L ≥ 1
cells. Each cell consists of K single-antenna user equipments
(UEs) that communicate simultaneously with an array of N
antennas, which can be either co-located at a macro BS or
distributed over multiple fully coordinated small BSs. The
analysis of our paper holds for any N and K, but we
are primarily interested in massive MIMO topologies, where
N K 1. The frequency-flat channel from UE k in cell
l to BS j is denoted as h
jlk
,
h
h
(1)
jlk
. . . h
(N)
jlk
i
T
∈ C
N×1
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 3
and is modeled as Rayleigh block fading. This means that
it has a static realization for a coherence block of T chan-
nel uses and independent realizations between blocks.
1
The
UEs’ channels are independent. Each realization is complex
Gaussian distributed with zero mean and covariance matrix
Λ
jlk
∈ C
N×N
:
h
jlk
∼ CN(0, Λ
jlk
). (1)
The covariance matrix Λ
jlk
, diag
λ
(1)
jlk
, . . . , λ
(N)
jlk
is
assumed to be diagonal, which holds if the inter-antenna
distances are sufficiently large and the multi-path scattering
environment is rich [27].
2
The average channel attenuation
λ
(n)
jlk
is different for each combination of cells, UE index, and
receive antenna index n. It depends, for example, on the array
geometry and the UE location. Even for co-located antennas
one might have different values of λ
(n)
jlk
over the array, because
of the large aperture that may create variations in the shadow
fading.
The received signal y
j
(t) ∈ C
N×1
in cell j at a given
channel use t ∈ {1, . . . , T } in the coherence block is conven-
tionally modeled as [9]–[12]
y
j
(t) =
L
X
l=1
H
jl
x
l
(t) + n
j
(t) (2)
where the transmit signal in cell l is x
l
(t) =
[x
l1
(t) . . . x
lK
(t)]
T
∈ C
K×1
and we use the notation
H
jl
= [h
jl1
. . . h
jlK
] ∈ C
N×K
for brevity. The scalar signal
x
lk
(t) sent by UE k in cell l at channel use t is either a
deterministic pilot symbol (used for channel estimation) or an
information symbol from a Gaussian codebook; in any case,
we assume that the expectation of the transmit energy per
symbol is bounded as E{|x
lk
(t)|
2
} ≤ p
lk
. The thermal noise
vector n
j
(t) ∼ CN(0, σ
2
I
N
) is spatially and temporally
independent and has variance σ
2
.
The conventional model in (2) is well-accepted for small-
scale MIMO systems, but has an important drawback when
applied to massive MIMO topologies: it assumes that the large
antenna array consists of N high-quality antenna branches
which are all perfectly synchronized. Consequently, the de-
ployment cost and total power consumption of the circuits
attached to each antenna would at least grow linearly with
N, thereby making the deployment of massive MIMO rather
questionable, if not prohibitive, from an overall cost and
efficiency perspective.
In this paper, we analyze the far more realistic scenario
of having inexpensive hardware-constrained massive MIMO
arrays. More precisely, each receive array experiences hard-
ware imperfections that distort the communication. The exact
distortion characteristics depend generally on which modula-
tion scheme is used; for example, OFDM [18], filter bank
multicarrier (FBMC) [28], or single-carrier transmission [15].
1
The size of the time/frequency block where the channels are static depends
on UE mobility and propagation environment: T is the product of the
coherence time ˜τ
c
and coherence bandwidth
˜
W
c
, thus ˜τ
c
= 5 ms and
˜
W
c
= 100 kHz gives T = 500.
2
The analysis and main results of this paper can be easily extended to
arbitrary non-diagonal covariance matrices as in [11] and [17], but at the cost
of complicating the notation and expressions.
