1
On the Capacity of Distributed
Antenna Systems
Lin Dai
City University of Hong Kong
Jan. 19, 2013
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Cellular Networks (1)
Base Station (BS)
Growing demand for high data rate
Multiple antennas at the BS side
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Cellular Networks (2)
•
Implementation
cost
Sum rate
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Co-located BS antennas
•
Distributed BS antennas
Lower
Higher?
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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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Single-user SIMO Channel
• Single user with a single antenna.
• L>1 BS antennas.
• Uplink (user->BS).
• Received signal:
• Channel gain :
(a) Co-located
BS Antennas
y  gs  z
g=γ h
(b) Distributed
BS Antennas
s CN (0, P): Transmitted signal
z  C L1
: Gaussian noise
zi CN (0, N0 ), i  1,..., L.
γ  C L1 : Large-scale fading
i
Log  CN (di /2 ,  i2 ), i  1,..., L.
h  C L1 : Small-scale fading
hi
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CN (0,1), i  1,..., L.
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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
• Ergodic capacity without channel state information at the
transmitter side (CSIT):

 P g

Co  Eh log 2 1 

N0



• Ergodic capacity with CSIT:
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2





2



P
g


Cw  max Eh log 2 1 

P: Eh [ P ] P

N0  




Function of
large-scale
fading vector 
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Co-located Antennas versus Distributed Antennas
• With co-located BS antennas:
 Ergodic Capacity without CSIT:
 
C  Eh log2 1   h
C
o
1
L
2

 is the average received SNR:
  P  γ / N0
2
• With distributed BS antennas:
 Distinct large-scale fading gains to
different BS antennas.
 Ergodic Capacity without CSIT:
 
CoD  Eh log2 1   g
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2

Normalized channel gain: g = β h
β
γ
γ
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Capacity of DAS
• 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 Nbk co-located
antennas.
o BS antennas are grouped into L clusters.
Each cluster has Nll co-located antennas.
o Asymptotic result as N goes to infinity and
bk and ll are fixed.
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• Implicit function
of  (need to solve
fixed-point
equations)
•
Computational
complexity
increases with L
and K.
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Capacity of DAS
• With random large-scale fading vector :
Average ergodic capacity (i.e., averaged over )
 Single user
 Without CSIT
 [Roh&Paulraj’02], [Zhang&Dai’04]: The user has identical access
distances to all the BS antennas.  i
Log  CN (1, 2 ), i  1,..., L.
 [Zhuang&Dai’03]: BS antennas are uniformly distributed over a
 /2
circular area and the user is located at the center.  i  i , i  1,..., L.
 [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.  i
Log  CN (di /2 ,  i2 ), i  1,..., L.
High computational complexity!
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Questions to be Answered
• How to characterize the sum capacity of DAS when there are
a large number of BS antennas and users?
 Large-system analysis using random matrix theory.
 Bounds are desirable.
• How to conduct a fair comparison with the co-located case?
 K randomly distributed users with a fixed total transmission power.
 Decouple the comparison into two parts: 1) capacity comparison and
2) transmission power comparison for given average received SNR.
• What is the effect of CSIT on the comparison result?
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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.
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Assumptions
• K users uniformly distributed within a circular cell. Each has a
single antenna.
• L BS antennas.
• Uplink (user->BS).
R
R
Random BS
antenna layout!
*: user
o: BS antenna
(a) Co-located Antennas (CA)
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(b) Distributed Antennas (DA)
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Uplink Ergodic Sum Capacity
• Received signal:
K
y   g k sk  z
k 1
• Uplink power control:
Pk  γ k
2
 P0 , k  1,..., K .
Ergodic capacity without CSIT



