Channel estimation for wireless cellular systems with massive linear receiving antennas

Full text

2015 10th
International
Conference on Communications
and
Networking
in
China (China Com)
Channel Estimation For Wireless Cellular Systems
with Massive Linear Receiving Antennas
Dian Fan t, Zhangdui Zhongt, Gongpu Wangt, and Feifei Gao+
t School
of
Computer and Information Technology, Beijing Jiaotong University, Beijing, China.
+ Tsinghua National Laboratory for Information Science and Technology, Beijing, China.
Email: tfandian@bjtu.edu.cn.tzhdzhong@bjtu.edu.cn.tgpwang@bjtu.edu.cn. + feifeigao@ieee.org.
Abstract-Massive Multiple
Input
Multiple
Output
(Massive
MIMO) system
can
be a promising technology for future fifth
generation (SG) wireless communication systems due to its
large gains in spectral-efficiency and energy-efficiency. One
well-known challenge for cellular systems with massive
MIMO
is
the estimation problem of large-scale channel parameters.
This
paper
investigates the channel estimation problem for
cellular systems with massive linear receiving antennas. Firstly,
a system model
is
built
up
based on a time-shifted pilots
and
time-division duplexing (TOO) cellular network with both intra-
cell
and
inter-cell interference taken into account. Secondly,
an
estimation algorithm is proposed to obtain all channel
parameters
through multiple signal classification (MUSIC)
method
and
reconstructing. Moreover,
Cramer
Rao Lower
Bound (CRLB)
is
found to evaluate the estimation performance.
It
is found
that
the suggested estimation algorithm works
well
and
can
obtain all channel
parameters
without requiring
a large
number
of pilots, which can effectively reduce the
pilot contamination. Finally, simulation results
are
provided to
corroborate
our
proposed studies.
Index
Termes
-Massive MIMO, pilot contamination, cellular
network, OOA, channel estimation
I.
INTRODUCTION
Massive Multiple Input Multiple Output (Massive MIMO)
is
a new wireless communication system where each base
station (BS)
is
equipped with a large antenna array and
serves multiple single-antenna user terminals simultaneously
[1]. Massive MIMO systems are capable
of
providing more
degrees
of
freedom than conventional MIMO systems and thus
can improve the system capacity and enhance the data rates
[1, 2]. The reference
[2]
shows that increasing the number
of
antennas
of
BS can overcome the effect
of
fast fading channel
and uncorrelated noise
in
a single-cell scenario.
It
is also shown
in
[2]
that
if
the number
of
BS antennas
tends to be infinity, the intra-cell interference and noise can
vanish, but the inter-cell interference still exits. Pilot con-
tamination
is
an inter-cell interference caused
by
re-use
of
the pilot sequences
in
other cells. It largely affects system
performance and can not be reduced by increasing the number
of
BS antennas only [3]. It also leads to quick saturation
of
the system performance. Thus, pilot contamination is the main
factor that limits the system capacity for wireless communi-
cation networks with massive MIMO
[4-6].
Some studies have been done about pilot contamination
[4-
6]. The reference
[4]
mathematically characterizes the impact
that uplink training has on the performance
of
multi-cell
95
UT
.'
.'
.'
.'
.'
.'
.'
.'
F
N:
Processing
F:
Forward link
R:
Reverse link
Uplink
Downlink
Fig.
l.
The communication process between the user terminal and the BS
in
TDD systems
massive MIMO systems. The reference
[5]
obtains an accurate
expression for signal to interference plus noise ratio (SINR)
and sum-rate lower bounds
of
both infinite and finite BS
antennas
in
the case
of
pilot contamination. The reference
[6]
proposes a time-shifted pilots scheme which can reduce pilot
contamination
by
rearranging reverse-link pilot sequences.
However, to our best knowledge, the estimation
of
all
channel parameters for cellular massive MIMO systems has
not been addressed, which motivates our present work. In
this paper, we proposed a channel estimation algorithm for
cellular systems with massive linear receiving antennas and
time-shifted pilots scheme . This algorithm only requires a few
pilots, and therefore can effectively reduce pilot contamina-
tion.
