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JWITC 2013Jan. 19, 2013
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On the Capacity of Distributed Antenna Systems
Lin Dai
City University of Hong Kong
JWITC 2013Jan. 19, 2013
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Cellular Networks (1)
Multiple antennas at the BS side
Base Station (BS)
Growing demand for high data rate
JWITC 2013Jan. 19, 2013
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Cellular Networks (2)
• Co-located BS antennas • Distributed BS antennas
Implementation cost
Sum rate
Lower
Higher?
JWITC 2013Jan. 19, 2013
A little bit of History of Distributed Antenna Systems (DAS)
• Originally proposed to cover the dead spots for indoor
wireless communication systems [Saleh&etc’1987].
• Implemented in cellular systems to improve cell
coverage.
• Recently included into the 4G LTE standard.
• Multiple-input-multiple-output (MIMO) theory has
motivated a series of information-theoretic studies on
DAS.
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JWITC 2013Jan. 19, 2013
Single-user SIMO Channel
• Single user with a single antenna.
• L>1 BS antennas.
• Uplink (user->BS). (a) Co-located BS Antennas
(b) Distributed BS Antennas
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s y g z• Received signal:
g = γ h• Channel gain : : Large-scale fading
: Small-scale fading
1LC γ
1LC h
: Transmitted signal: Gaussian noise
(0, )s PCN1LC z
0(0, ), 1,..., .iz N i LCN
(0,1), 1,..., .ih i LCN
/2 2Log ( , ), 1,..., .i i id i L CN
JWITC 2013Jan. 19, 2013
Single-user SIMO Channel
2
20
log 1P
N
g
• Ergodic capacity without channel state information at the transmitter side (CSIT):
• Single user with a single antenna.
• L>1 BS antennas.
• Uplink (user->BS).
• Ergodic capacity with CSIT:2
2: E [ ]
0
max E log 1wP P P
PC
N
hh
g
Function of large-scale fading vector
(a) Co-located BS Antennas
(b) Distributed BS Antennas
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EoC h
JWITC 2013Jan. 19, 2013
Co-located Antennas versus Distributed Antennas
• With co-located BS antennas:
Ergodic Capacity without CSIT:
212E log 1C
o LC h h is the average received SNR: 2
0/P N γ
• With distributed BS antennas:
Distinct large-scale fading gains to different BS antennas.
Ergodic Capacity without CSIT:
2
2E log 1DoC h g
Normalized channel gain:
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g = β h γ
βγ
JWITC 2013Jan. 19, 2013
Capacity of DAS
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• For given large-scale fading vector : [Heliot&etc’11]: Ergodic capacity without CSIT
o A single user equipped with N co-located antennas.
o BS antennas are grouped into L clusters. Each cluster has M co-located antennas.
o Asymptotic result as M and N go to infinity and M/N is fixed. [Aktas&etc’06]: Uplink ergodic sum capacity
without CSIT
o K users, each equipped with Nk co-located antennas.
o BS antennas are grouped into L clusters. Each cluster has Nl co-located antennas.
o Asymptotic result as N goes to infinity and k and l are fixed.
• Implicit function of (need to solve fixed-point equations)
• Computational complexity increases with L and K.
JWITC 2013Jan. 19, 2013
Capacity of DAS
[Choi&Andrews’07], [Wang&etc’08], [Feng&etc’09],
[Lee&ect’12]: BS antennas are regularly placed in a circular
cell and the user has a random location.
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Average ergodic capacity (i.e., averaged over )
[Zhuang&Dai’03]: BS antennas are uniformly distributed over a
circular area and the user is located at the center.
[Roh&Paulraj’02], [Zhang&Dai’04]: The user has identical
access distances to all the BS antennas.2Log (1, ), 1,..., .i i L CN
/2 , 1,..., .i i i L
/2 2Log ( , ), 1,..., .i i id i L CN
High computational complexity!
• With random large-scale fading vector :
Single user Without CSIT
JWITC 2013Jan. 19, 2013
Questions to be Answered
• How to characterize the sum capacity of DAS when there
are a large number of BS antennas and users?
• How to conduct a fair comparison with the co-located
case?
Large-system analysis using random matrix theory.
• What is the effect of CSIT on the comparison result?
Decouple the comparison into two parts: 1) capacity comparison and 2) transmission power comparison for given average received SNR.
K randomly distributed users with a fixed total transmission power.
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Bounds are desirable.
