ANALYSIS OF THE PILOT CONTAMINATION EFFECT IN VERY LARGE MULTICELL
MULTIUSER MIMO SYSTEMS FOR PHYSICAL CHANNEL MODELS
Hien Quoc Ngo Thomas L. Marzetta†
Erik G. Larsson
Department of Electrical Engineering (ISY), Linköping University, 581 83 Linköping, Sweden
†
Bell Laboratories, Alcatel-Lucent, Murray Hill, NJ 07974, USA
ABSTRACT
We consider multicell multiuser MIMO systems with a very large
number of antennas at the base station. We assume that the channel
is estimated by using uplink training sequences, and we consider a
physical channel model where the angular domain is separated into
a finite number of directions. We analyze the so-called pilot contamination effect discovered in previous work, and show that this
effect persists under the finite-dimensional channel model that we
consider. We further derive closed-form bounds on the achievable
rate of uplink data transmission with maximum-ratio combining, for
a finite and an infinite number of base station antennas.
Index Terms— Pilot contamination, very large MIMO systems.
1. INTRODUCTION
2. SYSTEM MODEL
Multiuser multiple-input multiple-output (MU-MIMO) systems can
offer a spatial multiplexing gain without the requirement of multiple antennas at the users [1]. Most studies have assumed that the
base station has some channel state information (CSI). The problem
of not having an a priori CSI at the base station has been considered
in [2,3], assuming that the channel estimation is done by using uplink
pilots. This requires time-division duplex (TDD) operation. References [2, 3] considered a single-cell setting. This is only reasonable
when the pilot sequences available in each cell are orthogonal to
those in other cells. However, in practical cellular networks, channel coherence times are not long enough to allow for orthogonality between the pilots in different cells. Therefore, non-orthogonal
training sequences must be utilized and hence, the multicell setting
should be considered.
In the multicell scenario with non-orthogonal pilots in different
cells, channel estimates obtained in a given cell will be impaired by
pilots transmitted by users in other cells. This effect, called “pilot
contamination” has been analyzed in [4]. Recently, [5] considered
the multicell MU-MIMO system with very large antenna arrays at
the base station and showed that when the number of antennas increases without bound, uncorrelated noise and fast fading vanish and
the pilot contamination effect dictates the ultimate limit on the system performance.
Most the studies referred to above assume that the channels
are independent [2–4] or the channel vectors for different users
are asymptotically orthogonal [5]. However, in reality, the MIMO
channel is generally correlated because the antennas are not sufficiently well separated or the propagation environment does not offer
This work was supported in part by ELLIIT and the Swedish Research
Council (VR). E. G. Larsson is a Royal Swedish Academy of Sciences (KVA)
Research Fellow supported by a grant from Knut and Alice Wallenberg Foundation.
978-1-4577-0539-7/11/$26.00 ©2011 IEEE
rich enough scattering. In this paper, we investigate the multicell
MU-MIMO with large antenna arrays assuming a physical channel
model. More precisely, we consider a finite-dimensional channel
model in which the angular domain is partitioned into a large, but
finite number of directions which is small relative to the number of
base station antennas. The channels are estimated by using uplink
training (assuming TDD operation, as in previous work). For such
channels, the number of parameters to be estimated is fixed regardless of the number of antennas. We show that the pilot contamination
effect persists under the finite-dimensional channel model. Furthermore, we derive a closed-form lower bound on the achievable rate
of the uplink transmission, assuming maximum-ratio combining at
the base station. This bound is valid for a large but finite number of
antennas.
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Consider L cells, where each cell contains one base station equipped
with M antennas and K single-antenna users. Assume that the L
base stations share the same frequency band. We consider uplink
transmission, where the lth base station receives signals from all
users in all cells. See Fig. 1. Then, the M × 1 received vector at
the lth base station is given by
yl =
√
pu
L
Υ ilx i + n l
(1)
i=1
where Υ il represents the M ×K channel matrix between the lth base
Υil ]m,k is the channel
station and the K users in the ith cell, i.e., [Υ
coefficient between the mth antenna of the lth base station and the
√
kth user in the ith cell; pux i is the K × 1 transmitted vector of
K users in the ith cell (the average power used by each user is pu );
and n l contains M × 1 additive white Gaussian noise (AWGN). We
assume that the elements of n l are Gaussian distributed with zero
mean and unit variance.
