lihua weng dept. of eecs, univ. of michigan

Lihua Weng

Dept. of EECS, Univ. of Michigan

Error Exponent Regions for

Multi-User Channels

2

Motivation: Downlink Communication

User 1: FTPhigh reliability

Pe = 10 -6

BaseStation

User 2: Multimedialow reliability

Pe=10-3

3

Motivation (cont.)

• Unequal error protection (ad hoc methods without systematic approach)

• Can reliability be treated as another resource (like power, bandwidth) that can be allocated to different users?

• Formulate this idea as an information theory problem, and study its fundamental limits.

4

Outline

• Background: Error Exponent

• Error Exponent Region (EER)

• Gaussian Broadcast Channel (GBC)

• Conjectured GBC EER Outer Bound

• Conclusion

5

• Error exponent: for a codeword of length N, the smallest possible probability of codeword error behaves as

where E(R) is the error exponent (as a function of the transmission rate R)- DMC (Elias55;Fano61;Gallager65; Shannon67)- AWGN (Shannon 59; Gallager 65)

Channel Capacity & Error Exponent: Single-User Channel

• Channel capacity: highest data rate for arbitrarily low probability of codeword error with long codewords

)( eP RENe

6

Error Exponent

R

E(R)

Rcrit C0

random coding

sphere packing

R

E(R)

Rcrit C0

random coding

sphere packing

expurgated

minimum distance

R

E(R)

Rcrit C0

random coding

sphere packing

expurgated

minimum distance

straight line

R

E(R)

Rcrit C0

random coding

sphere packing

expurgated

• We have a tradeoff between error exponent and rate

7

Capacity: Multi-User Channel

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

• Probability of system error: any user’s codeword is decoded in error

0 R1

R2

capacityregion

capacityboundary

8

Error Exponent: Multi-User Channel

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

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

- DMMAC/Gaussian MAC (Gallager 85)

- MIMO Fading MAC at high SNR (Zheng&Tse 03)

),( syse,

21P RRENe

N

RRNN

)),,(log(Plim)R,E(R 21syse,

21

9

Single Error Exponent: Drawback

• Multi-user channel – single error exponent- Different applications (FTP/multimedia)

• Our solution

- Consider a probability of error for each user, which implies multiple error exponents, one for each user.

10

Outline





• Conclusion

11

Multiple Error Exponents: Tradeoff 1

• We have tradeoff between error exponents (E1,E2) and rates (R1,R2) as in the single-user channel.

0 rate 1

rate 2

B

R1

R2

Large error exponents

0 rate 1

rate 2

A

R1

R2

Small error exponents

12

0 rate 1

rate 2

A

R1

R2

Multiple Error Exponents: Tradeoff 2

• Fix an operating point (R1,R2), which point from the capacity boundary can we back off to reach A?


• B A : E1 < E2

0

B

rate 1

rate 2

A

R1

R2

CB1

CB2


• B A : E1 < E2

D A : E1 > E2

0

D

rate 1

rate 2

A

R1

R2

CD2

CD1


• B A : E1 < E2

D A : E1 > E2

• Given a fixed (R1,R2), one can potentially tradeoff E1 with E2

0

B

D

rate 1

rate 2

A

R1

R2

13

Error Exponent Region (EER)

• Definition: Given (R1,R2), error exponent region is the set of all achievable error exponent pairs (E1,E2)

rate 1

rate 2

A

R1

R2 EER(R1,R2)

E2

E1

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

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

14

Outline



• Gaussian Broadcast Channel (GBC)- EER Inner Bound

• Single-Code Encoding• Superposition Encoding

- EER Outer Bound


• Conclusion

15

Gaussian Broadcast Channel

Y1

Y2

Z1

Z2

X

22

22

21

21

σ}E{Z

σ}E{Z

22

11

Y

Y

ZX

ZX

P}E{X2

16

Single-Code Encoding

CB = {Ck | k=(i-1)*M2+j; i = 1, … ,M1; j = 1, … , M2}

)},(),,(max{

)},(),,(max{

22

2122

212

21

2121

211

P

RREP

RREE

PRRE

PRREE

exr

exr

CB

1C

2C

21 MMC

17

Superposition Encoding

PPPPP ))(1()( 22122111

jiN CCX ,2,1

CB2CB1

1,1C

2,1C

1,1 MC

1,2C

2,2C

2,2 MC

k0 k0

CB2

P11

P12P21

P22

E{|C1,i(k)|2} E{|C2,j(k)|2}

N N)1(

CB1

1,1C

2,1C

1,1 MC

1,2C

2,2C

2,2 MC

N N)1(

18

Individual and Joint ML Decoding

• Individual ML Decoding (optimal)

1

2

1i ,1,2,12,22

1j ,2,2,11,11

)()|( maxarg)|( maxargj :2user

)()|( maxarg)|( maxargi : 1user M

ijiN

jj

N

j

M

jjiN

ii

N

i

CPCCYPCYP

CPCCYPCYP

• Joint ML Decoding

j)|( maxarg)j,i( :2user

i)|( maxarg)j,i( : 1user

,2,12),(

,2,11),(

jiN

ji

jiN

ji

CCYP

CCYP

),,,(

:1User

21

1221

111

,2',1,2,1

PP

RE

CCCC

np

jiji

- Type 1 error: one user’s own message decoded erroneously, but the other user’s message decoded correctly

19

Joint ML Decoding (cont.)

