1

Massive MIMO Has Unlimited Capacity

arXiv:1705.00538v4 [] 9 Nov 2017

Emil Björnson, Member, IEEE, Jakob Hoydis, Member, IEEE, Luca Sanguinetti, Senior Member, IEEE

Abstract—The capacity of cellular networks can be improved
by the unprecedented array gain and spatial multiplexing offered
by Massive MIMO. Since its inception, the coherent interference
caused by pilot contamination has been believed to create a
finite capacity limit, as the number of antennas goes to infinity.
In this paper, we prove that this is incorrect and an artifact
from using simplistic channel models and suboptimal precoding/combining schemes. We show that with multicell MMSE
precoding/combining and a tiny amount of spatial channel correlation or large-scale fading variations over the array, the capacity
increases without bound as the number of antennas increases,
even under pilot contamination. More precisely, the result holds
when the channel covariance matrices of the contaminating users
are asymptotically linearly independent, which is generally the
case. If also the diagonals of the covariance matrices are linearly
independent, it is sufficient to know these diagonals (and not
the full covariance matrices) to achieve an unlimited asymptotic
capacity.
Index Terms—Massive MIMO, ergodic capacity, asymptotic
analysis, spatial correlation, multi-cell MMSE processing, pilot
contamination.

I. I NTRODUCTION
The Shannon capacity of a channel manifests the spectral efficiency (SE) that it supports. Massive MIMO (multiple-input
multiple-output) improves the sum SE of cellular networks
by spatial multiplexing of a large number of user equipments
(UEs) per cell [1]. It is therefore considered a key timedivision duplex (TDD) technology for the next generation
of cellular networks [2]–[4]. The main difference between
Massive MIMO and classical multiuser MIMO is the large
number of antennas, M , at each base station (BS) whose
signals are processed by individual radio-frequency chains. By
exploiting channel estimates for coherent receive combining,
the uplink signal power of a desired UE is reinforced by
a factor M , while the power of the noise and independent
interference does not increase. The same principle holds for
the transmit precoding in the downlink. Since the channel
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E. Björnson is with the Department of Electrical Engineering (ISY),
Linköping University, 58183 Linköping, Sweden (emil.bjornson@liu.se).
J. Hoydis is with Nokia Bell Labs, Paris-Saclay, 91620 Nozay, France
(jakob.hoydis@nokia-bell-labs.com). L. Sanguinetti is with the University
of Pisa, Dipartimento di Ingegneria dell’Informazione, 56122 Pisa Italy
(luca.sanguinetti@unipi.it) and also with the Large Systems and Networks
Group (LANEAS), CentraleSupélec, Université Paris-Saclay, 3 rue JoliotCurie, 91192 Gif-sur-Yvette, France. The authors have contributed equally
to this work and are listed alphabetically.
This research has been supported by ELLIIT, the Swedish Foundation for
Strategic Research (SFF), the EU FP7 under ICT-619086 (MAMMOET), and
the ERC Starting Grant 305123 MORE.
Parts of this paper were presented at the International Conference on
Communications (ICC), 21–25 May, 2017, Paris, France.

estimates are obtained by uplink pilot signaling and the pilot
resources are limited by the channel coherence time, the
same pilots must be reused in multiple cells. This leads to
pilot contamination which has two main consequences: the
channel estimation quality is reduced due to pilot interference
and the channel estimate of a desired UE is correlated with
the channels to the interfering UEs that use the same pilot.
Marzetta showed in his seminal paper [1] that the interference
from these UEs during data transmission is also reinforced
by a factor M , under the assumptions of maximum ratio
(MR) combining/precoding and independent and identically
distributed (i.i.d.) Rayleigh fading channels. This means that
pilot contamination creates a finite SE limit as M → ∞.
The large-antenna limit has also been studied for other
combining/precoding schemes, such as the minimum mean
squared error (MMSE) scheme. Single-cell MMSE (S-MMSE)
was considered in [5]–[7], while multicell MMSE (M-MMSE)
was considered in [8], [9]. The difference is that with MMMSE, the BS makes use of estimates of the channels from
the UEs in all cells, while with S-MMSE, the BS only uses
channel estimates of the UEs in the own cell. In both cases,
the SE was proved to have a finite limit as M → ∞, under the
assumption of i.i.d. Rayleigh fading channels (i.e., no spatial
correlation). In contrast, there are special cases of spatially
correlated fading that give rise to rank-deficient covariance
matrices [10]–[12]. If the UEs that share a pilot have rankdeficient covariance matrices with orthogonal support, then
pilot contamination vanishes and the SE can increase without
bound. The covariance matrices R1 and R2 have orthogonal
support if R1 R2 = 0. To understand this condition, note that
for arbitrary covariance matrices




a c
d f
R1 = ?
R2 = ?
(1)
c b
f
e
every element of R1 R2 must be zero. The first element is
ad + cf ? . If we model the practical covariance matrices of
two randomly located UEs as realizations of a random variable
with continuous distribution, then ad + cf ? = 0 occurs with
zero probability.1 Hence, orthogonal support is very unlikely
in practice, although one can find special cases where it is
satisfied. The one-ring model for uniform linear arrays (ULAs)
gives orthogonal support if the channels have non-overlapping
angular support [10]–[12], but the ULA microwave measurements in [13] show that the angular support of practical
channels is highly irregular and does not lead to orthogonal
support. In conclusion, practical covariance matrices do not
have orthogonal support, at least not at microwave frequencies.
1 For any continuous random variable x, the probability that x takes a
particular realization is zero, while the probability that x takes a realization
in a certain interval can be non-zero. Hence, if x = ad + cf ? then x = 0
occurs with zero probability.

2

The literature contains several categories of methods for
mitigation of pilot contamination, also known as pilot decontamination. The first category allocates pilots to the UEs in an
attempt to find combinations where the covariance matrices
have relatively different support [10]–[12], [14]. This method
can substantially reduce pilot contamination, but can only
remove the finite limit in the unlikely special case when the
covariance matrices have orthogonal support. The second category utilizes semi-blind estimation to separate the subspace of
desired UE channels from the subspace of interfering channels
[15]–[19]. This method can fully remove pilot contamination
if M and the size of the channel coherence block go jointly to
infinity [18]. Unfortunately, the channel coherence is fixed and
finite in practice (this is why we cannot give unique pilots to
every cell), thus we cannot approach this limit in practice. The
third category uses multiple pilot phases with different pilot
sequences to successively eliminate pilot contamination [20],
[21], without the need for statistical information. However, the
total pilot length is larger or equal to the total number of UEs,
which would allow allocating mutually orthogonal pilots to all
UEs and thus trivially avoiding the pilot contamination problem. This is not a scalable solution for networks with many
cells. The fourth category is pilot contamination precoding that
rejects interference by coherent joint transmission/reception
over the entire network [22], [23]. This method appears to
achieve an unbounded SE, but this has not been formally
proved and requires that the data for all UEs is available at
every BS, which might not be feasible in practice.
In summary, it appears that pilot contamination is a fundamental issue that manifests a finite SE limit, except in unlikely
special cases. We show in this paper that this is basically
a misunderstanding, spurred by the popularity of analyzing
suboptimal combining/precoding schemes, such as MR and
S-MMSE, and focusing on unrealistic i.i.d. Rayleigh fading
channels (as in the prior work [8], [9] on M-MMSE). We prove
that the SE increases without bound in the presence of pilot
contamination when using M-MMSE combining/precoding, if
the pilot-sharing UEs have asymptotically linearly independent covariance matrices. Note that R1 and R2 in (1) are
linearly independent if [a b c]T and [d e f ]T are non-parallel
vectors, which happens almost surely for randomly generated
covariance matrices. Hence, our results rely on a condition
that is most likely satisfied in practice—it is the general case,
while prior works on the asymptotics of Massive MIMO have
considered practically unlikely special cases. In contrast to
prior work, no multicell cooperation is utilized herein and
there is no need for orthogonal support of covariance matrices.
In the conference paper [24], we proved the main result in a
two-user uplink scenario.2 In this paper, we prove the result
for both uplink and downlink in a general setting. Section II
proves and explains the intuition of the results in a two-user
setup, while Section III generalizes the results to a multicell
2 After

submitting our conference paper [24], the related work [25] appeared.
That paper considers the mean squared error in the uplink data detection of a
single cell with multiple UEs per pilot sequence. The authors show that the
error goes asymptotically to zero when having linearly independent covariance
matrices. However, the paper [25] contains no mathematical analysis of the
achievable SE.

setup. The results are demonstrated numerically in Section IV
and the main conclusions are summarized in Section V.
Notation: The Frobenius and spectral norms of a matrix X
are denoted by kXkF and kXk2 , respectively. The superscripts
T ?
, and H denote transpose, conjugate, and Hermitian transpose, respectively. We use , to denote definitions, whereas
NC (0, R) denotes the circularly symmetric complex Gaussian
distribution with zero mean and covariance matrix R. The
expected value of a random variable x is denoted by E{x}
and the variance is denoted by V{x}. The N × N identity
matrix is denoted by IN , while 0N is an N × N all-zero
matrix and 1N is an N × 1 all-one vector. We use an  bn
to denote an − bn →n→∞ 0 (almost surely (a.s.)) for two
(random) sequences an , bn .
II. A SYMPTOTIC S PECTRAL E FFICIENCY IN A T WO -U SER
S CENARIO
In this section, we prove and explain our main result in
a two-user scenario, where a BS equipped with M antennas
communicates with UE 1 and UE 2 that are using the same
pilot. This setup is sufficient to demonstrate why M-MMSE
combining and precoding reject the coherent interference
caused by pilot contamination. We consider a block-fading
model where each channel takes one realization in a coherence
block of τc channel uses and independent realizations across
blocks. We denote by hk ∈ CM the channel from UE k to the
BS and consider Rayleigh fading with hk ∼ NC (0, Rk ) for
k = 1, 2, where Rk ∈ CM ×M with3 tr(Rk ) > 0 is the channel covariance matrix, which is assumed to be known at the
BS. The Gaussian distribution models the small-scale fading
whereas the covariance matrix Rk describes the macroscopic
1
tr (Rk ) determines the
effects. The normalized trace βk = M
average large-scale fading between UE k and the BS, while the
eigenstructure of Rk describes the spatial channel correlation.
A special case that is convenient for analysis is i.i.d. Rayleigh
fading with Rk = βk IM [26], but it only arises in fully
isotropic fading environments. In general, each covariance
matrix has spatial correlation and large-scale fading variations
over the array, represented by non-zero off-diagonal elements
and non-identical diagonal elements, respectively.
A. Uplink Channel Estimation
We assume that the BS and UEs are perfectly synchronized
and operate according to a TDD protocol wherein the data
transmission phase is preceded by an uplink pilot phase for
channel estimation. Both UEs use the same τp -length pilot
sequence φ ∈ Cτp with elements such that kφk2 = φH φ = 1.
The received uplink signal Yp ∈ CN ×τp at the BS is given
by
p
p
(2)
Yp = ρtr h1 φT + ρtr h2 φT + Np
where ρtr is the normalized pilot power and Np ∈ CN ×τp is
the normalized receiver noise with all elements independently
distributed as NC (0, 1). The matrix Yp is the observation that
3 This assumption implies that there is non-zero energy received from and
transmitted to each UE.

