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Cell-Free Massive MIMO Versus Small Cells

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In this paper, the authors proposed a max-min power control algorithm to ensure uniformly good service throughout the area of coverage in a cell-free massive MIMO system, where each user is served by a dedicated access point.
Abstract
A Cell-Free Massive MIMO (multiple-input multiple-output) system comprises a very large number of distributed access points (APs), which simultaneously serve a much smaller number of users over the same time/frequency resources based on directly measured channel characteristics. The APs and users have only one antenna each. The APs acquire channel state information through time-division duplex operation and the reception of uplink pilot signals transmitted by the users. The APs perform multiplexing/de-multiplexing through conjugate beamforming on the downlink and matched filtering on the uplink. Closed-form expressions for individual user uplink and downlink throughputs lead to max–min power control algorithms. Max–min power control ensures uniformly good service throughout the area of coverage. A pilot assignment algorithm helps to mitigate the effects of pilot contamination, but power control is far more important in that regard. Cell-Free Massive MIMO has considerably improved performance with respect to a conventional small-cell scheme, whereby each user is served by a dedicated AP, in terms of both 95%-likely per-user throughput and immunity to shadow fading spatial correlation. Under uncorrelated shadow fading conditions, the cell-free scheme provides nearly fivefold improvement in 95%-likely per-user throughput over the small-cell scheme, and tenfold improvement when shadow fading is correlated.

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Cell-Free Massive MIMO Versus Small Cells
Ngo, H. Q., Ashikhmin, A., Yang, H., Larsson, E. G., & Marzetta, T. L. (2017). Cell-Free Massive MIMO Versus
Small Cells.
IEEE Transactions on Wireless Communications
,
16
(3), 1834-1850.
https://doi.org/10.1109/TWC.2017.2655515
Published in:
IEEE Transactions on Wireless Communications
Document Version:
Peer reviewed version
Queen's University Belfast - Research Portal:
Link to publication record in Queen's University Belfast Research Portal
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Download date:10. Aug. 2022