2.1. Physical Channel Model
Here we introduce the finite-dimensional channel model that is used
throughout the paper. The angular domain is divided into a large but
finite number of directions P . P is fixed regardless of the number
of base station antennas (P < M ). Each direction, corresponding to
the angle φk , φk ∈ [−π/2, π/2], k = 1, ..., P , is associated with an
M × 1 array steering vector a (φk ) which is given by
T
1 −jf1 (φk ) −jf2 (φk )
a (φk ) = √
e
,e
, ..., e−jfM (φk )
(2)
P
where fi (φ) is some function of φ. The channel vector from kth user
in the ith cell to the lth base station is then a linear combination of the
ICASSP 2011
(τ ≥ K), which satisfies Φ †Φ = I K , where pp = τ pu . From (6),
the received pilot matrix at the lth base station is
Y p,l =
√
pp A
L
1/2
H ilD il Φ T + N l
(7)
i=1
where N l ∼ ÑM,τ (00, I M , I τ ) is noise.1
3.1. Minimum Mean-Square Error (MMSE) Estimation
l-th cell
L-th cell
Fig. 1. Uplink transmission in multicell MU-MIMO systems.
steering vectors as follows: P
m=1 gilkma (φm ), where gilkm is the
propagation coefficient from the kth user of the ith cell to the lth base
station, associated with the physical direction m (direction of arrival
φm ). Let G il [gg il1 · · · g ilK ] be a P × K matrix with g ilk [gilk1 · · · gilkP ]T that contains the path gains from the kth user in
the ith cell to the lth base station. The elements of G il are assumed
to be independent. Then, the M × K channel matrix between the lth
base station and K users in the ith cell is
Υ il = AG il
βilk , m = 1, 2, ..., P
(4)
where hilkm is a fast fading
√ coefficient assumed to be zero mean and
have unit variance, and βilk models the path loss and shadowing
which are assumed to be independent of the direction m and to be
constant and known a priori. This assumption is reasonable since the
value of βilk changes very slowly with time. Then, we have
√
1/2
pp A
H ilD il + W l
L
Y p,l =
Ỹ
(8)
i=1
where W l N lΦ ∗ ∼ ÑM,K (00, I M , I K ). Since H ll has independent columns, we can estimate each column of H ll independently.
Y p,l . Then
Let ỹy p,ln be the nth column of Ỹ
(3)
a (φ1 ) · · · a (φP )] is a full rank M × P matrix.
where A [a
The propagation channel G il models independent fast fading,
geometric attenuation, and log-normal shadow fading. Its elements
gilkm are given by
gilkm = hilkm
We assume that the base station uses MMSE estimation. The received pilot matrix Y p,l can be represented by Y p,lΦ ∗ and Y p,lΦ ∗⊥ ,
where Φ ∗⊥ is the orthogonal complement of Φ ∗ . Since Y p,lΦ ∗⊥ only
includes the noise part, Y p,l Φ∗ is a sufficient statistic for the estimaY p,l Y p,lΦ ∗ . We have
tion of H ll . Let Ỹ
ỹy p,ln =
√
1/2
ppAh lln βlln +
√
pp A
L
1/2
h iln βiln + w ln
(9)
i=l
where h iln and w ln are the nth columns of H il and W l , respec
√
1/2
tively. Denote by z ln ppA L
i=l h iln βiln + w ln . Then the
MMSE estimate of h lln is given by
−1
1/2 √
hlln = βlln
ỹy p,ln
ppA † pp βllnAA † + Rz ln
(10)
ĥ
(5)
where Rz ln = E z lnz †ln . By using the matrix inversion lemma,
we obtain
−1
L
1/2 √
†
hlln = βlln pp ppA A
ĥ
βiln + I P
A †ỹy p,ln
(11)
where H il is the P ×K matrix of fast fading coefficients between the
H il ]km = hilkm ,
K users in the ith cell and the lth base station, i.e., [H
and D il is a K × K diagonal matrix whose diagonal elements are
D il ]kk = βilk . Therefore, (1) can be written as
given by [D
The kth diagonal element of ppA †A L
i=1 βiln in (11) equals
M pp L
β
.
Since
the
uplink
is
typically
interference-limited
i=1 iln
P
M pp L
h
[4], P
i=1 βiln 1. Therefore, ĥ lln can be approximated as
1/2
G il = H ilD il
yl =
√
pr A
L
i=1
√
1/2
G ilx i + n l = pr A
H ilD il x i + n l
L
(6)
i=1
hlln ≈
ĥ
1/2 √
βlln pp
†
pp A A
L
−1
βiln
A †ỹy p,ln
(12)
i=1
i=1
Thus, the MMSE estimate of H ll is
−1
1/2
Y p,lD −1
H ll = pp −1/2 A †A
A †Ỹ
Ĥ
l D ll
3. CHANNEL ESTIMATION
Channel estimation is performed by using training sequences received on the uplink. A part of the coherence interval is used for
the uplink training. All users in all cells simultaneously transmit pilot sequences of length τ symbols. The assumption on synchronized
transmission represents the worst case from the pilot contamination
point of view, but it makes no fundamental difference to assume unsynchronized transmission [5]. We assume that the same set of pilot
sequences is used in all L cells. Therefore, the pilot sequences used
√
√
in the lth cell can be represented by a τ ×K matrix ppΦ l = ppΦ
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(13)
where D l L
i=1 D il . Then, the estimate of the physical channel
matrix between the lth base station and the K users in lth cell is
given by
Υll = AĤ
Y p,lD −1
H llD 1/2
Υ̂
= pp −1/2ΠA Ỹ
l D ll
ll
(14)
1 Ñ
M , Σ , Ψ ) denotes a matrix-variate complex Gaussian distribum,n (M
tion with the mean matrix M ∈ Cm×n and the variance matrix Ψ T ⊗ Σ ,
where ⊗ denotes the Kronecker product, Σ ∈ Cm×m and Ψ ∈ Cn×n .
