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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 19, NO. 1, JANUARY 2020
77
Making Cell-Free Massive MIMO Competitive With
MMSE Processing and Centralized Implementation
Emil Björnson
,
Senior Member, IEEE
, and Luca Sanguinetti
,
Senior Member, IEEE
Abstract
—Cell-free
Massive
MIMO
is
considered
as
a
promising technology for satisfying the increasing number of
users and high rate expectations in beyond-5G networks. The key
idea is to let many distributed access points (APs) communicate
with all users in the network, possibly by using joint coherent
signal
processing. The aim
of
this paper is
to
provide
the
first comprehensive analysis of this technology under different
degrees of cooperation among the APs. Particularly, the uplink
spectral efficiencies of four different cell-free implementations
are analyzed, with spatially correlated fading and arbitrary
linear processing. It turns out that it is possible to outperform
conventional Cellular Massive MIMO and small cell networks
by a wide margin, but only using global or local minimum
mean-square error (MMSE) combining. This is in sharp contrast
to the existing literature, which advocates for maximum-ratio
combining. Also, we show that a centralized implementation
with optimal MMSE processing not only maximizes the SE but
largely reduces the fronthaul signaling compared to the standard
distributed approach. This makes it the preferred way to operate
Cell-free Massive MIMO networks. Non-linear decoding is also
investigated and shown to bring negligible improvements.
Index Terms
—Beyond 5G MIMO, cell-free massive MIMO,
cellular massive MIMO, uplink, AP cooperation, MMSE process-
ing, fronthaul signaling, non-linear decoding, small-cell networks.
I. I
NTRODUCTION
T
HE traditional way to cover a large geographical area
with wireless communication services uses the cellular
network topology in Fig. 1(a), where each base station (BS)
serves an exclusive set of user equipments (UEs) [2]. This
network topology has been utilized for many decades and
the spectral efficiency (SE) has been gradually improved by
reducing the cell sizes and applying more advanced signal
processing schemes for interference mitigation [3].
Recently, massive multiple-input multiple-output (mMIMO)
has become the key 5G physical-layer technology [4]–[7].
Manuscript received March 20, 2019; revised July 5, 2019 and September 3,
2019; accepted
September
9, 2019. Date
of publication
September
20,
2019; date of current version January 8, 2020. The work of E. Björnson
was supported by the Excellence Center at Linköping-Lund in Information
Technology (ELLIIT) and the Wallenberg AI, Autonomous Systems and
Software Program (WASP). The work of L. Sanguinetti was supported by the
University of Pisa through the Research Project CONCEPT under Grant PRA
2018-2019. This article was presented in part at the IEEE SPAWC 2019 [1].
The associate editor coordinating the review of this article and approving it
for publication was A. Zaidi.
(Corresponding author: Emil Björnson.)
E. Björnson is with the Department
of Electrical
Engineering
(ISY),
Linköping University, 58183 Linköping, Sweden (e-mail: emil.bjornson@
liu.se).
L. Sanguinetti is with the Dipartimento di Ingegneria dell’Informazione,
University of Pisa, 56122 Pisa, Italy (e-mail: luca.sanguinetti@unipi.it).
Color versions of one or more of the figures in this article are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TWC.2019.2941478
It can improve the SE by at least
10
×
over legacy cellu-
lar networks [3], by upgrading the BS hardware instead of
deploying new BS sites. The SE gain comes from that each
BS has a compact array with a hundred or more antennas,
which are used for digital beamforming and, particularly,
to spatially multiplex many user equipments (UEs) on the
same time-frequency resource [8]. The characteristic feature
of mMIMO, compared to traditional multi-user MIMO, is that
each BS has many more antennas than UEs in the cell.
Signal processing methods, such as minimum mean-squared
error (MMSE) combining in the uplink, can be used individ-
ually at each BS to suppress interference from both the same
and other cells [3], [9], [10], without the need for any BS
cooperation. The mMIMO theory also supports deployments
with spatially distributed arrays in each cell [11], [12], as also
illustrated in Fig. 1(a). This setup is essentially the same as
the Distributed Antenna System (DAS) setup in [13] and Coor-
dinated Multi-Point (CoMP) with static, disjoint cooperation
clusters [14], [15]. These are all different embodiments of
cellular networks.
An
alternative
network
infrastructure
was
considered
in [16], [17] under the name of
Cell-free mMIMO
. The idea is
to deploy a large number of distributed single-antenna access
points (APs), which are connected to a central processing
unit (CPU), also known as an edge-cloud processor [18] or
cloud radio access network (C-RAN) data center [19]. The
CPU operates the system in a Network MIMO fashion, with
no cell boundaries, to jointly serve the UEs by coherent
joint transmission and reception [15], [20]–[23]. Compared to
traditional Network MIMO, the outstanding aspect of Cell-free
mMIMO is the operating regime with many more APs than
UEs [16]. From an analytical perspective, an important novelty
was that imperfect channel state information (CSI) was consid-
ered in the performance analysis, while perfect CSI was often
assumed in the past [15]. The paper [16] advocated the use
of maximum ratio (MR) processing (a.k.a. matched filtering
or conjugate beamforming) locally at each AP, while [17],
[24] showed that partially or fully centralized processing at
the CPU can achieve higher SE.
A. Motivation
The focus in the early papers [16], [17] was on comparing
Cell-free mMIMO with a small-cell network; that is, the APs
are deployed at the same places, but each AP serves its own
exclusive set of UEs. Since small cells are a special case of
Cell-free mMIMO, they obviously provide lower performance.
Particularly, [16], [17] demonstrated large improvements in
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 19, NO. 1, JANUARY 2020
Fig. 1.
Comparison of different cellular and cell-free network topologies.
median and 95%-likely SE. In Section IV, we will show that
this is partially due to the fact that a poor implementation of
the small-cell network was considered in [16], [17]. In fact,
we will show that more sophisticated processing than MR is
needed in Cell-free mMIMO to always outperform small cells.
Unlike [16], [17], this paper aims at comparing Cell-free
mMIMO with conventional Cellular mMIMO and its primary
goal is to find the most competitive cell-free implementa-
tion.
1
Both network topologies are illustrated in Fig. 1. The
large differences make the comparison non-trivial and provide
interesting inputs into the design of beyond-5G networks.
Cellular mMIMO benefits from channel hardening and spatial
interference suppression, but cell-edge UEs can have bad
channel conditions. On the other hand, Cell-free mMIMO
benefits from strong macro diversity but its interference sup-
pression capability highly depends on how it is operated. The
early papers [16], [17] conjectured that channel hardening
also appears in Cell-free mMIMO, but it was later shown
that capacity bounds that presumes hardening can greatly
underestimate the practical SE [27]. To achieve a reasonably
fair comparison, we focus on the uplink and assume that
the data transmission is preceded by a pilot-based channel
1
Previous comparisons are found in [25], [26] but only for a single cell, so it
is not cellular, and only MR is used, which is known to perform badly [3].
estimation phase. All UEs transmit with equal powers for any
of the different levels and network topologies.
B. Contributions
The major contributions of this paper are two-fold. Firstly,
we introduce a taxonomy with four different implementations
of Cell-free mMIMO, which are characterized by different
degrees of cooperation among the APs. Secondly, we pro-
vide new achievable SE expressions, which are valid for
spatially correlated fading channels, imperfect CSI, APs with
an arbitrary number
N
of antennas, and heuristic or optimized
receive combining schemes. All this provides a common
analytical framework to numerically evaluate the benefits
and costs (in terms of fronthaul signaling) of the different
implementations and to understand how Cell-free mMIMO
should be operated and designed in order to get much higher
performance than conventional Cellular mMIMO and small
cells.
The four different levels of cooperation that we consider
in this paper are as follows. The so-called
Level 4
is a form
of Network MIMO and stands for a fully centralized network
in which the pilot and data signals received at all APs are
gathered (through the fronthaul links) at the CPU, which
performs channel estimation and data detection.
Level 3
relies
on the large-scale fading decoding (LSFD) strategy, which
was originally proposed for Cellular mMIMO in [28], [29].
Particularly, it operates in two stages. In the first stage, each
AP locally estimates the channels and applies an arbitrary
receive combiner to obtain local estimates of the UE data.
These are then gathered at the CPU where they are linearly
processed to perform joint detection. Only channel statistics
can be utilized in the second stage at the CPU since the
pilot signals are not shared over the fronthaul links.
Level 2
is a direct simplification of Level
3
in the sense that the
CPU performs detection in the second stage by simply taking
the average of the local estimates. This dispenses the CPU
from knowledge of the channel statistics and thus reduces
the amount of information to be exchanged. Finally,
Level 1
stands for a fully distributed network in which the detection
is performed locally at the APs by using only local channel
estimates and one AP serves each UE. This is a small-cell
network where nothing is exchanged with the CPU.