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. XX, NO. X, XXX 2017 1
Cell-Free Massive MIMO versus Small Cells
Hien Quoc Ngo, Alexei Ashikhmin, Hong Yang, Erik G. Larsson, and Thomas L. Marzetta
Abstract—A Cell-Free Massive MIMO (multiple-input
multiple-output) system comprises a very large number of
distributed access points (APs) which simultaneously serve a
much smaller number of users over the same time/frequency
resources based on directly measured channel characteristics.
The APs and users have only one antenna each. The APs
acquire channel state information through time-division duplex
operation and the reception of uplink pilot signals transmitted
by the users. The APs perform multiplexing/de-multiplexing
through conjugate beamforming on the downlink and matched
filtering on the uplink. Closed-form expressions for individual
user uplink and downlink throughputs lead to max-min power
control algorithms. Max-min power control ensures uniformly
good service throughout the area of coverage. A pilot assignment
algorithm helps to mitigate the effects of pilot contamination,
but power control is far more important in that regard.
Cell-Free Massive MIMO has considerably improved perfor-
mance with respect to a conventional small-cell scheme, whereby
each user is served by a dedicated AP, in terms of both 95%-
likely per-user throughput and immunity to shadow fading spatial
correlation. Under uncorrelated shadow fading conditions, the
cell-free scheme provides nearly 5-fold improvement in 95%-
likely per-user throughput over the small-cell scheme, and 10-fold
improvement when shadow fading is correlated.
Index Terms—Cell-Free Massive MIMO system, conjugate
beamforming, Massive MIMO, network MIMO, small cell.
I. INTRODUCTION
M
ASSIVE multiple-input multiple-output (MIMO),
where a base station with many antennas
simultaneously serves many users in the same time-frequency
resource, is a promising 5G wireless access technology that
can provide high throughput, reliability, and energy efficiency
with simple signal processing [2], [3]. Massive antenna
arrays at the base stations can be deployed in collocated or
distributed setups. Collocated Massive MIMO architectures,
where all service antennas are located in a compact area,
have the advantage of low backhaul requirements. In contrast,
in distributed Massive MIMO systems, the service antennas
are spread out over a large area. Owing to their ability
Manuscript received August 03, 2015; revised February 22, 2016, August
25, 2016, and December 16, 2016; accepted January 05, 2017. The associate
editor coordinating the review of this paper and approving it for publication
was Dr. Mai Vu. The work of H. Q. Ngo and E. G. Larsson was supported
in part by the Swedish Research Council (VR) and ELLIIT. Portions of this
work were performed while H. Q. Ngo was with Bell Labs in 2014. Part of
this work was presented at the 16th IEEE International Workshop on Signal
Processing Advances in Wireless Communications (SPAWC) [1].
H. Q. Ngo and E. G. Larsson are with the Department of Electrical
Engineering (ISY), Linköping University, 581 83 Linköping, Sweden (Email:
hien.ngo@liu.se; erik.g.larsson@liu.se). H. Q. Ngo is also with the School of
Electronics, Electrical Engineering and Computer Science, Queen’s University
Belfast, Belfast BT3 9DT, U.K.
A. Ashikhmin, H. Yang and T. L. Marzetta are with Nokia Bell Labo-
ratories, Murray Hill, NJ 07974 USA (Email: alexei.ashikhmin@nokia-bell-
labs.com; h.yang@nokia-bell-labs.com; tom.marzetta@nokia-bell-labs.com).
Digital Object Identifier xxx/xxx
to more efficiently exploit diversity against the shadow
fading, distributed systems can potentially offer much higher
probability of coverage than collocated Massive MIMO [4],
at the cost of increased backhaul requirements.
In this work, we consider a distributed Massive MIMO
system where a large number of service antennas, called access
points (APs), serve a much smaller number of autonomous
users distributed over a wide area [1]. All APs cooperate
phase-coherently via a backhaul network, and serve all users
in the same time-frequency resource via time-division duplex
(TDD) operation. There are no cells or cell boundaries. There-
fore, we call this system “Cell-Free Massive MIMO”. Since
Cell-Free Massive MIMO combines the distributed MIMO and
Massive MIMO concepts, it is expected to reap all benefits
from these two systems. In addition, since the users now
are close to the APs, Cell-Free Massive MIMO can offer a
high coverage probability. Conjugate beamforming/matched
filtering techniques, also known as maximum-ratio processing,
are used both on uplink and downlink. These techniques
are computationally simple and can be implemented in a
distributed manner, that is, with most processing done locally
at the APs.
1
In Cell-Free Massive MIMO, there is a central processing
unit (CPU), but the information exchange between the APs
and this CPU is limited to the payload data, and power
control coefficients that change slowly. There is no sharing
of instantaneous channel state information (CSI) among the
APs or the central unit. All channels are estimated at the
APs through uplink pilots. The so-obtained channel estimates
are used to precode the transmitted data in the downlink
and to perform data detection in the uplink. Throughout we
emphasize per-user throughput rather than sum-throughput. To
that end we employ max-min power control.
In principle, Cell-Free Massive MIMO is an incarnation of
general ideas known as “virtual MIMO”, “network MIMO”,
“distributed MIMO”, “(coherent) cooperative multipoint joint
processing” (CoMP) and “distributed antenna systems” (DAS).
The objective is to use advanced backhaul to achieve coherent
processing across geographically distributed base station an-
tennas, in order to provide uniformly good service for all users
in the network. The outstanding aspect of Cell-Free Massive
MIMO is its operating regime: many single-antenna access
points simultaneously serve a much smaller number of users,
using computationally simple (conjugate beamforming) signal
processing. This facilitates the exploitation of phenomena such
as favorable propagation and channel hardening which are
1
Other linear processing techniques (e.g. zero-forcing) may improve the
system performance, but they require more backhaul than maximum-ratio
processing does. The tradeoff between the implementation complexity and
the system performance for these techniques is of interest and needs to be
studied in future work.

