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Massive MIMO in the UL/DL of Cellular Networks:
How Many Antennas Do We Need?
Jakob Hoydis, S. ten Brink, Mérouane Debbah
To cite this version:
Jakob Hoydis, S. ten Brink, Mérouane Debbah. Massive MIMO in the UL/DL of Cellular Networks:
How Many Antennas Do We Need?. IEEE Journal on Selected Areas in Communications, Institute
of Electrical and Electronics Engineers, 2013, 31 (2), pp.160 - 171. �10.1109/JSAC.2013.130205�.
�hal-00925966�
1
Massive MIMO in the UL/DL of Cellular
Networks: How Many Antennas Do We Need?
Jakob Hoydis, Member, IEEE, Stephan ten Brink, Senior Member, IEEE,
and M
´
erouane Debbah, Senior Member, IEEE
Abstract—We consider the uplink (UL) and downlink (DL)
of non-cooperative multi-cellular time-division duplexing (TDD)
systems, assuming that the number N of antennas per base
station (BS) and the number K of user terminals (UTs) per cell
are large. Our system model accounts for channel estimation,
pilot contamination, and an arbitrary path loss and antenna
correlation for each link. We derive approximations of achievable
rates with several linear precoders and detectors which are
proven to be asymptotically tight, but accurate for realistic system
dimensions, as shown by simulations. It is known from previous
work assuming uncorrelated channels, that as N → ∞ while K is
fixed, the system performance is limited by pilot contamination,
the simplest precoders/detectors, i.e., eigenbeamforming (BF) and
matched filter (MF), are optimal, and the transmit power can
be made arbitrarily small. We analyze to which extent these
conclusions hold in the more realistic setting where N is not
extremely large compared to K. In particular, we derive how
many antennas per UT are needed to achieve η% of the ultimate
performance limit with infinitely many antennas and how many
more antennas are needed with MF and BF to achieve the
performance of minimum mean-square error (MMSE) detection
and regularized zero-forcing (RZF), respectively.
Index Terms—massive MIMO, time-division duplexing, chan-
nel estimation, pilot contamination, large system analysis, large
random matrix theory, linear precoding, linear detection
I. INTRODUCTION
V
ERY large multiple-input multiple-output (MIMO) or
“massive MIMO” time-division duplexing (TDD) sys-
tems [1], [2] are currently investigated as a novel cellular
network architecture with several attractive features: First, the
capacity can be theoretically increased by simply installing
additional antennas to existing cell sites. Thus, massive MIMO
provides an alternative to cell-size shrinking, the traditional
way of increasing the network capacity [3]. Second, large
antenna arrays can potentially reduce uplink (UL) and down-
link (DL) transmit powers through coherent combining and an
increased antenna aperture [4]. This aspect is not only relevant
from a business point of view but also addresses environmental
as well as health concerns related to mobile communications
Manuscript received January 28, 2012; revised June 7, 2012; accepted
September 4, 2012. Communicated by Thomas L. Marzetta, Guest editor.
Parts of this work have been presented at the Allerton Conference on
Communication, Control, and Computing, Urbana-Champaign, IL, US, Sep.
2011, and the IEEE International Conference on Communications, Ottawa,
Canada, Jun. 2012.
J. Hoydis is with Bell Laboratories, Alcatel-Lucent, Lorenzstr. 10, 70435
Stuttgart, Germany (email: jakob.hoydis@alcatel-lucent.com).
S. ten Brink is with Bell Laboratories, Alcatel-Lucent, Lorenzstr. 10, 70435
Stuttgart, Germany (email: stephan.tenbrink@alcatel-lucent.com).
M. Debbah is with the Alcatel Lucent Chair on Flexible Radio, Sup
´
elec,
91192 Gif-sur-Yvette, France (email: merouane.debbah@supelec.fr).
[5], [6]. Third, if channel reciprocity is exploited, the overhead
related to channel training scales linearly with the number K
of user terminals (UTs) per cell and is independent of the
number N of antennas per base station (BS). Consequently,
additional antennas do not increase the feedback overhead and,
therefore, “always help” [7]. Fourth, if N ≫ K, the simplest
linear precoders and detectors are optimal, thermal noise,
interference, and channel estimation errors vanish, and the
only remaining performance limitation is pilot contamination
[1], i.e., residual interference which is caused by the reuse of
pilot sequences in adjacent cells.
The features described above are based on several crucial
but optimistic assumptions about the propagation conditions,
hardware implementations, and the number of antennas which
can be deployed in practice. Therefore, recent papers study
massive MIMO under more realistic assumptions, e.g., a phys-
ical channel model with a finite number of degrees of freedom
(DoF) [8] or constant-envelope transmissions with per-antenna
power constraints [9]. Also first channel measurements with
large antenna arrays were reported in [10], [11], [12].