Nevertheless, the distortions can be classified into three dis-
tinct categories: 1) received signals are shifted in phase; 2)
distortion noise is added with a power proportional to the total
received signal power; and 3) thermal noise is amplified and
channel-independent interference is added. To draw general
conclusions on how these distortion categories affect massive
MIMO systems, we consider a generic system model with
hardware imperfections. The received signal in cell j at a given
channel use t ∈ {1, . . . , T } is modeled as
y
j
(t) = D
φ
j
(t)
L
X
l=1
H
jl
x
l
(t) + υ
j
(t) + η
j
(t) (3)
where the channel matrices H
jl
and transmitted signals x
l
(t)
are exactly as in (2). The hardware imperfections are defined
as follows:
1) The matrix D
φ
j
(t)
, diag
e
ıφ
j1
(t)
, . . . , e
ıφ
jN
(t)
de-
scribes multiplicative phase-drifts, where ı is the imag-
inary unit. The variable φ
jn
(t) is the phase-drift at the
nth receive antenna in cell j at time t. Motivated by the
standard phase-noise models in LOs [21], φ
jn
(t) follows
a Wiener process
φ
jn
(t) ∼ N(φ
jn
(t − 1), δ) (4)
which equals the previous realization φ
jn
(t − 1) plus
an independent Gaussian innovation of variance δ. The
phase-drifts can be either independent or correlated
between the antennas; for example, co-located arrays
might have a common LO (CLO) for all antennas
which makes the phase-drifts φ
jn
(t) identical for all
n = 1, . . . , N. In contrast, distributed arrays might have
separate LOs (SLOs) at each antenna, which make the
drifts independent, though we let the variance δ be equal
for simplicity. Both cases are considered herein.
2) The distortion noise υ
j
(t) ∼ CN(0, Υ
j
(t)), where
Υ
j
(t) , κ
2
L
X
l=1
K
X
k=1
E{|x
lk
(t)|
2
}diag
|h
(1)
jlk
|
2
, . . . , |h
(N)
jlk
|
2
(5)
for given channel realizations, where the double-sum
gives the received power at each antenna. Thus, the
distortion noise is independent between antennas and
channel uses, and the variance at a given antenna is
proportional to the current received signal power at this
antenna. This model can describe the quantization noise
in ADCs with gain control [19], approximate generic
non-linearities [4, Chapter 14], and approximate the
leakage between subcarriers due to calibration errors.
The parameter κ ≥ 0 describes how much weaker the
distortion noise magnitude is compared to the signal
magnitude.
3) The receiver noise η
j
(t) ∼ CN(0, ξI
N
) is independent
of the UE channels, in contrast to the distortion noise.
This term includes thermal noise, which typically is
amplified by LNAs and mixers in the receiver hardware,
and interference leakage from other frequency bands
and/or other networks. The receiver noise variance must
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 4
Pilot sequence Data symbols
Pilot sequenceData symbols
(a)
(b)
(c)
(d)
Coherence block
Data symbols
Fig. 2. Examples of different ways to distribute the B pilot symbols over the
coherence block of length T : (a) beginning of block; (b) middle of block; (c)
uniform pilot distribution; (d) preamble and a few distributed pilot symbols.
satisfy ξ ≥ σ
2
. If there is no interference leakage,
F =
ξ
σ
2
is called the noise amplification factor.
This tractable generic model of hardware imperfections at
the BSs is inspired by a plethora of prior works [3], [4],
[15], [18]–[24] and characterizes the joint behavior of all
hardware imperfections at the BSs—these can be uncalibrated
imperfections or residual errors after calibration. The model
in (3) is characterized by three parameters: δ, κ, and ξ. The
model is compatible with the conventional model in (2), which
is obtained by setting ξ = σ
2
and δ = κ = 0. The analysis in
this paper holds for arbitrary parameter values. Section IV
exemplifies the connection between imperfections in the main
transceiver circuits of the BSs and the three parameters. These
connections allow for circuit-aware design of massive MIMO
systems.
In the next section, we derive a channel estimator and
achievable UE rates for the system model in (3). By analyzing
the performance as N → ∞, we bring new insights into the
fundamental impact of hardware imperfections (in particular,
in terms of δ, κ, and ξ).
III. PERFORMANCE ANALYSIS
In this section, we derive achievable UE rates for the uplink
multi-cell system in (3) and analyze how these depend on
the number of antennas and hardware imperfections. We first
need to specify the transmission protocol.