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. XX, NO. X, XXX 2017 2
also key characteristics of cellular Massive MIMO [5]. In
turn, this enables the use of computationally efficient and
globally optimal algorithms for power control, and simple
schemes for pilot assignment (as shown later in this paper). In
summary, Cell-Free Massive MIMO is a useful and scalable
implementation of the network MIMO and DAS concepts
much in the same way as cellular Massive MIMO is a useful
and scalable form of the original multiuser MIMO concept
(see, e.g., [5, Chap. 1] for an extended discussion of the latter).
Related work:
Many papers have studied network MIMO [6], [8], [9] and
DAS [7], [10], [11], and indicated that network MIMO and
DAS may offer higher rates than colocated MIMO. However,
these works did not consider the case of very large numbers of
service antennas. Related works which use a similar system
model as in our paper are [12]–[18]. In these works, DAS
with the use of many antennas, called large-scale DAS or
distributed massive MIMO, was exploited. However, in all
those papers, perfect CSI was assumed at both the APs and
the users, and in addition, the analysis in [18] was asymptotic
in the number of antennas and the number of users. A realistic
analysis must account for imperfect CSI, which is an inevitable
consequence of the finite channel coherence in a mobile sys-
tem and which typically limits the performance of any wireless
system severely [19]. Large-scale DAS with imperfect CSI
was considered in [20]–[23] for the special case of orthogonal
pilots or the reuse of orthogonal pilots, and in [24] assuming
frequency-division duplex (FDD) operation. In addition, in
[20], the authors exploited the low-rank structure of users’
channel covariance matrices, and examined the performance
of uplink transmission with matched-filtering detection, under
the assumption that all users use the same pilot sequence.
By contrast, in the current paper, we assume TDD operation,
hence rely on reciprocity to acquire CSI, and we assume the
use of arbitrary pilot sequences in the network resulting in
pilot contamination, which was not studied in previous work.
We derive rigorous capacity lower bounds valid for any finite
number of APs and users, and give algorithms for optimal
power control (to global optimality) and pilot assignment.
The papers cited above compare the performance between
distributed and collocated Massive MIMO systems. An al-
ternative to (distributed) MIMO systems is to deploy small
cells, consisting of APs that do not cooperate. Small-cell sys-
tems are considerably simpler than Cell-Free Massive MIMO,
since only data and power control coefficients are exchanged
between the CPU and the APs. It is expected that Cell-Free
Massive MIMO systems perform better than small-cell sys-
tems. However it is not clear, quantitatively, how much Cell-
Free Massive MIMO systems can gain compared to small-cell
systems. Most previous work compares collocated Massive
MIMO and small-cell systems [25], [26]. In [25], the authors
show that, when the number of cells is large, a small-cell sys-
tem is more energy-efficient than a collocated Massive MIMO
system. By taking into account a specific transceiver hardware
impairment and power consumption model, paper [26] shows
that reducing the cell size (or increasing the base station
density) is the way to increase the energy efficiency. However
when the circuit power dominates over the transmission power,
this benefit saturates. Energy efficiency comparisons between
collocated massive MIMO and small-cell systems are also
studied in [27], [28]. There has however been little work that
compares distributed Massive MIMO and small-cell systems.
A comparison between small-cell and distributed Massive
MIMO systems is reported in [12], assuming perfect CSI at
both the APs and the users. Yet, a comprehensive performance
comparison between small-cell and distributed Massive MIMO
systems that takes into account the effects of imperfect CSI,
pilot assignment, and power control is not available in the
existing literature.
Specific contributions of the paper:
We consider a cell-free massive MIMO with conjugate
beamforming on the downlink and matched filtering on
the uplink. We show that, as in the case of collocated
systems, when the number of APs goes to infinity, the
effects of non-coherent interference, small-scale fading,
and noise disappear.
We derive rigorous closed-form capacity lower bounds for
the Cell-Free Massive MIMO downlink and uplink with
finite numbers of APs and users. Our analysis takes into
account the effects of channel estimation errors, power
control, and non-orthogonality of pilot sequences.
We compare two pilot assignment schemes: random as-
signment and greedy assignment.
We devise max-min fairness power control algorithms
that maximize the smallest of all user rates. Globally op-
timal solutions can be computed by solving a sequence of
second-order cone programs (SOCPs) for the downlink,
and a sequence of linear programs for the uplink.
We quantitatively compare the performance of Cell-Free
Massive MIMO to that of small-cell systems, under
uncorrelated and correlated shadow fading models.
The rest of paper is organized as follows. In Section II,
we describe the Cell-Free Massive MIMO system model. In
Section III, we present the achievable downlink and uplink
rates. The pilot assignment and power control schemes are
developed in Section IV. The small-cell system is discussed
in Section V. We provide numerical results and discussions in
Section VI and finally conclude the paper in Section VII.
Notation: Boldface letters denote column vectors. The su-
perscripts ()
, ()
T
, and ()
H
stand for the conjugate, transpose,
and conjugate-transpose, respectively. The Euclidean norm and
the expectation operators are denoted by k · k and E {·},
respectively. Finally, z CN
0, σ
2
denotes a circularly
symmetric complex Gaussian random variable (RV) z with
zero mean and variance σ
2
, and z N(0, σ
2
) denotes a real-
valued Gaussian RV.
II. CELL-FREE MASSIVE MIMO SYSTEM MODEL
We consider a Cell-Free Massive MIMO system with M
APs and K users. All APs and users are equipped with a
single antenna, and they are randomly located in a large area.
Furthermore, all APs connect to a central processing unit via
a backhaul network, see Figure 1. We assume that all M APs
simultaneously serve all K users in the same time-frequency
resource. The transmission from the APs to the users (down-
link transmission) and the transmission from the users to the

