Achilleas Anastasopoulos (joint work with Lihua Weng and Sandeep Pradhan) April 30 2004 A Framework...

Achilleas Anastasopoulos

(joint work with Lihua Weng and Sandeep Pradhan)

April 30 2004

A Framework for Heterogeneous

Quality-of-Service Guarantees in Wireless Networks:

A Communication-theoretic Approach

2

Outline

• Motivation

• Background: error exponents for single-user channels

• The concept of error exponent region (EER)

• Scalar Gaussian broadcast channel (SGBC)

• MIMO Fading broadcast channel

• Conclusions

3

Motivation: Scenario 1

User 1: FTP application

-High data rate

-High reliability User 2: Voice

-Low data rate

-Low reliability

Base Station

•Solution: allocate more resources (e.g., time slots, or BW) to user 1

4



-High data rate

-High reliability User 2: Telemetry data

-Low data rate

-High reliability

Base Station

•Solution: trade data rate for reliability for user 2 (e.g., using higher power and/or channel coding)

5



-High data rate

-High reliability User 2: Multi-media

-High data rate

-Low reliability

Base Station

•Solution 1: trade reliability for data rate for user 2 (e.g., no channel coding)•Solution 2: allocate more resources to user 1 (e.g., power, or BW to utilize in channel coding)

6

Comments/Questions

• An individual user can trade its own data rate for reliability (scenario 2, 3)

• There are several techniques (usually referred to as “unequal error protection”) that provide solutions through asymmetric resource allocation

What is the the best you can do for a given channel and given resources?

Can available reliability be treated as another resource (like power, or BW) that can be allocated to different users?

Can communication theory provide answers to these questions?

How do you do that in practice?

7

Basic result of this work

• As in single-user channels, there is a basic trade-off between data rate and reliability

• Multi-user channels provide an additional degree of freedom:

- Users can trade reliabilities with each other (even for fixed data rates)

- The above seems like an obvious statement…

• There is a way to formulate this problem as a communication theoretic problem and study its fundamental limits

8

Outline

• Motivation




• MIMO Fading multi-user channels

• Conclusions

9

Error exponent: Single-user channel

• Channel capacity, C: highest possible transmission rate that results in arbitrarily low probability of codeword error with long codewords

• Error Exponent, E: rate of exponential decay of codeword error probability

• For a codeword of length N, the probability of codeword error behaves as

where E(R) is the error exponent (as a function of the transmission rate R)- DMC (Gallager65; Shannon et al67)- AWGN (Shannon59; Gallager65)

)( errP RENe

10

Error exponent: Single-user channel

R

E(R)Er(R)Eex(R)Esp(R)Emd(R)Est(R)

Rcrit C

• Error exponent E(R) is an increasing function of the distance between R and C

• Only trade-off: increase E(R) by decreasing R, i.e, trade reliability for rate

• Upper bounds on Perr Lower bounds on E simple

- Random coding bound, expurgated bound

• Lower bounds on Perr Upper bounds on E not that simple

- Sphere packing bound, minimum distance bound, straight line bound

11

Error exponent: Multi-user channel

• Channel capacity region: all possible transmission rate vectors (R1,R2) for arbitrarily low system error probability

• System error probability: for correct transmission, all users have to be decoded correctly

0

Capacity boundary

Capacity region:

Achievable rate region

R2

R1

12

Error exponent: Multi-user channel

• Error Exponent: rate of exponential decay of system error probability

• For a codeword of length N, the probability of system error behaves as

where E(R1,R2) is the error exponent

- Gaussian MAC (Gallager85; Guess&Varanasi00)

- Wireless MIMO MAC at high SNR (Zheng&Tse03)

),( sys,err

21P RRENe

13

Error exponent: Multi-user channel (conclusions)

• We saw (scenario 1, 3) that different users might have different reliability requirements (e.g., FTP and multi-media)

• Based on a single probability of system error, a network can only be designed to satisfy the most stringent reliability requirement (equal QoS for all users), which might result in a suboptimum resource allocation

• Information/communication theory seems inadequate (so far) to address heterogeneous QoS requirements

14

Outline

• Motivation





• Conclusions

15

A straightforward extension

• Since a single system error probability is inadequate to characterize the requirements of multiple users, let us consider multiple error probabilities; one for each user

• Implication: multiple error exponents; one for each user

16

A straightforward extension

• We have trade-off between error exponents and rates (as in the single-user channel).

• Is there any other trade-off available for error exponents in a multi-user channel?

0 rate 1

rate 2

B

R1

R2

Large error exponents

0 rate 1

rate 2

A

R1

R2

Small error exponents

17

The concept of EER

• Fix an operating point (R1,R2)

0 rate 1

rate 2

A

R1

R2

18

The concept of EER


• Which point from the capacity boundary do we back off to reach A?