In this work, we provide a unified performance analysis of
the UL and DL of non-cooperative multi-cell TDD systems.
We consider a realistic system model which accounts for
imperfect channel estimation, pilot contamination, antenna
correlation, and path loss. Assuming that N and K are
large, we derive asymptotically tight approximations of the
achievable rates with several linear precoders/detectors, i.e.,
eigenbeamforming (BF) and regularized zero-forcing (RZF)
in the DL, matched filter (MF) and minimum mean-square
error detector (MMSE) in the UL. These approximations
are easy to compute and shown to be accurate for realistic
system dimensions. We then distinguish massive MIMO from
“classical” MIMO as a particular operating condition of cellu-
lar networks where multiuser interference, channel estimation
errors, and noise have a negligible impact compared to pilot
contamination. If this condition is satisfied or not depends on
several system parameters, such as the number of UTs per DoF
the channel offers (we denote by DoF the rank of the antenna
correlation matrices which might be smaller than N), the
number of antennas per BS, the signal-to-noise ratio (SNR),
and the path loss. We further study how many antennas per UT
are needed to achieve η% of the ultimate performance limit
with infinitely many antennas and how many more antennas
are needed with BF/MF to achieve RZF/MMSE performance.
Our simulations suggest that in certain scenarios, RZF/MMSE
can perform as well as BF/MF with almost one order of
magnitude fewer antennas.
2
Fig. 1. In each of the L cells is one BS, equipped with N antennas, and K
single-antenna UTs. We assume channel reciprocity, i.e., the downlink channel
h
H
jlk
is the Hermitian transpose of the uplink channel h
jlk
.
The paper is organized as follows: In Section II, we describe
the system model and derive achievable UL and DL rates
with linear detectors and precoders. Section III contains our
main technical results where we derive asymptotically tight
approximations of these rates. In Section IV, we apply the
asymptotic results to a simplified system model which leads
to concise closed-form expressions of the achievable rates.
This allows us to propose a precise definition of “massive”
MIMO and to investigate if sub-optimal signal processing can
be compensated for by the use of more antennas. We present
some numerical results in Section V before we conclude the
paper in Section VI. All proofs are deferred to the appendix.
Notations: Boldface lower and upper case symbols represent
vectors and matrices, respectively (I
N
is the size-N iden-
tity matrix). The trace, transpose, and Hermitian transpose
operators are denoted by tr (·), (·)
T
, and (·)
H
, respectively.
The spectral norm of a matrix A is denoted by ∥A∥. We
use CN (m, R) to denote the circular symmetric complex
Gaussian distribution with mean m and covariance matrix
R. E [·] denotes the expectation operator. lim
N
stands for
lim
N→∞
.
II. SYSTEM MODEL
Consider a multi-cellular system consisting of L > 1
cells with one BS and K UTs in each cell, as schematically
shown in Fig. 1. The BSs are equipped with N antennas,
the UTs have a single antenna. We assume that all BSs and
UTs are perfectly synchronized and operate a TDD protocol
with universal frequency reuse. We consider transmissions
over flat-fading channels on a single frequency band or sub-
carrier. Extensions to multiple sub-carriers, different numbers
of antennas at the BSs, or different numbers of UTs in each
cell are straightforward.
A. Uplink
The received base-band signal vector y
ul
j
∈ C
N
at BS j at
a given time instant reads
y
ul
j
=
√
ρ
ul
L
l=1
H
jl
x
ul
l
+ n
ul
j
(1)
where H
jl
= [h
jl1
···h
jlK
] ∈ C
N×K
, h
jlk
∈ C
N
is the channel from UT k in cell l to BS j,
x
ul
l
=
x
ul
l1
···x
ul
lK
T
∼ CN (0, I
K
), with x
ul
lk
the transmit
signal of UT k in cell l, n
ul
j
∼ CN (0, I
N
) is a noise vector,
and ρ
ul
> 0 denotes the uplink SNR. We model the channel
vectors h
jlk
as
h
jlk
=
˜
R
jlk
v
jlk
(2)
where R
jlk
△
=
˜
R
jlk
˜
R
H
jlk
∈ C
N×N
are deterministic and
v
jlk
∼ CN (0, I
N
) are independent fast-fading channel vec-
tors. Our channel model is very versatile as it allows us to
assign a different antenna correlation to each channel vector.