3

the BS utilizes to estimate h1 and h2 . We assume that channel
estimation is performed using the MMSE estimator given in
the next lemma (the proof relies on standard estimation theory
[27]).
Lemma 1. The MMSE estimator of hk for k = 1, 2, based
on the observation Yp at the BS, is
1
ĥk = p Rk Q−1 Yp φ?
(3)
ρtr
with Q = ρ1tr E{Yp φ? (Yp φ? )H } = R1 + R2 + ρ1tr IM
being the normalized covariance matrix of the observation
after correlating with the pilot sequence. The estimate ĥk
and the estimation error h̃k = hk − ĥk are independent
random vectors distributed as ĥk ∼ NC (0, Φk ) and h̃k ∼
NC (0, Rk − Φk ) with Φk = Rk Q−1 Rk .
Interestingly, the estimates ĥ1 and ĥ2 are computed in an
almost identical way in (3): the same matrix Qpis inverted and
multiplied with the same observation Yp φ? / ρtr . The only
difference is that for ĥk there is a multiplication with the UE’s
own channel covariance matrix Rk in (3), for k = 1, 2. The
channel estimates are thus correlated with correlation matrix
Υ12 = E{ĥ1 ĥH2 } = R1 Q−1 R2 . If R1 is invertible, then we
can also write the relation between the estimates as ĥ2 =
R2 R−1
1 ĥ1 . In the special case of i.i.d. fading channels with
R1 = β1 IM and R2 = β2 IM , the two channel estimates are
parallel vectors that only differ in scaling: ĥ2 = ββ12 ĥ1 . This
is an unwanted property caused by the inability of the BS to
separate UEs that have transmitted the same pilot sequence
over channels that are identically distributed (up to a scaling
factor). In the alternative special case of R1 R2 = 0M , the two
UE channels are located in orthogonal subspaces (i.e., have
orthogonal support), which leads to zero correlation: Υ12 =
0M . Consequently, it is theoretically possible to let two UEs
share a pilot sequence without causing pilot contamination, if
their covariance matrices satisfy the orthogonality condition
R1 R2 = 0M . As described in Section I, none of these special
cases occur in practice, therefore we will develop a general
way to deal with the correlation of channel estimates caused
by pilot contamination.
B. Uplink Data Transmission
During uplink data transmission, the received
p baseband
M
signal
at
the
BS
is
y
∈
C
,
given
by
y
=
ρul h1 s1 +
p
ul
ρ h2 s2 + n, where sk ∼ NC (0, 1) is the informationbearing signal transmitted by UE k, n ∼ NC (0, IM ) is the
independent receiver noise, and ρul is the normalized transmit
power. The BS detects the signal from UE 1 by using a
combining vector v1 ∈ CM to obtain v1H y. Using a standard
technique (see, e.g., [5], [26]), the ergodic uplink capacity of
UE 1 is lower bounded by




τp
SEul
E log2 1 + γ1ul
[bit/s/Hz]
(4)
=
1
−
1
τc
where the expectation is with respect to the channel estimates.
We refer to SEul
1 as an achievable SE. The instantaneous
effective signal-to-interference-and-noise ratio (SINR) γ1ul in
(4) is

γ1ul =
=

|v1H ĥ1 |2
o
n
E |v1H h̃1 |2 + |v1H h2 |2 + ρ1ul v1H v1 ĥ1 , ĥ2
|v1H ĥ1 |2

(5)




v1H ĥ2 ĥH2 + Z v1

P2
with Z = k=1 (Rk −Φk )+ ρ1ul IM . Since γ1ul is a generalized
Rayleigh quotient, the SINR is maximized by [8], [9]
!−1
2
X
v1 =
ĥk ĥHk + Z
ĥ1 .
(6)
k=1

This is called MMSE combining since (6) not only maximizes
the instantaneous SINR γ1ul , but also minimizes E{|x1 −
v1H y|2 |ĥ1 , ĥ2 } which is the mean squared error (MSE) in
the data detection (conditioned on the channel estimates).
Plugging (6) into (5) yields

−1
γ1ul = ĥH1 ĥ2 ĥH2 + Z
ĥ1 .
(7)
ul
We will now analyze the asymptotic behavior of SEul
1 and γ1
as M → ∞. To this end, we make the following technical
assumptions:

Assumption 1. For k = 1, 2, lim inf
M
lim sup kRk k2 < ∞.

1
M tr(Rk )

> 0 and

M

Assumption 2. For λ = [λ1 , λ2 ]T ∈ R2 and i = 1, 2,
lim inf
M

1
2
kλ1 R1 + λ2 R2 kF > 0.
{λ: λi =1} M
inf

(8)

The first assumption is a well established way to model that
the array gathers more energy as M increases and also that
this energy originates from many spatial dimensions [5]. In
particular, it is a sufficient condition for asymptotic channel
hardening; that is, khk k2 /E{khk k2 } → 1 in probability as
M → ∞. The second assumption requires R1 and R2 to
be asymptotically linearly independent, in the sense that if
one of the matrices is scaled to resemble the other one, the
subspace in which the matrices differ has an energy proportional to M . Note that this is a stronger condition than linear
independence, defined as inf {λ: λi =1} kλ1 R1 + λ2 R2 k2F > 0
for i = 1, 2, which is satisfied even if the matrices only differ
in one element. We will elaborate further on Assumption 2 in
Section II-D.
The following is the first of the main results of this paper:
Theorem 1. If MMSE combining is used, then under Assumptions 1 and 2, the instantaneous effective SINR γ1ul
increases a.s. unboundedly as M → ∞. Hence, SEul
1 increases
unboundedly as M → ∞.
Proof: The proof is given in Appendix B.
Remark 1. From the proof in Appendix B, we can see that
γ1ul /M has a non-zero asymptotic limit, which implies that the
SE grows towards infinity as log2 (M ). While Theorem 1 only
considers UE 1, one only needs to interchange the UE indices
to prove that the SE of UE 2 also grows unboundedly as

4

M → ∞. Hence, an unlimited asymptotic SE is simultaneously
achievable for both UEs. Since the SE is a lower bound on
capacity, we conclude that the asymptotic capacity is also
unlimited.
Observe that if R1 and R2 are linearly dependent, i.e.,
R1 = ηR2 , then Assumption 2 does not hold. Under these
circumstances, ĥ2 = η1 ĥ1 and by applying Lemma 5 in
Appendix A we obtain
γ1ul =

ĥH1 Z−1 ĥ1
1+

(9)

1 H −1
ĥ1
η 2 ĥ1 Z

from which, it is straightforward to show that γ1ul  η 2 (by
dividing and multiplying each term by M and using Lemma 3
in Appendix A). This implies that SEul
1 converges to a finite
quantity when M → ∞, as Marzetta showed in his seminal
paper [1] for the special case of R1 = ηR2 = IM .
C. Downlink Data Transmission
During the downlink data transmission, the BS
p transmits
M
ρdl w1 ς1 +
the
signal
x
∈
C
.
This
signal
is
given
by
x
=
p
ρdl w2 ς2 , where ςk ∼ NC (0, 1) is the information-bearing
signal transmitted to UE k, ρdl is the normalized downlink
transmit power, and wk is the precoding vector
associated

2
with UE k.
This
precoding
vector
satisfies
E
kw
= 1,
kk

2
dl
so that E kwk ςk k = ρ is the downlink transmit power
allocated to UE k. The received downlink signal z1 at UE 1
is4
p
p
z1 = ρdl hH1 w1 ς1 + ρdl hH1 w2 ς2 + n1
p
p
= ρdl E {hH1 w1 } ς1 + ρdl (hH1 w1 − E {hH1 w1 })ς1
p
(10)
+ ρdl hH1 w2 ς2 + n1
where n1 ∼ NC (0, 1) is the normalized receiver noise. The
first term in (10) is the desired signal received over the
deterministic average precoded channel E {hH1 w1 }, while the
remaining terms are random variables with unknown realizations. By treating these terms as noise in the signal detection
[5], [26], the downlink ergodic channel capacity of UE 1 can
be lower bounded by



τp
SEdl
=
1
−
log2 1 + γ1dl
[bit/s/Hz]
(11)
1
τc
with the effective SINR
γ1dl =

|E{hH1 w1 }|2
E {|h1 w2 |2 } + V {hH1 w1 } +
H

1
ρdl

.

(12)

Since UE 1 only needs to know E {hH1 w1 } and the total
variance of the second to fourth term in (10), the SE in (11)
is achievable in the absence of downlink channel estimation.
In contrast to the uplink, there is no precoding that is always
optimal [28]. However, motivated by uplink-downlink duality
4 For

H
notational convenience, we treat hH
1 and h2 as the downlink channels,
T
instead of hT
1 and h2 . This has no impact on the SE since the difference is
only in a complex conjugate.

[9], a reasonable suboptimal choice is the so-called MMSE
precoding
!−1
2
X
p
vk
ĥk ĥHk + Z
ĥk (13)
= ϑk
wk = p
E {kvk k2 }
k=1
P2
where vk= ( k=1 ĥk ĥHk +Z)−1 ĥk is MMSE combining and
ϑk = (E kvk k2 )−1 is a scaling factor. The following is the
second main result of this paper:
Theorem 2. If MMSE precoding is used, then under Assumptions 1 and 2 the effective SINR γ1dl increases unboundedly as
M → ∞. Hence, SEdl
1 increases unboundedly as M → ∞.
Proof: The proof is given in Appendix D.
This theorem shows that, under the same conditions as in
the uplink, the downlink SE (and thus the capacity) increases
without bound as M → ∞. The asymptotic SE growth
is proportional to log2 (M ), since the proof in Appendix D
shows that γ1dl /M has a non-zero asymptotic limit. UE 2 can
simultaneously achieve an unbounded SE, which is proved
directly by interchanging the UE indices.
D. Interpretation and Generality
Theorems 1 and 2 show that the SE (and thus the capacity) under pilot contamination is asymptotically unlimited if
Assumption 2 holds. To gain an intuitive interpretation of
this underlying assumption, recall from (3) that ĥ1 = R1 a
and ĥ2 = R2 a, where a = √1 tr Q−1 Yp φ∗ is the same
ρ

for both UEs. Hence, ĥ1 and ĥ2 are (asymptotically) linearly
independent when R1 and R2 are (asymptotically) linearly
independent, except for special choices of a. As illustrated in
Fig. 1, it is then possible to find a combining vector v1 (or
precoding vector w1 ) that is orthogonal to ĥ2 , while being
non-orthogonal to ĥ1 . Similarly, one can find v2 (and w2 )
such that v2H ĥ1 = 0 and v2H ĥ2 6= 0. For example, if we define
Ĥ = [ĥ1 ĥ2 ] ∈ CM ×2 , then the zero-forcing (ZF) combining
vectors
−1



(14)
v1 v2 = Ĥ ĤH Ĥ
satisfy these conditions. Note that ĤH Ĥ is only invertible if
the channel estimates (columns in Ĥ) are linearly independent.
Using ZF as defined in (14), we get v1H ĥ2 = 0 and v1H ĥ1 =
1. If the channel estimates are also asymptotically linearly
independent, it follows5 that kv1 k2 → 0 as M → ∞; that
is, we can reject the coherent interference and get unit signal
gain, while at the same time using the array gain to make
the noise term ρ1ul v1H v1 = ρ1ul kv1 k2 vanish asymptotically.
Since optimal MMSE combining (and also MMSE precoding)
provides a higher SINR than the heuristic ZF scheme in (14),
it also rejects the coherent interference while retaining an array
gain that grows with M .
To further explain the implications of Assumption 2, we
provide the following three examples.
5 Notice that, by applying Lemma 3 in Appendix A,
1
1
[ĤH Ĥ]nm  M
tr(Rn Q−1 Rm ). If the channel estimates
M
1
totically linearly independent, then M
ĤH Ĥ is invertible as M
 1 H −1
1
2
thus kv1 k = M ( M Ĥ Ĥ) ]11  0.

we have
are asymp→ ∞ and

5

v1
Orthogonal
only to ĥ2

Example 2. Channels with i.i.d. fading, where the covariance
matrices are R1 = β1 IM and R2 = β2 IM , are a notable case
when the covariance matrices are not linearly independent.
However, any such case is non-robust to perturbations of the
matrix elements. Suppose we replace R1 with

ĥ1
ĥ2

Fig. 1: If the pilot-contaminated channel estimates are linearly
independent (i.e., not parallel), there exists a combining vector v1 that
rejects the pilot-contaminated interference from UE 2 in the uplink,
while the desired
signal remains due to v1H ĥ1 6= 0. Similarly, if
p
w1 = v1 / E{kv1 k2 } is used as precoding vector, then no pilotcontaminated coherent interference is caused to UE 2 in the downlink.