The main contributions
of
this paper are summarized as
follows. The system model is build up based on time-shifted
pilots and grouped cells. A channel estimation algorithm with
low cost and high accuracy is proposed to estimate all channel
parameters. Moreover, Cramer-Rao lower bounds (CRLB) are
found to evaluate the performance
of
the suggested channel
estimation algorithm. Finally, simulation results are provided
to corroborate our proposed studies.
978-1-63190-077-8 © 2015
IEEE

Fig.
2.
A cellular system with four cell groups.
P
~
T
AJj\j
A,
I
A,
I
A,
I
A,
I
A,
I
A,
I
A,
I
A,
I
A.
I
A.
I
A.
I
A.
I
N
I
~
I
I A] I t
Al
' I
A"
I
R I I
A,
,,,,
I A
""I
F I
AU
I
Au4
I
AU4
I
A?u
l
Au
i
A1U
l
A1U
l
A1U
I A14 I
Al?4
I
AJ24
I
Al24
1
Al?
I
A1U
I
A1U
I
A1U
I
P:
Pilot N: Processing
F:
Forward
link
R: Reverse
link
Fig.
3.
The communication process between the user terminal and the BS in
a TDD system: the whole system into four exclusive cells groups denoted
by
Al,A2
,A3 , A4 . A
ijk
donates
Ai
,
Aj,Ak
transmit forward-link data
synchronous
ly.
II.
SYSTEM
MODEL
Consider a cellular network system with L hexagonal cells.
Each cell consists
of
one BS with M-antennas
in
the center
and K user terminals randomly distributed. Each user terminal
has single-antenna and all the K user terminals share the same
bandwidth. Furthermore, all user terminals and BSs operate on
a time-division duplexing (TDD) protocol.
The channel coefficient between the k-th user terminal
of
the
j-th
cell and the m-th BS antenna
of
the l-th cell
is
J
(3
lkj
gmlkj,
where
(3lkj
is
the large-scale fading accounting
for both geometric attenuation and shadow fading, and
gmlkj
indicates the fast fading coefficient. All fading coefficients are
independently and identically distribute (i.i .d.) and
gmlkj
rv
CN(O, 1). Therefore, the channel vector between the k-th user
terminal
of
the
j-th
cell and the BS
of
the l-th cell
is
(1)
where the channel coefficients
of
the BS
of
l-th cell form the
channel vector
h~j
=
(h
ll
kj,
h2
lkj,
...
,
hMlkj)
E
([
1 x
M,
and the corresponding fast fading coefficients form fast fading
vector
g~j
=
(gllkj,g2lkj,···
,g
M
lkj)
E
([
lx
M .
The TDD scheme shown
in
Fig. 1
is
described
as
follows. In
each cell, every coherence interval is divided into four parts:
•
CD
each terminal send pilot sequences
of
length T syn-
chronously, and these pilot are orthogonal;
• @ BSs use these pilot sequences to estimate the channel
vectors;
96
• @ BSs use the estimated channel vectors as beamforming
vectors to transmit forward-link data to their user termi-
nals;
• ® all the terminals
in
the cell transmit reverse link data
to their BSs.
The system with grouped cells works as follows. Suppose
there are totally
16
cells
in
the system . The
16
cells are divided
into four groups
(AI,
A2, A3) A4
),
each group has four cells,
and each cell has a BS and four user terminals (Fig. 2). When
user terminals
in
group Al send pilot sequence, BSs
in
group
A2, A3,
A4
will send forward link data to their user terminals.
Fig. 2 shows this process.
We
assume that user terminals and BS take a time-shifted
transmission scheme that
is
shown
in
Fig.
3.
This scheme
diminishes the severe interference due to simultaneous pilot
transmission. A time-shifted pilot scheme divide the whole
cellular system into R particular cells groups denoted
by
AI,
A2) ·
..
, AR. For example, the whole system is divided
into four cells groups
in
Fig.
3.
During each coherence interval,
there
is
only one group
of
cells transmitting reverse link pilot
sequences while others groups BS are transmitting forward-
link data.
At the beginning
of
every coherence interval
in
the time-
shifted pilots scheme, user terminals
in
one group transmit
orthogonal training sequences
of
length
T.