This is especially important for large antenna arrays with a
significant amount of antenna correlation due to either insuf-
ficient antenna spacing or a lack of scattering. The channel
model is also valid for distributed antenna systems since we
can assign a different path loss to each antenna. Moreover,
(2) can represent a physical channel model with a fixed
number of dimensions or angular bins P as in [8], by letting
˜
R
jlk
=
ℓ
jlk
[A 0
N×N−P
], where A ∈ C
N×P
, 0
N×N−P
is the N ×(N −P ) zero matrix, and ℓ
jlk
denotes the inverse
path loss from UT k in cell l to BS j.
B. Downlink
The received signal y
dl
jm
∈ C of the mth UT in the jth cell
is given as
y
dl
jm
=
√
ρ
dl
L
l=1
h
H
ljm
s
l
+ n
dl
jm
(3)
where s
l
∈ C
N
is the transmit vector of BS l, n
dl
jm
∼ CN(0, 1)
is receiver noise, and ρ
dl
> 0 denotes the downlink SNR. We
assume channel reciprocity, i.e., the downlink channel h
H
ljm
is the Hermitian transpose of the uplink channel h
ljm
. The
transmit vector s
l
is given as
s
l
=
λ
l
K
k=1
w
lk
x
dl
lk
=
λ
l
W
l
x
dl
l
(4)
where W
l
= [w
l1
···w
lK
] ∈ C
N×K
is a precoding matrix
and x
l
=
x
dl
l1
···x
dl
lK
T
∈ C
K
∼ CN(0, I
K
) contains the
data symbols for the K UTs in cell l. The parameter λ
l
normalizes the average transmit power per UT of BS l to
E
ρ
dl
K
s
H
l
s
l
= ρ
dl
, i.e.,
λ
l
=
1
E
1
K
tr W
l
W
H
l
. (5)
C. Channel estimation
During a dedicated uplink training phase, the UTs in each
cell transmit mutually orthogonal pilot sequences which allow
the BSs to compute estimates
ˆ
H
jj
of their local channels
H
jj
. The same set of orthogonal pilot sequences is reused
in every cell so that the channel estimate is corrupted by
pilot contamination from adjacent cells [1]. After correlating
the received training signal with the pilot sequence of UT k,
the jth BS estimates the channel vector h
jjk
based on the
3
observation y
tr
jk
∈ C
N
, given as
1
y
tr
jk
= h
jjk
+
l=j
h
jlk
+
1
√
ρ
tr
n
tr
jk
(6)
where n
tr
jk
∼ CN (0, I
N
) and ρ
tr
> 0 is the effective training
SNR. In general, ρ
tr
depends on the pilot transmit power and
the length of the pilot sequences. Here, we assume ρ
tr
to be a
given parameter. The MMSE estimate
ˆ
h
jjk
of h
jjk
is given
as [13]
ˆ
h
jjk
= R
jjk
Q
jk
y
tr
jk
= R
jjk
Q
jk
l
h
jlk
+
1
√
ρ
tr
n
tr
jk
(7)
which can be shown to be distributed as
ˆ
h
jjk
∼ CN (0, Φ
jjk
),
where we define
Φ
jlk
= R
jjk
Q
jk
R
jlk
, ∀j, l, k (8)
Q
jk
=
1
ρ
tr
I
N
+
l
R
jlk
−1
, ∀j, k. (9)
Invoking the orthogonality property of the MMSE es-
timate [13], we can decompose the channel h
jjk
as
h
jjk
=
ˆ
h
jjk
+
˜
h
jjk
, where
˜
h
jjk
∼ CN (0, R
jjk
− Φ
jjk
) is
the uncorrelated estimation error (which is also statistically
independent of
ˆ
h
jjk
due to the joint Gaussianity of both
vectors).
D. Achievable uplink rates with linear detection
We consider linear single-user detection, where the jth BS
estimates the symbol x
ul
jm
of UT m in its cell by computing
the inner product between the received vector y
ul
j
and a linear
filter r
jm
∈ C
N
. Two particular filters are of practical interest,
namely the matched filter r
MF
jm
and the MMSE detector r
MMSE
jm
,
which we define respectively as
r
MF
jm
=
ˆ
h
jjm
(10)
r
MMSE
jm
=
ˆ
H
jj
ˆ
H
H
jj
+ Z
ul
j
+ Nφ
ul
j
I
N
−1
ˆ
h
jjm
(11)
where φ
ul
j
> 0 and Z
ul
j
∈ C
N×N
is an arbitrary Hermitian
nonnegative definite matrix. This formulation of r
MMSE
jm
allows
us to treat φ
ul
j
and Z
ul
j
as design parameters which could be
optimized. One could choose for example φ
ul
j
=
1
ρ
ul
N
and Z
ul
j
to be the covariance matrix of the intercell interference and
the channel estimation errors, i.e.,
Z
ul
j
= E
˜
H
jj
˜
H
H
jj
+
l=j
H
jl
H
H
jl
=
k
(R
jjk
− Φ
jjk
) +
l=j
k
R
jlk
. (12)
1
For an integer variable s taking values in a set S, we use
∑
s
to denote
the summation over all s ∈ S and
∑
s=j
to denote the summation over
all s ∈ S \ {j}. Similarly, let s
′
be another integer variable taking values
in the set S
′
, we denote by
∑
(s,s
′
)=(j,j
′
)
the summation over all tuples
(s, s
′
) ∈ S × S
′
\ {(j, j
′
)}.