Example 1. Consider a two-user scenario with


2IN
0
R1 =
R2 = IM
0
IM −N

(15)

where the covariance matrices have full rank and are only
different in the first N dimensions. For any given M , we notice
that the argument of (8) for UE i = 1 becomes
1
kR1 + λ2 R2 k2F
inf
λ2 M
N (2 + λ2 )2 + (M − N )(1 + λ2 )2
(M − N )N
= inf
=
λ2
M
M2
(16)
where the infimum is attained by λ2 = −(M + N )/M .
Note that (16) goes to zero as M → ∞ if N is constant,
while it has the non-zero limit (1 − α)α if N = αM ,
for some 0 < α < 1. In the latter case, the matrices
{R1 , R2 } satisfy (8). Interestingly, although the covariance
matrices are diagonal, they are still asymptotically linearly
independent and the subspace in which they differ has rank
min(N, M − N ) = M min(α, (1 − α)), which is proportional
to M .
Let us further exemplify the interference rejection by considering ZF combining, which provides lower SINR than MMSE
combining, but gives more intuitive expressions. Assume for
the sake of simplicity that the channel realizations are such
that √1 tr Q−1 Yp φ∗ = 1M , which gives ĥ1 = R1 1M =
ρ

[21TN 1TM −N ]T and ĥ2 = R2 1M = 1M . The ZF combining
vectors are then given by


−1 
1


− N1 1N
N 1N
v1 v2 = Ĥ ĤH Ĥ =
.
1
2
− M −N
1M −N M −N
1M −N
(17)
If we set ρul = ρul = 1 for simplicity, the instantaneous
effective SINR in (5) for UE 1 becomes
γ1ul =
=

|v1H ĥ1 |2
|v1H ĥ2 |2 +

P2

k=1

v1H (Rk − Φk )v1 + kv1 k2

1
0+

7
4N

+

4
3(M −N )

+

M
N (M −N )

(18)

where the coherent interference from UE 2 is zero. The
remaining terms go asymptotically to zero if N = αM , for
0 < α < 1, in which case γ1ul grows without bound, as
expected from Theorem 1.
In the second example, we consider a scenario where
Assumption 2 is not satisfied.



1


R1 = β1 
 0
..
.

0
..
.
0

···




0 

M

(19)

where 1 , . . . , M are i.i.d. positive random variables. This
modeling is motivated by the measurement results in [29],
which shows that there are a few dB of large-scale fading
variations over the antennas in a ULA. For UE i = 1, we
have
1
kR1 + λ2 R2 k2F
M
M
1 X
(β1 m + λ2 β2 )2
= lim inf inf
M
λ2 M
m=1

lim inf inf
M

λ2

(a)

= lim inf
M

β12

M
M
1 X
1 X
m −
n
M m=1
M n=1

!2

(b)

= β12 E{(m − E{m })2 }

(20)

PM
1
where (a) is obtained from the fact that λ2 = − ββ12 M
n=1 n
PM
1
2
(β

+
λ
β
)
and
(b)
follows
from
the
minimizes M
2 2
m=1 1 m
strong law of large numbers. Note that E{(m − E{m })2 } in
the last expression is the variance of m . Since every random
variable has non-zero variance and β1 > 0, we conclude that
{R1 , R2 } satisfy (8) and thus Assumption 2 holds.
The key implication from Example 2 is that all cases where
R1 and R2 are equal (up to a scaling factor) are non-robust
to random perturbations and thus anomalies. Since practical
propagation environments are irregular and behave randomly
(see the measurements reported in [13], [29]), linearly dependent covariance matrices are not appearing in practice and
Assumption 2 is generally satisfied. In other words, it is fair
to say that the uplink and downlink SEs grow without bound as
M → ∞ in general, while the special cases when it does not
occur are of no practical importance. We end this subsection
with a comparison with related work and a remark regarding
acquisition of channel statistics.
Example 3 (Comparison with [22], [23]). Consider a BS with
two distributed arrays of M 0 = M/2 antennas that serve two
UEs having the covariance matrices

b I 0
R1 = 11 M
0

0
b12 IM 0




b I 0
R2 = 21 M
0

0
b22 IM 0


(21)

with b11 , b12 , b21 , b22 > 0. These covariance matrices are
(asymptotically) linearly independent if b11 b22 6= b12 b21 , in
which case the uplink and downlink SEs grow without bound
with MMSE or ZF.

6

The exemplified setup is equivalent to the multicell joint
transmission scenario considered in the pilot contamination
precoding works [22], [23] in which the heuristic vectors

−1
 1 p


b11 b12
y
0
v1 v2 = M 0 1
(22)
p
1
b21 b22
0
M 0 y2
0

are used for combining and precoding, and y1p , y2p ∈ CM are
obtained from the received pilot signals as [(y1p )T (y2p )T ]T =
p ?
tr
Y
 φ /ρ
H . These
 vectors are specifically designed to make
h1 h2 v1 v2  I2 as M → ∞, and thus this method
has the same asymptotic behavior as ZF in the special case
of block-diagonal covariance matrices where each block is a
scaled identity matrix. Note that the matrix inverse in (22)
only exists if b11 b22 6= b12 b21 , which is again the condition
for linear independence of the covariance matrices. Since
pilot contamination precoding can only be applied in special
multicell cooperation cases, MMSE combining/precoding is
generally the preferable choice.
Remark 2 (Acquiring Covariance Matrices). Theorems 1 and
2 exploit the MMSE estimator and thus the BS needs to know
the (deterministic) channel statistics. In particular, the BS can
only compute the MMSE estimate ĥk in Lemma 1 if it knows
Rk and also the sum R1 +R2 of the two covariance matrices.
In practice, Rk can be estimated by a regularized sample
covariance matrix, given realizations of hk over multiple
resource blocks (e.g., different times and frequencies) where
this channel is either observed in only noise [10], [30], [31]
or where some observations are regular pilot transmissions
containing the desired channel plus interference/noise and
some contain only the interference/noise [32]. It seems that
around M samples are needed to obtain a sufficiently accurate
covariance estimate [32]. The covariance estimation can be
further improved if the channels have a known structure.
For example, [33] provides algorithms for estimating the
covariance matrices of channels that have limited angle-delay
support that is also separable between users.

where Dk ∈ RM ×M and Λ ∈ RM ×M are diagonal matrices
with elements {[Rk ]ii : i = 1, . . . , M } and {[R1 ]ii + [R2 ]ii +
1
ρtr : i = 1, . . . , M }, respectively. Notice that Assumption 1
1
implies that6 lim inf M M
tr(Dk ) > 0 and lim supM kDk k2 <
∞ for k = 1, 2. To quantify the achievable SE when using
EW-MMSE, similar to the downlink we exploit the use-andthen-forget SE bound [26], which is less tight than (4) but
does not require the use of MMSE channel estimation. The
uplink ergodic capacity of UE 1 can be thus lower bounded
τp
ul
by SEul
1 = (1 − τc ) log2 (1 + γ 1 ) [bit/s/Hz] with
=
γ ul
1

|E{v1H h1 }|2
E {|v1H h2 |2 } + V{v1H h1 } +

.
1
E {kv1 k2 }
ρul

(25)

This bound is valid for any channel estimation and any combining scheme. A reasonable choice for v1 is the approximate
MMSE combining vector:
X
−1
2
v1 =
ĥk ĥHk + S
ĥ1
(26)
k=1

where ĥ1 , ĥ2 are computed
as in (24) and
 S is diagonal and
P2 
given by S = k=1 Dk − Dk Λ−1 Dk + ρ1ul IM . Note that
(26) is equivalent to the MMSE combining in (6) when the
covariance matrices are diagonal. We will now analyze how
γ ul
behaves asymptotically as M → ∞ when v1 is given by
1
(26). To this end, we impose the following assumption, which
states that D1 and D2 are asymptotically linearly independent
(i.e., the diagonals of R1 and R2 are asymptotically linearly
independent).
Assumption 3. For λ = [λ1 , λ2 ]T ∈ R2 and i = 1, 2,
lim inf
M

inf

{λ: λi =1}

1
2
kλ1 D1 + λ2 D2 kF > 0.
M

(27)

The following is the third main result of this paper:

E. Achievable SE with Partial Knowledge of Covariance Matrices

Theorem 3. If v1 in (26) is used with ĥ1 , ĥ2 given by (24),
then under Assumptions 1 and 3, the SINR γ ul
increases
1
unboundedly as M → ∞. Hence, SEul
increases
unboundedly
1
as M → ∞.

If the BS does not have full knowledge of the covariance
matrices, an alternative method for channel estimation is to
estimate each entry of hk separately, ignoring the correlation
among the elements. This leads to the element-wise MMSE
(EW-MMSE) estimator (called diagonalized estimator in [30])
that utilizes only the main diagonals of R1 and R2 . The
diagonals can be estimated efficiently using a small number
of samples, that does not need to grow with M [30], [32].

Proof: The proof is given in Appendix E.
As a consequence of this theorem, under Assumptions 1 and
3, the uplink SEs of UE 1 and UE 2 increase without bound as
M → ∞ even if the BS has only knowledge of the diagonal
elements of the covariance matrices. A similar result can be
proved for the downlink, using the methodology adopted in
Appendix D for proving Theorem 2. The details are omitted
for space limitations.

Lemma 2. Based on the observation [Yp φ∗ ]i , the BS can
compute the EW-MMSE estimate of the ith element of hk as
[Rk ]ii
1
[ĥk ]i = p
tr
[R
]
+
[R2 ]ii +
ρ
1 ii

1
ρtr

[Yp φ∗ ]i .

(23)

We may write ĥk in Lemma 2 in matrix form as
1
ĥk = p Dk Λ−1 Yp φ∗
ρtr

(24)

III. A SYMPTOTIC S PECTRAL E FFICIENCY IN M ULTICELL
M ASSIVE MIMO
We will now generalize the results of Section II to a Massive
MIMO network with L cells, each comprising a BS with M
antennas and K UEs. There are τp = K pilots and the kth
6 This easily follows by observing that tr(R ) = tr(D ) and also that
k
k
[Dk ]ii = [Rk ]ii ≤ kRk k2 since Rk is Hermitian.