JWITC 2013Jan. 19, 2013
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Part I. System Model and Preliminary Analysis
[1] L. Dai, “A Comparative Study on Uplink Sum Capacity with Co-located and Distributed Antennas,” IEEE J. Sel. Areas Commun., 2011.
JWITC 2013Jan. 19, 2013
Assumptions
• K users uniformly distributed within a circular cell. Each has a single antenna.
• L BS antennas.
• Uplink (user->BS).
R R
(a) Co-located Antennas (CA) (b) Distributed Antennas (DA)
*: user
o: BS antenna
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Random BS antenna layout!
JWITC 2013Jan. 19, 2013
Uplink Ergodic Sum Capacity
k k kg = γ h1
K
k kk
s
y g z
• Received signal:: Transmitted signal: Gaussian noise
(0, )k ks PCN1LC z
: Large-scale fading
: Small-scale fading
1Lk C γ
1Lk C h
0(0, ), 1,..., .iz N i LCN
: Channel gain
• Uplink power control:2
0 , 1,..., .k kP P k K γ
†_ 2
10
†02
10
1=E log det
=E log det
K
sum o L k k kk
K
L k kk
C PN
P
N
H
H
I g g
I g g
Ergodic capacity without CSIT
0
†_ 2
: E [ ]101,...,
†2
: E [ ]101,...,
1= max E log det
1= max E log det
k k k
k k
K
sum w L k k kP P P
kk K
K
L k k kP P P
kk K
C PN
PN
H
H
H
H
I g g
I g g
Ergodic capacity with CSIT
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, (0,1), 1,..., .i kh i LCN
/2, , , 1,..., .i k i kd i L
JWITC 2013Jan. 19, 2013
More about Normalized Channel Gain
• Normalized channel gain vector:
k k kg = β h : Normalized Large-scale fading
: Small-scale fading1Lk
kk
C γ
βγ
1Lk C h
With CA:
, , , .i k j k i j With DA:
1
1.k L
Lβ 1
o With a large L, it is very likely that user k
is close to some BS antenna :*kl *
kk lβ e
*1 if
0 otherwisek
l
l le
if .L 2
1k g
*
2 2
,| |
kk l k
hg
The channel becomes deterministic with a large number of BS
antennas L!
Channel fluctuations are preserved even
with a large L!
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1.k β
JWITC 2013Jan. 19, 2013
More about Normalized Channel Gain
• Theorem 1. For n=1,2,…,
: 1
2min
( 1)!E ,
( 1)!kk
k
n
k n
n L
L L
β βh g
:
2
1max E !,
kk k
n
k n
βh
βg
which is achieved when
which is achieved when
1
1.k L
Lβ 1
* .k
k lβ e
Channel fluctuations are minimized when
maximized when * .k
k lβ e
1
1,k L
Lβ 1
Channel fluctuations are undesirable when CSIT is absent,
desirable when CSIT is available.
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JWITC 2013Jan. 19, 2013
Single-user Capacity (1)
-10 -5 0 5 100.6
0.8
1
1.2
1.4
1.6
1.8
2
20
log
(1)
kC
0 (dB)
DA with
L=10
L=50
_Ck oC _
Ck wCL=10
L=50
*_Dk oC
*_Dk wC
with CSIT
without CSIT
CA
*k
k lβ e
(average received SNR)
0 0 0/ (dB)P N
• Without CSIT
-- Fading always hurts if CSIT is absent!
_ / 1k o AWGNC C
quickly approaches 1 as L grows.
_ /Ck o AWGNC C
*
_ _C Dk o k oC C
• With CSIT (when 0 is
small)_ / 1k w AWGNC C
*_ _D Ck w k wC C
at low
0 --“Exploit” fading
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JWITC 2013Jan. 19, 2013
Single-user Capacity (2)
The average received SNR . 0 0dB
• Without CSIT
• With CSIT
A higher capacity is achieved in the CA case thanks to better diversity gains.
A higher capacity is achieved in the DA case thanks to better waterfilling gains.
10 15 20 25 30 35 40 45 500.7
0.8
0.9
1
1.1
Sin
gle-
user
Cap
acity
Ck
(bit/
s/H
z)
Number of BS Antennas L
with CSIT
AWGN 2 0log (1 )C
without CSIT
CA DAAnalytical (Eq. (28))SimulationAnalytical (Eq. (29-30))
Simulation
Analytical (Eq. (21))SimulationAnalytical (Eq. (22-23))
Simulationx
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*k
k lβ eDA with
*k
k lβ eDA with
JWITC 2013Jan. 19, 2013
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Part II. Uplink Ergodic Sum Capacity
JWITC 2013Jan. 19, 2013
Uplink Ergodic Sum Capacity without CSIT
• Sum capacity without CSIT:
• Sum capacity per antenna (with K>L):
† †0_ 2 2 0
10
=E log det E log detK
sum o L k k Lk
PC
N
H HI g g I GG
†_ 2 0
2 01
1E log det
1E log 1
L o L
L
ll
CL
L
H
H
I GG
G = B H
1[ ,..., ]KB β β
1[ ,..., ]KH h h
where denotes the eigenvalues of .