The above levels have been partially analyzed before in
the literature, but not under the general and practical con-
ditions considered in this paper, which allow us to draw
conclusions that differ in several important ways—in par-
ticular, we show that MR combining performs terribly bad
in Cell-free mMIMO. Level
4
was considered in [24], [30],
[31] for
N
= 1
and in [27], [32] with
N
≥
1
but with
spatially uncorrelated channels. Level
3
was investigated in
[24], [33] for
N
=
1
and MR combining. Level
2
was
considered in [16], [31], [34]–[38] (among many others) but
only with MR combining. A suboptimal implementation of
Level
1
with
N
= 1
was considered in [16] (the suboptimality
is explained in detail in Section III-D). There are also previous
papers that consider various forms of Levels 1–4 under perfect
CSI; see the reference list of [16] for a good selection of
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BJÖRNSON AND SANGUINETTI: MAKING CELL-FREE MASSIVE MIMO COMPETITIVE WITH MMSE PROCESSING
79
such papers. In addition, there is previous research on BS
cooperation in cellular networks, where the received signals
and CSI are shared between BSs to cancel inter-cell interfer-
ence; see [39]–[41] and reference therein. These papers also
consider different levels of cooperation, but these are heavily
influenced by the cellular topology (e.g., BSs send signals to
each other, BSs are surrounded by UEs, and there exist cell
edges) and can thus not be applied to Cell-free mMIMO.
C. Paper Outline
The rest of this paper is organized as follows. Section II
defines
the
system
model
for
uplink
Cell-free
mMIMO
for both data transmission and channel estimation. Next,
Section III presents the four levels of receiver cooperation,
including achievable SE expressions for spatially correlated
fading, multi-antenna APs, and optimized receive combin-
ing. The four levels are numerically compared with Cellular
mMIMO in Section IV. This section also discusses the dif-
ferences and similarities with the previous results in [16].
Section V evaluates the potential benefit of using non-linear
decoding at the CPU, whereas the fronthaul signaling required
with the different implementations is quantified in Section VI.
Finally, the major conclusions and implications are drawn in
Section VII.
Reproducible research:
All the simulation results can be
reproduced using the Matlab code and data files available at:
https://github.com/emilbjornson/competitive-cell-free
Notation:
Boldface lowercase letters,
x
, denote column
vectors and boldface uppercase letters,
X
, denote matrices.
The superscripts
T
,
∗
and
H
denote transpose, conjugate, and
conjugate transpose, respectively. The
n
×
n
identity matrix
is
I
n
. We use
for definitions and
diag(
A
1
, . . . ,
A
n
)
for a
block-diagonal matrix with the square matrices
A
1
, . . . ,
A
n
on
the diagonal. The multi-variate circularly symmetric complex
Gaussian distribution with correlation matrix
R
is denoted
N
C
(
0
,
R
)
. The expected value of
x
is denoted as
E
{
x
}
.
II. C
ELL
-F
REE
N
ETWORK
M
ODEL
We consider a Cell-free mMIMO network consisting of
L
geographically distributed APs, each equipped with
N
antennas. The APs are connected via fronthaul connections to
a CPU, as illustrated in Fig. 1(b). There are
K
single-antenna
UEs in the network and the channel between AP
l
and UE
k
is denoted by
h
kl
∈
C
N
. We use the standard block fading
model where
h
kl
is constant in time-frequency blocks of
τ
c
channel uses [3]. In each block, an independent realization
from a correlated Rayleigh fading distribution is drawn
:
h
kl
∼ N
C
(
0
,
R
kl
)
(1)
where
R
kl
∈
C
N
×
N
is the spatial correlation matrix, which
describes the spatial properties of the channel and
β
kl
tr(
R
kl
)
/N
is the large-scale fading coefficient that describes
geometric pathloss and shadowing.
This paper considers the uplink, which consists of
τ
p
channel uses dedicated for pilots and
τ
c
−
τ
p
channel uses
for payload data. The two phases are described below. Notice
that the results of this paper apply to both systems oper-
ating in time-division duplex (TDD) and frequency-division
duplex (FDD) mode, since the uplink works the same in both
cases.
A. Pilot Transmission and Channel Estimation
We assume that
τ
p
mutually orthogonal
τ
p
-length pilot
signals
φ
1
, . . . ,
φ
τ
p
with
±
φ
t
±
2
=
τ
p
are used for channel
estimation. These pilots are assigned to the UEs in a determin-
istic but arbitrary way. The case of practical interest is a large
network with
K > τ
p
so that more than one UE is assigned
to each pilot. We denote the index of the pilot assigned to UE
k
as
t
k
∈ {
1
, . . . , τ
p
}
and call
P
k
⊂ {
1
, . . . , K
}
the subset of
UEs that use the same pilot as UE
k
, including itself.
When the UEs transmit their pilots, the received signal
Z
l
∈
C
N
×
τ
p
at AP
l
is
Z
l
=
K
i
=1
√
p
i
h
il
φ
T
t
i
+
N
l
(2)
where
p
i
≥
0
is the transmit power of UE
i
,
N
l
∈
C
N
×
τ
p
is the receiver noise with independent
N
C
(0
, σ
2
)
entries, and
σ
2
is the noise power. To estimate
h
kl
, the AP first correlates
the received signal with the associated normalized pilot signal
φ
t
k
/
√
τ
p
to obtain
z
t
k
l
1
√
τ
p
Z
l
φ
∗
t
k
∈
C
N
, which is given
by
z
t
k
l
=
K
i
=1
√
p
i
√
τ
p
h
il
φ
T
t
i
φ
∗
t
k
+
1
√
τ
p
N
l
φ
∗
t
k
=
i
∈P
k
√
p
i
τ
p
h
il
+
n
t
k
l
(3)
where
n
t
k
l
N
l
φ
∗
t
k
/
√
τ
p
∼ N
C
(
0
, σ
2
I
N
)
is the result-
ing noise. Using standard results from estimation theory
[3, Sec. 3], the MMSE estimate of
h
kl
is
ˆ
h
kl
=
√
p
k
τ
p
R
kl
Ψ
−
1
t
k
l
z
t
k
l
(4)
where
Ψ
t
k
l
=
E
{
z
t
k
l
z
H
t
k
l
}
=
i
∈P
k
τ
p
p
i
R
il
+
I
N
(5)
is the correlation matrix of the received signal in (3). The esti-
mate
ˆ
h
kl
and estimation error
˜
h
kl
=
h
kl
−
ˆ
h
kl
are independent
vectors distributed as
ˆ
h
kl
∼ N
C
(
0
, p
k
τ
p
R
kl
Ψ
−
1
t
k
l
R
kl
)
and
˜
h
kl
∼ N
C
(
0
,
C
kl
)
with
C
kl
=
E
{
˜
h
kl
˜
h
H
kl
}
=
R
kl
−
p
k
τ
p
R
kl
Ψ
−
1
t
k
l
R
kl
.
(6)
The mutual interference generated by the pilot-sharing UEs
in (3) causes the so-called
pilot contamination
that degrades
the
system
performance, similar
to
the
case
in
Cellular
mMIMO.
Remark 1
: The
computation
of
ˆ
h
kl
in
(4)
requires
knowledge of the correlation matrices
{
R
il
:
i
∈ P
k
}
, which
we assume to be locally available at AP
l
; see [3] for methods
to estimate them. To dispense with their full knowledge, the AP
can apply alternative channel estimation schemes as in Cellu-
lar mMIMO [3, Sec. 3.4]. One option is the so-called element-
wise MMSE estimator that uses only the main diagonals of
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 19, NO. 1, JANUARY 2020
{
R
il
:
i
∈ P
k
}
. Alternatively, the least-square estimator can
be used, which requires no prior statistical information and
computes the estimate of
h
kl
as
ˆ
h
kl
=
1
√
p
k
τ
p
z
t
k
l
; see [37].
B. Uplink Data Transmission
During the uplink data transmission, the received complex
baseband signal
y
l
∈
C
N
at AP
l
is given by
y
l
=
K
i
=1
h
il
s
i
+
n
l
(7)
where
s
i
∼ N
C
(0
, p
i
)
is the information-bearing signal trans-
mitted by UE
i
with power
p
i
and
n
l
∼ N
C
(
0
, σ
2
I
N
)
is the
independent receiver noise.
Remark 2
: The signal model in
(2)
and
(7)
implicitly
assumes that the entire network is synchronized in time. There
exist wired and over-the-air methods that can be used to
synchronize the clocks at the APs [19], [42], [43]. However,
the signal transmitted by a UE will never be synchronously
received by all the APs due to the largely different dis-
tances between the UE and different APs. In orthogonal
frequency-division multiplexing systems, a simple way to com-
pensate for that is to select the length of the cyclic prefix
so as to accommodate both the channel delay spread and
timing misalignments. This results in a quasi-synchronous sys-
tem [44]. For example, in the LTE standard, the cyclic prefix
is long enough to assume that a UE is quasi-synchronized to
all APs within a 1km radius. If the extended cyclic prefix is
used, the range increases up to 5km. Since the APs that are
further away will receive negligible signal power, the model
in
(2)
and
(7)
is accurate enough for the performance analysis
considered in this paper.
III. F
OUR
L
EVELS OF
R
ECEIVER
C
OOPERATION
All the APs are connected via fronthaul connections to a
CPU that has high computational resources.
2
Hence, the APs
can be viewed as remote-radio heads that cooperate to support
coherent communication with the UEs. The fronthaul can
consist of a mix of wired and wireless connections, organized
in a star or mesh topology [19]; the methods developed in
this paper can be applied with any fronthaul topology. AP
l
receives the signal
y
l
in (7) and can use the available channel
estimates
{
ˆ
h
kl
:
k
= 1
, . . . , K
}
to detect the data signals
locally, or can fully or partially delegate this task to the CPU.
The benefit of using the CPU is that it can combine the
inputs from all APs, but this must be balanced against the
required amount of fronthaul signaling. Four levels of receiver
cooperation are described below and compared with Cellular
mMIMO in Section IV by means of numerical results.