7

UE in each cell uses the same pilot. Following the notation
from [5], the received signal yj ∈ CM at BS j is
yj =

L X
K
X
√

ρhjli xli + nj

(28)

l=1 i=1

where ρ is the normalized transmit power, xli is the unitpower signal from UE i in cell l, hjli ∼ NC (0, Rjli ) is the
channel from this UE to BS j, Rjli ∈ CM ×M is the channel
covariance matrix, and nj ∼ NC (0, IM ) is the independent
receiver noise at BS j. Using a total uplink pilot power of ρtr
per UE and standard MMSE estimation techniques [5], BS j
obtains the estimate of hjli as

X
L
1
0
p
n
ĥjli = Rjli Q−1
h
+
ji ∼ NC (0, Φjli )
jl i
ji
ρtr
l0 =1
(29)
PL
where nji ∼ NC (0, IM ) is noise, Qji = l0 =1 Rjl0 i + ρ1tr IM ,
and Φjli = Rjli Q−1
ji Rjli . The estimation error h̃jli = hjli −
ĥjli ∼ NC (0, Rjli − Φjli ) is independent of ĥjli . However,
the estimates ĥj1i , . . . , ĥjLi of the UEs with the same pilot
are correlated as E{ĥjni ĥHjmi } = Rjni Q−1
ji Rjmi .
A. Uplink Data Transmission
We denote by vjk ∈ CM the receive combining vector
associated with UE k in cell j. Using the same technique
as in [5], [26], the uplink ergodic capacity is lower bounded
by




τp
ul
SEul
=
1
−
E log2 1 + γjk
[bit/s/Hz]
(30)
jk
τc
with the instantaneous effective SINR
H
|vjk
ĥjjk |2

ul
γjk
=

(
E

P
(l,i)6=(j,k)

H
H
|vjk
hjli |2 + |vjk
h̃jjk |2 +

H
|vjk
ĥjjk |2

=
vjk

P

ĥjli ĥjli + Zj
H

(l,i)6=(j,k)

)
ĥ(j)
(31)

!
H

H v
vjk
jk
ρul

vjk

where E{·|ĥ(j) } denotes the conditional expectation given
the MMSE channel estimates available at BS j and Zj =
PL PK
1
l=1
i=1 (Rjli − Φjli ) + ρul IM . As shown in [8], [9], the
instantaneous effective SINR in (31) for UE k in cell j is
maximized by
!−1
L X
K
X
H
vjk =
ĥjli ĥjli + Zj ĥjjk .
(32)
l=1 i=1

We refer to this “optimal” receive combining scheme as multicell MMSE (M-MMSE) combining. The “multicell” notion
is used to differentiate it from the single-cell MMSE (SMMSE) combining scheme [5]–[7], which is widely used in
the literature and defined as
!−1
K
X
H
v̄jk =
ĥjji ĥjji + Z̄j
ĥjjk
(33)
i=1

PK
PL
PK
with Z̄j = i=1 Rjji −Φjji + l=1,l6=j i=1 Rjli + ρ1ul IM .
The main difference from (32) is that only channel estimates
in the own cell are computed in S-MMSE, while ĥjli ĥHjli −
Φjli is replaced with its average (i.e., zero) for all l 6= j.
The computational complexity of S-MMSE is thus slightly
lower than with M-MMSE (see [9] for a detailed discussion).
However, both schemes only utilizes channel estimates that
can be computed locally at the BS and the pilot overhead is
identical since the same pilots are used to estimate both intracell and inter-cell channels. The S-MMSE scheme coincides
with M-MMSE when there is only one isolated cell, but it
is generally different and does not suppress interference from
interfering UEs in other cells. Plugging (32) into (31) yields
!−1
X
ul
γjk
= ĥHjjk
ĥjli ĥHjli + Zj
ĥjjk .
(34)
(l,i)6=(j,k)
ul
We want to analyze γjk
when M → ∞. To this end, we make
the following two assumptions.
1
M tr(Rjli )

Assumption 4. As M → ∞ ∀j, l, i, lim inf M
and lim supM kRjli k2 < ∞.

>0

Assumption 5. For any UE k in cell j with λjk =
[λj1k , . . . , λjLk ]T ∈ RL and l0 = 1, . . . , L
L
1 X
lim inf
inf
λjlk Rjlk
M
{λjk : λjl0 k =1} M
l=1

2

> 0.

(35)

F

The following is the fourth main result of the paper:
Theorem 4. If M-MMSE combining is used, then under
ul
Assumptions 4 and 5 the SINR γjk
increases a.s. unboundedly
ul
as M → ∞. Hence, SEjk increases unboundedly as M → ∞.
Proof: The proof is given in Appendix F.
This theorem proves the remarkable result that, under Assumptions 4 and 5, the uplink SE of a multicell Massive
MIMO network increases without bound as M → ∞, despite
pilot contamination. This is in sharp contrast to the finite
limit in case of MR combining [1] or any other singlecell combining scheme [5]–[7] and it is due to the fact that
M-MMSE rejects the coherent interference caused by pilot
contamination when Assumptions 4 and 5 hold. Note that these
are the natural multicell generalizations of Assumptions 1
and 2, respectively. In particular, the condition (35) says
that the covariance matrices {Rjlk : l = 1, . . . , L} of the
channels from the pilot-sharing UEs to BS j are asymptotically
linearly independent, which implies the same condition for
the estimated channels {ĥjlk : l = 1, . . . , L}. This condition
is used in Appendix F to prove Theorem 4 in a fairly
simple way. However, we stress that Theorem 4 is valid also
in a more general setting in which ĥjjk is asymptotically
linearly independent of the estimates of all pilot-interfering
UEs’ channels, but some of the interfering channel estimates
can be written as linear combinations of other interfering
channels. Let Sjk ⊆ {ĥjlk : ∀l 6= j} denote a subset of
the estimated interfering channels that form a basis for all
interfering channels. Under these circumstances, we only need
to take the estimates in Sjk into account in the computation of

8

the combining vector vjk in (32) and the same result follows.
To gain further insights into this, we notice (as done for the
two-user case in Section II-D) that one can find a receive
combining vector that is orthogonal to the subspace spanned
by Sjk . This scheme exhibits an unbounded SE when M → ∞
as it rejects the interference from all pilot-contaminating UEs
(not only from those in Sjk ), while retaining an array gain
that grows with M . We call this scheme
−1multicell ZF (M-ZF)
and define it as vjk = Ĥjk ĤHjk Ĥjk
e1 , where e1 is the
first column of I|Sjk |+1 (with |Sjk | being the cardinality of
Sjk ) and Ĥjk ∈ CN ×(|Sjk |+1) is the matrix with ĥjjk in the
first column and the channel estimates in Sjk in the remaining
columns. Since M-MMSE combining is the optimal scheme, it
has to exhibit an unbounded SE if this is the case with M-ZF.
B. Downlink Data Transmission
During downlink
p PKdata transmission, the BS in cell l transmits xl = ρdl l=1 wli ςli , where ςli ∼ NC (0, 1) is the data
signal intended for UE i in the cell and ρdl is the normalized
transmit power. This signal is assigned to a transmit precoding
vector wli ∈ CM , which satisfies E{kwli k2 } = 1, such that
E{kwli ςli k2 } = ρdl is the transmit power allocated to this UE.
Using the same technique as in [5], [26], the downlink ergodic
channel capacity of UE k in cell j can be lower bounded by
τp
dl
SEdl
jk = 1 − τc log2 (1 + γjk ) [bit/s/Hz] with
dl
γjk
=

|E{hHjjk wjk }|2
L P
K
P
l=1 i=1

E{|hljk wli
H

|2 }

− |E{hjjk wjk
H

. (36)
}|2

+

1
ρdl

ul
in (31), which only depends on the own combining
Unlike γjk
dl
vector vjk , γjk
depends on all precoding vectors {wli }.
The precoding should ideally be selected jointly across the
cells, which makes precoding optimization difficult in practice.
Motivated by the uplink-downlink duality [9], it is reasonable
to select {wli } based on the M-MMSE combining vectors
{vjk } given by (32). This leads to M-MMSE precoding
!−1
L X
K
X
p
p
H
wjk = ϑjk vjk = ϑjk
ĥjli ĥjli + Zj
ĥjjk
l=1 i=1

linearly independent from the channel estimates of pilotcontaminating UEs in other cells. Section IV demonstrates
numerically that the DL SE grows without bound as M → ∞.
C. Approximate M-MMSE Combining and Precoding
In Section II-E, we have shown that the SE with the approximate M-MMSE scheme (that only utilizes the diagonals
of the covariance matrices) grows unbounded as M → ∞,
in a two-user scenario. This result can be generalized to a
multicell Massive MIMO network. Due to space limitations,
we concentrate on the uplink. In particular, we assume that the
signal of UE k in cell j is detected by using the approximate
M-MMSE combining vector
!−1
L X
K
X
H
vjk =
ĥjli ĥjli + Sj
ĥjjk
(38)
l=1 i=1


1
D
where Sj = l=1 i=1 Djli − Djli Λ−1
jli + ρul IM is
ji
a diagonal matrix and the EW-MMSE estimate of hjli is
!
L
X
1
1
ĥjli = p Djli Λ−1
hjl0 i + p nji
(39)
ji
ρtr
ρtr
l0 =1
PL PK 

where nji ∼ NC (0, IM ) is noise and Djli ∈ RM ×M and
Λji ∈ RM ×M are diagonal with elements {[Rjli ]nn : n =
PL
1, . . . , M } and { l0 =1 [Rjl0 i ]nn + ρ1tr : n = 1, . . . , M }, respectively. Since Djli and Λji are diagonal, the computational
complexity of EW-MMSE estimation is substantially lower
than for MMSE estimation; see [30] for details. Notice that
the combining scheme in (38) can be applied without knowing
the full channel covariance matrices, as it depends only on the
diagonal elements of {Rjli : l = 1, . . . , L}. This is because
the elements of ĥjli are estimated separately, without exploiting the spatial channel correlation. By using the use-and-thenforget SE bound [26], the uplink ergodic capacity of UE k in
τp
ul
cell j can be lower bounded by SEul
jk = (1 − τc ) log2 (1 + γ jk )
[bit/s/Hz] with
γ ul
=
jk
H
|E{vjk
hjjk }|2

(37)

p
with the normalization factor ϑjk = ( E {kvjk k2 })−1 . This
is the fifth main result of the paper:
Theorem 5. If M-MMSE precoding is used, then under
dl
Assumptions 4 and 5 the SINR γjk
grows unboundedly as
dl
M → ∞. Hence, SEjk grows unboundedly as M → ∞.
Proof: Despite being much more involved, the proof
basically unfolds from the same arguments used for proving
Theorem 2 and by exploiting the results of Appendix F for
Theorem 4.
This theorem shows that an asymptotically unbounded
downlink SE is achieved by all UEs in the network, despite the
suboptimal assumptions of M-MMSE precoding, equal power
allocation, and no estimation of the instantaneous realization of
the precoded channels. The only important requirement is that
the channel estimates to the desired UEs are asymptotically

L P
K
P
l=1 i=1

E{|vjk hjli
H

|2 }

− |E{vjk hjjk
H

.
}|2

+

2
1
E{kvjk k }
ρul

(40)

We now want to understand how γ ul
behaves when M → ∞
jk
under the following assumption, which is the extension of
Assumption 5 to the case where only the diagonals of covariance matrices are used for channel estimation and receive
combining:
Assumption 6. For any UE k in cell j with λjk =
[λj1k , . . . , λjLk ]T ∈ RL and l0 = 1, . . . , L
L
1 X
λjlk Djlk
lim inf
inf
M
{λjk : λjl0 k =1} M
l=1

2

> 0.
F

The following is the last main result of the paper:

(41)

9

2

Theorem 6. If approximate M-MMSE combining is used,
then under Assumptions 4 and 6 the SINR γ ul
increases
jk

Proof: The proof is omitted for space limitations, but
follows along the lines of Theorem 3.
This theorem shows that it is sufficient that the diagonals of the covariance matrices are asymptotically linearly
independent and known at the BS to achieve an unbounded
uplink SE (and thus an unlimited capacity). This condition
is generally satisfied since small random variations in the
elements of the covariance matrices are sufficient to achieve
asymptotic linear independence, as illustrated by Example 2.
An unbounded SE can be also proved in the downlink using
similar methods (omitted for space reasons). This will be
demonstrated numerically in the next section.

