On the Capacity of Distributed Antenna Systems

JWITC 2013Jan. 19, 2013

1

On the Capacity of Distributed Antenna Systems

Lin Dai

City University of Hong Kong

JWITC 2013Jan. 19, 2013

2

Cellular Networks (1)

Multiple antennas at the BS side

Base Station (BS)

Growing demand for high data rate

JWITC 2013Jan. 19, 2013

3

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.

4

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

5

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

6

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:

7

g = β h γ

βγ

JWITC 2013Jan. 19, 2013

Capacity of DAS

8

• 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.

10

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

12

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

13

, (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!

14

1.k β

JWITC 2013Jan. 19, 2013


• 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

17

*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


• 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


0 0 0/ (dB)P N

21

JWITC 2013Jan. 19, 2013

Sum Capacity without CSIT (2)

The average received SNR 0 0dB.


10 15 20 25 30 35 40 45 5030

35

40

45

50

55

60

65

70

75


Sum

Cap

acity

Csum_o

(bi

t/s/

Hz)

CA

DA

Analytical (Eq. (31, 34))

Simulation


Simulation

The number of users K=100.

serves as an asymptotic lower-bound to .

*_

Dsum oC

_Dsum oC

22

* *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


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).

25

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 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


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

26

* *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

29

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


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.

30

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


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


Simulation


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?

33

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On the Capacity of Distributed Antenna Systems