• Joint ML Decoding- Type 3 error: both users’ messages are

decoded erroneously

),,,,,(

:1User

21

2221

2121

1221

11213

',2',1,2,1

PPPP

RRE

CCCC

npt

jiji

• Achievable Error Exponents

)],,,,,(),,,,(min[

)],,,,,(),,,,(min[

22

2222

2122

1222

112132

2

2222

2122

21

2221

2121

1221

112132

1

1221

1111

PPPPRRE

PPREE

PPPPRRE

PPREE

npt

np

npt

np

20

• Naïve Single-user decoding: Decode one user’s signal by regarding the other user’s signal as noise

Naïve Single-User Decoding

),,,(

),,,(

2212

222211

2122

2122

122121

1111

P

P

P

PREE

P

P

P

PREE

np

np

21

Special Case 1: Uniform Superposition

k0 k0

CB2CB1

`

E{|C1,i(k)|2} E{|C2,j(k)|2}

N N

1,1C

2,1C

1,1 MC

1,2C

2,2C

2,2 MC

P)1(

P

22

Special Case 2: On-Off Superposition (Time-Sharing)

k

E{|C1,i(k)|2}

0 k

E{|C2,j(k)|2}

0

CB1 CB2

P P

1,1C

2,1C

1,1 MC

1,2C

2,2C

2,2 MC

N N)1( N N)1(

23

0 1 2 3 4 5 6 7 8

x 10-3

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

E1

E2

Superposition

EER Inner Bound

R1 = 1

R2 = 0.1

SNR1 = 10

SNR2 = 5

0 1 2 3 4 5 6 7 8

x 10-3

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

E1

E2

Superposition

Uniform

On-off

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

R1

R2

24

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

E1

E2

Single-code

Superposition

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

E1

E2

Superposition

Uniform

On-off

EER Inner Bound

R1 = 0.5

R2 = 0.5

SNR1 = 10

SNR2 = 10

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

E1

E2

Superposition

Uniform

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

E1

E2

Superposition

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

R1

R2

25

k0 k0

P11P12

P21P22

E{|C1,i(k)|2} E{|C2,j(k)|2}

N N)1( N N)1(

Superposition vs. Uniform

k0 k0

P11

P12P21

P22

E{|C1,i(k)|2} E{|C2,j(k)|2}

N N)1( N N)1( k0 k0

P11

P12 P21

P22

E{|C1,i(k)|2} E{|C2,j(k)|2}

N N)1( N N)1( k0 k0

P11

P12 P21

P22

E{|C1,i(k)|2} E{|C2,j(k)|2}

N N)1( N N)1( k0 k0

P11

P12 P21

P22

E{|C1,i(k)|2} E{|C2,j(k)|2}

N N)1( N N)1(

26

Superposition vs. Uniform (cont.)

},min{)2/1,,,,,(

)2/1,,,(

13111

1111111121313

1111111

tt

nptt

npt

EEEPPPPPPRREE

PPPREE

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 100.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

P11

Err

or E

xpon

ent

Et11

Et13

Uniform On-Off 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

E1

E2

27

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

E1

E2

Joint ML vs. Naïve Single-User

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

E1

E2

E2sp,jm

E2sp,ns

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

E1

E2

"anticipated" EER using individual ML decoding

28

Outline



• Gaussian Broadcast Channel (GBC)- EER Inner Bound- EER Outer Bound

• Single-User Outer Bound• Sato Outer Bound


• Conclusion

29

EER Outer Bound: Single-User

(a)

P(Y1,Y2 |X)

D1

D2

E(m1,m2) X

Y1

Y2

w1

w2

(a)

P(Y1,Y2 |X)

D1

D2

E(m1,m2) X

Y1

Y2

w1

w2

(b)

P(Y1 |X) D1

D2X

Y1

Y2

w1

w2

E1X

E2 P(Y2 |X)

m1

m2

),( );,( 22

2221

11 P

REEP

REE susu

30

EER Outer Bound: Sato

22

212

12121 if ),(},min{

PRREEE su

P(Y1,Y2|X)

D1

D2

E(m1,m2) X

Y1

Y2

w1

w2

(a)