Normalized value

unboundedly as M → ∞. Hence, SEul
jk increases unboundedly
as M → ∞.

10

0

10

−2

10

One−ring model (∆=15◦)
Exponential correlation model (r = 0.5)
Uncorrelated, fading variations (2 dB)

−4

10

0

20

40
60
Eigenvalue (decreasing order)

where fm ∼ N (0, σ 2 ) and σ denotes the standard deviation.
Fig. 2 shows the eigenvalue distribution with the three
covariance models above, for M = 100 antennas, uniformly
distributed AoAs θ in [−π, +π), β = 1, ∆ = 15◦ , r = 0.5,
and σ = 2. All three models create eigenvalue variations, but
there are substantial differences. The one-ring model provides
rank-deficient covariance matrices, where a large fraction of
the eigenvalues is zero (this fraction is computed in [11]).
In contrast, the other two models provide full-rank covariance matrices with more modest eigenvalue variations. In the
remainder, we consider the latter two models to emphasize
that our main results only require linear independence between
the covariance matrices, not rank-deficiency (which in special
cases give rise to orthogonal covariance supports [10]).

100

Fig. 2: Average eigenvalue distribution with M = 100 and for
three different channel covariance models, whereof one gives a rankdeficient covariance matrix and the others have full rank.

BS
UE 1
UE 2

300 m

IV. N UMERICAL R ESULTS
The simulation results can be reproduced using the code
at https://github.com/emilbjornson/unlimited-capacity. In this
section, we will show numerically that an unlimited SE is
achievable under pilot contamination. To this end, we first
evaluate three ways to generate the channel covariance matrices and the resulting spatial correlation. For an arbitrary user,
the covariance matrix R can be modeled by:
1) One-ring model for a ULA with half-wavelength antenna
spacing and average large-scale fading β [11]. For an angleof-arrival (AoA) θ and many scatterers that are uniformly
distributed in the angular interval
∆, θ + ∆], the (m, n)th
R ∆[θ −πı(n−m)
β
sin(θ+δ)
e
dδ.
element of R is [R]m,n = 2∆
−∆
2) Exponential correlation model for a ULA with correlation
factor r ∈ [0, 1] between adjacent antennas, average largescale fading β, and AoA θ [34], which leads to [R]m,n =
βr|n−m| eı(n−m)θ .
3) Uncorrelated Rayleigh fading with average large-scale
fading β and independent log-normal large-scale fading variations over the array, which gives (similar to the perturbations
considered in Example 2)


R = βdiag 10f1 /10 , . . . , 10fM /10
(42)

80

Cell edge

300 m

Fig. 3: Multicell setup with two UEs per cell in the shaded cell-edge
area. All UEs have similar AoAs to all BSs, which typically leads to
similar covariance matrices and thus high pilot contamination.

A. Uplink
We consider the challenging symmetric setup in Fig. 3 with
L = 4 cells, K = 2 UEs per cell, pilots of length τp = K,
and coherence blocks of τc = 200 channel uses. The BSs are
located at the four corners of the area and the UEs are all
located at the cell edges and have similar but non-identical
AoAs and distances to the BSs. Thus, the pilot contamination
is very large in this setup. Note that the star-marked UEs share
a pilot, while the plus-marked UEs share another pilot.
The asymptotic behavior of the uplink SE is shown in Fig. 4
using the exponential correlation model (r = 0.5), with MMMSE, S-MMSE, MR, and M-ZF, where the latter cancels
interference between all UEs. The SE per UE is shown as
a function of the number of antennas, in logarithmic scale.
The average SNR observed at a BS antenna is set equal in the
pilot and data transmission: ρul tr(Rjli )/M = ρtr tr(Rjli )/M .
It is −6.0 dB for the intracell UEs and between −6.3 dB and
−11.5 dB for the interfering UEs in other cells. Fig. 4 shows
that S-MMSE provides slightly higher SE than MR, but both
converge to asymptotic limits of around 1 bit/s/Hz as M grows.
In contrast, M-MMSE provides an SE that grows without

10

Spectral efficiency [bit/s/Hz/user]

1.5

4

M−MMSE
S−MMSE
MRC
M−ZF
Time splitting

1

0.5

0
1
10

2

10
Number of antennas (M)

Fig. 4: Uplink SE as a function of M , for covariance matrices based
on the exponential correlation model (r = 0.5).

bound. The instantaneous effective SINR grows linearly with
M , which is in line with Theorem 4, as seen from the fact that
the SE grows linearly when the horizontal scale is logarithmic.
M-ZF performs poorly because the channel estimates are
so similar that full interference suppression removes most
of the desired signal. In contrast, M-MMSE finds a nontrivial tradeoff between interference suppression and coherent
combining of the desired signal, leading to superior SE. The
reference curve “time splitting” considers the case when the 4
cells are active in different coherence blocks, to remove pilot
contamination. MMSE combining is used and the SE grows
without bound, but at a slower pace than with M-MMSE, due
to the extra pre-log factor of 1/4. Hence, even for a small
system with L = 4, it is inefficient to avoid pilot contamination
by time splitting.
Next, we consider the uncorrelated Rayleigh fading model
in (42) with independent large-scale fading variations over the
array. The uplink SE with M = 200 antennas and varying
standard deviation σ from 0 to 5 is shown in Fig. 5(a).
M-MMSE provides no benefit over S-MMSE or MR in the
special case of σ = 0, where the covariance matrices are
linearly dependent (i.e., scaled identity matrices). This is a
special case that has received massive attention in academic
literature, mainly because it simplifies the mathematical analysis. However, M-MMSE provides substantial performance
gains over S-MMSE and MR as soon as we depart from
the scaled-identity model by adding small variations in the
large-scale fading over the array, which make the covariance
matrices linearly independent. This is in line with what we
demonstrated in Example 2. As the variations increase, the SE
with M-ZF improves particularly fast and approaches the SE
with M-MMSE. M-ZF will never be the better scheme since
M-MMSE is optimal. The motivation behind this simulation
is the measurement results reported in [29], which show largescale variations of around 4 dB over a massive MIMO array—
this corresponds to σ ≈ 4 in our setup.
Fig. 5(b) shows the received power (normalized by the noise
power) after receive combining for an arbitrary UE when σ =
4. It is divided into the desired signal power, the interference
from UEs using the same pilot, and the interference from UEs
using a different pilot. The figure shows that MR and S-MMSE
suffer from strong interference from the UEs that use the same

M-MMSE
M-ZF
S-MMSE
MR

3.5
3
2.5
2
1.5
1
0.5
0

3

10

0

1
2
3
4
Standard deviation of fading variations over the array

5

(a) Uplink SE
Received power over noise power [dB]

Spectral efficiency [bit/s/Hz/user]

2

25
MR
S−MMSE
M−MMSE
M−ZF

20
15
10
5
0

Desired signal

Interf: Same pilot

Interf: Diff. pilot

(b) Received signal power after receive combining
Fig. 5: Uplink with covariance matrices modeled by (42) for M =
200 and K = 2. (a) The SE as a function of the standard deviation
σ of the large-scale fading variations. (b) The received power after
receive combining with σ = 4 is separated into desired signal power
and interference from UEs with the same or different pilot than the
desired UE.

pilot, since these schemes are unable to mitigate the coherent
interference caused by pilot contamination. In contrast, MMMSE and M-ZF mitigate all types of interference and receive
roughly the same amount of interference from UEs with the
same or different pilots. Note that the price to pay for the
interference rejection is a reduction in desired signal power
when using M-MMSE and M-ZF.
B. Downlink
The setup in Fig. 3 is also used in the downlink wherein
we set ρdl = ρul to get the same SNRs as in the uplink. We consider a setup with both spatial channel correlation and large-scale fading variations over the array, such
that the EW-MMSE estimator is suboptimal but Assumption 6 is satisfied. More precisely, we consider a combination
of the exponential correlation model and (42): [R]m,n =
βr|n−m| eı(n−m)θ 10(fm +fn )/20 , where θ is the AoA, r = 0.5
is used as correlation factor, and f1 , . . . , fM ∼ N (0, σ 2 ) give
independent large-scale fading variations over the array with
σ = 4.
The downlink SE is shown in Fig. 6 as a function of
M , where Fig. 6(a) shows results with the MMSE estimator
that uses the full channel covariance matrices and Fig. 6(b)
shows results with the EW-MMSE estimator that only uses
the diagonals of the covariance matrices. When using the

11

4

5
M−MMSE
M−ZF
S−MMSE
MRT

Spectral efficiency [bit/s/Hz/user]

Spectral efficiency [bit/s/Hz/user]

5

3
2
1
0
1
10

2

10
Number of antennas (M)

4
3
2
1
0
1
10

3

10

(a) MMSE estimation
Approximate M−MMSE
M−ZF
Approximate S−MMSE
MRT

Spectral efficiency [bit/s/Hz/user]

Spectral efficiency [bit/s/Hz/user]

3

10

5

3
2
1
0
1
10

2

10
Number of antennas (M)

(a) MMSE estimation

5
4

M−MMSE
M−ZF
S−MMSE
MRT

2

10
Number of antennas (M)

3

10

4

Approximate M−MMSE
M−ZF
Approximate S−MMSE
MRT

3
2
1
0
1
10

2

10
Number of antennas (M)

3

10

(b) EW-MMSE estimation

(b) EW-MMSE estimation

Fig. 6: Downlink SE as a function of M for K = 2, when using either
the MMSE estimator (with full covariance knowledge) or the EWMMSE estimator (with known diagonals of the covariance matrices).
The exponential correlation model with r = 0.5 is used, but with
large-scale fading variations over the array with σ = 4.

Fig. 7: Downlink SE as a function of M for K = 10 UEs that are
uniformly distributed in the shaded cell edge area. The setup and
covariance model are otherwise the same as in Fig. 6.

EW-MMSE estimator, we consider the approximate M-MMSE
scheme in (38) and a corresponding approximation of SMMSE, while M-ZF and MR are as before. The results in
Fig. 6(a) with the MMSE estimator are similar to the uplink in
Fig. 5(a): M-MMSE and M-ZF provide SEs that grow without
bound, while the SEs with S-MMSE and MR converge to
finite limits. In contrast to the uplink, M-MMSE and M-ZF
precoding are both suboptimal in the downlink, but they can
be shown to be asymptotically equal.7 Interestingly, the same
behaviors are observed in Fig. 6(b) when using the EW-MMSE
estimator, which is a suboptimal estimator that neglects the offdiagonal elements of the covariance matrices. This result is in
line with Theorem 6. There is a small SE loss (2%–4% for MMMSE) compared to Fig. 6(a), but this is a minor price to pay
for the greatly simplified acquisition of covariance information
(estimating the entire diagonal is as simple as estimating a
single parameter [30], [32]).
We now increase the number of UEs per cell to K = 10,
which leads to more interference but the same pilot contamination per UE. The UEs are uniformly and independently
distributed in the cell-edge area, which is the shaded area in
7 For M-MMSE precoding in (37), Z has bounded spectral norm while
j
P P
H
l
i ĥjli ĥjli has LK eigenvalues that grow unboundedly as M → ∞.
As the impact of Zj vanishes, the approach in [28] can be used to prove that
M-MMSE approaches M-ZF asymptotically.