†GG{ }l
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JWITC 2013Jan. 19, 2013
More about Normalized Channel Gain
• Theorem 2. As and
E [ ] , H
when * *1
[ ,..., ].Kl l
B e e
,K L / ,K L
and 2 2E [ ] 2 , H B 0 . Bwith
0 B
B when1
.L KL
B 1
With DA and * *1
[ ,..., ] :Kl l
B e e
* ( )
0
( )( )
!( 1)!
kD x
k
xf x e
k k
as and
,K L / .K L
With CA:
2 21( ) (( 1) )( ( 1) )
2Cf x x x
x
1.L K
LB 1
as and
,K L / .K L
[Marcenko&Pastur’1967]
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JWITC 2013Jan. 19, 2013
Sum Capacity without CSIT (1)
*
_ _C DL o L oC C
Gap diminishes when is large -- the capacity becomes insensitive to the antenna topology when the number of users is much larger than the number of BS antennas.
0 2 4 6 8 100
1
2
3
4
5
6
7
Sum
Cap
acity
per
Ant
enna
CL_o
(bit/
s/H
z)
0 (dB)
=10
=5
=1
2 0log (1 )_CL oC
*_DL oC
(average received SNR)
0 0 0/ (dB)P N
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JWITC 2013Jan. 19, 2013
Sum Capacity without CSIT (2)
The average received SNR 0 0dB.
A higher capacity is achieved in the CA case thanks to better diversity gains.
10 15 20 25 30 35 40 45 5030
35
40
45
50
55
60
65
70
75
Number of BS Antennas L
Sum
Cap
acity
Csum_o
(bi
t/s/
Hz)
CA
DA
Analytical (Eq. (31, 34))
Simulation
Analytical (Eq. (31, 40))
Simulation
The number of users K=100.
serves as an asymptotic lower-bound to .
*_
Dsum oC
_Dsum oC
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* *1
[ ,..., ]Kl l
B e eDA with
CA
JWITC 2013Jan. 19, 2013
Uplink Ergodic Sum Capacity with CSIT
• Sum capacity with CSIT:
0
†_ 2
: E [ ]101,...,
1= max E log det
k k
K
sum w L k k kP P P
kk K
C PN
H
H I g g
2
k k kP P γ
With CA:
The optimal power allocation policy:
*
0 1† *
1 1k
k L j j j kj k
P NP
g I g g g
where is a constant chosen to meet the power constraint , k=1,…, K. [Yu&etc’2004]
0E kP PH
With DA and
The optimal power allocation policy:
*
* 2* 0 ,2
,
*
1 1 arg max | |
| |
0
i
i
i i kk
k i k
i
N k k hP h
k k
K
i=1,…,L, where is a constant chosen to meet the sum power constraint 01E .
K
kkP KP
H
* *1
[ ,..., ] :Kl l
B e e
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JWITC 2013Jan. 19, 2013
Signal-to-Interference Ratio (SIR)
(a) CA (b) DA with
The received SNR 0 0dB. The number of users K=100. The number of BS antennas L=10.
* *1
[ ,..., ]Kl l
B e e
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JWITC 2013Jan. 19, 2013
Sum Capacity (1)
-10 -5 0 5 10
0.8
1
1.2
1.4
1.6
1.8
2
0.6
with CSIT
without CSIT
0 (dB)
20
log
(1)
sum
K L
C
L
DA with
L=10
L=50
CA
L=10
L=50
L=10
L=50
L=10
L=50
CA
* *1
[ ,..., ]Kl l
B e e
DA with * *1
[ ,..., ]Kl l
B e e
(average received SNR)
0 0 0/ (dB)P N
• Without CSIT
• With CSIT
Gap between and
_Csum oC
*
_ _C Dsum o sum oC C
*_ _
D Csum o sum oC C
even at high SNR
(i.e., thanks to better multiuser diversity gains)
*_
Dsum oC is enlarged as L grows
(i.e., due to a decreasing K/L).
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The number of users K=100.