−1 †
where ΠA A A †A
A is the orthogonal projection onto A .
We can see that since post-multiplication of Y p,l with Φ∗ means
Y p,l is the
Φ†Φ = I K ), Ỹ
just multiplication with the pseudoinverse (Φ
conventional least-squares channel estimate. The channel estimator
that we derived thus performs conventional channel estimation and
then projects the estimate onto the physical (beamspace) model for
the array.
4. UPLINK DATA TRANSMISSION
We consider the uplink transmission represented by (1). The base
station uses its channel estimate obtained by uplink training as in
(14) to perform maximum-ratio combining.
The lth base station processes its received signal by multiplying it by
the conjugate-transpose of the channel estimate. From (6), (8) and
(14), we have
†
L
†
√
−1
−1/2
Υlly l = pp
r l = Υ̂
D llD l
pp A
G il + W l
× ΠA
i=1
√
pu A
L
G jlx j + n l
.
(15)
j=1
As M → ∞, the products of uncorrelated quantities can be
ignored [5]. Then (15) becomes
L
L
† A †A 1
−1
G il
G jlx j
(16)
r l → D llD l
√
pu M
M
i=1
j=1
We can see that for an unlimited number of antennas, the effect of
uncorrelated noise disappears. In particular, the pilot contamination
effect, which is due to the interference from users in other cells,
persists under the finite-dimensional channel model.
We now consider the special case of a uniform array. Then the
response vector is given by
T
(M −1)d
d
1
a (φk ) = √ 1, e−j2π λ sin φk , ..., e−j2π λ sin φk
(17)
P
where d is the antenna spacing, and λ is the wavelength. For k = l,
M −1
1 j2π λd (sin φk −sin φl )m
1 †
e
a (φk ) a (φl ) =
M
M P m=0
d
=
For k = l,
1 1 − ej2π λ (sin φk −sin φl )M
M P 1 − ej2π λd (sin φk −sin φl )
1 †
a
M
(φk ) a (φl ) =
1
P
To gain further insight into the pilot contamination effect, we derive
lower bounds on the achievable rate for a finite and an infinite number of base station antennas (the analysis requires that M > P ). To
obtain these lower bounds we use the techniques of [3, 4, 6]. We assume a uniform array at the base station, and the elements of H il
are i.i.d Gaussian radom variables. Let rln be thenth element of the
√
pp L
i=1 βiln
rln . Then
received K × 1 vector r l . Denote r̃ln βlln
from (15), we have
† L
L
√
√
A
g
w
Π
A
G
x
n
r̃ln = pp
pu
(21)
jl j +n l
A
iln +w ln
i=1
j=1
Equation (21) can be rewritten as
4.1. Maximum Ratio Combining
4.2. Analysis of the Pilot Contamination Effect
M →∞
→ 0.
L
α Tjnx j + zln .
(22)
j=l
√
√
†
†
pp pu L
puw †lnAG jl , and
where α Tjn i=1 g ilnA AG jl +
†
√
pp A L
ΠA n l . We have
zln i=1 g iln + w ln
√
(23)
E α Tln = pp pu E g †llnA †Ag lln e Tn
where e n is the nth column
of the K ×K identity matrix. By adding
and subtracting E α Tln from α Tln in (22), and using (23), we obtain
L
α Tjnx j + zln
r̃ln = E α Tln x l + α Tln − E α Tln x l +
j=l
√
= pp pu E g †llnA †Ag lln xln + α Tln − E α Tln x l
+
L
αTjn xj + zln .
(24)
j=l
Let C (x) log2 (1 + x). The nth user in the lth cell can achieve
an uplink rate of at least
⎞
⎛
2
†
†
p
p pu E g llnA Ag lln ⎟
⎜
Rln = C⎝
(25)
⎠
2
L
2
T 2
T
−E{α ln} +E |zln |
α jn
j=1 E
Proposition 1 The achievable uplink rate of the nth user in the lth
cell in (25) is lowerbounded by (26), shown at the top of the next
L
page, where βjl K
k=1 βjlk , β ln i=1 βiln , and
(18)
(A) τm (A)
A) =
f (A
. Therefore,
1 † M →∞ 1
A A →
IP .