A. Level 4: Fully Centralized Processing
The most advanced level of Cell-free mMIMO opera-
tion is when the
L
APs send their received pilot signals
2
In practice, cell-free systems will have more than one CPU and only a
subset of the APs will serve each UE [15], [19], [43]. The methods described
in this paper applies also to that case. The only requirement is that each UE is
assigned to one CPU that takes partial or full responsibility for the decoding
of the UE’s data and will then forward the decoded data to the core network.
TABLE I
N
UMBER OF
C
OMPLEX
S
CALARS TO
S
END
F
ROM THE
AP
S TO THE
CPU
VIA THE
F
RONTHAUL
, E
ITHER IN
E
ACH
C
OHERENCE
B
LOCK OR FOR
E
ACH
R
EALIZATION OF THE
U
SER
L
OCATIONS
/S
TATISTICS
.
{
z
tl
:
t
= 1
, . . . , τ
p
, l
= 1
, . . . , L
}
and received data signals
{
y
l
:
l
= 1
, . . . , L
}
to the CPU, which takes care of the
channel estimation and data signal detection. In other words,
the APs act as relays that forward all signals to the CPU [45].
In each coherence block, each AP needs to send
τ
p
N
complex
scalars for the pilot signals and
(
τ
c
−
τ
p
)
N
complex scalars
for the received signals. This becomes
τ
c
N
complex scalars
in total, which is summarized in Table I. Moreover, the spatial
correlation matrices
{
R
kl
:
k
= 1
, . . . , K, l
= 1
, . . . , L
}
are
assumed available at the CPU at Level 4, which are described
by
KLN
2
real scalars or
KLN
2
/
2
complex scalars.
3
The received signal available at the CPU is expressed as
⎡
⎢
⎣
y
1
.
.
.
y
L
⎤
⎥
⎦
y
=
K
i
=1
⎡
⎢
⎣
h
i
1
.
.
.
h
iL
⎤
⎥
⎦
h
i
s
i
+
⎡
⎢
⎣
n
1
.
.
.
n
L
⎤
⎥
⎦
n
(8)
or, in a more compact form, as
y
=
K
i
=1
h
i
s
i
+
n
.
(9)
The collective channel is distributed as
h
k
∼ N
C
(
0
,
R
k
)
where
R
k
=
diag(
R
k
1
, . . . ,
R
kL
)
∈
C
LN
×
LN
is
the
block-diagonal spatial correlation matrix. Notice that (9) is
mathematically equivalent to the signal model of a single-cell
mMIMO system with correlated fading [3, Sec. 2.3.1]. The
only difference is how the correlation matrices are generated
and how the pilots are allocated. In fact, in conventional
single-cell mMIMO orthogonal pilots are assigned to UEs
whereas the same pilot can be assigned to multiple UEs in
the investigated cell-free network. This leads to pilot contam-
ination between UEs served by the same AP antennas.
The CPU can compute all the MMSE channel estimates
{
ˆ
h
kl
:
k
=
1
, . . . , K, l
=
1
, . . . , L
}
using the received
pilot signals and channel statistics obtained from the APs.
The estimates can be computed separately without loss of
optimality. For UE
k
, the CPU can then form the collective
channel estimate
ˆ
h
k
⎡
⎢
⎣
ˆ
h
k
1
.
.
.
ˆ
h
kL
⎤
⎥
⎦
∼ N
C
(
0
, p
k
τ
p
R
k
Ψ
−
1
t
k
R
k
)
(10)
3
It is not strictly necessary for the CPU to know the spatial correlation
matrices, but it can use estimators that do not require that; see Remark 1.
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BJÖRNSON AND SANGUINETTI: MAKING CELL-FREE MASSIVE MIMO COMPETITIVE WITH MMSE PROCESSING
81
where
Ψ
−
1
t
k
= diag(
Ψ
−
1
t
k
1
, . . . ,
Ψ
−
1
t
k
L
)
. The estimation error is
˜
h
k
=
h
k
−
ˆ
h
k
∼ N
C
(
0
,
C
k
)
with
C
k
= diag(
C
k
1
, . . . ,
C
kL
)
.
Next, the CPU selects an arbitrary receive combining vector
v
k
∈
C
LN
for UE
k
based on all the collective channel
estimates
{
ˆ
h
k
:
k
= 1
, . . . , K
}
.
While the capacity of Level
4
networks with perfect CSI is
known in some cases [45], the ergodic capacity is generally
unknown in the considered case with imperfect CSI. However,
we can rigorously analyze the performance by using standard
capacity lower bounds [3], [46], which we refer to as achiev-
able SEs.
Proposition 1
: At Level 4, if the MMSE estimator is used
to compute channel estimates for all UEs, an achievable SE
of UE
k
is
SE
(4)
k
=
1
−
τ
p
τ
c
E
log
2
1 +
SINR
(4)
k
(11)
where the instantaneous effective signal-to-interference-and-
noise ratio (SINR) is
SINR
(4)
k
=
p
k
|
v
H
k
ˆ
h
k
|
2
K
∑
i
=1
,i
=
k
p
i
|
v
H
k
ˆ
h
i
|
2
+
v
H
k
K
∑
i
=1
p
i
C
i
+
σ
2
I
LN
v
k
(12)
and the expectation is with respect to the channel estimates.
Proof:
The proof follows the same steps as the proof of
[3, Th. 4.1] for Cellular mMIMO and is therefore omitted.
The pre-log factor
1
−
τ
p
/τ
c
in (11) is the fraction of
channel uses that are used for uplink data transmission. The
term
SINR
(4)
k
takes the form of an “effective instantaneous
SINR” [3], with the desired signal power received over the
estimated channel in the numerator and the interference plus
noise in the denominator.
4
We notice that the SE expression in (11) holds for any
receive combining vector
v
k
and is a multi-antenna generaliza-
tion of [24, Eq. (1)] and an extension of [27], [32] to spatially
correlated channels. The expression can be easily computed for
any
v
k
by using Monte Carlo methods, as done in Section IV.
A possible choice is to use the simple MR combining with
v
k
=
ˆ
h
k
, which has low computational complexity and
maximizes the power of the desired signal, but neglects the
existence of interference. Other heuristic combiners such as
zero-forcing (ZF) or regularized zero-forcing (RZF) can be
also applied. Instead of resorting to heuristics, we notice that
SINR
(4)
k
in (12) only depends on
v
k
and has the form of a
generalized Rayleigh quotient. Hence, the combining vector
that maximizes (12) can be obtained as follows.
Corollary 1
: The instantaneous SINR in
(12)
for UE
k
is
maximized by the MMSE combining vector
v
k
=
p
k
K
i
=1
p
i
ˆ
h
i
ˆ
h
H
i
+
C
i
+
σ
2
I
LN
−
1
ˆ
h
k
(13)
4
The word “effective” refers to the fact that
SINR
(4)
k
cannot be measured
in the system at any particular point in time, but the SE is the same as
that of a fading single-antenna point-to-point channel where
SINR
(4)
k
is the
instantaneously measurable SINR and the receiver has perfect CSI.
which leads to the maximum value
SINR
(4)
k
=
p
k
ˆ
h
H
k
⎛
⎝
K
i
=1
,i
=
k
p
i
ˆ
h
i
ˆ
h
H
i
+
K
i
=1
p
i
C
i
+
σ
2
I
LN
⎞
⎠
−
1
ˆ
h
k
.
(14)
Proof:
It follows from [3, Lemma B.10] since (11) is a
generalized Rayleigh quotient with respect to
v
k
.
It
can
be
shown
that
the
SINR-maximizing
combiner
in (13) minimizes the mean-squared error
MSE
k
=
E
{|
s
k
−
v
H
k
y
|
2
{
ˆ
h
i
}}
, which represents the conditional MSE between
the data signal
s
k
and the received signal
v
H
k
y
after receive
combining; see [3, Sec. 4.1] for details. This is why it is called
MMSE combining
. This type of receive combining normally
maximizes the mutual information of channels with multiple
receive antennas [47], but the particular expression in (13) is
unique for Cell-free mMIMO.
Compared to many heuristic solutions, MMSE combining
has higher computational complexity since it requires first the
computation of the
LN
×
LN
matrix inverse in (13) and then a
matrix-vector multiplication. However, this is not a major issue
since it has to be implemented at the CPU, which is assumed
to have high computational capability. If the complexity is a
concern, then ZF and RZF can be used instead since only
K
×
K
matrices need to be inverted. The price to pay is that
the UEs with low SNRs get an SE reduction, which may be
very large.
B. Level 3: Local Processing & Large-Scale
Fading Decoding
Instead of sending the
N
-dimensional vectors
{
y
l
:
l
=
1
, . . . , L
}
and the channel estimates to the CPU, each AP
can preprocess its signal by computing local estimates of the
data that are then passed to the CPU for final decoding. Let
v
kl
∈
C
N
be the local combining vector that AP
l
selects for
UE
k
. Then, its local estimate of
s
k
is
ˇ
s
kl
v
H
kl
y
l
=
v
H
kl
h
kl
s
k
+
K
i
=1
,i
=
k
v
H
kl
h
il
s
i
+
v
H
kl
n
l
.
(15)
Any combining vector can be adopted in the above expression.