The length T
is
chosen as T = l
~
J
,
where T is coherence interval length and
R is the number
of
groups. All the groups follow the same
transmission pattern. Therefore, if there are at most K user
terminals
in
each cell, the l-th BS received signal matrix
is
K
Yl
= L L
VPuThlkiSf{
iE
A r k = l
K
+ L L
Ck'lyPfVliQk'
i +
Nl
) (2)
i\i!
Ar
k'
= l
where
Pu
and
Pj
are the transmission powers
of
uplink and
forward-link respectively, Ckl indicates the fading coefficients
between l-th BS and k-th terminal,
Sk
denotes the mutually
independent T x 1 pilot sequences that are reused
in
each cell,
Vli indicates the M x M channel matrix between the BS
in
l-th cell and the BS
in
i-th cell,
Qk'
i
is
the M x T data
symbol matrix from i-th cell BS to
k'
-th terminal, and
Nl
is the additive noise with each element following CN(O,l)
distribution [9].
Our task
is
to
estimate
hlk
i from
Yl
(2). How
to
estimate
the channel state information
hlk
i quickly and effectively
is
a
challenge problem, which is our main focus
in
the next section.
III.
CHANNEL
ESTIMATION
In this section, we propose a channel estimation scheme for
the cellular network system described
in
Section II.

A.
Channel estimation algorithm
The fading coefficients related distance between l-th BS and
k-th terminal
Ckl
can be expressed as
[7]
A 2
Ckl
= GtGr
(-4
d
),
7r
kl
(3)
where Gr and Gt denote the gains
of
the transmitter and the
receiver respectively, A is the cellular carrier wavelength and
d
kl
indicates the distance between l-th BS and k-th terminal.
Since the distance from other cells user terminals
is
large,
and we obtain the following equation from (2), (3)
K
Yl
= L L VPuThlkiSk + Zl ,
iEAr k=1
(4)
where Zl
is
the sum
of
interference and noise given as
K
Zl = L L
Ck'
i
yPfVliqk'
i + Nl· (5)
i\l
Ar
k'
=1
In
this paper, we consider a simple situation that only
the LOS channel without reflections between BS and users.
Therefore, for the l-th BS with linear receiving antennas, the
channel vector hkl is expressed as
hlki =hlkd 1,
e-
j27r
!OTl(O,)
,
...
,e-
j27r!
OT
Nr(
O
Nr)]
T, (6)
where
fo
is
the cellular carrier frequency,
Ti(B
i)
is
the relative
delay for the k-th terminal to the l-th BS and the notation
[Y
is the transpose operator. Note that hkl is a vector
of
M x 1
dimensions.
Suppose the k-th user terminal transmits pilot sequences
Sk
to the l-th BS with the direction
of
arrival (DOA) B
lk
i. Clearly,
0::;
Blki
< 180°. Substituting (6) into (4) will produce
K
Yl
= L L hlki alki(8)Sk + Zl, (7)
iEAr k=1
where
alki
(8)
=[
1,
e-
j27rd
.
COS
~l
ki
,
•••
, e-
j27r
(
i-
l)d.
COS
~lk
i
]T
. (8)
Suppose each group has R cells, and we can further simplify
(7) as
where
Blr
= r
h~~lr
o
Bl2
o
o
o
hl2r
o
o
o
o
o
o
o
o
o
o
o
BLR
(9)
(10)
(11)
97
(12)
and
Al(8)
is
shown
in
(13) on the top
of
next page.
Multiplying
All
(8)
on both sides
of
(9) will produce
AII(8)Yl
= AII
(8)(Al(8)AlSl
+ Zt)
=
AlSl
+
AII(8)Zl.
(14)
Therefore, if we ignore the interference and noise, we can
obtain
(15)
Since
Al
is
a diagonal matrix, we can obtain the channel
state information if we get
Al
(8). So
Al
(8)
is
our main focus
in the next.
