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Exploiting Multiuser Diversity Using

Multiple Feedback Thresholds

Vegard Hassel 1 , Mohamed-Slim Alouini 2 , David Gesbert 3 and Geir E. Øien 1

1 Dept. of Electronics and Telecommunications

Norwegian University of Science and Technology, Norway

email: {vegard,oien}@iet.ntnu.no

2 Dept. of Electrical and Computer Engineering

University of Minnesota, USA

email: alouini@ece.umn.edu

3 Dept. of Mobile Communications

Eurecom Institute, France

email: gesbert@eurecom.fr

Department of

Electronics and Telecommunications

This presentation and the paper can also be found at http://www.tele.ntnu.no/projects/cuban/

Vegard Hassel @ VTC/Spring 01.06.2005

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Outline

• Background:

– System Model

– MUD: Multiuser Diversity

– The Feedback Problem

– SMUD: Selective Multiuser Diversity

Multiple Feedback Thresholds:

– Main Idea

– Why Should This Algorithm be Implemented?

– How Should This Algorithm be Optimized?

– Capacity having Quantized Feedback

– Number of Feedback Polls Before Transmission

– Scheduling Outage Probability Having a Deadline

Department of

Electronics and Telecommunications

Vegard Hassel @ VTC/Spring 01.06.2005

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System Model

• Time Division Multiplexed (TDM) system with N mobile users

• Carrier-to-noise-ratios (CNRs) of the users’ channels are i.i.d.

• To perform scheduling CNR feedback for every time slot is needed

• The time slots are smaller than a coherence time

Department of

Electronics and Telecommunications

Vegard Hassel @ VTC/Spring 01.06.2005

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Multiuser Diversity (MUD)

Diversity in wireless systems arises because of independently fading

channels

• Traditional forms of diversity: space, time, and frequency

• Viswanath and Tse: Multiuser diversity

• MUD Background: With many users in a cell, there is high probability

of finding a user with a good channel at any time

• Max CNR Scheduling: To obtain the highest system spectral efficiency,

the user with the best channel has to be chosen at all times

• Observe: While traditional forms of diversity give better link spectral

efficiency, multiuser diversity increases the system spectral efficiency

Department of

Electronics and Telecommunications

Vegard Hassel @ VTC/Spring 01.06.2005

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Department of

Electronics and Telecommunications

NTNU

Vegard Hassel @ VTC/Spring 01.06.2005

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The Feedback Problem

• IDEALLY: We only need feedback from the best user!

• BUT: The users do not know if they are the best

• HENCE: In conventional MUD systems all users need to feed back their

CNR values

Department of

Electronics and Telecommunications

Vegard Hassel @ VTC/Spring 01.06.2005

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SMUD: Selective Multiuser Diversity

• Algorithm to reduce the feedback load

• The scheduler asks for the instantaneous CNR level only from the users

that have CNR above a threshold, γ th

• If none of the users feed back their CNR, a random user is chosen

• The algorithm is not rate-optimal, but gives a significant reduction in

the feedback load

Department of

Electronics and Telecommunications

Vegard Hassel @ VTC/Spring 01.06.2005

7


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Main Idea

• Generalization of the SMUD algorithm:

We use multiple feedback thresholds

• First, the users are asked if they are above the highest threshold value

• If none of the users feed back their CNR value, the base station asks if

the users are above the second highest threshold value

• This process continues until one ore more users feed back their CNR

value

• If the lowest threshold value equals zero, the base station is guaranteed

that one or more users feed back their CNR

Department of

Electronics and Telecommunications

Vegard Hassel @ VTC/Spring 01.06.2005

8


NTNU

Why Should This Algorithm be Implemented?

• PROS:

– The MUD gain is maximized

– The time used to collect feedback is lower than for the full feedback

algorithm

– Reduced feedback from the mobile users will reduce their power

consumption

Department of

Electronics and Telecommunications

• CONS:

– Higher system complexity

– The time used to collect feedback is higher than for the SMUD

algorithm

Vegard Hassel @ VTC/Spring 01.06.2005

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How Should This Algorithm be Optimized?

• The feedback collection process should have the following properties:

– The duration of the process should be as short as possible

– In order to maximize the MUD gain we should collect the CNR value

of the best user

• The time used to collect feedback is a function of:

– The number of jumps before the successful interval is reached

– The number of users giving feedback from the successful interval

Department of

Electronics and Telecommunications

• Tradeoff between time it takes to look for the successful interval and

the time the users use to feed back their CNR values

Vegard Hassel @ VTC/Spring 01.06.2005

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How Should This Algorithm be Optimized?