Unlike at Level 4, however, AP
l
can only use its own local
channel estimates
{
ˆ
h
il
:
i
= 1
, . . . , K
}
for the design of
v
kl
.
The simplest solution is MR combining with
v
kl
=
ˆ
h
kl
as
in [16], [24] but preferably the AP should use its local CSI
to make
ˇ
s
kl
as close to
s
k
as possible. The combining vector
that minimizes the MSE,
MSE
kl
=
E
{|
s
k
−
v
H
kl
y
l
|
2
{
ˆ
h
il
}}
,
is
v
kl
=
p
k
K
i
=1
p
i
ˆ
h
il
ˆ
h
H
il
+
C
il
+
σ
2
I
N
−
1
ˆ
h
kl
(16)
which can be proved by computing the conditional expectation
and equating the first derivative with respect to
v
kl
to zero.
Notice that (16) is the combining vector that would maximize
the SE if AP
l
decoded the data signal
s
k
locally. We
call (16)
Local MMSE (L-MMSE) combining
to distinguish
it from the MMSE combining in (13) at Level 4, which is
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82
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 19, NO. 1, JANUARY 2020
applied at the CPU. A main benefit over MMSE combining
is that an
N
×
N
matrix is inverted in (16) instead of an
LN
×
LN
matrix. Importantly, even if
N
=
1
, (16) is
not equal to MR but differ by a non-deterministic scaling
factor.
The local estimates
{
ˇ
s
kl
:
l
= 1
, . . . , L
}
are then sent
to the CPU where they are linearly combined using the
weights
{
a
kl
:
l
= 1
, . . . , L
}
to obtain
ˆ
s
k
=
∑
L
l
=1
a
∗
kl
ˇ
s
kl
,
which is eventually used to decode
s
k
. From (15), we have
that
ˆ
s
k
=
L
l
=1
a
∗
kl
v
H
kl
h
kl
s
k
+
L
l
=1
a
∗
kl
K
i
=1
,i
=
k
v
H
kl
h
il
s
i
+
n
k
(17)
with
n
k
=
∑
L
l
=1
a
∗
kl
v
H
kl
n
l
. Let
g
ki
= [
v
H
k
1
h
i
1
. . .
v
H
kL
h
iL
]
T
be
the
L
-dimensional
vector
with
the
receive-combined
channels between UE
k
and each of the APs. Then, (17)
reduces to
ˆ
s
k
=
a
H
k
g
kk
s
k
+
K
i
=1
,i
=
k
a
H
k
g
ki
s
i
+
n
k
(18)
where
a
k
= [
a
k
1
. . . a
kL
]
T
∈
C
L
is the weighting coefficient
vector and
{
a
H
k
g
ki
:
i
= 1
, . . . , K
}
represent the effective
channels. Notice that
a
k
can be optimized by the CPU to
maximize the SE, but only channel statistics can be utilized
since the CPU does not have knowledge of the channel
estimates at Level 3. This approach is known as LSFD in
Cellular mMIMO [28], [29], and can be applied at Level
3
as
follows. Although the effective channel
a
H
k
g
kk
is unknown
at the CPU, we notice that its average
a
H
k
E
{
g
kk
}
is non-zero
(e.g., if L-MMSE or MR is used) and deterministic. Therefore,
it can be assumed known
5
and used to compute the following
achievable SE.
Proposition 2
: At Level 3, an achievable SE of UE
k
is
SE
(3)
k
=
1
−
τ
p
τ
c
log
2
1 +
SINR
(3)
k
(19)
with the effective SINR given by
SINR
(3)
k
=
p
k
|
a
H
k
E
{
g
kk
}|
2
K
∑
i
=1
p
i
E
{|
a
H
k
g
ki
|
2
} −
p
k
|
a
H
k
E
{
g
kk
}|
2
+
σ
2
a
H
k
D
k
a
k
(20)
where
D
k
= diag(
E
{±
v
k
1
±
2
}
, . . . ,
E
{±
v
kL
±
2
}
)
∈
C
L
×
L
and
the expectations are with respect to all sources of randomness.
Proof:
The proof is given in Appendix A.
The achievable SE above holds for any combining scheme.
Particularly, it is valid for both the L-MMSE combining in (16)
and the MR combining
v
kl
=
ˆ
h
kl
that was used in [24]. Unlike
the achievable SE in Proposition 1, it holds for any channel
5
When dealing with ergodic capacities, all deterministic parameters can be
assumed known without loss of generality, because these can be estimated
using a finite number of transmission resources, while the capacity is only
achieved as the amount of transmission resources goes to infinity. Hence,
the estimation overhead for obtaining deterministic parameters is negligible.
estimator (not only for the MMSE estimator (11)) but requires
channel hardening in order to approximate
a
H
k
g
kk
with its
mean value
a
H
k
E
{
g
kk
}
. However, this may not occur when
the number
N
of antennas at the APs is relatively small [27].
In that case, the SE expression in (20) underestimates the
achievable performance, but is anyway the best available
capacity bound.
The structure of (20) allows computing the deterministic
weighting vector
a
k
that maximizes
SINR
(3)
k
. This is given as
follows.
Corollary 2
: The effective SINR in
(20)
for UE
k
is maxi-
mized by
a
k
=
K
i
=1
p
i
E
{
g
ki
g
H
ki
}
+
σ
2
D
k
−
1
E
{
g
kk
}
(21)
which leads to the maximum value
SINR
(3)
k
=
p
k
E
{
g
H
kk
}
K
i
=1
p
i
E
{
g
ki
g
H
ki
}
+
σ
2
D
k
−
p
k
E
{
g
kk
}
E
{
g
H
kk
}
−
1
E
{
g
kk
}
.
(22)
Proof:
It follows from [3, Lemma B.10] by noting
that (20) is a generalized Rayleigh quotient with respect
to
a
k
.
Notice that Level
3
is an extension of the LSFD frame-
work in [24], [29], [33], [48], which has previously been
only used in Cell-free mMIMO along with MR combining.
In fact, the SE expressions provided in these papers only
apply for particular choices of receive combining and not for
arbitrary combining as (19). This makes Proposition 2 a novel
contribution of this paper.
The signaling required at Level
3
can be quantified as
follows. Each AP needs to send
(
τ
c
−
τ
p
)
K
complex scalars
(i.e.,
ˇ
s
kl
for all
k
) to the CPU per coherence block. In addi-
tion, the computation of
(21) requires knowledge of the
L
-dimensional complex vector
E
{
g
kk
}
, of the
L
×
L
Hermitian
complex matrix
E
{
g
ki
g
H
ki
}
, and of the real-valued
L
×
L
diagonal matrix
D
k
for
k, i
∈ {
1
, . . . , K
}
. Hence,
KL
+
(
L
2
K
2
+
KL
)
/
2
complex scalars are needed in total. These
values are summarized in Table I.
C. Level 2: Local Processing & Simple Centralized Decoding
Although the optimized LSFD step in Level
3
gives the
highest SE among schemes with local combining at each AP,
it requires knowledge of a number of statistical parameters
that grows quadratically with
L
and
K
, which can be very
large in Cell-free mMIMO. In practice, this large number
of parameters need to be jointly estimated by the APs and
sent to the CPU. This might not be feasible, especially if the
statistics vary with time. To overcome this issue, the CPU can
alternatively create its estimate of the signal
s
k
from UE
k
by
simply taking the average of the local estimates, as proposed
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83
in the early papers on the topic [16], [17].
6
This yields
ˆ
s
k
=
1
L
L
l
=1
ˇ
s
kl
(23)
where
ˇ
s
kl
is given in (15) and can be obtained by any
local combining vector. Since this is equivalent to setting
a
k
= [1
/L, . . .,
1
/L
]
T
in Proposition 2, the following result
is obtained.
Corollary 3
: At Level 2, an achievable SE of UE
k
is
SE
(2)
k
=
1
−
τ
p
τ
c
log
2
1 +
SINR
(2)
k
(24)
with the effective SINR given in (25) on the top of next page,
where the expectations are taken with respect to all sources
of randomness.
As for Proposition 2, the above SE can be utilized along
with any local combining vector and also channel estimator.
If MR is used with single-antenna APs (i.e.,
N
= 1
), then
Corollary 3 reduces to the case considered in [16] and can be
computed in closed form (similar results are found in [31],
[34]–[38]). The number of complex scalars to be exchanged
per coherence block is the same as at Level 3. The key
difference is that no statistical parameters are needed at the
CPU. This is summarized in Table I.
D. Level 1: Small-Cell Network
The simplest implementation level is when the signal from
UE
k
is decoded by using only the received signal from one
AP. In this case, the decoding can be done locally at the AP by
using its own local channel estimates without exchange any-
thing with the CPU.
7
This makes the network truly distributed
[15, Sec. 4.2] and essentially turns Cell-free mMIMO into a
small-cell network. The macro diversity achieved by selecting
the best out of many APs could potentially make it competitive
compared to conventional Cellular mMIMO with larger cells.
Cell-free mMIMO and small cells were compared in [16],
[17]
with
N
=
1
and
an
AP
selection
based
on
the
largest large-scale fading coefficient
β
kl
. In addition to this,
the authors impose that each AP can only serve one UE.
Unlike [16], [17], we remove all these restrictions by assuming
an arbitrary number of antennas per AP and letting the AP
that gives the highest SE to a specific UE be responsible
for decoding its signal. The latter makes the AP association
more complex than [16], [17], but the numerical results in
Section IV show that it vastly improves the performance.