JWITC 2013Jan. 19, 2013
Sum Capacity (2)
The average received SNR 0 0dB. The number of users K=100.
• Without CSIT
• With CSIT
A higher capacity is achieved in the CA case thanks to better diversity gains.
A higher capacity is achieved in the DA case thanks to better waterfilling gains and multiuser diversity gains.
10 15 20 25 30 35 40 45 5030
40
50
60
70
80
90
Number of BS Antennas L
Sum
Cap
acity
Csum (
bit/s
/Hz)
without CSIT
with CSIT
CA
DAAnalytical (Eq. (45-46))
SimulationAnalytical (Eq. (31, 40))
Analytical (Eq. (41-42))Analytical (Eq. (31, 34))Simulationx
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* *1
[ ,..., ]Kl l
B e eDA with
CA
JWITC 2013Jan. 19, 2013
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Part III. Average Transmission Power per User
JWITC 2013Jan. 19, 2013
Average Transmission Power per User
• Transmission power of user k: 02 , 1,..., .k
k
PP k K
γ
With DA:
What is the distribution of2?kγ
• Average transmission power per user:
1
1 K
kk
PK P
o Both users and BS antennas are uniformly distributed in the circular cell.
With CA:
o Users are uniformly distributed in the circular cell. BS antennas are co-located at cell center.2
k kL γ
2
2( )
k
xf x
R 02
2C P R
L
P
28
20
0( )
k
K Pf x dx
x
γ
JWITC 2013Jan. 19, 2013
Minimum Access Distance
• With DA, each user has different access distances to different BS antennas. Let
denote the order statistics obtained by arranging the access distances d1,k,…, dL,k.
(1) (2) ( )Lk k kd d d
• for L>1.
• An upper-bound for average transmission power per user with DA:
2 (1),
1
L
k l k kl
d d
γ
(1)0
|0 0( ) ( | )
k kk
R R yD DU
d
Pf y f x y dxdy
x
P P
2
2( )
k
yf y
R (1)
, ,
1| ||
( | ) (1 ( | )) ( | )l k k l k kkk
Ld dd
f x y L F x y f x y
2
2 2 2,
2
2
| 22
0( | )
arccosl k k
xR
d x y RxxyR
x R yf x y
R y x R y
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JWITC 2013Jan. 19, 2013
Average Transmission Power per User
CA: 1C O LP
10 20 30 40 50 60 70 80 90 10010
-5
10-4
10-3
10-2
10-1
Simulation
2C
LL
A
vera
ge
Tra
nsm
issi
on P
ow
er
Per
Use
r
Number of BS Antennas L
CA with
DA with0 0D CP P
10 0 5D CP P L DA with
Analytical
Eq. (53)Eq. (56)
1C L
DU
0 1CP
Path-loss factor =4.
DA: /2DU O L P
(path-loss factor >2)
With =4, C DP P if1
0 0 5 .D CP P L
For given received SNR, a lower total transmission power is required in the DA case thanks to the reduction of minimum
access distance.
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JWITC 2013Jan. 19, 2013
Sum Capacity without CSIT
10 0 5 .D C L The number of users
K=100.
• For fixed K and ( such that
Number of BS Antennas L
Sum
Cap
acity
Csum_o
(bi
t/s/H
z)
10 15 20 25 30 35 40 45 500
50
100
150
200
250
300
350CA
Analytical (Eq. (31, 34))
Simulation
Analytical (Eq. (31, 40))
Simulation
DA
=10dB0C
=0dB0C
31
_ 2 0log (1 / )C Csum oC L K L
_ ( )Dsum oC O L
0C
10 0 5D C L C DP P )
Given the total transmission power, a
higher capacity is achieved in the DA case. Gains
increase as the number of BS antennas grows.
0 2logL
CK e
JWITC 2013Jan. 19, 2013
Conclusions
• A comparative study on the uplink ergodic sum capacity with co-located and distributed BS antennas is presented by using large-system analysis.
– A higher sum capacity is achieved in the DA case. Gains increase with the number of BS antennas L.
– Gains come from 1) reduced minimum access distance of each user; and 2) enhanced channel fluctuations which enable better multiuser diversity gains and waterfilling gains when CSIT is available.
• Implications to cellular systems:
– With cell cooperation: capacity gains achieved by a DAS over a cellular system increase with the number of BS antennas per cell thanks to better power efficiency.
– Without cell cooperation: lower inter-cell interference with DA?
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JWITC 2013Jan. 19, 2013
Thank you!
Any Questions?
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