Fig. 3. The channel model is the same as in the previous figure.
The downlink SE per UE is shown in Fig. 7 when using either
MMSE or EW-MMSE estimation. The results resemble the
ones for K = 2, but the curves are basically shifted to the
right due to the additional interference. M-MMSE and M-ZF
provide SEs that grow without bound, while the SE with SMMSE and MR saturate, but more antennas are needed before
reaching saturation.
V. C ONCLUSIONS AND P RACTICAL I MPLICATIONS
We proved that the capacity of Massive MIMO systems
increases without bound as M → ∞ in the presence of pilot
contamination, despite the previous results that pointed toward
the existence of a finite limit. This was achieved by showing
that the conventional lower bounds on the capacity increase
without bound when using M-MMSE precoding/combining.
These schemes exploit the fact that the MMSE channel estimates of UEs that use the same pilot are linearly independent,
due to their generally linearly independent covariance matrices. For our results to hold, the covariance matrices can have
full rank and minor eigenvalue variations are sufficient. There
are special cases where the channel covariance matrices are
linearly dependent, but these are not robust to minor perturbations of the covariance matrices. Hence, they are anomalies
that will never appear in practice or be drawn from a random
distribution, although they have frequently been studied in
the academic literature. Since the SE of MR (also known as

12

conjugate beamforming or matched filtering) generally has a
finite limit, we conclude that this scheme is not asymptotically
optimal in Massive MIMO. Note that our results do not imply
that the pilot contamination effect disappears; there is still a
performance loss caused by estimation errors and interference
rejection, but there is no fundamental capacity limit.
Most of our results assume that the full covariance matrices of the channels are known, but this is not a critical
requirement. Theorems 3 and 6 proved that it is sufficient
that the diagonals of the covariance matrices are known and
linearly independent between pilot-sharing UEs; a condition
that has been shown to hold for practical channels by the
measurements in [29]. Such statistical information can be
accurately estimated from only some tens of channel observations [32], whereof some contain the desired signal plus
interference/noise and some contain only interference/noise.
The purpose of analyzing the asymptotic capacity when
M → ∞ is not that we advocate the deployment of BSs
with a nearly infinite number of antennas—that is physically
impossible in a finite-sized world and the conventional channel
models will eventually break down since more power is
received than was transmitted. The importance of asymptotics
is instead what it tells us about practical networks with finite
numbers of antennas. For example, consider a network with
any finite number of UEs that each have a finite-valued data
rate requirement. Our main results imply that we can always
satisfy these requirements by deploying sufficiently many
antennas, even in the presence of pilot contamination. In fact,
it is enough to have two channel uses per coherence block
(one for pilot, one for data) to deliver any capacity value to
any finite number of UEs. The linear M-MMSE scheme is
sufficient to achieve this in practice and interference can be
treated as noise in the receivers, because the capacity lower
bounds that we considered rely on such simplifications.

A PPENDIX B – P ROOF OF T HEOREM 1
By applying Lemma 5, we may rewrite γ1ul in (7) as
γ1ul

=M

Lemma 3 (Theorem 3.4, Corollary 3.4 [35]). Let A ∈ CM ×M
1
IM ). Assume that A has uniformly
and x, y ∼ NC (0, M
bounded spectral norm and that x and y are mutually in1
dependent and independent of A. Then, xH Ax  M
tr(A),
1
H
H
p
−p/2
x Ay  0 and E{|x Ax − M tr(A)| } = O(M
).
Lemma 4 ( [36]). For any positive semi-definite M × M
1
matrices A and B, it holds that M
tr (AB) ≤ kABk2 ≤
kAk2 kBk2 , tr (AB) ≤ kAk2 tr (B) and tr (I + A)−1 B ≥
1
1+kAk2 tr(B).
Lemma 5 (Matrix inversion lemma). Let A ∈ CM ×M be
a Hermitian invertible matrix, then for any vector x ∈ CM
and any scalar ρ ∈ C such that A + ρxxH is invertible
xH A−1
H −1
xH (A + ρxxH )−1 = 1+ρx
=
H A−1 x and (A + ρxx )
−1

H

−1

xx A
A−1 − ρA
1+ρxH A−1 x .
Let U, C, V be matrices of compatible sizes,
−1
then if C is invertible
(A + UCV)
= A−1 −

−1
−1
−1
−1
−1
A U C + VA U
VA .

1 H −1
ĥ2
M ĥ1 Z
1
M

+

2

!

1 H −1
ĥ2
M ĥ2 Z

(43)

by also multiplying and dividing each term by M . Under
Assumption 1 and using Lemma 3 we have, as M → ∞, that8
1 H −1
1
ĥ Z ĥ1 
tr(Φ1 Z−1 ) , β11
(44)
M 1
M
1 H −1
1
ĥ Z ĥ2 
tr(Φ2 Z−1 ) , β22
(45)
M 2
M
1
1 H −1
ĥ1 Z ĥ2 
tr(Υ12 Z−1 ) , β12 .
(46)
M
M
Note that β11 , β22 , and β12 are non-negative real-valued
scalars, since the trace of a product of positive semi-definite
matrices is always non-negative. Using this notation, it follows
from Assumption 1 that9 lim inf M β22 > 0 and we obtain
β2
γ1ul
 δ1 , β11 − 12 .
M
β22

(47)

To proceed, notice that Assumption 2 implies the following
result, as proved in Appendix C.
Corollary 1. If Assumption 2 holds, then for λ = [λ1 , λ2 ]T ∈
R2 and i = 1, 2,
lim inf inf
M
{λ: λi =1}


1  −1
tr Q
λ1 R1 + λ2 R2 Z−1 λ1 R1 + λ2 R2 > 0.
M
(48)
By expanding the condition in Corollary 1 for i = 1, we
have that

lim inf inf β11 + λ22 β22 + 2λ2 β12 > 0.
(49)
M

A PPENDIX A – U SEFUL R ESULTS

1 H −1
ĥ Z ĥ1 −
M 1

λ2

By the definition of the lim inf M operator, lim inf M β22 >
0 holds if and only if every convergent subsequence has a
non-zero limit, i.e., limM β22 > 0. This ensures that, for an
arbitrary convergent subsequence,

β2
inf β11 + λ22 β22 + 2λ2 β12 = β11 − 12 = δ1
λ2
β22

(50)

where the infimum is attained by λ2 = β12 /β22 . Substituting
(50) into (49), implies that lim inf M δ1 > 0. Therefore, we
have that γ1ul grows a.s. unboundedly and, thus, the first part
of the theorem follows.
Since γ1ul grows a.s. unboundedly and the logarithm is a
strictly increasing function, it follows that log2 (1 + γ1ul ) also
grows a.s. without bound. Moreover, since the almost sure
divergence of a sequence of non-negative random variables
implies the divergenceof its expectedvalue, it follows that also
ul
SEul
grows without bound.
1 = (1 − τp /τc )E log2 1 + γ1
8 Under Assumption 1, Q−1 R Z−1 R has uniformly bounded spectral
i
k
norm, which can be easily proved using Lemma 4.
9 This can be proved by similar arguments as in Appendix C, since
tr(A2 ) ≥ (tr(A))2 /rank(A) if A is Hermitian and A 6= 0.

13

A PPENDIX C – P ROOF OF C OROLLARY 1 IN A PPENDIX B
Consider i = 1 and notice that the argument on the left-hand
side of (48) is lower bounded as
1
M kR1

( ρ1tr

+ λ2 R2 k2F
P2
+ kR1 + R2 k2 )( ρ1ul + k k=1 (Rk − Φk )k2 )

(51)

by applying Lemma 4 twice. The denominator of (51) is
bounded from above due to Assumption 1 and independent
of λ2 . This proves that Assumption 2 is sufficient for (48) to
hold for i = 1. The result for i = 2 follows by interchanging
the indices in the proof.
A PPENDIX D – P ROOF OF T HEOREM 2
We begin by plugging (13) into (12) to obtain
γ1dl =

|E {hH1 v1 } |2
ϑ2
H
H
2
ϑ1 E {|h1 v2 | } + V{h1 v1 } +

1
ρdl ϑ1

.

(52)

We need to characterize all the terms in(52) and begin with
E {hH1 v1 }. Notice that E {hH1 v1 } = E ĥH1 v1 since v1 is
independent of the zero-mean error h̃1 . Then, we can express
ĥH1 v1 as

−1
ĥH1 ĥ2 ĥH2 + Z
ĥ1
γ1ul
ĥH1 v1 =
(53)
=

−1
1 + γ1ul
1 + ĥH1 ĥ2 ĥH2 + Z
ĥ1
by first applying Lemma 5 and then identifying γ1ul in (7) in the
γ ul
numerator and denominator. Theorem 1 proves that M1  δ1
and applying this result to (53) yields ĥH1 v1  1. By the
dominated convergence theorem and the continuous mapping
theorem [35], we then have that |E{hH1 v1 }|2  1.
2
1k }
Consider now the noise term ρdl1ϑ1 = E{kv
where ϑ1 =
ρdl

(E kv1 k2 )−1 . By applying Lemma 5 twice, we may rewrite
kv1 k2 as

−2
ĥH1 ĥ2 ĥH2 + Z
ĥ1
kv1 k2 =

2
1 + γ1ul

−2
1 H
H
ĥ
ĥ
ĥ
+
Z
ĥ1
2
1
2
1 M
=
.
(54)

2
1
M
+ 1 γ ul
M

M

1

Let Re(·) denote the real-valued part of a scalar. The numerator in (54) can be expressed as
1 H −2
ĥ Z ĥ1 −
M 1
+

1 H −2
1 H −1
Re( M
ĥ1 Z ĥ2 M
ĥ2 Z ĥ1 )
2
1
1 H −1
ĥ2
M + M ĥ2 Z
1 H −2
1 H −1
ĥ2 | M ĥ2 Z ĥ1 |2
M ĥ2 Z
2
1
1 H −1
ĥ2
M + M ĥ2 Z

(55)

by applying again Lemma 5 twice. Under Assumption 1 and
by applying Lemma 3,
1 H −2
1
0
ĥ Z ĥ1 
tr(Φ1 Z−2 ) , β11
(56)
M 1
M
1 H −2
1
0
ĥ Z ĥ2 
tr(Φ2 Z−2 ) , β22
(57)
M 2
M
1 H −2
1
0
ĥ Z ĥ2 
tr(Υ12 Z−2 ) , β12
(58)
M 1
M

0
0
0
where β11
, β22
, and β12
are non-negative real-valued scalars,
since the trace of a product of positive semi-definite matrices
is always non-negative. Therefore, we obtain
−2
0
1 H
β12 β12
β2 β0
0
ĥ1 ĥ2 ĥH2 + Z
ĥ1  β11
−2
+ 12 122 , δ10 .
M
β22
(β22 )
(59)
γ ul