8
Within the above setting, the following result is obtained.
6
Level
2
also includes other cases where the weight
a
kl
is selected based
only on the statistical information available at AP
l
. For example, we have
tried
a
kl
=
β
ν
lk
for different exponents
ν
but the performance gap to Level
3
remained to be large. Further research in this direction is needed.
7
In all the four levels, the
K
data streams need to be transmitted to the
core network after decoding. This requires a backhaul load proportional to
the sum SE, which is not included in Table I but is different for each level.
8
In practice, selecting the AP that maximizes the SE can be replaced by
selecting the AP that maximizes some kind of approximate closed-form SINR.
Such a selection rule has the same implementation complexity as selecting
the AP with the largest large-scale fading coefficient. However, the challenge
is that the SINR is affected by the transmit powers, so if these powers are
optimized, the optimization must also involve the AP selection.
Corollary 4
: At Level 1, an achievable SE of UE
k
is
SE
(1)
k
=
1
−
τ
p
τ
c
max
l
∈{
1
,...,L
}
E
log
2
1 +
SINR
(1)
kl
(26)
where the instantaneous effective SINR at AP
l
is
SINR
(1)
kl
=
p
k
|
v
H
kl
ˆ
h
kl
|
2
K
∑
i
=1
i
=
k
p
i
|
v
H
kl
ˆ
h
il
|
2
+
v
H
kl
K
∑
i
=1
p
i
C
il
+
σ
2
I
N
v
kl
(27)
and the expectation is with respect to the channel estimates.
The maximum value in
(27)
is achieved with the L-MMSE
combining in
(16)
and is given by
SINR
(1)
kl
=
p
k
ˆ
h
H
kl
⎛
⎜
⎝
K
i
=1
i
=
k
p
i
ˆ
h
il
ˆ
h
H
il
+
K
i
=1
p
i
C
il
+
σ
2
I
N
⎞
⎟
⎠
−
1
ˆ
h
kl
.
(28)
Proof:
For each AP, the SE is computed in the same way
as in Proposition 1 and the maximum SINR is achieved as in
Corollary 1.
The SE expression above is more general than the one
considered for small cells in [16], where
N
= 1
is considered
and each AP only estimates the channel of the UE it serves.
When considering that special case, the following result is
obtained instead.
Proposition 3
: At Level
1
with
N
= 1
, if AP
l
decodes
the signal from UE
k
using only its local estimate
ˆ
h
kl
,
an achievable SE is
e
1
ω
kl
(1+
A
kl
)
E
1
1
ω
kl
(1+
A
kl
)
−
e
1
ω
kl
A
kl
E
1
1
ω
kl
A
kl
ln(2)
(29)
where
A
kl
=
∑
i
∈P
k
\{
k
}
p
i
β
il
p
k
β
kl
2
is due to pilot contamination,
ω
kl
=
p
2
k
τ
p
β
2
kl
Ψ
t
k
l
∑
i
∈P
k
p
i
β
il
+
∑
i
∈P
k
p
i
C
il
+
σ
2
,
(30)
E
1
(
x
) =
∞
1
e
−
xu
u
du
denotes the exponential integral, and
ln(
·
)
denotes the natural logarithm.
Proof:
The proof is given in Appendix B.
Comparing the achievable SE in (29) with [16, Eq. (47)]
we notice that, despite the different notation, the equivalence
only holds when
A
kl
= 0
; that is, in the absence of pilot
contamination. Although [16] states the result without proof,
it seems that the paper has neglected the conditioning on the
local channel estimate
ˆ
h
kl
when computing the interference
power; see (47) in Appendix B. This leads to an approximate
SE rather than an exact expression. This is why we included
Proposition 3 in this paper and will use it for numerical
comparison in Section IV.
Remark 3
: We noticed that the expression in
(29)
is numer-
ically unstable when
ω
kl
(1 +
A
kl
)
and/or
ω
kl
A
kl
are small.
This is because
e
1
/x
→
∞
and
E
1
(1
/x
)
→
0
when
x
→
0
. When this happens, one can instead utilize the bounds
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 19, NO. 1, JANUARY 2020
SINR
(2)
k
=
p
k
L
∑
l
=1
E
{
v
H
kl
h
kl
}
2
K
∑
i
=1
p
i
E
L
∑
l
=1
v
H
kl
h
il
2
−
p
k
L
∑
l
=1
E
{
v
H
kl
h
kl
}
2
+
σ
2
L
∑
l
=1
E
{±
v
kl
±
2
}
(25)
x
1+
x
≤
e
1
/x
E
1
(1
/x
)
≤
x
in [49, Eq. (5.1.19)] to realize that
e
1
/x
E
1
(1
/x
)
≈
x
when
x
→
0
.
IV. C
ELL
-F
REE
V
ERSUS
C
ELLULAR M
MIMO
In this section, we compare the uplink performance of
Cell-free mMIMO, with the different cooperation levels and
either
MR
or
MMSE/L-MMSE
combining,
and
Cellular
mMIMO. We first briefly describe the cellular setup that is
considered.
A. Cellular mMIMO Setup
We
consider
a
cellular
network
with
L
c
=
4
cells,
M
c
= 100
antennas per cellular BS, and
K
c
= 10
UEs per
cell. The block-fading channel from BS
j
to UE
k
in cell
l
is
modeled as
h
j
lk
∼ N
C
0
,
R
j
lk
(31)
where
R
j
lk
∈
C
M
c
×
M
c
is the spatial correlation matrix with
large-scale fading coefficient
β
j
lk
tr(
R
j
lk
)
/M
c
describing
the geometric pathloss and shadowing. The uplink transmit
power of UE
k
in cell
l
is denoted by
p
lk
≥
0
.
We assume there are
τ
p
=
K
c
mutually orthogonal pilots
and that UE
k
in every cell uses the same pilot (i.e., pilot reuse
one). When using standard MMSE estimation [3, Th. 3.1],
the MMSE estimate of
h
j
lk
∈
C
M
c
is given by
ˆ
h
j
li
∼ N
C
0
,
R
j
li
−
C
j
li
(32)
and the independent estimation error
˜
h
j
li
∈
C
M
c
is
˜
h
j
li
h
j
li
−
ˆ
h
j
li
∼ N
C
0
,
C
j
li
(33)
with
C
j
li
=
R
j
li
−
p
li
τ
p
R
j
li
L
c
l
=1
p
l i
τ
p
R
j
l i
+
σ
2
I
M
c
−
1
R
j
li
.
(34)
An achievable SE of UE
k
in cell
j
is [3, Th. 4.1]
SE
(c)
jk
=
1
−
τ
p
τ
c
E
log
2
1 +
SINR
(c)
jk
(35)
where the effective SINR,
SINR
(c)
jk
, is maximized by multi-cell
MMSE (M-MMSE) combining [9]. This gives
SINR
(c)
jk
=
p
jk
(
ˆ
h
j
jk
)
H
⎛
⎜
⎝
L
c
l
=1
K
c
i
=1
(
l,i
)=(
j,k
)
p
li
ˆ
h
j
li
(
ˆ
h
j
li
)
H
+
L
c
l
=1
K
c
i
=1
p
li
C
j
li
+
σ
2
I
M
c
−
1
ˆ
h
j
jk
.
(36)
Other combining schemes can be used but they provide lower
SEs. By considering M-MMSE, we thus compare Cell-free
mMIMO with the most competitive form of Cellular mMIMO.
B. Simulation Setup and Propagation Model
The cellular network has
4
square cells in a
1
×
1
km
area, as in Fig. 1, with 100 co-located antennas per BS. The
cell-free network is deployed in the same area and has either
400 single-antenna APs (i.e.,
N
= 1
) or 100 four-antenna
APs (i.e.,
N
= 4
). Hence, all the network configurations have
the same number of antennas. To make a fair comparison,
the APs are deployed on a square grid (we consider random
deployment later in this section) and the same propagation
model is used in all cases. We anticipate that the APs in
Cell-free mMIMO will be deployed in urban environments
with high user loads, roughly 10m above the ground. This
matches well
with the
3GPP Urban Microcell model in
[50, Table B.1.2.1-1] with a 2GHz carrier frequency and
β
kl
[
dB
] =
−
30
.
5
−
36
.
7 log
10
d
kl
1
m
+
F
kl
(37)
where
d
kl
is the distance between UE
k
and AP
l
(computed
as the minimum over different wrap-around cases, and taking
the 10m height difference into account) and
F
kl
∼ N
(0
,
4
2
)
is the shadow fading. The shadowing terms from an AP to
different UEs are correlated as [50, Table B.1.2.2.1-4]
E
{
F
kl
F
ij
}
=
4
2
2
−
δ
ki
/
9
m
,
l
=
j
0
,
l
²
=
j
(38)
where
δ
ki
is the distance between UE
k
and UE
i
. The second
row in (38) accounts for the correlation of shadowing terms
related to two different APs, which is negligible since we have
at least 50m between adjacent APs in the simulation setup
(notice that
2
−
50
/
9
≈
0
.
02
).
Since the propagation model from [50] is designed for
cellular networks, we use the same propagation model for
Cellular mMIMO by simply adding an additional index to
all the parameters to specify in which cell a particular UE
resides. By having a common model for cell-free and cellular
networks, we can be sure that the performance differences
that we observe are caused by differences in technology
characteristics, and not by the propagation model. There are
K
= 40
UEs in the simulation setup, whereof ten are uni-
formly dropped in each cell and assigned to unique pilots with
random indices.