M
P
Substitution of (19) into (16) yields
L
L
†
1
M →∞ 1
G
G
x
.
rl →
D llD −1
√
jl j
l
il
pu M
P
i=1
j=1
r̃ln = α Tlnx l +
Xm,n (A) n (n + 1) λ2m
(27)
m=1 n=1
(20)
where A diag (λ1 , λ2 , ..., λP ), with λk , k = 1, ..., P , is the
kth eigenvalue of A †A , (A) is the number of distinct eigenvalues, λ1 , ..., λ(A) are the distinct eigenvalues in decreasing order, τm (A) is the multiplicity of λm and Xm,n (A) is the (m, n)th
characteristic coefficient of A which is defined in [7].
Since the elements of G il are independent, the above result reveals
that the performance of the system under the finite-dimensional
channel model with P angular bins and with an unlimited number
of base station antennas is the same as the performance under an
uncorrelated channel model with P antennas.
Corollary 1 For an unlimited number of base station antennas, the
lower bound on the achievable uplink rate of the nth user in the lth
cell becomes
2
βlln
∞
Rln = C L
.
(28)
L
1
2
j=l βjln + P β ln
j=1 βjl
(19)
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⎞
⎛
2
⎟
⎜
pp pu βlln
⎟
Rln= C⎜
2
2
⎠
⎝L †
†
A)−Tr(A A )
pp pu Tr(A A )
pp
pu
P
2 f (A
2
p
β
+
β
β
+
β
p
β
+
β
+
p
−p
p
u
p
u
jl
jl
ln M
jln
lln M ln M 2
j=1
M2
M2
12.0
16.0
11.0
M=20
M=50
M=100
14.0
10.0
9.0
Sum Rate (bits/s/Hz)
Sum Rate (bits/s/Hz)
(26)
R = 11
8.0
7.0
6.0
5.0
pu = -10, -5, 0, and 10 dB
4.0
3.0
12.0
10.0
8.0
6.0
4.0
2.0
2.0
1.0
0.0
20
40
60
0.0
0.0
80 100 120 140 160 180 200 220 240 260 280 300
Number of Base Station Antennas (M)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Cross Gain (a)
Fig. 2. Lower bound on the sum-rate of uplink transmission as a
function of the number of base station antennas M .
Fig. 3. Lower bound on the sum-rate of uplink transmission versus
the cross gain.
Remark 1 If P also goes to infinity,
the lower
bound on the achiev-
6. CONCLUDING REMARKS
This paper has analyzed the pilot contamination effect in multicell
MU-MIMO systems with very large antenna arrays for a physical
channel model with a finite number of scattering centers visible from
the base station. We showed that the pilot contamination effect discovered in [5] persists under the finite-dimensional channel model.
We also derived a closed-form lower bound on the achievable uplink
rate for finite and infinite number of base station antennas.
∞
able rate becomes Rln
= C
β2
L lln 2
β
j=l jln
This equals the exact
value for the rate obtained in [5]. This is due to the fact that when P
is large, things that were random before become deterministic and
hence, the lower bound approaches the exact value.
5. NUMERICAL RESULTS
We consider a system with 4 cells, and K = 10 users per cell. The
training sequence length is τ = K, and the number of physical directions is P = 20. We consider a uniform array at the base station
with λd = 0.3 and φk = −π/2 + (k − 1) π/P . We further assume
that all direct gains are equal to 1 and all cross gains are equal to a,
i.e, βllk = 1, and βjlk = a, ∀j = l, k = 1, ..., K. The lower bound
on the sum-rate of lth cell is defined as Rl = K
n=1 Rln .
Figure 2 shows the lower bound on the sum-rate of the uplink
transmission in the lth cell versus the number of antennas M , at
a = 0.1 and for different average transmit powers per user pu =
−10, −5, 0, and 10 dB. We see that at low SNR (since the noise
power is unity, the SNR is equal to pu ), using a large number of base
station antennas significantly improves the achievable rate. Furthermore, we can see that as M → ∞, the sum-rates approach Rl∞ .
This is the asymptotic value of sum-rate with an unlimited number
of base station antennas and it is independent of the SNR (cf. (28)).
Figure 3 depicts the lower bound on the sum-rate of uplink transmission versus the cross gain at an SNR of pu = 0 dB for different
M = 20, 50 and 100. We can see that the effect of pilot contamination can be very significant if the value of the cross gain is close to
the value of the direct gain, regardless of M .
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