P(Y1,Y2|X)

D1

D2

E(m1,m2) X

Y1

Y2

w1

w2

(a)

P(Y1,Y2|X)E(m1,m2) X

Y1

Y2

D(w'1,w'2)

(b)

Y1 Y2

Z1 Z2'

X

P(Y1,Y2|X)

D1

D2

E(m1,m2) X

Y1

Y2

w1

w2

(a)

P(Y1,Y2|X)E(m1,m2) X

Y1

D(w'1,w'2)

(b)

31

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

E1

E2

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

E1

E2

EER Inner & Outer Bounds

R1 = R2 =0.5

SNR1 = SNR2 =10

• This is a proof that the true EER implies a tradeoff between users’ reliabilities

impossiblevalid

32

Outline





• Conclusion

33

),,,,(

),,,,(

122

21

2122

222

21

2111

EPP

RRfE

EPP

RRfE

• Each outer bound is based on single-user error exponent upper bounds. The right hand side of the inequalities depends only on R1 and R2

Review: GBC EER Outer Bound

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

),(

),(

22

2121

2121

22

22

21

11

PRRE

PRREEE

PREE

PREE

susu

su

su

34

Gaussian Single-User Channel (GSC) with Two Messages

121, },...,1;,...,1|{ N

ji SMjMiCCB

222 }E{Z Z;XY P;}E{X

O

SN-1

Y

Z

X DecoderEncoder(i,j) (i',j')

35

Background: Minimum Distance Bound

CCd

CdRd

RdSNR

SNRRE

C

md

code of distance minimum :)(

)( max)(

)(8

),(

min

mincode spherical

min

2min

22

82

2/

, )2

( 2

dSNR

NxSNR

N

d

paire ed

SNRNQdxeSNR

NP

36

C1,1

C1,2

C1,M2

C2,1

C2,2

C2,M2

CM1,1

CM1,2

CM1,M2

C1,1

C1,2

C1,M2

C2,1

C2,2

C2,M2

CM1,1

CM1,2

CM1,M2

d

GSC EER Outer Bound - Partition

SNR

EddeeP E

dSNR

NNEe

181,

81

2

1

C1,1

C1,2

C1,M2

C2,1

C2,2

C2,M2

CM1,1

CM1,2

CM1,M2

dd '

C1,1

C1,2

C1,M2

C2,1

C2,2

C2,M2

CM1,1

CM1,2

CM1,M2

21E

d

37

Union of Circles

SNR

EdE

181

C1,1

C1,2

C1,M2

21E

d

C1,i

C1,j

1El

22

82, 1

212

8 E

lSNR

NNEe l

SNREeeP

E

38

Union of Circles

• C = {C1, C2, …, CM}

• A(C,r): area of the union of the circles with radius r

C1

C2

CM

Ci

Cj

r

39

Minimum-Area Code

1. What is the maximum of dmin(C) under the constraint A(C,r) is at most A’?

2. What is the minimum of A(C,r) under the constraint dmin(C) is at least d’?

d'

d'd'

d'

40

Intuition: Surface Cap

O

SN-1

41

Conjectured Solution

O

SN-1

SN-2

sin

O

SN-1

SN-2

O

SN-1

D = {D1,D2,...,DM}

')(sin min dRd

)( max)( mincode sphere

min CdRdC

SN-2

O

SN-1

D = {D1,D2,...,DM}

')(sin min dRd

0),(

)*,(log

1 ),()*,(

rDA

rCA

NrDArCA

42

Conjectured GSC EER Outer Bound

),( )],,,([sin

),( )'(sin

)(8

)'(sin

)(8

8

22

121

22

22min

2

2min

22 1

SNRRESNRERR

SNRRE

RdSNR

CdSNR

lSNR

E

md

md

E

O

SN-1

'

What is the maximum of dmin(C) under the

constraint A(C,r) is at most A’?

)('sin)( 2minmin RdCd

),( )],,,([sin 22

1212 SNRRESNRERRE md

43

Conjectured GBC EER Outer Bound

),( )],,,([sin 222

11212 SNRRESNRERRE md

R1 = 0.5

R2 = 2.4

SNR1 = 100

SNR2 = 1000

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

E1

E2

New EER outer bound

44

Conclusion

• EER for Multi-User Channel- The set of achievable error exponent pair (E1,E2)

• Gaussian Broadcast Channel- EER inner bound : single-code, superposition- EER outer bound : single-user, Sato


• Gaussian Multiple Access Channel- EER is known for some operating points

• MIMO Fading Broadcast ChannelMIMO Fading Multiple Access Channel- Diversity Gain Region

lihua weng dept. of eecs, univ. of michigan

Documents

probability of error

error exponentwe

error exponents e1

multiple error exponents

unequal error protection

operating point r1

rate pair r1

tradeoff e1