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. XX, NO. X, XXX 2017 3
terminal 1
terminal k
terminal K
AP1
AP2
APm
gmk
APM
CPU
Fig. 1. Cell-Free Massive MIMO system.
APs (uplink transmission) proceed by TDD operation. Each
coherence interval is divided into three phases: uplink training,
downlink payload data transmission, and uplink payload data
transmission. In the uplink training phase, the users send pilot
sequences to the APs and each AP estimates the channel to all
users. The so-obtained channel estimates are used to precode
the transmit signals in the downlink, and to detect the signals
transmitted from the users in the uplink. In this work, to avoid
sharing of channel state information between the APs, we
consider conjugate beamforming in the downlink and matched
filtering in the uplink.
No pilots are transmitted in the downlink of Cell-Free
Massive MIMO. The users do not need to estimate their
effective channel gain, but instead rely on channel hardening,
which makes this gain close to its expected value, a known
deterministic constant. Our capacity bounds account for the
error incurred when the users use the average effective channel
gain instead of the actual effective gain. Channel hardening in
Massive MIMO is discussed, for example, in [2].
Notation is adopted and assumptions are made as follows:
The channel model incorporates the effects of small-scale
fading and large-scale fading (that latter includes path
loss and shadowing). The small-scale fading is assumed
to be static during each coherence interval, and change
independently from one coherence interval to the next.
The large-scale fading changes much more slowly, and
stays constant for several coherence intervals. Depending
on the user mobility, the large-scale fading may stay
constant for a duration of at least some 40 small-scale
fading coherence intervals [29], [30].
We assume that the channel is reciprocal, i.e., the channel
gains on the uplink and on the downlink are the same.
This reciprocity assumption requires TDD operation and
perfect calibration of the hardware chains. The feasibility
of the latter is demonstrated for example in [31] for
collocated Massive MIMO and it is conceivable that the
problem can be similarly somehow for Cell-Free Massive
MIMO. Investigating the effect of imperfect calibration
is an important topic for future work.
We let g
mk
denote the channel coefficient between the
kth user and the mth AP. The channel g
mk
is modelled
as follows:
g
mk
= β
1/2
mk
h
mk
, (1)
where h
mk
represents the small-scale fading, and β
mk
represents the large-scale fading. We assume that h
mk
,
m = 1, . . . , M, K = 1, . . . K, are independent and iden-
tically distributed (i.i.d.) CN (0, 1) RVs. The justification
of the assumption of independent small-scale fading is
that the APs and the users are distributed over a wide area,
and hence, the set of scatterers is likely to be different
for each AP and each user.
We assume that all APs are connected via perfect back-
haul that offers error-free and infinite capacity to the
CPU. In practice, backhaul will be subject to significant
practical constraints [32], [33]. Future work is needed to
quantify the impact of backhaul constraints on perfor-
mance.
In all scenarios, we let q
k
denote the symbol asso-
ciated with the kth user. These symbols are mutually
independent, and independent of all noise and channel
coefficients.
A. Uplink Training
The Cell-Free Massive MIMO system employs a wide
spectral bandwidth, and the quantities g
mk
and h
mk
are de-
pendent on frequency; however β
mk
is constant with respect to
frequency. The propagation channels are assumed to be piece-
wise constant over a coherence time interval and a frequency
coherence interval. It is necessary to perform training within
each such time/frequency coherence block. We assume that
β
mk
is known, a priori, wherever required.
Let τ
c
be the length of the coherence interval (in samples),
which is equal to the product of the coherence time and
the coherence bandwidth, and let τ
cf
be the uplink training
duration (in samples) per coherence interval, where the su-
perscript cf stands for “cell-free”. It is required that τ
cf
<
τ
c
. During the training phase, all K users simultaneously
send pilot sequences of length τ
cf
samples to the APs. Let
τ
cf
ϕ
ϕ
ϕ
k
C
τ
cf
×1
, where kϕ
ϕ
ϕ
k
k
2
= 1, be the pilot sequence
used by the kth user, k = 1, 2, ··· , K. Then, the τ
cf
× 1
received pilot vector at the mth AP is given by
y
p,m
=
q
τ
cf
ρ
cf
p
K
X
k=1
g
mk
ϕ
ϕ
ϕ
k
+ w
p,m
, (2)
where ρ
cf
p
is the normalized signal-to-noise ratio (SNR) of
each pilot symbol and w
p,m
is a vector of additive noise at
the mth AP. The elements of w
p,m
are i.i.d. CN (0, 1) RVs.
Based on the received pilot signal y
p,m
, the mth AP
estimates the channel g
mk
, k = 1, ..., K. Denote by ˇy
p,mk
the projection of y
p,m
onto ϕ
ϕ
ϕ
H
k
:
ˇy
p,mk
= ϕ
ϕ
ϕ
H
k
y
p,m
=
q
τ
cf
ρ
cf
p
g
mk
+
q
τ
cf
ρ
cf
p
K
X
k
0
6=k
g
mk
0
ϕ
ϕ
ϕ
H
k
ϕ
ϕ
ϕ
k
0
+ ϕ
ϕ
ϕ
H
k
w
p,m
. (3)