Without loss
of
generality, we can assume
R zl =
E{ZlZf}
=
a;Im'
(16)
noting that the uncorrelation between the noise and the pilot
sequences, we can obtain the received covariance
Ryl
as
Ryl
=
E{Yl
Yf}
= Af
Al(8)E{SlSf}Af
(8)
+
a;Im
= Af
Al(8)PAf
(8)
+
a;Im,
(l7)
where
(18)
Clearly, all eigenvalues
of
the
Ryl
are positive. Suppose the
descending order
of
M positive eigenvalues
of
Ryl
is Al
;:::
A2
;:::
A3
;:::
...
;:::
Ar
>
Ar
+l =
Ar
+2 =
...
=
AM.
Clearly,
the top K eigenvalues are the intra-cell channel gains.
We can decompose
Ryl
into two orthogonal subspace:
intra-cell subspace
Us,
and inter-cell interference and noise
subspace
Un,
therefore we can obtain through singular value
decomposition (SVD)
Ryl
= [
Vs
Vn
J[
~OK
a2I~
_
K]
[
~~
]
(19)
where V s and V n are orthogonal.
Multiplying both side
of
(17) and (19) with V
n,
we can
obtain
RylVn
= Af
Al(8)PAn8)V
n +
a;Vn,
(20)
Ryl
Vn
=
a;V
n. (21)
Therefore, we can find
(22)
Following the similar steps with MUSIC algorithm [8], we
can have
V;;
Al(8)PAf
(8)Vn
= 0 (23)
Thus, we can estimate DOA by searching for the maximum
values
of
the following expression
1
Pt(8) = H . (24)
al
(8
i
)V
n
V{!al(8
i)

AI(8)
= [
al11
(8)
,
al21
(8)
,
...
,
alKl
(8),
al12(8),
a122(8),
...
,
aIK2(8),
...
,al1R(8)
, aI2R(8),
...
,aIKR(8)
]. (13)
Its maximum value 8i
is
the estimated value
of
the DOA
8i. Note, we can reconstruct
AI(8)
from (13) and
Al
from
(15). Finally, the estimate
of
channel matrix H can be found
as
(25)
B.
CRLB
of
channel estimation
Now, we analyze the Cramer Rao Lower Bound (CRLB)
of
channel state information for Massive MIMO system.
To
simplify our analysis, we assume that every group has r cells
and each cell has single user terminal.
In
this case,
Yl
(9) can
be simplified as
y =
A(8)D(h)s
+
n,
(26)
where y
is
the l-th complex BS received signal,
A(
8)
is
M x r
direction matrix, S = [Sl,
S2
, ·
..
,
sr
]
T,
n
is
a complex M
xl
WGN vector
of
variance (Y2 and
o
o
o
o
o
We
can rewrite the BS received signal vector y
as
y =
Fh
+ n ,
(27)
(28)
where F =
A(8)D(s),
D(s) =
diag(sl,···
,sr) and h =
[hI , h2, ·
..
, h
rV
·
The likelihood function is
1 H - 1
p(y
;
h)
=
JrM
de
t(C)
exp
[-
(y -
Fh)
C (y -
Fh)
],
where C is the covariance matrix
of
y.
To
check the equality condition we have
8Inp(y
;h)
8h
*
8(y
-
Fh)H
C - 1(y -
Fh)
8h
*
= _
~
(
8(y
-
Fh)H
C - 1(y -
Fh)
2
8~{h}
,
v------'
PI
(29)
.
8(y
-
Fh)HC
-1(y -
Fh))
+ J 8
'S
{h} , (30)
..
~
PJJ
where PI and
PI
I defined
as
the corresponding stems can be
further found
as
PI
= - yHC - 1F -FHC - 1y
+ h HF H C - 1F + F H C - 1
Fh
, (31)
P
II
= -
jyHC
-1F +
jFHC
-1y
+
jh
HFHC - 1F -
jF
HC - 1
Fh.
(32)
98
TABLE I
SIMULATION
PARAMETERS
PARAMETERS VALUES
Cell radi us r c 2 km
Protection distance r h 0.15 km
Carrier frequency F 2.4 GHz
Bandwidth B 20 MHz
Number
of
ceUs L 9
Number
of
user termina ls per cell K 3
Number
of
groups R 3
Number
of
BS antennas M 64
Adjacent antenn as distance d
>.