Plugging (59) into (54) and using M1  δ1 yields M kv1 k2 
δ10
such that
δ12

E kv1 k2
1 δ10
1

.
(60)
=
ρdl ϑ1
ρdl
M ρdl δ12

Consider now the two terms V{hH1 v1 } and ϑϑ21 E |hH1 v2 |2 .
H
Similar to [5, Eq. (47)], we can upper bound
1 v1 } as
 H V{h
2
H
H
H
V{h1 v1 } ≤ 2E {|h1 v1 − E {h1 v1 }|} + E h̃1 v1 . Notice
that (by using E {hH1 v1 }  1 and the dominated convergence
theorem) E {|hH1 v1 − E {hH1 v1 }|}  0 and


2
= E v1H (R1 − Φ1 )v1
E h̃H1 v1
(a)

(b)
≤ kR1 − Φ1 k2 E kv1 k2  0
(61)

where (a) and (b) follow from Lemma 4 and E kv1 k2  0
0
1 δ1
(since, as shown above, kv1 k2  M
 0), respectively.
δ12
H
Therefore,
we
have
that
V{h
v
}

0.
Finally,
we consider
1
1
 H 2
ϑ2
E
|h
v
|
.
By
using
(45),
(46),
and
lim
inf
β11 > 0 (as
2
M
1
ϑ1
follows from Assumption 1), we have that

−1
hH1 ĥ1 ĥH1 + Z
ĥ2
(a)
hH1 v2 =

−1
1 + ĥH2 ĥ1 ĥH1 + Z
ĥ2
1 H −1
h Z ĥ2
(b) M 1

=

(c)



β12 −

β11 β12
β11

δ2

−1
1
hH
ĥ1 Z−1 ĥH
ĥ1 M
1 Z
2
1
1
H
−1 ĥ
1
M + M ĥ1 Z
1 ul
1
M + M γ2

−

1
M

=0

(62)

where (a) and (b) follow from Lemma 5 after identifying10
−1
ĥH2 ĥ1 ĥH1 + Z
ĥ2 as γ2ul (by also dividing and multiplying
by M ), and (c) follows by using (44), (46) and the fact that
β2
γ2ul
 δ2 , β22 − 21
M
β11

(63)

with lim inf M δ2 > 0 (which follows from the proof of
Theorem 1 by interchanging UE indices). By applying Lemma

δ0 δ2
3, this implies E |hH1 v2 |2  0. Observe now that ϑϑ12  δ12 δ20
1 2
where δ20 is obtained from δ10 by interchanging UE indices.
Since all the quantities in δ10 are uniformly bounded (due to
Assumption 1), lim inf M δ1 > 0 (as proved in Appendix B)
and lim inf M δ2 < ∞ (since from (63) δ2 < β22 and
lim inf M β22 < ∞ due to Assumption 1), we eventually have
that ϑϑ12 E |hH1 v2 |2  0.
Combining all the above results yields
γ1dl
δ2
 ρdl 10 .
M
δ1

(64)

10 The uplink SINR γ ul of UE 2 is obtained from (7) by interchanging UE
2
indices.

14

Since all the quantities in δ10 are uniformly bounded and
lim inf M δ1 > 0, it follows that γ1dl grows unboundedly as
M → ∞. This implies that also SEdl
1 grows unboundedly as
M → ∞, which can be proved by the same arguments as in
the last paragraph of Appendix B.

A PPENDIX E – P ROOF OF T HEOREM 3

Corollary 2. If Assumption 3 holds, then for λ = [λ1 , λ2 ]T ∈
R2 and i = 1, 2,
lim inf inf
M
{λ: λi =1}


1  −1
tr Λ QΛ−1 λ1 D1 + λ2 D2 S−1 λ1 D1 + λ2 D2 > 0.
M
(69)
As for µ̃ul
1 in (65), we have that
−1
1 H
1 ul
µ̃1 =
ĥ1 ĥ2 ĥH2 + S
h1
M
M
1 H −1
1 H −1
ĥ S ĥ2 M
ĥ2 S h1
1 H −1
 υ1
=
ĥ1 S h1 − M 11
1
H
−1 ĥ
M
2
M + M ĥ2 S
(70)

The EW-MMSE estimate ĥk and the estimation error h̃k =
hk − ĥk are random vectors distributed as ĥk ∼ NC (0, Σk )
and h̃k ∼ NC (0, Σ̃k ) with Σk = Dk Λ−1 QΛ−1 Dk and
Σ̃k = Rk − Dk Λ−1 Rk − Rk Λ−1 Dk − Σk . Unlike with
MMSE estimation, the vectors ĥk and h̃k are correlated with
E{ĥk h̃Hk } = E{ĥk (hk − ĥk )H } = Dk Λ−1 Rk − Σk . Hence,
v1 and h̃1 are also correlated. For later convenience, we also
notice that E{h1 ĥH1 } = R1 Λ−1 D1 , E{h1 ĥH2 } = R1 Λ−1 D2 ,
and E{ĥ2 ĥH1 } = D2 Λ−1 QΛ−1 D1 = Θ21 . We need to
characterize all the terms in (25) and begin with E {v1H h1 }.
By applying Lemma 5 and by dividing and multiplying by M ,
we can express v1H h1 as

since the diagonal structure of the matrices Λ, D1 , D2 , and
S implies that

−1
ĥ2 ĥH2 + S
h1
=

−1
1 H
+M
ĥ1 ĥ2 ĥH2 + S
ĥ1

1
1 H −1
ĥ2 S h1 
tr(R1 Λ−1 D2 S−1 )
M
M
M
1 X [R1 ]i,i [R2 ]i,i
=
= α12 .
M i=1 [S]i,i [Λ]i,i

v1H h1 =

1 H
M ĥ1
1
M



1 ul
M µ̃1
1
1 ul .
M + M µ1

(65)
ul
Notice that µul
1 has the same form as γ1 in (7), but with
{ĥk : k = 1, 2} now given by (24). Under Assumption 1 and
by Lemma 3,11
M
1
1 X [R1 ]2i,i
1 H −1
ĥ1 S ĥ1 
tr(Σ1 S−1 ) =
, α11
M
M
M i=1 [S]i,i [Λ]i,i
(66)
M
2
X
[R2 ]i,i
1 H −1
1
1
ĥ S ĥ2 
tr(Σ2 S−1 ) =
, α22
M 2
M
M i=1 [S]i,i [Λ]i,i
(67)
1 H −1
1
ĥ S ĥ2 
tr(Θ21 S−1 )
M 1
M
M
1 X [R1 ]i,i [R2 ]i,i
, α12 .
(68)
=
M i=1 [S]i,i [Λ]i,i

By applying the same line of reasoning as when analyzing γ1ul
in Appendix B and exploiting the fact that lim inf M α22 >
µul
0 (which follows from Assumption 1), we obtain M1 =
−1

α2
1 H
H
h1  υ1 , α11 − α12
. Note that
M ĥ1 ĥ2 ĥ2 + S
22
lim inf M υ1 > 0 under Assumption 3. This can be proved,
as done in Appendix B for δ1 , by expanding the condition
reported in the corollary below (the proof unfolds from the
same arguments as in Appendix C).

11 The expressions in (66)–(68) have been simplified by utilizing the fact
that Q and Λ have the same diagonal elements and Rk and Dk have the
same diagonal elements, for k = 1, 2.

1
1 H −1
ĥ S h1 
tr(R1 Λ−1 D1 S−1 )
M 1
M
M
1 X [R1 ]2i,i
=
= α11
M i=1 [S]i,i [Λ]i,i

(71)

(72)

Applying these results to (65) yields v1H h1  1 from which it
follows that |E{v1H h1 }|2  1.

Next, consider the noise term ρ1ul E ||v1 ||2 for which

−2
ĥH1 ĥ2 ĥH2 + S
ĥ1
kv1 k2 =
(76)

2
1 + µul
1

−2
1 H
H
ĥ
ĥ
ĥ
+
S
ĥ1
2
2
1 M 1
=
.
(77)

2
1
M
+ 1 µul
M

M

1

Under Assumption 1 and by Lemma 3,
M
1 H −2
1
1 X [R1 ]2i,i
0
ĥ1 S ĥ1 
tr(Σ1 S−2 ) =
, α11
M
M
M i=1 [S]2i,i [Λ]i,i
(78)
M
2
1 H −2
1
1 X [R2 ]i,i
0
ĥ2 S ĥ2 
tr(Σ2 S−2 ) =
, α22
M
M
M i=1 [S]2i,i [Λ]i,i
(79)
1 H −2
1
−2
ĥ S ĥ2 
tr(Θ21 S )
M 1
M
M
1 X [R1 ]i,i [R2 ]i,i
0
=
, α12
(80)
M i=1 [S]2i,i [Λ]i,i
0
0
0
, α22
, and α12
are non-negative real-valued scalars.
where α11
By applying Lemma 5 twice to the numerator in (77) and by
using the above results, we obtain
−2
0
1 H
α12 α12
α2 α0
0
ĥ1 ĥ2 ĥH2 + S
ĥ1  α11
−2
+ 12 22
, υ10 .
M
α22
(α22 )2
(81)

15



Aj,\k + Ĥjk,\j ĤHjk,\j

−1


−1
−1
−1
H
= A−1
ĤHjk,\j A−1
j,\k − Aj,\k Ĥjk,\j IL−1 + Ĥjk,\j Aj,\k Ĥjk,\j
j,\k .

ul
γjk
1 H
1 H
=
ĥ A−1 ĥjjk −
ĥ A−1 Ĥjk,\j
M
M jjk j,\k
M jjk j,\k



1 H
1
IL−1 +
Ĥ
A−1 Ĥjk,\j
M
M jk,\j j,\k

−1

1 H
Ĥ
A−1 ĥjjk .
M jk,\j j,\k

−1
−1
1 −1
1 XX
1 
Aj,\k =
ĥjli ĥHjli + Zj
Ĥj,\k ĤHj,\k + Zj
=
M
M
M
l i6=k

−1
1 −1
1 −1
1 H
1
1 H
=
Zj −
Zj Ĥj,\k
IL(K−1) +
Ĥj,\k Z−1
Ĥ Z−1
Ĥ
j,\k
j
M
M
M
M
M j,\k j

υ0

Plugging (81) into (77) yields M kv1 k2  υ12 such that
1

υ0
1
E ||v1 ||2  M1ρul υ12 .
ρul
1
As for V{v1H h1 }, it can be easily proved (using the above
results and Lemma 3), that V{v1H h1 }  0. Consider now the
interference term E |v1H h2 |2 . Using (67) and (68), we have
(by applying Lemma 5 and dividing and multiplying by M )
that

−1
1 H
H
ĥ
ĥ
ĥ
+
S
h2
2
2
M 1
v1H h2 =
1
1 ul
M + M µ1
1 H −1
h2
M ĥ1 S

=

α12 −



α12 α22
α22

υ1

−1
−1
1
ĥH
ĥH
ĥ2 M
h2
1 S
2 S
1
1
H S−1 ĥ
+
ĥ
2
M
M 2
1
1 ul
M + M µ1

−

1
M

=0

(82)

1 H −1
ĥ1 S h2  α12 and
where we have used the fact that M
1 H −1
h2  α22 . Applying Lemma 3 to (82), we obtain
Mĥ2 S
E |v1H h2 |2  0.
γ ul