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. XX, NO. X, XXX 2017 4
Although, for arbitrary pilot sequences, ˇy
p,mk
is not a suf-
ficient statistic for the estimation of g
mk
, one can still use
this quantity to obtain suboptimal estimates. In the special
case when any two pilot sequences are either identical or
orthogonal, then ˇy
p,mk
is a sufficient statistic, and estimates
based on ˇy
p,mk
are optimal. The MMSE estimate of g
mk
given
ˇy
p,mk
is
ˆg
mk
=
E
n
ˇy
p,mk
g
mk
o
E
n
|ˇy
p,mk
|
2
o
ˇy
p,mk
= c
mk
ˇy
p,mk
, (4)
where
c
mk
,
q
τ
cf
ρ
cf
p
β
mk
τ
cf
ρ
cf
p
P
K
k
0
=1
β
mk
0
ϕ
ϕ
ϕ
H
k
ϕ
ϕ
ϕ
k
0
2
+ 1
.
Remark 1: If τ
cf
K, then we can choose ϕ
ϕ
ϕ
1
,ϕ
ϕ
ϕ
2
, ··· , ϕ
ϕ
ϕ
K
so that they are pairwisely orthogonal, and hence, the second
term in (3) disappears. Then the channel estimate ˆg
mk
is inde-
pendent of g
mk
0
, k
0
6= k. However, owing to the limited length
of the coherence interval, in general, τ
cf
< K, and mutually
non-orthogonal pilot sequences must be used throughout the
network. The channel estimate ˆg
mk
is degraded by pilot signals
transmitted from other users, owing to the second term in (3).
This causes the so-called pilot contamination effect.
Remark 2: The channel estimation is performed in a decen-
tralized fashion. Each AP autonomously estimates the channels
to the K users. The APs do not cooperate on the channel
estimation, and no channel estimates are interchanged among
the APs.
B. Downlink Payload Data Transmission
The APs treat the channel estimates as the true channels,
and use conjugate beamforming to transmit signals to the K
users. The transmitted signal from the mth AP is given by
x
m
=
q
ρ
cf
d
K
X
k=1
η
1/2
mk
ˆg
mk
q
k
, (5)
where q
k
, which satisfies E
|q
k
|
2
= 1, is the symbol in-
tended for the kth user, and η
mk
, m = 1, . . . , M, k = 1, . . . K,
are power control coefficients chosen to satisfy the following
power constraint at each AP:
E
|x
m
|
2
ρ
cf
d
. (6)
With the channel model in (1), the power constraint
E
|x
m
|
2
ρ
cf
d
can be rewritten as:
K
X
k=1
η
mk
γ
mk
1, for all m, (7)
where
γ
mk
, E
n
|ˆg
mk
|
2
o
=
q
τ
cf
ρ
cf
p
β
mk
c
mk
. (8)
The received signal at the kth user is given by
r
d,k
=
M
X
m=1
g
mk
x
m
+ w
d,k
=
q
ρ
cf
d
M
X
m=1
K
X
k
0
=1
η
1/2
mk
0
g
mk
ˆg
mk
0
q
k
0
+ w
d,k
, (9)
where w
d,k
is additive CN (0, 1) noise at the kth user. Then
q
k
will be detected from r
d,k
.
C. Uplink Payload Data Transmission
In the uplink, all K users simultaneously send their data
to the APs. Before sending the data, the kth user weights its
symbol q
k
, E
|q
k
|
2
= 1, by a power control coefficient
η
k
,
0 η
k
1. The received signal at the mth AP is given by
y
u,m
=
q
ρ
cf
u
K
X
k=1
g
mk
η
k
q
k
+ w
u,m
, (10)
where ρ
cf
u
is the normalized uplink SNR and w
u,m
is additive
noise at the mth AP. We assume that w
u,m
CN (0, 1).
To detect the symbol transmitted from the kth user, q
k
, the
mth AP multiplies the received signal y
u,m
with the conjugate
of its (locally obtained) channel estimate ˆg
mk
. Then the so-
obtained quantity ˆg
mk
y
u,m
is sent to the CPU via a backhaul
network. The CPU sees
r
u,k
=
M
X
m=1
ˆg
mk
y
u,m
=
K
X
k
0
=1
M
X
m=1
q
ρ
cf
u
η
k
0
ˆg
mk
g
mk
0
q
k
0
+
M
X
m=1
ˆg
mk
w
u,m
. (11)
Then, q
k
is detected from r
u,k
.
III. PERFORMANCE ANALYSIS
A. Large-M Analysis
In this section, we provide some insights into the perfor-
mance of Cell-Free Massive MIMO systems when M is very
large. The convergence analysis is done conditioned on a
set of deterministic large-scale fading coefficients {β
mk
}. We
show that, as in the case of Collocated Massive MIMO, when
M , the channels between the users and the APs become
orthogonal. Therefore, with conjugate beamforming respec-
tively matched filtering, non-coherent interference, small-scale
fading, and noise disappear. The only remaining impairment
is pilot contamination, which consists of interference from
users using same pilot sequences as the user of interest in
the training phase.
On downlink, from (9), the received signal at the kth user
can be written as:
r
d,k
=
q
ρ
cf
d
M
X
m=1
η
1/2
mk
g
mk
ˆg
mk
q
k
| {z }
DS
k
+
q
ρ
cf
d
M
X
m=1
K
X
k
0
6=k
η
1/2
mk
0
g
mk
ˆg
mk
0
q
k
0
| {z }
MUI
k
+w
d,k
, (12)