1 K

† 
Csum _ o =E H log 2 det  I L 
Pk g k g k  

N 0 k 1








P0 K

† 
=E H log 2 det  I L 
g
g

k k 
N


0 k 1



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sk CN (0, Pk ) : Transmitted signal
: Gaussian noise
z  C L1
zi CN (0, N0 ), i  1,..., L.
g k = γ k hk
γ k  C L1
hk  C L1
: Channel gain
: Large-scale fading
 i,k  di,k /2 , i  1,..., L.
: Small-scale fading
hi,k CN (0,1), i  1,..., L.
Ergodic capacity with CSIT


1
Csum _ w = max E H log 2 det  I L 
Pk : E H [ Pk ] Pk
N0


k 1,..., K

† 
P
g
g

k k k 
k 1
 

1

= max E H log 2 det  I L 
Pk : E H [ Pk ] P0
N0


k 1,..., K

† 
P
g
g

k k k 
k 1
 
K
K
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More about Normalized Channel Gain
hk  C L1: Small-scale fading
γ
β k  k  C L1 : Normalized
γk
Large-scale fading
βk  1.
• Normalized channel gain vector:
g k = βk hk
 With CA:
gk
 With DA:
1
1L1.
L
βk 
2
The channel becomes
deterministic with a
large number of BS
antennas L!
 1 if L  .
bi,k  b j ,k , i  j.
o With a large L, it is very likely that user k is
close to some BS antenna lk* :
gk
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2
| hl* ,k |
2
k
βk  el*
k
1 if l  lk*
el  
0 otherwise
Channel fluctuations
are preserved even
with a large L!
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More about Normalized Channel Gain
• Theorem 1. For n=1,2,…,
min Ehk  g k

β k : β k 1
2n
max Ehk  gk

βk : βk 1
  (n  L  1)! , which is achieved when β k  1 1L1.
 Ln ( L  1)!
L
2n
  n!,

which is achieved when βk  el* .
k
1
1L1 ,
L
maximized when βk  el* .
 Channel fluctuations are minimized when β k 
k
 Channel fluctuations are undesirable when CSIT is absent,
desirable when CSIT is available.
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Single-user Capacity (1)
• Without CSIT
2
CA
CkC_ o
1.8
L=10
L=50
L=10
L=50
with CSIT
 Ck _ o / CAWGN  1 -- Fading
always hurts if CSIT
is absent!
CkC_ w
1.6
Ck
log 2 (1  0 )
DA with
β k  el *
 CkC_ o / CAWGN quickly
approaches 1 as L grows.
k
CkD_*o
CkD_*w
1.4
C
D*
 Ck _ o  Ck _ o
1.2
• With CSIT (when 0 is small)
1
 Ck _ w / CAWGN  1 at low 0
--“Exploit” fading
0.8
without CSIT
0.6
-10
-5
0
N0 (dB)
(average received SNR) 0  P00 /(dB)
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5
10
D*
C
 Ck _ w  Ck _ w
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Single-user Capacity (2)
1.1
with CSIT
DA with βk
Single-user Capacity Ck (bit/s/Hz)
CAWGN  log 2 (1  0 )
• Without CSIT
 el*
 A higher capacity is
achieved in the CA
case thanks to better
diversity gains.
k
1
0.9
• With CSIT
without CSIT
0.8
CA
x
0.7
DA with βk
10
DA
20
25
k
Analytical (Eq. (28))
Simulation
Analytical (Eq. (29-30))
Simulation
Analytical (Eq. (21))
Simulation
Analytical (Eq. (22-23))
Simulation
15
 A higher capacity is
achieved in the DA case
thanks to better
waterfilling gains.
 el*
30
35
40
45
50
Number of BS Antennas L
The average received SNR 0  0dB.
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Part II. Uplink Ergodic Sum Capacity
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Uplink Ergodic Sum Capacity without CSIT
• Sum capacity without CSIT:


P

Csum _ o =E H log 2 det  I L  0
N0





†
g
g

E
log
det
I


GG



k
H
2
L
0
k 1


K
†
k



G=B H
• Sum capacity per antenna (with K>L):
CL _ o


1
 E H log 2 det I L  0GG †
L
1
L

 E H  log 2 1  0 ll  
L
 l 1


B  [β1 ,..., β K ]
H  [h1 ,..., hK ]
where {ll } denotes the eigenvalues of GG† .
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More about Normalized Channel Gain
• Theorem 2. As K , L   and K / L   ,
EH [l ]  ,
and EH [l 2 ]  2  2  B , with 0  B   .
1
1LK .
L
B   0 when B  [el* ,..., el* ].
B    when B 
1
 With CA: B 
f l ( x) 
C
1
1
1LK .
L
 With DA and B  [el* ,..., el* ]:
1
((   1)  x)( x  (   1) )
2
2 x
as K , L   and K / L   .
[Marcenko&Pastur’1967]
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K
2
f l ( x)   e
D*
 ( x  )
K
( x )k

k 0 k !(k  1)!

as K , L   and K / L   .
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Sum Capacity without CSIT (1)
Sum Capacity per Antenna CL_o (bit/s/Hz)
7
6
CLC_ o  CLD_*o
log 2 (1  0 )
CLC_ o

CLD_*o
 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.
5
4
=10
3
=5
2
1
=1
0
0
2
4
6
(average received SNR) 0 0P(dB)
0 / N0 (dB)
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8
10
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Sum Capacity without CSIT (2)
75
Sum Capacity Csum_o (bit/s/Hz)
 A higher capacity is
achieved in the CA case
thanks to better
diversity gains.
CA
70
65
60
DA with
B  [el* ,..., el* ]
55
1
50
D*
 Csum
_ o serves as an
asymptotic lower-bound
D
to Csum
_ o.
K
CA
Analytical (Eq. (31, 34))
45
Simulation
40
DA
Analytical (Eq. (31, 40))
35
Simulation
30
10
15
20
25
30
35
40
45
50
Number of BS Antennas L
The average received SNR 0  0dB. The number of users K=100.
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Uplink Ergodic Sum Capacity with CSIT
• Sum capacity with CSIT:



1 K

† 
Csum _ w = max E H log 2 det  I L 
Pk g k g k  

Pk : E H [ Pk ] P0
N0 k 1





k 1,..., K
 With CA:
2
 With DA and B  [el* ,..., el* ]:
1
The optimal power allocation policy:

1
1
Pk*  N 0  
  g†k I L   j  k Pj*g j g j





1

gk 

where  is a constant chosen to meet
the power constraint E H Pk  P0 ,
k=1,…, K. [Yu&etc’2004]
 
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Pk  Pk γ k
K
The optimal power allocation policy:


 
1
1 


N  
Pk*   0   | h * |2 
i , ki

 
0


k  ki*  arg max | hi ,k |2
kK i
k  ki*
i=1,…,L, where  is a constant chosen to
meet the sum power constraint
EH
 P   KP .
K
k 1 k
0
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Signal-to-Interference Ratio (SIR)
(a) CA
(b) DA with B  [el1* ,..., elK* ]
The received SNR 0  0dB. The number of users K=100. The number of BS antennas L=10.
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Sum Capacity (1)
2
with CSIT
CA
1.8
B  [el* ,..., el* ]
1
K
L=10
L=50
L=10
L=50
1.6
Csum
L log 2 (1  KL 0 )
DA with
• Without CSIT
C
D*
 Csum _ o  Csum _ o
C
 Gap between Csum
_ o and
D*
Csum
_ o is enlarged as L grows
1.4
(i.e., due to a decreasing K/L).
1.2
• With CSIT
1
CA
0.8
DA with
B  [el* ,..., el* ]
1
without CSIT
0.6
-10
-5
K
even at high SNR
L=10
L=50
L=10
L=50
0
/ N0 (dB)
(average received SNR) 0 P00(dB)
The number of users K=100.
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D*
C
 Csum _ o  Csum _ o
5
10
(i.e., thanks to better
multiuser diversity gains)
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Sum Capacity (2)
90
• Without CSIT
DA with
B  [el* ,..., el* ]
1
 A higher capacity is
achieved in the CA
case thanks to better
diversity gains.
K
80
Sum Capacity Csum (bit/s/Hz)
with CSIT
CA
70
60
• With CSIT
CA
50
x
DA
40
Analytical (Eq. (45-46))
Analytical (Eq. (31, 40))
Simulation
without CSIT
30
 A higher capacity is
achieved in the DA case
thanks to better
waterfilling gains and
multiuser diversity gains.
Analytical (Eq. (41-42))
Analytical (Eq. (31, 34))
Simulation
10
15
20
25
30
35
40
45
50
Number of BS Antennas L
The average received SNR 0  0dB. The number of users K=100.
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Part III. Average Transmission Power
per User
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Average Transmission Power per User
• Transmission power of user k: Pk 
P0
γk
2
, k  1,..., K .
• Average transmission power per user:
K   P
1 K
0