9
The same UE locations and pilot assignments
are used in the cell-free case, but the shadowing is generated
independently.
9
Each UE in the Cellular mMIMO case is connecting to the BS providing
the largest large-scale fading coefficient; that is,
β
j
jk
= max
l
β
j
jk
. Due to
the shadowing, this might not be the geographically closest BS.
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85
Fig. 2.
Comparison of cellular mMIMO and cell-free mMIMO when using
MMSE or L-MMSE combining.
The cellular BSs and multi-antenna APs are equipped with
half-wavelength-spaced uniform linear arrays. The spatial cor-
relation is generated using the Gaussian local scattering model
with
15
◦
angular standard deviation [3, Sec. 2.6]. All UEs
transmit with power
p
k
=
p
jk
= 100
mW, the bandwidth is
20MHz, the noise power is
σ
2
=
−
96
dBm, and the coherence
blocks contain
τ
c
= 200
channel uses (e.g., achieved by 2ms
coherence time and 100kHz coherence bandwidth).
Remark 4
: The early Cell-free mMIMO papers [16], [24]
used another propagation model, which has since then become
standard in the field. However, that model is based on the
COST-Hata model from [51] for macro-cells, where the APs
are at least 30m above the ground and the UEs are at least
1km from the AP. This is very different from the micro-cell-like
deployment we anticipate for Cell-free mMIMO and it should
be noted that the model creators themselves specified that it
“must not be used for micro-cells” [51, Ch. 4]. Moreover,
the model in [16], [24] has no shadowing when a UE is
closer than 50m from an AP, which is often the case in
Cell-free mMIMO deployments. When the distance is larger,
the shadowing decorrelation distance is
10
×
larger than in
the 3GPP model [50]. For all these reasons, we believe that
the propagation model used in this paper is a better baseline
for evaluating Cell-free mMIMO systems.
C. Numerical Comparisons
Fig. 2(a) compares Cellular mMIMO and Cell-free mMIMO
with
L
= 400
and
N
= 1
. The figure shows the cumulative
Fig. 3.
Comparison of cellular mMIMO with cell-free (
L
= 400
,
N
= 1
)
when using MR combining.
distribution function (CDF) of the SE of a randomly located
UE,
when
using
MMSE
or
L-MMSE
combining in
the
cell-free cases. At the 90% or 95% likely SE points (i.e.,
where the vertical axis is 0.1 or 0.05), the cell-free cases
perform according to their level: Level
4
provides by far
the highest SEs, while Level
1
gives the lowest SEs but is
anyway preferable as compared to Cellular mMIMO. Looking
at the complete CDF curves, the situation is more complicated
since the Level
1
and Cellular mMIMO curves are crossing
the Level
2
and Level
3
curves. Hence, UEs with good
channel conditions get better performance with these methods.
However, Level
4
performs better than Cellular mMIMO for
every UE.
Fig. 2(b) considers the same setup but with fewer APs that
are equipped with multiple antennas:
L
= 100
and
N
= 4
. The
general trends are the same as in Fig. 2(a) but Level
4
loses
in SE due to the reduced macro diversity; that is, the average
distance from a UE to an AP is increased. In contrast, Level
1
gains in performance since each AP can now suppress
interference locally, by using its four antennas. In fact, Level
1
is now comparable to Level
2
for the weakest UEs and
substantially better for the strongest UEs.
Next, Fig. 3 considers the case
L
= 400
,
N
= 1
and
MR combining, which is the receiver processing advocated
in the early papers on Cell-free mMIMO. More precisely,
Level
2
was considered in [16] and Level 3 in [24]. The poor
processing leads to a large SE loss, compared to Fig. 2(a), for
all levels of Cell-free mMIMO receiver cooperation, except
Level 1. In fact, Level
2
is outperformed by both small
cells (Level 1) and Cellular mMIMO for every single UE.
Note that we are considering single-antenna APs in this
figure, so MR processing is suboptimal even in that basic
case, and the use of LSFD in Level
3
cannot make up for
the performance loss. This is because L-MMSE and MR
differ by a non-deterministic scalar and LSFD only involves
deterministic scalars. Not even Level
4
performs better than
Cellular mMIMO or small cells when using MR, so we can
conclude that Cell-free mMIMO should never use the MR
scheme.
D. Revisiting “Cell-Free Massive MIMO Versus Small Cells”
Interestingly, our observations in Fig. 3 contradict the
previous
results
in
[16],
where
Cell-free
mMIMO
with
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‘Level 2 (MR)’ was shown to perform much better than
small cells, in terms of both 95%-likely and median SE. The
reason for the differences is explained in this subsection by
reproducing [16, Fig. 4, Fig. 6] and adding some additional
curves to them. The following three-slope propagation model
was used in [16]:
β
kl
[
dB
]
=
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
−
81
.
2
,
d
kl
<
10
m
−
61
.
2
−
20 log
10
d
kl
1
m
,
10
m
≤
d
kl
<
50
m
−
35
.
7
−
35 log
10
d
kl
1
m
+
F
kl
, d
kl
≥
50
m
(39)
where
d
kl
is the horizontal distance between UE
k
and AP
l
(i.e., ignoring the height difference). The shadowing term
F
kl
∼ N
(0
,
8
2
)
only appears when the distance is larger than
50m and the terms are correlated as
E
{
F
kl
F
ij
}
=
8
2
2
2
−
δ
ki
/
100
m
+ 2
−
lj
/
100
m
(40)
where
δ
ki
is the same as in (38) and
lj
is the distance
between AP
l
and AP
j
. The maximum UE power is
100
mW,
the bandwidth is 20MHz, the noise power is
σ
2
=
−
92
dBm,
and the coherence blocks are determined by
τ
c
= 200
.
We consider the same setup as in [16] with
L
= 100
uniformly distributed APs in a
1
×
1
km area,
N
=
1
antenna per AP,
K
= 40
uniformly distributed UEs, and
τ
p
= 20
orthogonal pilots. The pilots are assigned to the UEs
according to the greedy algorithm described in [16, Sec. IV.A],
with the only difference that we use the uplink SE as the
metric in Step
2
of the algorithm (instead of the downlink SE).
Moreover, we use Proposition 3 to accurately compute the SE
with small cells, but this has little impact on the results. The
thick lines in Fig. 4 correspond to the original curves from [16]
with correlated shadowing. Fig. 4(a) considers the case when
the UEs transmit at full power (as in [16, Fig. 6]) and Fig. 4(b)
considers the case when the UEs transmit pilots at full power
but reduce the power during the data transmission to optimize
the network-wide max-min fairness (as in [16, Fig. 4]). To this
end, we use the same optimization algorithms as in [16].
In the full power case, in Fig. 4(a), it is clear from the
thick curves that Cell-free mMIMO at Level
2
with MR gives
the UEs with the 50% worst channel conditions substantially
higher SE than with small cells. The remaining UEs get better
SEs with small cells, which indicates that the considered
Cell-free system is not well implemented—since the cell-free
network has access to more APs and signal observations when
decoding a UE’s signal, the performance should be better for
everyone
. Moreover, the comparison in [16, Fig. 6] is based
on a suboptimal assignment of UEs to small cells; the UEs are
sequentially selecting the AP that has the largest large-scale
fading coefficient
β
kl
(i.e., the best channel), but only among
those that are not already serving another UE. If we change
that to let each UE being served by the AP giving the highest
SE, represented by the curve ‘Ref. [14] (Improved)’, then the
performance gap between Cell-free mMIMO and small cells
diminishes. The reason is that around 40% of the UEs prefer
Fig. 4.
Comparison of cell-free mMIMO at Level
2
and small cells, using
different SE expressions and AP assignments. The UEs either transmit with
full power or optimizes the power as described in [16]. The thick lines
correspond to the curves in [16, Fig. 4, Fig. 6].
to be served by another small cell. Additionally, if we use the
new improved SE expression in Corollary 4, represented by
the curve ‘L1 (Small cells)’, all the UEs get higher SE with
small cells than with Cell-free mMIMO.
Does this mean that small cells are actually better than
Cell-free mMIMO? The answer is no. Indeed, as observed
in the last subsection, the problem is that MR combining
performs badly in Cell-free mMIMO, even if single-antenna
APs are used. By simply changing to Level
2
with L-MMSE
combining, the rightmost curve in Fig. 4(a) is achieved, which
gives uniformly higher SE to all the UEs than when using
small cells. Even higher SE can be achieved by considering
Level
3
or Level
4
implementations.
The results in Fig. 4(b) with max-min power control are
different and more in line with the observations made in [16]:
Level
2
with MR gives much higher SE than small cells,
but the gap can be reduced by selecting APs based on the
maximum SE rather than the maximum
β
kl
(represented by
the curves ‘Ref. [15] (Improved)’ and ‘L1 (Small cells)’). The
benefit of max-min power control can be seen by considering
the two thick lines (obtained as in [16]): the lower end of the
CDF curves are shifted to the right as compared to the full
power case in Fig. 4(a), yielding a higher guaranteed SE level.