• We cannot find sensible threshold values which minimizes the number

of jumps

• We therefore choose find the threshold values which minimizes the feedback

load

• The optimization process can now be done in two stages:

– Find optimal threshold values for a given L

– Find optimal L

• The optimal L can be found for a set of system parameters. The

optimum will be where the advantage of introducing a new threshold is

less than the disadvantage

Department of

Electronics and Telecommunications

Vegard Hassel @ VTC/Spring 01.06.2005

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Notation and Mathematical Framework

• L + 1 thresholds:

γ th,L >γ th,L−1 > · · ·>γ th,0

• To cover the whole CNR range: γ th,0 = 0

• To make the optimization easier: γ th,L = ∞

• The base station initially requests feedback from those users whose CNR

is above γ th,L−1

• Cumulative Distribution Function for the CNR of one user: P γ (·)

Department of

Electronics and Telecommunications

Vegard Hassel @ VTC/Spring 01.06.2005

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Feedback: Mathematical Analysis

• Normalized Feedback load (NFL) is defined as the average percentage

of users that give feedback for every feedback request

• For the new algorithm the normalized feedback load (NFL) can be expressed

as:

L−1 ∑ N∑

( ) N

n (P γ (γ

n th,l+1 ) − P γ (γ th,l )) n · Pγ N−n (γ th,l )

¯F = 1 N

l=0 n=1

Using the binomial expansion formula, it can be shown that this expression

can be written as:

¯F =

L−1 ∑

l=0

(

Pγ (γ th,l+1 ) − P γ (γ th,l ) ) · P N−1 γ (γ th,l+1 )

Department of

Electronics and Telecommunications

Vegard Hassel @ VTC/Spring 01.06.2005

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Feedback: Mathematical Analysis

• Taking the gradient of this expression with regard to the thresholds γ th,l

and setting the result equal to zero yields:

γ ∗ th,l = P −1 γ

(

Sl · P γ (γ th,l+1 ) ) , l = 1, 2, 3, · · ·, L−1,

where Pγ

−1 (·) is the inverse CDF of the CNR for a single user

• The expression for S l is:


⎨N 1

1−N , l = 1

S l =

1

,

⎩[N − (N − 1)S l−1 ] 1−N ,l = 2, 3, · · ·, L−1

Department of

Electronics and Telecommunications

where N ≥ 2.

Vegard Hassel @ VTC/Spring 01.06.2005

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Electronics and Telecommunications

NTNU

Vegard Hassel @ VTC/Spring 01.06.2005

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Department of

Electronics and Telecommunications

NTNU

Vegard Hassel @ VTC/Spring 01.06.2005

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Capacity having Quantized Feedback

• If the feedback is quantized, it can be shown that the overall capacity

can be given as:

L·J−1 ∑

C QUANT =

×

j=0

∫ qj+1

q j

P N γ (q j+1 ) − P N γ (q j )

2(P γ (q j+1 ) − P γ (q j ))

log 2 (1 + γ)p γ (γ) dγ

− N · ¯F · log 2 (J)

N S

,

• J is the number of quantization regions employed

• q j and q j+1 are the lower and upper quantization region thresholds,

respectively

• N S is the number of symbols per time slot

Vegard Hassel @ VTC/Spring 01.06.2005

Department of

Electronics and Telecommunications

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Number of Feedback Polls Before Transmission

Pr[M = k] =

P N γ (γ th,L+1−k ) − P N γ (γ th,L−k ), k = 1, 2, · · ·L.

Department of

Electronics and Telecommunications

Vegard Hassel @ VTC/Spring 01.06.2005

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Department of

Electronics and Telecommunications

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Vegard Hassel @ VTC/Spring 01.06.2005

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Scheduling Outage Probability Having a

Deadline

P OUT =1 − P M (D)

=1 −

D∑

(1 − SL−l N ) · P γ N (γ th,L+1−l )

l=1

Department of

Electronics and Telecommunications

Vegard Hassel @ VTC/Spring 01.06.2005

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Department of

Electronics and Telecommunications

NTNU

Vegard Hassel @ VTC/Spring 01.06.2005

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Conclusion

By using multiple feedback thresholds, we obtain the same spectral efficiency

and a significant reduction in the feedback load compared to full

feedback

Department of

Electronics and Telecommunications

Vegard Hassel @ VTC/Spring 01.06.2005

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