f γ 2 ( x)dx
P


k
0 x
k
K k 1
P
 With CA:
 With DA:
o Users are uniformly distributed
in the circular cell. BS antennas
are co-located at cell center.
γk
2
 L  k
2x
f k ( x )  2
R
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P
C
2 P0 R


 2 L
o Both users and BS antennas
are uniformly distributed in
the circular cell.
What is the distribution of γ k
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?
29
Minimum Access Distance
• With DA, each user has different access distances to different
BS antennas. Let
dk(1)  dk(2)   dk( L)
denote the order statistics obtained by arranging the access
distances d1,k,…, dL,k.
•
γk
2
   dl , k 
L
l 1

  d k(1) 

for L>1.
• An upper-bound for average transmission power per user with DA:
R
R y P
D
DU
0
P P
  f k ( y ) 
 f d (1) | ( x | y )dxdy


0
0
k
k
x
2y
f k ( y )  2
R
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f d (1) | ( x | y )  L(1  Fdl ,k |k ( x | y)) L 1 f dl ,k |k ( x | y )
k
k
2x

0 x  R y
R2

f dl ,k |k ( x | y)  
x2  y 2  R2
2x
R y  x  R y
arccos

2 xy
  R2
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Average Transmission Power per User
-1
10
Analytical
Average Transmission Power Per User
C
L1
Eq. (53)
Eq. (56)
-2
10
C
 DA: P
DU
 O  L /2 
(path-loss factor >2)
For given received SNR, a
lower total transmission
power is required in the DA
case thanks to the reduction
of minimum access distance.
DU
-3
10
C
L
L2
-4
Simulation
10
CA with P0C  1
DA with P0D  P0C
-5
10
 O  L1 
 CA: P
10
D
0
DA with P
P  L
30
40
50
20
C
0
 With =4, P
1
5
60
70
80
90
100
C
P
D
if
P0D  P0C  15 L.
Number of BS Antennas L
Path-loss factor =4.
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Sum Capacity without CSIT
350
C
• For fixed K and 0
D
C
(0  0  15 L such that
P C  P D)
CA
Analytical (Eq. (31, 34))
Simulation
Sum Capacity Csum_o (bit/s/Hz)
300
DA
Analytical (Eq. (31, 40))
Simulation
250
200
C
C
 Csum _ o  L log2 (1  0 K / L)
L 

 0C K log 2 e
C
0 =10dB
D
 Csum _ o  O( L)
150
100
50
0C=0dB
0
10
15
20
25
30
35
40
45
Number of BS Antennas L
50
Given the total transmission
power, a higher capacity is
achieved in the DA case.
Gains increase as the number
of BS antennas grows.
0D  0C  15 L. The number of users K=100.
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Conclusions
• A comparative study on the uplink ergodic sum capacity with colocated and distributed BS antennas is presented by using largesystem 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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Thank you!
Any Questions?
Jan. 19, 2013
JWITC 2013