Using a standard bound based on the worst-case uncorre-
lated additive noise [14] yields the ergodic achievable uplink
rate R
ul
jm
of UT m in cell j:
R
ul
jm
= E
log
2
1 + γ
ul
jm
(13)
where the associated signal-to-interference-plus-noise ratio
(SINR) γ
ul
jm
is given by (14) on the top of the next page
and where we have used E [·|·] to denote the conditional
expectation operator. We will denote by γ
MF
jm
and γ
MMSE
jm
the
SINR with MF and MMSE detection, respectively.
E. Achievable downlink rates with linear precoding
Since the UTs do not have any channel estimate, we provide
an ergodic achievable rate based on the techniques developed
in [15]. To this end, we decompose the received signal y
dl
jm
as
y
dl
jm
=
ρ
dl
λ
j
E
h
H
jjm
w
jm
x
dl
jm
+
ρ
dl
λ
j
h
H
jjm
w
jm
− E
h
H
jjm
w
jm
x
dl
jm
+
(l,k)=(j,m)
ρ
dl
λ
l
h
H
ljm
w
lk
x
dl
lk
+ n
dl
jm
(16)
and assume that the average effective channels
λ
j
E
h
H
jjm
w
jm
can be perfectly learned at the UTs.
Thus, an ergodic achievable rate R
dl
jm
of UT m in cell j is
given as [15, Theorem 1]
R
dl
jm
= log
2
1 + γ
dl
jm
(17)
where the associated SINR γ
dl
jm
is given by (15) on top of the
next page.
2
We consider two different linear precoders W
j
of practical
interest, namely eigenbeamforming (BF) W
BF
j
and regularized
zero-forcing (RZF) W
RZF
j
, which we define respectively as
W
BF
j
=
ˆ
H
jj
(18)
W
RZF
j
=
ˆ
H
jj
ˆ
H
H
jj
+ Z
dl
j
+ Nφ
dl
j
I
N
−1
ˆ
H
jj
(19)
where φ
dl
j
> 0 is a regularization parameter and Z
dl
j
∈ C
N×N
is an arbitrary Hermitian nonnegative definite matrix. As the
choice of Z
dl
j
and φ
dl
j
is arbitrary, they could be further
optimized (see, e.g., [15, Theorem 6]). This is outside the
scope of this paper and left to future work. We will denote by
γ
BF
jm
and γ
RZF
jm
the SINR with BF and RZF, respectively.
Remark 2.1: Under a block-fading channel model with co-
herence time T , one could account for the rate loss due to
channel training by considering the net ergodic achievable
rates κ(1 − τ/T )R
ul
jm
and (1 − κ)(1 − τ/T )R
dl
jm
for a given
training length τ ∈ [K, T ] and some κ ∈ [0, 1] which
determines the fraction of the remaining time used for uplink
transmissions.
2
We denote by var [x]
△
= E[(x − E[x]) (x − E[x])
H
] for some random
variable x.
4
γ
ul
jm
=
r
H
jm
ˆ
h
jjm
2
E
r
H
jm
1
ρ
ul
I
N
+
˜
h
jjm
˜
h
H
jjm
− h
jjm
h
H
jjm
+
l
H
jl
H
H
jl
r
jm
ˆ
H
jj
(14)
γ
dl
jm
=
λ
j
E
h
H
jjm
w
jm
2
1
ρ
dl
+ λ
j
var
h
H
jjm
w
jm
+
(l,k)=(j,m)
λ
l
E
h
H
ljm
w
lk
2
(15)
III. ASYMPTOTIC ANALYSIS
As the ergodic achievable rates R
ul
jm
and R
dl
jm
with both
types of detectors and precoders are difficult to compute for
finite system dimensions, we consider the large system limit,
where N and K grow infinitely large while keeping a finite
ratio K/N. This is in contrast to [1] where the authors assume
that the number of UTs remains fixed while the number of
antennas grows without bound. We will retrieve the results of
[1] as a special case. In the following, the notation “N → ∞”
will refer to K, N → ∞ such that lim sup
N
K/N < ∞.