0 rate 1

rate 2

A

R1

R2

19

The concept of EER



• B A : E1 < E2

0

B

rate 1

rate 2

A

R1

R2

CB1

CB2

20

The concept of EER



• B A : E1 < E2

D A : E1 > E20

D

rate 1

rate 2

A

R1

R2

CD1

CD2

21

The concept of EER



• B A : E1 < E2

D A : E1 > E20

D

rate 1

rate 2

A

R1

R2

CD1

CD2

• In addition to error exponent/rate trade-off, given a fixed (R1,R2), one can potentially trade-off E1

with E2

22

The concept of EER: Definition

• Definition: The error exponent region (EER) is the set of all achievable error exponent pairs (E1,E2)

• Careful!- Channel capacity region: one for a given channel

- EER: numerous, i.e., one for each pair of (R1,R2)

rate 1

rate 2

A

R1

R2 EER(R1,R2)

E2

E1

Possible shape

for EER

23

Outline

• Motivation





• Conclusions

24

• Scalar Gaussian Broadcast Channel

• Observe: two messages; joint encoder; separate decoders

• This is a degraded broadcast channel (i.e., if then, Y2=X+N1+N’2=Y1+ N’2, with E{(N’2)2}=

)

SGBC definitions

2i

2i σ}E{N

22

11

Y

Y

NX

NX

P}E{X2

25

• Achievable EER by time-sharing:

where E(R,SNR) is any of the error exponent lower bounds for a single-user AWGN channel

SGBC EER Inner Bound: Time-sharing

),1

()1(

),(

22

22

21

11

PREE

PREE

ts

ts

N N)1(

User 1 User 2

N

10

26

SGBC EER Inner Bound: Time-sharing

• Indeed, there is a trade-off for error exponents, given a fixed pair of rates for time-sharing

R1 = R2 =0.5

P/12 = P/2

2 =10

27

SGBC EER Inner Bound: Superposition

• Superposition encoding:

- Generate two independent codebooks i, each of size and power

- Select a codeword from each codebook based on the individual messages and transmit their sum

- Note: this is a capacity-achieving strategy for any degraded broadcast channel

PXEP

PXEP

XXX

)1(}{

,}{222

211

21

10

iNR2iP

28


• Decoding: two options (at least)- Individual ML decoding (optimal)

- Joint Maximum-Likelihood (ML) decoding

)},|(max{ maxargj :2user

)},|(max{ maxargi :1user

212

211

jiij

jiji

P

P

XXY

XXY

iiji

jj

j

jjji

ii

i

PPP

PPP

)()(maxarg)(maxargj :2user

)()(maxarg)(maxarg i :1user

121222

221111

XX,X|YX|Y

XX,X|YX|Y

29

• Upper bound derivation for joint ML decoding- Let us look at user 1:

- Type 1: M1 is decoded erroneously, but M2 is decoded correctly same as if only user 1 was present in the channel

- Type 3: both messages are decoded erroneously (similar bound as in Gallager85 for MAC channels)

},min{

error 3 type

21

211

error 1 type

21

211

111,

3131 2

}ˆ&ˆPr{}ˆ&ˆPr{

}ˆPr{

EENENEN

uu

err

eee

MMMMMMMM

MMP


30

• Superposition Inner Bound with joint ML decoding

where E(R,SNR) is any of the error exponent lower bounds for a single-user AWGN channel, and Et3(R,SNR1,SNR2) is a slightly more complicated expression (for type 3 errors)

))1(

,,(),)1(

,(min

))1(

,,(),,(min

22

22

21322

22

21

21

21321

11

PPRRE

PREE

PPRRE

PREE

ts

ts


31

SGBC EER Inner Bound

• Observation: although superposition achieves capacity (while time-sharing does not always achieve it), time sharing can help in expanding the EER. Why?

R1 = R2 =0.5

P/12 = P/2

2 =10

32

Time-Sharing vs. Superposition

• Three possible reasons:- The superposition EER is derived based on

joint ML decoding, but the optimum decoder is individual ML decoding

- Joint ML decoding might be still a good strategy, but Et3 is a loose bound

- Time-Sharing can sometimes indeed expand the EER obtained by superposition: when we need very high reliability for one user, it might be better to separate the users

33

SGBC EER Inner Bound: Summary

• We can keep expanding the inner bound by finding better and better strategies

• It is not clear yet that the exact EER implies a trade-off between users’ reliabilities

Possible true EER

• We need an outer bound for the EER

34

SGBC EER Outer Bound: Single-user

),(21

11 P

REE su

P(Y1,Y2|X)

D1

D2

E(M1,M2) X

Y1

Y2

where Esu (R,SNR) is any error exponent upper bound for the AWGN channel

is always worse than two separate single-user channels with same marginals

Any broadcast channel

P(Y1|X) D1

D2X

Y1

Y2

E1X

E2 P(Y2|X)

M1

M2

thus ),(22

22 P

REE suand

35

SGBC EER Outer Bound: Sato

)},(),,(max{}, min{22

2121

2121 P

RREP

RREEE susu

P(Y1,Y2|X)

D1

D2

E(M1,M2) X

Y1

Y2

For any Q(Y1,Y2|X) with the same marginals as P(Y1,Y2 |X)

is always worse than

Q(Y1,Y2|X)E(M1,M2) X

Y1

Y2

D

By choosing the worst-case Q(Y1,Y2 |X)