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Space-Time Block Coding for Wireless Communications

E. Masoud
TL;DR: In this paper, the authors propose a solution to solve the problem of the problem: this paper ] of the "missing link" problem, i.i.p.II.
Journal ArticleDOI

6G and Beyond: The Future of Wireless Communications Systems

TL;DR: Significant technological breakthroughs to achieve connectivity goals within 6G include: a network operating at the THz band with much wider spectrum resources, intelligent communication environments that enable a wireless propagation environment with active signal transmission and reception, and pervasive artificial intelligence.
Journal ArticleDOI

Precoding and Power Optimization in Cell-Free Massive MIMO Systems

TL;DR: Cell-free Massive MIMO is shown to provide five- to ten-fold improvement in 95%-likely per-user throughput over small-cell operation and a near-optimal power control algorithm is developed that is considerably simpler than exact max–min power control.
References
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Book

Convex Optimization

TL;DR: In this article, the focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them, and a comprehensive introduction to the subject is given. But the focus of this book is not on the optimization problem itself, but on the problem of finding the appropriate technique to solve it.
Journal ArticleDOI

Noncooperative Cellular Wireless with Unlimited Numbers of Base Station Antennas

TL;DR: A cellular base station serves a multiplicity of single-antenna terminals over the same time-frequency interval and a complete multi-cellular analysis yields a number of mathematically exact conclusions and points to a desirable direction towards which cellular wireless could evolve.
Journal ArticleDOI

A Tractable Approach to Coverage and Rate in Cellular Networks

TL;DR: The proposed model is pessimistic (a lower bound on coverage) whereas the grid model is optimistic, and that both are about equally accurate, and the proposed model may better capture the increasingly opportunistic and dense placement of base stations in future networks.
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Energy and Spectral Efficiency of Very Large Multiuser MIMO Systems

TL;DR: In this paper, the tradeoff between the energy efficiency and spectral efficiency of a single-antenna system is quantified for a channel model that includes small-scale fading but not large scale fading, and it is shown that the use of moderately large antenna arrays can improve the spectral and energy efficiency with orders of magnitude compared to a single antenna system.
Journal ArticleDOI

How much training is needed in multiple-antenna wireless links?

TL;DR: This work compute a lower bound on the capacity of a channel that is learned by training, and maximize the bound as a function of the received signal-to-noise ratio (SNR), fading coherence time, and number of transmitter antennas.
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