/ 2
Coherence interval T 14 symbols
Process ing phase length B 1 symbol
Substituting (33) and (32) into (30) will produce
8Inp(y
;
h)
= F H C
-1
- F HC
-1
Fh
(33)
8h
* y
= F H C - 1(y -
Fh)
= F H
C - 1F [
(F
H
C - 1F) -
lF
H
C - 1y - h ].
Thus, we can obtain
(34)
is efficient estimate. Moreover, we can find the covariance
matrix
of
the channel estimates h
as
(35)
Finally, the CRLB
of
the channel h
is
the trace
of
the
following matrix
rl(h)
= C;::l =
FH
C - 1F
[
2~{FH
C - 1
F}
- 2'S
{F
HC - 1
F}
IV.
SIMULATION
RESULTS
In this section, we evaluate the estimation algorithm for
our model through computer simulations. The simulation
parameters are given
in
Table I and will remain the same if
they are not specified otherwise.
Fig. 4 shows the estimated DOAs when there is only
one group cell transmitting pilot sequences from terminals .
The transmitting signal to noise radio (SNR)
of
each ter-
minal is 30dB and the angle
Bil
can be divided into two-
part: one consists
of
the intra-cell user terminals DOAs
(20°
,40
°
,60
°) and the other contains the inter-cells user
terminals DOAs (10°
,15
°
,30
°
,50
°,
70
°,
80°).
We
estimate
these DOAs through (24).
It
can be found from Fig. 4 that
there exist seven peaks. That corresponds
to
the angles
of
DOA.
We
choose three DOA informations that have the three
largest value in y-axis.
Fig. 5 depicts the MSE performance
of
the DOA estima-
tion algorithm with different target SNRs using different BS

&
0:-
'0
Cl>
"
~
Cl>
,s
105
10'
10'
102
10'
100
10-1
1L
2
'---~
10
20
30
40
50
60 70
80
90
the value of 8il
Fig.
4.
DOA estimation.
1~r-----~------~----~------~--r=~====~
-+-
M=16
----B--- M=32
--B---
M=64
M=128
~M=256
10
-'
L------'---------L------'---------'---------'----------'
-10
-5
0 5
10
15
20
SNR(dB)
Fig.
5.
MSE
of
DOA estimation algorithm .
antennas schemes.
It
can
be
found from Fig. 5 that larger
SNR can result
in
better estimates.
It
can also
be
shown that
more antennas
in
the BS can increase estimation performance.
This figure shows that a significant performance gain can be
obtained
by
using our proposed DOA estimation algorithm.
Fig. 6 demonstrates the MSE performance
of
proposed
channel estimation algorithm with different target SNRs using
different BS antennas schemes. For comparison, the CRLB
of
the channel estimates are also plotted
on
case
of
different BS
antennas. This figure further confirms the effectiveness
of
our
proposed channel estimation algorithm.
V.
CONCLUSION
In
this paper, we investigated the channel estimation prob-
lem for cellular systems with massive linear receiving anten-
nas. First, a time-shifted cellular system model with grouped
cells was build
up.
Second,
an
algorithm was proposed
to
estimate the massive channel parameters. Moreover, CRLB
was found to evaluate the estimation performance.
It
has been
found that the suggested estimation algorithm works well
99
w
Cf)
:2
1 0 -
1
r-----~------~----~----____;;=====~=======;_J
-+-
M=64
----B--- CRLB(M=64)
--B---
M=128
CRLB(M=128)
~M=256
-+-
CRLB(M=256)
10
-'
L-
-----'---------L------'---------'---------'----------'
-10
-5
o 5
SNR(dB)
10
15
Fig.
6.
MSE of channel estimation algorithm.
20
and can obtain all channel parameters without requiring large
number
of
pilot sequences, which can effectively reduce pilot
contamination.
ACKNOWLEDGEMENT
This research was supported
by
the Natural Science Foun-
dation
of
China (Grant No. : U1334202), the
Key
grant
Project
of
Chinese Ministry
of
Education (No.313006), and
the Fundamental Research Funds for the Central Universities
(No. 2014JBZ003).
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