Nevertheless, the use of L-MMSE combining is more appeal-
ing than the use of max-min power control, as can be seen
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87
from the rightmost curve in Fig. 4(b) that considers Level
2
with L-MMSE and full power transmission. This approach
gives the same performance as ‘L2 MR’ with max-min fairness
for the 2% weakest UEs, but higher SE for all other UEs;
for example, it achieves a 40% higher 95%-likely SE and a
3
×
higher median SE. Hence, if L-MMSE processing is used,
advanced power control is not needed in Cell-free mMIMO to
give good performance to the weakest UEs.
V. L
EVEL
4
W
ITH
N
ON
-L
INEAR
D
ECODING
Section IV-C showed that Level
4
can provide vastly higher
SE than the other cooperation levels in Cell-free mMIMO.
The comparison is based on using linear receive combining,
but another benefit of centralizing the signal processing at a
CPU is that more advanced decoding methods can potentially
be used, since network-wide CSI and high computational
resources are available. In this section, we investigate the
potential benefits of the non-linear successive interference can-
celation (SIC) method [52, Sec. 8.3.4] in Cell-free mMIMO,
which means that the CPU decodes one UE signal at a time,
and then sequentially subtracts interference that the decoded
signal caused to the remaining signals. The interference cannot
be fully canceled since the CPU has imperfect CSI, but it can
still improve the SE of the UEs compared to linear combining.
Proposition 4
: At Level 4, if the MMSE estimator is used
to compute channel estimates for all UEs and the signals are
decoded using MMSE combining and SIC (MMSE-SIC), then
for any decoding order an achievable sum SE is
SSE
(SIC)
=
1
−
τ
p
τ
c
E
log
2
det
I
K
+
P
ˆ
H
H
E
−
1
ˆ
H
(41)
where
P
= diag(
p
1
, . . . , p
K
)
,
ˆ
H
= [
ˆ
h
1
. . .
ˆ
h
K
]
∈
C
LN
×
K
,
E
=
∑
K
i
=1
p
i
C
i
+
σ
2
I
LN
, and the expectation is with respect
to the channel estimates.
Proof:
The proof is given in Appendix C.
Proposition 4 provides the sum SE of the Cell-free mMIMO
network, and not the individual SEs of the UEs. The reason is
that the latter depends on the decoding order; that is, the later
a UE is decoded, the less interference it will be affected by
and thereby it will gain more in SE compared to using linear
combining. Irrespective of the decoding order, all UEs get at
least as high SE with MMSE-SIC as with MMSE combining.
Fig. 5 revisits the scenario in Fig. 2(a) by considering
Cell-free mMIMO with
L
= 400
and
N
= 1
. The CDF
of the sum SE over different random realizations of the
UE locations is plotted when using either Levels 1-4 with
MMSE/L-MMSE combining or Level
4
with MMSE-SIC,
based on Proposition 4. The MMSE-SIC method improves the
sum SE, but the average gain over ‘L4 (MMSE)’ is only 1%.
The reason for such modest gain is the favorable propagation
phenomenon that makes the UEs’ channels nearly orthogonal
[27], meaning that the inter-user interference is effectively
canceled by the MMSE processing described in Section III.
Hence, we conclude that non-linear processing is not needed
in Cell-free mMIMO. This is also the reason why we did not
present the detailed per-user SEs in this section.
Fig. 5.
CDF of the sum SE over different random user locations with
L
= 100
,
N
= 4
,
K
= 40
,
τ
p
= 10
. The four cooperation levels are
compared with MMSE-SIC, based on Proposition 4.
Another observation that can be made from Fig. 5 is that
Level
1
and Level
3
provide roughly the same sum SE, while
Level
2
is far behind in performance. The large gap to Level
4
further reinforces the point that a centralized implementation
is strongly preferred in Cell-free mMIMO.
VI. A L
OOK AT THE
F
RONTHAUL
S
IGNALING
L
OAD
The reported results show that a Level
4
implementation is
strongly preferred. The counterargument might be that such an
implementation would require much more fronthaul signaling
than Level
2
and Level 3, but we will now show that it is not
necessarily the case. By using the formulas in Table I, Level
4
requires less signaling if
τ
c
NL
(
τ
c
−
τ
p
)
KL
=
τ
c
τ
c
−
τ
p
N
K
<
1
.
(42)
Since
τ
c
τ
c
−
τ
p
≈
1
and
K
³
N
are typical for Cell-free
mMIMO, Level
4
actually requires
much less
signaling.
Fig. 6 shows how many complex scalars need to be sent
from an AP to the CPU per channel use, as a function
of the coherence block length
τ
c
. We consider the same
setup as in Fig. 2(b):
L
= 100
,
N
= 4
,
K
= 40
, and
τ
p
= 10
. Level
4
requires more signaling if
τ
c
≤
11
, while
much less signaling is required when
τ
c
becomes a hundred,
as in practical systems. As
τ
c
→ ∞
, Level
2
and
3
require
K/N
= 10
times more fronthaul signaling than Level 4. The
reason is that the received data signals constitute a much
larger number of scalars than the channel estimates. Since
K
≥
N
is typically the case in Cell-free mMIMO, Level
2
and Level
3
increase the fronthaul signaling by processing
the
N
-dimensional vector
y
l
into the
K
-dimensional vector
[ˇ
s
1
l
, . . . ,
ˇ
s
Kl
]
T
. In practice, an AP will not serve all the UEs in
the network but only those with a good channel. Nevertheless,
as long as each AP serves more UEs than it has antennas (e.g.,
more than one UE in conventional Cell-free mMIMO with
N
= 1
), Level
4
is preferable in terms of fronthaul signaling.
Admittedly, this comparison assumes that all scalars are
shared with infinite precision, while in practice it is plausible
that the pilot signals require higher bit-resolution when sent
to the CPU than the data signals. On the other hand, the pilot
signals constitute only a minor fraction of the total signaling
and [32], [53] recently showed that the estimates can be
compressed rather well.
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 19, NO. 1, JANUARY 2020
Fig. 6.
Number of complex scalars that needs to be shared between an AP
and the CPU per channel use (
L
= 100
,
N
= 4
,
K
= 40
,
τ
p
= 10
).
A. Serial Fronthaul
Since Level
2
and Level
3
necessarily provide lower SE than
Level 4, these levels are only practically interesting if they
require lower fronthaul capacity. The previous example shows
that this is not the case when each AP transmits individually
over the fronthaul, but there are alternative solutions that
reduce the fronthaul capacity requirement. In particular, this
happens when several APs are deployed along the same wired
connection, as illustrated in the lower right corner of Fig. 1(b).
Suppose AP
1
and AP
2
share a fronthaul connection in this
way. When the locally estimated signal
ˇ
s
k
1
at AP
1
is sent
over the fronthaul to AP
2
, this AP will compute
ˇ
s
k
1
+ ˇ
s
k
2
.
The result is then sent to the CPU, which can still form its
signal estimate in (23) at Level 2, since it is the summation of
the local estimates at all the APs. By instead transmitting the
weighted local estimates
a
kl
ˇ
s
kl
over the fronthaul, Level
3
can
be implemented in the same sequential fashion (assuming that
a
kl
can be computed locally at AP
l
).
Since only one scalar per UE is transmitted over each
segment of the fronthaul, the capacity requirement does not
grow with the number of APs that are sharing the wired
connection. In the extreme case when all APs are deployed
along the same wire, the number of complex scalars sent over
the fronthaul per coherence block reduces from
(
τ
c
−
τ
p
)
KL
in Table I to
(
τ
c
−
τ
p
)
K
. This type of serial fronthaul is needed
for Level
2
and Level
3
to make practical sense, which is why
it is adopted by the radio stripes concept described in [43].
VII. C
ONCLUSION
This paper introduced a taxonomy for Cell-free mMIMO
with four different implementation levels, from fully central-
ized to fully distributed, and generalized previous results to
account for multi-antenna APs, spatially correlated fading, and
arbitrary receive combining. The majority of previous papers
on this topic relied upon a distributed implementation with
local MR processing. Remarkably, we discovered that this is
basically the worst way to operate cell-free networks.
Firstly,
local
MMSE
processing
provides
substantially
higher SE than MR, and is the key prerequisite for Cell-free
mMIMO to outperform conventional Cellular mMIMO and
small-cell networks. Importantly, this is the case even if
each AP is equipped with only one antenna; local MMSE
processing can roughly double the SE per UE.
Secondly, we showed that a centralized implementation,
with all the signal processing taking place at an edge-cloud
processor (a.k.a. CPU in the cell-free literature), is highly
preferable
compared
to
distributed
alternatives.
In
fact,
the centralized Level
4
implementation can simultaneously
increase the SE and reduce the fronthaul signaling. Linear
processing is sufficient at Level
4
since non-linear processing
provides negligible gains due to the favorable propagation
property [27]. The pCell technology [19] is an example of
a centralized cell-free system, which demonstrates that it is
practically feasible. A serial fronthaul is needed to make a dis-
tributed implementation competitive in terms of the fronthaul
capacity requirements, and an improved version of Level
3
needs to be developed to reduce the performance gap to
Level 4. Non-linear processing can be useful at Level
3
and the
compute-and-forward relaying framework can potential guide
the development of such methods [47], [54], [55].