36

SGBC EER Outer Bound

R1 = R2 =0.5

SNR1 = SNR2 =10

• This is a proof that the true EER implies a trade-off between users’ reliabilities

impossiblevalid

37

Outline

• Motivation





• Conclusions

38

Background: Single-user channel

• MIMO Fading Single-user Channel (Tse, 2003) : block fading

- X: m x t channel input matrix- Y: n x t channel output matrix- Z: n x t noise matrix; i.i.d. with CN(0,1)- H: n x m fading matrix; i.i.d. with CN(0,1)

Assume H is known at receiver, but not at transmitter

ZHXY m

SNR

39


• MIMO fading single-user channel (Zheng&Tse03)

- Diversity and Multiplexing trade-off (high SNR)

• r: multiplexing gain

• d: diversity gain

)(

logrd

e SNRP

SNRrR

SNR

Prd

SNR

Rr e

SNRSNR log

loglim)(;

loglim

40


41

Multiplexing Gain Region (MGR)Diversity Gain Region (DGR)

• MIMO fading multi-user channel

- Multiplexing Gain Region: the set of all achievable multiplexing-gain vector (r1,…,rK)

- Diversity Gain Region: the set of all achievable diversity-gain vector (d1,…,dK), given a multiplexing-gain vector.

SNR

RrSNRrR i

SNRiii log

lim ;log

SNR

PrSNRP ei

SNRi

rdei

i

log

loglim)(d ;)(

42

MIMO Fading Broadcast Channel

• MIMO Fading Broadcast Channel (MFBC) : block fading

- X : m x t channel input matrix

- Yi : ni x t channel output matrix

- Zi : ni x t noise matrix; i.i.d. element CN(0,1)

- Hi : ni x m fading matrix; i.i.d. element CN(0,1)

Assume Hi is known at receivers, but not at transmitter

iii m

SNRZXHY

43

MFBC Multiplexing Gain Region

• Proposition: For a MIMO fading broadcast channel, the multiplexing gain region is the same region achieved by time-sharing.

K

i i

iK nm

rrr

11 1

),min(:),...,(MGR

44

MFBC DGR Inner Bound: Time-Sharing

• Time-Sharing

10

)1

(

)(

2,2

1,1

2

1

p

p

rdd

p

rdd

nmts

nmts

pl lp)1(

User 1 User 2

l

45

MFBC DGR Inner Bound: Superposition

• Superposition: X = X1 + X2

X1 : m x l matrix with i.i.d. element CN(0,1)

X2 : m x l matrix with i.i.d. element CN(0,SNR-(1-p))

- Joint Maximum-Likelihood (ML) decoding

Note : The role of user 1 and user 2 can be exchanged

)(),(min

)()}(),(min{

21,2

,2

21,21,1,1

22

111

rrdp

rpdd

rrdrrdrdd

nmnms

nmnmnms

46

MFBC DGR Inner Bound: Superposition

• Superposition: X = X1 + X2

X1 : m x t matrix with i.i.d. element CN(0,1)

X2 : m x t matrix with i.i.d. element CN(0,SNR-(1-p))

- Joint ML and naïve single-user decoding

Note : The role of user 1 and user 2 can be exchanged.

)(),(min

)}(),(max{

21,2

,'2

1,,,21,'1

22

11

rrdp

rpdd

rdrrdd

nmnms

nsptnmnm

s

47

Naïve Single-user Diversity Gain Region

r

dm,n(r)

1-p 1-p 1-p

(0,mn)

(1,(m-1)(n-1))

(min(m,n),0)

0

)(,,, rd ns pnm

r

(0,mn)

(1-p,(m-1)(n-1))

01-p

)(,,, rd ns plnm

r

(0,mn)

(1-p,(m-1)(n-1))

01-p

48

MFBC DGR Outer Bound

• MFBC DGR Outer Bound

)}(),(max{},min{

)(

)(

21,21,21

2,2

1,1

21

2

1

rrdrrddd

rdd

rdd

nmnm

nm

nm

49

Diversity Gain Region Inner/Outer Bound

• Observation: For a symmetric MFBC, inner and outer bounds are tight at d1 = d2

• Observation: For a MFBC, either user 1 (or user 2) can achieve his maximum (single-user) diversity gain if r1+r2 < 1

0 2 4 6 8 10 12 140

2

4

6

8

10

12

14

d1

d 2

0 2 4 6 8 10 12 140

2

4

6

8

10

12

14

d1

d 2

m = n1 = n2 =4

t = 120

r1 = r2 = 0.5

50

Conclusions

- The concept of error exponent region for multi-user channels was presented

- Inner (time-sharing/superposition) and outer (single-user/Sato) bounds were derived for the SGBC EER

- Implication: Users can trade reliability between each other even for a fixed set of transmission rates

Ongoing Work- Tighten EER inner/outer bounds for SGBC

- EER for Gaussian multiple-access channels

- Diversity/multiplexing trade-off region for wireless MIMO BC/MAC

- Practical schemes that achieve EER

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Achilleas Anastasopoulos (joint work with Lihua Weng and Sandeep Pradhan) April 30 2004 A Framework...

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