Combining all the above results together yields M1 
υ2
ρul υ10 . Since all the components of υ10 in (81) are uniformly
1
bounded and lim inf M υ1 > 0 (under Assumption 3), it
follows that γ ul
grows unboundedly as M → ∞. Hence, SEul
1
1
also grows without bound.
A PPENDIX F – P ROOF OF T HEOREM 4
ul
We start by rewriting γjk
in (34) as

ul
γjk

H

= ĥjjk

XX
l

|

H

ĥjli ĥjli + Zj +

X

!−1
H

ĥjlk ĥjlk

ĥjjk

l6=j

i6=k

{z

Aj,\k

}

|

{z

}

Ĥjk,\j ĤH
jk,\j

(83)
P P

H
where Aj,\k =
l
i6=k ĥjli ĥjli + Zj is independent of {ĥjlk : l = 1, . . . L} and Ĥjk,\j =
[ĥj1k . . . ĥjj−1k ĥjj+1k . . . ĥjLk ] ∈ CM ×(L−1) collects all
vectors ĥjlk with l 6= j (i.e., the channels of UEs that cause
pilot contamination). By Lemma 5, we obtain (73) at the top of
the page. Plugging (73) into (83) and dividing both sides by M
leads to (74). By applying Lemma 5 once again, (75) follows
where Ĥj,\k ∈ CM ×L(K−1) denotes the matrix collecting all

(73)

(74)

(75)

vectors ĥjli with i 6= k, which is independent of ĥjlk for any
j and l. Therefore, it follows that the first term in (74) is such
that
(a) 1
1 H
ĥjjk A−1
ĥjjk 
ĥH Z−1 ĥjjk
j,\k
M
M jjk j
(b) 1

tr(Φjjk Z−1
(84)
j ) , βjk,jj
M
where (a) follows from Lemma 3 since ĥjjk and Ĥj,\k are
1 H
independent and thus M
ĥjjk Z−1 Ĥj,\k  0L(K−1) (remember that Ĥj,\k collects the L(K − 1) vectors {ĥjli } with
i 6= k), and (b) follows from Lemma 3 by recalling that
ĥjjk ∼ NC (0, Φjjk ) where the matrices Φjjk can be proved
(using Lemma 4) to have uniformly bounded spectral norm due
to Assumption 4. Using similar arguments, we have that the
1 H
1×(L−1)
ĥjjk A−1
lth element of the row vector M
j,\k Ĥjk,\j ∈ C
is such that


1 H
1 H −1

Ĥ
ĥjjk A−1
ĥjjk Zj ĥjlk
j,\k jk,\j
M
M
l
1
−1
tr(Rjlk Q−1
(85)

jk Rjjk Zj ) , βjk,lj
M
for l = 1, 2, . . . , L − 1. Furthermore, the (n, m)th element of
−1
1
H
M Ĥjk,\j Aj,\k Ĥjk,\j is
i
1 h H
1 H
Ĥjk,\j A−1
Ĥ

ĥjnk Z−1
jk,\j
j ĥjmk
j,\k
M
M
n,m
1
−1
tr(Rjmk Q−1
(86)

jk Rjnk Zj ) , βjk,mn .
M
For notational convenience, let us define bjk ∈ RL−1 and
Cjk ∈ R(L−1)×(L−1) with entries

H


1
−1/2
−1/2
bjk l = βjk,lj = vec √ Zj
Rjlk Qjk
M


1
−1/2
−1/2
vec √ Zj
Rjjk Qjk
(87)
M
and

H


1
−1/2
−1/2
Cjk l,n = βjk,ln = vec √ Zj
Rjlk Qjk
M


1
−1/2
−1/2
vec √ Zj
Rjnk Qjk
(88)
M
where we have used the fact that tr(AB) = vec(AH )H vec(B).
In Appendix G, it is shown that, under Assumption 5, the
following corollary holds.

16

Corollary 3. If Assumption 5 holds, then for any UE k in cell
j with λjk = [λj1k , . . . , λjLk ]T ∈ RL and l0 = 1, . . . , L
lim inf
M

inf

{λjk :λjl0 k =1}
L

L

l=1

l=1

X

X

1
−1
tr Q−1
λ
R
Z
λ
R
jlk
jlk
jlk
jlk
j
jk
M

!
>0

which implies that the vectors {ujlk : ∀l} are asymptotically linearly independent. Since any subset of a finite set
with linearly independent vectors is also linearly independent,
(94) ensures that {ujlk : ∀l 6= j} are also asymptotically
linearly independent. This proves that, under Assumption 5,
the Gramian matrix Cjk is invertible as M → ∞ and this
completes the proof.

(89)
and the matrix Cjk is invertible as M → ∞.

R EFERENCES

Since Cjk is invertible as M → ∞ under Assumption 5,
we have that

ul
γjk
M

in (74) is such that

ul
γjk
 δjk , βjj,jk − bHjk C−1
(90)
jk bjk .
M
Expanding condition (89) in Corollary 3 for l0 = j and using
the definitions of bjk and Cjk yield


T
T
lim inf inf βjj,jk + 2λjk bjk + λjk Cjk λjk > 0 (91)
M

λjk

with λjk = [λj1k , . . . , λj(j−1)k , λj(j+1)k , . . . , λjLk ]T ∈
RL−1 . The invertibility of Cjk as M → ∞ ensures that the
infimum exists for sufficiently large M and that it is given by

T
T
inf βjj,jk + 2λjk bjk + λjk Cjk λjk
λjk

= βjj,jk − bTjk C−1
jk bjk = δjk

(92)

where the infimum is attained by λjk = C−1
jk bjk . Substituting
ul
(92) into (91) implies that lim inf M δjk > 0. Therefore, γjk
ul
grows a.s. unboundedly and this implies that SEjk grows
unboundedly as M → ∞, which can be proved as done in
the last paragraph of Appendix B.
A PPENDIX G – P ROOF OF C OROLLARY 3 IN A PPENDIX F
The argument of the left-hand side of (89) can be lower
bounded by
PL
2
1
l=1 λjlk Rjlk F
M
 
 1
PL
PL
1
+
l=1 Rjlk − Φjlk
l=1 Rjlk 2
ρtr +
ρul
2
(93)
by applying Lemma 4 twice. Notice that the denominator is
bounded due to Assumption 5 and independent of {λljk }.
Therefore, if (35) holds, it follows from (93) that (89) also
holds.
We now exploit (89) to prove that Cjk is invertible for
sufficiently large M . To this end, observe that Cjk with entries
given by (88) is a Gramian matrix obtained as the inner
−1/2
−1/2 
Rjlk Qjk
:
products of the vectors {ujlk = vec √1M Zj
∀l 6= j}. Therefore, as M grows large the matrix Cjk is
invertible if and only if the vectors {ujlk : ∀l 6= j} are
asymptotically linearly independent. Notice that the condition
(89) in Corollary 3 for l0 = j can be rewritten in compact
form as

H
X
ujjk +
λjlk ujlk
lim inf
inf
l6=j
M
{λjk :λjjk =1}


X
ujjk +
λjlk ujlk > 0
(94)
l6=j

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Emil Björnson (S’07, M’12) received the M.S. degree in Engineering Mathematics from Lund University, Sweden, in 2007. He received the Ph.D. degree
in Telecommunications from KTH Royal Institute
of Technology, Sweden, in 2011. From 2012 to mid
2014, he was a joint postdoc at the Alcatel-Lucent
Chair on Flexible Radio, SUPELEC, France, and at
KTH. He joined Linkping University, Sweden, in
2014 and is currently Senior Lecturer and Docent
at the Division of Communication Systems.
He performs research on multi-antenna communications, Massive MIMO, radio resource allocation, energy-efficient communications, and network design. He is on the editorial board of the IEEE
T RANSACTIONS ON C OMMUNICATIONS and the IEEE T RANSACTIONS ON
G REEN C OMMUNICATIONS AND N ETWORKING. He is the first author of
the textbooks Massive MIMO Networks: Spectral, Energy, and Hardware
Efficiency (2017) and Optimal Resource Allocation in Coordinated Multi-Cell
Systems (2013). He is dedicated to reproducible research and has made a large
amount of simulation code publicly available.
Dr. Bjrnson has performed MIMO research for more than ten years and
has filed more than ten related patent applications. He received the 2016 Best
PhD Award from EURASIP, the 2015 Ingvar Carlsson Award, and the 2014
Outstanding Young Researcher Award from IEEE ComSoc EMEA. He has
co-authored papers that received best paper awards at WCSP 2017, IEEE
ICC 2015, IEEE WCNC 2014, IEEE SAM 2014, IEEE CAMSAP 2011, and
WCSP 2009.

Jakob Hoydis (S’08–M’12) received the diploma
degree (Dipl.-Ing.) in electrical engineering and information technology from RWTH Aachen University, Germany, and the Ph.D. degree from Supélec,
Gif-sur-Yvette, France, in 2008 and 2012, respectively. He is a member of technical staff at Nokia
Bell Labs, France, where he is investigating applications of deep learning for the physical layer.
Previous to this position he was co-founder and
CTO of the social network SPRAED and worked for
Alcatel-Lucent Bell Labs in Stuttgart, Germany. His
research interests are in the areas of machine learning, cloud computing, SDR,
large random matrix theory, information theory, signal processing, and their
applications to wireless communications. He is a co-author of the textbook
Massive MIMO Networks: Spectral, Energy, and Hardware Efficiency (2017).
He is recipient of the 2012 Publication Prize of the Supélec Foundation, the
2013 VDE ITG Förderpreis, and the 2015 Leonard G. Abraham Prize of the
IEEE COMSOC. He received the IEEE WCNC 2014 best paper award and has
been nominated as an Exemplary Reviewer 2012 for the IEEE Communication
letters.

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. Since 2005 he has been with the
Dipartimento di Ingegneria dell’Informazione of the
University of Pisa. In 2004, he was a visiting Ph.D.
student at the German Aerospace Center (DLR),
Oberpfaffenhofen, Germany. During the period June
2007 - June 2008, he was a postdoctoral associate in
the Dept. Electrical Engineering at Princeton. During
the period June 2010 - Sept. 2010, he was selected for a research assistantship
at the Technische Universitat Munchen. From July 2013 to October 2017 he
was with Large Systems and Networks Group (LANEAS), CentraleSupélec,
Gif-sur-Yvette, France.
Dr. Sanguinetti is currently serving as an Associate Editor for the IEEE
S IGNAL P ROCESSING L ETTERS. He served as an Associate Editor for IEEE
T RANSACTIONS ON W IRELESS COMMUNICATIONS, and as Lead Guest Editor of IEEE J OURNAL ON S ELECTED A REAS OF C OMMUNICATIONS Special
Issue on “Game Theory for Networks” and as an Associate Editor for IEEE
J OURNAL ON S ELECTED A REAS OF C OMMUNICATIONS (series on Green
Communications and Networking). Dr. Sanguinetti served as Exhibit Chair
of the 2014 IEEE International Conference on Acoustics, Speech, and Signal
Processing (ICASSP) and as the general co-chair of the 2016 Tyrrhenian
Workshop on 5G&Beyond. He is a co-author of the textbook Massive MIMO
Networks: Spectral, Energy, and Hardware Efficiency (2017).
His expertise and general interests span the areas of communications
and signal processing, game theory and random matrix theory for wireless
communications. He was the co-recipient of two best paper awards: IEEE
Wireless Commun. and Networking Conference (WCNC) 2013 and IEEE
Wireless Commun. and Networking Conference (WCNC) 2014. He was also
the recipient of the FP7 Marie Curie IEF 2013 “Dense deployments for green
cellular networks”.