An interesting analogy can be made between the results in
this paper and recent developments in the Cellular mMIMO
area. The seminal paper [4] advocated for using MR process-
ing, based on asymptotic arguments. MR was known to be
suboptimal when having a small number of antennas, but
anyway became the most well-studied method in the literature
since the SE can be computed in closed-form, even with more
complicated system models containing spatially correlated fad-
ing and/or hardware impairments [3]. However, recent works
have shown that M-MMSE processing greatly outperforms
MR even in the asymptotic regime [9]. Similarly, the main con-
clusion of this paper is that it is time to forget about MR also
in Cell-free mMIMO and instead consider only MMSE-based
schemes—irrespective of the level of cooperation among the
APs and the number of antennas used at each one.
A
PPENDIX
A
P
ROOF OF
P
ROPOSITION
2
Since the CPU does not have knowledge of the channel
estimates, it needs to treat the average channel gain
a
H
k
E
{
g
kk
}
as the true deterministic channel. Hence, the signal model is
ˆ
s
k
=
a
H
k
E
{
g
kk
}
s
k
+
υ
k
(43)
which is a “deterministic” channel with the additive interfer-
ence plus noise term
υ
k
=
(
a
H
k
g
kk
−
a
H
k
E
{
g
kk
}
)
s
k
+
K
i
=1
,i
=
k
a
H
k
g
ki
g
H
ki
a
k
s
i
+
n
k
.
(44)
The interference term
υ
k
has zero mean and is uncorrelated
with the signal term in (43) since
E
{
a
H
k
g
kk
−
a
H
k
E
{
g
kk
}}
=0
E
{|
s
k
|
2
}
= 0
.
(45)
Therefore, we can apply [3, Cor. 1.3] to obtain the achievable
SE in (19).
A
PPENDIX
B
P
ROOF OF
P
ROPOSITION
3
In this proof, we drop the bold face to emphasize that
all parameters are scalars. Using the capacity lower bound
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BJÖRNSON AND SANGUINETTI: MAKING CELL-FREE MASSIVE MIMO COMPETITIVE WITH MMSE PROCESSING
89
in [3, Cor. 1.3] with
ˆ
h
kl
as the known channel realization,
an achievable SE is
E
log
2
1 +
p
k
|
ˆ
h
kl
|
2
E
{|
υ
|
2
|
ˆ
h
kl
}
+
σ
2
(46)
where
υ
=
˜
h
kl
s
k
+
∑
i
=
k
h
il
s
i
and
E
{|
υ
|
2
|
ˆ
h
kl
}
=
i
∈P
k
\{
k
}
p
2
i
β
2
il
p
k
β
2
kl
|
ˆ
h
kl
|
2
+
i
∈P
k
p
i
β
il
+
i
∈P
k
p
i
C
il
(47)
by exploiting the fact that
ˆ
h
il
and
ˆ
h
kl
are independent for all
i
²∈ P
k
and
ˆ
h
il
=
√
p
i
β
il
√
p
k
β
kl
ˆ
h
kl
for all
i
∈ P
k
. By inserting (47)
into (46), we can expand the expression as
E
⎧
⎪
⎨
⎪
⎩
log
2
⎛
⎜
⎝
1 +
|
ˆ
h
kl
|
2
p
k
(1 +
A
lk
)
∑
i
∈P
k
p
i
β
il
+
∑
i
∈P
k
p
i
C
il
+
σ
2
⎞
⎟
⎠
⎫
⎪
⎬
⎪
⎭
−
E
⎧
⎪
⎨
⎪
⎩
log
2
⎛
⎜
⎝
1+
|
ˆ
h
kl
|
2
p
k
A
lk
∑
i
∈P
k
p
i
β
il
+
∑
i
∈P
k
p
i
C
il
+
σ
2
⎞
⎟
⎠
⎫
⎪
⎬
⎪
⎭
(48)
and compute each of the expectations using [23, Lemma 3]
and
ˆ
h
kl
∼ N
C
(0
, p
k
τ
p
β
2
kl
/
Ψ
t
k
l
)
to obtain the final expression
in (29).
A
PPENDIX
C
P
ROOF OF
P
ROPOSITION
4
The received signal in (9) for Level
4
can be expressed as
y
=
K
i
=1
ˆ
h
i
s
i
+
e
(49)
where
e
n
+
∑
K
i
=1
˜
h
i
s
i
has zero mean and correlation
matrix
E
. Since the MMSE channel estimates are known and
e
is uncorrelated with
ˆ
h
i
s
i
for all
i
, (49) can be treated as a
multiple access channel with colored noise. In the worst case,
in terms of mutual information, the colored noise is indepen-
dent of the desired signals and Gaussian distributed. Hence,
we can apply pre-whitening followed by standard results
on MMSE-SIC receivers to obtain the achievable sum SE
[52, Sec. 8.3.4]
E
{
log
2
det(
I
NM
+
A
−
1
/
2
ˆ
HP
ˆ
H
H
A
−
1
/
2
)
}
.
This expression reduces to (41) by utilizing the fact that
det(
I
+
BC
) = det(
I
+
CB
)
for any matrices
B
,
C
of
compatible sizes, and by including the pre-log factor
1
−
τ
p
/τ
c
that is the fraction of channel uses used for data. Note that
the sum SE expression is independent of the decoding order.
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Emil Björnson
(S’07–M’12–SM’17) received the
M.S. degree in engineering mathematics from Lund
University, Sweden, in 2007, and the Ph.D. degree
in telecommunications from the KTH Royal Institute
of Technology, Sweden, in 2011.
From 2012 to 2014, he held a joint post-doctoral
position at the Alcatel-Lucent
Chair on Flexible
Radio, SUPELEC, France, and the KTH Royal Insti-
tute of Technology. He joined Linköping University,
Sweden, in 2014, where he is currently an Asso-
ciate Professor and a Docent with the Division of
Communication Systems. He has authored the textbooks
Optimal Resource
Allocation in Coordinated Multi-Cell Systems
in 2013 and
Massive MIMO
Networks: Spectral, Energy, and Hardware Efficiency
in 2017. He is dedi-
cated to reproducible research and has made a large amount of simulation
code publicly available. He performs research on MIMO communications,
radio resource allocation, machine learning for communications, and energy
efficiency.
Dr. Björnson was a recipient of the 2014 Outstanding Young Researcher
Award from the IEEE ComSoc EMEA, the 2015 Ingvar Carlsson Award,
the 2016 Best Ph.D. Award from EURASIP, the 2018 IEEE Marconi Prize
Paper Award in Wireless Communications, the 2019 EURASIP Early Career
Award, and the 2019 IEEE Communications Society Fred W. Ellersick Prize.
He also coauthored articles that received the Best Paper Awards at the
conferences, including WCSP 2009, the IEEE CAMSAP 2011, the IEEE
WCNC 2014, the IEEE ICC 2015, WCSP 2017, and the IEEE SAM 2014.
Since 2017, he has been on the Editorial Board of the IEEE T
RANSACTIONS
ON
C
OMMUNICATIONS
and the IEEE T
RANSACTIONS ON
G
REEN
C
OM
-
MUNICATIONS AND
N
ETWORKING
since 2016. He has performed MIMO
research for more than ten years and has filed more than ten MIMO related
patent applications.
Luca
Sanguinetti
(SM’15) received
the
Laurea
Telecommunications Engineer degree
(cum laude)
and the Ph.D. degree in information engineering
from the University of Pisa, Italy, in 2002 and 2005,
respectively.
In
2004,
he
was
a Visiting
Ph.D. Student
at
the German Aerospace Center (DLR), Oberpfaffen-
hofen, Germany. From June 2007 to June 2008,
he was a Postdoctoral Associate with the Depart-
ment of Electrical Engineering, Princeton University.
From July 2013 to October 2017, he was with
Large Systems and Networks Group (LANEAS), CentraleSupélec, France.
He is currently an Associate Professor with the ’Dipartimento di Ingegneria
dell’Informazione’, University of Pisa. He has coauthored the textbook
Mas-
sive MIMO Networks: Spectral, Energy, and Hardware Efficiency
in 2017. His
expertise and general interests span the areas of communications and signal
processing.
Dr. Sanguinetti was a recipient of the 2018 Marconi Prize Paper Award in
Wireless Communications and coauthored an article that received the Young
Best Paper Award from the ComSoc/VTS Italy Section. He was the recipient
of the FP7 Marie Curie IEF 2013 “Dense deployments for green cellular
networks”. He was also a co-recipient of the two best conference paper
awards: IEEE WCNC 2013 and IEEE WCNC 2014. He served as an Associate
Editor for the IEEE T
RANSACTIONS ON
W
IRELESS COMMUNICATIONS
and
the IEEE J
OURNAL ON
S
ELECTED
A
REAS OF
C
OMMUNICATIONS
(series on
Green Communications and Networking) and as a Lead Guest Editor for the
IEEE J
OURNAL ON
S
ELECTED
A
REAS OF
C
OMMUNICATIONS
Special Issue
on “Game Theory for Networks”. He is currently serving as an Associate
Editor for the IEEE S
IGNAL
P
ROCESSING
L
ETTERS
, the IEEE T
RANSAC
-
TIONS ON
C
OMMUNICATIONS
. He is also a member of the Executive Editorial
Committee of the IEEE T
RANSACTIONS ON
W
IRELESS
C
OMMUNICATIONS
.
Authorized licensed use limited to: Concordia University Library. Downloaded on October 28,2023 at 23:12:16 UTC from IEEE Xplore.
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2. Design the pull-up network.
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Note:
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1. Design CMOS circuit for F = (AB) + (CD) using PUN and PDN. Do not simplify the function.
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o vdd and ground rail heights: 1.8 microns
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