3
The T channel
uses of each coherence block are split between transmission of
uplink pilot symbols and uplink data symbols. It is necessary
to dedicate B ≥ K channel uses for pilot transmission if
the receiving array should be able to spatially separate the
different UEs in the cell. The remaining T − B channel
uses are allocated for data transmission. The pilot symbols
can be distributed in different ways: for example, placed in
the beginning of the block [17], in the middle of the block
[29], uniformly distributed as in the LTE standard [30], or a
combination of these approaches [22]. These different cases
are illustrated in Fig. 2. The time indices used for pilot
transmission are denoted by τ
1
, . . . , τ
B
∈ {1, . . . , T }, while
D , {1, . . . , T } \ {τ
1
, . . . , τ
B
} are the time indices for data
transmission.
3
We assume that the same protocol is used in all cells, for analytic
simplicity. It was shown in [12, Remark 5] that nothing substantially different
will happen if this assumption is relaxed.
A. Channel Estimation under Hardware Imperfections
Based on the transmission protocol, the pilot sequence of
UE k in cell j is
˜
x
jk
, [x
jk
(τ
1
) . . . x
jk
(τ
B
)]
T
∈ C
B×1
. The
pilot sequences are predefined and can be selected arbitrarily
under the power constraints. Our analysis supports any choice,
but it is reasonable to make
˜
x
j1
, . . . ,
˜
x
jK
in cell j mutually
orthogonal to avoid intra-cell interference (this is the reason
to have B ≥ K).
Example 1: Let
e
X
j
, [
˜
x
j1
. . .
˜
x
jK
] denote the pilot
sequences in cell j. The simplest example of linearly inde-
pendent pilot sequences (with B = K) is
e
X
temporal
j
, diag(
√
p
j1
, . . . ,
√
p
jK
) (6)
where the different sequences are temporally orthogonal since
only UE k transmits at time τ
k
. Alternatively, the pilot
sequences can be made spatially orthogonal so that all UEs
transmit at every pilot transmission time, which effectively
increases the total pilot energy by a factor K. The canonical
example is to use a scaled discrete Fourier transform (DFT)
matrix [31]:
e
X
spatial
j
,
1 1 . . . 1
1 W
K
. . . W
K−1
K
.
.
.
.
.
.
.
.
.
.
.
.
1 W
B−1
K
. . . W
(B−1)(K−1)
K
e
X
temporal
j
(7)
where W
K
, e
−ı2π/K
.
The pilot sequences can also be jointly designed across
cells, to reduce inter-cell interference during pilot transmis-
sion. Since network-wide pilot orthogonality requires B ≥
LK, which typically is much larger than the coherence block
length T , practical networks need to balance between pilot
orthogonality and inter-cell interference. A key design goal is
to allocate non-orthogonal pilot sequences to UEs that have
nearly orthogonal channel covariance matrices; for example,
by making tr(Λ
jjk
Λ
jlm
) small for any combination of a UE
k in cell j and a UE m in cell l, as suggested in [32].
For any given set of pilot sequences, we now derive esti-
mators of the effective channels
h
jlk
(t) , D
φ
j
(t)
h
jlk
(8)
at any channel use t ∈ {1, . . . , T } and for all j, l, k. The
conventional multi-antenna channel estimators from [33]–[35]
cannot be applied in this paper since the generalized system
model in (3) has two non-standard properties: the pilot trans-
mission is corrupted by random phase-drifts and the distortion
noise is statistically dependent on the channels. Therefore, we
derive a new LMMSE estimator for the system model at hand.
Theorem 1: Let ψ
j
,
y
T
j
(τ
1
) . . . y
T
j
(τ
B
)
T
∈ C
BN
denote the combined received signal in cell j from the pilot
transmission. The LMMSE estimate of h
jlk
(t) at any channel
use t ∈ {1, . . . , T } for any l and k is
ˆ
h
jlk
(t) =
˜
x
H
lk
D
δ(t)
⊗ Λ
jlk
Ψ
−1
j
ψ
j
(9)