From now on, all vectors and matrices must be understood as
sequences of vectors and matrices of growing dimensions. For
the sake of simplicity, their dependence on N and K is not
explicitly shown. The large system limit implicitly assumes
that the coherence time of the channel scales linearly with
K (to allow for orthogonal pilot sequences of the UTs in a
cell). However, as we use the asymptotic analysis only as a
tool to provide tight approximations for finite N, K, this does
not pose any problem.
3
In a realistic deployment, one could
expect BSs equipped with several hundred antennas serving
each tens of UTs simultaneously [1].
In what follows, we will derive deterministic approxima-
tions ¯γ
ul
jm
(¯γ
dl
jm
) of the SINR γ
ul
jm
(γ
dl
jm
) with the MF and the
MMSE detector (BF and RZF precoder), respectively, such
that
γ
ul
jm
− ¯γ
ul
jm
a.s.
−−−−→
N→∞
0, γ
dl
jm
− ¯γ
dl
jm
−−−−→
N→∞
0 (20)
where “
a.s.
−−−−→
N→∞
” denotes almost sure convergence. One can
then show by the dominated convergence [18] and the con-
tinuous mapping theorem [19], respectively, that (20) implies
that
R
ul
jm
− log
2
1 + ¯γ
ul
jm
−−−−→
N→∞
0
R
dl
jm
− log
2
1 + ¯γ
dl
jm
−−−−→
N→∞
0. (21)
These results must be understood in the way that, for each
given set of system parameters N and K, we provide approx-
imations of the SINR and the associated rates which become
increasingly tight as N and K grow. We will show later by
simulations that these approximations are very accurate for
realistic system dimensions. As we make limiting considera-
tions, we assume that the following conditions hold:
A 1: lim sup
N
∥R
jlk
∥ < ∞ ∀j, l, k
3
Note that similar assumptions have been made in [16], [17].
A 2: lim inf
N
1
N
tr R
jlk
> 0 ∀j, l, k
A 3: lim sup
N
∥
1
N
Z
ul
j
∥ < ∞, lim sup
N
∥
1
N
Z
dl
j
∥ < ∞ ∀j
Before we continue, we recall two related results of large
random matrix theory which will be required for the asymp-
totic performance analysis of the MMSE detector and t he RZF
precoder.
Theorem 1 ([20, Theorem 1]): Let D ∈ C
N×N
and
S ∈ C
N×N
be Hermitian nonnegative definite and let
H ∈ C
N×K
be random with independent column vectors
h
k
∼ CN
0,
1
N
R
k
. Assume that D and the matrices R
k
,
k = 1, . . . , K, have uniformly bounded spectral norms (with
respect to N). Then, for any ρ > 0,
1
N
tr D
HH
H
+ S + ρI
N
−1
−
1
N
tr DT(ρ)
a.s.
−−−−→
N→∞
0
where T(ρ) ∈ C
N×N
is defined as
T(ρ) =
1
N
K
k=1
R
k
1 + δ
k
(ρ)
+ S + ρI
N
−1
and the elements of δ(ρ) , [δ
1
(ρ) ···δ
K
(ρ)]
T
are defined as
δ
k
(ρ) = lim
t→∞
δ
(t)
k
(ρ), where for t = 1, 2, . . .
δ
(t)
k
(ρ) =
1
N
tr R
k
1
N
K
j=1
R
j
1 + δ
(t−1)
j
(ρ)
+ S + ρI
N
−1
with initial values δ
(0)
k
(ρ) = 1/ρ for all k.
Remark 3.1: The fixed-point algorithm in Theorem 1 to
compute the quantities δ
k
(ρ) can be efficiently numerically
solved and is proved to converge. In some cases, closed-form
solutions for δ(ρ) exists. An example will be shown later in
Corollary 3.
Theorem 2 ([21], see also [20]): Let Θ ∈ C
N×N
be Her-
mitian nonnegative definite with uniformly bounded spectral
norm (with respect to N). Under the conditions of Theorem 1,
1
N
tr D
HH
H
+ S + ρI
N
−1
Θ
HH
H
+ S + ρI
N
−1
−
1
N
tr DT
′
(ρ)
a.s.
−−−−→
N→∞
0 (22)
where T
′
(ρ) ∈ C
N×N
is defined as
T
′
(ρ) = T(ρ)ΘT(ρ) + T(ρ)
1
N
K
k=1
R
k
δ
′
k
(ρ)
(1 + δ
k
(ρ))
2
T(ρ)
