Multiuser Diversity in Delay-LimitedCellular Systems

Ralf R. Müllerralf@iet.ntnu.no

Department of Electronics & TelecommunicationsNorwegian University of Science & Technology, Trondheim

Giuseppe Caire & Raymond Knopp{caire,knopp}@eurecom.fr

Institut EurècomSophia-Antipolis, France

Disclaimer

This is NOT a talk on fair scheduling of users.

Here:

• Fairness is not enough. • Each user has to get their message instantaneously with probability going to 1.

Ergodic vs. Delay-Limited Capacity

1. The channel changes very often within a single codeword.

2. The channel is constant within a single codeword.

Two extreme cases of fading channels:

Ergodic capacity

Delay-Limited capacity

Delay-Limited Capacity for a Single User

Let the fading d be known at both transmitter and receiver,

AWGN channel with energy per symbol E and noise

density N0:

C =d:

dEf (d)≤1inf log

21 + f(d)

|d |2 EN

0

⎛

⎝⎜

⎞

⎠⎟

if E |d|-2 exists, C > 0 and f (d) = |d|-2.

C = 0 for Rayleigh fading.

Delay-Limited Capacity for Many Users

The result is given as a multi-dimensional optimization problem in

Stephen V. Hanly, David N.C. Tse:Multiaccess fading channels. Part II: Delay-limited capacitiesIEEE Trans. Inform. Theory, vol. 44, no. 7, pp. 2816 - 2831, Nov. 1998.  

Can we obtain a gain by multiuser diversity without any drawback in quality of service, e.g. no additional delay?

Gaussian Multiple-Access Channel without Fading

Dual Representation of GMAC without Fading

Transmit Power vs. Receive Power

Rescaling of axis in power region

Transmit Power vs. Receive Power

2

GMAC with Fading

Let the attenuations be random from codeword to codeword.

if E |d|-2 exists, C > 0

C = 0 for Rayleigh fading.

%Eb

N0

= ln(2) 2Rxdx0

1

∫ ⋅dF

|d|2x( )

x∫

Single User vs. Infinite Users

Single User:

Infinite Users:

%Eb

N0

= ln(2) 2R F

|d|2x( )

⋅dF

|d|2x( )

x∫

jiR

R

j

i ,0 ∀∞<<for all rates

Path Loss |di|2 ~ |ri|-4

no shadowing no Rayleigh fading

Signal Attenuation

r1

r2

r3

Delay-Limited Capacity for Path Losssp

ec.

effic

ienc

y [b

its/s

ec/H

z]

infin

ite u

sers

[dB]

single

use

rin

f. or

th. u

sers

0

2

~

Nd

Eb

−

2

There is a gain by multiuser diversity without any drawback in quality of service,

Without constraints to orthogonal separation of users, the gain is greater.

if E |d|-2 exists.

Multiuser Diversity + Frequency Diversity

Let each user have M parallel channels.

Theorem 1:Subject to some technical conditions on the rates and the fading, each user uses only that channel of theirs which has the best propagation conditions, as the number of users approaches infinity.

For Rayleigh fading, DL-capacity is positive if M > 1.

Remark 1:Theorem 1 does not hold for a finite number of users, in general.

Multiuser Diversity + Frequency Diversity

Corollary 1:Subject to some technical conditions on the rates and the fading, frequency diversity only re-shapes the fading distribution, as the number of users approaches infinity.

d = max{|d1|; |d2|; ··· ; |dM|}

Path Loss |di|2 ~ |ri| -4

2nd order frequency diversity

no shadowing

Signal Attenuation

r1

r2

r3r1

r2

r3

Rayleigh fading

0

2

~

Nd

Eb

−

Delay-Limited Capacitysp

ec.

effic

ienc

y [b

its/s

ec/H

z]

chan

nel c

apac

ity

[dB]

orth

ogon

al li

mit

without Rayleigh fadingwith Rayleigh fading

single u

ser

The mean fading changes as well.

2

Uplink vs. Downlink

Theorem 2:Thanks to multiple-access broadcast duality on Gaussian channels, all results for multiple-access channels and transmit power, also hold for the Gaussian broadcast channel.

Cellular Systems

Corollary 2:Subject to some technical conditions on the rates and the fading, minimizing interference power onto base stations of other cells instead of transmit power is equivalent to re-shaping the fading distribution as the number of users approaches infinity.

Cellular systems are interference limited.You want to minimize interference into other cells instead of transmit power.

Open question: Does MAC broadcast duality apply here?

Cellular Systems (cont’d)

Consider a linear cellular system w.l.o.g.You want to minimize interference into other cells instead of transmit power.

I ∝

dn

2

n=−∞n≠0

+∞

∑

d0

2

Large number of users makes interference symmetric.

I ∝

d2n

2

n=−∞n≠0

+∞

∑

d0

2

Fd0

2(x) a Fd0

2

dn

2

n=−∞n≠0

+∞

∑

(x)

Fd0

2(x) a Fd0

2

d2n

2

n=−∞n≠0

+∞

∑

(x)Reuse factor 2:

%Eb

N0

→ ∞

Delay-Limited Capacity of Cellular System

spec

. ef

ficie

ncy

[bits

/sec

/Hz]

path loss exponent

capacity

single user

Spectral effciency can be doubled.

reuse 2

reuse 2

reuse 3

reuse 3

reuse 4

reuse 4

Coming Soon ...

Hexagonal cells ... Rayleigh fading

Just one more thing:

R =1

2Rxg(x)dx0

1

∫ g(x) =

π 2

4(1 −x)2 1 + tan2 π

2x( )( ) −1

path loss exponent 2:

with

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